Nuclear Engineering Handbook (Etherington).pdf

Nuclear engineering handbook. Etherington, Harold New York, McGraw-Hill, 1958.

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SECTIONS OF THE HANDBOOK Section

L

Mathematical Data and General Tables

Section

2.

Nuclear Data

Section

3.

Mathematics

Section

4.

Nuclear Physics

Section

6.

Experimental Techniques

Section

6.

Reactor Physics

Section

7.

Radiation and Radiological Protection

Section

8.

Control of Reactors

Section

9.

Fluid and Heat Flow

Section 10.

Reactor Materials

Section 11.

Chemistry and Chemical Engineering

Section 12.

Nuclear-power-plant Selection

Section 13.

Mechanical Design and Operation of Reactors

Section 14.

Isotopes

HOW TO USE THE HANDBOOK The handbook is divided into fourteen major sections listed on the The first page of each section gives a table of contents of

opposite page. the section.

Selection of Data and Formulas

Two methods are available for finding data and formulas — the index and the set of tables listed below.

These guide tables indicate the con


tents of the most frequently used tables and, by showing contextual mate

rial, provide guidance that may be inherently difficult to give in an index. Judicious use of both the index and the guide tables is recommended. . The guide tables are followed, in Section 1-1, by articles summarizing frequently used data and formulas, and giving references to pertinent sections of the handbook. Descriptive Matter information is best found from the index, but the tables of of each section provide a convenient guide to information on a broad topic. Specific

contents at the beginning

GUIDE TABLES FOR SELECTION OF FREQUENTLY USED DATA

Table

Tables of Mathematical Functions '2 Mathematical Formulas Engineering Conversion Factors 3 4 General Atomic and Nuclear Data Nuclear Data used in Reactor Neutron Physics 5 6 Reactor Theory 7 Calculation of Radioactivity 8 Health Physics 9 Shielding 10 Physical Properties of Fluids 11 Fluid Flow and Heat Flow 12 Thermal Stress 13 of Physical and Mechanical Properties Materials Corrosion and Wear Resistance of Structural Materials. .14 15 Radiation Damage 16 Chemistry 1

Page

1-2 1-3 1-3 1 -3 1-4

1-5 1-5 1-6 1-7 1-8 to 1-10 1-10 1-10 1-10 1-11

l-U

I -U


NUCLEAR ENGINEERING HANDBOOK

McGRAW-HILL HANDBOOKS

Abbott and Smith

• National Electrical Code Handbook, 9th ed. Purchasing Handbook American Institute ok Physics • American Institute of Physics Handbook American Society of Mechanical Engineers ASME Handbook: Engineering Tables ASME Handbook: Metals Engineering — Design ASME Handbook: Metals Engineering — Processes ASME Handbook: Metals Properties American Society of Tool Engineers • Die Design Handbook American Society of Tool Engineers • Tool Engineers Handbook Beeman • Industrial Power Systems Handbook Berry, Bollay, and Beers • Handbook of Meteorology Brady • Materials Handbook, 8th ed. Cockrell • Industrial Electronics Handbook Compressed Air and Gas Institute • Compressed Air Handbook, 2d ed. Condon and Odishaw • Handbook of Physics Considine ■Process Instruments and Controls Handbook Crocker • Piping Handbook, 4th ed. Croft • American Electricians' Handbook, 7th ed. Davis • Handbook of Applied Hydraulics, 2d ed. Etherington • Nuclear Engineering Handbook Factory Mutual Engineering Division • Handbook of Industrial Loss Prevention Fink • Television Engineering Handbook Harris • Handbook of Noise Control Henney • Radio Engineering Handbook, 5th ed. Hunter • Handbook of Semiconductor Electronics Johnson and Auth • Fuels and Combustion Handbook Juran • Quality-control Handbook Ketchum • Structural Engineers' Handbook, 3d ed. King ■Handbook of Hydraulics, 4th ed. Knowlton • Standard Handbook for Electrical Engineers, 9th ed. Kurtz • The Lineman's Handbook, 3d ed. Labberton and Marks ■Marirle Engineers' Handbook Landee, Davis, and Albrecht • Electronic Designers' Handbook Laughner and Hargan • Handbook of Fastening and Joining of Metal Parts Le Grand • The New American Machinist's Handbook Liddell • Handbook of Nonferrous Metallurgy, 2 vols., 2d ed. Magill, Holden, and Ackley ■ Air Pollution Handbook Manas ■National Plumbing Code Handbook Mantell • Engineering Materials Handbook Marks and Baumeister • Mechanical Engineers' Handbook, 6th ed. Markus and Zeluff ■Handbook of Industrial Electronic Circuits Markus and Zeluff • Handbook of Industrial Electronic Control Circuits Maynard ■Industrial Engineering Handbook Merritt • Building Construction Handbook Morrow • Maintenance Engineering Handbook O'Rourke • General Engineering Handbook, 2d ed. Pacific Coast Gas Association • Gas Engineers' Handbook Perry • Chemical Business Handbook Perry • Chemical Engineers' Handbook, 3d ed. Shand ■Glass Engineering Handbook, 2d ed. Staniar • Plant Engineering Handbook, 2d ed. Terman • Radio Engineers' Handbook Truxal • Control Engineers' Handbook Urquhart • Civil Engineering Handbook, 4th ed. Voder, Heneman, Titrnbull, and Stone ■Handbook of Personnel Management and Labor Relations


Auian

•


NUCLEAR ENGINEERING HANDBOOK HAROLD ETHERINGTON, Editor Vice President Nuclear Products-Erco, Division of ACF Industries formerly Director, Naval and Reactor Engineering Divisions Argonne National Laboratory

FIRST EDITION

New York

Toronto

London

McGRAW-HILL BOOK COMPANY, INC. 1958

fclgin

library

T/<

NUCLEAR ENGINEERING HANDBOOK Printed in the United Copyright © 1958 by the McGraw-Hill Book Company, Inc. All rights reserved. This book, or parts thereof, may not be States of America. reproduced in any form without permission of the publishers.


Library of Congress Catalog Card Number:

THE MAPLE PRESS

COMPANY,

57-8000

TORE, *A.

CONTRIBUTORS

Paul C. Aebersold, M.A., Ph.D., Assistant Director for Isotopes, Office of Indus trial Development, U. S. Atomic Energy Commission. H-l. Isotopes and Their Use.


Robert C. Allen, Director of Mechanical Engineering, Industries Group, AllisChalmers Manufacturing Company. 12-3. Power Generating Equipment. Alfred Amorosi, B.S.M.E., B.S.Ch., Technical Director, Atomic Power Develop ment Associates, Inc. (on loan from Detroit Edison Company), formerly Director, Reactor Engineering Division, Argonne National Associate Laboratory. 12-1. Selection of Reactors. George A. Anderson, B.M.E., Manager, Reactor Engineering, Nuclear Products -Erco, Division of ACF Industries, Incorporated, formerly Project Engineer, Argonne National Laboratory. 18-7. Reactor Construction. Arthur H. Barnes, A.B., A.M., Ph.D. (deceased), formerly Director, Reactor Engineering Division, Argonne National Laboratory. 13-3. Liquid-metal Reactor Systems.

Franklin T. Binford, B.S., Senior Development Engineer, Reactor Operations Department, Oak Ridge National Laboratory. 14-3. Production of Radio isotopes.

Everitt P. Blizard, B.A., M.A., Director, Applied Nuclear Physics Division, Oak Ridge National Laboratory. 7-3. Nuclear Radiation Shielding. Charles F. Bonilla, Ch.E., Ph.D., Professor of Chemical Engineering, Columbia University. 9-2. Fluid Flow in Reactor Systems. 9-3. Heat Removal from Nuclear Reactors. Dwain B. Bowen, M.S., Group Leader, Solid State Physics, Atomics Interna 10-1. The Metallic State. tional, North American Aviation, Inc. R. B. Briggs, B.S.Ch.E., Director of Homogeneous Reactor Project, Oak Ridge National Laboratory. 13-2. Homogeneous Aqueous Reactor Systems. Thomas

J. Burnett, B.A., Health Physicist, Oak Ridge National Laboratory,

Safe Handling of Fuel (in Section 7-2).

Leslie Burns, Jr., B.S., M.S., Associate Chemical Engineer, Laboratory. 11. Chemistry and Chemical Engineering.

Argonne

National

Vincent P. Calkins, B.S., M.S., Ph.D., Manager, Applied Materials Research, Aircraft Nuclear Propulsion Department, General Electric Company. 10-6. Radiation Damage to Liquids and Organic Materials.

A. B.

Carson, A.B., Manager, Process Design Operation, Irradiation Processing Department, General Electric Compaay, Hanford Works. Pressurized-tube Graphite-moderated Reactors (in Section 13-1).

vi

CONTRIBUTORS

Glen Clewett, A.B., Group Leader, Sterling Forest Laboratory, Union Ca Nuclear Co., formerly Director of Materials Chemistry Division, Oak R. National Laboratory. 14-2. Isotope Separation by Chemical Exchange l. Related Processes.

L. F. Coleman, B.S., Associate Chemical Engineer, Argonne National Laboratory. Instrumentation

(in Section 11).

Charles E. Crompton, Ph.D., Associate Technical Director, National Lead Com pany of Ohio, formerly Deputy Director, Isotopes Division, U. S. Atomic Energy Commission. 14-1. Isotopes and Their Use.

Robert Daane, B.S., M.S., Research Engineer, Beloit Iron Works, formerly Senior Mechanical Engineer, Nuclear Development Corporation of America, Mechanical Engineer, Associate Argonne National Laboratory. 9-4Thermal Stress and Distortion.

Frank B. Daniels, B.S., Assistant Manager, Research and Development, Nuclear Products-Erco, Division of ACF Industries, formerly Incorporated, Assistant to Director of Mechanical Engineering, Industries Group, AllisChalmers Manufacturing Company. 12-3. Power Generating Equipment.


Russell W. Dayton, Ch.E., M.S., Ph.D., Assistant Technical Director, Battelle Memorial Institute. 10-7. Recent Developments in Reactor Materials.

J.

R. Dietrich, Ph.D., Vice President, General Nuclear Engineering Corporation, formerly Associate Director, Reactor Engineering Division, Argonne National Laboratory. 6-2. Reactor Calculations.

William B. Doe, B.S., Associate Chemical Engineer, Argonne National Laboratory. 7-4- Mechanical Handling of Radioactive Materials. Elesa R. Etherington, B.Sc, Housewife.

1-2. Mathematical

Tables.

B.Sc,

Harold Etherington, Assoc. Roy. Sch. Mines, Vice President, Nuclear Products-Erco, Division of ACF Industries, Incorporated, formerly Direc tor, Naval and Reactor Engineering Divisions, Argonne National Laboratory. 1-1. Selected Data and Formulas, 1-2. Mathematical Tables, IS. Units and Conversion Tables.

Myron F. Fair, B.S., M.S., Health Physicist, Oak Ridge National Laboratory. Beryllium Toxicity (in Section 7-2). Kenneth R. Ferguson, A.B., M.S., Associate Physicist, Argonne Laboratory. 7-4. Mechanical Handling of Radioactive Materials.

National

George A. Garrett, Ph.D., Superintendent of Operations Analysis Division, Oak Ridge Gaseous Diffusion Plant. 14-4- Gaseous Diffusion Separation Process.

Division, Remote Control Engineering Raymond C. Goertz, B.S., Director, Handling of Radioactive Argonne National Laboratory. 7-4- Mechanical Materials. G. K. Green, Ph.D., Senior Physicist,

Brookhaven National Laboratory.

6-4.

Accelerators.

Joseph M. Harrer, B.S.E.E., M.S.E.E., Associate Director, Reactor Engineering Division, Argonne National Laboratory. 8-3. Control Instruments and Drives.

John A. Harvey, B.Sc, Ph.D., Senior Physic ist, Oak Ridge National Laboratory, formerly Brookhaven National Laboratory. 6-2. Experimental Neutron Physics.

Vll

CONTRIBUTORS

Alston S. Householder, Ph.D., Head of Mathematics Panel, Oak Ridge National 3-3. Principles of High-speed Laboratory. 3-1. Algebra and Geometry, Computing Machinery.

John P. Howe, B.S., Ph.D., Chief, American Aviation, Inc.

Research, Atomics International, 10-4- Radiation Damage to Solids.

North

Frank C. Hoyt, Ph.D.,

Manager, Nuclear and General Physics Division, Lock heed Aircraft Corporation, Missile Systems Division, formerly Alternate Division Leader, Los Alamos Scientific Laboratory. 4- Nuclear Physics.

John R. Huffman, Ph.D., Assistant Phillips Petroleum Company.

Donald

J.

Manager, Technical, Atomic Energy Division, 5-5. Engineering Testing in Reactors.

Hughes, Ph.D., Senior Physicist, Brookhaven National Laboratory.

5-2. Experimental

Neutron Physics.

Herbert S. Isbin, Ph.D., Associate Professor, Department of Chemical Engineer ing, LTniversity of Minnesota. 13-5. Kinds of Reactors. Wayne H. Jens, B.S., M.S., Ph.D., Assistant Technical Director, Atomic Power Development Associates, formerly Senior Engineer, Nuclear Development 9-1. Physical Properties of Heat Transfer Mediums. Corporation of America. Generated for wjivans (University of Florida) on 2015-09-23 02:45 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

A A. Jonke, tory. C.

B.S., M.S., Associate Chemical Engineer, Argonne National Labora

Precipitation

(in Section 11).

P. Keim, A.B., M.Sc, Ph.D., Director, Technical Information Division, formerly Director, Stable Isotopes Division, Oak Ridge National Laboratory. 14-5. Electromagnetic Separation of Stable Isotopes.

Alf Kolfiat,

Degree in Mechanical Engineering, Senior Partner, Sargent & Lundy,' 13-6. Reactor Building Design.

Engineers.

Herbert Kouts, Ph.D., Experimental Reactor Physics Group National Laboratory. Sidney

Krasik, Ph.D.,

Leader, Brookhaven 5-3. Measurement of Reactor Constants.

Westinghouse

Electric

Corporation.

Control.

8-1. Physics of

Stephen Lawroski, B.S., M.S., Ph.D., Director, Chemical Engineering Division, Argonne National Laboratory. 11. Chemistry and Chemical Engineering.

M.

Levenson, B.S., Associate Chemical Engineer, Argonne National Laboratory. Auxiliary Operations (in Section 11). Miles C. Leverett, B.S., M.S.E., Sc.D., Manager, Development Laboratories, Aircraft Nuclear Propulsion Department, General Electric Company. 13-4- Gas-cooled Reactor Systems.

L. K. Link, B.S.,

Argonne National Laboratory.

Associate Chemical Engineer, 12-2. Economics of Nuclear Power.

C. Rogers McCullough, Ph.D., Chairman, Advisory Committee on Reactor Safe Project Specialist, Monsanto guards, U. S. Atomic Energy Commission, Chemical Company. 8-4- Reactor Safeguards.

Warren

J. McGonnagle, A.B., M.S., Ph.D., Group Leader, Nondestructive Testing, Argonne National Laboratory. 10-6. Nondestructive Testing.

F. L. McMillan, M.S., Assistant

Superintendent,

Operations Engineering Branch, 5-5. Engineering Company.

Atomic Energy Division, Phillips Petroleum Testing in Reactors.

viii

CONTRIBUTORS C. Moise, Ph.D., Assistant Principal Engineer, Liquid Rocket Plant, Aerojet-General Corporation, formerly Assistant Project Engineer, Fox and Whitney Aircraft. 8-5. Reactor-system Dynamic Project, Pratt

John

Simulation.

Karl Ziegler Morgan, A.B., M.A., Ph.D., Director Oak Ridge National Laboratory.

of Health Physics Division, 7-2. Health Physics.

J. Mover, Ph.D., Professor of Physics, University of California Berkeley, and Physicist, University of California Berkeley Radiation Laboratory. 5-1. Instruments for Measuring Radiation.

Burton

C. E. Normand, Ph.D., Stable Isotope

Ridge National Laboratory.

Research and Production Division, Oak ljf.S. The Electromagnetic Separation of Stable

Isotopes.

David Okrent, M.E., A.M., Ph.D., Manager, Fast Reactor Safety Project, MuUigroup Calculations (in Section 6-2). Argonne National Laboratory. Walton A. Rodger, B.S.Ch.E., B.S.Met.E., M.S.Ch.E., Ph.D., Associate Director, 11. Chem Chemical Engineering Division, Argonne National Laboratory.


istry and Chemical Engineering.

Arthur F. Rupp, B.S.Ch.E., Laboratory Services Superintendent, National Laboratory. 14-3. Production of Radioisotopes.

H. A. Sailer, B.S.,

Oak

Ridge

(deceased), formerly Assistant Technical Director, Battelle 10-2. Properties of Reactor Materials.

Memorial Institute.

Ward Conrad Sangren, A.B., M.A., Ph.D., Chief of Computing, General Atomic, General Dynamics Corporation, formerly Chief of Computing and Mathe matics, Curtis- Wright Research, Assistant Chief of Mathematics Panel, Oak Ridge National Laboratory. 3-2 . Analysis, 3-3. Principles of High-speed Computing Machinery.

M. A.

Schultz, B.S.E.E., Project Manager, Westinghouse Testing Reactor, 8-2. The Control System. Westinghouse Electric Company.

Otto Schulze, B.S., M.S., Manager, Nuclear Development, American Machine and Foundry Company, formerly Associate Physicist, Argonne National Laboratory. Matrix Solutions for Reflected Reactors and Evaluation of Material Constants of t/ie Reactor (in Section 6-2).

W. B. Seefeldt, B.S., M.S., Associate Laboratory. Corrosion (in Section

Chemical

Engineer,

Argonne National

11).

Sidney Siegel, A.B., Ph.D., Technical Director, Atomics International, North American Aviation, Inc. 10-4- Radiation Damage to Solids. Harry Soodak,

B.S., M.A., Ph.D., Assistant Professor

College of New York.

2. Nuclear

of Physics,

The City

Data.

Roger Sutton, B.Ch.E., M.S., Metallurgist, International Nickel Company, Inc., formerly Senior Metallurgist, Argonne National Laboratory. 10-3. Struc tural Materials in High-temperature Water Reactor Systems. Untermyer II, B.S., Consulting Engineer on Reactor Technology, Vallecitos Atomic Laboratory, Atomic Power Development Department General Electric Company. 13-1. Water-cooled Reactor Systems. ,

Samuel

R. C. Vogel, B.S., M.A., M.S., Ph.D., Associate Director, Chemical Engineering Division, Argonne National Laboratory. Precipitation (in Section 11).

CONTRIBUTORS John W.

Weil, B.S., Ph.D., Manager, Power Reactor Physics, Atomic Power

Equipment Department,

John

IX

General

Electric Company.

6-1. Reactor Theory.

M. West, B.S., M.S., Vice President, General Nuclear Engineering Cor poration, formerly Associate Director, Reactor Engineering Division, Argonne National Laboratory. 7-1. Calculation of Radioactivity.


Walter H. Zinn, Ph.D., President, General Nuclear Engineering Corporation, 12-2. Economics formerly Director, Argonne National Laboratory. of Nuclear Power.


PREFACE

The

"Nuclear Engineering Handbook"

has been prepared to satisfy a

for an authoritative single-volume compilation of nuclear information and data. More than 70 contributors, each a specialist in his pressing need

icular area of nuclear science or engineering, have contributed sections satisfy the needs of engineers, scientists, students, research workers, and others interested in nuclear science and technology. The handbook presents the basic data used in all phases of nuclear part

to

Science and technology, and theory and practice are covered, with special emphasis on reactor engineering. thoroughly Reactor designers will be particularly interested in the sections on reactor Formulas, physics, radioactivity, shielding, fluid flow, and heat transfer.


engineering.

comprehensive

tables,

and methods

of calculation make this material

useful. Among the special features of the book are a series of important contributions on the present state of reactor technology and the economics of nuclear power. The book has been planned for easy reference with an introductory especially

This section also con

section

which serves

tains a

collection of the most frequently used data and formulas.

E.

as a guide to the handbook.

J.

R. Dietrich, Dr. Walter H. Zinn, and other contributing authors have offered important advice concerning the of particular sections of the handbook. Dr. Norman constitution Hilberry and Mr. Neal F. Lansing have offered many valuable sug gestions, and several officials of the United States Atomic Energy Com mission have been most helpful in promptly reviewing handbook material This assistance and the many long hours of relative to classification. careful work of the contributing authors are gratefully acknowledged. Dr. Charles

Crompton,

Dr.

Harold Etherington

xi


CONTENTS

For the detailed contents of any section consult the first page of that section.

Contributors

v

Preface

xi

Section

1.

Mathematical Data and General Tables

Section 1-1.

Selected Data and Formulas and Guide to the

Handbook


Section 1-2. Section 1-3.

1-1

Mathematical Tables Units and Conversion Tables

....

1-2 1-75 1-134

2.

Nuclear Data

2-1

Section 3.

Mathematics

3-1

Section

3-2


Analysis

3-64

Section 3-3.

Computers

3-142

Algebra and Geometry

Section 4.

Nuclear Physics

Section 5.

Experimental

Section 5-1. Section 5-2. Section 5-3. Section 5-4. Section 5-5. Section 6.

Section Section Section Section

Techniques

.... ....

Instruments for Measuring Radiation Experimental Neutron Physics Measurement of Reactor Constants . Accelerators Engineering Testing in Reactors

.

.

.

.

Reactor Physics

Section 6-1. Section 6-2. Section 7.

4-1

7-2. 7-3. 7-4.

5-2 5-53 5-101 5-123 5-140 6-1

6-2

Reactor Theory Reactor Calculations

Radiation and Radiological 7-1.

5-1

6-29

Protection

.

....

Calculation of Nuclear Radiation Health Physics Nuclear Radiation Shielding Mechanical Handling of Radioactive

Materials

.7-1 7-2 7-22 7-60 7-117

xiii

xiv

CONTENTS

Section 8. Section Section Section Section Section Section 9.

Control of Reactors

8-1

....

8-1. 8-2.

Physics of Control The Control System

8-3.

Control Instruments and Drives Reactor Safeguards Reactor System Dynamic Simulation

8-4. 8-5.

Physical

Properties

Section 9-2. Section 9-3. Section 9-4.

of

Heat

.... ....

Fluid Flow in Reactor Systems Heat Removal from Nuclear Reactors Thermal Stress and Distortion

.

.

....

and Chemical Engineering

Section 11.

Chemistry

Section 12.

Nuclear -power -plant Selection

Section Section Section Section Section Section Section Section 14.

Selection of Reactors Economics of Nuclear Power

Power Generating Equipment

.... .... .... .... ....

13-2. 13-3. 13-4. 13-5. 13-6. 13-7.

Reactor Systems Aqueous Homogeneous Reactor Systems Liquid-metal Reactor Systems Gas-cooled Reactor Systems Kinds of Reactors . Water-cooled

Reactor Building Design Reactor Construction

Isotopes and Their Use Isotope Separation by Chemical and Related Processes

10-2 10-17 10-73 10-83 10-126 10-149 10-168

.

12-1 12-2 12-78 12-114 13-1 13-3 13-49 13-80 13-105 13-123 13-160 13-177 14-1

Isotopes


9-24 9-50 9-110

11-1

.

Mechanical Design and Operation of Reactors 13-1.

9-2

10-1

The Metallic State . . . Properties of Reactor Materials . in Structural Materials High-temperature Water Reactor Systems Section 10-4. Radiation Damage to Solids Section 10-5. Radiation Damage to Liquids and Organic Materials Section 10-6. Nondestructive Testing Section 10-7. Recent Developments in Reactor Materials Section 10-1. Section 10-2. Section 10-3.

Section 13.

8-76 8-89

Transfer

Reactor Materials

Section 12-1. Section 12-2. Section 12-3.

8-21 8-49

9-1

Mediums


.

Fluid and Heat Flow

Section 9-1.

Section 10.

.

8-2

14-2 Exchange 14-15


CONTENTS

Section 14-3. Section 14-4. Section 14-5. Isotopes

Index

follows

Production of Radioisotopes

Gaseous-diffusion

The Electromagnetic

Section 14.

Separation Process Separation of Stable

XV

14-26

14-38

14-44



NUCLEAR ENGINEERING HANDBOOK


SECTION

1

MATHEMATICAL DATA AND GENERAL TABLES BY

ETHERINGTON, ASSOC. ROY. SCH. MINES, B.Sc, Vice President, Nuclear Products-Erco, Division of ACF Industries, Inc., formerly Director,' Naval and Reactor Engineering Divisions, Argonne National laboratory. ELESA R. ETHERINGTON, B.Sc, Housewife.

HAROLD


CONTENTS SELECTED DATA AND FORMULAS AND GUIDE TO THE HANDBOOK

1-1

BY HAROLD ETHERINGTON Guide to Handbook Tabulated Data. . Unite and Constanta The Elementa and Nuclides Nuclear Data Used in Reactor Calculations 5 Constants and Formulas Relating Power, Flux, and Fuel Consumption.. 6 Numerical Integration by Simpson's Rule 7 Reactor Calculations 8 Calculation of Nuclear Radiation 9 Health Physics 10 Nuclear Radiation Shielding 11 Physics of Reactor Control 12 Fluid Properties 13 Fluid Flow 14 Heat Transfer 15 Thermal Stress and Distortion 16 Radiation Damage to Liquids and Organic Materials 1 2 3 4

PAOF. 1-63 1-71 1-74

17 Solution of Equations 18 Atomic Energy Commission Literature. References PAGE

1-2 1-12 1-13

1-2

MATHEMATICAL TABLES

BY HAROLD ETHERINGTON AND ELESA R. ETHERINGTON 1 Tables of Functions 2 Bibliography of English-language

1-19

Tables

1-28

1-3

1-28 1-29 1-30 1-42 1-43 1-45 1-48 1-50 1-53 1-61

1-75 1-127

UNITS AND CONVERSION FACTORS

BY HAROLD ETHERINGTON 1 Dimension and Unit Systems 2 Fundamental Standards and Exact

Equivalents

1-62

1-1

3 Conversion Factors 4 Atomic Units, Constants, and Conver sion Factors References

1-134 1-139 1-142 1—153 1—156

1-1

SELECTED DATA AND FORMULAS AND GUIDE TO THE HANDBOOK BY Harold Etherington

The tables of Art. 1 provide a systematic guide to the more important tabulated and graphic data of the handbook. Subsequent articles contain collections of fre quently used data and formulas largely selected from other sections of the handbook, together with references to pertinent sections.

GUIDE TO HANDBOOK TABULATED DATA

1

Tables

facilitate selection and location of important data. These tables the same sequence as they occur in the handbook, but with no correspondence between table numbers and handbook section The sequence is as follows: numbers. 1

to

18


are arranged by subject, in approximately

Table 1. Tables of Mathematical Functions Table 2. Mathematical Formulas Table 3. Engineering Conversion Factors Table 4. General Atomic and Nuclear Data Table 5. Nuclear Data Used in Reactor Neutron Physics Table 6. Reactor Theory Table 7. Calculation of Radioactivity Table 8. Health Physics Table 9. Shielding Table 10. Physical Properties of Fluids Table 11. Fluid Flow and Heat Flow Table 12. Thermal Stress Table 13. Physical and Mechanical Properties of Materials Table 14. Corrosion and Wear Resistance of Structural Materials Table 15. Radiation Damage Table 16. Chemistry Table 17. Particulars of Reactors Table 18. The Greek Alphabet Table

section and page

Transcendental constants Logarithms: Natural logarithms and sines

Tangents Exponential and functions:

Tables of Mathematical

Handbook

Function

Cosines

1.

hyperbolic

Hyperbolic cosines and sine* Hyperbolic tangenta

Table

1-75

1

1-76 to 77 1-78 to 79

2 3

1-80 to 83 1-84 to 85

4 5

1-86 to 89 1-90 to 92 1-93

6 7 8

Functions*

Function

Bessel functions: Yo(x) and YUz) h(x) and /.(*) Ko(x) and Ki(x)

Auxiliary functions Other functions: l'(x) and x! polynomials Error integral Factorials and reciproculs. Bibliography of tables Legendre

* For mathematical formulas see Table 2.

1-2

Handbook section and page

Table

1-94 to 99 1-100 to 105 1-106 to 109 1-110 to 113 1-114 to 116

9 10

1-117 1-119 1-123 1 124 1 126 1-127

14 15 16 17 19

to 118 to 122 to 126 to 133

II

12 13

Art. 2

SELECTED DATA AND FORMULAS AND GUIDE

SEC. 1-1]

Table 2.

Mathematical

Formulas Handbook section and page

Table

3-5 3-18 3-69 3-69 to 74 3-65 3-68 3-99 to 100 3-128 3-130 1-75

Art. 1.13 Art. 2.2

Subject

Polynomial products Derivatives (list) Integrals (list)

Table 3.

section and page

Table

I-M3

4 S 6 7 a » 10


Fundamental and mechanical: Length Area Volume Mass Time Force Angular Velocity. Flotr. . Density. Pressure. Viscosity

3 4 I 2

i

6 7

Engineering Conversion Factors

Handbook

Property

1-3

1-143 1-144 1-145 1-145 1-145 1-145 I- 146 1-146 1-146 1-147 1-149

Handbook |

Property

section and page

Thermodynamic :

II

12 13 14 17

Energy,

work,

heat of

Energy units

1-147 atomic

physics Power

-I5S -146

Internal heat generation and power density Temperature Specific heat Heat capacity

Thermal conductivity. . . . Heat flux and conductance

Electrical

-151 -150 -151 -151 1-151 -151 1-152

Miscellaneous:

Concentration Corrosion

Summary

Table

4.

table

General Atomic and Nuclear Data* Subjeot

Basic constants: Physical constants Mass-energy equivalents Rest mass of particles Wavelength and energy Summary table

Atomic weights and atomic numbers of the elements: Alphabetical list Periodic

-152 -152 -12

table

Handbook | section and page

1-154 1-155 1-155 1-156 1-13

1-14 11-4

Collected properties of the elements and nuclides: Isotopic abundance and nuclear properties of elements and nuclides (from

BNL

325)

Nuclide chart Fission energy equivalents * For specific nuclear data, see Table 5.

2-15 to 23 1-15 to I9| 1-26

GENERAL DATA

1-4 Table

6.

Nuclear Data Used in Reactor Neutron Physics* Handbook

Subject

-


U»"

Natural-uranium equilibrium spectrum Fast reactors Delayed-neutron data: Fast fission of V"', U">, U'", Pu>», Pu»«, Th«": riit, ffi/ff, 0,., 0*

f.

Thermal fission of U»", U">, Pu»»: ryu, Pi/0, gi,, 0, H

Thermal fission of U"»: Tlfc, Ti, Xi, ffi/fi, Pi, 0,r, Photoneutrons (DjO reactor) Thermal-neutron properties: o-o, »/, , Um, Pu»» (Westcott values) tj, y, a for U18B, U"1, Pu11*, natural uranium L and X/r for Th, ThO«, U, UK). Resonance-neutron properties: Absorption integrals and parameters for Ula8 and Th1*1 Resonance constants A and p for uranium, compounds, and mixtures. Resonance constants A. pi A, and Xo for Th13s and uranium xi for II and UbOs Intermediate- and fast-neutron properties: and Pu"» v(E) for U"!. V". Pu"» Fast a/ and o> for fuels at of nonthermally fissionable nuclei fission of nuclides

Fast-fission effect « Moderators: P, N, 2., {2., Dn, L,r P. N, 2„.n, 2..,*, Xi,,ik, D,k, Lih, {, 2../, £2,./, \i,,l, D/, t L, Xlr r for moderators and mixtures Time and number of collisions to thermalize Resonance constants {a» and ttt/p Photoneutrons (DtO reactor) P for HjO as a function of temperature p for DiO as a function of temperature Fission products : ia(E) for Xe>" Yield and aa of Xelu, Sin1*', and long-lived fission product*. aa{E) for fission product* Distribution and decay — see Table 7 * For data of a more general

character,

Table

scotion

and page

Element! and nuclides in general : Isotopic abundance, atomic and isotopic oross sections (from BNL 325) . . . Thermal-neutron properties: A, p, N, 1 ?-,. {, ya(2,200), «-.(<«) for the element* A, p, JV, t — no, a ,, 7a,
Spontaneous

[SEC. 1

see Table 4.

2-15to23

19

2-13 to M 1-24 to 25 7-8 2-25 to 26

27 I 20

1-24 to 25 2- 13 to 14 2-28 2-29 2-30 2-30, 31 2-33 2-35

27 18 24 25 26 27. 28 33 34

(2-2 14-85 1-26

18

Art. 4.31 29

7-91 2-3 8-3 2-32 2-31

15 2 I 31 30

8-3

2

8-5

4

8-4

3

8-5

4

1-46

39

2-3

3

2-5 1-21 1-26 6-86

6 25 28 13

2-26 6-82 2-27 2-27 6-83

21 9 23 22 10

2-6

7

2-6 2-7 2-6 to 8 2-8 2-5 6-85

8 9 7-11 12(11) 5 12. Art. 9.5

2-9

13

1-20 6-86 6-88 2-10

24 13 14 14

6-83 2-3 i 9-10 ! 1-49 9-16

3 20 41 28

2-11 i 2-11 I 1-23 2-11

15 16 26 17

For radioactivity data, Bee Table 7.

II

SEC. 1-1]

Table 6.

Reactor Theory Handbook

Subject

section and page

Kernels: Diffusion in infinite media Gaussian slowing down in infinite media Transforms Solutions of steady-state diffusion equations: One-dimensional media with localized sources Wave equation for homogeneous bare reactors Equations for reflected reactors Matrix solutions for reflected reactors Lumped gray absorbers Lumped black absorbers Disadvantage-factor constants E and F Control-rod equivalent radii and absorption areas for noncircular rods Power, flux, and fuel consumption: Flux-power relation Fuel consumption, g/Mwd

Table 7.


section and page

Units of radiation dose Radioactivity of the heavy nuclides: 4n, 4n + lf An + 2, An + 3 deoay U isotopes

1-42 chains.

...

isotopes

Tfe isotopes

Neutron activation

and decay data: Cross sections, half-lives, decay 7-ray energy General nuclear data (see Table 5)

Reactor coolants Irradiation and decay formulas: Irradiation and decay Activity of reactor coolants Fission : prompt 7-ray distribution Fission-product

4-35 to 38 11-34 11-38 11-41 7-8

Heat rate nuclides,

6-40 6-58 6-60 6-70 6- 106 6-107 6-78 6-17 to 19

2 5

see Table

16.

Art. 7 Art. 8 17 18 8

5-8 30 31

Table

36

Figs. 10-13 8

II

14 I 34

1-38 1-41 7-72

32 35 8

11-25 7-73 1-39 11-26

Fig. 3 Fig. 4

decay:*

Activity Energy groups

* For decay data by specific

1 3 4

1-27 1-27

Handbook

Table

6-39 6-46 6-56

Calculation of Radioactivity

Subject

Pu

1-5


33

Fig. A

1-6

GENERAL DATA Table 8.

Health Physics

Subject

Ionization, stopping power, etc., in tissue: Specific ionization and range as a function of energy Stopping power as a function of energy Relative biological effectiveness, RBE: General values for particles Heavy-ionizing particles as a function of specific ionization Effect of radiation Dose and dose rate: Maximum permissible, various parts of body Natural background Clinical X-ray exposures Flux for maximum permissible exposure: Various particles Neutrons of various energy Summary table Photon flux for 1 rad/hr. Absorbed and biological dose from neutron beams Maximum permissible concentrations in air and water: General radioisotopes Maximum permissible concentrations in the body


Specific

[SEC.


7-34 7-36 7-32 7-27 7-28

Table

3

Fig. 4 2

Fig. 1

7-30. 31 7-35 7-37

Figs. 2,3

7-32 17-3S I 7-84 1-43 7-66 7-31 to 44

2 4 13 37 3 Figs. 5-13

7-44 11-141 11-141

5 6

7 39 39

1

SEC.

1-1]

Table 9.

Shielding

Subject


1-7


Gummashielding: Absorption coefficients: Total mass attenuation coefficients for elements, air, Nal, HtO, concrete tissue Total mass and linear attenuation coefficients for metallic shield materials . Energy-absorption mass attenuation coefficients for elements, air, Nal. HiO, concrete, tissue Dose build-up factors: Point source Plane source Constants for analytic functions Energy absorption build-up factors: Point source Gamma sources and energy distribution: Fission prompt y rays Fission- product- decay 7-ray energy groups Capture 7 rays Fission-product 7-ray yield, half-life, and energy Attenuation (illustrative and specific cases); Lead for Co*0 and Cs117sources Relaxation lengths Tenth-thicknesses for l-Mev 7 rays Neutronshielding: Dominant energy in hydrogenous shield Removal cross sections for fission neutrons Microscopic removal cross sections for hydrogenous materials Neutron sources and energy distribution: lTm fission Delayed neutrons • Alpha neutron sources Photoneutron sources Attenuation (illustrative and specific data): Fist neutrons by water t Relaxation lengths for fast neutrons Boron, lithium, and nitrogen reaction data Shieldmaterials (including attenuation properties): Compositions and properties of concretes Aggregates Grouts Ordinary concrete Barytes High-density Physical properties Cost Properties of transparent shield materials


7-62 7-112 7-63 7-67 7-68 7-100 7-69 7-72 7-73 7-74 7-16

1 21 2 4 5 19 7 8

Fig. 4 9 2

7-68 2-35 11-136

6 36 37

7-80 7-82 7-1 II

II 12 20

7-91 7-92 7-92 7-92

15 16 17 18

7-85 2-35 7-77

14 36 10

7-113 7-113 7-112 7-112 7-114 7-115 7-115 7-122

23 24 22 22 25 26 27 1

1-8

GENERAL

DATA

[Sec.

Physical Properties of Fluids

Table 10.

Handbook Properties'^

Fluid

Parameters

Coolants (general).

section and page 12-58 to 59

Gases:

Air

Molliere chart Enthalpy

Pressure

Cp, ii, k, Pr

c

Temperature Temp., press. Temperature

k

Pressure

9-3 9-3 9-25 9-4


9-4 9-4 9-25

Eq. (5) Eq. (7)

Pr

f


Fig. 14

Cp

f

Carbon dioxide.

29

Usual Temperature Temperature Temperature

Cp, ii, k,

Argon .

Table

Cp, n, k,

Pr

12-129 9-2 9-25 9-2 9-3

I

Eq. (3) 2 3 4 5

Eq. (4) 6

7 8

Helium.

c„ M, *, Pr

Temperature Temperature

9-5 9-25

Hydrogen.

c„ ii, k, Pr


9-5 9-6 9-25

Eq. (10)

Pressure

Pressure

9-6 9-7 9-25 9-3

12 13 Eq. (2) 6

Temperature

9-25

Eq. (6)

Nitrogen.

cp, ii, k, Pr

f

Temp., press. Temperature

Oxygen. Steam and water: Steam and water.

Molliere chart Saturated-Bteam

Steam.

Superheated-steam

tables

tables

Cp

Pr Light water.

Critical properties r Cp k

f

p(0-IOO°C) Pr Heavy-water vapor. Heavy water

Critical properties (temp., press., density) Vapor pressure Latent heat of vaporisa tion e(IO-250°C) x e,

Usual Usual

12-124, 125 12-119 to 120

Usual

12-121 to 122 9-7 9-8 9-25 9-8 9-9 9-9

Temp., press. Temp., press. Temperature Temp., press. Temp., press. Temp., press.

9

10

II

Fig. 4 I

14 15

Eq. (8) 16 17 18

Temp., press. Temp., press. Temp., press. Temp., press. Temperature Temp., press.

9-9 9-10 9-11 9-12 9-13 1-49 9-14

19 20 21 22 25 41 24

Temperature

9-25

Eq. (9)

Temperature

9-15 9-15

25 26

Temperature Temperature Temperature Temperature Temperature

27 28 29 30 31

1

1-9


SEC. 1-1]

Table

10.

Physical Properties of Fluids.

Handbook

Properties*t

Fluid

(Continued)

Parameters

section and page

Table

13-81 9-18 9-18 9-19 9-19 9-19 9-20 9-20 9-20 9-21 9-21 9-22

1 33 34 35 36 37 38 39 40 41 42, 43 44

Molten

metals:} General table Vap. press., Vap. press., Vap. press., P. Cp, it, k Vap. press., Vap. press., Vap. press., Vap. press., Vap. press., Vap. press., Vap. press.,

Lead

Lead-bismuth eutectic Magnesium Mercury Potassium Sodium-potassium alloys Tin

p. Cp, it, k p, cp, it, k p, Cp, it, k p, p, p. p. p. p. p,

c„

Temperature Temperature

Temperature Temperature Temperature Temperature Temperature Temperature Temperature Temperature

n, k

Cp eP, it, Cp, it, cP, it, cp, it, c„, a,

k k k k

Temperature

*

Miscellaneous: Molten sodium hydroxide. . P. e,, a, * Diphenyl. liquid} Vap. press., p. cp, a

Temperature Temperature Temperature

P „

Comparison of reactor

cool


ants

9-17 9-22 9-22 9-26, 27 See index

Physical snd nuclear prop

* s>= density, cp =»specific heat at constant pressure, viscosity, Pr — Prandtl number, v = specific volume.

where T is the absolute t Tables for molten points and latent heats.

of expansion.

11.

k — thermal

thermal

data

pM

including

Handbook

Dimensionless

section and page

ratios

9-32

-

Orifices

Equivalent diameter of noncirculur Turbulent flow: Long channels Fanning factor Pipe lit rum-

duct

of cross section and nozilcs in pipes Equivalent diameters of noncircular ducts Liquid-gas mixtures Heat flow: Thermal conduction: Plates, cylinders, and spheres Packed beds with stagnant fluid Forced convection in uniform channels: Turbulent flow: Nonmetallic fluids Metallic fluids Laminar flow Forced convection outside of channels Natural convection Combined natural and forced convection. Condensation Change Orifices

Boiling Hot-channel factors

For gases 0 = \/T, melting

and boiling

Fluid Flow and Heat Flow

Subject

Fluid friction and pressure drop: Streamline flow: Long channels Redistribution loss

29

conductivity, n ** absolute —.

For any fluid, 0 — ~ — = v M

temperature. metals and diphenyl give other

Table

Eqs. (11-13)

12-58. 59

erties

t Volumetric coefficient

32 45 45

.

9-30 9-30 9-37. 30 1-51 9-30. 33 9-33 9-37 1-52 1-52 1-51 9-42

1-54 to 59 9-54

9-59 9-59 9-61. 62 9-63 9-64 9-65 9-65 9-66 9-87

Table

3

1. Eq. (20) 2 4 (with Table 1) 42 1. Eq. (28) FiK. 2 4 44 45 43 5

46-50 2

3 4 5-7 8

9, Fig. 3 10 Art. 4 Art. 5 12

GENERAL

1-10

Table 12.

DATA

Thermal

Stress Handbook

Subject

Stress Constanta

Table 13.

section and page 1 1-61, 62 9-1 10 9-113

\

of materials

Physical and Mechanical

51

Art. 1

Handbook

2 1

11-5 to 7 10-3 10-7 10-17 to 27 10-22 11-35 10-23 10-24 to 28 10-22, 24 10-29 to 32 10-32 to 37 : 10-22 1 11-41 10-33 to 35 10-36. 37 ' 10-22. 37 ,11-38

Fig. 3

Art.

'

Physical properties. Coefficient of expansion. . . Mechanical properties. . . . Uranium oxide Uranium alloys (Al and Zr) . Thorium Physical properties . . Mechanical properties. Thorium alloys Plutonium. Solid moderators: General table Beryllium Graphite Chemical composition Physical properties Mechanical properties Structural materials: Aluminum Physical properties Alloys Stainless steels and high-tem|>erature Zirconium and alloys Control materials

12-70 10-39, 51. 52 10-55 to 58 10-55 10-56 10-57 to 59

alloys.

* For resistance to chemical attack see Table 14. t Sec also Sec. 10-7. X See also Table I of Sec. 10-2 (p. 10-18).

10-47 10-39 10-47 10-60 10-39 10-70

to 66 to 43 to 71

For radiation

damage

9 3 5-13 2. 4t 14-20*. Art. 1.28 Art. 1.3 21 15 21-26 27. Art. 1.38 2. 28t 12 33

29.

Various media on graphite See also specific

material in Sec. 10-2

Handbook

4I-43J, Art. Art. 3.2

2.4

46 47* 48-50

Art. 2.3 29J

34-40 52-59. Art. 4.2 29-33J, Art. 2.2 64-66 see Table

Corrosion and Wear Resistance of Structural Medium

1.2

n

to 51

to 51

f

Table

section and page

The elements: Physical properties Electron configuration. . . Atomic radii Fuels and fertile materials: Uranium


Table

Properties of Materials*

Material and property

Table 14.

[Sec.

15.

Materials

section and page

Table

10-20. 21 10-75 to 82 f 10-59 I 12-74

Figs. 1. 2 1-8 51 34

1

SELECTED DATA AND FORMULAS AND GUIDE Table 15.

Radiation

Damage Handbook

Material

Table

section

and page Graphite Graphite-uranium Bervllium oxide

10-110 to 116 10-122 10-116 10-118 to 122 10-106 to 108 10-124 10-133 10-136 10-140 10-142 10-144

oxide

Water and aqueous solutions Elastomers Plastics

Table 16.

Handbook

By nuclides Table 17.

I 2

7-16 11- 12 to 24 11-28 to 31 11-25

2 4 5

Handbook

Capital

Lower case

A B

a, a ft s

Gamma Delta Epsilon

A

o, d

Z«ta

Z

Eta Theta Iota Kappa Lambda Mu

H

f

7

E «,

i

e

K A

M

Fig. 4 33 6. 7

section and page

Table

13-124 to Ml 10-84 to 91

1-7 1-7

The Greek Alphabet

Table 18.

r

Fig. 3

Particulars of Reactors

Subject

Beta

Table

11-4 11-5 to 7

11-26 1-39 11-32

Gross

Name

Capital

Nu

N

Omicron

0

Xi

Pi Rho

Lower case

{

n P 2

1

Tau

T

0, <>

Upsilon

T

1 X JC,


section and page

Periodic chart of the elements Physical properties of the elements Fission-product nuclides: Half-life and energy of fission nuclides Yield and decay characteristics Composition of fission products as a function of cooling time Fission-product activity Fission-product heat:

Alpha

17. 18. Figs. 8-17 24. 25 19 21. 22. 24. 25. Figs. 20. 21 14. 15 Art. 5.7 1 2 3 4 5

Chemistry

Subject

Name

1-11

X

Phi

*

Chi Psi Omega

*

T

X *

SEC. 1-1]

1-12

GENERAL DATA

[SEC.

UNITS AND CONSTANTS

2

Mathematical

2.1

Constants

Table 19 gives some important mathematical constants. stants are given to greater accuracy in Table 1 of Sec. 1-2. Table 19.

■/« 0.36788

.1/

log e

Id 10

2.3026

These and other con

Constants 1.1284

0.31331 I 111

0.43429

* Factor used in correcting 2.2

Mathematical

9.8696

3. 14159

1

2.71828

In 2

First zero of Jt(x)

0.69314

2.4048

for Maxwell distribution.

Engineering

Units and Conversion Factors

The column "More Table 20 gives conversion factors for engineering units. exact value" includes definitions (marked with asterisks) and equivalents calculated from definitions. Table 20.

Conversion Factors for Engineering

Units Reference

Usual Unit

engineering

approximation

More exact value

to Sec. 1-3

Other units Definition

and

conversion


factors 1 cm 1 in

0.3937 in.

Art. 2.11 Art. 2.11

2.54 cm

0. 393 7*t 2 540005

0. 001076 ft< 6.45 «'in' 929 cm1

0.001076387 6.451626 929.0275

1 ft' 11

3.53 X l0-«ft» 2.83 X 10' cm' 1.000 fin • 231 in.'

3.531445 28.317.02 1.000.028 231*

1 kg 1 lb

2.205 1b 453. 6 g

2.204622

453.5924277*t

Art. 2.12 Art. 2.12

1 day 1 year

86,400 aec. 3. 16 X 10' sec

86.400* (mean solar day) 3. 155695 X 10'

Art. 2.13 Art. 2.13

Standard gravity, go Standard gravity, yo

32. 2 ft/seo> 981 cm/sec»

32. 1740

Art. 2.14 Art. 2.14

1 lb/ft>

62. 4 lb /ft'

0.0160 g/cm'

62.42833 0.01601837

1 atmosphere

14.7 psi

1.013.250* dynes /om»

Table 14

tr

Hie + H(tr

- 273°C 32) - 460°F

-273.16* -459.69

Art. 2.21 Art. 2.21

10' ergs 4. 187 joules 0.00397 Btu 252 cal 1.60 X 10" joule

10'* 4. 18684 0.00396832 251.996 1.60206 X 10-"

Art. Art. Art. Art. Art.

Table 4 Table 5

1 in." 1 ft'

Time

tc

Absolute zero Absolute zero 1 joule = 1 watt-sec 1 I.T. cal 1 I.T. oal 1 Btu

X

Table 6 10>

980.665*

Table 6 Table 6 Table 7 Table 8 Table 9 Table 13 Table 14 Table 18

32

2.3 2.23 2.23 2.23 4.32

Table 15

Table 28 Table 16

1 kw 3412. 10 Art. 2.3 34l2Btu/hr Heat flux and power density Table 22 13.270 Btu/(hr)(ft») 13.272. 1 Table 22 3.170 Btu/(hr) (ft") 3.170.20 96,600 Btu/(hr)(ft«) 1 watt/cm' a lkw/1 Table 23 96.620.4 * Exact value by definition. f United States definition. Other measures involving this unit are based on this definition.

1-13


SEC. 1-1]

2,3

Physical

Constants and Conversion Factors

For more exact values and additional data, see Art. 4 of Sec. 1-3. 2.31 Physical Constants. Table 21 gives the physical constants most used in nuclear engineering (see Art. 4.2 of Sec. 1-3). 2.32

Mass and Energy and Wavelength 1 amu = 1.660 X 10_M g = 931 Mev Mass of electron = 0.00055 g = 0.51 Mev

For masses of particles, energy equivalents, and wavelengths, Table

Avogadro's number N, chemical scale


Physical

wavelength of

Art.

Best value

0.602 X 10" (g-mole)-'

0.602322

2.86 X 10-»/«M cm

2 86005 X 10 >/c7W

(£ in

4.32 of Sec. 1-3.

Constants

Usual engineering approximation

Constant

De Hroglie neutron

21.

see

X 10"

Reference to Sec. 1-3

Table 27 Table 30

ev)

Bultf manna constant *

8.62 X 10 »ev/°C 4.79 X 10 *ev/°F

8.6167 4.7871

Electronic charge e

1.60 X 10" coulomb 4.80 X I0'°esu

1.60206 4.80286

Energy of 2.200-m/sec neutron

0.0253ev

0.0252973

Table 30

20°C (69°F)

20.426°C(68.767°F)

Table 30

Molar volume of ideal gas at 1 atm.. chemical scale

22.4 X lO'cm'-atm. /g-mole 359 ft'-atm./lb-mol

22.4146 358.746

Physical scale /chemical

1.000

1.000272

3.00 X 10" cm /sec

2.997930

kT temperature for ni/sec neutrons

2.200-

X 10-« X 10' X 10 -'• X 10-"

X 10"

3 3.1

Table 27 Table 27

Table 27 Table 27 Table 27 (footnote) and Art. 4.31

scale

Velocity of light c

Table 27 Table 27

X 10"

Table 27

THE ELEMENTS AND NUCLIDES Atomic Weights and Atomic Numbers

Table 22 gives atomic weights and atomic numbers of the elements. Tables 1 and 2 of Sec. 11 give, respectively, the periodic chart and important physical properties of the elements.

GENERAL DATA

1-14

Actinium Aluminum. . . . Americium. . . Antimony Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium . . . Carbon Cerium


Cesium

Chlorine Chromium Cobalt Copper Curium Dysprosium . . Einsteinium . . Erbium Europium. . . Fermium Fluorine Francium Gadolinium . . Gallium Germanium

. .

Gold Hafnium Helium Holmium .... Hydrogen. . . . Indium Iodine Iridium Iron Krypton Lanthanum. . . Lead Lithium Lutetium Magnesium. . . Manganese.

..

Mendelevium,

1

Atomic Weights and Atomic Numbers of the Elements

Table 22. Element

[SEC.

Symbol

Atomic

Atomic

number

weight*

Ac

Al Am SI,

A Aa

At Ba Hit Be Bi B

Br Cd C»

Ci

C Ce Cs

CI Cr

Co Cu Cm

Dy Es F.r

Eu Fin F

Fr Gd Ga Qe

Au

lit

He Ho

II In I

lr Fe

Kr

La Pb l.i

Lu Ms Mn Mil

• A value given in parentheses

89 13 95 SI 18 33 B5 36 97 4 83 3 33 48 20 98 6 58 55 17 24 27 29 98 66 99 68 63 100 9 87 64 31 32 79 72 2 67 I 49 53 77 26 36 57 82 3 71 12 25 101

227 26. 89 (243) 121.76 39.944 74.91 (210) 137.36 (249) 9.013 209 00 10. 82 79.916 I 12 41 40.08 (249) 12 01 1 140. 13 132.91 35.457 52.01 58.94 63.54 (245) 162. 51 (254) 167 27 152 0 (252) 19.00 (223) 157.26 69.72 72.60 197.0 178. 50 4.003 164.94 1.0080 114.82 126. 91 192.2 55.85 83.80 138.92 207. 21 6.940 174 99 24 32 54. 94 (256)

Mercury Molybdenum. . Neodymium. . . Neon

Neptunium. . . . Nickel Niobium

Hg Mo Nd Ne Np Ni Nb N No Oe

Nitrogen

Nobelium Osmium Oxygen

o

Palladium Phosphorus.

...

Platinum. ..... Plutonium .... Polonium Potassium Praseodymium

. Promethium. . . Protactinium. .

Pd P Pt Pu Po

K Pr Pm Pa

Radium Radon Rhenium Rhodium Rubidium Ruthenium. . . .

Ra Rn

Samarium Scandium Selenium

Sin Sc Be Si Ag

Silicon Silver

Re Rh Rb

Ru

Sa

Sodium

Strontium Sulfur Tantalum Technetium

Tellurium Terbium Thallium Thorium Thulium Tin Titanium

Sr S Ta .. .

Tc Te Tb Tl Th Tm Sn

Ti

Tungsten

W

Uranium Vanadium. ... Xenon Ytterbium Yttrium

u

Zinc Zirconium

denotes the mass number

3.2

Symbol

Element

V

Xc

Yb

y

Zu

Atomic number

80 42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 11 38 16 73 43 52 65 BI 90 69 50 2 7 9: 2: 5 70 39 30 40

of the isotope of longest known

half-life

The Nuclide Chart

Tables giving important properties of the nuclides are listed in Tables 4 and 5 of this section. Table 23 is a chart giving selected properties of the naturally occurring nuclides and their neutron-activation products. 3.21 Use of Table 23. This chart gives the atomic number, atomic weight, and neutron absorption cross sections of the elements; the isotopic abundance and neutron absorption cross section of the naturally occurring isotopes; and the half-life and modes of decay of naturally radioactive and neutron-activated product nuclides.

SEC.

1-1]

1-15

SELECTED DATA AND FORMULAS AND GUIDE A Chart of Nuclide Properties

Table 23.

mscngi &Kr« rvFtwdmtew Co"tu"gtphoWiCtm oftlemynH (ToMe 22).

DOW Onrod ho»op< oh

00119" # TO 1.3X I0*j h fl .*"H

- Additonoi flown supptamtntory bo* - Horf-**» h'hourt, iy"ytar», d'ooyl, m«mmitt, *' - Mo*ofOtcot

I2.4f.

Cum wgarj* AHen

Holf-M■ Mootof(Mcoy

rcbdri rj }. crouMC«n(r\yCictpfwn«nt othwwiM Itartd).r( •fiUW crottMCl-0n(r '■■;.-s or* for2200m/MCOictpt0) notedin Art.3.21Oftail. PN'pilt ntulrofit.

m

IT

* .B.

0

».0C7 «t40

r

»O0O57 OOO'S He MOO

4

99965 0015 1226, ■ 35 4003

9L9

OMo

900 # .033 U4XC*hIrOO SHOT Be

* T *Oi3

*oo

c«

" *n»r eh*

PLC

I

■3

1

"99.53|037 0

# us 000021 99T59 0037

204

I*- .5 .00022. 00002]

k

.',N .o

rO0O2

|r.0038UO009|

009 2C.4I3

r

.P

5600

!20n »■ 0052 MOOf * 1.9

\:

2299 *

55 2432 » .27

* ots 2596

2J0m A*

23 n.09

»

,4

r*3

2«f>

3

P.

20 950

0.75

0(7 5.0a

870

0

» -20 12OM . 4»

|
:p

»

.Si

•S


H

0'

1-16

GENERAL DATA Table 23. G

*s

3 33.4 S•

rC!

» 35 ■ 35.94

..A

5 39■, )

■K

r

?. 40Ot

*0Co

» .4 4
„Sc

» 2 47.K

«TI

» ex SCLK

oV

#■ 4J MO

„Cr

r J 34.t4

»Mn

r 13.3

3i»

■*


itCo

» ST 5671

MNi

U.34

„Cu

r 17

joZn

• HO n.72

„Ga uGt

»A; «St „Br HKr .rRb »Sr »v «»Zr «Nb „Mo „Tc

72*0 V ?3 74* r 43 7«SC # 13 739* V 6C •ISO •&4fl # 7 •7*3 « IS M»2 » 1.3 >?2 * * • I M»5 r

U

A Chart of Nuclide Properties.

[Sec. (Continued)

1

A Chart of Nuclide Properties.

Table 23. On r 15 Rh ■> LO

* 00 06.4

r

•

Ru

•01 104 WO ■

I

V 44

190 174 110

Cd


W44 HAS • u

1X4

129

m

J

r

117 M

124 127 00*2 36A4 K

IIS Ml

»«

HO

, 1

»J r

* .7 -WW

..

J

129 <30 29.4 4J

13) 2L2

r

132 (33 i34 29l9 92TeJ KX4

v < 9 # 49 w < B # no w < 3

Bo Wfl—

[5«**2794 iT]

T. Notmrt batin*, it.

r

ISO ami r

9

r<9 2Jl

i33

r

139 64

7

l-.sl

Ct 139 134 <37 136 139 131 192 r 2* 113 1 717 99w 44 74 242 IL64 0.097 •» 1 r <4 r 9 * < 1 r 4 1* 49 «X r 3

•Mr

r

»«1

J

-2»IO»,[

147 «c.«

144

149

144

I4T

144

144

130

149

149

I-9XC7*

t 8290 1123

922

En TP" 9000

r 4VX

6d

040

2344 2J9

W.T

204

137 199 19.7 24.9

199 m

WO 2L9

r M fi' 724 too

-wopoc

r <129 K

Tb '37 198 w «3 fiI9« 0092 64ft aow 1544 229

» 43

K

* «00

,.H,

l«2 0J3C

' 194 «3 79m L94 R

199 o> K

144 534

167 224

51

r

Nd

» 49

* 94 _ *727 MEr « 170

™'

f

145

■Or

r

'U. L

123 124 129 UI 4i 4* • X v AO" ao. ..2 «.«- f.O04

124 127 129 129 130 131 344 WW,23m 314 19.7 * M IK)dLt. r .019334IT. W 14 ♦4 94*fi' ♦.13 72- fi' * .2 ■30 29.0m '"^ « 64 »:w

129 tJt

- .QL-—121

■4*rr

r 3400C

m

121

123 7J0

. •. ■

r « t«a;3

I99J

J*

oo

■ a 3739

4.4m I.T.42. fi- |

1

IT, ,22m 1Pd1" 33* 1 » uO HQ*Ptl.TJ I lf>0f» *t .10 | 1 116 H7 at •13 114 H5 HZ 7« 3h IT. 12.8 244 12.3 28.8 r J4 *3d0" SOW 93hfi~ » .03+27.00C fl-

r .001 ♦J m

r «»

T

.4 fi

K.L •*

1

■98 mM9 94«fir 13. 0.39 14.3

r 16 132*

103 ..Si. - •03 4CM <9h

«■ 1.2 «"

Ok K

r .006

*10

-F'

'94 12.7 12.7 17.0 31.9

Afl KM * 30 («,*♦. * MS 12.4 1.3 L22 6.7K an T

122 123 '24 90. 37 244 43 Sb on r 7* f»Jfi r 3.5 fi~ aow 604 2.9 0.99 48 XX • TO'74 K v 2.7* 400 # 7

» u <2T«

(Continued)

»

M 1.9

r

UJ

.Ce

97

*- II fi' » 91.4 2-3>n■464 ■

"44 In 12 * 91 1.02 B2< aw

» L2 1642

r

K

*S300

,Cl

96 18

in '04 >09 1.0 106 ior 27.3 7XK3*y 26,7 I3.CH ■4 H

•03 22.2

** to 0241

»*4

1-17


SEC. 1-1]

9.7m

r

(84

w 2

170 M9

169 944 fi'

163 2\0

164 262 2700

«t.L 168 271

162 299

r

9

171

f

Ho

163 2.32h fi166 100 2T.2h l>30|) m 64 fi'

GENERAL DATA

1-18 Table 23. •67.27 HEr „Tm *,*» „Lu

„Hf

Yb

174 018

179 70d

r K>6 I0O95

*~O00

K

r

1

r


.T1

nBI

IT* 52

#
» i«.a

a -20

188 L39

* 98 200.61

197

189 199 13-3 l*J

» 380 204.39

190 284 r

*L

PI

pt<*0~©«, «|

196 197 0.0 •Ml 3M30 K

199 O0

8ITI 172 173 14.3 21.6 162

IT4 3(8

179 4.24

199

•J

too 20* 23,1 a.2

r

39.3

tot 294 S Tl

K

r

203 494

"

» 033

r

194 198

8U

•

II

K

8

r

.4

r.« 3

197 198

(98 29.4

v L2 » 27 «

71

* 4 *4 100 2.70

U

- A*

t*t SOn 0-

w 98

0208 703 4,20*

r

*■Jl »«S

L3 *

199 S3.T

209 5.2*

204 88

0~ r 293

Pb

• I* 744

199 ,»

193 38)

r t» 190 V 990 "O0I2 104 0.79
2500 r <50 #<90 •

* 13 207.21

192 41.0

191 184

.~90

ITT 2.0*

v T 0$' 97.40"280 684 m 3.7h*-| r 35 4000 0~

I8i 464

6

IT6 12.T

*~*0

M.OOO K

Ir

1

7.9*

9 9 1294

90 • 89 w 10 #" 182 O0i2 99.999 1124 To m 22 $183 184 •83 1*6 *7 ■0J4 •406 26.2 14.3 30.7 744 267 24h W # to 9 II * 2.0 *" r 38 r
r 480 m K) 197.0

MS

$too

r

w« «.-,?. |

184 185 0.018 9M

{Continued)

9.4d

Tm 168 r 125 0'4 324 303

178 179 190 17.1 13.9 382

ITT AS

M

1902

i*Hfl

■>2

• 37 174.99 W O* 17850

Properties.

of Nuclide

167 '66 33.4 22.9 27J

125 173.04

r 16 18622

_Aj| rtAU

165 KM K

K

» 170 168 M

r 21 .8338

„lr

A Chart

162 163 164 0.136 75m 156

1

[Sec.

L

26

tor 21

209

92

r .03 #■.73 00043

r

M

2j»»o«r»un

mPo «At •«Rn

«Fr 228OS MRa

«M

*«. »

„Ac N>Th

■

Ac

>22H

!H_. 233

j09 232

Th

w 7.5

„Np

» 73 Pa

•tPo

«u

"100

238.07 » 7.68 [w,4J*>

lulSJ

IN |

nan™

BMBaM

*>n3T

j

W

r

It.."-444

«.»>

"2T4fl 234

238 • 0Ti4 6.T54 •9*26 213" ]j-391"0^ 694I KJ3 »2.3M

!»>■»,7

"I

H

«"

|

;

99i

2« HI a«oo, IVI, ■•III

SEC. 1-1]


1-19


The data presented and the method of using the chart are explained in the key at the The following limitations of Table 23 should be noted. top of the chart. Cross Sections. Cross sections are for 2,200-m/sec neutrons with the following exceptions. 1. Where cross sections are given for the spectrum of pile neutrons, this is denoted by the subscript PN. 2. Where the absorption (or fission) cross section is not l/t»-dependcnt, the value has been multiplied by the appropriate factor,1'* so that a correct average cross section for the Maxwell distribution will be obtained when the Maxwell-distribution

The factors are: Cd and Cd11', 1.3; Xe1", 1.16; factor V»72 = 1/1.128 is applied. Sm, 1.5; Eu, 0.95; Gd, 0.85; Hg, 0.95; U, 0.99 for o-„ and 0.981 for 07; U"s, 0.981 for o-„ and a/. For consistent treatment, the cross sections for Pu"9 should be multiplied hy the factor 1.075, but this factor is highly energy-dependent, increasing to over 2.0 in the range of temperatures used in power reactors. The cross sections of this nuclide (and other fuel nuclides also) are therefore read from curves or tables as a function of energy or temperature. The energy of emitted particles is not given. Energy of Decay. Gamma Emission. Decay and neutron capture are usually accompanied by gamma Also, no notice is taken of exceptions, emission, which is not specifically indicated. such as decay unaccompanied by y rays, or emission, by internal conversion, of orbital electrons instead of y rays. Isomeric states are indicated only where they arise from Isomeric Transition.^ neutron activation and have a long enough half-life to affect significantly the activity of the irradiated material. The principal mode is given first; modes accounting More Than One Mode of Decay. for less than 10 per cent of the decay are given in parentheses. 3.22

Important

Charts of Nuclides.

Charts of the nuclides usually include all

known nuclides, stable and unstable, and may give as much nuclear data as con sistent with readability. The General Electric "Chart of the Nuclides" (1966). Copies of this important wall chart plus an accompanying booklet may be obtained free by writing to Dept. 2-119, General Electric Co., Schenectady, N.Y. This chart was used as the primary reference in preparing Table 23. This is a detailed large-scale accordion-folded Trilinear Chart of Nuclides (1957). strip chart by W. H. Sullivan, for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

4

NUCLEAR

DATA USED IN REACTOR

CALCULATIONS

For references to nuclear by H. Soodak, and Sec. 6-2, by J. R. Dietrich. data presented in this handbook, see Table 5 of this section. See also Ref . 3. See Sec. 2,

4.1

Cross Sections and Related Data

Nuclear properties of moderators are given in Table 13 of Moderators. and Tables 13 and 14 of Sec. 6-2. Table 24 of Sec. 1-1 gives a collection of these For age of metal-water mixtures, see Table 14 of Sec. 6-2. properties. The absorption coefficient and diffusion length of heavy water of Heavy Water. commercial For isotopic purity are greatly decreased by light-water impurity. heavy water of ordinary purity 4.11

Sec. 2

£„

m 0.000033(1

+ 6.56i)

* Superscript numbers refer to References at end of subsection. t Isomeric states (excited states of appreciable half-life) are common in nuclides formed by neutron capture: in the majority of cases, the higher isomer reaches the ground state by gamma emission (iso meric transition) with a half-life measured in minutes or less, although sometimes extending to many months.

*

Mr.

3.51 0.480 0.86 0.385

0.0195 0.0000335 0.00109 0.000342"

0.48 2.86 1.43 2.75

X,, (cm)

0. 159 0.953 0.481 0.917

D (cm)

of Moderators

(cm)

1

2

it

a

J.

tXf ,

~ U.)

- du

by Soodak

(Sec.

2.57 3.70 1.47 2.75

5.6 1.12 9.85 19.08

Vr

and Dietrich

131F 1.291 0.562 1.0161

D

properties

2)

\_ 3

'

Jutk fS.(u) The values tire considerably lower for commercial I)»0 containing fraction of per cent of H2O (see Art. 4.1 ))• ** Based on commercial graphite having aa ~ 0.0048 barn. tt Values of 2.73 and 2.76, and a compromise value of 2.80 are often used for L, with corresponding values of 7.45. 7.62. and 7.84 for L*. — Ed.

a

/ ■/■!!>t"[Z.(u))'(l /"0-5 ro.5

is

~

in selections

1.28} 0. 179J 0. 153 0.612

Epithermal

8. 12ft 28900{ 441 2680

1.40{ 0.351$ 0.737 0.385

(68°F)

{2.

0.920 0.509 0.2078 0. 1589

at 20°C

L' (cm«)

Templin*

2.85ft 170| 21.0 51.8

L.

properties

By

Properties

1.

Collection of these data into necessary in some cases to resolve differences single table has made L. Templin has kindly undertaken the selection of values in this table. reference density of Calculated g/cm*. Actually 0.998 at 20°C. quantities are for Mev to thermal less, Values are from about 0.01 Mev to thermal. 2# averaged from from the formula Calculated

Graphite

0.0334 0.0332 0. 1229 0.0802

2. (cm"')

2. (cm-')

Thermal

Nuclear

L

l.OOt 1.105 1.84 1.60

density, g/cm3

Nuclei per cm3

24.

J.

H,0 D.O Be

Moderator

Reference

Table


(Sec. 6-2)*

31 125 97 364

T

a

a

}

0

r Cross Section

-

697.8

-

Thermal

Reactor

a, = 581.6 2.070

540.7 539.5 538.5 537.4 536.4

540.4 539.2 538.2 537.1 536.1

Values for Well- moderated

r.

650.2 649.1 648,0 647.0 646.1

,

"Effective

532.0

644.5 643.3 642.1 641.0 640.0

Spectra."

2.083 2.0825 2.082 2.0815 2.081

2.085 2.085 2.084 2.084 2.0835

».

0.07

1398.6 1431.5 1466.5 1503.8 1543.4 1584.9 1628.2 1673.3 1719.0 1765.2 1811. 1858.0 1903.6 1948.4 1992.4

1424.1 1473.2 1525.1 1579.4 1636.1 1694.8 1755.5 1817.7 1881.2 1945.8

1270.3 1291.9 1315.1 1340.4 1368.2

-

1225.0 1258.4 1295.0 1334.8 1378.0

1103.2 1122. 1143.5 1167.7 1194.7

-

See Art.

Temperature*

*/

1099.5 1133.4 1168.3 1204.0 1240.4

-

1.9915 1.984 1.976 1.968 1.960 1.952 1.943 1.935 1.927 1.918 1.910 1.902 1.894 1.8865 1.879 1.8725 866 1.860 1.854 1.849

1.998 1.986 1.975 1.963 1.951 1.940 1.929 1.918 1.907 1.897 1.888 879 1.870 1.862 1.855

0.07

2.049 2.040 2.030 2.020 2.009

=

»/ - 738.02 - 2.0906

1165.8 1191. 1216.7 1241.6 1266.2

1040.3 1064.3 1089.1 1114.4 1140.

938.0 956.0 975.2 995.7 1017.4

869. 880.8 893.2 906.7 921.6

= 0.07

pu«»

949.3 976.4 1005.0 1035.2 1066.8

841.0 859.0 878.8 900.4 924.0

776. 786.6 797.8 810.5 824.8

-

of

4.12 and Ref.

C. H. Westcott:

»/ -

579.1 580.2 581.4 582.4 583.4

547.5 546.0 544.5 543.1 541.8

547.6 546.0 544.5 543.0 541.6

2.083 2.083 2.084 2.084 2.0845

Function

7

*a = 590.0

531.5 531.7 532.2 532.5 532.8

657.3 655.6 654.2 652.8 651.4

555.8 554.0 552.2 550.5 548.9

556.8 554.8 552.8 550.9 549.2

as

4

From

1

646.3 647.5 648.9 650.1 651.3

652.5 650.6 648.9 647.3 645.8

574.0 575.1 576.1 577.1 578.1

667.7 665.4 663.0 661.0 659.1

2.076 2.078 2.079 2.0805 2.082

-

Pu"',

1

1.

2.200 m/8ec

r 589.4 589.7 590.2 590.5 590.9

1

320 340 360 380 400

r 530.3 530.6 530.8 531.0 531.2

r

640.2 641.5 642.7 643.9 645.

r 0 1

588.1 588.4 588.6 588.9 589.

r

8

220 240 260 280 300

r

664.2 661.6 659.0 656.7 654.5

0

569.2 570.1 571.0 571.9 572.9

r

530.0 530.0 530.0 530.0 530.1

r

4

634.3 635.4 636.6 637.7 638.9

0

587.8 587.8 587.8 587.8 587.9

0.07

r

566.7 564.4 562.1 559.9 557.9

-

and

0

568.6 566.1 563.6 561.2 559.1

-

U"»,

r

681.9 678.9 675.7 672.8 670.1

0.07

U"',

r

680.4 676.9 673.4 670.1 667.0

-

*/

of

0

565.3 566.0 566.8 567.6 568.4

-

U«»

Sections

r

531.4 531.0 530.7 530.5 530.3

0.07

#„

Cross

r 0

120 140 160 180 200

-

Fission

r

629.4 630.3 631.3 632.3 633.4

-

and

0

589.4 588.9 588.6 588.3 588.1

0.07

*/

Absorption

1

r

20 40 60 80 100

-

Average

j

-

26.

a

Temperature, °C

Table


9

1

.

„ 2.

*

1-22

GENERAL DATA

where x is the weight per cent of light water.

[SEC.

1

Other properties are affected in smaller

degree. 4.12

In reactor calculations, the average absorption cross section of a 1/v Fuel. absorber is calculated from the 2,200-m/sec reference value by applying factors for the Maxwell distribution and for temperature (See Sec. 6-2 and Art. 7.12 of Sec. 1-1). The marked departure from 1 /u-dependence exhibited by fuel nuclides is most simply allowed for by substituting adjusted cross sections that will give the correct reaction rate when 1/w-dependence is assumed. Table 25 gives the Westcott* adjusted values for IP" Um, and Pu2». The Maxwell distribution factor (0.886 or 1/1.128) and the temperature factor [V293/(273 + l°C)] must still be applied, since Table 25 gives only the secondary corrections necessary to give 1/v-equivalent cross sections.

_

9

a

V293/(273 +

t)

1.128

where t = temperature, °C

The parameter r in Table 25 takes into account the density of epithermal neutrons in the \/E "tail" from slowing down of fission neutrons. neutron density in

X

"tail"

(total neutron density)

2

is

it is

is,

The table assumes the tail to be superimposed on the Maxwell distribution, with a cutoff at bkT [4.317'(°K) X 10~4 ev) as the lower limit. The condition r = 0 corre sponds to a pure Maxwell distribution, and the values for this condition may be used for calculations of well-moderated reactors. For other reactors, the parameter r must be estimated or calculated from reactor theory, and Table 25 may then be read by Westcott states that r for the NRX reactor is 0.03 in the moderator interpolation. and 0.07 in the fuel. Since reactor calculations are usually based on the thermal-neutron density and the Westcott cross sections are based on total neutron density, some further adjustment is indicated for consistent treatment of diffusion-dependent terms in reactor theory. however, dependent on the particular model used in calculation, The adjustment and, since usually not very important, generally ignored. 4.13 Fission Products. Table 26 gives absorption cross sections of fission products and yields of xenon and samarium. See Tables 15 to 17 of Sec. 2. 4.14 Other Materials. Table 27 gives data for other common constituents of Table 19 of Sec. reactor cores. gives data for all naturally occurring elements.

it


1.01

SEC.


1-1]

Table

26.

Fission-product

Thermal

Cross Sections

Cross section,

barns

Yield per

Fission product

fission, per cent

1-23

2. 200 m /sec

Maxwell distribution

Handbook reference

at 20°C

6.9

Samarium1***

1.3

Sec.

2.72 X 10'

2.80 X 10"

6.6

5.0 X 10*

X

Total

10*

products:

At lero burnup

After 10 per cent burnup at

>t>

Other fission

Tables ! 5 and 16 of

b b

100 100

=* I0H

7I«

80 65

2.

See

Table 16 of Sec. Table 19 of Sec.

Art. 4.3 of Sec.

58ft

also

Table 4 of Sec. 11. 2 2.

0.3

Indirect

2

Xenon1"*: Direct from fission

*

The tabulated yields (or yields close to them) are commonly used in reactor calculations. Xenon given by different observers cover a wide range. The cross section data are from BNL 325, Supplement . Two sets of data for Xe1" (S. Bernstein and E. C. Smith), combined with two assumptions of spin factor, give four possible values, ranging from 2.19 to 2.76 megabarns for 2,200-m/sec neutrons. The four values for the average cross section over a Maxwell distribution would, according to Westcott's Weatcott prefers the highest values, which are in close analysis.1 range from 2.17 to 2.83 megabarns. agreement with those of BNL 325. The values for Xe1" given by Westcott are:

1

Temperature and Maxwell distribution factors lations (see Art. 4.12). Cross section per fission. Assuming /e dependence.

300

400

500

600

3.208

3.388

3.419

...

must be applied to

3

Temperature, °K

X t


$

1

yields

700 3.251

for conventional reactor calcu

GENERAL DATA

1-24

[SEC. 1 Macroscopic Properties

Table 27.

Thermal General properties Microscopic crosssections Element and atomic number Atomic weight

A.t

Reference density P. g/cm"

Atoms per cm" 0.6023X I0»

„ N-

A cm-1

,

era em'

1 — itn

X gC

12 01 22 997 24 32 26 98 28 09

1.600 0.97 1.74 2 699 2 33

.0802 .0254 .0431 .0603 .0500

.9440 .9708 .9724 .9751 .9761

wCr

30.975 32.066 39.100 47 90 52 01

1.82 2.07 0.86 4.54 7.19

.0354 .0389 .0132 .0571 .0833

9783 .9790 .9828 .9860 .9871

.19 .49 1.97 5.6 2 9

!sMn »Ke nPo a*.\i ■Co

54 93 55 85 58 94 58 69 63.54

7.43 7 87 8.9 8 9 8 96

.0815 .0849 .0909 .0913 .0849

.9878 .9880 .9886 .9885 .9894

13.2 2 53 37 4.6 3.62

91.22 92.91 95.95 118.7 183.92

6 5 8 55 10.2 7 298 19 3

.0429 .0554 .0640 .0370 .0632

.9926 .9928 .9930 .9943 .9963

,!Pb .slii »Th

207.21 209 00 232.12

11.34 9.8 11.5

.0330 .0282 0298

.9968 .9968 .9971

uNs uAl uSi nP >&

,.K „Ti


10 " .0032 .505 .063 .230 .13

.180

I.I

2.5 .60 19.2 .170 .032 7.45

»• cm'

X IO-m .00284 .448 .0558 .204 .115 168 434 1.75 4.96 2.57 11.7 2.24 32.8 4 08 3.27 .160 .97 2.22 532 17.0 .151 .0284 6.2

X 10 M 4.8 4.0 3 6 1.4 1.7 5 1.1 1.5 4 3 2.3

II

7 17.5 7.2 8 5 7 4 5

II 9 12 5

• Table 27 Rives data for elementslikely to be presentas important constituentsof reactors. Table 24 BhoukJ be used for

SEC. 1-1]

of Elements for Thermal

Reactors*

properties

Epithermal properties

Microscopic properties

Macroscopic properties

Z. cm"1

1-25


-

cm *

2«,

-

cm"1

1/Z.r em

ft cm*

t

Macroscopic properties Ele ment 2

=

cmr1

«3. cm "'

Z.(l

- SO

X„~

cm"1

cm


X 10" 000228 .0114 .00241 0123 .0058

385 .102 .155 084 .085

.364 .099 .151 .082 .083

2.75 10. 1 6.6 12.2 12.1

4.8 3.1 3 4 1.4 2 2

.1589 0852 0807 .0730 .0702

.385 .079 .147 .084 .110

.061 0067 .0118 .0062 .0077

.364 .076 .142 .082 .107

2.75 13.1 7.0 12 2 9.3

0060 .0169 .0231 .283 .214

.18 043 .0199 .23 .25

.17 .042 .0195 .23 25

5.8 24 51 4.4 4.1

3.4 1.1 2.1 4.2 3.9

.0637 .0616 .0507 .0415 .0383

.120 .043 .028 .2+0 .325

.0077 0026 .00141 .0100 .0124

.118 .042 .027 .240 .321

8.5 24 37 4 23 3.12

1.9 11.4 5 8 17.4 7.7

0363 .0357 .0338 .0340 0314

.155 .968 .528 1.59 .654

.0056 .0345 .0178 .0540 .0205

.153 .956 .521 1.57 .647

.953 .190 2 98 .372 .273

.187 .93 .64 1.60 .612

.185 .92 .63 1.58 .605

00685 .054 .142 .0197 1 08

34 .28 .45 .15 .32

.34 .28 .45 .15 .31

2 9 3.6 2.2 6.8 3.2

6.2 6.5 6.0 4.8 5 6

0220 .0216 .0209 .0169 .0109

.266 .360 .384 .178 .354

.00585 .00778 00803 00300 .00386

.00497 00080 .222

.36 25 373

.36 25 .372

2.8 3.9 2.69

11.3 9.28 12 5

.0097 .0096 .0087

.372 .262 .373

O036I 00252 .00324

moderators.

5.4 1 08 1.6 .633 1.65

C Na Mg

Al Si P s

E Ti

a

6.5 1.05 1.92 .637 1.55

Mo Fe Co Ni Cn

.264 .358 .381 .177 .353

3.79 2.80 2.62 5 66 2.84

Zr Nb Mo tin W

.371 .261 .372

2.69 3.83 2.69

Pb Bi Th

Table 18 of Sec. 2 gives the basic data for all the elements, and Table 20 of See. 2 gives resonance integrals.

1-26

GENERAL DATA

4.21

Neutron Additional

Table 28 gives neutron-regeneration Regeneration. data are given in Table 6 of Sec. 2.

-

-

« =

1

Fission Data

4.2

fuels.

[Sec.

1

v

V

- '/d

data

for

+ a)

4.22 Other Fission Data. References to other data (including fission-neutron spectrum, energy dependence of nuclear properties, spontaneous fission rates, photoneutron yield, fission-product distribution, and dclayed-neutron data) are given in Table 5 of this section.

Table 28.

Neutron-regeneration

U'"

Neutrons per thermal absorption ij


Captures per fission at/af = a

*-

1

— \

Data* U'»

Pu"»t

Natural uranium

2.52 2.28

2.47 2.07

2.91 2.09

2.47 1.34

0. 105

0. 192

0.39

0.85

•"World consistent" data for 2.200-m/aec neutrons, Ref. 1, 1957 Supplement. These data differ slightly from those of Sec. 2. Table 6. For the neutron spectrum of a thermal reactor, a and ij can be calculated from Table 25 and v of this table. In thermal reactors, the strong resonance of t All values in the table are for 2.200-m/seo neutrons. Pu!3v causes a to be considerably higher and y considerably lower than the tabulated values. See Table 25 of this section and Art. 6.5 of See. 6-1. 6

CONSTANTS AND FORMULAS RELATING POWER, FLUX, AND FUEL CONSUMPTION

See Sec. 6-2, by

J. R.

Dietrich, 5.1

and Sec. 12-2, by L. E. Link and Walter H. Zinn. Energy

from Fission

The available energy per fission is usually taken as 200 Mev.* The energy dis tribution for fission of U"' by thermal neutrons is given in Table 1 of Sec. 2. Table 29 See also Table 28 of gives some frequently used energy and fission equivalents. Sec. 1-3.

Table 29.

Fission Energy Equivalents*

= 3. 20 X 10-11 joules (or watt-sec) 1 fission I joule 3. 12 X 10'° fissions = 3. 12 X 10'" fissions /sec I watt I Mwd (megawatt-day) — 2. 70 X 10" fissions * Bused on 200 Mev per fission and 1 ev «= 1.60203 X I0~" ergs.

-

6.2 6.21

Average Thermal Flux and Power *!* =

Mw whore

Flux-Power Relations

t\

Mw

— =

-

i

K X x

10"

i*.

X

specific power

X kK

average thermal flux,t n/(rm1)(sec) power, Mw

* Formulas in this article are based on 200 Mev /fission for the total beat generation in the reactor and shielding. A small adjustment must be made to apply the formulas to heat generation in the core alone. t *ia is the average thermal flux in fuel. For an effectively homogeneous reactor this ia the same as the average over the core.

SEC. 1-1]


--

kg specific power

= mass of fissionable material, kg = Mw/kg of fissionable nuclide

K "

constant as given below

General Formula for

Constant

the

K

A' =

where A 9/

E

1-27

Eq.

1'036(^)

(96) of Sec. 6-2

= atomic weight of isotope, g = microscopic fission cross section (averaged over the Maxwell

distribution at operating temperature, with allowance for deviation from 1/v depend ence), barns = energy per fission, Mev

E

= (8.617

X 10-')r(°K)

X 10-')r(°R)

= (4.787

For a Mixture of Fuels i

E

i

IK

Value of K and I for (/"*. Table 30 gives = 200 Mev, a, (r=0) from Table 25.

K


Table 30.

and

K and 1 /K for U"6 baaed 1/K

on A = 235 g,

for U»s

Temperature, °C

50

20

K MK 6.22

100

2.55 0.392

2.41

0.414

Fast Flux.

2.77 0.361

150

200

250

2.98 0.335

3.18 0.315

3.36 0.297

300

350

400

3.54 0.282

3.72 0.269

3.88 0.257

Virgin Flux
3.12

X

10'" v

X

=

(watts /cm')

«■= number of neutrons per fission (2.47 for Um) 2 = total macroscopic cross section of reactor material for virgin neutrons Tico-group Fast Flux. See Art. 6.5 of Sec. 6-2 (also Art. 7.71 of this section).

where

6.3

Consumption of Fissionable

Material

Art. 1.1 of Sec. 12-2. One gram of fuel fissioned yields approximately 0.95 Mwd. Table 31 gives fuel consumption (including nonfission capture) based on 200 Mev per fission. See

Table

31.

Consumption Rate of Fuel, Grams per Megawatt-day* Thermal reactors General formula (j4 =» atomic weight of nuclide,

rjm

a

Consumption, grama /Mwd * For highly enriched reactors

0.00445AO + o)/« t — t; for large natural-uranium

-

0.105 a

1.15/. graphite

reactors

-

puin 0.192

l.25/< «

*

1.04.

a

-

0.39

1. 48/«

GENERAL DATA

1-28 and Burnup

5.31 Exposure 1,000 kw of fission

Units.

Based on the rough approximation

heat corresponds to fission of 1 g of fuel per day:

«

of fuel fissioned

1 per cent

10,000

More exactly: 1

= 9,500 = 8,600

per cent of fuel fissioned

1

[SEC.

that

Mwd/tonne

Mwd/tonne Mwd/ton (2,000 lb) or of

For 1 per cent of atoms consumed, including radiative capture, divide the 9,500 8,600 by (1 + a). For nominal values of a (Table 31), the Mwd/tonne equivalents 1 per cent burnup of Um, U"», Pu2" arc 8,600, 8,000, and 6,800, respectively. For average burnup based on heat developed in the core, multiply these numbers

by

Heat retained in core /(heat in core + heat outside core) Conversion and Breeding

5.4

5.41 Available Neutrons. The parameters n, /, P, R2, and t have their customary meanings (sec Art. 7 of this section). Values of t) at 2,200 m/soc: U"1, 2.28; U"6, 2.07; Pu"9, 2.09.

Neutrons available for breeding or converting in core Neutrons absorbed in fissionable


where

L P

= leakage = probability

material

(1

L)

that a neutron will not leak from the core

^

P

/ (see

^

/

Art. 2.31 of

Sec. 12-2)

Conversion Ratio

5.42

C ,

.

where

_

- - -I q

1

atoms generated s ; atoms destroyed I = neutrons lost (by leakage and capture in all but fissionable materials) per neutron absorbed in fissionable material (see Art. 2.33 of Sec. 12-2)

C

Initial

.

„. = conversion ratio

Conversion

=

Ratio TOa

where r = fraction of 5.43

U"* in uranium mixture

Breeding or Conversion Gain (C > G = C

6.44

-

I

1)

= r,

(see

- -I 2

Doubling Time ,. .. Doubling time in uuys = ,

T.

.

.

Fuel Utilization (C <

Art. 2.34 of Sec. 12-2)

:

G (daily fuel consumption, grams) 31 and

the megawatts of heat.

1)

Upper limit where 7? = fraction of fissionable (see Art. 2.34 of Sec. 12-2) 6

(see

inventory of fissionable material, grams —- — —

The daily fuel consumption is obtained from Table 6.45

Art. 2.33 of Sec. 12-2)

R

1

-C

material in mixture of fuel and fertile material

NUMERICAL INTEGRATION BY SIMPSON'S RULE

Various methods of numerical integration, including Simpson's rule, are discussed in Art. 1.4 of Sec. 3-2. Simpson's rule is widely used for integrating nonanalytic


Sec. 1-1]

1-29

functions, including determination of reactor constants from empirical data. tical application is summarized below.

To evaluate

Jb

Prac

y dx, where y is a function of x:

If n is the 1. Divide the interval ab into an even number of equal intervals h. number of intervals, h = (6 — a)/n. 2. Evaluate y (analytically, from tables, by measurement, or from empirical data) = a, = a + h, x = a + 2h, . . . , x = b. at h fb 3. / y dx = {y„ + 4ya+k + 2j/„+2a + iya+3\ + 2j/„+,» + • • • + 4//»_* + j/i] ■/» 3

i

i

-

Example: To evaluate

y

xJo(x) dx by Simpson's rule, divide the interval

1

to

2

into four equal parts, h = 0.25

From

x tables,

J,(x) xJa{x)

xJo(x)

dx =

— [0.7652

- 0.7134 3

1.25 0.6459 0.8074

1

0.7652 0.7652


2.0 0.2239 0.4478

+ 3.2294 + 1.5354 + 2.5828 + 0.4478]

The correct solution, using the analytic relation 7

1.75 0.3600 0.6457

1.5 0.5118 0.7677

REACTOR

JxJo(x)

dx =

xJi(x),

is 0.7133.

CALCULATIONS

6-2, by J. R. Dietrich. This article, like others of Sec. 1-1, is a guide to a major section of the handbook and should not be used alone. Formulas have been selected from Sec. 6-2 to highlight the steps of criticality calculation and to serve as a framework for references to the detailed instructions contained in Sec. 6-2. For additional tabulated data, see also Table 5 of Sec. 1-1. Article 11 of Sec. 6-2 gives examples showing application of See Sec.

formulas.

Nomenclature

kCm

A = atomic weight of element, g. Mass number of nucleus relative to mass of neutron, dimensionless B* = buckling, cm-! D = diffusion coefficient, cm. E = neutron energy, ev. Also a dimensionless constant F = dimensionless constant f = thermal utilization, dimensionless k = multiplication constant in infinite medium, dimensionless kT) = Boltzmann constant, ev/deg L = diffusion length, cm L, = slowing-down length, cm .v/= = migration area, cm* N = number of nuclei per cubic centimeter, em"' N = 0.6023 X 10" = Avogadro's number, atoms per gram atomic weight, chemical scale, g-1 P = resonance escape probability, dimensionless T = absolute temperature, °K or °R t = temperature, °C or °F V = volume, cm3 V = neutron velocity, cm/sec ( = fast fission factor, dimensionless n = regeneration factor, dimensionless X = reciprocal of diffusion length, cm"'

1-30

GENERAL X = Ao = v =

=

£ p =

S = a = t = =

DATA

[SEC.

1

mean free path of neutron, cm average cosine of scattering angle (numerically mo = 2/3/1), dimcnsionless average number of neutrons emitted per fission, dimensionless average logarithmic change of energy per collision, dimensionless density, g/cm* macroscopic cross section, cm-1 microscopic cross section, cm* Fermi age, cm2 neutron flux, cm-1 sec-1

Subscripts 0 = initial or standard condition 1 = fuel region 2 = moderator region 3 =

clad region

a — absorption = fast (when used with a denotes fission) t = t'th component of a mixture s = scattering th = thermal

v

/

tr = transport 7.1

Average Thermal

Microscopic

Absorption

Cross Section

aa

The reference energy used in selection of Energy of Thermal Neutrons. cross-section data is the energy corresponding to the most probable velocity in the Maxwell distribution at the operating temperature [see Eq. (63) of Sec. 6-2].


7.11

kT

X lO-'T

= 8.617 = 4.787

X 10-T

T T

ev ev

= =

°K °R

In thermal reactors, T may exceed the moderater Effective Neutron Temperature T. temperature by about 50°C. See Art. 4.4 of Sec. 6-2 and formula in Art. 5.3 of Sec. 2. 7.12 See Eqs. (65) and (66) of Sec. 6-2. 1/v Absorbers with Maxwell Distribution. From 2,200-m/sec Data 293

««0

1.128

!

+ 273)

!

+ 460)

where

°C

529

ffoO

1.128

t =

ffaO

.0253

1.128

E

E

= ev

aaa =

absorption cross section at standard (monocnergetic) neutron velocity of 0.0253 ev 2,200 m/sec (corresponding to an energy of approximately and a kT temperature of approximately 20.4°C or 68.8°F) 9« = average cross section to be applied in conjunction with total thermal flux

2/\/r

= 1.128 is the correction for Maxwell The number The distribution. square-root factor in the formulas is the temperature correction for reactors in which the neutron temperature is not 20°C (69°F). Table 19 of Sec. 2 gives o-„0 for elements and nuclides, and Table 27 of this section gives 9ao — o-„/1.128 for some important elements. From Curves of a„ as a Function of the Energy kT: For curves of aQ for monoenergotic neutrons of energy kT, see "Neutron Cross Sections."1 So

7.13 Non-l/w Absorbers The temperature-correction

=
See Art. 4.3 of Sec. 6-2. with Maxwell Distribution. factor and the 1.128 distribution factor used for l/v

SEC. 1-1] are

absorbers

SELECTED DATA AND FORMULAS AND GUIDE In

not applicable to non-l/v absorbers.

general,

1-31

from Eq. (66) of

Sec. 6-2,

».

- fg'-'.E exp (-E/kT) dE/lkTKEi

(-E,/kT)

+ kT) exp

- (Ei +kT)exp

(~E2/kT)]}

integral can be evaluated (over a sufficient range Ei to Et) from Rcf. 1, example, Simpson's rule (see Art. 6). For practical application the follow ing ampler processes are generally adequate. U«u, u«« and Pu»' {non-\/v absorbers) Determine 9. or 9, for U»", U"«, or Pu"» from 6 in Table 25 of Sec. 1-1. Other non-1 /v A bsorbers 1. For a temperature of 20°C, the cross section is read from Table 18 or 19 of Sec. 2, the 1/1.128 factor is applied, and the result is multiplied by the correction factor given in the table (Cd, 1.3; Xe»», 1.16; Sm, 1.5; Eu, 0.95; Gd, 0.85; Hg, 0.95). 2. For temperatures up to about 250°F (120°C), the cross section for the cor responding kT energy may be read from curves' and the factors used in (a) may be At still higher temperatures the correction may be increasingly in error. applied. 3. Westcott* gives 9a for Mn, Co, Rh, In115, Gd, and Au, in addition to the fuel and fission product nuclides discussed in Arts. 4.12 and 4.13. 7.14 The 1/E "Tail." The departure from Maxwell distribution at the highenergy end of the spectrum, due to slowing-down neutrons, is discussed in Art. 4.4 of Sec. 6-2; it can generally be ignored (Table 25 and other data from Westcott2 take the l/E "tail" into account). where the

Macroscopic Cross Sections 2

7.2 7.21

Single Nuclide or Element (All Cross Sections)

A If Z*

u known for a material at standard density p0 (e.g., Table 27 of Sec. 1-1 and Sec. 2) 2 at some other density p is given by 2 = 20(p/po).

Table 18 of 7.22

Homogeneous Mixtures (All Cross Sections) Eq. (3) of Sec. 6-2 the partial density or g/cm' of the ith material. Mean Free Path X. Single Nuclides or Homogeneous Mixtures

»here p, is 7.23

Sectioas): X = 1/2. 7.24 Average Macroscopic

(All Cross

Absorption Cross Section. 2„ must be averaged The Maxwell neutron energy distribution and corrected for temperature. corrections described for microscopic absorption cross sections may be applied either to tr, or to 2a. Table 27 of Section 1-1 gives 2„ already corrected for the Maxwell distribution at room temperature; for other temperatures, the temperature correction must be applied. 7.26 Special Cases. Other macroscopic cross sections are discussed in Art. 7.5.

over the

7.3 7.81

Average Logarithmic Change of Energy per Collision. £ =

*here

and Scattering Constants

Slowing-down

In

^ E

r =

(AM


using, for

\A

1

~

+

—!— 1 —

'V

+ 1/

r

In r

Sec Art . 3. 1 of Sec. 6-2.

Eq.

(36) of Sec. 6-2

Eq. (35) of Sec. 6-2

GENERAL DATA

1-32 Values of

{

for the elements are given in Table

I

=

[SEC. For

of Sec. 2.

18

~

1

a mixture of nuclides,

Eq.

(37) of Sec. 6-2

Z2For isotropic scattering in the center-

7.32 Average Cosine of Scattering Angle. of-gravity system, fio = cos 0

- 2/ (3 A)

Art. 1 — Ao is

where 0 is the scattering angle in the laboratory system. in Table 18 of Sec. 2.

k = r,tpf

For calculation of p and

given as

1 —

cos

8

Constant*

Infinite Multiplication

7.4

2.13 of Sec. 6-2

Eq. (168) of Sec. 6-2

/


in fuel-moderator lattices, the unit cell is idealized (usually to cylindrical geometry), maintaining true fuel and moderator volumes (see Arts. 9.3 and 9.4 of Sec. 6-2). 7.41 Regeneration Factor r>. See Art. 9.2 of Sec. 6-2. If applied to the fissionable isotope, v =

If

applied to a homogeneous

KoY/Va)

fuel mixture

- T viNiv/i

1

i

7.42

See

Homogeneous

(169) of Sec. 6-2

Art.

v and i\. For a heterogeneous fuel element, 11.21 of Sec. 6-2 for a numerical example.

/

Thermal Utilization

/

Eq.

N>
i

gives values of

Table 28 of Sec. see Eq. (170) of Sec. 6-2. 1-1

Jy

= absorption in fuel /total absorption

Mixture

Z.i

Z.i

2a(t,,tiil)

Sal + 2 o(other)

1 ^

^q(otiter)

Eq. (171) of Sec. 6-2

2.,

The subscript "other" means materials other than fuel. Fuel-Moderator Lattice

I[

/

E and F

.

+

rVtS.i L

+ V i2„i

J

F + {E_l)

are calculated from Table 8 of Sec. 6-2, and xt and 13 of Sec. 6-2.

from Table

Disadvantage factor = See

Art.

I?

=

(1

Eq.

(190) of gee.

xj (equals 1/Li and

- l)

Eq.

1

/Li)

(177) of Sec. 6-2

11.22 of Sec. 6-2 for numerical example.

* The macroscopic cross sections entering into reactor theory are usually average valuea for mixtures of materials over particular flux distributions. Where confusion is unlikely, these average cross sections are represented

by X instead of 2.


SEC. 1-1]

Resonance Escape Probability

7.43

p = exp

I

where

=

A

^

^rom

^Q"

See Art. 9.4 of Sec. 6-2.

p.

Mixture

Homogeneous

(

1-33

- I— /)

Eqs. (57) and (192) of Sec. 6-2

°* êc- ^"2, is calculated from £ and ff#((W in

$7)

i

»

Table 22 of Sec. 2

Nt

_

number of absorber atoms/cm* total macroscopic scattering cross section of mixture

=

resonance integral from Fig. 14 (see also Arts. 3.51 and 3.53 of Sec. 6-2). Article 6 of Sec. 2 also gives absorption integrals

2.

I

Fuel-Moderator

Lattice

"

p = exp

f

. If region 2 is a

"

1

— -=

I

N.V,(A + nS/M)

pure moderator,

[ "


Eqs. (206) and (200) of Sec. 6-2

p = exp

1

NaVM

.

"

+ nS/M)

atoms of resonance absorber (U"s) per cubic centimeter of fuel; E and F from Table 8 of Sec. 6-2 using xi and x2 from Tables 10 and 11 respec tively; S/M = (surface /mass) of fuel lump; A and /j are found from Table 9 of Sec. 6-2 (for natural uranium A = 9.25, /i = 24.7). Modified parameters for uranium and thorium, and their extension to diluted fuel lumps, are given in Tables 21 and 22 of Sec. 2 and are discussed in Art. 6.2 of that where

N

=

are calculated

action.

j.srj for a pure moderator is found from Table 11 of Sec. 6-2 (H20, 38.5; D20, 5.28; C, 0.76). See Art. 11.23 of Sec. 6-2 for a numerical example.

Be, 1.26; 7.44

Fast-fission

Effect

«

t -

fhere cross <= 1 +

(, ~ 1

1

+

-.

- Z./S,) ^ P

-

sections refer to fuel and are defined on p. 6-84.

Eq. (207) of Sec. 6-2

For natural uranium

0.09P/(l 0.52P') (see Table 12 of Sec. 6-2). Pand P' are found from Fig. 15 of Sec. 6-2 and xi (fuel, thermal) = \/L\ from Table 13 of Sec. 6-2. P' corresponds to xi/2( = 0. For close-packed lattices of slightly uranium, see Table

enriched

12

of Sec. 6-2.

See Art. 11.24 of Sec. 6-2 for numerical

examples.

7.6 7.51

Other Material Constants of Reactors

Transport Mean Free Path and Transport Cross Section Arr =

of Sec. 6-2

Values.

l/2(r

For experimental values of X,r for moderators, see Table 13 and Table 24 of Sec. 1-1 (H20, 0.48; D20, 2.80; Be, 1.43; graphite, 2.75).

Experimental

GENERAL DATA

1-34 Calculated

Values.

If

[SEC.

1

experimental data are lacking use

- A.)

2,r = 2.(1

= 2.[1

- (2/3A)]

Values for important elements are given in Table 27 of Sec. 1-1. Mixture

Homogeneous

2(r —

2|r,;

^ i

Experimental values of 2,r.i should be used wherever available: if pa is the standard density of the ith constituent and —!—

—

p,- is

grams of material in

1

cm',

2,,,,-

= 2,r,o,— = Pw

where X,r can be taken from Table 13 of Sec. 6-2 or Table 24 of Sec. 1-1.

Xfr.Oi POi

For

constituents of unknown transport cross section, use 2er.i = 2,v(l — 7.62 Thermal Diffusion Coefficient A*. Homogeneous Medium, Weak Absorber, For a pure medium: Experimental Values.

^■T-^-S^T^ For moderators

X(r is

given in Table

13

D,»

=

l/

of Sec. 6-2; \„,a, and /),* in Table 24 of Sec. 1-1

.

Medium, General Case, Calculated

[32(1

-

-

(l

Ao)

-5

|°

+

|°

+•

•

•)]

Homogeneous

(10) of Sec. 6-2

x„ = 1/2,,

E<1' (9)

°f

Bec" 6"2

d«~

-

j_r 32M

i+™ Ll +m(WSirt)-l

1

is

2

and 2„ are total and absorption macroscopic cross sections, respectively. Fuel- Moderator Lattice. If the ratio of moderator volume to fuel volume large, use Dtk of the moderator. Otherwise,

where

"

£[)

\

D,

s«

0

W -M -

2„ =

^

/)](2.,/2.i) (see Art. 9.6 of Sec. 6.2) where m = [//(l 7.63 Fast Diffusion Coefficient D/. Homogeneous If an average Medium. value of a. can be chosen for the moderator constituents over the energy range from fission to thermal

l/32,r

=

To evaluate for variation of Fuel-Moderator Lattice. medium.

a, with E, see Art. 9.8 of Sec. 6-2. Lumping can usually be neglected; treat as homogeneous

Thermal Diffusion Area

7.64

L»

- l/x« - Z>tt/2,.tt

Eq.

(208) of Sec. 6-2

L

2

is

Experimental Values for Moderators. given in Table 13 of Sec. 6-2, Table 24 of Sec. 1-1, and Table 13 of Sec. (H20, 2.85; pure DjO, 170; Be, 21.0; graphite 52). Homogeneous Mixture. Calculate from above formula. Use experimental values available; otherwise calculate from of

if


For a mixture:

Eo.

L»

- }i\,rK -

l/(32,r2„)

1-35


SEC. 1-1]

By combination of Eqs.

Lattice.

Moderator-Fuel

(l-Z + g/)/*-

-

-

«

(177), (208), and (209) of Sec. 6-2

2«i L* £s'(l /) [Eq. (211) of Sec. 6-2), where hi (moderator) is found Table 13 of Sec. 6-2 or an equivalent source. 7.65 t is given in Table 14 of Age t. Experimental Values for Moderators, Sec. 6-2, Table 24 of Sec. 1-1, and Table 13 of Sec. 2 (H20, 31; D20, 125; Be, 97; p&phite, 364).

If T.t from

Calculated Values

for

of Higher Atomic Weight than Beryllium

Moderators

-

"*>

/.*s§>¥

f

from fission source: See Art. 9.9 and Eq. (48) of Sec. 6-2. Moderator-F uel Lattices. Use r for moderator if fuel is uranium (volume fuel)/(volume moderator) is small. (See Art. 9.9 of Sec. 6-2).

Effective age

Two-group

7.66

Area

Slowing-down

L/1 = For hydrogen-moderated reactors,

V

=

(«'*'

—

Eq.

*/'

**

= t

-

(84) of Sec. 6-2

Art. 5.6 of Sec. 6-2

(see Arts. 5.6 and 9.9 of Sec. 6-2)

1)/B«

the buckling according to Fermi theory obtained either from the char equation (1 + LtB1)e,B' = k (if the core material has been selected) or the equivalent bare-reactor equation (Table 5 of Sec. 6-2), if the geometry has

where B* is acteristic from

been chosen.

Fictitious Fast Absorption Cross Section S«/

7.67

2./

Extrapolation Distance For plane black boundary

7.68

=

D,/L,'

« = 0.71X,,

For other cases see Art. 2.22 of Sec. 6-2.

Characteristic Equations

7.6 Fermi Age Theory

(1

Modified One-group Theory 1 where M 1 =

LJ

tj=J M*

Eq.

(93) of Sec. 6-2

+ a

is,

r(Fermi) or L*

Eq. (89) of Sec. 6-2

L/1 (two-group).

Bare Reactor. of Sec. 6-2.

B'

=

Formulas

V«0

-

x'4> =

0

V*

0

Solution of Wave Equations for Critical Reactors +

+

is

+ L»B»)(1 + L/'B») = k

(89) for core and reflector are given in Eqs. (109) to (112) of Sec. 6-2. in all cases, first approximation of the fundamental buckling,

l)/M* L'

7.7

7.71

=

Eq. (82) of Sec. 6-2

+ L/»

Eq.

Solutions of

Table

-k

B'

+ M»B» = k

(1 B* = (4 — where M*

+ L'B*)er'"

Theory

Tito-group

5


For other moderators: for example,

L,*

metal and

Eqs. (76) and (13) of Sec. 6-2

for flux and critical

dimensions

are

given in

1-36

GENERAL DATA

[Sec.

1

Two-group Fast Flux

*/ _

Z..n + DikB1

*' _

v

_

k

2^

n vf"' V'\ X iissions/(cm,)(sec) 2G/(1

1 1

+ L''Bt

Eqs. (92), (208), and (89) of Sec. 6-2

+ L,*B>)

Leakage formulas are given in Art. 6.6 of Sec. 6-2. 7.72

Reflected Reactors, Two-region,

S

-

*±-

Two-group. Z° '*

tk

Coupling Coefficients

+ D'"B'

Eq. (92) of Sec. 6-2

V^'f

Other coupling coefficients are given in Eqs. (127) to (130) of Sec. 6-2. Solutions. Solutions are given in Sec. 6-2 as follows: Cylinder :

Art. Art. Art. Art. Art.

Radial reflector

7-1 7-2 7-3 7-4 7-5 Art. 7-7

End reflector


Rectangular parallelepiped reflected on one pair of faces Sphere Bare cylindrical reactor with central region Reflectors in more than one direction: reflector-savings method See also example of complete reactor solution in Art. 1 1 . Solution by Matrix Method (for two or more regions). See Art. Art. 11.7 of Sec. 6-2. 8 See Sec. 7-1, by

CALCULATION

J. M.

8

and example in

RADIATION

OF NUCLEAR

West. Nomenclature

A

Mi

= activity, disintegrations per second. For reactor coolant, A = disintegrations per second per cubic centimeter of coolant = atomic mass of original nuclide, Mj = atomic mass of daughter or product

N

nuclide

= number of atoms of nuclide at time t. For reactor coolant, N = number, at time I, per cm' of coolant. Ni refers to original nuclide, Nt to daughter or product nuclide, No (abbreviation of Ari„) = number of original atoms at time 1 = 0 T = total operating time of reactor, sec I = time, sec. U = time for a single circulation of coolant through a closed reactor circuit, t, - time for a single passage through the reactor core

Q = volume rate of flow of coolant from reactor, cm'/'sec a =

fluid weight of— — ; in reactor core pweight of fluid in the system

„

For

.

,

..

a constant-density

...

„

.

fluid, this is the same

as the volume ratio disintegration constant, sec-1. Xi refers to original nuclide, Xs to daughter or product nuclide = neutron flux, neutrons/(cm,)(sec) a = microscopic absorption cross section, cm!. a\ and aaci are the absorption
8.1 8.11

7-1).

Half-life

Units

TW = (In 2)/X = 0.693/X

X = 0.693/7^4

(see

Art

1.22 of Sec.


Sec. 1-1]

1-37

8.12 Mean lifetime = 1/X, sec. 8.13 1 curie = 3.7 X 1010 disintegrations per second — more precisely, the quan tity of a radioactive material that undergoes this rate of disintegration. 8.2

Burnup, Activation, and Decay

Activation and Decay Formulas. Table 32 summarizes formulas for simple of activation and decay. These and other cases are given in Art. 2 of Sec. 7-1. Cross sections and half-lives are given in Table 1 of Sec. 7-1 and Table 19 of Sec. 2.

8.21 cases

8.22

Activity

of Radioactive Parent

Activity

= /liVoXi disint./sec = /iiVoXi/(3.7 X 1010) curies Activity per gram of parent nuclide initially present = 0.602 X 10"/iXi/A/i disint./sec = 1.63 X 1013/i\i/A/, curies 8.23

Activity of Radioactive Daughter or Product Activity

= /2AToX2 disint./sec = /jJVoX2/(3.7 X 10l°) curies Activity per gram of original nuclide initially present = 0.602 X 10,4/jX2/Afi disint./sec = 1.63 X 10"/2X2/A/l curies


8.3 See 8.31

Art.

Fission-product

Decay Heat

3 of See. 7-1, also Art. 2.4 of Sec. 11 and Art. 1.7 of Sec. 2.

Way-Wigner Formula

£

Po

= 6.22

X 10-»[r0!

-

(To + O-0-']

= decay heat rate as a fraction of operating heat rate To = time fuel is in reactor, sec I = time after shutdown or removal from reactor, sec Approximately half the heat appears as (S-particle energy and half as 7-ray energy. 8.32 Untermyer-Weills Formula. Table 33 gives the decay heat rate according to the Untermyer-Weills formula (see Art. 3.2 of Sec. 7-3). 8.33 See Art. 2.4 of Sec. 11. Decay Heat of Specific Fission Nuclides.

where

P/Po

8.4

Coolant Activity

8.41 Activation and Activity Data. Table 34 summarizes data for dominant activities of short half-life. For data on other short-half-life nuclides, see Art. 4 of Sec. 7-1. Table 35 gives formulas for calculating coolant activity from the data of Table 34. 8.42 Activity of Coolant. Table 35 summarizes formulas for ATt, the number of activated atoms per cubic centimeter

Activity

per cubic centimeter = A'2X2 disint./sec =» iV2X2/(3.7 X 10lu)

curies

nuclide

by neu

Xi are for the activation

product,

+

containing

+ /.ATo

-

—

yi(.o\

>
HXil)

are not

synonymous.

»i)*<] — "\M

X,)l) «Xi«

» Xi(

and activation

X,([l

Xil(l

absorption

»!**.)

with av meaning
AT. -

(1

exp (-Xi/i)

(I

Xi)(a])

-

(1

formulas

J) exp [-(
X,)l)|

-

X,.)

Xi«

+

Neutron-absorption

(-

exp (-(»,«

{exp (—
-

(exp (-»i«0

-

(I

JVi -

(

exp

+

— texp (— aidtt) exp (— at4*t)]

exp (-Xs<)]

-

(I

/iATo

— (710(a)

exp (-en**)

exp (— ai4>t)

Xj

of stable or long-lived nuclide for Activation time ta followed by additional decay time td

unstable

Product

(

stable

(exp (-Xil)

-

decay or burnup •>/

Product atoms, - Nt/N,

+

Product

Burnup of stable or long-lived tron al>sorption:

[I

exp

-

/i

Original atoms remaining, - Ni/N,

Simplified formula for small (all values of X< nd

/.

Daughter

unstable

fx

— Xi()

/,

exp (-Xif)]

Formulas*

a

Xi Xi -

-

atoms,

Decay

<•

exp (— XiO

or first daughter - Ni/No

Product

and

formula

Activation

General

Simple

<

nuclide:

Original atoms remaining, - Ni/No

32.

Ii

Decay of radioactive Daughter stable

Case

Table


0

*

SEC. 1-1]


Decay Heat Rate Exposure

Cooling time

1M*

1 min

0. 0193

1 rain

10 min

0.0549 (0.0513)

0. 0132

0 0355

(0 0132)

(0 0334)

0. 0446 (0.0418) 0.0325 (0. 0297)

0. 0050

0 0234

(• 0049)

(0 0214)

0. 00052

0. 00006


1day

1week

0 00001 (0 00001)

of Fission Products

time

30 days

90 days

1 year

2 years

Infinite

0. 0566

0.0575

0.0582

0.0584

0.0589

(0.0528)

(0.0537)

(0.0544)

(0.0546)

(0.0551)

0.0496 (0. 0459)

0.0505 (0. 0468)

0.0512 (0.0475)

0.0514

(0.0443)

(0.0476)

0.0519 (0.0481)

0.0358 (0.0322)

0.0375 (0.0337)

0.0384 (0. 0346)

(0.0353)

0.0393 (0.0355)

0.0397 (0.0360)

0.0242 (0. 021 1)

0. 0247 (0.0216)

0.0479

0.0390

0.0174

0.0207

0.0224

0.0234

(0.0152)

(0.0177)

(0.0193)

(0.0202)

0. 0240 (0. 0209)

0. 0092 (0.0085)

0.0124 (0.0109)

0.0141 (0.0124)

0.0151 (0.0133)

(0.0140)

0.0159 (0.0142)

0.0164 (0.0147)

0. 0037 0 00041 (0 00040) (0. 0034)

0.0066 (0.0056)

0. 0083 (0.0071)

0.0092 (0.0080)

0.0099 (0. 0087)

0.0101 (0. 0089)

(0.0094)

0.0013 0 00008 (0 00007) (0. 001 1)

0.0035

0. 0050

(0.0026)

(0.0041)

0.0059 (0.0050)

0. 0066 (0.0057)

0.0068 (0.0058)

0.0073 (0.0063)

0.00018 0 00001 (0 00001) (0.00014)

0. 00089 0. 0019 (0. 00074) (0.0017)

0.0027 (0.0025)

0.0033 (0.0032)

0.0035

0.0040

(0.0033)

(0.0038) 0.0023 (0.0023)

0 0089

0 0025

(0 00005) (0 0021) 6hr

1 week

0.0516 (0.0488)

(0 00046) (0 0073) 1 hr

1 day

0 0425 (0 0404)

(0 0192) 10tee

1 hr

*

1-39

0.0157

0.0106

30 day.

0. 00003 (0.00003)

0. 00020 0. 00061 (0. 00020) (0.00061)

0.0011 (0.0011)

0.0016 (0.0016)

0.0018 (0.0018)

90 days

0.00001 (0.00001)

0.00005 (0.00005)

0. 00018 (0.00018)

0.00041 (0.00041)

0. 00076 (0.00076)

0.00090

0.0014

(0.00090)

(0.0014)

0. 00001 (0.00001)

0. 00003 (0.00003)

(0.00007)

1year

0. 00007

0. 00018 0.00026 (0. 00018) (0.00026)

5years 1 1 * The tabulated quantity ia P/P%, the decay heat rate or power aa a fraction according to the Untcrmyer-Weills formula (see Art. 3.2 of Sec. 7-1). The first beat ratio for irradiated natural uranium. The quantities in parentheses are alone, after allowing for heat of decay of U,,t and NpMt; for practical purposes irradiated U*»*,

0. 00003 (0.00003)

0.00067 (0. 00067) 0.00035 (0.00035)

of the operating power, quantities give the total for the fission products this is the heat ratio for

D(n,T)T

0

1 Bi

Air

K 10 10' 10

X X X

1.07 1.33 5.48 Thermal

Thermal

Thermal

Thermal

1.9 0-3.6. 71.51 0-1.2. 7137 0-1.17 a5.30. 7O.8O

I0'« 10-* 10" 10"

1.60 5.80

days days

1.06

109 ruin

5 138.

1.55

12.4 hr

3

'

*

is

is

is

1

IX

vP

H

"

10"P B the calculated

7.6

activity

by 30 per cent.

,

B reactors,

6.7 X 10"), $* - 1.14 X 10'»P. use the same constants but increase

(p

-

9

(N ordinary temperatures In heavy-water moderated

where

t0»»*P Na*a

X

At

3.1

X

■»number of neutrons per fission — power density, watts /cm1 = number of hydrogen atoms per cubic centimeter Xg — scattering cross section of moderator for neutrons above a Mev. For hydrogen.
t

*• -

of coolant; must be adjusted

centimeter number

Bismuth

Air

Potassium

and its

alloys

for

temperature,

the coolant

and its alloys

if

This

per cubic

4.5

2.5

1.39, 4.2 72.755. 1.38. 3.7

10"

1.28 X

and its alloys

Sodium

0-3.8, 4.3, 10.3 76.13, 7.10

water Heavy

Water, water, heavy other coolants contain ing oxygen

0-0.019

0.0936

XX

15.0 hr

see

10"

1.77

12.4 years

in

X

For luiuid coolants, N\o*ct the tabulated quantity multiplied by the number of grams of reference material the density of the coolant in grama per cubio oentimcter. the pure reference material, the multiplier For air (0.94 per cent argon), the tabulated centimeter of air at 0°C and 76 cm Ug. oubic Nivi per quantity with other gases. pressure, and dilution Virgin flux:

19 mb

0. 53 barn

barn

•

100

99.60

A«(n,7)A
.

'

Bi'°»(n,7)Bi»>° — Po"1"

6.91

1.47 X l0-»

Virgint

Thermal

Particle and energy, Mev

Reactor coolants which activity important

X

0

K"(n,7)K"

Na

10"« 10"

10

X

0. 56 barn

1.33 X 1.21

3.4

Disint. const. At, sec-1

Decay of product

Data

Half-life

7.

100

HiO DsO

DiO

Applicable flux

Coolant-activity

4

Na"(n,7)Na"

mb

mb

Factor for determining

X

0.04

0.57

Reference material

Reactor

'

99.76

100

Reaction

Activation cross section ff
34.

X

0"(n/,p)N"

Iso topic abundance, per cent

Table


is

is

1-41


SEC. 1-1]

Table 35.

Activity of Reactor Coolants* Activated atoms per cubic centimeter,

At

Closed (recirculating) systems:

ATi
General case Very short-lived

etc.):

activitieat (N11,

Xj + 0t+

At exit from core. At entrance to core

Xj

- exp

[1

-exp(-X«a/e)](expI-Xj(l

(-XM/e)]/Il

(-XjM]

-a)M|/[1 -exp(-Xrfe)J

N\aaet
(A« Na«, etc.) Long-lived activities purities.) :%

(most

im N\Oaet
In general


- exp

[1

short-lived activities}

Moderately

(1

Xi + wtja

Special case

1.

Xj X>tr*tfr

Special case

2.

ujo

Upper

- exp [-(X» + «Fi^)qf,]tl 1 - exp {-(X, + «7a)T]} I 1 - exp I- (Xi 4- «*a)/<)]

limit

»

N\Oact

X»

-

[I

-

- exp [-(At exp

+fft*a)71)

(-X«D]

exp (-
of activity}

Total number of accumulated undecayed effluent atoms after time T

Activated atoms Ni per cubic centimeter

Open (once- through) Systems:

N\9*ei

General case

Xi +
Very short-lived

activities! (N11,

etc.)

[1

Moderately

activities^!

purities) Cpper

-

exp

- exp

(-(Xi

(-Xtf,)]

+

^[1 Xi

«*)M) '

short-lived activities}

(A«, Na", etc.)

Long-lived

f1

of effluent

limitf

im

Nitr

(-X.D]

- exp

(-XiDl

JV,Q/Xi ATiQ/X,

Nlffatt&r (most

- exp

I

XiQ

[1

N*QT

* Burnup and decay of the initial nuclide are ignored in all cases. Ni and Nt are numbers per cubic centimeter of coolant. Nivi and Xi are derived from Table 34 or sources given in Art. 8.21. t Burnup of activated nuclide during a few half-lives is assumed negligible; i.e., < O.lXi. proach to saturation is assumed, i.e., steady operation for more than five half-lives. Circulation time U of fluid through the system (or passage time U through the reactor for an open system) is assumed to be short compared with the half-life, i.e., U or U < 0.2?^. 1 Burnup and decay of the activated nuclide during a few circulations are assumed negligible, i.e., (Xt + m)te < 0. 1. For open circuit, (Xi + «0)(r < 0. 1. } This upper limit of activation for a finite time is approached if burnup and decay of the activated nuclide are negligible in the time T, i.e., if (Xa +
\iT

1-42

GENERAL DATA

K.

Z. Morgan. 9.1

Units Used in Dosimetry. Art. 1.2 of Sec. 7-2.

9.11

details,

Units

Table 36 describes

units of radiation dose.

For

see

Table 36. Type of radiation

Unit

Units of Radiation Dose Medium to

Basis of definition

which applicable

measured

X-ray T-ray of less than 3 Mev.

Air

Roentgen equivalent physical, rep

Any

Usually soft

Rad

Any

Any

(98 ergs /g based on 34 ev per ion pair) 100 ergs/g

Roentgen equivalent man, rem

Any

Any body tissue (including bone)

(Number of rads) X RBE

Roentgen,


1

HEALTH PHYSICS

9

See Sec. 7-2, by

[SEC.

r

1 esu/cins of air at 0°C 760 mm Hg ( 1 cma of air 0.001 293 g)

-

tissue

Usually 93 ergs/g

Remarks

Equiva Exposure dose. lent to 83.9 ergs/g air based on 32.5 ev per ionpair, or 87.7 ergs/g based on 34 ev per ion-pair Absorbed dose (obso lescent) dose. Stand Absorbed ard unit for all classes of radiation work Standard unit for biologi cal effects

Dose. Gram-roentgen, gram-rep, and gram-rad are the doses Volume-integrated (in r, rep, and rad, respectively) integrated to give total energy absorption over the entire body (see Art. 1.25 of Sec. 7-2). 9.12 Relative Biological Effectiveness, RBE. See Art. 1.4 of Sec. 7-2.

j^gg

_

Co607 dose in rads to produce a biological change

actual dose in rads to produce the same change

For RBE of heavy ionizing particles, see Table Table 2 of Sec. 7-2 (see also Table 37 of Sec. 1-1). 9.2

Art.

1

of Sec. 7-2; for other particles,

Radiation Dose from External Sources

1.6 of Sec. 7-2. Factors Affecting Maximum Permissible Dose. Figure 1 of Sec. 7-2 shows linear and threshold radiation-damage rates. Figures 2 and 3 show maximum per missible dose at various parts of the body. 9.22 Maximum Permissible Dose Rate. Sec Art. 1.6 of Sec. 7-2. Basic rate If averaged over several years, 0.1 rem/week. for adults 0.3 rem/week. For minors under 18 and others living in the neighborhood of the controlled area, one-tenth of the above rates. These dose rates may be increased considerably as prescribed by Figs. 2 and 3 of Sec. 7-2. 9.23 Planned emergency dose for adult male (or woman over 45), 6.25 rem. This dose may be increased considerably as prescribed by Fig. 3 of Sec. 7-2. 9.24 Calculation of Dose Rate. Table 37 of this section, based principally on Table 2 of Sec. 7-2, summarizes the particle fluxes corresponding to maximum per missible dose rate. Footnotes indicate other sources of data. See 9.21

1-43


SEC. 1-1] Table 37.

Flux for 7.6 mrem/hr at Surface of Body (0.3 rem per 40-hr Week) Approximate flux, parti cles/(cmJ) (sec)

RBE

Type of radiation

1 (2.5) Fist neutrons*

(10) 1 10 10 20

Heavy ions

4.200/E (Mev)

1.930 (2,000*) >3 3

Mev

2

1

0.5

0.1

0.01

Flux 30 30 40 60 80 200 1.000 4,340/S., S- from Table 3 of Sec. 7-2 400/S., S« from Table 3 of See. 7-2 1.3 X I0'/(stopping power from Fig. 4 of Sec. 7-2) Oxygen ion: 6.5 X 10'/(atopping power from Fig. 4 of Sec. 7-2)

* International values Actual biological dose and energy deposition in (see Table 4 of Sec. 7-2). d»ue as a function of depth are given by Snyder, Figs. 5 to 13 of Sec. 7-2. Table 4 of Sec. 7-2 gives maximum permissible neutron flux based on these curves. t Flux is also given for electrons by 1.3 X I0*/(stopping power) and for a particles by 1.3 X 10V stopping power). Obtain stopping power from Fig. 4 of Sec. 7-2. The results are only approximately equal. Equations (12) to (19) with Table 3 permit calculation of dose in soft tissue or bone at any depth, or averaged over the range of a particle of given initial energy. 9.26


of Sec.

Natural Background

and

Common X-ray Exposures.

See Tables 5 and 6

7-2. 9.3

Maximum Permissible Internal Dose

The maximum permissible concentration in air or water is determined by the maximum permissible body burden for the particular nuclide to produce 0.3 rem /week st the critical organ. Article 1.7 of Sec. 7-2 describes methods of calculation for For periods not exceeding specific nuclides, for both continuous and single exposures. a few months, it is usually safe to use the following "general maximum permissible concentrations" (from Table 7 of Sec. 7-2). /J or 7 emitters or Dose

emitters

5

X

10-9 f
jic/ml air

from Internally Absorbed Discrete Particles. 10

See Sec. 7-3,

NUCLEAR RADIATION

10-7 /ic/ml water 10~7

^c/ml water

See Eqs. (33) to (36) of Sec. 7-2.

SHIELDING

by E. P. Blizard. Nomenclature

S.

absorption build-up factor, a function of geometry, material, energy, and number of relaxation lengths (Table 7 of Sec. 7-3) Br = dose build-up factor, a function of geometry, material, energy, and number of relaxation lengths (Tables 4 and 5 of Sec. 7-3) D = dose rate (including build-up), rad/hr £<>= photon energy of source, Mev. B = rate of heat deposition, ergs/(sec)(g) H' = rate of heat deposition, watts/g R = distance from point source, cm S = source strength for point source, photons/sec t = thickness of attenuating material for point source, cm = thickness of attenuating material for collimatcd plane source, cm m = linear attenuation coefficient of uncollided flux, cm"1 =■energy

j

p =

density of material, g/cm'

1-44

GENERAL DATA

[Sec.

1

ti/p = total mass attenuation coefficient, a function of material and energy (Table 1 of Sec. 7-3), cm'/g a function of material and ita/p = energy-absorption mass attenuation coefficient, energy (Table 2 of Sec. 7-3), cm'/g r =■uncollided gamma flux at penetration t or x, photons/(cms)(sec) To = incident gamma flux for plane source, photons/(cm,)(sec) Relaxation Length, Half-thickness,

10.1

and Tenth-thickness

The thickness of material to attenuate a y dose rate by a factor of 2, e, or 10 is The thicknesses are called convenient for rule-of-thumb estimate of attenuation. the relaxation length, half-thickness, and tenth-thickness, respectively, and are interconvertible in accordance with Table 38. These characteristic thicknesses are a function of y energy and also vary with actual thickness because of the build-up factor. Relaxation Length, Half-thickness,

Table 38.

Relaxation lengths

Half-thicknesses


1 0.6931 2.303

1.

Tenth-thicknesses

0.4343 0.3010 1

1.443 1 3.322

-y-ray Attenuation from Point and Plane Sources

10.2

Step

and Tenth-thickness

Calculate uncollided flux

Point source:

r

=

-4-

4irR3

r

X e-f

Eq.

(2) of Sec. 7-3

Plane source:

r

= Toe-"*

Eq.

(2a) of Sec. 7-3

Determination of p.. Table 1 of Sec. 7-3 gives (m/p). m = (p/p)pTable 21 of Sec. 7-3 gives both (jjl/p) and p. for selected heavy elements in a more limited range than Table 1. Number of relaxation lengths* = pi or px. Calculate •y-radiation dose rate D or rate of heat deposition H at penetra Step 2. tion t for a point source

D

- 5.767

X

10-'(p.a/p)E<,rBr

rad/hr

Eqs. (3) and (4) of Sec. 7-3

Simplified solution for tissue:

d

E0VBr (Table 3, column 3) H = 1.602 X l0-*(pJp)E„rBa ergs/(sec)(g) H' 1.602 X 10-ls(M„/p)tfcirB„ watts/g

=

(Table 3, column 2)

-

Determination of E0. * Relaxation length

rad /hr Eq. (5) of Sec. 7-3

See Art. 2 of Sec. 7-3, also Tables 1 and 2 of Sec. 7-1.

(thickness

to produce attenuation

by factor «)

*

Sec. 1-1]


1-45

(mo/p) from Table 2 of Sec. 7-3; B, from Dttermination of (na/p), BT, and Ba. of Sec. 7-3 (for point source) and Table 5 (for plane source); B„ from Table 7 [for point source).

Table 4

7-Ray Attenuation, Special Cases

10.3

Lead Shield for Co60 and Cs137 Sources. Thickness for 7.5 mrem/hr at surface (see Table 6 of Sec. 7-3). 10.32 Approximate Unshielded y Dose from a Point Source. See Art. 9.2 of

10.31 outer

Sec. 11.

D

« 7CJW(ff')s

D = dose rate, r/hr ~ rad/hr C = activity of point source, curies JJo = photon energy, Mev R' = distance from source, ft 10.33 Shields containing a homogeneous Composite Shields. and laminated shields (see Art. 1.4 of Sec. 7-3).

where

mixture of elements

Sources of y and X Rays

10.4

spectrum of prompt y rays from fission of U*" is given in Table 8 of decay energy of fission products as a function of time and energy groups of Sec. 7-3; and y energy from neutron capture in Table 9 of Sec. 7-3.

The energy Sec. 7-3; the


in Fig. 4

10.5

Dose from Point Source of Fission Neutrons. Hydrogenous (13) and (14) in Art. 3.4 of Sec. 7-3 give solutions in terms of thickness, hydrogen density and thickness (Fig. 5), and removal cross sections 12 or Fig. 6). Separate solutions are given for nonaqueous and aqueous Biological

10.61 Skidds. shield iTable

Neutron Attenuation Principles

Equations

shields.

shields (less than 50 atomic per cent hydrogen). The dose is by Eq. (15) or (16) as a function of biological effect of neutrons (Table 13), distance from source, shield thickness, and total removal cross section. Smhydrogenous

calculated

Sources of Neutrons

10.6 The delayed neutron

U*" fission spectrum of prompt neutrons

is given in Table 15 of Sec. 7-3, neutron data in Table 16 (see also Table 39 of Sec. 1-1), and a- and photosources in Tables 17 and 18. 10.7

Geometry

y and neutron attenuation for with curves and tables of pertinent functions.

Article 5 of Sec. 7-3 gives methods of calculating various geometries,

10.8

Shield Materials

Article 6 of Sec. 7-3 contains nuclear and engineering data of common shielding For list of tables, see Table 9 of Sec. 1-1.

materials.

11 See Sec. 8-1, by » bare reactor.*

PHYSICS OF REACTOR CONTROL

S. Krasik.

The section is based on one-group diffusion theory for

* Two-group theory for a bare reactor and the effect of a reflector ">Arte. 13 »nd H of Sec. 6-2.

on neutron

lifetime

are discusnod

1-46

GENERAL DATA

[Sec.

1

Nomenclature

B2 = buckling (geometric), cm"1 k = multiplication constant in infinite medium, dimensionless; k,n is "k effective"; k,x is "excess k" L = diffusion length, cm T = solutions of the reactor kinetics equation, sec; T, is the stable period (e-folding

time) after transients have disappeared = multiplication «S = source strength, neutrons /sec V = volume of reactor core, cm1

M

v — 18 =

A = X = p = Z„ = t = 0 =

neutron velocity, cm/sec fraction of delayed neutrons per neutron emitted in fission; 0, refers to ith group effective neutron lifetime, sec. /* is frequently used for A decay constant of delayed neutrons, sec1; X, refers to t'th group, reactivity, dimensionless macroscopic absorption coefficient, cm"1 mean life of delayed neutrons, sec; n refers to ith group; f is average mean life neutron flux, neutrons/(cm2)(sec);

Group

55.72 22.72 6. 22 2.30 0. 610 0. 230

lOOpVp* percentage of delayed neutrons

ri

X.

80. 39 32 78 8. 97 3.32

0.0124 0.0305 0. 111 0. 301 1. 14 3.01

half-life, sec

1 2 3 4 5 6

Data for U"s*

Delayed-neutron

0.88 0.33

3. 3 21.9 19. 6 39. 5 11.5 4. 2

Totals

100 0

* Data of Keepin,

0ITI

0.00021 0 00140 0.00125 0.00253 0.00074 0.00027

0.0170 0.0459 0.0113 0.0084 0.00065 0. 000089

0.0064

0.0833

Wimett, and Zeigler, Tables 2 to 4 of Sec. 8-1. 11.1

Delayed-neutron

Data

Table 1 of Sec. 8-1 gives the data of Hughes et al. for thermal fission of U236 and Tables 2 to 4 give later data by Keepin, Wimett, and Zeigler for thermal and fast fission of Th232, U233, U23S, U2™, Pu230, and Pu2<0. Table 39 of this article summarizes data for U236 from Tables 2 to 4 of Sec. 8-1; and numerical values in this article are based on these data. Table 40.

Reactivity Conversion Factors for

Data of Hughes, Dobbs, Cahn, and Hall

Data of Keepin and Wimett

Reactivity, p

Inhoura

Dollars

Cents

1 ~2.3 X I0 » 0.0064 6 4X 10 >

~4.35X 10<

156 ~3.6 X IO-» 1 0.01

I.56X 10'

1

-280 ~2.8

U23S*

~0 36 100

Reactivity, p

1

~2.6X 10* 0.00755 7 55 X 10-*

Inhours

Dollars

~3 .84 X 10' 132 1 ~3 45 X 10 ' 290 1 0.01 2.9

Cents

1.32 X 10' ~0.345 100 1

• One dollar of reactivity correspondsto prompt critical, i.e., p equal to 0. Reactivity in inhours = p/(p for stable period of I hr) In both casesp may be calculatedby the inhour equation (Art. 11.41below), but the table is basedon the approximation that p for a stable period of I hr is equal to 0f/3,6OO. For long periods, reactivity in inhours is approximately 3.600/ 7Y


Sec. 1-1]

11.2 ,

t./f

=

neutrons

Reactivity

generation born in (n + l)th 2 k ■ = -— , : ; T._. born in nth generation 1 + L'B1

neutrons

= 1

-

+ k„

- -—

11.3

Effective Neutron Lifetime* A

1 — p

=

-

klx

.

.

Art.

. „ , _ 1.2 of Sec. 8-1

"

~ ^ —

—

k.u

1-47

k.si

1 =

p 1 — p

1/[»2.(1 + L'B1)] sec

where » = neutron velocity, cm /sec See Art. 1.2 of Sec. 8-1, also Eq. (245) of Sec. 6-2. 11.4

It

is assumed that there is no loss of delayed neutrons.

Inhour Equation

11.41

-


Reactor Kinetics Equation

k.,, k,/f

—

1

A

Tk,,// =

+

) V t

,

1

?\

Eq. (4) of Sec. 8-1; also Eq. (241) of Sec. 6-2

m + A,/

Eq. (5) of Sec. 8-1

!+7^+?rfc]

For UMS, the data of Keepin and Wimett give:

J

A

T

[A

IT

0.00021

, 1

+ 0.01247'

_

0.00140 1

0.00125

0.005 100253

f + O.llir

+ 0.03057'

+

1

4-0.301 T 0.00027

0.00074 1

+ 1.14T

1

+

1

3.0irJ

11.42 Guide to Solution of the Inhour Equation. See Art 1.2 of See. 8-2. The Stable Iteaclor Period is the e-folding time after transients have disappeared. For positive reactivity, T, can have any positive value. For negative reactivity, T, can have any negative value numerically greater than r, (80.39 sec for U"«). The inhour equation is solved by trial, making use of simplified approximate

formulas (below) where applicable. The Six Transient Hoots. These are all real solved by trial, and are located by the sequence

-80.39

< T7, <

-32.78

< T2 <

-8.97

<

T,

<

and negative.

-3.32

<

T,

< <

They

-0.88 -0.33

are

usually

< T, < 7', <

-A

See 11.43 Approximate Formulas for Stable Period with Positive Reactivity. Art. 1.3 of Sec. 8-1. Very hug periods (small reactivity): p < 0.0005, T, > 200 sec (useful also as first approximation for shorter periods — the approximate value of T, is too large). • The following is for hare-reactor one-group theory. The contribution of slowing down to neutron lifetime (two-group theory) is diacus»ed by Dietrich in Arts. 13.3 and 14.2 of Sec. r>.2.

1-48

GENERAL DATA

For UJ",

T.

=

T.

=

-

A

=

k./f(p

(see

P

Art.

1

1.32 of Sec. 8-1)

=

(A + 0.0833)

P

Very short periods (reactivity exceeding equal to at least several times A).

T'

«

(A + jSf)

p

[Sec.

p

prompt critical by an amount numerically

-

_ A(l (p

0)

See Art. 1.33 of Sec. 8-1; also Eq. (242) of Art. 6-2. 11.44 Approximate Formulas for Stable Period Art. 1.5 of Sec. 8-2.

-

p)

- 0)

with Negative Reactivity.

See

Very long periods (small negative reactivity), T. negative but numerically greater than 200 sec. Same as for positive reactivity. Shorter Periods. The period is always numerically larger than the mean life of the longest-lived delayed neutrons. For U"> t his is 80.39 sec. The limit T, = — 80.39 sec is approached for very large negative reactivity. 11.45 Rough Approximation of Time Behavior for Step Change of Reactivity 1. Prompt jump (or drop)


± «

o

M

~

"> 0 — p

occurring in a small fraction of a second.

See Art.

1.6 of Sec. 8-2 and Eq. (244) of

Sec. 6-2.

2. The flux increases from the level of the prompt jump, with a stable period T, calculated from Eq. (4) of Sec. 8-1 or from approximations.

11.5

Steady-state Subcritical Reactor (fc«// <

See Art. 1.2 1.2 of Sec. 8-2, Art.

1

M

Total neutron production: Total neutron population:

MS

, Neutron a flux in core: xt

MSAv — I

12

= 1

1 —

k,/t

n/sec

-—— *.//

-

neutrons

SAv

=

with Source

g of Sec. 8-3, and Art. 13.1 of Sec. 6-2

1.1

Source multiplication:

MSA

1)

I'd

- k,/f)

..

...

n/(cm,)(sec)

.

FLUID PROPERTIES

See Sec. 9-1, by Wayne H. Jens. 12.1

Units

Nonsystemntic units (see Art. 1.15 of Sec. 1-3) are used in fluid-flow and heattransfer calculations. 12.11 Fluid Flow. The units are foot, second, pound-mass, and pound-force. 12.12 Heat Flow. The units arc foot, hour, pound-mass, Btu, and degree Fahrenheit.


SEC. 1-1] 12.13 as used

1-49

Fluid Properties. Properties are expressed in Sec. 9-1 in the same units for heat flow, i.e., foot, hour, pound-mass, Btu, and degree Fahrenheit. Dimensionless

12.2

Numbers or Moduli

Dimensionless numbers must be calculated from properties given in consistent units. Thus in fluid-flow problems velocity, which is expressed in feet per second for all other calculations, must be expressed in feet per hour in calculating dimensionless numbers such as Reynolds number from data of Sec. 9-1. Viscosity data from other sources are often expressed in other units. If it is con venient to express the other quantities in consistent units (as in the metric system), the data can be used directly; otherwise it is suggested that Table 17 of Sec. 1-2 be used to effect conversion. The most frequently used dimensionless numbers are: Reynolds number, Re

_ DVP A"

Nusselt number, Nu =


Prandtl number, I'r =

ftP k k

For meaning of symbols, see nomenclature tables in Arts. gives a list of dimensionless numbers.

13

and

14.

Table 3 of

Sec. 9-2

Density of Water Table 41 gives the density of water in grams per cubic centimeter at atmospheric from 0 to IOO°C. The density maximum of 0.99997 g/cm* at approximately 4*C corresponds to 1 g/ml exactly (see definition of milliliter in Table 6 of Sec. 1-3). For temperatures above 100°C, Table 19 of Sec. 9-1 gives the specific volume v in cubic feet per pound over a range of temperature (°F) and pressure (psi). The density is given by: pressure

1

0.01602

62.428i>

v

s

.

,

The density of heavy water is given in Table 27 of Sec. 9-1. Table 41. °c

0

2

Density of Water, g/cm' 3

4

5

6

J_ _'_

9

0 10 20 }0 40

0.99

984 970 821 565 222

990 961 799 534 183

994 950 777 503 144

996 938 754 471 104

997 925 730 438 063

996 910 705 404 021

994 895 679 369 »979

990 878 652 333 •937

985 860 624 297 •893

978 841 595 260 *849

50 60 70 80 90

0. 98

0.96

804 321 778 180 531

759 269 720 118 464

712 217 662 055 396

666 164 604 •991 327

618 110 545 •927 258

570 056 486 •862 189

522 002 426 •797 119

472 •947 365 •731 049

422 •891 304 •665 •978

372 •835 242 •598 *907

100

0.95

S35

0.97

GENERAL DATA

1-50

[Sec.

1

FLUID FLOW

13

See Sec. 9-2, by Charles F. Bonilla.

Nomenclature cp = specific

D

= D. = e = fr = G = gc =

V v to

0

K L


AP Q

S

ft p

heat at constant pressure, Btu/(lb.
in heat flow, units are

lb.„/(ft!)(hr)

constant numerically equal to gravitational acceleration, 32.17 ft /sec*. Usu Sometimes ally regarded as a dimensional constant (ft)(lb.v)/(secJ)(lbf). regarded as dimensionless = average velocity, ft/sec = specific volume, ft'/lb = rate of mass flow, lb/sec — volumetric coefficient of thermal expansion, (°F)_1 = number of velocity heads lost, dimensionless =• length, ft = pressure drop, Ibf/ft* — volume rate of fluid flow, ft'/scc = cross-sectional area of fluid stream, ft' = viscosity = absolute is viscosity, viscosity = dynamic lbv/(hr)(ft). viscosity of fluid at wall temperature, in is viscosity at bulk temperature = density, lbw/ft' 13.1 General Equations

13.11

Continuity Equation G = Vp

13.12

w

-

Eq. (44) of Sec. 9-2

Velocity Pressure Velocity pressure

13.2 13.21

Qp = VpS = GS

=

— 2gc

=

—

2pgc

2gc

Friction in a Straight Pipe

Pressure Drop by Friction

A/'

fr

=

4/F

•

A

•

p~

=

4/K

■~ •

^-

[See Eq. (28) of Sec. 9-2]

is the Fanning factor, given in Fig. 2 of Sec. 9-2. Many authors eliminate the 4 by using & friction factor equal to 4fr. 13.22 Streamline Flow, Re < 2,100. See Art. 3 of See. 9-2. For isothermal flow in long channels,

f

fr For

=

Hi/Re

AP

= 4/K ■— • — - = -

Dr

2f/c

—

D,'gc

short channels, sec Table 2 of Sec. 9-2. D, for streamline flow is given in Table 42 below (see Table 1 and Art. 3.12 of Sec. 9-2, in which other cross sections are also discussed). For nonisothermal flow (fluid and walls at different temperatures), calculate AP (or fr) as above, and multiply by (ji»//i*)0M if the fluid is being cooled, and biv/iu)0** if the fluid is being heated. (See Art. 3.2 of Sec. 9-2 which also gives alternate treatments.)

SEC. 1-1]


Shape

Streamline Flow Parallel slot

Square

Circular

e

♦ a

»a-»

1-51 Narrow annulus

Annulus

T

f

b

—02-

W

I.12az

V*tf-ttk ****

Turbulent Flow, Re

> 3,500. See Art. 4.2 of Sec. 9-2. Factor /f. Calculate Re, estimate e/D from Art. 4.2 of Sec. 9-2, read Fig. 2 of Sec. 9-2.

13.23

Fanning from

/*

Examples:

drawn tubing, 0.000,005 ft; for commercial steel pipe, 0.00015 ft. /* i» reactor applications is typically 0.006, but may range from 0.015 to 0.003 or less. Isothermal Flow. Apply the equation of Art. 13.21. If fluid and wall are at different temperatures, calculate/],- as Xonisolhermal Flow. t for

but

above,

use m and p at the average of bulk gas temperature and wall tempera alternate method see Art. 4.21 of Sec. 9-2). liquids, use bulk liquid temperature and multiply fr by 0.98 0„/w)0,1J. The bulk density is used in calculating pV*/2gr. See Art. 4.2 of Sec. 9-2. For a general approximation, D.for Turbulent Flow.


1. for gases,

ture (for 2. for

D.

4

X

area of cross section

wetted perimeter

(from Art. 4.2 of Sec. 9-2) gives Dt for special cases; the formulas conform the above formula in all cases except for the annulus. If the pressure drop is small compared with the total pressure, Compressible Fluids. use average properties; otherwise see Art. 4.22 of Sec. 9-2. Table

43

exactly to

Circular

Rectangular

Table 43.

Special Cases

Square

Parallel slot

Annulus

Shop* ♦

*-a—

h-Di-H

— D2— 2ob a+b

13.3

2b

Friction by Change of Direction or Pipe Cross Section in Turbulent

2f7e 13.31

2D2tnD2/D|

Pipe Fittings and Bends.

See Table 4 of Sec. 9-2.

Narrow annulus

Mi

— Dz-H D2-0,

Flow

GENERAL DATA

1-52 K

Table 44.

s./s,

[SfiC.

1

for Sudden Change of Cross Section

0

0. 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.81

0.64 0.34

0.49 0.31

0.2S 0.22

0. 16

0.09 0. II

0.04 0.03

0.01

0.36

0.36

(0.4)

0.02

0 0

Enlargement

0.27

0.16

* K for contraction varies appreciably with conditions. The tabulated values are the lowest reported contraction — to be conservative 0. 1 should be added throughout. For ( Weisbach) for a sharp-cornered rounded edges the values are much lower.

Sudden Change of Cross Section. 13.32 and sudden contraction of a pipe.

S\/Si

Table 44 gives

K for

sudden enlargement

= (smaller cross-sectional area) /(larger cross-sectional area)

K is to be used with the higher velocity, i.e., the velocity in the smaller of the two cross sections. For more accurate treatment, see Art. 4.3 of Sec. 9-2. For a well-rounded entrance to a contraction, K = 0.05. 13.33 Orifices and Nozzles in Pipes. The over-all pressure-loss factor, based on the velocity at the orifice or nozzle, is given in Table 45. K

Table 46. Orifice

or nozzle diam.


Pipe inside diameter

K

for sharp-edged

K

for nozzle

orifice

for Orifices and Nozzles in Pipes

0. 1

0.2

0. 3

0. 4

0.5

n u . t. 0

It / U. 7

0.8

0.9

2.68

2.62

2.38

2. 16

1.87

1.50

1. 10

0.66

0.40

0.98

0.92

0.83

0.70

0.56

0.41

0.26

0.13

0.04

of the orifice system, general, K = I/Cd1, where Co is the discharge coefficient including a sufficient length of pipe to permit contraction and subsequent expansion of the stream.

In

External Flow

13.4

For more than five rows of tubes, K per row Flow across Tube Banks. on maximum velocity): 0.72 for triangular pitch, 0.32 for rectangular pitch. Article 8.2 of Sec. 9-2 gives a more accurate treatment. See Art. 8.3 of Sec. 9-2. Flow through Beds of Particles. 13.42 13.41

(based

13.5

Pressure Changes Other Than Friction Losses

The subscripts 1 and 2 refer to the upstream and down Acceleration. 13.61 means a pressure drop in the A positive value of stream side, respectively. direction of flow, a negative value means a pressure gain. Acceleration causes a loss (AP positive), deceleration causes a gain (AP negative).

\P

Pressure Drop Due

to Change

of Velocity at Constant

Density

--îr^-t' [©'"-I

'['-©']

Since this drop for a closed system usually does not exceed one high-velocity it is usually ignored in rough calculations.

head,

of Density in a Channel of Constant

Cross

Acceleration Section

Pressure Drop Due

AP

=

to Change

P2IV

—

---

P1V1

O

For a more exact formula see Sec. 9-2, Eq. (30). disregarded except for a boiling liquid.

(Vt

-

Vt)

This drop can

also generally

be


Sec. 1-1] Change

1-53

In rough calculations, it is usual to ignore the of Velocity Distribution. in pressure that accompany velocity redistribution across a channel. Difference of Elevation. See Eqs. (15) and (16) of Sec. 9-2.

changes

small

13.62

13.6

Total Pressure Drop

The total pressure drop between two points is the algebraic sum of losses by friction, If parameters vary, drop by acceleration and difference of elevation. sections are subdivided and formulas are evaluated over the subdivisions, using In some cases analytic solutions are available average values for the parameters. for cases of varying parameters (see Art. 4 of Sec. 9-2). and pressure

14 See Sec.

HEAT TRANSFER

9-3, by Charles F. Bonilla. Nomenclature

A = area normal to heat flow in slab geometry, ft1

width, ft

a =

H = heat generation per unit volume per unit time, to be constant unless otherwise stated Generated for wjivans (University of Florida) on 2015-09-23 02:45 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

*/

Btu/(ft*)(hr).

k =

thermal conductivity,

L = length of cylinder, q —

r = =

( = =

i

is assumed

Mm coefficient, Btu/(ft,)(hr)(°F)

=

kt = radiation coefficient, Btu/(ft»)(hr)(°F) hr.g = gas radiation coefficient, Btu/(ft')(hr)(°F)

T

H

ft

Btu/(ft)(hr)(°F).

k is assumed

to be constant

It, is assumed that heat flow is normal total heat flow per unit time, Btu/hr. to the surface in slab geometry and radial in cylindrical and spherical geometries radius, ft absolute temperature, °K I temperature, °F distance from left face of a slab or plate, ft

1, 2, 3, — Subscripts: constant for a layer (a, H, which vary across a layer values at the left side of a layer (Tables 46 to 49). Surface film coefficients adjacent wall temperature,

temperature. Variables (x, r, and

= first, second, third, — layer. For quantities which are For quantities k) the subscript refers to the entire layer. (q/A, q/2-rL,
and fluid temperatures take the same subscript as the the fluid temperature being primed to distinguish it from

the wall

<) are

indicated by absence of a subscript.

14.1

Steady-state Conduction

Conduction formulas for important simple cases are collected and tabulated in article. For other formulas and for unsteady-state conduction, see Arts. 2 and 7

this

of Sec. 9-3.

Algebraic Signs. In all tables in this section, the positive direction of distance and flow is to the right in slab geometry, and radially outward in cylindrical and spherical geometries. If heat flow is in the negative direction, algebraic formulas remain unchanged, but when numerical values are substituted, the negative direction is recognized by change of sign. Negative values arising in the course of computation heat

retained. Conduction in Homogeneous Solids with Internal Heat Generation. or Plate. Table 46 gives formulas for heat flow normal to the surface. must

be

14.11

Slab

1-54


[Sec.

1

Conduction in Homogeneous Slabs or Plates Nomenclature and General Procedure* Calculate qi/A if not already known: (a) If q* is known, q\/A = qt/A — H\a\ (6) If l\ — fa is known, qx/A is given in each particular case by the first equation for that case. hi «■thermal conductivity H\ ■=> heat generation per unit time per unit volume For sign conventions see text.

q/A-

General Solutiont Sen above figure. Criterion for an internal temperature maximum: cooling from both sides, or ai//i > (-qi/A) > 0

[

qi/A

ti ti
-U= - = - tit -

-= (i]

- t*)k\/a\ -

Hiat/2

(qi/A + Hia,/2)ai/k\ (qi/A + Hix/2)T/ki (qi/Ay/{2Hxki)

Heat Through-flow with No Internal Heat Generation


See Eq. (8) of Sec. 9-3. Hi = 0

q/A

-

- -

(ii

- (i)*i/ai

fi tt = {qi/A)aifk\ (qi/A)x/ki t ti Highest temperature: fi or

fiat

a:

Oor

r

= ft], respectively.

Internal Heat Generation with No External Heat Input Case t.

Heat remolded at right-hand

See Eq. (9) of Sec. 9-3.

Case 2.

of Case

Case 3.

»(

* Solutions

I.

- -

surface

0

ri ti = //iuiV2Jfci t ti Ifix*/2ki J*max ™ 0 'max =

- -

-

Heat removed at both surfaces, surface temperatures equal Q\/A ■* — Hiat/2 ,. t ti = //i(a. x)x/2ki r
-

^\

in this table are for uniform

H the general equation

-

Heat removed at left-hand surface Qi/A — — H\ai t, (i tfioi'/2*i ( ti = Hi(oi x/l)x/ki XmKx — di tuutx = fi

This case becomes redundant if it is regarded as a mirror reflection

qi/A

-

heat generation.

-

- -

For exponential

heat generation

of the form

a exp ( — paj) + 0 exp ( — ftgx)

is

To obtain ti — tt write ai for x. t To take into account unequal surface cooling or heating conditions of a single-layer slab, use Table 49, but reject all terms applying to second and third layers.


SEC. 1-1]

1-55

Hollow Cylinder. Hollow-cylinder formulas contain terms of the type In r2/ri. When Tz/ti is not large, it is convenient to write i / In T\fT\

™

p..

r

x

radial thickness of tube ■ —■ wall — . average radius of tube

r2/n


F

r2 —

« t„

(r2

+

n

n)/2

1

1.5

2

3

4

1

1.01

1.04

1.10

1.15

„ 2(r»/n

t

=■

—

r2/ri +

—1)■ 1

Table 47 gives formulas for radial heat flow. When r2/n < 2, it is also satis factory to use slab formulas, with oi equal to the wall thickness of the tube and the flow area equal to the cylindrical surface at average radius, i.e., x(ri + r2)L. Table 48 gives formulas for radial heat flow in spheres. Sphere. 14.12 Conduction in Multilayer Solids with Internal Heat Generation and Con vection Cooling. Table 49 gives solutions for the temperature differences across layers for multilayer slabs, cylinders, and spheres, with heat generation uniform in each layer and with convection cooling on both sides of the assembly. Solutions are given for three-region solids, but they can be extended to any number of regions by including terms for additional regions. Procedure. The specified parameters are evaluated in sequence for the geometry in question. Usually the temperature difference Att* between the coolant streams is known* and in this case q\ is determined from the second of the two formulas for g'i. If the heat flow (at either surface) is known instead of the temperature difference, g'i is determined from the first of the two formulas.! If the left-hand or inner surface is cooled, the numerical value of q\ is negative. Cylindrical or Spherical Assembly with Solid Central Region. For a solid central region the solution is modified as follows: 1. Continue to designate the solid inner core as region 1, with radius r2. 2. For a cylinder /i = H,n*/2 and mt = Hirt'/4ki. For a sphere fi = Htri'/3 %nd mi = Hir^l&ki. The parameters Q\, l/o, h, and q\ have no meaning. 3. Calculate /, g, I, and m for the surrounding hollow regions as usual. 4. For both cylinder and sphere Ati = mi, A<2 = m2, At3 = ms, At/, — m/,. Heal Transfer across a Fluid-filled Gap. A gap can be treated as an additional layer, say the ith layer. Usually there is no heat generation in the layer and fi, gi, and m.i are all zero. The value of U is as follows:

Convection

Conduction

Slab

Cylinder

!/«' + \/h"

l/(r,*') + l/(n+,A")

Oi/*

(In

ruifn)/ki

Sphere

l/O-i'V)

O/n

+ l/(ri+,«»")

-

l/n+i)/fa

where h' and h" are the convection coefficients on the two sides of the gap. Distribution within a Region. The temperature distribution within Temperature the first layer is given by the general case in Tables 46 to 48. With proper change of subscripts the same formulas, including the criterion for a temperature maximum, apply to any layer. The parameter Meaning of Parameters. represents! the total heat generation within a layer, / represents the temperature difference across a layer caused by unit rate of constant heat through-flow, gl is the temperature difference across a layer caused by the internal heat generation within the layer, and m is the temperature difference caused by heat generation within the layer and all interior layers. Table 50 gives formulas for simple cases, Simplified Formulas for Fuel Elements. ignoring heat generation in the clad. For less simple cases, use the general multi

/

layer solution. • Aft* may be positive, zero, or negative. It is positive if the temperature at the left-hand side of a ?Ub is higher than that at the right-hand side, or if the temperature inside a cylinder or sphere is higher than the temperature outside. t See second footnote in Table 49. X The characteristic geometric divusorto A, 2wL. and 4jr have been applied to simplify the arithmetic.

GENERAL DATA

1-56 Table 47.

[SEC.

Conduction in Homogeneous

1

Cylinders

Nomenclature and General Procedure* Calculate gi/2xL if not already known: (a) If f/j is known q\/2vL = qt/lirL — Hi(ti* — n!)/2 (6) If t\ — tt is known, qi/2rL is given in each particular < by the first equation for that case, fci = thermal conductivity = heat generation per unit time per unit volume Hi For sign conventions, see text. General Solution for Hollow Cylinder! figure.

See above

Criterion for internal temperature maximum: cooling from both Hides or > 0 n> > -(q,/2rL)(2/Hi) ri"

-

?./2t£

- li

-

-

(«■

<»)*i

- HiHn' - r,') -

In ri/ri In ri/ri + tfi(r.»

2/-11In

ri/n]/4

- - -

2, ,' In ri/n)/4]/Jti = (W./2.-Z,) ri> (((7i/2xf,) In r/ri + Hi(r' ri» iVi" In r/n )/4]/fci n« (oi/2rL)(2/ffi) fl — (mav • /iiria(a — 1 — a In a) /4ki where a ** (rmax/n)1

li

- -

tl I rmw'

-

-

Hollow Cylinder with Heat Through-flow and No Internal Heat Generation

qi/2rL

= (li —

-
- -

ti)k\/\ll rj/n

[(7,/2»L) In r,/r,]/*, K«,/2tL) In r/nj/ti d I Highest temperature: d or tt at ri or rj, respectively. (1

t; r2


H,= 0

Hollow Cylinder with Internal Heat Generation and No External Heat Input Caeel. Orttward heat flow 0 9I/2W, 2/-,' In n/ri]/4*i d (i = ffi|(ri' ri») (i — * — ffil(r» 2n« In r/n]/4Jbi ri») Maximum temperature: li at ri

-

-

Cage

f.

-

-

-

-

Inward heat floic = n»)/2 = tf i|2vi« In r-i/ri Maximum temperature: (i at rj

-//,(ri'

q,/2*L ll tl

-

-

Com 3.

Outward

-

-

n')]/4*i

and inward heat flow, surface temperature* ?i/2xL = -/Y.(r,„a*> n«)/2

-

-

-

-

equal

rro.x' (r.« n")/(21n r./n) a + a In a)/4Ar, li (max ffiri«(l (rm„/r,)' = |(r,/n)' where a l]/(2 In — For ti t use the general formula.

-

fl

'

-

i

-

rj/ri)

* Solid Cylinder with Internal Heat Generation tl (l

- II - -

* Solutions are for uniform heat generation.

// fi - t

J

= (a/r) exp

(qt/2wL) In r/r, + — (exp

//ir2V4*l

« Hir'/Aki 'mux = 'l

For exponential heat generation

(-nar)

(-^n)

In

+

r/n

(£/')

exp

+ ffi(Ma')

4- — [exp

of the form

(-^r)

-

Ei(narx)}

(-ppn)

In

r/n

+ Bitjipr)

-

Bifofin)]

I

integral (Table 15 of Sec. 1-2). where Ei is the exponential t To take into account unequal surface cooling or heating conditions of a single-layer hollow cylinder, u»e Table 49 but reject all terms applying to second and third layers.


SEC. 1-1]

Table 48.

1-57

Conduction in Homogeneous Spheres

Nomenclature and General Procedure*

Calculate q\ /4r if not already known fa) If qi is known, qi/A* = gi/4r — H\(ri* — na)/3 is given in each particular case by (6) If d — ft is known, the first equation for that case. For sign conventions see text.

General Solution for Hollow Spheret

-

-

-

-

-

-

-

-

l/r,) qx/4r n)(n + 2r,)n/6 ffi(ri (li h>*i/(l/ri l/r,) + ffi(n Criterion for internal temperature £i — f* — [(fli/4r)(l/n n)»(n + 2n)/bn]/ki maximum: I tx l/r) + Wi(r n)»(r + 2ri)/6r)/*i [(fli/4r)(l/n eooling from both sides n> rm„J («i/4r)(3/.Yi) -(7,/4»)(3///0 or > 0 Fur fm„x evaluate rmax and substitute for r in t\ — I formula. rt* n» >

-See above figure.

- -

-

-

-

Hollow Sphere with Heat Through-flow and No Internal Heat Generation

-

-

-

-

1/rt) (ti (t)*i/(1/n (fli/4r)(l/ri l/n)/*i » 1 (oi/4ir)(l/ri l/r)/ifci Highest temperature: ti or ij at rt

Ai/4r

ii-/i=« Generated for wjivans (University of Florida) on 2015-09-23 02:45 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

-

or rt, respectively.

Hollow Sphere with Internal Heat Generation and No External Heat Input Caee 1.

Outward

heat flow

-

- tt - 0>7i(ri - n)Hn + 2n)/6r./fe. h - t - /7i(r - n)«(r + 2ri)/6r*i qi/Ax

I (i

Maximum temperature:

fi at rt

Inward heat flow

Cane 2.

-

-

-//.(rj> n")/3 »i/4* H,(n it n)>(2r! + n)/6tin li //i(r ri')/ri n)[2(rs> Muxiinuin temperature: fi at ri

tt /

- -

-

-

-

-

(r

-

n)(r

+ 2n)]/6*ir

•Safid Sphere with Internal Heat Generation

(i ti

- (i -timix

Hir,'/bki Hn'/bki

m ti at center

* Solutions are for uniform heat generation. t To take into account unequal surface cooling or heating conditions in a single-layer k Table 49 but reject all terms applying to second and third layers.

hollow sphere,


1-58


Sf.C. 1-1] Table

60.

1-59

Simplified Conduction Formulas for Fuel Elements Temperature

differences

Cue

Symmetrically clad plate: Cooling symmetric Cylinder: Solid core, gap. clad

Fuel

Internal gap

H,
No gap No gap

Clad

// fa/af/2ke

0, IIfa}at/kt

Hfrfag/2ko

Surface

film

///o//2A

H,af/h

H ft ftlrflrtke

Hi = heat (feneration per unit time per unit volume of fuel ay, at, a9 = thickness (radial in cylindrical geometry) of fuel, clad, and gap, respectively h = film coefficient at surface rj. Tt, and R = radius of fuel, mean radius of clad, and outside radius of clad respectively 14.2

Convection, Condensation, and Boiling


Articles 3, 4, and 5 of Sec. 9-3 give comprehensive compilations of formulas for convection, condensation, and boiling, respectively. Table 11 of Sec. 1-1 may be used as an index to these formulas. For turbulent flow of nonmetallic fluids in long circular pipes See Table 3 of Sec. 9-3 Nu = 0.023 Re" 8 Pr« Radiation

14.3

In the high-heat-flux regions of reactors, radiation is usually negligible compared with conduction or convection. 14.31 Surface Radiation. For black-body conditions:

^=0171[(iS)4-(lS,)1 hR

«

6.84

(— V VlOOO/

q/A

~

M*i

<•)

-

Bt«/(ft.)(hr) Btu/(ftl)(hr)(°F) Btu/(ft«)(hr)

and Tt = absolute temperatures of the two surfaces, °R jH.v = average absolute temperature of the two surfaces, °R ii and tj = temperatures of the two surfaces, °F In cases where surface radiation is important, emissivity and geometric factors must be introduced.* Gas Radiation. Carbon dioxide and steam arc the only gaseous reactor 14.32 coolants that have strong absorption (and radiation) wavelength bands. The coeffi cient of heat transfer is usually at least an order of magnitude lower than for surface radiation, and can be evaluated as a rough approximation from Fig. 1. If the value from Fig. 1 shows gas radiation to be important, a more accurate evaluation should be made.* Example: For a layer 0.01 ft thick of a mixture of 50 per cent COj with nonabsorbing and average gas-wall temperature of 1500°F: gas, at a pressure of 100 atmospheres,

where

Ti

DP From Fig.

1,

has

"

5

= 0.01

X

50

X

100

= 50

Btu/(ft!)(hr)(°F).

•See, for example, W. H. McAdams, "Heat Transmission," 3d ed., McGraw-Hill Inc., New York, 1954.

Book Company,

GENERAL DATA

1-60

[Sec.

1

sc

||

40


t- SO

Fio. 1. Nomograph to evaluate heat transfer coefficient by gas radiation. (From G. Etherington and H. Etherington, "Modern Furnace Technology," 3d. ed., Charles Griffin Jt Co., Ltd., London, in press.) 14.4

Over-all Coefficient of Heat Transfer, Conductance

14.41 Plane Geometry with Heat Flow Normal to Surface. The over-all coeffi cient of heat transfer U for constant heat through-flow q/A is defined by

q/A

-

U(l,

where ti and /> are the terminal temperatures of the system. Series Resistances to Heal Flow. For a series of parallel slabs of uniform thickness

U =

1

2(1 /A) + 2(a A)

where the first term of the denominator is the sum of the reciprocals of all surface coefficients in series and the second term is the sum of the thickness/conductivity ratios of all solid layers in series. Heat Flow Processes in Parallel. Heat transfer coefficients by independent processes in parallel are additive. For example, the heat transfer coefficient h from an external surface to surroundings at constant temperature is the sum of the convection and

radiation coefficients: h = hc

+

hR

If

he is a conductance across a transparent layer involving coefficients two heat-exchanging surfaces, hc = 1/(1/Ai + 1/hi).

14.42

Cylindrical and Spherical Geometries Cylindrical: U = 1-xL/CLl, + 2/) Spherical: U = 4x/(2t/ + XI)

where // and I are defined

in Table 49.

?i = UAt qi = UAt

hi and A2 at


SEC. 1-1]

15 See Sec. 9-4, by

1-61

THERMAL STRESS AND DISTORTION*

R. A. Daane. Nomenclature

and Units!

a = thickness of plate, inside radius of cylinder, in. 6 = outside radius of cylinder, in. d = diameter

E

—

k

«

H

=

t —

a = v =
of cylinder or sphere, in. modulus of elasticity, lb/(in.2) heat generation per volume, Btu/(hr)(in.*) thermal conductivity, Btu/(hr)(in.)(°F);fc = k'/V2 whereA;'isinBtu/(hr)(ft)(0F) temperature at a point, °F coefficient of linear thermal expansion, (°F)~l Poisson's ratio unit normal stress, tension when positive, lb/(in.*)

Table

61a.

Maximum

Thermal

Stress for Simple Shapes* Formula for maximum

Case

Component for uniform

Location of maximum stress.. Character of maximum stress Equality of surface stress Sign of surface stress

Component for temperature difference

between

surfaces

At surface At surface Biaxial, equal stresses | Biaxial, equal stresses Magnitude same at both surfaces except as modified by factors for thick hollow cylinders Tensile at both surfaces Tensile at colder surface, compressive at hotter surface

EaHd'

Solid sphere *

- ») 32*(l EaHa' I2i(l -

Long solid cylinder a-

')

Flat plate, free to bend a

EaHd'

■•)

60*(1

, Ha'

Ea 2(1

-

Atk

,)

-

Ea

-

2(1

, Ha'

.)

►)

-

2(1

Ea 2(1

I

Outride strews a\,

Ea 2(1

bk

y)

Inside stress a*

Ha'

Ea -

-)

Hollow cylinder!

-

- Il)t

•««

Mtt

-

/•(to

- (.)

(to

-

u)

U)

f

2(1

,)

Ea

Flat plate, not free to bend a.

J

/i

15.1

lh

t

is

a

* Stress components are algebraically additive; positive value indicates tension at the surface and For calculation of temperature difference, see Tables 46 to 48. a negative value indicates compression. The tensile (positive) stress at the colder surface. ii and 11 are the surface temperatures. are the temperatures at the inner and outer surfaces, /i. /». and are given in Table 516. la and respectively.

Order of Magnitude of Thermal Stress Calculated by Elastic Theory

*

the greatest temperature difference in the body and

For significance of calculated thermal stress see Art. 1.5 of Sec. 9-4. The formulas are valid for any consistent set of units, e.g., CGS°C.

is

V

—

K

where At

is

1

_ KEaAt

t


heat generation

stress

a function

of

GENERAL DATA

1-G2 Table Bib.

Factors

ID/OD

h- 6' 2b' - a'

1 (In ft/a)

2o» 6«

- a»

Ka (I

-

»)

I (In b/a)

-1,-1

[SeC.

1

/ for

Use with Hollow-cylinder

0.1

0.2

0 3

0.4

0.5

0.6

0 7

0.8

0.9

1.0

1.07

I 04

1.02

I 01

1.01

1.00

1.00

1.00

1.00

1.00

1.59

I 46

I 37

I 29

1.22

1.17

1.12

1.07

I 04

1.00

-0

- 0 63 - 0.71

41 -0.54

Formulas

-0.78 -0.83 -0.88

- 0.93 - 0 96 -1.00

Uranium (cast) 230, Beryllium (extruded) 300. Graphite 1.8.

at room temperature (see Table I of Sec. 9-4):

Austcnitic stainlesssteel 370. Aluminum 210. Zirconium 60. Structural steel 260, Cadmium 210, Concrete 30.

shape and temperature distribution.

and

is usually between extreme

1; also

K

For simple shapes,
=

Fct&t' 1 — v

where

for homogeneous

materials

At' is the difference between

and average temperatures in the body.

Exception: Axial Temperature

Gradient

in Long Bar


Generally, owi < \$Ea(M)d where (M)d is the maximum temperature difference cross-sectional dimension of the bar. 15.2 See 16

Art.

2

in a length equal to the major

Thermal Distortion of Bars and Rods

of Sec. 9-4.

RADIATION DAMAGE TO LIQUIDS AND ORGANIC MATERIALS

See Sec. 10-5, by V. 16.1

P. Calkins.

Calculation of Radiation Damage to Specific Materials

See Art. 3 of Sec. 10-5. Damage to a specific material is estimated from Table 1 of Sec. 10-5 (water, aqueous solutions, and fused salts), Table 2 (organic fluids), Table 3 (elastomers), or Table 4 (plastics). The procedure is as follows: Calculate the rate of energy deposition Ei for the given flux of each kind Step 1. of particle, using the right-hand part of the appropriate tabic:

Ei

=

(given particle flux) X 10" Particle dosages each equivalent to 1 X 10" rads

rads/sec

For Tables 1 and 2, n = !); for Tables 3 and 4, n = 7. Determine the total rate of energy deposition by adding the values of Ei for each kind of particle. Find the acceptable dose in rads from the damage indicated in the table. Step 2. Iêt this be D. Useful life of material = D/E sec, = D/Z.W0E hr = 0/(3.10 X lO'-E) Step 3. A numerical example is given in Art. 3.2 of Sec. 10-5. years. 16.2

Estimate of Fraction of Molecules Affected

In absence of data, the fraction of molecules affected may be estimated by formulas for energy absorption given in Art. 2 of Sec. 10-5.


SEC. 1-1]

Fraction of affected molecules

0

1-63

~ 10~1SG.1//J

= molecules affected

per 100 ev absorbed [commonly about four for covalent materials (see Art. 2.4 of Sec. 10-5)] M = molecular weight of compound D = absorbed dose in rads For example, if O = 4, M = 100, and D = 10" rad, the fraction of molecules affected = 0.04 or 4 per cent.

where

16.3 See

Radiation Damage to Electrical, Electronic, and Mechanical

Systems

Table 5 of Sec. 10-5. 17

SOLUTION OF EQUATIONS

by Alston S. Householder, and Sec. 3-2, by Ward Conrad Sangren. The theory and solution of equations are discussed in Arts. 4.2 and 4.3 of Sec. 3-1. Practical application of some of the methods of solution is illustrated here. See Sec. 3-1,

17.1 17.11

Polynomial Equations

Quadratic Equations


ax* + bx + r = 0 X

-6

±

y/b*

-

4ac

2a

Cubic equations can be solved algebraically Cubic and Quartic Equations. trigonometrically, and the method can be extended algebraically to solution of quartic (biquadratic) equations that have at least one pair of real roots. However, Horner's method and Graeffe's method, described below, can be conveniently applied to these equations. See Art. 4.3 of Sec. 3-1. 17.13 Graeffe's Method for Polynomial Equations. Graeffe's method is direct, requires no trial solutions, and solves for complex roots as well as real roots. Complications which may develop are discussed at the end of the example. The Method. To solve the equation x* + bx' + cx* + dx' + ex' + fx + <7= 0, tabulate and proceed as described at the top of page 1-64. Criteria for Terminating the Process and Formulas for Roots of the Original Equation. case 1. Each of the coefficients bm, cm, . . . is equal to the square of the correspond ing coefficient for the preceding equation; i.e., the product terms become negligible. This is the case when all roots of the equation are real and unequal. In absolute magnitude, xx = \/bm/l, it = \Zcm/b„, x* - y/dm/e„, etc. The sign for each root must be found by trial, or by applying rules from theory of 17.12

or

equations.

case 2. The coefficients in a given column, say the c coefficients, become and remain always half the square of the corresponding coefficients of the preceding equa tions, while the coefficients on either side of this column behave normally, i.e., become the square of the preceding coefficients. This indicates two numerically equal roots, although the signs may be different. The absolute magnitude of each of the equal roots is \/dm/bm, i.e., the 2mth root of the ratio of the normal coefficients. case 3. The coefficients of one column, say the c coefficients, continue to behave This indicates irregularly, while the coefficients on either side behave normally. a pair of complex roots x = u ± iv. A negative coefficient at any stage (except in the original equation) indicates the presence of such a pair of roots.

GENERAL DATA

1-64 First

stage of solution

fc —

1

lac

^f

^d^

—,2bd

1

Description of process

—

2ce-p2df

^g

= coefficients of original equation = squares of coefficients

—2eg

(algebraic sums of above terms)

_

g'

products as alternating

indicated,

signs

= Coefficients of transformation

equation

Process is repeated until crite ria below are satisfied. For successive values of m, write m = 1, 2, 4, 8, 16,

4

8 Generated for wjivans (University of Florida) on 2015-09-23 02:45 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

[Sec.

16

Qm

Modulus /dm/b„, —

Ji(6 \/r2

-

+ algebraic sum of all roots) u*

u ± iv

other cases. Cases of more than two roots equal in absolute magnitude, or more than one pair of complex roots between two real roots can also be solved by Graeffe's method, as shown on page 1-65.* Columns representing multiple roots may con Limitations of Graeffe's Method. verge slowly, and initial convergence may even give place to divergence as the method seeks to find roots which, in the higher-order equations, have been rendered unequal Roots of approximately equal absolute by accumulation of approximation errors. There are simple algebraic processes (see Art. value also lead to slow convergence. 4.2 of Sec. 3-1) for detecting and eliminating multiple roots (see standard works on Horner's method (Art. 17.14) is useful to find and remove theory of equations). troublesome roots. Numerical Example from Page 1-65. first hoot: x, = y/bn/\ = v'l.6148 X 10". Logi, 33.20812 -i- 64 = 0.51888 = rules from theory or by applying 3.3028. The can be found often by trial, |x,| sign of equations; for example, it is shown in Art. 17.14 that there is a root between —3 and —4, and it can easily be shown that there is no root greater than 3. X\ = —3.3028. pair of equal roots (shown by column r):

-

=

It

- tydm/bm

= !x^(8.75!)6 'V(8.7596

48 X 10") X 1077)/(1.01 10'')/(1.0148

-

2.2361

is found that both positive and negative values satisfy the equation.

* See, for example, Doherty, R. E.. and E. G. Keller, " Mathematics of Modem Engineering.' pp. 109 128, John Wiley A Sons, Inc., New York. 1930.

vol.

1,

**

irt "N

o e Vft

© s «\ 1ft ft* o m

ft* 4i

oo o*n

^*

«ftj u-ir*. ft*ft* ■ 1

rft ©

Q © •ft m* (N •e'V LA »r»f*> r-*.o ft*—

o

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•ft ft* •ft. ~~

1ft CN •ft ■"

©

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OO fNl rft ft*

cr. (ft ■»■ ift ftj ftl ■♦ © tft

o o

© ~

XX

X XX o> in o>"r ft*© ♦ ft* c r-i ■♦o

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oo

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oo oo

©

oo

X

XX

1

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JJ ©

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a

OO r>. (ft.-* 0 ©♦ o> ©♦ ft)o *o ("NO

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—

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7

X

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"T cc O S •ft. ♦ ■OO Of*. -ft | IN— © 1

XXX

— %Oe© rft — tft (NINO

moo

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rft, aOO 1 o* r>.cc ♦ fft ♦ ft* 1 1

t. «

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o —

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m O

OO

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m

X

XX

X

CO ■»(ft 1 ift ftlÔ 1 a(ft« c rvi Nf<^ 1

•ft

s O

5 o

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X

rft 0> ■♦ ft*

V

..

©o

XX

T O

X

«0(ft 1 oo^ 1 (ft©

r*.

1

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fftrftO©

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X =C rN -oft* ■ft, »ft ■ft o (ft

CO CO -IsOOO o. ""*

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ai*\ o*

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=■■ •ft ft. cO

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-

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OOO ___

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X

5438 7295 2126 X 0012

■o

272 25 -44.00 -35.00 -5.00

i

i

■ft

■ft ft* X X CO cc

—

2 X

022 .749 016

w« • * ooo© ™JJ1IJ"_

1

1

K -v «*

I I

!

XX

X

TM.

©

O

XX

X

XX

X

-c

ooft* Ift —

— ■«■ rftrft
rvi

cc ©

«(ft

ft*—

1-65

1

s

©

XX

x

x

o X ao -r

o

'

-

-

-2 —

r-i ro

-J*o II

1

-r

"

oo

E

cc

■-4

C ft*

o

D

H II

- -

—

ft*

s

1

-

"

X

— ft* Oft* Iftft* —o

z 1

OO

1

'

X

2

O

H «

a

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O

XX

1

i

oo

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4.0498 0.0313

©— — ■*• ♦ ft*

3!2

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1^.0 rsir* rft—

-V

oo

II

-N

oo

r-* jS

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1

■o

*ft «*\

e

■ft -oo

°2 ~2

©

£

OO

u-lO ft*© ft*oo

OOO

XX

« cc O o —

' i

*o fftOO coo — cc (*M"NO ■ft. rftift.O -3 ♦ ftl©

1

ft*— »ft OÔ* ft*ft*© •ft«©rft ■♦r*.© * — OO

oo

'1

X

1

XXX

***.

X

5

XXX

11

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1 1

1

ft*

oo

d

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0

. ~otr\—

O w>0 . .

1

«

©

<=■OOO

1

>-. ■<

y

C

-

oo©

ooo

d

I

>

~

*'


:

.ft.

OOO

X

u-vOO *rs ft*0© *ft) ©»ft© fft-eft* 1 i

+ s

e

a

*l

„

©o©

X

1 •ft.

X

X

O

X x . «ft (ftl o^ o* r^ NT ^

o

5 o

CM (ft o* u-i ft*

a

© —

x

X

*

j

o

GENERAL DATA

1-66

[SEC.

1

REMAINING REAL ROOT:

VW/«

x, =

= ^(2.9392

X 10»)/(4.7493 X 10")

= 0.30277

x6 = +0.30277. PAIR OF COMPLEX ROOTS:

By trial, r! = u =

= \V(4.7493 X 10S8)/(8.7596 X 10") = 0.49994 —}>4(b + algebraic sum of all real roots) = —1,^(3.5 2.2361 + 0.30277) 3.3028 + 2.2361

VU/dm

- V0.49994 - 0.06249

-

t, =

u» v/r» x,.s = u ± iv = -0.24999

+ 0.66140;

=

-

=

-0.24999

= 0.66140

0.24999(-l

± 2.6457t)

+2.2361, The six roots of the equation are: -3.3028, 0.2500(-l ± 2.6457t), +0.3028. See Art. 4.3 of Sec. 3-1. 17.14 Horner's Method for Polynomial Equations. This method is usually used in conjunction with rules for exploring the character of roots and assisting in approximate location of roots. The method is illustrated, without such aids, using the same equation as in the illustration by Graeffe's method. Divide through by the coefficient of the highest power of x, write Step 1.

f(x)

= xs + 3.5x5

-

4x<

-

-

16.5x3

- ox

5.5xs

+ 2.5


and evaluate f(x) for a series of values of x: x

-4

-3

-2

-1

m

478.5

-32

10.5

12

1 1

0

1

2

j

2.5

-24

-49.5

3 |

748

f(x) can be evaluated by direct substitution, but (except for small integral values of x) This is illustrated for it can be evaluated more easily by the remainder theorem. x = 3:

Add (vertical arrow) and multiply by 3 (diagonal arrow) as indicated.

fix)

-

748

From the change in sign of tabulated values, fix) becomes zero between x — — 4 and x = — 3, between x = — 3 and x = — 2, between x = 0 and x = 1, and between x = 2 and x = 3. Arbitrarily, the equation will be solved for the root between 2 and 3. Step 2. A new equation is formed with all its roots 2 less than those of the original equation. The process starts like the method described in Step 1 for evaluating fix), but is repeated in the manner shown. 1

3.5 2

1

1

I 1

5.5

7

-16.5 14

-2.5

-5.5

-5

2.5

-10.5

-21 -26

-52 -49.6

-5

15

44

83

145

7.5

22

41.5

72.5

119

2

19

82

975

41

123.5

2

23

128

64 27

251.6

11 .5

13.5 2

1

11

2

2 1

-4

15.6

91

247

319.5


SEC. 1-1]

-

1-67

The "remainders" (boldface) are the coefficients of the transformed equation, 0, whose roots are 49.5 x* + 15.5xs + 91x* + 251.5xs + 319.5xJ + 119x /
-

15.5

1

OA^

15.9

1

251.5

91

6. 36

38.94

97.36

290.44

"

-

-49.5

319.5

119

116.18

174.27

435.68

293.27

[0^4

117.31

67.81

Higher values of x give larger positive values of f(x); so the root must be less than = 0.4. Trial shows that for x = 0.3, /(x) = 22.52, and for x = 0.2, /(x) = -10.76; All roots of the equation are next i.e., the root lies between x = 0.2 and x = 0.3. decreased by 0.2.

i

1

15.5

91

0.2

3

319.5 54.07 373.57 57.96

193.71

280.02

15.7

94 14 18

270.33 19.46

1

15.9

97 32 3 22

289.79 20.11

431.53 61.98

1

16.1

100 54 3 26

309.90 20.76

493.51

80

S30.66

3

0.2 0.2

16.3

1

103

0_2_

1

16.5 (K2

1

16.7

The transformed

+

x«

119

18.83

1

0.2


251.5 14

74.71

-49.5 38.74

-10.76

86.31

3.30 107.10

equation with roots 2.2 less than those of the original equation is

16.7xs + 107.10x<

+ 330.66xJ + 493.51xs + 280.02x

-

10.76 = 0

To continue the process, locate the root lying between 0 and 0.1. As a first approxi mation, x = 10.76/280.02 = 0.038, and evaluation confirms that the root lies between The equation is therefore transformed by reducing all roots by 0.03, 0.03 and 0.04. giving x«

+

16.88x5 + 109.61x« + 343.66x3 + 523.85x2 + 310.54x

-

1.91

= 0

The roots of this equation are now 2.23 less than those of the original equation, and continuation of the process establishes the root of the new equation as being close to 0.006; hence the root of the original equation is x = 2.236 The syn Reduce the original equation by dividing by (x — 2.236). Step 3. thetic division is identical with Step 1, with the root being used as the multiplier 1

1

3.5

-4

2.236

12.8257

5.736

8.8257

-16.5 19.7343

3.2343

-5.5

-5

7.2319

3.8725

1.7319

-1.1275

2.5

-2.5211 -0.0211

|2.236

GENERAL DATA

1-68

[Sec.

- 1.1275

1

= 0. The reduced equation is xs + 5.7360x« + 8.8257x» 4- 3.2343xs + 1.7319x The quantity —0.0211, interpreted in Step 1 as the value of f(x) when x = 2.236, Ideally the remainder is alternatively the remainder after division by (x — 2.236). should be zero, and the division serves as a check on the accuracy of the solution. The reduced equation is solved for another root, and the process can be continued If t here is only one pair of complex roots, these until all real roots have been found. can be obtained from the residual quadratic equation. Solution for the root between 0 and 1 gives x = 0.304 and a reduced equation x> + 6.0400x3 + 10.6619x2 + 6.4755x + 3.7005 = 0. Solution of this equation for the root between —2 and —3 gives x = —2.236 and a reduced equation xs + 3.8040x2 + 2.1562x

Solution for the root between + 0.5014 = 0.

x* 4- 0.5010x

+ 1.6542 =

0

—3 and —4 gives x = —3.303 and a reduced equation Solution of this quadratic equation gives x =

0.2505(-l

± 2.6439t)


The roots of the equation are thus -3.303, ±2.236, 0.2505(-l ± 2.6439t), +0.304. If the equation contains more than one pair of Limitation of Horner's Method. complex roots, the residual equation after eliminating real roots is of fourth or higher The residual equation can, however, be solved by Graeffe's method. degree. 17.2

Solution of Transcendental Equations

The following methods are applicable for finding real roots of any equation in one variable, including transcendental equations. The methods are illustrated by solution of the equation x — sin x = 1. 17.21 Let ;/ = sin x, then Approximate Solution by the Two-graph Method. also y = x — 1. Plot the two equations with y as ordinate and x as abscissa. Points In this case, the only of intersection of the two curves give roots of the equation. A large-scale plot in the region of x = 1.9 will give real root is at x » 1.9 (Fig. 2). a closer approximation. 3 2 y i 0

Fig.

-I 2.

Two-graph method to solve transcendental

equations.

This method is useful as a starting point for the following two numerical methods. 17.22 Method of Successive Linear Interpolation. (See also Method of False Position in Art. 4.3 of Sec. 3-1.) Write f(x) = x — sin x — 1 and find two values of x close together, say a and b, such that f(a) = a — sin a — 1 and f(b) = b — sin 6 — 1 differ in sign. Then linear interpolation will give as a first approximation 11

°

[/('')- m)/(b

-

a)

°r

*'

[/(6)

-f(a)]/{b

-

a)

The process is repeated using values of a and 6 closer to Xi, but still yielding a difference in sign between f(a) and/(6). Further repetition gives increasingly close approximation.


SEC. 1-1]

1-69

For the illustrative equation,

/(a)

=

- - 0.8415 -

/(D

1

/HO = /(2) = 0,

e,

= 1

-

= 2

-

2

1 =

-0.8415

1

+0.0907

- 0.9093 - -0.8415

(0.090 7 + 0.8415)/(2 0.0907 (0.0907

+ 0.8415)/(2

_ d

1-w-'

— 1 90" >7

- -

1.9027

1)

For the next approximation try a = 1.90, /(a) = —0.0463, and (after inspecting in the first approximation) b = 1.95, fib) = +0.0210.

the values developed

6i = 1.9

For the difference

-°-°f3

+ 0.0463)/(1.95

next approximation, try a = 1.934, /(a) of sign) b = 1.935, /(6) = 0.0006. e»


(0.0210

0 0008

= 1.934

0.0014/0.001

-

= 1.9344 1.90)

= —0.0008,

1.934 + 0.0006

and (maintaining

a

= 1.9346

The next approximation gives 0» = 1.9345. 17.23 Newton's Method. Newton's method is a refinement of the method of It is illustrated by solution of the equation used in successive linear interpolation. the linear-interpolation method above. — cos 9, Write f(9) = 6 — sin 6 — 1 and the first and second derivatives. f'{8) = 1 /"(») = sin 0. Find As before, locate the root approximately :/(l) 0.8415, /(2) = +0.0907. = 0.8415, /"(2) = +0.9031. For a first approximation, 9\, use the value of 9 in which /(9) and /"(9) have the same sign, i.e., 6 = 2.

f(l)

e2 =

Successive

approximations

9l_/W=2-0^07 1.4161 /'(9i)

are made without

= 1.9359

reference to f"($).

93_,,_ZW _19M_O0020 1.3571 f'(82)

6t =

1.9346

-0-0001= 1.3558

1.9345

method is convenient only if the derivative is a simple expression; also, is contingent on specific conditions being satisfied. The less elegant linear-interpolation method is not so restricted. Newton's

convergence

17.3 17.31 in

Art.

by the

Solution of Systems of Linear Algebraic Equations

Several variants of this method arc discussed Solution by Elimination. of Sec. 3-1, including the Doolittle method. This method is illustrated following numerical example, incorporating an arithmetic check at each step: 4.1

1-70

GENERAL 4.15x 3.213

2.9U

- 3.17y

DATA

[Sec.

-

- 3.3U

+ 2.412

1

7.61 = 2.41 + 1.65y + 2.332 = 8.41

+2.12y

Check sum* Operation

V

I. Coefficients of equations

Referring to

of

-3

4. 15 3.21 2.91

by the

I

17 2 12 1 65

764 0 660 0 567

-1 -1

subtract second line from first, and third from

each line coefficient of

of

by

3

Divide

the

4

6.

Divide

from

By operation

By addition

(11.00) (4.43) (15.30)

11.00 4.43 15.30

0.581 1.031 0. 801

1.834 0.751 2.890

2.651 1.380 5.258

2.651 1.380 5.258

1.612

-0.220

1.083 1.056

-2.607

-2.607

-1.132

-0.761

-0.893

-0.893

0.165

0.793

1.959

1.958

1.297

-1.554

-2.852

-2.851

.198

2.199

1.271

1.271

z

- 1.834 (0.581

0.581* 0.595)

-

0.761

1.698

X 1.198)

+

(0.764

1.198 0.761 = (1.132 X 1.198)

.'

IMS

-

0.69

+

0.764»

-

-

6

z

-

4

1.132*

X

From operation

of

2

5

by coefficient

From operation

1.834

*

The numbers in the first column (except numbers in parentheses) are obtained by applying the The numbers in the second column (and the numbers indicated operation to the preceding check sums. in parentheses in the first column) are obtained by direct addition of the coefficients in the same hori zontal row. The numbers must check at each line.

-

+

d, d,

is

is

+

Check by substituting in one of the original equations. For example, a check using the first equation gives (4.15 X 1.593) (3.17 X 0.595) (2.41 X 1.198) = 7.012. It possible to solve the problem with one less row of figures, but the procedure somewhat less straightforward. 17.32 Solution by Determinants. Sec Art. 1.4 of Sec. 3-1. The solution to the simultaneous equations Oil ftil/ + f|2 = ail + bty + c»z — asj + b,y + c,z — d,

Hi

C

a,

C\

<■■:

Of

C|

a.

c*

«...

C|

a.

//,

c» Cl

"

a, Ol

a",

(1,

Cj

d,

C\

,1,

h. !>■: fci ft, ft, ft,

Ql

a,

b, ft. d,

"i

a,

c. e«

y

d, d,

d.

d,

written directly in determinant form as follows: b, ft, bt ft, /,, ft,

is

d. 0\ c» C|

is

The denominator the same in each case, and the determinant formed by the coefficients of the unknowns. The numerators are identical with the denominators except for one column, in which the quantities on the right-hand side of the equation replace the coefficients of the particular unknown expressed. is


4

5.

Subtract second line of first line of

424 331

1 1

y

4.

first

7.61 2.41 8.41

2.41

-3.31 2.33

-0

I

2,

3.

2. Divide each line coefficient (if x

Right siilc

SEC. 1-1]


To evaluate

1-71

the denominator Oj

61

Ci

ai

6S

c2

at

63

c3

6j

C2 &3

^1

+

6l

«3

Cl I

(In this expansion the elements of the first column become the coefficients of the Each minor is obtained from the determinant minors, with the signs alternating. by suppressing the column and row containing the particular a coefficient.)

-

ai(biC,

— 6,c2)

- as(6,Cj

— 6jCi)

— bjCi)

+ os(6iCS

iTo evaluate a second-order determinant, multiply the elements diagonally in pairs, downward to the right positive, upward to the right negative.) For n simultaneous equations, the nth-order determinant is expanded in terms of its n minors (determinants of order n — 1), and so on. The properties of deter minants permit manipulation to simplify the work. Numerical example: 4.15x 3.17j/ + 2.41z = 7.61 3.2Lr + 2.12s/ 3.31z = 2.41 2.91* + 1.65i/ + 2.33.J = 8.41


-

7.61 2.41 8.41

-3.17 2.12 1.65

-3.31

4.15 3.21 2.91

-3.17

2.41

2.41

-

2.33 !l

-3.31

2.12 1.65

2.33

z =

4.15 3.21 2.91

4.15 3.21 2.91

-3.17

2.41

2.12 1.65

-3.31

2.12 3.31 1.65 2.33| 4.15(4.940 + 5.462)

-

3.21

8.41

2.33

95.311

"

= » = !/

L593

I

-3.17

I

1.65

(-7.386

56.774 95T3H

2.33

2.33

2Al

I

,on, i jy

2.33 ! 3.977)

-

i I

2.12

-

114J94

0.596

-3-17

2-41

-3.31

I

4- 2.91(10.493 5.109) * 95.311

By similar evaluation of the numerators 151.823

2.41

-3.31

7.61 2.41 8.41

2.12 1.65

•J" 3.21

-3.31

2.12 1.65

-3.17

2.41

7.61 2.41

-3.17

4.15 3.21 2.91

4.15 3.12 2.91

= 4.15

Denominator

-

95.3lT

= 1.198

in original equations

Substituting

(4.15 (3.21 (2.91 18

X X X

-

1.593) (3.17 1.593) + (2.12 1.593) + (1.65

X X X

0.596) 0.596) 0.596)

+ (2.41 X 1.198) = 7.609

-

(3.31

X

1.198)

+ (2.33 X 1.198)

= 2.413 = 8.410

ATOMIC ENERGY COMMISSION LITERATURE

is intended as an aid in locating and identifying the AEC literature at the end of most sections of the handbook. References to commercial publications and publications of scientific and technical societies ne«d no clarification.

This article

referenced

18.1

Availability of AEC Literature

Unclassified and declassified AEC Depository Libraries. AEC reports 18.11 are available in approximately 70 depository libraries throughout the United States.

GENERAL DATA


1-72

[Sec.

1

Approximately the same number of depository libraries have been established in foreign countries. The locations of these libraries are given in most of the general and informational AEC publications, e.g., Refs. 1, 2, and 5 of Art. 18.2.* A few of the depository libraries are designated "industrial information depository These have supplementary technical reports and abstracts and collections libraries." of unclassified AEC drawings. 18.12 The Office of Technical Services, U.S. Department of Commerce, Wash Many unclassified and declassified reports can be purchased from ington 25, D.C. this agency (abbreviated OTS in subsequent references) in printed, photostatic, or microcopy form. List of reports free on request (Ref. 3 of Art. 18.21). 18.13 Superintendent of Documents, U.S. Government Printing Office, Wash ington 25, D.C. Some of the more substantial AEC documents can be purchased List of reports, from this agency only (abbreviated GPO in subsequent references). "Atomic Energy and Civil Defense Price List," free on request. AEC Technical Information Service Extension (TISE), P.O. Box 1001, Oak 18.14 Some documents are obtainable from this agency (abbreviated TISE Ridge, Tenn. in subsequent references). 18.16 Microcopy. Microcopy is available from OTS, from Microcard Foundation, P.O. Box 2145, Madison 5, Wis., and from Readex Microprint Corporation, 115 University Place, New York 3, N.Y., the two latter being commercial enterprises. Atomic Energy Publications by Other Public Agencies. United States 18.16 Bureau of Standards handbooks relating to radiological protection may be obtained from GPO at prices ranging from SO. 15 to $0.25. Nuclear Science Series Reports are available from the National Academy of Sciences — National Research Council, Washington, D.C. 18.2

Guides to AEC Literature

The following AEC publications indicate content, source, price, and other pertinent data relating to official publications. 18.21 Consolidated Listings (NSA), issued twice monthly, GPO, $7.50 per year, 1. Nuclear Science Abstracts Abstracts of pertinent scientific and technical also by purchase of separate issues. Also periodic cumulated and supplementary literature, official and unofficial. indexes, the latter serving to supplement TID-4000 (see next item). 2. TID-4000, "Cumulated Numerical List of Available Unclassified U.S. Atomic Gives report Energy Commission Reports," revised from time to time, OTS, $1.25. numbers (not titles), readily available sources (official and other), reference to Nuclear Science Abstracts, and prices if for sale. 3. "List of AEC Research Reports for Sale," revised twice a year, OTS, free. Gives report numbers, titles, etc., and prices. 4. TID-4022, "Consolidated Availability Listing of Confidential Reports Announced to the Civilian Application Program," revised quarterly, TISE, free to L-and Q-cIeared TID-4022 is unclassified. Formerly the listings appeared access permit holders. in the abstract journal, Confidential Reports for Civilian Applications (RCCA). Gives report number (not title), category, abstract reference, prices of printed reports and microcopy. Availability Listing of Secret Reports Announced 5. TID-4023, "Consolidated to the Civilian Application Program," revised quarterly, TISE, free to Q-cleared access permit holders. TID-4023 is unclassified. Formerly the listings appeared in TID-3063, "Bibliography of Secret U.S. Atomic Energy Reports Available through the Civilian Gives report number (not title), category, Application Program." abstract reference, prices of printed reports and microcopy. 6. TID-4100, "Unclassified Engineering Materials List" (UEML), with supple ments, issued from time to time, Technical Information Service Extension (TISE), P.O. Box 1001, Oak Ridge, Tenn., free. * An up-to-date

list should be consulted,

since the number of libraries

is continually increasing.

SEC. 1-1]


1-73

This document is "A Catalog of Drawings, Photographs, and Specifications Released The listed items are for sale by the United States Atomic Energy Commission." and the price and source are stated. Additional information from TISE. 7. TID-4550, ''What's Available in the Unclassified Atomic Energy Literature," revised from time to time, GTS, free. Gives general information on AEC literature. Lists some AEC-sponsored series, manuals, and handbooks. Less detailed than


TID-4575. 8. TID-4563, "Special

Sources of Information on Isotopes," September, 1957, Isotopes Extension Division of Civilian Application, U.S. AEC, Oak Ridge, Tenn., free. A guide to sources of information on radioisotopes. 9. TID-4575, "Guide to Atomic Energy for the Civilian Application Program," revised from time to time, OTS, free. Describes in considerable detail the various categories of AEC publications, including availability and means of locating material. 10. "Selected Readings on Atomic Energy," revised from time to time, GPO, $0.25. Describes important official and commercial books, periodicals, and bibliographies. 11. Classified Reports. Abstracts and consolidated availability listings of con fidential and secret reports are available to qualifying organizations and personnel These include TID-3063 (bibliography, through the Civilian Application Program. see Item 5); TID-3080, Nuclear Technology (bibliography); TID-3081, Feed Materials (bibliography); Bulletin (confidential); Confidential Reports Report Announcement for Civilian Applications, CRCA (abstract journal, formerly Civilian Applications of Atomic Energy). Changes of classification from secret or confidential to unclassified are listed in TID-4035, "AEC Reports Declassified," and supplements. TID-4035 gives report number, title, author, date, and classification status. Catalog cards are available (including abstracts) from TISE 12. Card Catalog. for reports abstracted in the various AEC journals. 18.22 Additions to the Literature 1. Nuclear Science Abstracts Each semimonthly issue (see Ref. 1 of Art. 18.21). gives abstracts of additions to the technical literature, and announces new nuclear data. 2. Monthly listings and abstracts of new literature (unclassified and classified) are available on a subscription basis. 3. Press releases. Important announcements are made by AEC in the form of Many press releases, which are also given wide distribution in mimeographed form. of these announcements are of economic or technical importance. 18.3

References Available in Alternative Forms

Several publications are referred to frequently in the handbook, hut not always in identical terms, since some are available in alternative forms. The following arc the more important cases of dual publication. 18.31 Nuclear Data. The standard reference for nuclear data other than data on neutron cross sections has been : K. Way, et al, "Nuclear Data," Nat. Bur. Standards (U.S.) Circ. 499 (1950) and supplements. This important reference is being revised and expanded, under the sponsorship of the U.S. AEC, by the Nuclear Data Group of the National Academy of Sciences — National Research Council. It is now issued as; a. "Nuclear Level Schemes," published in sections, each section being released when completed, available from GPO. b. Nuclear Data Cards, available on an annual subscription basis from Publications Office, National Academy of Sciences — National Research Council, Washington, D.C. c. New Nuclear Data, cumulated from Nuclear Data Cards quarterly, semiannually, and annually, and published in Nuclear Science Abstracts; the same medium was used formerly to supplement Nat. Bur. Standards (U.S.) Circ. 499. Several important books have been sponsored by the 18.32 Reference Books. AEC and published both commercially in cloth covers and also officially in paper

GENERAL DATA

1-74

[SEC.

1

The Government Printing Office editions are often referred to by code numbers instead of titles. The most important, books in this category are given below with the name of the commercial publisher and the AEC document number. 1. D. J. Hughes and J. A. Harvey, "Neutron Cross Sections," McGraw-Hill Book The preceding edition appeared as Company, Inc., New York, 1955. BNL-325. covers.

AECU-2040. McGraw-Hill Book 2. "The Reactor Handbook," published in three volumes. Company, Inc., New York, 1953. AECD-3645, AECD-3046, and AECD-3047. 18.4

The United States Atomic Energy Commission

1. Semiannual Reports of the U.S. Atomic Energy Commission to the Congress of the United States, Washington, D.C., republished for sale under a variety of titles,* GPO, various prices, $0.35 to $1.25. Appendixes give much general information and describe the organization, policies, operations, etc., of AEC.

REFERENCES and J. A. Harvey: "Neutron Cross Sections," McGraw-Hill Book Com pany, Inc., New York, 1955. Also Supplement 1 by D. J. Hughes and R. B. Schwartz (1957). 2. Westcott, C. H.: "Effective Cross Section Values for Well-moderated Thermal Reactor Spectra," CRRP-680, TNCC(Can)-7, (also AECL-407), January 25, 1957. Revised as AECL No. 670(CRRP-787), August 1, 1958. 3. ANL 5800: "Reactor Physics Constants," August, 1958. 1. Hughes,


D. J.,

* In recent years, for the designated " Programs has been used.

ft-month

period the title " Major Activities in the Atomic Energy

MATHEMATICAL TABLES

1-2

BY


Harold Etherington and Elesa R. Etherington Reactor calculations, as made by organized computer groups, are often carried out on desk calculating machines or by digital computers to eight or more significant The nuclear data entering into the calculations are, however, seldom accu figures. rate to more than two or three significant figures, and if calculations are to be made without the services of a computer group, slide-rule accuracy in conjunction with fourfigure tables is satisfactory for most purposes for which such calculations would be For the most part, the tables of this section are confined to those commonly made. used in nuclear engineering.

TABLES

1

OF FUNCTIONS

Tables of the circular, exponential, hyperbolic, modified Bessel (/0, I\, Ko, K{), and factorial, exponential integral (argument 0 to 5), and error (argument 0 to 1) functions have been prepared, with permission of the publishers, from E. Fliigge, "Four Place Tables of Transcendental Functions," McGraw-Hill Book Company. Tables of the other Bessel functions (J0, Jt, Ya, Y,), Iêgendre polynomials, and error function (argument 1 to 3) have been prepared, with permission of the publishers, from E. Jahnke and F. Emde, "Tables of Functions with Formulae and Curves," Dover Publications, New York.* The tables have been modified to a consistent form, and explanatory notes have been added. In many cases, these two references and other four-figure tables provide alternative sources over some part of the range

r

covered.

Tables of the functions En(x) and Ez(z) have been compiled from the six-figure tables of the National Research Council of Canada, t Auxiliary functions for par ticular ranges of the Bessel functions have been prepared from the eight-figure tables of the British Association^ the tables for the modified functions have been further extended by computation for values of the argument from 20 to 40. Table

r l/x

= 3.14159 26536 =- 0.31830 98862 = 1.77245 38509

l/V»

= 0.56418 95835

VI

Euler or Euler-Mascheroni

Transcendental

1.

I/t' -

M

conxtant y In y

-

0.10132 11836

2.50662 82746 = 0.39894 22804 = 0.49714 98727 2.71828 18285 log « 0.43429 4481 = 0.57721 56649 ■=0.54953 93130

\/Vu

log w e

-

Constants

»' = 9.86960 44011

-

-

* Jahnke and Emde use the nomenclature t See bibliography.

31.00627 66803

V7h

In p > 1.14472 98858 = 0.36787 94412 - In 10 2.30258

-

-

Nv(z) and JVi(x) for

1-75

1.25331 41373

Jod)

and

1.78107 24180

Ji(t).

respectively,

50930

GENERAL

1-76

Table 2.

[Sec.

1

2

3

4

5

6

7

8

9

0.0000 0414 0792 1139 1461

0043 0453 0828 1173 1492

0086 0492 0864 1206 1523

0128 0531 0899 1239 1553

0170 0569 0934 1271 I5B4

0212 0607 0969 1303 1614

0253 0645 1004 1335 1644

0294 0682 1038 1367 1673

0334 0719 1072 1399 1703

0374 0755 1106 1430 1732

1761 2041 2304 2553 2788

1790 2068 2330 2577 2810

1818 2095 2355 2601 2833

1847 2122 2380 2625 2856

1875 2148 2405 2648 2878

1903 2175 2430 2672 2900

1931 2201 2455 2695 2923

1959 2227 2480 2718 2945

1987 2253 2504 2742 2967

2014 2279 2529 2765 2989

2 4

3010 3222 3424 3617 3802

3032 3243 3444 3636 3820

3054 3263 3464 3655 3838

3075 3284 3483 3674 3856

3096 3304 3502 3692 3874

3118 3324 3522 3711 3892

3139 3345 3541 3729 3909

3160 3365 3560 3747 3927

3181 3385 3579 3766 3945

3201 3404 3598 3784 3962

2.5 2.6 2.7 2.8 2.9

3979 4150 4314 4472 4624

3997 4166 4330 4487 4639

4014 4183 4346 4502 4654

4031 4200 4362 4518 4669

4048 4216 4378 4533 4683

4065 4232 4393 4548 4698

4082 4249 4409 4564 4713

4099 4265 4425 4579 4728

4116 4281 4440 4594 4742

4133 4298 4456 4609 4757

3.0 3.2 3.3 3.4

4771 4914 5051 5185 5315

4786 4928 5065 5198 5328

4800 4942 5079 5211 5340

4814 4955 5092 5224 5353

4829 4969 5105 5237 5366

4843 4983 5119 5250 5378

4857 4997 5132 5263 5391

4871 5011 5145 5276 5403

4886 5024 5159 5289 5416

4900 5038 5172 5302 5428

3.5 3.6 3.7 3.8 3.9

5441 5563 5682 5798 5911

5453 5575 5694 5809 5922

5465 5587 5705 5821 5933

5478 5599 5717 5832 5944

5490 5611 5729 5843 5955

5502 5623 5740 5855 5966

5514 5635 5752 5866 5977

5527 5647 5763 5877 5988

5539 5658 5775 5888 5999

5551 5670 5786 5899 6010

4.0

6021 6128 6232 6335 6435

6031 6138 6243 6345 6444

6042 6149 6253 6355 6454

6053 6160 6263 6365 6464

6064 6170 6274 6375 6474

6075 6180 6284 6385 6484

6085 6191 6294 6395 6493

6096 6201 6304 6405 6503

6107 6212 6314 6415 6513

6117 6222 6325 6425 6522

6532 6628 6721 6812 6902

6542 6637 6730 6821 6911

6551 6646 6739 6830 6920

6561 6656 6749 6839 6928

6571 6665 6758 6848 6937

6580 6675 6767 6857 6946

6590 6684 6776 6866 6955

6599 6693 6785 6875 6964

6609 6702 6794 6884 6972

6618 6712 6803 6893 6981

6990 7076 7160 7243 7324

6998 7084 7168 7251 7332

7007 7093 7177 7259 7340

7016 7101 7185 7267 7348

7024 7110 7193 7275 7356

7033 7118 7202 7284 7364

7042 7126 7210 7292 7372

7050 7135 7218 7300 7380

7059 7143 7226 7308 7388

7067 7152 7235 7316 7396

1. 1 1.2 1.3 1.4 1.5

1.6 1.7 1.8 1.9 2 0 2 1

2.2 2.3

3.1

4.1

4.2 4.3 4.4 4 5

4.6 4 7

4.8 4.9 5.0 5.1

5.2 5.3 5.4

Examples: 1. log 5.576 0.7463. 2. log 557.6 = log (5.576 X

-

-

1

Common Logarithms

0

1.0


DATA

2.7463. 10') 10, etc. 3 = 7.7463 log (5.576 X 10"') 3.7463 = 0.7463 -2.2537. 3. log 0.005576 The mantissa (decimal part of the logarithm) is independent of the position of the decimal place in the number. The char act eristic (integer part of the logarithm) can bIbo be found by inspection. It is positive for numbers having significant figures to the left of the decimal place and is one less than the number of It is negative for numbers less than unity and is numerically equal such figures (Examples I and 2). to the number of decimal places up to and including the first significant figure to the right of the decimal place (Example 3).

-

-

-

-

-

mathematical tables

Sec. 1-2]

Table 2.

(Continued)

0

1

2

3

4

5

6

7

8

9

0 7404 7482 7559 7634 7709

7412 7490 7566 7642 7716

7419 7497 7574 7649 7723

7427 7505 7582 7657 7731

7435 7513 7589 7664 7738

7443 7520 7597 7672 7745

7451 7528 7604 7679 7752

7459 7536 7612 7686 7760

7466 7543 7619 7694 7767

7474 7551 7627 7701 7774

7782 7853 7924 7993 6062

7789 7860 7931 8000 8069

7796 7868 7938 8007 8075

7803 7875 7945 8014 8082

7810 7882 7952 8021 8089

7818 7889 7959 8028 8096

7825 7896 7966 8035 8102

7832 7903 7973 8041 8109

7839 7910 7980 8048 8116

7846 7917 7987 8055 8122

5 6 7 8

8129 8195 8261 8325 8388

8136 8202 8267 8331 8395

8142 8209 8274 8338 8401

8149 8215 8280 8344 8407

8156 8222 8287 8351 8414

8162 8228 8293 8357 8420

8169 8235 8299 8363 8426

8176 8241 8306 8370 8432

8182 8248 8312 8376 8439

8189 8254

7.0 7. 7 4

8451 8513 8573 8633 8692

8457 8519 ' 8579 8639 8698

8463 8525 8585 8645 8704

8470 8531 8591 8651 8710

8476 8537 8597 8657 8716

8482 8543 8603 8663 8722

8488 8549 8609 8669 8727

8494 8555 8615 8675 8733

8500 8561 8621 8681 8739

8506 8567 8627 8686 8745

5 6 7 8 9

8751 8808 8865 8921 8976

8756 8814 8871 8927 8982

8762 8820 8876 8932 8987

8768 8825 8882 8938 8993

8774 8831 8887 8943 8998

8779 8837 8893 8949 9004

8785 8842 8899 8954 9009

8791 8848 8904 8960 9015

8797 8854 8910 8965 9020

8802 8859 8915 8971 9025

8 0 8 1 8 3 8 4

9031 9085 9138 9191 9243

9036 9090 9143 9196 9248

9042 9096 9149 9201 9253

9047 9101 9154 9206 9258

9053 9106 9159 9212 9263

9058 9112 9165 9217 9269

9063 9117 9170 9222 9274

9069 9122 9175 9227 9279

9074 9128 9180 9232 9284

9079 9133 9186 9238 9289

« 5 6 7 8 9

9294 9345 9395 9445 9494

9299 9350 9400 9450 9499

9304 9355 9405 9455 9504

9309 9360 9410 9460 9509

9315 9365 9415 9465 9513

9320 9370 9420 9469 9518

9325 9375 9425 9474 9523

9330 9380 9430 9479 9528

9335 9385 9435 9484 9533

9340 9390 9440 9489 9538

9 0 9.1 9 2

9.3 9.4

9542 9590 9638 9685 9731

9547 9595 9643 9689 9736

9552 9600 9647 9694 9741

9557 9605 9652 9699 9745

9562 9609 9657 9703 9750

9566 9614 9661 9708 9754

9571 9619 9666 9713 9759

9576 9624 9671 9717 9763

9581 9628 9675 9722 9768

9586 9633 9680 9727 9773

9 9 9 9 9

9777 9823 9868 9912 9956

9782 9827 9872 9917 9961

9786 9832 9877 9921 9965

9791 9836 9881 9926 9969

9795 9841 9886 9930 9974

9800 9845 9890 9934 9978

9805 9850 9894 9939 9983

9809 9854 9899 9943 9987

9814 9859 9903 9948 9991

9818 9863 9908 9952 9996

5 5

5.6 5 7

5.8

5 9 6 0 6 1 6 2

6.3 6.4 6 6 6 6

6.9

7 1

7.2

J


Common Logarithms.

1-77

7 7 7 7 7

8.2

8 * i? 1

5 6 7 8 9

log ab

"

log a + log b

log a" ■»n log a log

(The cotogarithm

of

*

M

I

0

Va

= log a — log 6 »= log a'/"

=

- log

a

is the logarithm of its reciprocal.)

log « -0.4343

log a - M In n

log*

log

-

log a =« colog a

number

-

log

= 0.4971

0.4343 In a

- In 10 - 2.3026 In a - — log a - 2.3026 —

log a

8319 8382 8445

1-78

GENERAL

[Sec.

Natural or Naperian Logarithms

Table 3. 0

1

2

3

4

5

6

7

8

9

0.0000 0953 1823 2624 3365

0100 1044 1906 2700 3436

0198 1133 1989 2776 3507

0296 1222 2070 2852 3577

0392 1310 2151 2927 3646

0488 1398 2231 3001 3716

0583 1484 2311 3075 3784

0677 1570 2390 3148 3853

0770 1655 2469 3221 3920

0862 1740 2546 3293 3988

0.4055 4700 5306 5878 6419

4121 4762 5365 5933 6471

4187 4824 5423 5988 6523

4253 4886 5481 6043 6575

4318 4947 5539 6098 6627

4383 5008 5596 6152 6678

4447 5068 5653 6206 6729

4511 5128 5710 6259 6780

4574 5188 5766 6313 6831

4637 5247 5822 6366 6881

2.2 2.3 2.4

0.6931 7419 7885 8329 8755

6981 7467 7930 8372 8796

7031 7514 7975 8416 8838

7080 7561 8020 8459 8879

7129 7608 8065 8502 8920

7178 7655 8109 8544 8961

7227 7701 8154 8587 9002

7275 7747 8198 8629 9042

7324 7793 8242 8671 9083

7372 7839 8286 8713 9123

2.5 2.6 2.7 2.8 2.9

0 9163 9555 0.9933 1. 0296 0647

9203 9594 9969 0332 0682

9243 9632 •0006 0367 0716

9282 9670 *0043 0403 0750

9322 9708 *0080 0438 0784

9361 9746 •0116 0473 0818

9400 9783 •0152 0508 0852

9439 9821 •0188 0543 0886

9478 9858 •0225 ' 0578 0919

9517 9895 *0260 0613 0953

3.0

3 4

1. 0986 1314 1632 1939 2238

1019 1346 1663 1969 2267

1053 1378 1694 2000 2296

1086 1410 1725 2030 2326

1119 1442 1756 2060 2355

1151 1474 1787 2090 2384

1184 1506 1817 2119 2413

1217 1537 1848 2149 2442

1249 1569 1878 2179 2470

1282 1600 1909 2208 2499

3.5 3.6 3.7 3.8 3.9

1.2528 2809 3083 3350 3610

2556 2837 3110 3376 3635

2585 2865 3137 3403 3661

2613 2892 3164 3429 3686

2641 2920 3191 3455 3712

2669 2947 3218 3481 3737

2698 2975 3244 3507 3762

2726 3002 3271 3533 3788

2754 3029 3297 3558 3813

2782 3056 3324 3584 3838

4.0 4.2 4.3 4.4

1. 3863 4110 4351 4586 4816

3888 4134 4375 4609 4839

3913 4159 4398 4633 4861

3938 4183 4422 4656 4884

3962 4207 4446 4679 4907

3987 4231 4469 4702 4929

4012 4255 4493 4725 4951

4036 4279 4516 4748 4974

4061 4303 4540 4770 4996

4085 4327 4563 4793 5019

4.5 4.6 4.7 4.8 4.9

1.5041 5261 5476 5686 5892

5063 5282 5497 5707 5913

5085 5304 5518 5728 5933

5107 5326 5539 5748 5953

5129 5347 5560 5769 5974

5151 5369 5581 5790 5994

5173 5390 5602 5810 6014

5195 5412 5623 5831 6034

5217 5433 5644 5851 6054

5239 5454 5665 5872 6074

5.0

1 6094 6292 6487 6677 6864

6114 6312 6506 6696 6882

6134 6332 6525 6715 6901

6154 6351 6544 6734 6919

6174 6371 6563 6752 6938

6194 6390 6582 6771 6956

6214 6409 6601 6790 6974

6233 6429 6620 6808 6993

6253 6448 6639 6827 7011

6273 6467 6658 6845 7029

1.7047 7228 7405 7579 7750

7066 7246 7422 7596 7766

7084 7263 7440 7613 7783

7102 7281 7457 7630 7800

7120 7299 7475 7647 7817

7138 7317 7492 7664 7834

7156 7334 7509 7681 7851

7174 7352 7527 7699 7867

7192 7370 7544 7716 7884

7210 7387 7561 7733 7901

1.0

I.I

1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9

2.0 2.1


DATA

3.1 3 2

3.3

4. 1

5.1

5.2 5 3

5.4 5 5

5.6 5.7 5 8

5.9

In x

-

2(a +

aJ/3

+

a'/5

+

•) where a

-

(x

-

l)/(i

+ I)

1

Sec.

mathematical tables

1-2] Table

Natural or Naperian Logarithms.

(.Continued)

0

1

2

3

4

5

6

7

8

9

6.4

1.7918 8083 8245 8405 8563

7934 8099 8262 8421 8579

7951 8116 8278 8437 8594

7967 8132 8294 8453 8610

7984 8148 8310 8469 8625

8001 8165 8326 8485 8641

8017 8181 8342 8500 8656

8034 8197 8358 8516 8672

8050 8213 8374 8532 8687

8066 8229 8390 8547 8703

6 6 6 6 6

5 6 7 8 9

1 8718 8871 9021 9169 9315

8733 8886 9036 9184 9330

8749 8901 9051 9199 9344

8764 8916 9066 9213 9359

8779 8931 9081 9228 9373

8795 8946 9095 9242 9387

8810 8961 9110 9257 9402

8825 8976 9125 9272 9416

8840 8991 9140 9286 9430

8856 9006 9155 9301 9445

7 0 7. 1 7 2 7.

7.4

1.9459 9601 9741 1 9879 2 0015

9473 9615 9755 9892 0028

9488 9629 9769 9906 0042

9502 9643 9782 9920 0055

9516 9657 9796 9933 0069

9530 9671 9810 9947 0082

9544 9685 9824 9961 0096

9559 9699 9838 9974 0109

9573 9713 9851 9988 0122

9587 9727 9865 •0001 0136

7 5

2.0149

0162 0295 0425 0554 0681

0176 0308 0438 0567 0694

0189 0321 0451 0580 0707

0202 0334 0464 0592 0719

0215 0347 0477 0605 0732

0229 0360 0490 0618 0744

0242 0373 0503 0631 0757

0255 0386 0516 0643 0769

0268 0399 0528 0656 0782

0919 1041 1163 1282

0807 0931 1054 1175 1294

0819 0943 1066 1187 1306

0832 0956 1078 1199 1318

0844 0968 1090 1211 1330

0857 0980 1102 1223 1342

0869 0992 1114 1235 1353

0882 1005 1126 1247 1365

0894 1017 1138 1258 1377

0906 1029 1150 1270 1389

2. 1401 1518 1633 1748 1861

1412 1529 1645 1759 1872

1424 1541 1656 1770 1883

1436 1552 1668 1782 1894

1448 1564 1679 1793 1905

1459 1576 1691 1804 1917

1471 1587 1702 1815 1928

1483 1599 1713 1827 1939

1494 1610 1725 1838 1950

1506 1622 1736 1849 1961

2. 1972 2083 2192 2300 2407

1983 2094 2203 2311 2418

1994 2105 2214 2322 2428

2006 2116 2225 2332 2439

2017 2127 2235 2343 2450

2028 2138 2246 2354 2460

2039 2148 2257 2364 2471

2050 2159 2268 2375 2481

2061 2170 2279 2386 2492

2072 2181 2289 2396 2502

9 9

2 2513 2618 2721 2824 2925

2523 2628 2732 2834 2935

2534 2638 2742 2844 2946

2544 2649 2752 2854 2956

2555 2659 2762 2865 2966

2565 2670 2773 2875 2976

2576 2680 2783 2885 2986

2586 2690 2793 2895 2996

2597 2701 2803 2905 3006

2607 2711 2814 2915 3016

10 0

2 3026

6 0

6. 1 6 2 6 3

J

7.6 7.7

0281 0412 0541 0669

7 8

7.9


3

1-79

8 0 8. 1

2.0794

8.2

8 3

8.4

8 5

8.6 8 7 8 8 8 9

9.0 9.1 9 2

9.3 9 4 9 5

9.6 9 7

9.8

In In In In In

10 10' 10' 10< I0>

-

2.3026 4.6052 6.9078 9.2103 11.5129

In 10"

-

In In In In In

-

13.8155 16.1181 18.4207 20.7233 23.0259

2.302585 X n

For negative exponents of 10. the logarithms are negative; In 10-"

-

10« 10' 10s 10' = 10" =

= -2.302585

Examples:

e.g., In 10 * = —6.9078,

X n

1. In 5.576 = 0.7185. 5.3237. 2. In 557.6 = In (5.576 X I0») = 0.7185 + 4.6052 7, etc. 0.7185 6.9078 = -6.1883 = 0.8117 3. In 0.005576 = In (5.576 X I0>) The general relations stated for common logarithms are applicable to natural logarithms.

-

-

In

c

-

,in .

-

Also,

1-80


Circular Functions

: cos

[Sec.

1

x and sin x

COS X 7

8

9

9988 9888 9689 9394 9004

9982 9872 9664 9359 8961

9976 9856 9638 9323 8916

9968 9838 9611 9287 8870

9960 9820 9582 9249 8823

8577 8021 7385 6675 5898

8525 7961 7317 6600 5817

8473 7900 7248 6524 5735

8419 7838 7179 6448 5653

8365 7776 7109 6372 5570

8309 7712 7038 6294 5487

5148 4267 3342 2385 1403

5062 4176 3248 2288 1304

4976 4085 3153 2190 1205

4889 3993 3058 2092 1106

4801 3902 2963 1994 1006

4713 3809 2867 1896 0907

4625 3717 2771 1798 0807

0508 0492 1487 2466 3421

0408 0592 1585 2563 3515

0308 0691 1684 2660 3609

0208 0791 1782 2756 3702

0108 0891 1881 2852 3795

OOOB 0990 1979 2948 3887

•0092 1090 2077 3043 3979

•0192 1189 2175 3138 4070

1

2

3

1 1 3 4

1. 0000 0.9950 9801 9553 9211

1. 0000 9940 9780 9523 9171

9998 9928 9759 9492 9131

9996 9916 9737 9460 9090

9992 9902 9713 9428 9048

5 6 7 B 9

8776 8253 7648 6967 6216

8727 8196 7584 6895 6137

8678 8139 7518 6822 6058

8628 8080 7452 6749 5978

1.0 1 2 3 4

5403 4536 3624 2675 1700

5319 4447 3530 2579 1601

5234 4357 3436 2482 1502

5 6 7 8 9

+ 0707

0608 0392 1388 2369 3327

0.0


6

0

X

-0292 -1288 -2272 -3233

4

5

-4161

1 2 3 4

-5048 -5885 -6663 -7374

4252 5135 5966 6737 7441

4342 5220 6046 6811 7508

4432 5305 6125 6883 7573

4522 5390 6204 6956 7638

4611 5474 6282 7027 7702

4699 5557 6359 7098 7766

4787 5640 6436 7168 7828

4875 5722 6512 7237 7890

4962 5804 6588 7306 7951

5 6 7 8 9

-8011

8071 8620 9083 9455 9733

8130 8670 9124 9487 9755

8187 8720 9165 9518 9777

8244 8768 9204 9549 9797

8301 8816 9243 9578 9817

8356 8863 9281 9606 9B36

8410 8908 9318 9633 9853

8464 8953 9353 9660 9870

8517 8998 9388 9685 9885

9914 9995 9977 9859 9642

9926 9998 9969 9841 9615

9938 9999 9961 9823 9587

9948 1. 0000 9952 9804 9558

9958 1.0000 9941 9784 9528

9967 9998 9930 9762 9497

9974 9996 9918 9740 9466

9981 9993 9904 9717 9433

9987 9988 9890 9693 9399

9329 8923 8428 7848 7190

9293 8877 8373 7786 7120

9255 8831 8318 7723 7050

9217 8783 8262 7659 6978

9178 8735 8206 7594 6907

9137 8686 8148 7529 6834

9096 8636 8090 7462 6761

9054 8585 8030 7395 6686

9011 8534 7970 7328 6612

6460 5666 4815 3916 2978

63B4 5583 4727 3824 2882

6306 5500 4639 3731 2787

6229 5416 4550 3638 2690

6150 5332 4461 3545 2594

6071 5247 4371 3451 2497

5991 5162 4281 3357 2400

5911 5076 4190 3263 2303

5830 4990 4099 3168 2206

1912 0923 •0076 1074 2061

1814 0823 •0176 1173 2159

1715 0723 •0276 1273 2257

1617 0623 •0376 1372 2354

1518 0524 •0476 1471 2451

1419 0424 •0576 1570 2548

1320 0324 •0676 1668 2644

1221 0224 •0775 1767 2741

3028

3123

3218

3312

3407

3500

3594

3687

2.0

-8569 -9041

-9422 -9710 -9900

3.0 1 2 3 4

-9991

5 6 7 8 9

-9365 -8968

4.0

-6536 -5748

-9983 -9875 -9668

-8481

-7910 -7259

1 2 3 4

-4903

-4008 -3073

5 6 7 8 9

-2108

+ 0875 1865

2010 1022 0024 0975 1963

5.0

2837

2932

- 1122 -0124

I

Iw

-

6.2832; 4*

-

12.5664; 6w

-

cos ( — x) *» COS X

+

18.8495; 8ir

-

25.1327;

I0x

-

31.4159.

mathematical

Sec. 1-2] Table 4.

0

2

: cos sin x

1-81

i and sin x.

3

4

5

6

{Continued)

7

8

9

2 3 «

0 0000 0998 1987 2955 3894

0100 1098 2085 3051 3986

0200 1197 2182 3146 4078

0300 1296 2280 3240 4169

0400 1395 2377 3335 4259

0500 1494 2474 3429 4350

0600 1593 2571 3523 4439

0699 1692 2667 3616 4529

0799 1790 2764 3709 4618

0899 1889 2860 3802 4706

7 g 9

4794 5646 6442 7174 7833

4882 5729 6518 7243 7895

4969 5810 6594 7311 7956

5055 5891 6669 7379 8016

5141 5972 6743 7446 8076

5227 6052 6816 7513 8134

5312 6131 6889 7578 8192

5396 6210 6961 7643 8249

5480 6288 7033 7707 8305

5564 6365 7104 7771 8360

8415 8912 9320 9636 9854

8468 8957 9356 9662 9871

8521 9001 9391 9687 9887

8573 9044 9425 9711 9901

8624 9086 9458 9735 9915

8674 9128 9490 9757 9927

8724 9168 9521 9779 9939

8772 9208 9551 9799 9949

8820 9246 9580 9819 9959

8866 9284 9608 9837 9967

9975 9996 9917 9738 9463

9982 9992 9903 9715 9430

9987 9988 9889 9691 9396

9992 9982 9874 9666 9362

9995 9976 9857 9640 9326

9998 9969 9840 9613 9290

9999 9960 9822 9585 9252

1. 0000 9951 9802 9556 9214

1. 0000 9940 9782 9526 9174

9998 9929 9761 9495 9134

9093 8632 8085 7457 6755

9051 8581 8026 7390 6681

9008 8529 7966 7322 6606

8964 8477 7905 7254 6530

8919 8423 7843 7185 6454

8874 8369 7781 7115 6378

8827 8314 7718 7044 6300

8780 8258 7654 6973 6222

8731 8201 7589 6901 6144

8682 8143 7523 6828 6065

5985 5155 4274 3350 2392

5904 5069 4183 3255 2295

5823 4983 4092 3161 2198

5742 4896 4001 3066 2100

5660 4808 3909 2970 2002

5577 4720 3817 2875 1904

5494 4632 3724 2779 1806

5410 4543 3631 2683 1708

5325 4454 3538 2586 1609

5240 4364 3444 2489 1510

1411 + 0416

1312 0316 0684 1676 2652

1213 0216 0783 1775 2748

1114 01 16 0883 1873 2844

1014 0016 0982 1971 2940

0915 •0084 1082 2069 3035

0815 *0 184 1181 2167 3131

0715 *0284 1281 2264 3225

0616 *0384 1380 2362 3320

0516 •0484 1479 2459 3414

-6119 -6878

3601 4515 5383 6197 6950

3694 4604 5467 6276 7021

3787 4692 5550 6353 7092

3880 4780 5633 6430 7162

3971 4868 5716 6506 7232

4063 4955 5797 6582 7301

4154 5042 5879 6657 7369

4245 5128 5959 6731 7436

4335 5213 6039 6805 7502

-8183 -8716 -9162 -9516

7633 8240 8764 9201 9546

7697 8296 8812 9240 9576

7761 8352 8859 9278 9604

7823 8406 8905 9315 9631

7885 8460 8950 9351 9658

7946 8513 8994 9386 9683

8007 8565 9037 9420 9708

8066 8616 9080 9453 9731

8125 8666 9121 9485 9754

-9775 -9937

9796 9948 1.0000 9952 9805

9816 9957 1. 0000 9942 9785

9834 9966 9998 9931 9764

9852 9974 9996 9919 9742

9868 9981 9993 9905 9719

9884 9986 9989 9891 9695

9899 9991 9983 9876 9670

9912 9995 9977 9860 9644

9925 9997 9970 9843 9617

-9589

9560

9531

9500

9468

9435

9402

9367

9332

9295

0 0

1.0 2 3 4 5 7 g 9


1

Circular Functions

tables

2 0 1 3

5 7 g 9 3.0 1 2 3 4 5 5 g 9

4.0 I 2 3

* s 6 7 g •

i.t

-0584

- 1577 - 2555 - 3508 -4425

- 5298

- 7568

- 9999 -9962 - 9825

■in

i - x - x'J; + I*t- - x> 77 +

' * '

em

=

-

sin

i

GENERAL DATA

1-82 Table

z

Circular Functions

0

1

4

+ 0.2837 3780 4685 5544 6347

2932 3872 4773 5627 6424

3028 3964 4861 5709 6500

5 6 7 S 9

7087 7756 8347 8855 9275

7157 7818 8402 8901 9312

6.0 1 2 3 4

9602 9833 9965 9999 9932

5 6 7 8 9

7.0

5.0 1 2

i


4.

1 2

J

4 5 6 7 8 9

6

7

3123 4056 4948 5791 6576

3218 4147 5035 5872 6651

3312 4238 5121 5953 6725

3407 4328 5206 6033 6799

3500 4418 5292 6112 6872

3594 4508 5376 6191 6944

3687 4597 5460 6269 7016

7226 7880 8456 8946 9348

7295 7942 8509 8991 9383

7363 8002 8561 9034 9417

7430 8061 8612 9076 9450

7497 8120 8662 9118 9482

7563 8178 8712 9158 9514

7628 8235 8761 9198 9544

7692 8292 8808 9237 9573

9629 9850 9973 9996 9920

9656 9867 9980 9993 9907

9681 9883 9986 9989 9892

9706 9898 9991 9984 9877

9729 9911 9994 9978 9861

9752 9924 9997 9971 9844

9774 9936 9999 9962 9826

9794 9947 1.0000 9953 9807

9814 9957 1.0000 9943 9787

9766 9502 9144 8694 8157

9744 9471 9103 8644 8099

9721 9438 9061 8593 8040

9697 9405 9018 8542 7980

9672 9370 8975 8489 7919

9646 9335 8930 8436 7858

9619 9298 8885 8382 7796

9591 9261 8838 8327 7733

9563 9223 8791 8271 7669

9533 9184 8743 8215 7604

7539 6845 6084 5261 4385

7473 6772 6004 5175 4295

7406 6698 5924 5090 4205

7339 6624 5843 5003 4114

7270 6548 5761 4916 4023

7201 6473 5679 4829 3931

7132 6396 5597 4741 3839

7061 6319 5514 4653 3746

6990 6241 5430 4564 3653

6918 6163 5346 4475 3560

3466 2513 1534 0540

3372 2416 1435 0440 0560

3278 2319 1336 0340 0660

3183 2221 1237 0240 0759

3088 2124 1137 0140 0859

2993 2026 1038 0040 0959

2898 1928 0938 •0060 1058

2802 1829 0839 *0160 1158

2706 1731 0739 *0260 1257

2609 1632 0639 •0360 1356

1554 2532 3485 4404 5278

1653 2629 3579 4493 5363

1751 2725 3672 4582 5447

1849 2821 3765 4671 5530

1948 2917 3857 4759 5614

2046 3013 3950 4847 5696

2143 3108 4041 4934 5778

2241 3203 4132 5021 5859

2338 3297 4223 5107 5940

6100 6860 7552 8169 8704

6179 6933 7618 8226 8753

6257 7004 7682 8283 8801

6335 7075 7746 8338 8848

6412 7146 7808 8393 8894

6488 7215 7871 8447 8939

6564 7284 7932 8500 8984

6639 7352 7992 8552 9027

6713 7420 8052 8604 9070

9192 9539 9791 9945 1.0000

9231 9569 9811 9955 1.0000

9269 9597 9830 9964 9999

9306 9625 9848 9972 9997

9342 9652 9865 9979 9994

9377 9677 9880 9985 9990

9412 9702 9895 9990 9985

9445 9726 9909 9994 9979

- 1455 2435 - 3392

1 2 3 4

-4314 -5193 -6020

- 6787

5 6 7 8

(Continued)

5

-

8.0

-7486 -8111 -8654

3

and sin x,

1

4

-0460

2

: cos x COS X

[SEC.

8

9

9 0 1 2 i 4

-9111 -9477 -9922 -9997

9152 9509 9770 9934 9999

S 6 7 8 9

-9972 -9847 -9624 -9304 -8892

9964 9829 9596 9267 8846

9955 9810 9567 9229 8799

9945 9790 9538 9190 8751

9934 9769 9507 9150 8702

9922 9747 9476 9109 8652

9909 9725 9443 9068 8602

9895 9701 9410 9025 8550

9880 9676 9376 8982 8498

9864 9650 9340 8937 8445

10 0

-8391

8336

8280

8224

8166

8108

8049

7990

7929

7868

- 9748

i

For > 10. subtract any multiple of 2» (6.28318) the table. Example: cos 20 coa (20 far) = coa 1.15105

-

-

-

that will bring the argument 0.4080.

within the range of

Sec. 1-2] Table 4.

z

0

5 0

-


6 0

7 0

8.0

i> 0

10 0

For z

table.

tables

mathematical Circular Functions

: cos x sin x

1-83

and sin x.

1

2

3

4

0 . 9589 9258 8835 8323 7728

-

--

9560 9220 8787 8267 7664

9531 9181 8739 8210 7599

9500 9141 8690 8153 7534

9468 9100 8640 8094 7468

9435 9058 8589 8035 7401

— 7055 — 6313 — 5507 4646 — 3739

-

6984 6235 5423 4557 3646

6912 6156 5339 4468 3553

6840 6077 5254 4378 3459

6766 5997 5169 4288 3365

— 2794 — 1822 — 0831 + 0168 1165

2698 1723 0731 0268 1265

2602 1625 0631 0368 1364

2505 1526 0532 0468 1463

2151 3115 4048 4941 5784

2249 3210 4140 5028 5866

2346 3305 4231 51 14 5946

6570 7290 7937 8504 8987

6645 7358 7997 8557 9030

9380 9679 9882 9985 9989

5

(Continued)

6

7

8

9402 9015 8538 7975 7333

9367 8971 8485 7915 7265

9332 8926 8432 7853 7196

9295 8881 8378 7791 7126

6692 5917 5083 4198 3271

6618 5836 4996 4107 3176

6542 5755 4910 4015 3081

6467 5673 4822 3924 2986

6390 5590 4734 3831 2890

2408 1427 0432 0568 1562

2311 1328 0332 0668 1660

2213 1229 0232 0767 1759

2116 1129 0132 0867 1857

2018 1030 0032 0967 1955

1920 0931 •0068 1066 2053

2443 3399 4321 5200 6026

2540 3493 4411 5285 6106

2637 3586 4500 5369 6185

2733 3680 4590 5454 6263

2829 3772 4678 5537 6341

2925 3865 4766 5620 6418

3020 3957 4854 5703 6494

6719 7425 8057 8608 9073

6793 7492 81 16 8658 9115

6866 7558 8174 8708 9155

6938 7623 8231 8757 9195

7010 7687 8287 8805 9234

7081 7751 8343 8851 9272

7151 7813 8397 8898 9309

7221 7875 8451 8943 9345

9414 9704 9897 9990 9984

9447 9728 9910 9994 9978

9480 9750 9923 9997 9971

9511 9772 9935 9999 9963

9542 9793 9946 1. 0000 9954

9571 9812 9956 1.0000 9944

9599 9831 9965 9999 9933

9627 9849 9973 9997 9921

9654 9866 9980 9994 9908

9894 9699 9407 9022 8546

9879 9674 9373 8978 8494

9863 9648 9338 8934 8440

9845 9621 9301 8888 8386

9827 9594 9264 8842 8331

9808 9565 9226 8795 8276

9789 9535 9187 8747 8219

9768 9505 9147 8698 8162

9746 9473 9106 8648 8104

9723 9441 9064 8597 8045

7985 7344 6630 5849 5010

7924 7276 6554 5768 4923

7863 7207 6479 5686 4836

7801 7137 6402 5603 4748

7738 7067 6325 5520 4660

7674 6996 6247 5436 4571

7610 6924 6169 5352 4482

7544 6851 6090 5268 4393

7478 6778 6010 5182 4303

741 1 6704 5930 5097 4212

4121 3191 2229 1245 + 0248

4030 3096 2131 1145 0148

3938 3001 2033 1046 0048

3846 2905 1935 0946 •0052

3754 2809 1837 0847 ♦0152

3661 2713 1739 0747 •0252

3567 2617 1640 0647 •0352

3474 2520 1542 0548 •0452

3380 2423 1443 0448 •0552

3286 2326 1344 0348 •0652

-0752 -1743 -2718 -3665 -4575

0851 1842 2814 3758 4664

0951 1940 2910 3850 4752

1050 2038 3005 3942 4840

1150 2136 3100 4034 4927

1249 2233 3195 4125 5014

1348 2331 3290 4216 5100

1447 2428 3384 4307 5186

- 5440

1546 2525 3478 4397 5271

1645 2621 3572 4486 5356

5524

5607

5689

5771

5853

5934

6014

6093

6172

> 10. subtract any multiple of 2r (6.283 18) that will bring the argument

Sec example for cosine.

9

within the range of the

GENERAL DATA

1-84 Table 5.

0.0


'

0

X

2

| 2 3 4

0.0000 1003 2027 3093 4228

0100 1104 2131 3203 4346

5 6 7 8 9

5463 6841 8423 1.030 1.260

1.0 1 2 3 4

1.557 1.965 2.572 3.602

5 6 7 8 9

+ 14.10

2.0

-2.185 -1.710

1 2 3 4 5 6 7 8 9

3.0

- 1. 119

-0.9160 -7470

-6016 -4727 -3555

-2464

- 1425

5

6

7

8

9 ~

0400 1409 2447 3537 4708

0500 1511 2553 3650 4831

0601 1614 2660 3764 4954

0701 1717 2768 3879 5080

0802 1820 2876 3994 5206

~~0902 1923 2984 4111 5334

5594 6989 8595 1.050 1.286

5726 7139 8771 1.072 1.313

5859 7291 8949 1.093 1.341

5994 7445 9131 1. 116 1.369

6131 7602 9316 1.138 1.398

6269 7761 9505 1.162 1.428

6410 7923 9697 1. 185 1.459

6552 8087 9893 1.210 1.491

6696 8253 1.092 1.235 1.524

1.592 2.014

1.628 2.066 2.733 3.903 6.581

1.665 2. 120 2.820

1.704 2. 176 2.912

1.784 2.296 3. 113

1.827 2.360 3.224

4.072

4.256

1.743 2 234 3.010 4.455

7.055

7.602

8.238

4.673 8.989

4.913 9.887

1.871 2.427 3.341 5.177 10.98

1.917 2.498 3.467 5.471 12.35

1256 10.05

•108.6

4.953 3.242 2.370

4.710 3.130 2.306

•52.07 8.349 4.489 3.026 2.244

3.747

- 1. 374

4

tan x

0300 1307 2341 3425 4586

6.165

-34.23 -7.697 -4.286 -2.927

3

:

1

1206 2236 3314 4466

2.650

5.798

0200-

Circular Functions

[Sec.

16.43 25.49 7.137 4. 100 2.834

19.67 20.31

24.50 16.87

32.46

48.08

92.62

6.228

3.929 2.746

3.771 2.663

5.854 3.624

12.60 5.520

18

6.652

14.43

3.488

2.584

2.509

3.361 2.438

2.129 1.671 1.345 1.097 8978

2.074 1.634 1.318 1.075 8799

2.022 1.598 1.291 1.054 8623

1.973 1.563 1.264 1.033 8450

1.925 1.529 1.239 1.013 8280

1.878 1.496 1.214 •9924 8113

1.834 1.464 1.189 •9728 7948

1.791 1.433 1.165 •9535 7787

1.750 1.403 1.142 •9346 7627

7316 5881 4606 3443 2358

7163 5747 4485 3332 2253

7013 5615 4365 3221 2148

6865 5484 4247 3111 2044

6719 5354 4129 3001 1940

6574 5226 4013 2893 1836

6432 5100 3897 2785 1733

6292 4974 3782 2677 1630

6153 4850 3668 2570 1528

1222 0216 0786 1803 2858

1121 0116 0886 1907 2967

1019 0016 0987 2011 3076

0918 •0084 1088 2115 3186

0818 •0184 1190 2219 3296

0717 •0284 1291 2325 3407

0617 •0384 1393 2430 3519

0516 •0484 1495 2536 3632

II.

5.222

9.121

1 2 3 4

+ 0585 1597 2643

1324 0316 0685 1700 2750

5 6 7 8 9

3746 4935 6247 7736 9474

3860 5060 6387 7897 9666

3976 5186 6529 8060 9861

4092 5313 6673 8227 1 006

4209 5442 6818 8396 1.026

4327 5573 6966 8568 1.047

4447 5704 7115 8743 1 068

4567 5838 7267 8921 1.090

4688 5973 7421 9102 1.112

4811 6109 7577 9286 1. 135

4.0

1.158 1.424 1.778 2.286 3.096

1. 182 1.454 1.820

1.206 1.486 1.864 2.416 3.322

1.231 1.518 1.910 2.486 3.447

1.256 1.552 1.957 2.560

1.282 1.587

1.309 1.622 2.058 2.719

1.365 1.697 2.167

3.580

1.336 1.659 2. 2.806

1.394 1.737 2.225 2.994

5. 134 10.79

5.422

5.743

•131.4 9.257

•56.78 8.463 4.523

1 2 3 4

-0416

4.637 8.860

5 6 7 8 9

+ 80.71

5.0

-3.381

-11.38 -5.267

2.350 3.206 4.873 9.733 418.6 10.21

4.994 3.260 2j->

4.747 3. 148

12.

II

3.042

\7x>

TTT cot x

*

tan x

-

— tan (x ± 1.S708)

13.79 •36.21 7.794 4.317 2.942

2.006 2.638 3.723

3.878

6. 104 16.01

6.511 19.07

•26.58 4.129

•20.99 6.725 3.956

2.849

2.760

7.221

"JO tan ( — x) — — tan x

Ill

2.897 4.225

4.422

6.975 23.58 •17.34 6.292 3.796

7.509 30.86 •14.77

44.66 •12.86

5.910 3.647

5.571 3.509

2 676

2 597

2.521

4.044

r' 1

2x» 945

-r

8.130

-■■•)
cot ( — x) — — cot x

For x > 10, subtract a multiple of w to bring argument within range of table.

mathematical tables

Sec. 1-2] Table

tan x.

:

(Continued)

--

0

1

2

3

4

5

6

7

8

9

5 0 1

— 3.381

3.260

3.148

3.042

2.942

2.849

886 - 1.1.501

-1.218

2.316 1.798 1.438 1. 169

2.254 1.756 1.408 1. 146

2.194 1.716 1.378 1.123

2. 137 1.677 1.350 1.100

2.760 2.083 1.640 1.322 1.079

2.676

2.381 1.841 1.469 1.193

2.031 1.604 1.295 1.057

2.597 1.980 1.568 1.268 1 036

2.521 1.932 1.534 1.243 1.016

-0.9956 -8139 -6597 -5247 -4031

9759 7975 6455 5120 3915

9565 7812 6314 4994 3800

9376 7652 6175 4870 3686

9189 7495 6038 4747 3573

9007 7340 5902 4625 3461

8827 7187 5768 4504 3349

8651 7037 5635 4384 3238

8477 6888 5504 4266 3128

8307 6742 5375 4148 3019

-2910 -1853 -0834 1173

2802 1749 0733 0266 1275

2694 1646 0633 0368 1377

2587 1544 0532 0468 1479

2481 1442 0432 0569 1581

2375 1340 0332 0669 1684

2270 1238 0232 0770 1787

2165 1137 0132 0870 1890

2060 1036 003 2 0971 1994

1956 093 5 •0068 1072 2098

2203 3279 4428 5683 7091

2308 3390 4548 5816 7242

2413 3502 4669 5951 7396

2520 3614 4791 6087 7552

2626 3728 4915 6225 7710

2733 3842 5040 6365 7871

2841 3957 5166 6506 8034

2949 4073 5293 6650 8200

3058 4190 5422 6795 8369

3168 4308 5552 6942 8540

8714 1.065 1.305 1.617

8892 1.086 1.332 1.653 2. 102

9073 1.109 1.360 1.691 2.158

9257 1.131 1.389 1.731 2.216

9444 1.154 1.419 1.771 2.276

9635 1.178 1.449 1.813 2.339

9830 1.202 1.461 1.857 2.405

1.003 1.227 1.513 1.902 2.475

1.023 1.252 1.547 1.949 2.548

1.044 1.278 1.581 1.998 2.625

2.706 3.852

2.792 4.017 6.897 22.72

2.978

3.080

3. 188

4.390 8.024 41.69

3.303 5.090 10.61

3.558 5.690

3.700

71.52

4.834 9.582 251.2

3.426

4 602 8.735

17.83

2.882 4. 196 7.419 29.42 15.13

13.13

11.60

10.38

6.357 3.820

5.968

5.622

3.670

3.530

2.689 2.039 1.609

2.609 1.988 1.574

2.533 1.940 1.540

5.314 3 400 2.461 1.893 1.506

1.299 1.061 8679 7061 5656

1.273 1.040 8505 6912 5525

1.247 1.019 8334 6765 5395

4403 3 256 2181 1153 0148

4284 3146 2077 1052 0048

9

1770 2824 3939 5146

0854 1874 2932 4055 5273

0.0

6484

6627

6.0

--2.449

+ 0168

; Generated for wjivans (University of Florida) on 2015-09-23 02:45 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Circular Functions

6.

1-85

o

2.049

r' ft 0 i 2 3 4 5 0 7 8 » 9 0 1 2 3 4 5 6 7

8

6.443 + 18.51 -21 72

-6.800 -3.982 -2.774 -2.091

-

1. 646

-

1.326

-1.082 -0.8856 -7211

-5789 -4523 -3367 -2286

-

1254

-0248

+ 0754

For rapidly changing

Example: Un 4.705 =

6.043 15.61 ♦27.75

8.581

13.49 •38.43 7.893

4.557

4.349

3.058

4.158

2.263 1.763 1.413

2.958 2.204 1.722 1.383

2.863 2.146 1.683 1.354

•62.42

7.307

2.392 1.848 1.474

4.785 3. 165 2.326 1.805 1.443

1.222 •9988 8166 6620 5267

1.197 •9790 8001 6477 5140

1.173 •9596 7838 6336 5014

1. 149 •9406 7678 6197 4890

1. 126 •9219 7520 6059 4766

1.104 •9036 7365 5924 4644

4167 3036 1973 0951 *0052

4050 2927 1869 0850 •0152

3934 2819 1766 0749 •0252

3819 2711 1663 0649 •0352

3705 2604 1560 0548 •0453

3591 2498 1458 0448 •0553

3479 2392 1356 0348 •0653

0955 1977 3041 4172 5401

1056 2082 3151 4290 5531

1157 2186 3261 4409 5662

1259 2291 3372 4529 5795

1361 2397 3484 4650 5930

1463 2503 3596 4772 6066

1565 2609 3710 4895 6203

1667 2716 3824 5020 6342

6771

6918

7067

7218

7371

7527

7685

7845

values

-

•166.2 9.397

5.374 11.86

5.036

3.279

of tan x (when x approaches an odd multiple of I tan x tan (r ± 1.5708) I I 135.

-

t/2),

-

-„_„„„ ta^6T758 For rapidly changing values of cot x (when x approaches

a multiple of

t),

use cot x —

l/tan

x.

1-86

GENERAL DATA Table

X

Exponential Functions

6.

1

2

3

4

5

6

7

8

9

1 2 3 4

1 000 1.105 1.221 1.350 1.492

1.010 1. 116 1.234 1.363 1.507

1.020 I. 127 1.246 1.377 1.522

1.030 1.139 1.259 1.391 1.537

1.041 1.150 1.271 1.405 1.553

1.051 1. 162 1.284 1.419 1.568

1.062 1.174 1.297 1.433 1.584

1.073 1.185 1.310 1.448 1.600

1.083 1. 197 1.323 1.462 1.616

1.094 1.209 1.336 1.477 1.632

5 6 7 8 9

1.649 1.822 2.014 2.226 2.460

1.665 1.840 2.034

1.699 1.878 2.075 2.293

1.716 1.896

1.733 1.916 2.117 2.340 2.586

1.751 1.935

2.248 2.484

1.682 1.859 2.054 2.270 2.509

1.768 1.954 2. 160 2.387

1.786 1.974 2.181 2.411

1.804 1.994 2.203 2.435 2.691

1.0 1 2 3 4

2.718 3.004 3.320 3.669 4.055

2.746 3.034

2.773 3.065

3.353 3.706

3.387 3.743 4. 137

4 482 4.953 5.474

4.527

6.050 6.686

1 2 3 4

7.389 8. 166 9.025 9.974 11.02

5 6 7 8 9

3.0

5 6 7 8 9

2.0

1 2 3 4

2.535

2.560

2.829 3.127 3.456

2.801 3.096 3.421 3.781 4.179

2.858 3.158

3.819 4.221

2.138 2.363 2.612 2.886

2.638 2.915

2.664

3.190

3.222

2.945

3.490 3.857

3.525

4.263

3.896

3.561 3.935

4.306

4.349

4.393

2.974 3.287 3.633 4.015 4.437

4.711 5.207

4.759 5.259

5.755 6.360

5.812 6.424 7.099

4.807 5.312

4.855 5.366

4.904 5.419

5.871

5.930

5.989

6.488

6.554

7.171

7.243

6.619 7.316

7.846

7.925

8.758

8.004 8.846

9.679 10.70 11.82

9.777 10.80 11.94

8.085 8.935 9.875 10.9! 12.06

3.254 3.597 3.975

4.572 5.053 5.585

4.618

6.110 6.753

6.172

6.234 6.890

6.297 6.959

7.463 8.248

7.538 8.331

7.614

7.691

8.415

9.116

9.207

9.300

8.499

10.18 11.25

10.28 11.36

9.393 10.38 11.47

9.488

10.07 11.13

10.49 11.59

8.671 9.583 10.59 11.70

12.18 13.46 14.88 16.44 18.17

12.30 13.60 15.03 16.61 18.36

12.43 13.74 15.18 16.78 18.54

12.55 13.87 15.33 16.95 18.73

12.68 14.01 15.49 17.12 18.92

12.81 14. 15 15.64 17.29 19.11

12.94 14.30 15.80 17.46 19.30

13.07 14.44 15.96 17.64 19.49

13.20 14.59 16.12 17.81 19.69

13.33 14.73 16.28 17.99 19.89

20.09 22.20 24.53 27.11 29.96

20.29 22.42

20.49 22.65 25.03

20.70 22.87 25.28 27.94

20.91 23. 10 25.53 28.22 31. 19

21. 12 23.34 25.79 28.50

21.33 23.57 26.05 28.79

21.54 23.81 26.31 29.08

21.76

21.98

24 05 26.58 29.37

24.29 26.84 29.67

32.46

32.79

35.87 39.65 43.82 48.42 53.52

36.23

59. 15

59.74

65.37 72.24 79.84

66.02

5 6 7 8 9

33.12 36.60 40.45

4.0

54.60 60.34

1 2 3 4

4.096

2.096 2.316

1

: e*

0

0.0


[Sec.

44.70 49.40

66.69 73.70 81.45

5 6 7 8 9

90.02 99.48

5.0

148.4

109.9 121 5 134.3

5.003 5.529

24.78

6.821

27.39 30.27

27.66 30.57

30.88

33.45 36.97

33.78

34.12

37.34 41.26 45.60 50.40

37.71

55.70 61.56 68.03

56.26

62.18 68.72

82.27

75. 19 83.10

75.94 83.93

90.92

91.84

92.76

100.5

Mil

122.7 135 6

101.5 112.2 124.0 137.0

102.5 113.3 125.2 138.4

149.9

151.4

152.9

40.85 45.15 49.90 55.15

60.95 67.36 74.44

4.665 5.155 5.697

5. 104 5.641

41.68 46.06 50.91

I

<*-|+7l

For very small values of x, e* « I + x. Larce values of x: Method I. e* («*/■)«, e.g., e" («•«)*

-

-

7.029 7.768 8.585

31.50

31.82

32.14

34.47 38.09 42.10 46.53 51.42

34.81

35.16 38.86 42.95

35.52

47.47 52.46

47.94

56.83 62.80

57.40 63.43

57.97 64.07

69.41 76.71

58.56 64.72

70.

84.77

77.48

70.81 78.26

85.63

86.49

71.52 79.04

87.36

93.69

94.63

95.58

103.5 114.4 126.5 139.8

104.6 115.6 127.7 141.2

105.6 116.7 129 0 142.6

154.5

156.0

157.6

+

I»

38.47 42.52 46.99 51.94

I"

2i+3i

518.0*

-

II

39.25

43.38 52.98

88.23

72 97 80.64 89. 12

96 54 106.7 117.9 130.3 144.0

97.51 107.8 119. 1 131.6 145.5

98 49 108.9 120.3 133 0 146.9

159.2

160.8

162.4

+

7.200 X

I0">.

40 04 44. 2>'. 48.91 54.05

{Continued

on p. 88)

mathematical tables

Sec. 1-2] Table 6.


X

0

1

Functions:

Exponential 2

4

3

e~*.

5

1-87 (Continued) 6

7

8

9

0 0 1 2 3 4

1. 0000 0.9048 8187 7408 6703

•9900 8958 8106 7334 6637

•9802 8869 8025 7261 6570

•9704 8781 7945 7189 6505

•9608 8694 7866 7118 6440

•9512 8607 7788 7047 6376

•9418 8521 7711 6977 6313

•9324 8437 7634 6907 6250

•9231 83 53 7558 6839 6188

•9139 8270 7483 6771 6126

5 6 7 8 9

6065 5488 4966 4493 4066

6005 5434 4916 4449 4025

5945 5379 4868 4404 3985

5886 5326 4819 4360 3946

5827 5273 4771 4317 3906

5769 5220 4724 4274 3867

5712 5169 4677 4232 3829

5655 5117 4630 4190 3791

5599 5066 4584 4148 3753

5543 5016 4538 4107 3716

1.0 1 2 3 4

3679 3329 3012 2725 2466

3642 3296 2982 2698 2441

3606 3263 2952 2671 2417

3570 3230 2923 2645 2393

3535 3198 2894 2618 2369

3499 3166 2865 2592 2346

3465 3135 2837 2567 2322

3430 3104 2808 2541 2299

3396 3073 2780 2516 2276

3362 3042 2753 2491 2254

5 6 7 8 9

2231 2019 1827 1653 1496

2209 1999 1809 1637 1481

2187 1979 1791 1620 1466

2165 1959 1773 1604 1451

2144 1940 1755 1588 1437

2122 1920 1738 1572 1423

2101 1901 1720 1557 1409

2080 1882 1703 1541 1395

2060 1864 1686 1526 1381

2039 1845 1670 1511 1367

2 0 1 2 3 4

1353 1225 1108 1003 0.09072

1340 1212 1097 •9926 8982

1327 1200 1086 •9827 8892

1313 1188 1075 •9730 8804

1300 1177 1065 •9633 8716

1287 1165 1054 •9537 8629

1275 1153 1044 •9442 8543

1262 1142 1033 •9348 8458

1249 1130 1023 •9255 8374

1237 1119 1013 ♦9163 8291

5 6 7 8 9

8208 7427 6721 6081 5502

8127 7353 6654 6020 5448

8046 7280 6587 5961 5393

7966 7208 6522 5901 5340

7887 7136 6457 5843 5287

7808 7065 6393 5784 5234

7730 6995 6329 5727 5182

7654 6925 6266 5670 5130

7577 6856 6204 5613 5079

7502 6788 6142 5558 5029

1 2 3 4

4979 4505 4076 3688 3337

4929 4460 4036 3652 3304

4880 4416 3996 3615 3271

483 2 4372 3956 3579 3239

4783 4328 3916 3544 3206

4736 4285 3877 3508 3175

4689 4243 3839 3474 3143

4642 4200 3801 3439 3112

4596 4159 3763 3405 3081

4550 4117 3725 3371 3050

5 6 7 8 »

3020 2732 2472 2237 2024

2990 2705 2448 2215 2004

2960 2678 2423 2193 1984

2930 2652 2399 2171 1964

2901 2625 2375 2149 1945

2872 2599 2352 2128 1925

2844 2573 2328 2107 1906

2816 2548 2305 2086 1887

2788 2522 2282 2065 1869

2760 2497 2260 2045 1850

4.0

1832 1657 1500 1357 1228

1813 1641 1485 1343 1216

1795 1624 1470 1330 1203

1777 1608 1455 1317 1191

1760 1592 1441 1304 1180

1742 1576 1426 1291 1168

1725 1561 1412 1278 1156

1708 1545 1398 1265 1145

1691 1530 1384 1253 1133

1674 1515 1370 1240 1122

1067 •9658 8739 7907 7155

1057 •9562 8652 7828 7083

1046 •9466 8566 7750 7013

1036 •9372 8480 7673 6943

1025 •9279 8396 7597 6874

1015 •9187 8312 7521 6B06

6474

6409

6346

6282

6220

6158

3.0

1 2 3 4

llll

5 6 7 8 9

1005 0.009095 8230 7447

1100 •9952 9005 8148 7372

1089 •9853 8067 7299

1078 •9755 8826 7987 7227

5 0

6738

6671

6605

6539

89IS

-

-

I

+

+

I! 2! 3! For very small values of i. e * * I i. Large values of x: The formulas for e* apply; alio e~* = l/e*. Method I. «-» 1.39 X 10". («-•»)« = (0.001930)' (Continued

-

-

on ,,. SB)

GENERAL DATA

1-88 Table 0

X


1

2

3

: e*.

1

(Continued)

4

5

6

7

8

9

1 2 3 4

148.4 164.0 181.3 200.3 221.4

149.9 165.7 183. 1 202.4 223.6

151.4 167.3 184.9 204.4 225.9

152.9 169.0 186.8 206.4 , 228. 1

154.5 170.7 188.7 208 5 230.4

156.0 172.4 190.6 210.6 232.8

157.6 174.2 192.5 212.7 235. 1

159. 2 175.9 194.4 214.9 237.5

160.8 177.7 196.4 217.0 239.8

162. 4 179.5 198.3 219. 2 242.3

5 6 7 8 9

244.7 270.4 298.9 330.3 365.0

247.2 273. 1 301.9 333.6 368.7

249.6 275.9 304.9 337.0 372.4

252. 1 278.7 308.0 340.4 376.2

254.7 281.5 311. 1 343.8 379.9

257.2 284.3 314.2 347.2 383.8

259.8 287. 1 317.3 350.7 387.6

262.4 290.0 320.5 354.2 391.5

265. 1 292.9

323.8

267. 7 295 . 9 327 0

357.8 395.4

361.4

432.7 478.2 528.5 584. 1 645.5

437.0 483.0 533.8 589.9 652.0

441.4 487.8 539. 2 595.9 658.5

5.0

399 4

1 2 3 4

403.4 445.9 492.7 544.6 601.8

407.5 450.3 497.7 550.0 607.9

411.6 454.9 502.7 555.6 614.0

415. 7 459.4 507.8 561.2 620.2

419.9 464. 1 512.9 566.8 626.4

424. 1 468. 7 518.0 572.5 632.7

428. 4 473.4 523.2 578.2 639. 1

5 6 7 1 9

665. 1 735. 1 812.4 897.8 992.3

671.8 742.5 820.6 906.9 1002

678.6 749.9 828.8 916.0 1012

685.4 757.5 837. 1 925.2 1022

692.3 765. 1 845.6 934.5 1033

699.2 772.8 854. 1 943.9 1043

706.3 780.6 862.6 953.4 1054

713.4 788.4 871.3 962.9 1064

720.5 796.3 880. 1 972.6 1075

727.8 804.3 888. 9 982.4 1086

7.0

4

1097 1212 1339 1480 1636

1108 1224 1353 1495 1652

1119 1236 1366 1510 1669

1130 1249 1380 1525 1686

1141 1261 1394 1541 1703

1153 1274 1408 1556 1720

1164 1287 1422 1572 1737

1176 1300 1437 1588 1755

1188 1313 1451 1604 1772

1200 1326 1466 1620 1790

5 6 7 S 9

1808 1998 220S 2441 2697

1826 2018 2231 2465 2724

1845 2039 2253 2490 2752

1863 2059 2276 2515 2779

1882 2080 229B 2540 2807

1901 2101 2322 2566 2836

1920 2122 2345 2592 2864

1939 2143 2368 2618 2893

1959 2165 2392 2644 2922

1978 2186 2416 2670 2951

8.0

2981 3294 3641 4024 4447

3011 3328 3678 4064 4492

3041 3361 3715 4105 4537

3072 3395 3752 4146 4583

3103 3429 3790 4188 4629

3134 3463 3828 4230 4675

3165 3498 3866 4273 4722

3197 3533 3905 4316 4770

3229 3569 3944 4359 4817

3262 3605 3984 4403 4866

4915 5432 6003 6634 7332

4964 5486 6063 6701 7406

5014 5541 6124 6768 7480

5064 5597 6186 6836 7555

5115 5653 6248 6905 7631

5167 5710 631 1 6974 7708

5219 5768 6374 7044 7785

5271 5B25 6438 7115 7864

5324 5884 6503 7187 7943

5378 5943 6568 7259 8022

8103 8955 9897 I0« X 1.094 I0< X 1.209

8185 9045 9997 1. 105 1.221

8267 9136 * 1. 0 10 1. 116 1.233

8350 9228 •1.020 1. 127 1.246

8434 9321 •1.030 1. 138 1.258

8519 9414 •1.040 1. 150 1.271

1.336 1.476 1.632 1.803 1.993

1.349 1.491 1.648 1.821 2.013

1.363 1.506 1.665 1.840 2.033

1.377 1.521 1.681 1.858 2.054

1.390 1.537 1.698 1.877 2.074

1.404 1.552 1.715 1.896 2.095

1.419 1.568 1.733 1.915 2. 116

1.433 1.584 1.750 1.934 2. 138

1.447 1.599 1.768 1.954 2. 159

1.462 1.616 1.785 1.973 2. 181

10' X 2. 203

2. 225

2. 247

2. 270

2.293

2.316

2.339

2.362

2.386

2.410

6.0


6.

[SEC.

1 2

J

1 3 4

, 5 7 g 9

9.0 2 3 4

j (, 7

10.0

I0< I0< 10* I0« I0«

X X X X

X

Method 2. Method 3. Method 4.

-

f

f,

-

e.g., e» «•*•»*• X t* X By natural logarithm., e.g.. In By common luiiarithma, e.g., log

c"

t"

f"

8604 8691 9509 9605 •1.051 •1.061 1. 161 1. 173 1.284 1.296

8778 8866 9701 9799 •1.072 •1.083 1. 197 1. 185 1.310 1.323

-

7.202 X 10". X e10 X e> = (2.203 X I0«)» X 148.4 = 25 =■23.0259 + 1.9741 e» = 7.200 X 10".

= 25 log a = 25 X 0.4343 = 10.8575

e"

-

7.205 X 10"

mathematical tables

Sec. 1-2] Table


s

1

0


6.

: e~*.

1-S9 (Continued)

2

3

4

5

6

7

8

9

5.0

0.006738

1 2 3 4

6097 5517 4992 4517

6671 6036 5462 4942 4472

6605 5976 5407 4893 4427

6539 5917 5354 4844 4383

6474 5858 5300 4796 4339

6409 5799 5248 4748 4296

6346 5742 5195 4701 4254

6282 5685 5144 4654 4211

6220 5628 5092 4608 4169

6158 5572 5042 4562 4128

5 6 7 8 9

4087 3698 3346 3028 2739

4046 3661 3313 2997 2712

4006 3625 3280 2968 2685

3966 3589 3247 2938 2658

3927 3553 3215 2909 2632

3887 3518 3183 2880 2606

3849 3483 3151 2851 2580

3810 3448 3120 2823 2554

3773 3414 3089 2795 2529

3735 3380 3058 2767 2504

6.0 1 2 3 4

2479 2243 2029 1836 1662

2454 2221 2009 1818 1645

2430 2198 1989 1800 1629

2405 2177 1969 1782 1612

2382 2155 1950 1764 1596

2358 2133 1930 1747 1581

2334 2112 1911 1729 1565

2311 2091 1892 1712 1549

2288 2070 1873 1695 1534

2265 2050 1855 1678 1519

5 6 7 8 9

1503 1360 1231 1114 1008

1488 1347 1219 1103 •9978

1474 1333 1207 1092 •9878

1459 1320 1195 1081 *9780

1444 1307 1183 1070 •9683

1430 1294 1171 1059 •9586

1416 1281 1159 1049 •9491

1402 1268 1148 1038 •9397

1388 1256 1136 1028 •9303

1374 1243 1125 1018 •9210

7.0 0.0009119 1 2 3 4

8251 7466 6755 6113

9028 8169 7392 6688 6052

893 8 8088 7318 6622 5991

8849 8007 7245 6556 5932

8761 7928 7173 6491 5873

8674 7849 7102 6426 5814

8588 7771 7031 6362 5757

8502 7693 6961 6299 5699

8418 7617 6892 6236 5643

8334 7541 6823 6174 5586

5 6 7 8 9

5531 5005 4528 4097 3707

5476 4955 4483 4057 3671

5421 4905 4439 4016 3634

5367 4857 4394 3976 3598

5314 4808 4351 3937 3562

5261 4760 4307 3898 3527

5209 4713 4265 3859 3492

5157* 4666 4222 3820 3457

5106 4620 4180 3782 3422

5055 4574 4139 3745 3388

1 2 3 4

3355 3035 2747 2485 2249

3321 3005 2719 2460 2226

3288 2975 2692 2436 2204

3255 2946 2665 2412 2182

3223 2916 2639 2388 2161

3191 2887 2613 2364 2139

3159 2859 2587 2340 2118

3128 2830 2561 2317 2097

3097 2802 2535 2294 2076

3066 2774 2510 2271 2055

5 6 7 8 9

2035 1841 1666 1507 1364

2014 1823 1649 1492 1350

1994 1805 1633 1477 1337

1975 1787 1617 1463 1324

1955 1769 1601 1448 1310

1935 1751 1585 1434 1297

1916 1734 1569 1420 1284

1897 1717 1553 1405 1272

1878 1700 1538 1391 1259

I860 1683 1522 1378 1247

9.0

1234 1117 1010 .00009142 8272

1222 1106 1000 9051 8190

1210 1095 ♦9904 8961 8109

1198 1084 •9805 8872 8028

1186 1073 •9708 8784 7948

1174 1062 •9611 8697 7869

1162 1052 •9516 8610 7791

1151 1041 •9421 8524 7713

1139 1031 •9327 8440 7636

1128 1021 •9234 8356 7560

7485 6773 6128 5545 5017

7411 6705 6067 5490 4968

7337 6639 6007 5435 4918

7264 6573 5947 5381 4869

7192 6507 5888 5328 4821

7120 6443 5829 5275 4773

7049 6378 5771 5222 4725

6979 6315 5714 5170 4678

6910 6252 5657 5119 4632

6841 6190 5601 5063 4586

.00004540

4495

4450

4406

4362

4319

4276

4233

4191

4149

8.0

1

10.0

«" -

-

«-«• X •"> X «• X 0.006738 = 1.388 X 10-". (0.00004540)' Method 2. Methods 3 and 4 are not convenient. Method 5. Any of the four methods can be used in conjunction with e~z = l/e*. e.g.,

—

-

1/(7.202

X

10")

1.388

X 10-"

1-90

GENERAL Table 7.

Hyperbolic

DATA

[Sec.

1

Functions: cosh x and sinh x cosh x

X

0.0

'

2

3

4

5

6

7

8

9

1. 0000

1.0002 1.007 1.024 1.052 1.090

1.0005 1.008 1.027 1.055 1.094

1 0008 1.010 1.030 1.058 1.098

1.0013 1.011 1.031 1.062 1.103

1.0018 1.013 1.034 1.066 1. 108

1.0025 1.014 1.037 1.069 1.112

1.0032 1.016 1.039 1.073 1.117

1.0041 1.018 1.042 1.077 1.123

1 2 3 4

I.00S 1.020 1.045 1.081

1 0001 1.006 1.022 1.048 1.085

5 6 7 8 9

1. 128 1.185 1.255 1.337 1.433

1.133 1.192 1.263 1.346 1.443

1.138 1.198 1.271 1.355 1.454

1.144 1.205 1.278 1.365 1.465

1.149 1.212 1.287 1.374 1.475

1.155 1.219 1.295 1.384 1.486

1. 161 1.226 1.303 1.393 1.497

1.167 1.233 1.311 1.403 1.509

1.173 1.240 1.320 1.413 1 520

1.179 1.248 1.329 1.423 1.531

1.543 1.669

1.567 1.696 1.841 2.005 2.189

1.579 1.709 1.857 2.023 2.209

1.591 1.723 1.872 2.040 2.229

1.604 1.737 1.888

1.616 1.752 1.905 2.076 2.269

1.629 1.766 1.921 2.095 2.290

1.642 1.781 1.937 2.113 2.310

1.655 1.796 1.954

1.971 2.151

1.555 1.682 1.826 1.988 2. 170

2.439 2.675 2.936

2.462 2.700 2.964

2.484 2.725 2.992

2.530 2.776

2.554 2.802

3.228

3.259 3.585

3.290 3.620

2.507 2.750 3.021 3.321

3.655

3.049 3.353 3.690

3.078 3.385 3.726

3.948 4.351 4.797 5.290 5.837

3.987 4.393 4.844 5.343 5.895

4.026 4.436

4.065 4.480

4.891 5.395 5.954

4.939 5.449

4.104 4.524 4.988

6.013

6.072

6.443

6.507

6.636

6.702

7.327 8.091 8.935

7.400 8.171

1.0 1 2 3 4


0

I.8M

5 6 7 8 9

2.352 2.577 2.828

2.374 2.601 2.855

2.395 2.625 2.882

3.107 3.418

2.417 2.650 2.909

3.137

3.167 3.484

3.197 3.517

2.0

3.835 4.226

3.873 4.267 4.704 5. 188 5.723

3.910 4.309 4.750

6.379 7.042 7.776

3.451

3.762

3.799

1 2 3 4

4.144

4.185 4.613

5.037 5.557

5.087

5 6 7 8 9

6.132

6.193

4.568

5.612

4.658 5.137 5.667 6.255 6.904 7.623

6.769 7.473 8.253

6.836

9.115

9.206

8.418 9.298

6.317 6.973 7 699 8.502 9.391

1 2 3 4

10.07 11.12 12.29 13.57 15.00

10.17 11.23 12.41 13.71 15.15

10.27 11.35 12.53 13.85 15.30

5 6 7 8 9

16.57 18.31 20.24 22.36 24.71

16.74 18.50 20.44 22.59 24.96

27.31 30. 18

27.58 30.48 33.69

3.0

4.0 1 2 3 4 5 6 7 8 9

5.0

7.548 8.335

3.551

5.239

5.780

5.503

8.587 9.484

7.932 8.759

9.579

9.675

9.772

9.869

9.024 9.968

10.37 11.46 12.66 13.99 15.45

10.48 11.57 12.79 14.13 15.61

10.58 11.69 12.91 14.27 15.77

10.69 11.81 13.04 14.41 15.92

10.79 11.92 13.17 14.56 16.08

10.90 12.04 13.31 14.70 16.25

11.01 12. 16 13.44 14.85 16.41

16.91 18.68 20.64 22.81 25.21

17.08 18.87 20.85 23.04 25.46

17.25 19.06

21.06 23.27 25.72

17.42 19.25 21.27 23.51

17.60 19.44

17.77 19.64

18.13 20.03 22.14 24.47 27.04

27.86

28.14

30.79

28.42 31.41 34.71

37.60 41.55

41.97

42.39

45.01 49.75 54.98

45.47

46.38

60.76

51.26 56.65 62.61

46.85 51.78 57.22

67.15

61.37 67.82

45.92 50.75 56.09 61.99

68.50

74.21

74.96

75.71

50 25 55.53

2.331

7.853 8.673

37.23 41. 14

40.73

2.132

6.571 7.255 8.011 8.847

31.10 34.37 37.98

33.35 36.86

2.058 2.249

34.02

OOBh X

38.36

7.112

21.49

21.70

17.95 19.84 21.92

23.74 26.24

23.98 26.50

26.77

28.71 31.72 35.06 38.75 42.82

29.00 32.04

29.29

29.58

29.88

32 37

35.41 39.13 43.25

35.77

32.69

43.68

36.13 39.93 44. 12

33.02 36.49

47.32

47.80 52.82 58.38

48.28

48.76 53.89

59.56 65.82

71.30

53 35 58.96 65 16 72.02

72.74

66.48 73.47

78.80

79.59

80.39

81.20

25.98

69.19

63.24 69.89

52.30 57.80 63.87 70.59

76.47

77.24

78.01

«* + «'

1 +

7.183

2!

+

41

64.52

+

39.53

24.22

40.33

44.57 49.25 54.43 60. 15

Sec. 1-2]

MATHEMATICAL

Table

7.

Hyperbolic

Functions

:

TABLES

1-91

cosh x and sinh x.

sinh x

I


0.0

0

1

2

3

0.0000

4

(Continued)

5

6

7

8

9

1 2 3 4

1002 2013 3045 4108

0100 1102 2115 3150 4216

0200 1203 2218 3255 4325

0300 1304 2320 3360 4434

0400 1405 2423 3466 4543

0500 1506 2526 3572 4653

0600 1607 2629 3678 4764

0701 1708 2733 3785 4875

0801 1810 2837 3892 4986

0901 1911 2941 4000 5098

5 6 7 8 9

5211 6367 7586 6881 1.027

5324 6485 7712 9015 1.041

5438 6605 7838 9150 1.055

5552 6725 7966 9286 1.070

5666 6846 8094 9423 1.085

5782 6967 8223 9561 1.099

5897 7090 8353 9700 1.114

6014 7213 8484 9840 1.129

6131 7336 8615 9981 1.145

6248 7461 8748 •0122 1.160

1.0 1 2 3 ♦

1.175 1.336 1.509 1.698 1.904

1.191 1.352 1.528 1.718 1.926

1.206 1.369 1.546 1.738 1.948

1.222 1.386 1.564 1.758 1.970

1.238 1.403 1.583 1.779 1.992

1.254 1.421 1.602 1.799 2.014

1.270 1.438 1.621 1.820 2.037

1.286 1.456 1.640 1.841 2.060

1.303 1.474 1.659 1.862 2.083

1.319 1.491 1.679 1.883

5 6 7 8 9

2.129 2.376 2.646 2.942 3.268

2.153 2 401 2.674 2.973 3.303

2. 177 2.428 2.703 3.005

2.201 2.454 2.732

2.225 2.481 2.761

3.069 3.408

2.274 2.535 2.820 3. 134

2.299 2.562 2.850

3.037 3.372

2.250 2.507 2.790 3.101 3.443

2.324 2.590 2.881 3.200

2.0

3.627 4.022 4.457 4.937 5.466

3.665

3.741 4. 148

4.988 5.522

3.703 4. 106 4.549 5.039 5.578

3.780 4. 191 4.643

3.820

1 2 3 4

3.859 4.278 4.739 5.248

5 6 7 8 9

6.050 6.695 7.406 8.192 9.060

6. 112 6.763 7.481 8.275 9.151

6.174 6.831 7.557 8 359 9.244

6.237

6.300

6.901 7.634 8.443

9.337

6.971 7.711 8.529 9.431

3.0

10.02 11.08 12.25 13.54 14.97

10. 12 II. 19 12 37 13.67 15.12

10.22 11.30 12.49 13.81 15.27

10.32 11.42 12.62 13.95 15.42

16.54 18.29

16.71 18.47 20.41 22.56 24.94

16.88 18.66 20.62 22.79 25. 19

17.05 18.84 20.83

27.56

27.84

30.47 33.67

30.77

28.12 31.08 34.35 37.97 41.96

1

20 21

22.34 24.69

4.0

5.0

27.29 30.16 33.34 36.84 40.72

4.064 4 503

3.337

34.01

4.596 5.090 5.635

23.02 25.44

37.21 41.13

37.59 41.54

45 00 49 74

45.46

46.37

54.97

55.52 61.36

45.91 50.74 56.08

62.60

50.24

60 75 67. 14

67 82

61.98 68.50

74.20

74.95

75.70

sinh :

5.142 5.693

4.234 4.691 5.195 5.751

6.365 7.042

2.617 2.911

3.234 3.589

3.899 4.322 4.788

3.940

3.981

5.810

5.302 5.869

5.356 5.929

6.429

6.495

6.561

7.113

7.185 7.948 8.790

4.367

4.837

4.412 4.887 5.411 5.989

6.627 7.332 8.110 8.969 9.918

7.789 8.615 9.527

8.702 9.623

9.720

7.258 8.028 8.879 9.819

10.43 11.53 12.75 14.09 15.58

10.53 11.65 12.88 14.23 15.73

10.64 11.76 13.01 14.38 15.89

10.75 11.88 13.14 14.52 16.05

10.86 12.00 13.27 14.67 16.21

10.97 12. 12 13.40 14.82 16.38

17.22 19.03 21.04 23.25 25.70

17.39 19.22

17.57 19.42 21.46 23.72 26.22

17.74 19.61 21.68 23.96

17.92 19.81 21.90 24.20 26.75

18. 10 20.01 22. 12 24.45 27.02

28.98

29.27 32 35 35.75

28.40

31.39 34.70

21.25 23.49 25.96

28.69 31.71

35.05 38.73

7.868

32.03 35.40

26.48

42.81

39. 12 43.24

39.52 43.67

46.84 51.77 57.21 63.23

47.31

47.79

52.29

52.81

48.27 53.34

69. 19

69.88

76.46

77.23

51.25 56.64

2.350

3.552

3.479

3.167 3.516

2.106

38 35

42.38

29.56

29.86

32.68

33.00 36.48

36. II 39.91 44. II

58.37

58.96

70.58

64.51 71.29

65. 16 72.01

48.75 53.88 59.55 65.81 72.73

78.01

78.79

79.58

80.38

57.79

63.87

40.31

44.56 49.24 54.42 60. 15

66.47 73.46 81.19

1-92

GENERAL

DATA

[SEC.

1

Table 7. Hyperbolic Functions : cosh x and sinh z. (Continued) (Table gives cosh x. Sinh x is the same except that the last digit is to be diminished by one ' appears.) where X

0

5.0

74.21'

2

3

4

5

6

7

8

9

74.96' 82.84' 91.55'

75.71'

76.47' 84.51

86.22' 95.29'

78.80' 87.09' 96.24

79.59'

80.39'

93.40'

77.24' 85.36' 94.34'

78.01

83.67' 92.47'

III.

102.2 112.9

103.2 114. 1

104.3 115.2

105.3 116.4

106.4 117.6'

88.84 98. 19' 108.5 119.9

81.20' 89.74'

101.2 8

87.96' 97.21 107.4 118.7

127.3 140.7 155.5 171.9 190.0

128.6 142.1 157.1 173.6 191.9

129.9 143.6 158.7 175.4 193.8

131.2 145.0 160.3 177. 1 195.8

132.5 146.5 161.9 178.9 197.7

133.9 147.9 163.5 180.7 199.7

209.9

212.1

232.0

234.4

214.2

218.5 241.5

220.7 243.9 269.6 297.9 329.3

1 2 3 4

82.01 90.64' 100.2 110.7

5 6 7 8 9

122.3 135.2 149.4 165.2' 182.5

123.6 136.6 150.9 166.8 184.4

124.8 137.9 152.5 168.5 186.2

126.1 139.3 154.0 170.2 188. 1

201.7 222.9 246.4

205.8

207.9

227.4 251.4

272.3

203.7 225.2 248.9' 275.0

300.9

303.9

277.8 307.0

229.7 253.9 280.6 310.1

332.6 367.5 406.2 448.9

335.9 371.2

496.1

501.1

339.3 375.0 414.4 458.0 506.2

342.7 378.7 418.6 462.6 511.2

548.3

553.8

559.4 618.2 683.2

565.0 624.4

6.0 1 2 3 4 5 6 7 8 9


1

7.0 1 2 3 4

606.0 669.7 740.2' 818.0

5 6 7 8 9

904.0

8.0

410.3

453.4

612.1 676.4 747.6

755.1

690.1

762.7 842.9

99. 17 109.6 121.1

256.4

259.0

283.4 313.2

286.2 316.4

289.1

216.3 239.1 264.2 292.0

319.5

322.7

295.0 326.0

346.1

349.6 386.4 427.0 471.9 521.6

353.1 390.3 431.3

356.7 394.2 435.7

360.3 398.2 440.0

363.9

476.7 526.8

481.5 532.1

486.3

491.2

382.5 422.8 467.2 516.4

570.7 630.7 697.0

236.7

261.6

266.9

537.5

576.4

582.2

588.1

594.0

643.5 711.1 785.9

649.9

770.4

637.1 704.1 778.1

851.4

859.9

868.6

793.8

656.5 725.5 801.8

877.3

886.1

718.3

402.2 444.5 542.9

600.0 663.1 732.8 809.9 895.0

826.2

834.5

913. 1 1009 1115 1233 1362

922.3 1019 1126 1245 1376

931.6

940.9

950.4

959.9

969.6

1030 1138 1257 1390

1040 1149 1270 1404

1050 1161 1283 1418

1061 1172 1296 1432

1072 1184 1309 1446

979.3 1082 1196 1322 1461

989.2

999. 1 1104 1220 1349

1 2 3 4

1490 1647 1820 2012 2224

1505 1664 1839 2032 2246

1521 1681 1857 2053 2268

1536 1697 1876 2073 2291

1551 1714 1895 2094 2314

1567 1732 1914 2115 2338

1583 1749 1933 2136 2361

1599 1767 1952 2158 2385

1615 1784 1972 2180 2409

1631 1802 1992 2201 2433

5 6 7 8 9

2457 2716 3001 3317 3666

2482 2743 3032 3350 3703

2507 2771 3062 3384 3740

2532 2799 3093 3418 3778

2558 2827 3124 3452 3616

2583 2855 3155 3487 3854

2609 2884 3187 3522 3893

2636 2913 3219 3558 3932

2662 2942 3251 3593 3971

2689 2972 3284 3630 4011

9.0

4

4052 4478 4949 5469 6044

4092 4523 4998 5524 6105

4133 4568 5049 5579 6166

4175 4614 5099 5636 6228

4217 4660 5151 5692 6291

4259 4707 5202 5749 6354

4302 4755 5255 5807 6418

4345 4802 5307 5866 6482

4389 4851 5361 5925 6548

4433 4899 5415 5984 6613

5 h 7 » 9

6680 7382 8159 9017 9965

6747 7457 8241 9107

6815 7532 8324 9199

7093 7839 8663 9574

7164 7918 8750 9671

7236 7997 8838 9768

7309 8078 8927 9866

•1.017

•1.027

6952 7684 8492 9385 •1 037

7022 7761 8577 9479

"1.005

6883 7607 8407 9291

•1.048

•1.058

•1.069

•1.080

•1.090

1. 101

1. 112

1.124

1. 135

1.146

1. 169

1. 181

1. 193

1.205

1 2

I

10 0

I0< X

Where x is large, sinh x = cosh x

»

H«*>

1.158

1093 1208 1335 1476

MATHEMATICAL TABLES

3

4

5

6

7

8

0300 1293 2260 3185 4053

0400 1391 2355 3275 4136

0500 1489 2449 3364 4219

0599 1586 2543 3452 4301

0699 1684 2636 3540 4382

0798 1781 2729 3627 4462

0898 1877 2821 3714 4542

ìflmẵlnäm

4621 5370 6044 6640 7163

4699 5441 6107 6696 7211

4777 5511 6169 6751 7259

4854 5581 6231 6805 7306

4930 5649 6291 6858 7352

5005 5717 6351 6911 7398

5080 5784 6411 6963 7443

5154 5850 6469 7014 7487

5227 5915 6527 7064 7531

5299 5980 6584 7114 7574

Ahvlh̉lợ-O

7616 8005 8337 8617 8854

7658 8041 8367 8643 8875

7699 8076 8397 8668 8896

7739 8110 8426 8692 8917

7779 6144 8455 8717 8937

7818 8178 8483 8741 8957

7857 8210 8511 8764 8977

7895 8243 8538 8787 8996

7932 8275 8565 8810 9015

7969 8306 8591 8832 9033

`00\IO~VI

9051 9217 9354 9468 9562

9069 9232 9366 9478 9571

9087 9246 9379 9488 9579

9104 9261 9391 9498 9587

9121 9275 9402 9508 9595

9138 9289 9414 9517 9603

9154 9302 9425 9527 9611

9170 9316 9436 9536 9618

9186 9329 9447 9545 9626

9201 9341 9458 9554 9633

IMIJN-Ó

9640 9705 9757 9801 9837

9647 9710 9762 9805 9840

9654 9716 9767 9809 9843

9661 9721 9771 9812 9846

9667 9727 9776 9816 9849

9674 9732 9780 9820 9852

9680 9737 9785 9823 9855

9687 9743 9789 9827 9858

9693 9748 9793 9830 9861

9699 9753 9797 9833 9863

Ồữklơvh́

9866 9890 9910 9926 9940

9869 9892 9912 9928 9941

9871 9895 9914 9929 9942

9874 9897 9915 9931 9943

9876 9899 9917 9932 9944

9879 9901 9919 9933 9945

9881 9903 9920 9935 9946

9884 9905 9922 9936 9947

9886 9906 9923 9937 9949

9888 9908 9925 9938 9950

Ißulh̉l-(D

9

2 0200 1194 2165 3095 3969

9951 9959 9967 9973 9978

9952 9960 9967 9973 9978

9952 9961 9968 9974 9979

9953 9962 9969 9974 9979

9954 9963 9969 9975 9979

9955 9963 9970 9975 9980

9956 9964 9971 9976 9980

9957 9965 9971 9976 9981

9958 9965 9972 9977 9981

9959 9966 9972 9977 9981

9982 9985 9988 9990 9992

9982 9985 9988 9990 9992

9982 9986 9988 9990 9992

9983 9986 9988 9991 9992

9983 9986 9989 9991 9992

9984 9986 9989 9991 9993

9984 9987 9989 9991 9993

9984 9987 9989 9991 9993

9984 9987 9990 9991 9993

9985 9988 9990 9992 9993

9993 9995 9996 9996 9997

9993 9995 9996 9996 9997

9994 9995 9996 9996 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9994 9995 9996 9997 9997

9998 9998 9998 9999 9999

9998 9998 9998 9999 9999

9998 9998 9998 9999 9999

9998 9998 9998 9999 9999

9998 9998 9998 9999 9999

9998 9998 9999 9999 9999

9998 9998 9999 9999 9999

9998 9998 9999 9999 9999

9998 9998 9999 9999 9999

9998 9998 9999 9999 9999

9999

9999

9999

9999

9999

9999

9999

9999

9999

9999

Ze

Zz-12. coth 2 =:

24!-I-'.

When

:is

u

Ì 3'

verylarlze,

tanh

=¦cot1l 1 :S

1

.

et

671' 2,

1

1

=

+

Ị

27-' 8

I

et

+

2

Et

1

-

cosh Z

~1†

r.-J

I

2

when zỉllm-¡e_ t́ßnh̉

sinh _.___

I

1

0

1

- I I _ỢẠẠ__

I' - ,,,.__ I c°t́hz=_L=ẾỂ-L_'1ị-|+ - rt tanh: slnhz tanl 1; =

e

U A vl Q


N

“

ẶWÑ-IQ

,

`

1 0100 1096 2070 3004 3885

Nůùẵlflh́lll

0000 0997 1974 2913 3799

tanh

J›UIN-Q

,

Hyperbolic Functions:

'0G`\4Ơ^V! .

Z

0

Table 8.

1-93 :I:

SI-:C. 1-2]

1-94

GENERAL Table 9.

X

0

1

0 0 1 2 3 4

+ 1.0000 + 0.9975 9900 9776 9604

1 . 0000 9970 9890 9761 9584

5 6 7 9

9385 9120 8812 8463 8075

1.0 1 2 ) 4


[Sec.

Bessel Functions: Jt>(x) and Jt(x)

1

Ji(x)

3

4

5

6

7

8

0 9999 9964 9879 9746 9564

9998 9958 9868 9730 9543

9996 9951 9857 9713 9522

9994 9944 9844 9696 9500

9991 9936 9832 9679 9478

9988 9928 9819 9661 9455

9984 9919 9805 9642 9432

9980 9910 9791 9623 9409

9360 9091 8779 8426 8034

9335 9062 8745 8388 7993

9310 9032 8711 8350 7952

9284 9002 8677 8312 7910

9258 8971 8642 8274 7868

9231 8940 8607 8235 7825

9204 8909 8572 8195 7783

9177 8877 8536 8156 7739

9149 8845 8500 8116 7696

7652 7196 6711 6201 5669

7608 7149 6661 6149 5614

7563 7101 6611 6096 5560

7519 7054 6561 6043 5505

7473 7006 6510 5990 5450

7428 6957 6459 5937 5395

7382 6909 6408 5884 5340

7336 6860 6356 5830 5285

7290 6810 6305 5777 5230

7243 6761 6253 5723 5174

5 6 7 8 9

5118 4554 3980 3400 2818

5062 4497 3922 3342 2760

5006 4440 3864 3284 2702

4950 4383 3806 3225 2644

4894 4325 3748 3167 2586

4838 4268 3690 3109 2528

4781 4210 3632 3051 2470

4725 4153 3574 2993 2412

4668 4095 3516 2934 2354

4611 4038 3458 2876 2297

2.0

2239 1666 1104 0555 + 0025

2181 1609 1048 0502 ♦0027

2124 1553 0993 0448 •0079

2066 1496 0937 0394 •0130

2009 1440 0882 0341 •0181

1951 1383 0827 0288 •0232

1894 1327 0773 0235 •0283

1837 1271 0718 0182 •0334

1780 1215 0664 0130 •0384

1723 1159 0609 0077 •0434

0533 1015 1469 1891 2280

0583 1062 1512 1932 2317

0632 1108 1556 1972 2354

0681 1154 1599 2012 2390

0729 1200 1641 2051 2426

0778 1245 1684 2090 2462

0826 1291 1726 2129 2497

0873 1336 1768 2167 2532

0921 1380 I80'J 2205 2566

2634 2951 3228 3465 3661

2668 2980 3253 3486 3678

2701 3009 3278 3507 3695

2733 3038 3303 3528 3711

2765 3066 3328 3548 3727

2797 3094 3351 3568 3743

2829 3122 3375 3587 3758

2860 3149 3398 3606 3773

2890 3176 3421 3625 3787

-4018

3815 3927 3997 4027 4015

3828 3936 4002 4027 4012

3841 3944 4007 4028 4008

3853 3953 4011 4027 4004

3865 3960 4014 4027 4000

3876 3967 4017 4026 3995

3887 3974 4020 4025 3990

3898 3981 4022 4023 3984

3908 3987 4024 4021 3978

-3971 -3887 -3766 -3610 -3423

3965 3876 3752 3593 3402

3958 3865 3737 3575 3381

3950 3854 3722 3557 3360

3942 3842 3707 3539 3339

3934 3831 3692 3520 3318

3925 3818 3676 3501 3296

3916 3806 3660 3482 3274

3907 3793 3644 3463 3251

3897 3779 3627 3443 3228

-3205 -2961 -2693 -2404 -2097

3182 2936 2665 2374 2066

3159 2910 2637 2344 2034

3135 2883 2609 2314 2002

-

3111 2857 2580 2283 1970

3087 2830 2551 2253 1938

3062 2803 2522 2222 1906

3037 2776 2493 2191 1874

3012 2749 2464 2160 1841

2987 2721 2434 2129 1809

1743

1710

1677

1644

1611

1578

,544

1511

1477

I

1 2 3 4 5 6 7 8 9

-0484 -0968

3 0 1 2

-2601 -2921

i

4 5 6 7 8 9

4 0 1 2 1 4 s (. 7 8 9

5.0

- 1424 -1850 -2243

-3202 -3443 -3643

- 3801 -3918 -3992 -4026

1776

2

DATA

2i(ll)i J'rix)

-Ji(x)

2<(21)'

2'(3!)>

9

mathematical

Sec. 1-2] Table 9.

t

I.I

1 2 3 4

J0(x)

and

1-95

J i(x).

(Continued)

0

1

2

3

4

S

6

7

8

9

+ 0 00000 0.04994 0 09950 0. 1483

I960

00500 05492 10442 1531 2007

01000 05989 1093 1580 2054

01500 06486 1142 1628 2101

02000 06983 1191 1676 2147

02499 07479 1240 1723 2194

02999 07974 1289 1771 2240

03498 08469 1338 1819 2286

03997 08964 1386 1866 2332

04495 09457 1435 1913 2377

2423 2867 3290 3688 4059

2468 2910 3331 3727 4095

2513 2953 3372 3765 4130

2558 2996 3412 3803 4165

2603 3039 3452 3840 4200

2647 3081 3492 3878 4234

2692 3124 3532 3915 4268

2736 3166 3572 3951 4302

2780 3207 361 1 3988 4335

2823 3249 3650 4024 4368

5 6 7 »


Bessel Functions:

tables

1.0 1 2 3 *

4401 4709 4983 5220

J4I9

4433 4738 5008 5242 5437

4465 4767 5033 5263 5455

4497 4795 5058 5284 5472

4528 4823 5082 5305 5488

4559 4850 5106 5325 5504

4590 4878 5130 5344 5520

4620 4904 5153 5364 5536

4650 4931 5176 5383 5551

4680 4957 5198 5401 5565

} 6 7 g 9

5579 5699 5778 5815 5812

5593 5709 5783 5817 5809

5607 5718 5788 5818 5806

5620 5727 5793 5818 5803

5632 5735 5798 5819 5799

5644 5743 5802 5818 5794

5656 5751 5805 5818 5790

5667 5758 5808 5817 5785

5678 5765 5811 5816 5779

5689 5772 5813 5814 5773

5767 5683 5560 5399 5202

5761 5672 5545 5381 5180

5754 5661 5530 5362 5158

5746 5650 5515 5343 5136

5738 5638 5500 5324 5113

5730 5626 5484 5305 5091

5721 5614 5468 5285 5067

5712 5601 5451 5265 5044

5703 5587 5434 5244 5020

5693 5574 5416 5223 4996

4971 4708 4416 4097 3754

4946 4680 4385 4064 3719

4921 4652 4354 4030 3683

4895 4624 4323 3997 3647

4870 4595 4291 3963 3611

4843 4566 4260 3928 3575

4817 4536 4228 3894 3538

4790 4507 4195 3859 3502

4763 4477 4163 3825 3465

4736 4446 4130 3790 3428

3391 3009 2613 2207 1792

3353 2970 2573 2165 1751

3316 2931 2533 2124 1709

3278 2892 2492 2083 1667

3240 2852 2452 2042 1625

3202 2813 241 1 2000 1583

3164 2773 2370 1959 1541

3125 2733 2330 1917 1500

3087 2694 2289 1876 1458

3048 2654 2248 1834 1416

1374 0955 0538 + 0128

1332 0913 0497 0088 0312

1290 0871 0456 0047 0351

1248 0829 04145 00069 0390

1206 07876 0373 "0033 0429

1164 0746 0332 •0074 0468

1122 0704 0291 *0 114 0507

1080 0663 0250 0546

1038 0621 0210 •0193 0584

0996 0580 0169 •0233 0622

0698 1069 1421 1751 2057

0736 1105 1455 1783 2086

0774 1141 1489 1814 2115

081 1 1177 1522 1845 2144

0849 1212 1556 1876 2173

0886 1247 1589 1907 2201

0923 1282 1622 1938 2229

0960 1317 1654 1968 2256

0996 1352 1687 1998 2284

-3147

2337 2589 2812 3003 3161

2364 2613 2832 3020 3175

2390 2636 2852 3037 3189

2416 2659 2872 3054 3202

2442 2682 2892 3070 3216

2467 2704 291 1 3086 3228

2492 2726 2930 3102 3241

2517 2748 2949 3117 3253

2541 2770 2967 3132 3264

-3276

3287

3298

3308

3318

3328

3337

3346

3355

3363

I.t 1 2 3 4

J

6 7 8 »

I.I 2

I

4

I

b 1 6

'

4.1 2 ) 4 5 6 7

i

50

-0272 -0660

-

- 1033 1386 -1719 -2028 -2311 -2566

- 2791 - 2985

S.M

/iW

-

X

T 0! • 1! /'•(*)

*•

-

+ 2'- It - 21 V • 2!

31

*0I53

GENERAL DATA

1-96 Table9. X

0

5.0 1 2 3 4

-0.

J

-0758 -0412 -0068

Jo(x) and

Ji (i).

1

(Continued)

1

2

3

4

5

6

7

8

9

1743 1410 1069 0723 0378

1710 1376 1034 0689 0343

1677 1342 1000 0654 0309

1644 1308 0965 0620 0274

1611 1274 0931 0585 0240

1578 1240 0896 0550 0205

1544 1206 0862 0516 0171

151 1 1171 0827 0481 0137

1477 1137 0793 0447 0103

6 7 6 9

+ 0270 0599 0917 1220

0034 0303 0632 0948 1250

0000 0336 0664 0979 1279

•0034 0370 0696 1010 1308

•0068 0403 0728 1040 1337

•0102 0436 0760 1071 1366

•0135 0469 0792 1101 1394

•0169 0501 0823 1131 1423

•0203 0534 0855 1161 1451

•0236 0567 0886 1191 1479

6.0

1506 1773 2017 2238 2433

1534 1798 2041 2259 2451

1561 1824 2064 2279 2469

1589 1849 2086 2299 2486

1616 1873 2109 2319 2504

1642 1898 2131 2339 2521

1669 1922 2153 2358 2537

1695 1947 2175 2377 2554

1721 1970 2196 2396 2570

1747 1994 2217 2415 2585

2601 2740 2851 2931 2981

2616 2753 2860 2937 2984

263 1 2765 2869 2943 2987

2646 2777 2878 2949 2990

2660 2788 2886 2955 2993

2674 2799 2895 2960 2995

2688 2810 2902 2965 2997

2702 2821 2910 2969 2998

2715 2831 2917 2973 2999

2728 2841 2924 2977 3000

4

3001 2991 2951 2882 2786

3001 2988 2945 2874 2775

3001 2985 2939 2865 2764

3001 2982 2933 2856 2752

3000 2978 2927 2847 2740

2999 2974 2920 2837 2728

2998 2970 2913 2828 2715

2997 2966 2906 2818 2703

2995 2961 2898 2807 2690

2993 2956 2890 2797 2677

5 6 7 8 9

2663 2516 2346 2154 1944

2650 2500 2327 2134 1922

2636 2484 2309 2113 1899

2622 2467 2290 2093 1877

2607 2451 2271 2072 1855

2593 2434 2252 2051 1832

2578 2416 2233 2030 1809

2563 2399 2214 2009 1786

2547 2381 2194 1987 1763

2532 2364 2174 1965 1740

8 0 1 2 3 4

1717 1475 1222 0960 0692

1693 1450 1196 0934 0665

1669 1425 1170 0907 0637

1645 1400 1144 0880 0610

1622 1375 1118 0853 0583

1597 1350 1092 0826 0556

1573 1325 1066 0800 0529

1549 1299 1039 0773 0501

1524 1274 1013 0745 0474

1500 1248 0987 0719 0447

5 6 7 8 9

0419 + 0146

0392 0119 0152 0419 0678

0365 0092 0179 0445 0704

0337 0065 0206 0471 0729

0310 0037 0233 0497 0754

0283 0010 0260 0524 0779

0255 0286 0549 0804

0228 •0044 0313 0575 0829

0201 •0071 0339 0601 0854

0928 1166 1389 1597 1786

0952 1189 1411 1616 1804

0976 1211 1432 1636 1821

1000 1234 1453 1655 1839

1024 1257 1474 1674 1856

1048 1279 1495 1694 1873

1072 1302 1516 1712 1890

1096 1324 1536 1731 1907

1119 1346 1556 1749 1923

1955 2104 2230 2332 2410

1971 2117 2241 2341 2417

1987 2131 2252 2350 2423

2002 2144 2263 2358 2429

2017 2157 2273 2366 2434

2032 2169 2284 2374 2440

2047 2182 2294 2382 2445

2061 2194 2304 2389 2450

2076 2206 2313 2396 2455

2475

2478

2481

2484

2486

2488

1 2

J

4

J

6 7 8 »


- 1776

1443 - 1103

Bessel Functions:

[SEC.

7.0 1 2

J

9.0 1 2 3 4 5 6 7 8 9

-0125 -0392 -0653 -0903 -1142

- 1367 - 1577 - 1768

- 1939 - 2090

- 2323 -2218

- 2403

10.0

- 2459

Zeros of

JaW

x=

12

2464

2468

2471

2.4048

5.5201

3 8.6537

4 11.7915

*00I7

5 14.9309

6 18.0711

7 21.2116

0174

*0I00 0366 0627 0879

8 24.3525

mathematical

Sec. 1-2] Table 9.

z

It

1 2 J 4

Jt>(x) and

1-97

Ji{x).

(Continued)

0

1

2

3

4

5

6

7

8

9

-0.3276 -3371

3287 3379 3436 3460 3451

3298 3386 3440 3461 3448

3308 3393 3444 3461 3445

3318 3400 3447 3461 3442

3328 3406 3450 3461 3438

3337 3412 3453 3460 3434

3346 3417 3455 3459 3430

3355 3423 3457 3457 3425

3363 3428 3458 3456 3420

-3432 -3460

-3453

5 6 7 8

-3414 -3343

3409 3335 3230 3096 2934

3403 3325 3218 3081 2917

3396 3316 3205 3065 2899

3390 3306 3192 3050 2881

3383 3296 3179 3034 2862

3376 3286 3166 3018 2844

3368 3275 3153 3002 2825

3360 3264 3139 2985 2806

3352 3253 3125 2969 2786

'. 0 1 ; 5

--2767 2559

2747 2537 2305 2055 1789

2727 2514 2281 2029 1762

2707 2492 2257 2003 1734

2686 2469 2232 1977 1707

2666 2446 2207 1950 1679

2645 2423 2182 1924 1651

2623 2400 2157 1897 1623

2602 2377 2132 1870 1595

2580 2353 2106 1843 1567

s 6 7 1 9

-1538

1510 1220 0923 0622 0319

1481 1191 0893 0592 0288

1453 1162 0863 0561 0258

1424 1132 0833 0531 0228

1395 1102 0803 0501 0198

1366 1073 0773 0470 0167

1337 1043 0743 0440 0137

1308 1013 0713 0410 0107

1279 0983 0682 0379 0077

*0I63 0457 0742 1016 1277

*0192 0486 0770 1043 1302

•0222 0514 0798 1070 1328

,


Bessel Functions:

tables

M

-3241 -3110 -2951

-2329

-

- 2081 1816 - 1250 -0953 -0652 -0349 -0047

I 2 } 4

+ 0251 0543 0826 1096

0017 0282 0572 0853 1123

•0013 0310 0601 0881 1149

"0043 0340 0629 0908 1175

♦0073 0369 0658 0935 1201

•0103 0398 0686 0963 1226

•0133 0428 0714 0990 1252

) t 7 8

1352 1592 1813 2014 2192

1377 1615 1834 2032 2208

1402 1638 1855 2051 2225

1426 1660 1875 2069 2241

1450 1683 1896 2088 2257

1475 1705 1916 2106 2272

1498 1727 1936 2123 2287

1522 1749 1956 2141 2303

1546 1771 1975 2158 2317

1569 1792 1994 2175 2332

2346 2476 2580 2657 2708

2360 2488 2589 2664 2711

2374 2499 2598 2670 2715

2388 2510 2606 2675 2718

2401 2521 2614 2681 2720

2414 2531 2622 2686 2723

2427 2542 2630 2691 2725

2440 2552 2637 2696 2727

2452 2561 2644 2700 2729

2464 2571 2651 2704 2730

'

2731 2728 2697 2641 2559

2732 2726 2693 2634 2550

2733 2724 2688 2626 2540

2733 2721 2683 2619 2530

2733 2719 2678 2611 2519

2733 2716 2672 2603 2509

2732 2713 2666 2595 2498

2731 2709 2660 2566 2487

2730 2705 2654 2577 2476

2729 2701 2648 2568 2465

1 2 3 *

2453 2324 2174 2004 1816

2441 2310 2158 1986 1797

2429 2296 2142 1968 1777

2417 2281 2125 1950 1757

2404 2267 2108 1931 1737

2391 2252 2091 1912 1716

2378 2237 2074 1893 1696

2365 2221 2057 1874 1675

2352 2206 2040 1855 1655

2338 2190 2022 1836 1634

7 ; 4

1613 1395 1166 0928 0684

1591 1373 1143 0904 0659

1570 1350 1119 0880 0634

1549 1328 1096 0856 0609

1527 1305 1072 0831 0585

1506 1282 1048 0807 0560

1484 1259 1025 0782 0535

1462 1236 10006 0758 0510

1440 1213 0977 0733 0485

1418 1190 0953 0708 0460

DO

+ 0435

0410

0385

0360

0334

0309

0284

0259

0234

0209

If rata!

J Ax)

'

It 2 3 4 j

i 7

J I

i -

1 3.8317

2 7.0156

3 10.1735

4 13.3237

5 16.4706

6 19.6159

7 22.7601

8 25.9037

1-98


X

:o.o

Bessel Functions: Jv(x) and ./,( x).

4

1

(Continued)

0

1

2

-0.2459

2464 2492 2495 2474 2428

2468 2493 2494 2470 2422

2471 2495 2493 2467 2416

2475 2496 2492 2463 2410

2478 2496 2490 2458 2403

2481 2497 2488 2454 2396

2484 2497 2485 2449 2389

2486 2497 2483 2444 2382

2488 2497 2480 2439 2374

3

6

5

7

8

9

i 2 3 4

-2490

5 6 7 8 9

-2366 -2276 -2164 -2032

2358 2266 2152 2018 1865

2350 2256 2140 2003 1848

2342 2245 2127 1989 1832

2333 2234 2114 1974 1815

2324 2223 2101 1959 1798

2315 2212 2087 1943 1781

2306 2200 2074 1928 1764

2296 2188 2060 1912 1747

2286 2177 2046 1897 1730

II. 0

-1712 -1528

1694 1508 1309 1099 0880

1676 1489 1289 1078 0858

1658 1470 1268 1056 0835

1640 1450 1247 1034 0813

1622 1430 1227 1012 0790

1603 1411 1206 0991 0767

1584 1391 1185 0969 0745

1566 1370 1163 0946 0722

1547 1350 1142 0924 0699

+ 0020 0250

0654 0423 0190 0043 0273

0631 0400 0167 0066 0296

0608 0376 0143 0089 0319

0585 0353 0120 0112 0342

0562 0330 0097 0135 0364

0539 0307 0073 0159 0387

0516 0283 0050 0182 0410

0493 0260 0027 0205 0432

0469 0237 0004 0228 0455

12.0 1 2 3 4

0477 0697 0908 1108 1296

0499 0718 0928 1127 1314

0521 0740 0949 1147 1331

0544 0761 0969 1166 1349

0566 0782 0989 1185 1367

0588 0803 1009 1203 1384

0610 0824 1029 1222 1401

0632 0845 1049 1241 1418

0653 0866 1069 1259 1435

0675 0887 1088 1277 1452

5 6 7 8 9

1469 1626 1766 1887 1988

1485 1641 1779 1898 1997

1502 1655 1792 1909 2006

1518 1670 1804 1920 2015

1534 1684 1817 1930 2023

1550 1698 1829 1940 2031

1565 1712 1841 1950 2039

1581 1726 1853 I960 2047

1596 1739 1864 1970 2055

1611 1753 1876 1979 2062

13.0 1 2 3 4

2069 2129 2167 2183 2177

2076 2134 2169 2183 2175

2083 2138 2172 2184 2173

2089 2143 2174 2184 2171

2096 2147 2176 2183 2169

2102 2151 2178 2183 2166

2108 2154 2179 2182 2163

2113 2158 2180 2181 2160

2119 2161 2182 2180 2157

2124 2164 2182 2179 2154

5 6 7 8 9

2150 2101 2032 1943 1836

2146 2095 2024 1933 1824

2142 2089 2016 1923 1812

2138 2083 2008 1913 1800

2133 2076 1999 1903 1788

2128 2069 1990 1892 1775

2123 2062 1981 1881 1763

2118 2055 1972 1870 1750

2113 2048 1963 1859 1737

2107 2040 1953 1847 1724

14.0 1 2 3 4

1711 1570 1414 1245 1065

1697 1555 1397 1227 1046

1684 1539 1381 1210 1028

1670 1524 1364 1192 1009

1656 1509 1348 1174 0990

1642 1493 1331 1156 0971

1628 1478 1314 1138 0952

1613 1462 1297 1120 0933

1599 1446 1280 1102 0914

1584 1430 1262 1083 0895

5 6 7 8 9

0875 0679 0476 0271 + 0064

0856 0659 0456 0250 0043

0837 0639 0436 0229 0023

0817 0618 0415 0209 + 0002

0798 0598 0394 0188

0778 0578 0374 0167

0758 0558 0353 0147

0738 0538 0333 0126

0719 0517 0312 0105

0699 0497 0291 0085

1 2 3 4 5 6 7 8 9


[Sec.

For

-2496 -2477 -2434

- 1881 — 1330

-1121 -0902 -0677

-0446 -0213

i>

15 ■fi(z)

■v/—

[ cos (z

VI

-

-0019

-)

+ — sin

0.7979

-0039

(x

-0060

- l) 1 . error

0.7854

-0081

< 0.000 1

-0101

-0122

Sec. 1-2]

mathematical Table 9.

1-99

J0(x) and ./i(x).

(Continued)

/id)

X

0

1

2

3

4

5

6

7

10.0 4 2 3 4

+ 0.0435

0410 0159 0091 0338 0578

0385 0134 0116 0362 0602

0360 0109 0141 0386 0626

0334 0084 0165 041 1 0649

0309 0059 0190 0435 0673

0284 0034 0215 0459 0696

0259 0009 0240 0483 0719

5 6

-0789

1422 — 1603

0811 1034 1244 1441 1621

0834 1056 1265 1459 1638

0857 1077 1285 1478 1655

0879 1099 1305 1496 1671

0902 1120 1325 1515 1688

0924 1141 1344 1533 1704

1768 1913 2039 2143 2225

1783 1927 2050 2152 2231

1798 1940 2061 2161 2238

1814 1953 2072 2169 2245

1828 1966 2083 2178 2251

1843 1979 2093 2186 2257

-2284 2320 2333 2323 2290

2288 2322 2333 2321 2285

2293 2324 2333 2318 2281

2297 2326 2332 2315 2276

2301 2328 2332 2312 2270

2234 2157 2060 — 1943 — 1807

2228 2149 2049 1930 1793

2221 2140 2038 1917 1778

2214 2130 2027 1904 1763

1655 — 1487 — 1307 1114 — 0912

1639 1470 1288 1095 0892

1623 1452 1269 1075 0871

— — — —

0703 0489 0271 0052 0166

0682 0467 0249 0030 0188

0380 0590 0791 0984 1165

14.0 1 2 3

5 6 7 g

I

8 9 1 10

I

2

J

4 S 7 9


Bessel Functions:

tables

12.0 2 3 4 5 6 7 9 13.0 j 2 3 4

For x

9

0234 0264 0507 0742

0209 •0041 0289 0531 0766

0946 1162 1364 1551 1720

0968 1183 1383 1568 1736

0990 1203 1403 1586 1752

1857 1991 2104 2194 2263

1872 2003 2114 2202 2268

1886 2015 2123 2210 2274

1900 2027 2133 2217 2279

2305 2329 2331 2309 2265

2308 2331 2330 2306 2259

2312 2332 2328 2302 2253

2315 2332 2327 2298 2247

2317 2333 2325 2294 2241

2206 2121 2015 1891 1748

2199 2111 2004 1877 1733

2191 2101 1992 1863 1718

2183 2091 1980 1850 1702

2175 2081 1968 1836 1687

2166 2070 1955 1821 1671

1606 1435 1250 1055 0850

1590 1417 1231 1035 0830

1573 1399 1212 1014 0809

1556 1380 1192 0994 0788

1539 1362 1173 0974 0767

1522 1344 1154 0954 0746

1505 1325 1134 0933 0724

0661 0445 0227 0008 0209

0639 0423 0205 ♦0014 0231

0618 0402 0183 ♦0036 0252

0596 0380 0161 •0057 0274

0575 0358 0139 ♦0079 0295

0553 0336 0117 •0101 0317

0532 0314 0096 •0123 0338

0510 0293 0074 •0144 0359

0402 0610 081 1 1003 1183

0423 0631 0831 1021 1200

0444 0651 0850 1040 1217

0465 0671 0870 1058 1234

0486 0692 0889 1076 1251

0507 0712 0908 1094 1268

0528 0732 0927 1112 1285

0548 0752 0946 1130 1301

0569 0772 0965 1148 1318

1334 1488 1626 1747 1850

1350 1502 1639 1758 I860

1366 1517 1652 1769 1869

1382 1531 1664 1780 1878

1397 1545 1677 1791 1886

1413 1559 1689 1801 1895

1428 1573 1701 181 1 1903

1443 1586 1713 1821 1911

1458 1600 1724 1831 1919

1473 1613 1736 1841 1927

1934 1999 2043 2066 2069

1942 2004 2046 2067 2068

1949 2009 2049 2068 2067

1956 2014 2052 2069 2066

1962 2019 2054 2070 2064

1969 2023 2057 2070 2062

1975 2027 2059 2070 2061

1981 2031 2061 2070 2058

1987 2035 2063 2070 2056

1993 2039 2065 2069 2054

+ 0184

-0066 -0313 -0555

-

- 1012 1224

--

----

--

-

+

fo 7 8 9

9

8

+

> 15

JiW " 'J—

[«in

(i -

^)

-

= 0.7979

^

cos

^

(r

-

-

^) ].

2.3562

error

< 0,0001

*00I6

1-100


1

Yt(x) and Y,(x)

2

3

4

5

6

7

8

3.005 1.473 1.049 7847 5884

2.564 1.416 1.0175 7627 5712

2.305 1.364 0.9877 7414 5542

2.122 1.316 0.9591 7206 5377

1.979 1.271 0.9316 7003 5214

1.863 1. 228 0.9050 6806 5055

1.764 1. 189 0.8794 6613 4898

1.678 1. 151 0.8546 6424 4745

0.8306

+ 0056

4299 2960 1797 0771 0143

4156 2837 1689 0675 0229

4015 2715 1582 0580 0314

3876 2595 1476 0486 0398

3739 2476 1372 0393 0481

3604 2359 1269 0301 0563

3472 2244 1167 0210 0644

3341 2130 1066 0120 0725

3212 2018 0966 0032 0804

0883 1622 2281 2865 3379

0960 1691 2343 2920 3427

1037 1760 2404 2974 3473

1113 1828 2464 3027 3520

1188 1895 2523 3079 3565

1262 1961 2582 3131 3610

1336 2026 2640 3162 3654

1409 2091 2698 3232 3698

1480 2155 2754 3282 3741

1551 2218 2810 3331 3783

3824 4204 4520 4774 4968

3865 4239 4548 4796 4984

3906 4273 4576 4818 5000

3945 4306 4603 4839 5015

3984 4338 4629 4859 5029

4022 4370 4655 4879 5043

4060 4401 4680 4898 5056

4097 4432 4705 4916 5069

4133 4462 4728 4934 5081

4169 4491 4752 4951 5093

1 2 3 4

5104 5183 5208 5181 5104

5114 5188 5207 5175 5094

5124 5192 5207 5169 5083

5133 5196 5205 5163 5072

5142 5199 5203 5156 5060

5150 5202 5201 5148 5048

5158 5204 5198 5141 5036

5165 5206 5194 5132 5022

5172 5207 5190 5123 5009

5177 5208 5186 5114 4995

5 6 7 8 9

4981 4813 4605 4359 4079

4966 4794 4582 4333 4049

4951 4775 4559 4306 4019

4935 4755 4535 4279 3989

4919 4735 4511 4251 3958

4902 4714 4487 4223 3927

4885 4693 4462 4195 3896

4868 4672 4437 4167 3865

4650 4650 4411 4138 3833

4832 4628 4385 4109 3801

3.0 1 2 3 4

3769 3431 3071 2691 2296

3736 3396 3033 2652 2256

3703 3361 2996 2613 2216

3670 3325 2958 2574 2175

3637 3289 2921 2535 2135

3603 3253 2883 2495 2094

3569 3217 2845 2456 2054

3535 3181 2807 2416 2013

3500 3144 2768 2376 1972

3466 3108 2730 2336 1931

5 6 7 B 9

1890 1477 1061 0645 + 0234

1849 1436 1019 0604 0193

1808 1394 0977 0562 0152

1767 1352 0936 0521 0112

1726 1311 0894 0480 0071

1684 1269 0853 0439 0031

1643 1227 081 1 0397 •0009

1602 1186 0769 0356 *0050

1560 1144 0728 0315 *0090

1519 1102 0686 0275 •0130

1 2

-0169 -0561 -0938

4

-1633

0209 0599 0974 1331 1666

0249 0638 101 1 1365 1698

0288 0676 1047 1400 1730

0328 0714 1083 1434 1762

0367 0751 1119 1467 1793

0406 0789 1155 1501 1825

0445 0826 1191 1535 1856

0484 0864 1226 1568 1886

0522 0901 1261 1601 1917

1977 2262 2518 2744 2939

2007 2289 2542 2765 2956

2036 2315 2566 2786 2973

2065 2342 2589 2806 2990

2094 2368 2612 2826 3007

2123 2394 2635 2845 3023

2151 2419 2658 2865 3039

2179 2444 2680 2884 3055

2207 2469 2702 2902 3070

3100

3114

3128

3142

3155

3168

3180

3193

3204

M

0.0 1 2

)

0

— 00

-1.534

- 1.081

- 0.8073

4

-6060

5 D 7 B »

-4445 -3085 -1907 -0868

1.0 1 2 3

* 5

(

7 8 9


Bessel Functions:

[Sec.

2.0

4.0

i

5 6 7 8 9

5.0

- 1296 - 1947 - 2235 - 2494 2723 -2921

- 3085

Linear interpolation is inaccurate for low values of the argument. use the auxiliary functions. Table 13, J"o(x) = -3/i(x)

9

1.602 15

I.J

6240 4594

For greater accuracy in this range,

SEC. 1-2J

MATHEMATICAL TABLES Table 10.

X

Ko(x) and Y^x). Kid)

(Continued)

0

1

2

3

4

5

6

7

8

9

— so — 6.459

31.86 5.409

3.042

1 .781

63.68 5 886 3. 176 2.227 1.743

2. 165 1.708

21.26 5.007 2.919 2. 107 1.673

15.96 4.662 2.807 2.052 1.641

12.79 4.364 2.704 2.000 1.610

10.68 4. 103 2.609 1.952 1.580

9. 167 3.873 2.521 1.906 1.551

8.038 3.670 2.440 1.862 1.523

7.160 3.487 2.364 1.820 1.497

1.471 1 . 260 1 . 103 9781 8731

1.447 1.243 1.090 9669 8634

1.423 1.226 1.076 9558 8539

1.401 1.209 1.063 9449 8444

1.378 1. 193 1.050 9342 8351

1.357 1. 177 1.038 9236 8258

1.337 1. 161 1.025 9132 8167

1.317 1. 146 1.013 9030 8077

1.297 1. 132 1.0013 8929 7988

1.279 1.117 •9896 8829 7900

-6211 -5485 -4791

7726 6902 6137 5415 4724

7640 6823 6063 5344 4656

7555 6745 5990 5274 4589

7471 6667 5916 5204 4521

7388 6590 5844 5135 4454

7305 6513 5771 5066 4388

7223 6437 5699 4997 432!

7142 6361 5628 4928 4255

7061 6286 5556 4860 4189

4123 3476 2847 2237 — 1644

4057 3412 2785 2177 1586

3992 3349 2724 2117 1528

3927 3285 2662 2057 1470

3862 3222 2601 1997 1412

3797 3159 2540 1938 1355

3732 3096 2479 1879 1297

3668 3034 2418 1820 1240

3604 2972 2357 1761 1184

3540 2909 2297 1702 1127

-

1070 0517 0015 0523 1005

1014 0463 0067 0572 1052

0958 0409 0118 0621 1098

0902 0355 0170 0670 1144

0846 0301 0221 0719 1190

0791 0248 0272 0767 1236

0736 0195 0323 0815 1281

0681 0142 0373 0863 1326

0626 0090 0423 091 1 1371

0571 0037 0473 0958 1415

1459 1884 2276 2635 2959

1503 1924 2314 2669 2990

1547 1965 2351 2703 3020

1590 2005 2388 2736 3050

1633 2045 2424 2769 3079

1675 2084 2460 2802 3108

1718 2123 2496 2834 3136

1760 2162 2531 2866 3164

1801 2200 2566 2897 3192

1843 2239 2601 2929 3220

3247 3496 3707 3879 4010

3273 3519 3726 3893 4021

3300 3542 3745 3908 4032

3326 3564 3763 3922 4042

3351 3585 3780 3936 4052

3376 3607 3798 3949 4061

3401 3627 3815 3962 4070

3425 3648 3831 3975 4079

3449 3668 3847 3987 4087

3473 3688 3863 3999 4095

''

4102 4154 4167 4141 4078

4109 4157 4166 4137 4070

4115 4160 4165 4132 4061

4122 4162 4163 4126 4052

4127 4164 4161 4120 4043

4133 4165 4159 4114 4033

4138 4166 4156 4108 4023

4142 4167 4153 4101 4013

4147 4167 4149 4094 4002

4150 4167 4145 4086 3991

4 0 [ 2 3 4

3979 3846 3680 3484 3260

3967 3831 3662 3463 3236

3955 3815 3643 3441 3212

3943 3800 3624 3420 3187

3930 3783 3605 3397 3163

3917 3767 3586 3375 3138

3903 3750 3566 3353 3113

3889 3733 3546 3330 3087

3875 3716 3525 3307 3062

3861 3698 3505 3283 3036

5 6 7 3 9

3010 2737 2445 2136 1812

2984 2709 2415 2104 1780

2957 2680 2384 2072 1746

2930 2652 2354 2040 1713

2904 2623 2323 2008 1680

2876 2594 2292 1976 1647

2849 2564 2261 1943 1613

2821 2535 2230 191 1 1580

2794 2505 2199 1878 1546

2766 2475 2167 1845 1512

+1479

1445

1411

1377

1343

1309

1275

1240

1206

1172

0 0 1 2 i 3 4 5 6 7 S 9

—3.324 — 2.293

-

--

-0.

-

-

1.0

-7812

1 2 3 4

6981

---

6 7 8 9


Bessel Functions:

1-101

-

2 0

+

2 3 4 5 6 7

a •

3.0 1 2 3

*

s 6 7

1

Linear interpolation is inaccurate for low values of the argument. w the auxiliary functions, Table 13. Yi(x)

- - J".(x)

For greater accuracy in this range,

1-102

GENERAL Table 10.

DATA

[Sec.

Bessel Functions: Yo(x) and Yt(x).

(Continued)

Y^x)

X

0

2

3

4

5

6

7

8

9

I 2 3 4

-0.3085 -3216 -3313 -3374 -3402

3100 3227 3320 3379 3403

3114 3238 3328 3383 3403

3128 3249 3335 3386 3403

3142 3259 3341 3389 3403

3155 3269 3348 3392 3402

3168 3278 3354 3395 3402

3180 3287 3359 3397 3400

3193 3296 3365 3399 3399

3204 3304 3370 3400 3397

5 6 7 8 9

-3395 -3354 -3282 -3177 -3044

3392 3349 3273 3165 3029

3389 3342 3263 3153 3013

3386 3336 3254 3140 2998

3383 3329 3244 3127 2982

3379 3322 3233 3114 2966

3375 3315 3223 3101 2950

3370 3307 3212 3087 2933

3365 3299 3201 3073 2916

3360 3290 3189 3058 2899

6.0

5.0


1

1

-2882

1 2 3 4

-2694 -2483 -2251 -1999

2864 2674 2461 2226 1973

2846 2654 2438 2202 1947

2828 2633 2415 2177 1921

2810 2613 2393 2152 1894

2791 2592 2369 2127 1868

2772 2570 2346 2102 1841

2753 2549 2322 2077 1814

2734 2527 2299 2051 1787

2714 2505 2275 2025 1760

5 6 7 e 9

-1732 -1452 -1162

1705 1424 1132 0834 0532

1677 1395 1103 0804 0502

1650 1366 1073 0774 0472

1622 1337 1044 0744 0441

1594 1308 1014 0714 0411

1566 1279 0984 0684 0381

1538 1250 0954 0653 0350

1509 1221 0924 0623 0320

1481 1191 0894 0593 0290

7.0

-0259

I 2 3 4

+ 0042 0339 0628 0907

0229 0072 0368 0656 0934

0199 0102 0397 0684 0961

0169 0131 0426 0713 0988

0139 0161 0455 0741 1015

0108 0191 0484 0769 1042

0078 0221 0513 0797 1068

0048 0250 0542 0824 1095

0018 0280 0571 0852 1121

•0012 0309 0599 0879 1147

5 6 7 8 9

1173 1424 1658 1872 2065

1199 1448 1680 1893 2083

1225 1472 1702 1913 2101

1250 1496 1724 1932 2119

1276 1520 1746 1952 2136

1301 1543 1768 1972 2153

1326 1567 1789 1991 2170

1351 1590 1810 2010 2187

1375 1613 1831 2028 2203

1400 1635 1852 2047 2219

8.0 I 2 3 4

2235 2381 2501 2595 2662

2251 2394 2512 2603 2667

2266 2407 2522 2611 2672

2282 2420 2532 2618 2677

2296 2432 2542 2625 2681

2311 2444 2551 2632 2686

2326 2456 2561 2639 2689

2340 2468 2570 2645 2693

2354 2479 2578 2651 2696

2367 2490 2587 2657 2699

5 6 7 8 9

2702 2715 2700 2659 2592

2705 2714 2697 2653 2583

2707 2714 2694 2647 2575

2709 2713 2690 2641 2566

2710 2712 2687 2635 2558

2712 2711 2683 2628 2549

2713 2709 2678 2621 2539

2714 2707 2674 2614 2530

2714 2705 2669 2607 2520

2715 2703 2664 2599 2510

9.0 | 2 3 4

2499 2383 2245 2086 1907

2489 2371 2230 2069 1889

2478 2357 2215 2052 1870

2467 2344 2199 2034 1851

2456 2331 2184 2017 1831

2444 2317 2168 1999 1812

2433 2303 2152 1981 1792

2421 2289 2136 1963 1772

2408 2274 2119 1945 1752

2396 2260 2103 1926 1732

5 6 7 8 9

1712 1502 1279 1045 0804

1692 1480 1256 1021 0779

1671 1458 1233 0998 0755

1650 1436 1210 0974 0730

1630 1414 1186 0949 0705

1609 1392 1163 0925 0681

1588 1369 1140 0901 0656

1566 1347 1116 0877 0631

1545 1324 1093 0853 0606

1523 1302 1069 0828 0582

0557

0532

0507

0482

0457

0432

0407

0382

0357

0332

10.0 Zeros of z =

-0864 -0563

YM

1 0.8936

2 3.9577

3 7.0861

4 10.2223

5 13.3611

6 16.5009

7 19.6413

22.7820

Sec. 1-2]

mathematical Table 10.

r

0

1

2

3

5 0 | 2

+ 0. 1479

1445 1103 0757 041 1 0067

1411 1069 0723 0376 0033

1377 1034 0688 0342

1-103

l'o(i) and Yi(x). 4

6

5

(Continued) 7

8

9

1343 1000 0653 0307 •0035

1309 0965 0619 0273 »0069

1275 0930 0584 0238 •0103

1240 0896 0549 0204 •0137

1206 0861 0515 0170 •0170

1172 0827 0480 0136 •0204

«

1137 0792 0445 + 0101

s

-0238

7 8 »

0568 — 0887 — 1192 1481

-

0271 0601 0918 1222 1509

0304 0633 0949 1251 1536

0338 0665 0980 1281 1564

0371 0697 101 1 1310 1591

0404 0729 1042 1339 1618

0437 0761 1072 1368 1645

0470 0793 1102 1396 1671

0503 0824 1133 1425 1698

0535 0856 1163 1453 1724

1750 — 1998 — 2223 2422 2596

1776 2022 2244 2441 2611

1801 2045 2265 2459 2627

1827 2068 2285 2477 2642

1852 2091 2306 2495 2657

1877 2114 2326 2512 2672

1902 2136 2346 2530 2686

1926 2158 2365 2547 2700

1950 2180 2385 2563 2714

1974 2201 2404 2580 2728

2741 2857 2945 3002 — 3029

2754 2868 2952 3006 3030

2767 2877 2958 3010 3031

2779 2887 2965 3013 3032

2791 2896 2971 3016 3032

2803 2905 2977 3019 3032

2814 2913 2983 3022 3031

2826 2922 2988 3024 3031

2836 2930 2993 3026 3030

2847 2937 2997 3028 3028

3025 2990 2927 2836 2718

3023 2985 2919 2825 2705

3020 2980 291 1 2814 2692

3017 2974 2902 2803 2678

3014 2968 2893 2792 2664

3011 2962 2885 2780 2650

3007 2955 2875 2768 2636

3003 2949 2866 2756 2621

2999 2942 2856 2744 2606

)

6.0 2 3 ♦ 5 6 7 8 9


Bessel Functions:

tables

7 0

-

----

-3027 2995 — 2934 — 2846 2731

*000l

2 3

-

5

— 2591 2428 2243 2039 1817

2576 2410 2224 2017 1794

2560 2393 2204 1996 1771

2545 2375 2184 1974 1748

2529 2357 2164 1952 1724

2512 2338 2143 1930 1701

2496 2320 2123 1908 1677

2479 2301 2102 1885 1653

2462 2282 2081 1863 1629

2445 2263 2060 1840 1605

1581 1331 — 1072 — 0806 — 0535

1556 1306 1046 0779 0508

1532 1280 1020 0752 0460

1507 1255 0993 0725 0453

1482 1229 0967 0698 0426

1457 1203 0940 0671 0398

1432 1177 0913 0644 0371

1407 1151 0887 0617 0344

1382 1125 0860 0589 0316

1357 1099 0833 0562 0289

— 0262 001 1 0280 0544 0799

0234 0038 0307 0569 0824

0207 0065 0333 0595 0849

0180 0092 0360 0621 0873

0152 01 19 0386 0647 0898

0125 0146 0413 0672 0922

0098 0173 0439 0698 0947

0071 0200 0465 0723 0971

0043 0227 0491 0748 0995

0016 0253 0518 0774 1019

1043 1275 1491 1691 1871

1067 1297 1512 1710 1888

1091 1319 1532 1728 1905

1114 1341 1553 1747 1922

1137 1363 1573 1765 1938

1161 1385 1593 1783 1954

1184 1406 1613 1801 1970

1207 1428 1633 1819 1986

1229 1449 1652 1837 2001

1252 1470 1671 1854 2017

2032 2171 2287 2379 2447

2047 2183 2297 2387 2452

2061 2196 2307 2394 2458

2076 2208 2317 2402 2463

2090 2220 2326 2409 2467

2104 2232 2336 2416 2472

2118 2243 2345 2423 2476

2131 2254 2354 2429 2480

2145 2265 2362 2435 2484

2158 2276 2371 2441 2487

2490

2493

2496

2498

2500

2502

2504

2506

2507

2508

2.1971

5.4297

8.5960

7 g 9 8 0 2 3

5

---

--

+

7 g 9 9 0 2 3 4 5 (, 7 9

10.0

+

11.7492

14.8974

18.0434

21.1881

24.3319

GENERAL

1-104 Table 10.


X

0

1

DATA

[Sec.

Bessel Functions: Yo(x) and Yi(x). rod) 2

4

3

5

1

{Continued)

6

7

8

9

10.0 + 0.0557 1 0307 2 + 0056 3 -0193 4 -0437

0532 0281 0031 0218 0462

0507 0256 0006 0242 0486

0482 0231 •0019 0267 0510

0457 0206 •0044 0291 0534

0432 0181 •0069 0316 0557

0407 0156 •0094 0340 0581

0382 0131 •0119 0365 0605

0357 0106 •0143 0389 0628

5 6 7 8 9

-0675 -0904 -1122 -1326 -1516

0699 0926 1143 1346 1534

0722 0949 1164 1366 1552

0745 0971 1185 1385 1569

0768 0993 1205 1404 1587

0791 1015 1226 1423 1604

0814 1036 1246 1442 1622

0837 1058 1267 1461 1639

0859 1079 1287 1479 1655

0882 110 1 1307 1498 1672

11.0 1 2 3 4

-1608

-2183

1705 1857 1990 2101 2191

1721 1871 2002 2111 2199

1737 1885 2014 2121 2206

1752 1899 2025 2130 2213

1768 1913 2037 2140 2220

1783 1926 2048 2149 2227

1798 1939 2059 2158 2234

1813 1952 2070 2166 2240

1828 1965 2081 2175 2246

5 6 7 8 9

-2252 -2299 -2322 -2322 -2298

2258 2302 2323 2320 2295

2263 2305 2324 2319 2291

2269 2308 2324 2317 2287

2274 2311 2325 2315 2283

2278 2313 2325 2313 2278

2283 2315 2324 2310 2273

2287 2317 2324 2308 2269

2291 2319 2324 2305 2263

2295 2321 2323 2302 2258

12.0 1 2 3 4

-2252 -2184 -2095 -1986 -1858

2247 2176 2085 1974 1844

2241 2168 2075 1962 1830

2234 2160 2064 1949 1816

2228 2151 2054 1937 1802

2221 2142 2043 1924 1787

2214 2133 2032 1911 1772

2207 2124 2021 1898 1758

2200 2115 2009 1885 1743

2192 2105 1998 1871 1727

t

-1712 -1375 -1187 -0989

1697 1534 1357 1168 0968

1681 1517 1338 1148 0948

1665 1499 1320 1129 0927

1649 1482 1301 1109 0907

1633 1464 1282 1089 0886

1617 1447 1264 1069 0866

1601 1429 1245 1049 0845

1584 1411 1226 1029 0824

IS67

6 7 8 9

1393 1206 1009 0803

0740 0526 0309 0090 0128

0719 0505 0287 0068 0150

0698 0483 0265 0046 0172

0676 0461 0243 0024 0193

0655 0439 0221 0002 0215

0634 0418 0199 •0019 0236

0612 0396 0177 •0041 0258

0591 0374 0156 •0063 0279

-1843

-1977 -2091

-1551

0332 0081 •0168 0413 0652

13.0 1 2 3 4

-0782

+ 0085

0761 0548 0331 0112 0107

5 6 7 8 9

0301 0512 0717 0913 1099

0322 0533 0737 0932 1117

0343 0554 0757 0951 1134

0365 0574 0777 0970 1152

0386 0595 0796 0989 1169

0407 0615 0816 1007 1187

0428 0636 0836 1026 1204

0449 0656 0855 1044 1221

0470 0677 0875 1062 1238

0491 0697 0894 1081 1255

14.0 1 2 3 4

1272 1431 1575 1703 1812

1289 1446 1589 1715 1822

1305 1461 1602 1726 1832

1321 1476 1615 1738 1842

1337 1491 1628 1749 1851

1353 1505 1641 1760 I860

1369 1520 1654 1771 1869

1385 1534 1666 1781 1878

1401 1548 1679 1792 1886

1416 1562 1691 1802 1895

5 6 7 8

1903 1974 2025 2056 4 2065

1911 1980 2029 2058 2065

1919 1986 2033 2059 2065

1926 1992 2036 2061 2064

1934 1997 2040 2062 2064

1941 2002 2043 2063 2063

1948 2007 2046 2064 2061

1955 2012 2049 2065 2060

1962 2017 2051 2065 2058

1968 2021 2054 2065 2057

'

-0569 -0352

-0134

For x > 15

Y,(r)

-

\j-2- [sin (x

^.

-

-J - — 0.7979

cos

(i -

M J.

0.7854

error < 0.0001

mathematical tables

Sec. 1-2] Table 10.

r

Bessel Functions: Y0(x) and Y,(x).

(Continued)

0

1

2

3

4

5

6

7

8

+ 0.2490 2508 2502 2471 2416

2493 2509 2500 2466 2409

2496 2509 2498 2462 2402

2498 2509 2495 2457 2394

2500 2509 2492 2451 2387

2502 2508 2489 2446 2379

2504 2507 2486 2440 2371

2506 2506 2483 2435 2363

2507 2505 2479 2428 2355

2508 2504 2475 2422 2346

2337 2236 2114 1973 1813

2328 2225 2101 1958 1796

2319 2214 2088 1942 1779

2309 2202 2074 1927 1762

2299 2190 2060 1911 1745

2289 2178 2046 1895 1727

2279 2166 2032 1879 1709

2269 2153 2017 1863 1692

2258 2140 2003 1846 1674

2247 2128 1988 1830 1655

1637 1446 1243 1029 0807

1619 1427 1222 1008 0785

1600 1407 1201 0986 0762

1581 1387 1180 0964 0740

1562 1366 1159 0941 0717

1543 1346 1137 0919 0694

1524 1326 1116 0897 0671

1505 1305 1095 0875 0648

I486 1285 1073 0852 0625

1466 1264 1051 0830 0602

0579 0348 0114

-0118 -0347

0556 0324 0091 0141 0370

0533 0301 0068 0164 0392

0510 0278 0045 0187 0415

0487 0254 0021 0210 0437

0464 0231 •0002 0233 0460

0441 0208 *0025 0256 0482

0417 0184 •0048 0279 0505

0394 0161 •0072 0302 0527

0371 0138 •0095 0324 0549

-0571 -0787 -0994 -1189 -1371

0593 0809 1014 1208 1389

0615 0830 1034 1227 1406

0637 0851 1054 1246 1423

0659 0871 1074 1264 1440

0681 0892 1093 1282 1457

0702 0913 1113 1300 1474

0723 0933 1132 1318 1490

0745 0954 1151 1336 1506

0766 0974 1171 1354 IS22

-1538 -1689 -1821 -1935 -2028

1554 1703 1834 1945 2036

1570 1717 1846 1955 2044

1585 1730 1857 1965 2052

1601 1744 1869 1975 2060

1616 1757 1880 1984 2067

1631 1771 1892 1993 2074

1645 1783 1903 2002 2081

1660 1796 1914 2011 2088

1675 1809 1924 2020 2095

-2101 -2152 -2182 -2190 -2176

2107 2156 2183 2189 2173

2113 2160 2185 2188 2170

2118 2163 2186 2188 2167

2124 2167 2187 2187 2164

2129 2170 2188 2185 2161

2134 2172 2189 2184 2157

2139 2175 2189 2182 2153

2144 2178 2190 2180 2149

2148 2180 2190 2178 2145

-2140 -2084

2136 2077 1999 1901 1785

2131 2070 1990 1890 1773

2126 2063 1981 1879 1760

2120 2056 1971 1868 1747

2115 2048 1962 1857 1734

2109 2040 1952 1845 1721

2103 2032 1942 1834 1707

2097 2024 1932 1822 1694

2090 2016 1922 1810 1680

-1666 -1520 -1359 -1186

1652 1504 1342 1168 0984

1638 1489 1325 1150 0965

1624 1473 1308 1132 0946

1610 1457 1291 1114 0927

1595 1441 1274 1096 0907

1580 1425 1257 1077 0888

1565 1409 1239 1059 0869

1550 1392 1222 1040 0849

1535 1376 1204 1021 0830

5 -0810 6 1 -0612 7 -0408 8 -0202

0791 0591 0387 0181 0026

0771 0571 0367 0160 0047

0751 0551 0346 0140 0067

0732 0531 0326 0119 0088

0712 0510 0305 0098 0108

0692 0490 0284 0077 0129

0672 0469 0264 0057 0149

0652 0449 0243 0036 0170

0632 0428 0222 0015 0190

10.0

II 0 1


1-105

12 0 1 2

13.0 1

-

2007

-1912 -1798 U 0

<

- 1003

+ 0005

Fori

> 15

™ - \S [■'■(■-

7)

+

0.7979

e~(-t)]2.3562

error < 0.0001

9

GENERAL DATA

1-106 Table 11.

Bessel Functions: /o(z) and

1

Ii(x)

0

1

2

3

4

5

6

7

8

9

1 2 3 4

1.000 1.003 1 OKI 1.023 1.040

1 000 1.003 1.024 1.042

1.000 1.004 1.012 1.026 1.045

1 000 1 004 1.013 1.027 1.047

1.000 1.005 1.014 1.029 1.049

1 001 1.006 1.016 1.031 1.051

1. 001 1.006 1.017 1.033 1.054

1.001 1.007 1.018 1.035 1.056

1.002 1.008 1.020 1.036 1.058

1.002 1.009 1.021 1.038 1.061

5 6 7 8 9

1.063 1.092 1.126 1. 167 1.213

1.066 1.095 1.130 1.171 1.218

1.069 1.098 1.134 1.175 1.223

1.071 1.102 1. 138 1. 180 1.228

1.074 1.105 1.142 1.184 1.233

1.077 1 108 1.146 1.189 1 239

1.080 1.112 1.150 1.194 1.244

1.083 1.115 1. 154 1. 198 1.249

1.086 1.119 1.158 1.203 1.255

1.089 1.123 1. 162 1.208 1.260

1 2 3 4

1.266 1.326 1.394 1.469 1.553

1.272 1.333 1.401 1.477 1.562

1.278 1.339 1.408 1.485 1.571

1.283 1.346 1.416 1.494 1 580

1.289 1.352 1.423 1.502 1.590

1.295 1.359 1 430 1.510 1.599

1.301 1.366 1.436 1.519 1.608

1.307 1.373 1.446 1.527 1.618

1.314 1.380 1.454 1.536 1.627

1.320 1.387 1.461 1.545 1.637

5 6 7 8 9

1.647 1.750 1.864 1.990 2.128

1.657 1.761 1.876 2.003 2.142

1.667 1.772 1.888

1.687 1.794 1.913 2.043 2. 187

1.697 1.806 1.925 2.057

2.202

1.707 1.817 1.938 2.071 2.217

1.718 1.829 1.951 2.085 2.233

1.728 1.840 1.963 2.099

2.157

1.677 1.783 1.900 2.030 2. 172

1.739 1.852 1.976 2.113 2.264

2.280

2.296 2.464 2.648

2 312

2.328

2.344 2.517 2.707 2.915 3.143

2.361

2.378 2.554 2.747 2.959 3. 191

2.395 2.573 2.768 2.981 3.215

2.412

2.536

3.445 3.723

3.472 3.752

4.028 4.361 4.725

4.396

X

0.0

1.0


[Sec.

2.0 1 2 3 4 5 6 7 8 9

3.0 1 2 3 4 5 6 7 8 9

2.446 2.629 2.830

1011

2.851

3.049

3.072

3.290

3.315 3.581

2.016

2.482 2.668 2.872 3.096

2.499 2.687 2.893 3. 119

3.366

3.842

3.872

3.341 3.609 3.903

4.157 4.503

4. 190

4.224

3.553

4.881 5.294 5.747 6.243 6.785

4.539 4.921 5.338 5.795

6.295 6.842

4.576

3.637 3.933 4.256 4.613

4.961

5.001

5.382

5.426

5.843 6.347 6.899

6.400 6.957

7.378

7.441

8.028

8.096

8.739 9.517 10.37

8.813 10.46

7.503 8. 165 8.889 9.681 10.55

1 2 3 4

11.30 12.32 13.44 14.67 16.01

11.40 12.43 13.56 14.80 16.15

5 6 7 8 9

17.48 19.09 20.86 22.79 24.91

17.64 19.26 21.04 23.00 25. 14

4.0

5.0

27.24

9.599

27.48

5.891

3.392 3.666 3.965

4.292 4.650

3.419 3.694 3.996 4.326

4.688

5.125 5.561

5.471 5.940

5.083 5.516 5.989

6.454

6.508

6.039 6.562

7.016

7.075

7. 134

7.631

7.696

8.304

9.764

9.848

10.64

11.50 12.54 13.68 14.93 16.29 17.79 19.43 21.23

21.42

25.36 27.73

7»(r)

3.167

5.042

7.567 8.234 8.965

23.20

2.727 2.937

4.431 4.802 5.209 5.653

6.140 6.672

3.526 3.812 4.124 4.467 4.841 5.251 5.700 6.191

6.728

7.255

7.316

9.356

9.436

10.11 11.01

10. 19 11.11

10.28 11.20

11.90 12.98 14.16 15.46 16.88

12.01 13. 10 14.29 15.59 17.03

12.11 13.21 14.41 15.73 17.18

12.22 13.33 14.54 15.87 17.33

18.27 19.96 21.80 23.83 26.05

18.43 20.13 22.00 24.04

26.28

18.59 20.31 22.19 24.26 26.52

18.76 20.49 22.39 24.47

26.76

18.92 20.67 22.59 24.69 27.00

28.49

28.74

29.00

29.26

29.52

11.60 12.65 13.80 15.06 16.44

11.70 12.76 13.92 15. 19 16.58

11.80 12.87 14.04 15.32 16.73

17.95 19.61 23.41 25.59

18.11 19.78 21.61 23.62 25.82

27.98

28.23

2<(2I)»

5.607 6.089 6.617 7.195

3.499 3.782 4.092

3.026 3.265

7.960 8.665

10.73

2>(ll)«

5.166

2.788 3.004 3.240

2.429 2.610 2.809

7.893 8.591

8.447

= i +

4.764

2.591

7.827 8.519 9.276

7.761

8.375 9. 119 9.933 10.82

9.041

4.060

2.248

9. 197 10.02 10.92

2«t31J« •

Sec.

mathematical tables

1-2] Table 11.

4

6

8

9

0.0000

1 2 3 4

0501 1005 1517 2040

0050 0551 1056 1569 2093

0100 0601 1107 1621 2147

0150 0651 1158 1673 2200

0200 0702 1209 1725 2254

0250 0752 1260 1777 2307

0300 0803 1311 1829 2361

0350 0853 1362 1882 2415

0400 0904 1414 1935 2470

0450 0954 1465 1987 2524

5 t> 7 8 9

2579 3137 3719 4329 4971

2634 3194 3778 4391 5038

2689 3251 3838 4454 5104

2744 3309 3899 4518 5171

2800 3367 3959 4581 5239

2855 3425 4020 4646 5306

2911 3483 4081 4710 5375

2967 3542 4142 4775 5443

3024 3600 4204 4840 5512

3080 3659 4266 4905 5582

5652 6375 7147 7973 8861

5722 6450 7227 8059 8953

5793 6525 7308 8146 9046

5864 6601 7389 8233 9140

5935 6677 7470 8321 9235

6008 6754 7553 8409 9330

6080 6832 7636 8498 9426

6153 6910 7719 8588 9522

6227 6988 7803 8678 9620

6300 7067 7888 8769 9718

9817 1.085 1. 196 1.317 1.448

9916 1.096 1.208 1.330 1.462

1.002 1.106 1.220 1.343 1.476

1.012 1.117 1.232 1.355 1.490

1.022 1.128 1.244 1.368 1.504

1.032 1.139 1.256 1.381 1.518

1.043 1.151 1.268 1.395 1.532

1.053 1.162 1.280 1.408 1.547

1.064 1.173 1.292 1.421 1.561

1.074 1.185 1.305 1.435 1.576

1.591 1.745 1.914

1.606 1 762 1.932 2.117 2.319

1.621 1.778 1.950 2.136 2.340

1.636 1.795 1.968

1.666 1.828 2.004

1.682 1.845

1.713 1.879 2.060 2.257 2.471

1.729 1.897 2.079

2.156 2.362

1.651 1.811 1.986 2. 176 2.383

2.540 2.780 3.043 3.331

2.563 2.806 3.071 3.361

2.586 2.831

2.610 2.857

2.706 2.962

3.613

3.645

3.678

2.731 2.989 3.271 3.580

3.953 4.326 4.734

3.989

4.025 4.405 4.820 5.275 5.773

4.062

4.098

4.445 4.864 5.323 5.826

4.485

4.136 4.526

4.908 5.371

4.953 5.420

5.879

5.932

5.469 5.986

6.319 6.917 7.572 8.289

6.376 6.979

6.434

6.493

7.107

6.552

9.076

7.640 8.365 9. 159

7.043 7.710 8.441 9.242

8.518 9.326

7.171 7.851 8.595 9.411

9.938 10.88 11.92 13.06 14.30

10.03 10.98 12.03 13.18 14.44

10.12 11.08 12.14 13.30 14.57

10.21 18 12.25 13.42 14.70

15.82 17.33 18.99 20.82 22.82

15.96 17.49 19.17 21.01 23.03

25.02

25.25

5 6 7 8

9 2 0 | 2 3 4 5

*

7 8 9

J

0 | 2 3 4

J

to 7 8 9

4.0 1 2 3 4 5 6 7 8 9

5.0

2.098 2 298

2.517 2.755 3.016 3.301

5.181

5.670 6.206 6.793 7.436 8. 140

8.913 9.759 10.69 11.71

12.82 14.05 15.39 16.86 18.48 20.25

4.365 4.777 5.228 5.722

6.262 6.854 7.503

8.215 8.994 9.848 10.79 11.81 12.94 14.17 15.53 17.02 18.65

22 20

20.44 22.40

15.67 17.17 18.82 20.63 22.61

24.34

24.56

24.79

/i(x)

3.099 3.392 3.712

2- 0!l!

5

7

0.0

t 2 3 4

3

(Continued)

1

10

2

Ii(x).

0

J-


Bessel Functions: h(x) and

1-107

3.127 3.422 3.745

ll-

2.196

2.022 2.216

2.405

2.427

1.698 1.862 2.041 2.236 2.449

2.633 2.883 3.155 3.453 3.779

2.657 2.909 3. 184

2.682 2.935 3.213

3.485 3.813

3.516 3.848

4.173

4.211

4.567 4.997

3.242 3.548

2.277 2.494

3.883

3.918

4.608

4.249 4.650

4.287 4.692

5.043 5.519

5.088

5.569

5.134 5.619 6. 150

6.040

6.095

6.732

9.497

6.671 7.302 7.994 8.753 9.584

10.31 11.29 12.36 13.54 14.84

10.40 11.39 12.48 13.67 14.97

10.50 11.49 12.59 13.79 15.11

10.59 11.60 12.71 13.92 15.25

16.11 17.65 19.35 21.20 23.24

16.26 17.81 19.52 21.40 23.46

16.41 17.98 19.70 21.60 23.67

16.56 18.14 19.88 21.80 23.89

16.71 18.31 20.07 22.00 24.11

25.48

25.72

25.95

26.19

26.44

7.780

II.

I'

■+ ■ 2'- 2!3! 112!

6.611 7.237 7.922 8.674

7.369

8.067 8.832 9.671

1-108


X

0

5.0 1 2 3 4 5 6 7 8 9

32.58 35.65 39.01

42.69

27.48

30.06 32.88 35.97 39.36 43.08

46.74

47.16

51.17

51.64 56.55

27.73 30.33 33.17

27.98

28.23

28.49

30.60

30.88

33.47

33.78

36.30 39.72

36.62 40.08

36.96 40.44

31.15 34.08 37.29

43.47 47.59 52.11

43.87

44.27

48.03

52.59

48.46

49.35 54.04

58.65 64.24

64.83

63.65

68.47 75.02

69.10 75.71 82.97 90.93

84.50

96.96

81.46 89.28 97.86

5 6 7 8 9

106.3 116.5 127.8 140.1 153.7

7.0

28.74 31.43 34.39 37.63

44.67 48.90 53.55

45.08

59. 18

7

29.00

31.72 34.70 37.97 41.55

45.49 49.80 54.53

8

29.26 32.00 35.01 38.31 41.93 45.90 50.25 55.03

9

29.52 32.29 35.33

38.66 42.31

46.32 50.71 55.53

59.72 65.42

60.27 66.02

60.82 66.62 72.99

71.02 77.82 85.28 93.47

71.67 78.53

72.33

86.06 94.33

86.85

87.65

95.20

96.08

102.5

103.4

104.4

105.3

98.76

99.67

69.73 76.41 83.73 91.77 100.6

107.3 117.6 129.0 141.4 155.1

108.3 118. 7 130.2 142.7 156.6

109.3 119.8 131.4 144.1 158.0

110.3 120.9 132.6 145.4 159.5

111.3 122.0 133.8 146.8 161.0

112.3 123.2 135.1 148. 1 162.5

113.4 124.3 136.3 149.5 164.0

114.4 125.5 137.6 150.9 165.5

115.5 126.6 138.8 152.3 167.0

168.6 185.0

170.2 186.7

173.3 190.2

204.8 224.7 246.6

176.6 193.7 212.6 233.2 256.0

178.2 195.5 214.5 235.4

179.9 197.4 216.5

222.7 244.3

175.0 191.9 210.6 231.1 253.6

181 6 199.2

202.9

171.7 188.4 206.7 226.8

183.2 201.0 220.6 242.1

268.2

270.7 297.1 326.1

278.3

280.9 308.4 338.5

283.6 311.3 341.7

88.46

294.3 323.1

354.7 389.4 427.6 469.5

358.0 393.1

431.6 473.9 520.4 571.6 627.8

82.21

90.10

248.9 273.2

299.9 329.2 361.4 396.8 435.6 478.4

208.6 229.0 251.3

275.8 302.7 332.3

364.8 400.5

305.5 335.4 368.2 404.2

525.3

439.7 482.9 530.3

577.0 633.7

582.4 639.7

696.1 764.7 840.1

702.7 771.9 848.0

779.2

923.0

931.7

1005

1014

1094 1202 1321 1451 1595

1104 1213 1333 1465 1610

1753 1927 2119 2329 2561 2816

1 2 3 4

621.9

5 6 7 8 9

683.2 750.5 824.4 905.8 995.2

689.6 757.5 832.2 914.4

9.0 I 2 3 •t 5 6 7 8 9

515.6 566.3

443.9

487.4 535.3 587.9

645.7

70.37 77.11 92.61 101.5

371.6

258.4

375.1

408.0

411.9

448.0 492.0 540.3

452.2 496.6 545.4 599.0 658.0

593.4 651.8

1

(Continued)

6

41.18

57.59

57.07

Ii(x).

40.81

63.08

74.34

10.0

5

62.51

67.85

8.0

4

61.94

56.04

73.66 80.72

5 6 7 8 9

3

and

61.38 67.23

1 2 3 4

2

h(x)

53.06 58.11

1 2 3 4

6.0


27.24 29 79

1

Bessel Functions:

[Sec.

286.2

314.2 344.9 378.6 415.7 456.5 501.3 550.6

604.7 664.2

218.6 239.8

79.98

263.2

265.7

288.9 317.1 348.1

320. 1

291.6

382.2 419.6

351.4 385.8 423.6

460.8

465.1

506.0

510.8 561.0

555.7 610.4 670.5

616.1

676.8

940.6

716.0 786.6 864.2 949.5

958.4

729.6 801.5 880.6 967.5

1024

1033

1043

1053

1063

976.7 1073

1114 1225 1346 1479 1626

1125 1236 1359 1493 1641

1136 1248 1371 1507 1657

1146 1260 1384 1522 1673

1157 1272 1398 1536 1688

1168 1284 1411 1551 1704

1179 1296 1424 1565 1721

1190 1308 1438 1580 1737

1770 1946 2139 2352 2585

1787 1964 2159 2374 2610

1804 1983 2180 2397 2635

1821 2002 2201 2419 2660

1838 2021 2222 2442 2685

1856 2040 2243 2466 2711

1874 2060 2264 2489 2737

1891 2079 2286 2513 2763

1909 2099 2307 2537 2789

2843

2870

2897

2925

2952

2981

3009

3038

3067

For larger values uf the argument,

709.3 856.1

722.8 794.0

237.6 260.8

79.25

872.3

use tin* auxiliary functions. Table 13.

736.5 809. 1

888.9

743.4 816.7 897.3 985.9 1083

mathematical tables

Sec. 1-2] Table

Bessel Functions: 7o(i) and I\{x).

.'

0

1

2

3

S 0 1 2 3 4

24.34 26.68 29.25 32.08 35.18

24.56

24.79 27.18 29.80

25.02

25.25

27.43

27.68

35.51

s 6 7 8 9

38.59 42.33 46.44 50.95 55.90 61.34 67.32 73.89

i.O 1 2 3 4 5 6 7 8 9


11.

7.0 1 2 3 4 5 6 7 3 J

26.93

32.68 35.84

36.51

38.95 42.72 46.87 51.42 56.42

39.31 43. 12 47.30 51.90 56.95

39.67 43.52 47.74 52.38 57.48

40.04 43.93 48.19 52.87 58.02

61.91

62.49 68.58

63.08 69.22 75.98 83.40 91.55

63.67 69.87

89.03

67.95 74.58 81.86 89.86

97.74

98.65

99.58

107.3

142.1

108.3 118.9 130.6 143.4

109.3 120.0 131.8 144.8

156.0 171.4 188.3

157.5 173.0 190.0

159.0 174.6 191.8 210.7 231.5

81. 10

117.8 129.4

296.8 227.2

208.7 229.3

249.6 274.2 301.3

251.9 276.8 304.2 334.2

331. 1 363 9

4

30.07 32.98 36.17

29.53 32.38

367.3

1-109

30.35

33.29

(Continued)

5

6

25.48

25.72 28.20 30.92

27.94

30.63 33.59 36.85 40.41 44.33

48.64 53.37 58.56 64.26 70.53 77.41

33.91

37.19 40.79 44.75 49.09 53.86 59. 10

64.86 71.18

7

8

9

25.95

26. 19 28.72

26.44

31.49 34.54 37.88

31.79 34.86

41.55 45.58

41.94

50.01 54.87 60.21

50.48 55.38 60.77

66.08 72.52

66.70 73.20

79.60

28.46 31.20 34.22 37.53 41. 17

45.16 49.55 54.36 59.66 65.47 71.85

38.23

46.01

92.41

84.97 93.28

85.77 94.15

78.86 86.57 95.04

87.38 95.93

80.35 88.20 96.83

100.5 110.4 121.2 133.1 146.1

101.5 4 122.3 134.3 147.5

102.4 112.4 123.5 135.6 148.9

103.4 113.5 124.6 136.9 150.3

104.3 114.6 125.8 138.1 151.7

105.3 115.6 127.0 139.4 153.1

106.3 116.7 128.2 140.8 154.6

160.5 176.3 193.6

163.5 179.6 197.3

165.1 181.3 199.2

166.6 183.0 201.0

168.2 184.7 202.9

169.8 186.5

212.7 233.7

162.0 177.9 195.5 214.7 235.9

216.7

218.8

222.9

238.1

240.4

220.9 242.6

244.9

204.9 225.0 247.2

254.3 279.4

256.7

259.2

261.6

264.1

266.6

269.1

271.7

282.1

284.8

287.4

307.0 337.4 370.8

310.0

315.9

340.6 374.3

312.9 343.8 377.9

347.1

324.9 357.1

381.4

328.0 360.4

385.1

321.9 353.7 388.7

415.3

419.2

423.2

460.7 506.4 556.7

465.1

511.2 562.0 617.8

427.2 469.5

431.3 474.0

516.1

521.0

567.3 623.6

572.7 629.6

679.1

685.6

692. 1 760.9

75.27 82.63 90.70

76.69

84.18

III.

8.0 I 2 3 4

399.9 439.5 483.0 531.0 583.7

403.7 443.7 487.6 536.0 589.2

407.5 447.9 492.3 541.1

411.4

594.8

600.5

5 8 7 8 9

641.6

647.7

653.9

705 4

712.1

718.9

666.4 732.6

775.5 852.7 937.5

782.9 860 .8 946.5

790.4

660.1 725.7 797.9

805.5

672.7 739.6 813.2

955.5

877.3 964.6

885.6 973.8

894.1 983.1

».o 1 2 3 4

1031 1134 1247 1371 1508

1041 1144 1259 1384 1522

1051 1155 1271 1397 1537

1061 1166 1283 1411 1552

1071 1178 1295 1424 1566

1081 1189 1307 1438 1581

5 6 7 8 1

1658 1824 2006 2207 2428

1674 1842 2026 2228 2451

1690 1859 2045 2250 2475

1707 1877 2065 2271 2498

1723 1895 2084 2293 2522

10 0

2671

2697

2722

2749

2775

869.0

452.1

496.9 546.2

456.4 501.7

551.4 606.2

611.9

78.13

290.2 318.8 350.4

746.7 820.9 902.6 992.5

295.7

292.9

392.4

753.8 828.7

values of the argument,

298.5

396.1

435.4 478.5

526.0 578.2 635.6 698.7 768.2 844.6

911.2

836.6 919.9

928.7

1002

1012

1021

1091 1200 1320 1452 1596

1102 1212 1332 1465 1612

1112 1223 1345 1479 1627

1123 1235 1358 1494 1643

1739 1913 2104 2315 2547

1756 1931 2125 2337 2571

1773 1950 2145 2359 2596

1790 1969 2165 2382 2621

1807 1987 2186 2405 2646

2802

2828

2856

2883

2911

i For larger

28.99

use the auxiliary functions, Tabic

13

GENERAL DATA

1-110 Table 12.

0

X

1

Ku(x) and K,(x)

Bessel Functions:

2

3

4

5

6

7

8

9

DO 2.427 1.753 1.372 1. IIS

4.721 2.333 1.706 1.342 1.093

4.028 2.248 1.662 1.314 1.072

3.624

1 2 3 4

2. 170 1.620 1.286 1.052

3.337 2.097 1.580 1.259 1.032

3. 114 2.030 1.542 1.233 1.013

2.933 1.967 1.505 1.208 0.9943

2.780 1.909 1.470 1. 183 9761

2.647 1.854 1.436 1. 160 9584

2.531 1.802 1.404 1.137 9412

5 6 7 8 9

0.9244 7775 6605 5653 4867

9081 7646 6501 5568 4796

8921 7520 6399 5484 4727

8766 7397 6300 5402 4658

8614 7277 6202 5321 4591

8466 7159 6106 5242 4524

8321 7043 6012 5165 4459

8180 6930 5920 5088 4396

8042 6820 5829 5013 4333

7907 6711 5740 4940 4271

1.0 1 2 3 4

4210 3656 3185 2782 2437

4151 3605 3142 2746 2405

4092 3556 3100 2709 2373

4034 3507 3058 2673 2342

3977 3459 3017 2638 2312

3922 3411 2976 2603 2282

3867 3365 2936 2569 2252

3813 3319 2897 2535 2223

3760 3273 2858 2502 2194

3707 3229 2820 2469 2166

$ 6 7 8 9

2138 1880 1655 1459 1288

2111 1856 1634 1441 1273

2083 1832 1614 1423 1257

2057 1809 1593 1406 1242

2030 1786 1573 1388 1226

2004 1763 1554 1371 1211

1979 1741 1534 1354 1196

1953 1719 1515 1337 1182

1928 1697 1496 1321 1167

1904 1676 1478 1305 1153

2.0 1 2 3 4

1139 1008 0.08927 7914 7022

1125 •9956 8820 7820 6939

•9836 8714 7726 6856

1098 •9717 8609 7634 6775

1084 *9600 8506 7544 6695

1071 •9484 8404 7454 6616

1058 •9370 8304 7365 6538

1045 •9257 8204 7278 6461

1033 •9145 8106 7191 6384

1020 •9035 8010 7106 6309

5 6 7 8 9

6235 5540 4926 4382 3901

6161 5475 4868 4331 3856

6089 541 1 481 1 4281 3811

6017 5348 4755 4231 3767

5946 5285 4700 4182 3724

5877 5223 4645 4134 3681

5808 5162 4592 4086 3638

5739 5102 4538 4039 3597

5672 5042 4485 3992 3555

5606 4984 4433 3946 3514

3.0 1 2 3 4

3474 3095 2759 2461 2196

3434 3060 2728 2433 2171

3395 3025 2697 2405 2146

3356 2990 2666 2378 2122

3317 2956 2636 2351 2098

3279 2922 2606 2325 2074

3241 2889 2576 2298 2051

3204 2856 2547 2272 2028

3168 2824 2518 2246 2005

3131 2791 2489 2221 1982

> 6 7 8 9

I960 1750 1563 1397 1248

1938 1730 1546 1381 1234

1916 1711 1528 1366 1221

1894 1692 1511 1350 1207

1873 1673 1494 1335 1194

1852 1654 1477 1320 1180

1831 1635 1461 1306 1167

1810 1617 1445 1291 1154

1790 1599 1428 1277 1141

1770 1581 1412 1262 1129

I 2 3 4

1116 0.009980 8927 7988 7149

1104 9869 8829 7900 7070

1091 9760 8731 7813 6992

1079 9652 8634 7726 6915

1067 9545 8539 7641 6839

1055 9439 8444 7557 6764

1044 9334 8351 7473 6689

1032 9231 8259 7391 6616

1021 9128 8167 7309 6543

1009 9027 8077 7229 6471

S 6 7 1 9

6400 5730 5132 4597 4119

6329 5668 5076 4547 4074

6260 5605 5020 4497 4030

6191 5544 4965 4448 3986

6123 5483 4911 4399 3942

6056 5423 4857 4351 3899

5989 5363 4804 4304 3857

5923 5305 4751 4257 3814

5858 5246 4699 4210 3773

5794 5189 4648 4164 3732

3691

3631

3611

3572

3533

3494

3456

3419

3382

3345

0.0


1

[Sec.

4.0

5.0

llll

A''oU)

-

-£i(>)

mathematical tables

Sec. 1-2] Table 12.

X

1 0 1


: o

7

8

9

19.91

16.56

6.477 3.747

6.053 3.588 2.476

14.17 5.678

12.37 5.345

3.440

3.303

5.046

2.323 1.745

2.252

1.840

2.397 1.792

1.464 1.167 9496 7847 6560

1.429 1.142 9311 7704 6447

1.396 1. 118 9130 7564 6336

1.364 1.095 8955 7428 6228

1.333 1.072 8784 7295 6122

5627 4779 4084 3508 3026

5534 4703 4021 3455 2982

5443 4629 3960 3404 2939

5354 4556 3900 3354 2897

5267 4485 3841 3305 2855

5181 4415 3782 3256 2814

2657 2307 2009 1754 1534

2620 2275 1982 1730 1514

2583 2244 1955 1707 1494

2546 2213 1928 1684 1474

2510 2182 1902 1662 1455

2475 2152 1876 1640 1436

2440 2123 1851 1618 1417

1362 1196 1052 9261 8165

1345 1181 1038 9144 8063

1327 1166 1025 9029 7963

1310 1151 1012 8916 7864

1293 1136 •9993 8804 7767

1276 1121 •9867 8694 7670

1260 1107 •9742 8586 7575

1244 1093 •9620 8478 7482

7298 6448 5704 5050 4474

7208 6369 5634 4989 4421

7119 6292 5566 4929 4368

7031 6215 5498 4869 4316

6945 6139 5432 4811 4264

6859 6064 5366 4753 4213

6775 5990 5301 4696 4163

6692 5917 5237 4639 4113

6609 5845 5174 4584 4064

2812 2500

3968 3521 3127 2779 2471

3921 3480 3090 2746 2442

3874 3438 3054 2714 2414

3828 3398 3018 2682 2385

3782 3358 2983 2651 2358

3738 3318 2948 2620 2330

3693 3279 2913 2589 2303

3649 3240 2879 2559 2276

3606 3202 2845 2529 2250

2224 1979 1763 1571 1400

2198 1957 1743 1553 1384

2173 1934 1722 1535 1368

2147 1912 1703 1517 1353

2123 1890 1683 1500 1337

2098 1868 1664 1483 1322

2074 1846 1645 1466 1307

2050 1825 1626 1449 1292

2026 1804 1607 1432 1277

2003 1783 1589 1416 1263

1248 1114 8872 7923

1234 1101 9826 8772 7834

1220 1089 9715 8674 7746

1206 1076 9605 8576 7659

1193 1064 9497 8479 7573

1179 1052 9390 8384 7488

1166 1040 9284 8290 7404

1152 1028 9179 8196 7321

1139 1017 9076 8104 7239

1126 10052 8973 8013 7158

7078 6325 5654 5055 4521

6999 6254 5591 4999 4471

6920 6185 5529 4943 4421

6843 6116 5467 4889 4372

6766 6047 5406 4834 4324

6691 5980 5346 4781 4276

6616 5913 5286 4727 4229

6542 5847 5228 4675 4182

6469 5782 5169 4623 4136

6397 5717 5112 4572 4090

4045

4000

3956

3912

3869

3826

3784

3742

3700

3660

2

3

4

5

m

9.854 4.776 3.056

99.97 8.935 4.532 2.944

33.27 7.519

24.92 6.962 3.919

2. 184

2. 120

49.95 8. 169 4.309 2.839 2.059

2.647 1.945

2.559 1.892

1.656

7165

1.615 1.274 1.029 8456 7039

1.575 1.246 1.008 8298 6915

1.536 1.219 9882 8144 6794

1.499 1.192 9686 7993 6675

6019 5098 4346 3725 3208

5918 5016 4279 3670 3161

5819 4935 4212 3615 3115

5722 4856 4147 3561 3070

2774 2406 2094 !>-26 1597

2734 2373 2065 1802 1575

2695 2340 2037 1777 1555

1399 1227 1079 8372

1380 1212 1065 9379 8268

7389 6528 5774 5111 4529 4016 3563

0.09498

3.0

3164

4.0 1

0.009938

t.l

(Continued)

6

1

1.050 0.8618

8

Bessel Functions: Kn(x) and A',(r).

0

1 303

0

1-111

4. 106 2.740 2.001

Jti(x)

-

-K'.(x)

10.97 3.175 1.700

GENERAL

1-112 Table 12.

DATA

[Sec.

Bessel Functions: K0(x) and Ki(x).

1

(.Continued)

JCo(x)

*

1

0

3

4

5

6

7

8

9

1 2 3 4

0.003691 3308 2966 2659 2385

3651 3272 2934 2630 2359

3611 3237 2902 2602 2333

3572 3202 2870 2574 2308

3533 3167 2839 2546 2283

3494 3132 2808 2518 2258

3456 3098 2778 2491 2234

3419 3065 2748 2464 2210

3382 3031 2718 2437 2186

3345 2998 2688 2411 2162

5 6 7 8 9

2139 1918 1721 1544 1386

2116 1898 1703 1528 1371

2093 1877 1684 1511 1356

2070 1857 1666 1495 1342

2048 1837 1648 1479 1327

2026 1817 1630 1463 1313

2004 1798 1613 1447 1299

1982 1778 1595 1432 1285

1961 1759 1578 1416 1271

1939 1740 1561 1401 1258

1244 1117 1 2 1002 3 0.0009001 8083 4

1231 1105 •9918 8905 7997

1217 1093

8810 7911

1204 1081 •9706 8715 7827

1191 1070 •9602 8622 7743

1179 1058 •9499 8530 7660

1166 1047 ♦9398 8438 7578

1153 1035 •9297 8348 7497

1141 1024 •9197 8259 7417

1129 1013 •9099 8171 7338

5 6 7 8 9

7259 6520 5857 5262 4728

7182 6451 5795 5206 4677

7105 6382 5733 5150 4627

7029 6314 5672 5095 4578

6954 6246 5611 5041 4529

6880 6180 5551 4987 4481

6806 6114 5492 4934 4434

6734 6048 5434 4882 4386

6662 5984 5376 4830 4340

6591 5920 5318 4778 4294

7.0 1 2 3 4

4248 3817 3431 3084 2772

4203 3777 3394 3051 2742

4158 3737 3358 3019 2713

4114 3697 3323 2987 2685

4070 3658 3287 2955 2656

4027 3619 3253 2924 2628

3984 3580 3218 2893 2600

3942 3542 3184 2862 2573

3900 3505 3150 2832 2545

3858 3468 3117 2802 2518

5 6 7 8 9

2492 2240 2014 1811 1629

2465 2216 1993 1792 1611

2439 2193 1972 1773 1594

2413 2170 1951 1754 1578

2388 2147 1930 1736 1561

2363 2124 1910 1717 1545

2338 2102 1890 1699 1528

2313 2079 1870 1681 1512

2288 2057 1850 1664 1496

2264 2036 1830 1646 1480

1465 1317 1 1185 2 3 1066 4 .00009588

1449 1303 1172 1055 9487

1434 1290 1160 1043 9387

1419 1276 1148 1032 9288

1404 1263 1136 1022 9191

1389 1249 1124 1011 9094

1374 1236 1112 10002 8998

1360 1223 1100 •9897 8904

1346 1210 1089 •9793 8810

1331 1198 1077 •9690 8717

5 6 7 8 9

8626 7761 6983 6283 5654

8535 7679 6909 6217 5595

8445 7598 6837 6152 5536

8356 7519 6765 6088 5478

8269 7439 6694 6024 5420

8182 7361 6624 5961 5364

8096 7284 6554 5898 5307

8011 7208 6485 5836 5252

7926 7132 6417 5775 5197

7843 7057 6350 5714 5142

9.0 1 2 5 4

5088 4579 4121 3710 3339

5035 4531 4078 3671 3304

4982 4484 4036 3632 3270

4930 4437 3993 3594 3235

4878 4390 3951 3557 3202

4827 4344 3910 3519 3168

4776 4299 3869 3483 3135

4726 4254 3829 3446 3102

4677 4209 3789 3410 3070

4628 4165 3749 3374 3038

5 6 7 8 9

3006 2706 2436 2193 1975

2974 2678 2411 2170 1954

2943 2650 2385 2148 1934

2912 2622 2360 2125 1913

2882 2595 2336 2103 1894

2852 2567 2311 2081 1874

2822 2541 2287 2059 1854

2793 2514 2263 2038 1835

2763 2488 2240 2017 1816

2734 2462 2216 1995 1797

1778

1759

1741

1723

1705

1687

1670

1652

1635

1618

5.0

6.0


2

8.0

10.0

»98ll

For larger values of the argument,

use the auxiliary functions, Table 13.

mathematical tables

Sec. 1-2] Table 12.

Bessel Functions:

A". >(x)

and

A",

1-113

(x).

(Continued)

KiUO

I

1

2

3

4

5

6

7

8

4000 3579 3204 2868 2568

3956 3540 3168 2836 2540

3912 3501 3133 2805 2512

3869 3462 3099 2774 2485

3826 3424 3065 2744 2457

3784 3386 3031 2714 2430

3742 3349 2998 2684 2404

3700 3312 2965 2655 2377

3660 3275 2932 2625 2351

2326 2083 1866 1673 1499

2300 2060 1846 1654 1483

2275 2038 1826 1636 1467

2250 2016 1806 1619 1451

2225 1994 1786 1601 1435

2201 1972 1767 1584 1419

2177 1950 1748 1566 1404

2153 1929 1729 1549 1389

2130 1908 1710 1532 1374

2106 1887 1691 1516 1359

6 0 1344 1 1205 2 1081 3 0.0009691 4 8693

1329 1192 1069 9586 8599

1315 1179 1057 9483 8506

1301 1166 1046 9380 8414

1286 1154 1034 9279 8324

1273 1141 1023 9178 8234

1259 1129 1012 9079 8145

1245 1116 1001 8981 8057

1232 1104 •9904 8884 7970

1218 1092 •9797 8788 7884

0

5.0 0.004045 1 3619 3239 2 2900 3 4 2597


5 t> 7 3 9

9

5 6 7 6 9

7799 6998 6280 5636 5059

7715 6922 6212 5576 5005

7632 6848 6145 5516 4951

7549 6774 6079 5456 4898

7468 6701 6014 5398 4845

7387 6629 5949 5340 4793

7308 6558 5885 5282 4742

7229 6487 5822 5226 4691

7151 6417 5759 5170 4641

7074 6348 5697 5114 4591

7 0 1 2 3 4

4542 4078 3662 3288 2953

4493 4034 3623 3253 2922

4445 3991 3584 3219 2891

4397 3948 3545 3184 2860

4350 3906 3508 3150 2829

4304 3864 3470 3116 2799

4257 3823 3433 3083 2769

4212 3782 3396 3050 2740

4167 3741 3360 3018 2710

4122 3701 3324 2985 2682

5 6 7 8 9

2653 2383 2141 1924 1729

2625 2358 2118 1903 1710

2597 2333 2096 1883 1692

2569 2308 2074 1863 1674

2542 2283 2051 1843 1656

2514 2259 2030 1824 1639

2488 2235 2008 1804 1621

2461 2211 1987 1785 1604

2435 2188 1966 1766 1587

2409 2164 1945 1747 1570

».0 1 2 3 4

1554 1396 1255 1128 1014

1537 1382 1242 1116 10036

1521 1367 1229 1105 *9930

1505 1352 1216 1093 *9825

1489 1338 1203 1081 •9721

1473 1324 1190 1070 •9618

1457 1310 1177 1058

«95I6

1442 1296 1165 1047 •9415

1427 1282 1153 1036 •9316

1411 1269 1140 1025 •9217

5 .00009120 6 8200 7 7374 8 6631 9 5964

9023 8113 7296 6561 5901

8928 8028 7219 6492 5838

8833 7943 7142 6423 5777

8740 7859 7067 6355 5716

8648 7776 6992 6288 5656

8556 7694 6918 6222 5596

8466 7612 6845 6156 5537

8376 7532 6773 6091 5479

8288 7452 6702 6027 5421

<

1 2 3 4

5364 4825 4340 3904 3512

5307 4774 4294 3863 3476

5251 4723 4249 3822 3439

5196 4674 4204 3782 3403

5141 4624 4160 3742 3367

5087 4576 4116 3703 3332

5033 4528 4073 3664 3297

4980 4480 4030 3626 3262

4928 4433 3988 3587 3228

4876 4386 3946 3550 3194

5 6 7 8 9

3160 2843 2559 2302 2072

3127 2814 2532 2278 2050

3094 2784 2505 2254 2029

3062 2755 2479 2231 2008

3029 2726 2453 2207 1987

2998 2697 2427 2184 1966

2966 2669 2402 2161 1945

2935 2641 2377 2139 1925

2904 2613 2352 2116 1905

2874 2586 2327 2094 1885

10.0

1865

1845

1826

1807

1788

1769

1751

1732

1714

1696

For larger values of the argument,

use the auxiliary functions. Tabic

13.

GENERAL DATA

1-114 Table 13.

Bessel Functions

:

[Sec.

1

Auxiliary Functions

Auxiliary Functions Yo(x) and Yi(x) for Small Values of Argument For small values of the argument, Fo(-z) and Yi(x) are rapidly changing functions and linear inter polation is inaccurate. These tables of auxiliary functions can be used to give accurate interpolated For values of the argument above 0. 1 the main tables arc satisfactory if interpolation formulas values. are used.

KoU) = Co + Do log x Yi(x) (Ci/x) +Z>ilogx

-

Co

0.0

0

1

2

3

4

5

6

7

8

9

-0.0738

0738 0717 0660 0569 0444

0737 0713 0652 0558 0430

0736 0708 0645 0547 0415

0735 0703 0636 0535 0400

0734 0698 0628 0523 0385

0732 0693 0619 051 1 0369

0729 0687 0609 0498 0353

0727 0681 0600 0485 0337

0724 0674 0590 0472 0321

-0720 -0667 -0579

1 2 3 4

-0458

Do

0.0


I 2 3 4

0

1

2

3

4

5

6

7

8

9

1.4659 1.4622 1.4512 1.4331 1. 4078

4658 4614 4498 4309 4049

4657 4606 4482 4286 4019

4655 4597 4465 4262 3989

4653 4587 4448 4238 3958

4650 4576 4431 4213 3926

4646 4565 4412 4188 3893

4641 4553 4393 4161 3860

4635 4540 4373 4134 3826

4629 4527 4352 4107 3792

C,

0.0

0

'

2

3

4

5

t

7

8

9

-0.6366

6366 6390 6452 6550 6681

6367 6394 6460 6561 6695

6368 6399 6468 6573 6710

6369 6404 6477 6586 6726

6371 6410 6487 6598 6741

6373 6416 6496 6611 6757

6376 6422 6506 6624 6773

6379 6429 6517 6638 6789

6382 6436 6527 6652 6806

0

1

2

3

4

5

6

7

8

9

0.0000 0732 1459 2174 2873

0073 0805 1531 2245 2942

0146 0878 1603 2316 301 1

0220 0951 1675 2386 3079

0293 1024 1746 2456 3148

0366 1096 1818 2526 3215

0440 1169 1890 2596 3283

0513 1241 1961 2666 3351

0586 1314 2032 2735 3418

0659 1386 2103 2804 3485

- 6386 - 6444

1 2 3 4

0.0 1 2 3 4

-6538 -6666

Auxiliary Functions Ko(x) and Ki(x) for Small Values of Argument For small values of the argument, Ko(x) and Ki(x) are rapidly changing functions and linear inter polation is inaccurate. These tables of auxiliary functions can be used to give accurate interpolated values. For values of the argument above 0. 1 the main tables are satisfactory if interpolation formulas are used.

Ko(x) = Eo + Fo log x

Ki(x)

-

(Ei/x)

+

Fi log*

Eo(x)

0.0 1 2 3 4

0

1

2

3

4

5

6

7

8

9

0. 1159 1187 1271 1412 1612

1160 1193 1283 1430 1635

1160 1200 1295 1448 1659

1162 1207 1308 1466 1684

1164 1214 1321 1485 1709

1166 1222 1335 1505 1735

1169 1231 1349 1525 1761

1173 1240 1364 1546 1788

1177 1250 1380 1567 1816

1182 1260 1396 1590 1844

Sec. 1-2]

MATHEMATICAL

Table

0 0 1 2 3 4

13.

Bessel Functions

TABLES Functions.

Auxiliary

:

1-115 (Continued)

0

1

2

3

4

5

6

7

8

9

1 0000 0.9969 9875 9716 9485

1 . 0000 9963 9863 9696 9458

9999 9955 9849 9676 9430

9997 9948 9835 9654 9401

9995 9939 9820 9633 9371

9992 9930 9804 9610 9341

9989 9921 9788 9586 9310

9985 9910 9771 9562 9278

9980 9899 9753 9537 9245

9975 9888 9735 9512 9211

F,W

0 0 1 2 3 4

0

'

2

3

4

5

6

7

8

9

-2.3026 -2.3083

3026 3096 3280 3582 4004

3028 3109 3305 3619 4053

3031 3123 3331 3657 4103

3035 3139 3359 3696 4154

3040 3156 3387 3736 4206

3047 3173 3417 3778 4260

3054 3193 3447 3821 4315

3063 3213 3479 3865 4371

3073 3234 3513 3910 4429

-2 -2

3257 3547

-2.3956


F,(x)

0 0 1 2 3 4

0

1

2

3

4

5

6

7

8

9

0.0000

0115 1268 2431 3612 4820

0230 1384 2548 3731 4943

0345 1500 2666 3851 5066

0461 1616 2783 3971 5189

0576 1732 2901 4092 5313

0691 1848 3019 4212 5437

0806 1964 3137 4333 5562

0922 2081 3255 4454 5687

1037 2197 3374 4576 5812

1153 2314 3493 4698

Examples

of use of auxiliary functions for small values of argument:

-

0.0715 + 1.4610 X 7.0607 Example I. 0.0715 1.4610 + 0.0887 = K«(0.II5) Linear interpolation from the direct-reading table of Ya would give the less accurate value

-

K«(0.I15)

Example

2.

-0.6392

yi
0.115 compared with the less accurate

+ 0.08415

X

1.0607

1.4438.

-1.4444

-

-5.558

-

0.084 + 0.005

-

-5.637

value of — 5.648 obtained by linear interpolation of the tablefor

Yi(z).

Auxiliary functions /o(x), /i(x). K«(s). Kttx) for large values of argument

0

1

2

3

4

5

6

7

8

9

10.0

0.1278

12.0 13 0 14 0

1217 1164 1118 1076

1272 1212 1159 1113 1072

1265 1206 1154 1109 1068

1259 1201 1150 1105 1065

1253 1195 1145 1100 1061

1247 1190 1140 1096 1057

1241 1185 1136 1092 1053

1235 1180 1131 1088 1050

1229 1174 1126 1084 1046

1223 1170 1122 1080 1043

15 0 16.0 17.0 18.0 19.0

1039 1005 0975 0950 0921

1035 1002 0972 0944 0919

1032 0999 0969 0942 0917

1029 0996 0966 0940 0914

1025 0993 0963 0937 0912

1022 0990 0961 0934 0909

1018 0987 0958 0931 0907

1015 0984 0955 0929 0905

1012 0981 0952 0926 0902

1009 0978 0950 0924 0900

20 30

0898 0731

0676 0719

0856 0708

0836 0697

0819 0687

0802 0677

0786 0667

0771 0658

0757 0649

0744 0641

no

It(x) s= tabulatt'd number X «*. For greater values of x, e~*Io(x)

(i

+

e)/ve.

GENERAL DATA

1-116 Table 13.

Bessel Functions

:

[SEC.

Auxiliary Functions.

(Continued)

«"'/i(x) 0 0. 1213 1161

3

4

5

6

7

8

9

1202 1151 1106 1066 1030

1196 1146 1102 1062 1027

1191 1142 1098 1059 1023

1186 1137 1094 1055 1020

1181 1132 1090 1051 1017

1175 1128 1086 1048 1013

1170 1123 1082 1044 1010

1165 1119 1078 1040 1007

M.O

1074 1037

1207 1156 1110 1070 1034

15.0 16.0 17 0 18.0 19.0

1004 0973 0946 0920 0897

1001 0971 0943 0918 0895

0997 0968 0941 0915 0892

0994 0965 0938 0913 0890

0991 0962 0935 0911 0888

0988 0959 0933 0908 0886

0985 0957 0930 0906 0884

0982 0954 0928 0904 0881

0979 0951 0925 0901 0879

0976 0948 0923 0899 0877

20 30

0875 0719

0855 0708

0836 0697

0818 0687

0801 0677

0786 0667

0771 0658

0757 0649

0744 0641

0731 0633

10.0 11.0 12.0 13.0

Ii(x)

HIS

™ tabulated

For greater

number

X e*. «

values of x, e~*I\(x)

1

0

Ojt/ '

N

2

3

4

5

6

7

8

9

12.0 13.0 14.0

0.3916 3738 3582 3444 3321

3897 3721 3567 3431 3309

3879 3705 3553 3418 3298

3860 3689 3539 3406 3286

3842 3673 3525 3393 3275

3824 3657 3511 3381 3264

3806 3642 3497 3368 3253

3789 3627 3484 3356 3242

3772 3612 3470 3344 3231

3755 3597 3457 3333 3221

15.0 16.0 17.0 18.0 19.0

3210 3110 3018 2934 2857

3200 3100 3009 2926 2850

3189 3091 3001 2918 2842

3179 3081 2992 2910 2835

3169 3072 2984 2903 2828

3159 3063 2975 2895 2821

3149 3054 2967 2887 2813

3139 3045 2959 2879 2806

3129 3036 2950 2872 2799

3119 3027 2942 2864 2792

20 30

2785 2279

2719 2242

2658 2207

2599 2174

2545 2142

2494 2111

2446 2082

2401 2054

2358 2027

2318 2001

10.0


2

It.O

A'o(j-) — tabulated

number

X

For greater values of x, e*Kn(x)

=

e'Kx(x) 0

1

2

3

4

5

6

7

8

9

10.0 0 12.0 13.0 14.0

0.4108 3904 3728 3574 3437

4086 3886 3712 3560 3425

4064 3867 3696 3545 3412

4043 3849 3680 3531 3399

4023 3831 3664 3518 3387

4002 3813 3649 3504 3375

3982 3796 3633 3490 3363

3962 3779 3618 3477 3351

3943 3762 3603 3464 3339

3923 3745 3589 3450 3327

15.0 16.0 17.0 18.0 19.0

3315 3205 3106 3015 2931

3304 3195 3096 3006 2923

3292 3185 3087 2997 2915

3281 3174 3077 2989 2907

3270 3164 3068 2980 2900

3259 3154 3059 2972 2892

3248 3144 3050 2964 2884

3237 3135 3041 2955 2877

3226 3125 3032 2947 2869

3216 3115 3023 2939 2862

20 30

2854 2317

2783 2278

2717 2241

2655 2206

2598 2173

2544 2141

2493 2110

2445 2081

2400 2053

2357 2026

II.

Ki(x)

= tabulated

number

For greater values of t.

X e'*.

fA'iW

m

Example of use of auxiliary functions for large values of argument: /0(25)

-

7.202 X

I0,B X 0.0802

-

5.776 X I0»

1

mathematical tables

Sec. 1-2]

Gamma and Factorial Functions: r(x)

Table 14.


X

1-117 = y\

0

1

2

3

4

5

6

7

8

9

1 . 0000

0.9994

9988 9932 9878 9825 9774

9983 9927 9872 9820 9769

9977 9921 9867 9815 9764

9971 9916 9862 9810 9759

9966 9910 9856 9805 9755

9960 9905 9851 9800 9750

9954 9S79 9846 9794 9745

9949 9894 9841 9789 9740

!/

1.00 1 2 3 4

o.oo i 2 3 4

0.9943 9888 9835 9784

9938 9883 9830 9779

5 6 7 8 9

5 6 7 8 9

9735 9687 9642 9597 9555

9730 9683 9637 9593 9550

9725 9678 9633 9589 9546

9721 9673 9628 9584 9542

9716 9669 9624 9580 9538

9711 9664 9619 9576 9534

9706 9660 9615 9571 9530

9702 9655 9610 9567 9526

9697 9651 9606 9563 9522

9692 9646 9602 9559 9518

1.10 1 2 3 4

0. 10 1 2 3 4

9514 9474 9436 9399 9364

9509 9470 9432 9396 9361

9505 9466 9428 9392 9357

9501 9462 9425 9389 9354

9498 9459 9421 9385 9350

9494 9455 9417 9382 9347

9490 9451 9414 9378 9344

9486 9447 9410 9375 9340

9482 9443 9407 9371 9337

9478 9440 9403 9368 9334

5 b 7 8 9

5 6 7 8 9

9330 9298 9267 9237 9209

9327 9295 9264 9234 9206

9324 9292 9261 9231 9203

9321 9289 9258 9229 9201

9317 9285 9255 9226 9198

9314 9232 9252 9223 9195

9311 9279 9249 9220 9192

9308 9276 9246 9217 9190

9304 9273 9243 9214 9187

9301 9270 9240 9212 9184

1 20 1 2 3 4

0.20 I 2 3 4

9182 9156 9131 9108 9085

9179 9153 9129 9105 9083

9176 9151 9126 9103 9081

9174 9148 9124 9101 9079

9171 9146 9122 9098 9077

9169 9143 9119 9096 9074

9166 9141 9117 9094 9072

9163 9138 9114 9092 9070

9161 9136 9112 9090 9068

9158 9133 9110 9087 9066

5 6 7 8 V

5 6 7 8 9

9064 9044 9025 9007 8990

9062 9042 9023 9005 8989

9060 9040 9021 9004 8987

9058 9038 9020 9002 8986

9056 9036 9018 9000 8984

9054 9034 9016 8999 8982

9052 9032 9014 8997 8981

9050 9031 9012 8995 8979

9048 9029 9011 8994 8978

9046 9027 9009 8992 8976

1.30 1 2 3 4

0.30 I 2 3 4

8975 8960 8946 8934 8922

8973 8959 8945 8933 8921

8972 8957 8944 8931 8920

8970 8956 8943 8930 8919

8969 8954 8941 8929 8918

8967 8953 8940 8928 8917

8966 8952 8939 8927 8916

8964 8950 8937 8926 8915

8963 8949 8936 8924 8914

8961 8948 8935 8923 8913

5 b 7 8 9

5 6 7 8 9

8912 8902 8893 8885 8879

8911 8901 8892 8885 8878

8910 8900 8892 8884 8877

8909 8899 8891 8883 8877

8908 8898 8890 8883 8876

8907 8897 8889 8882 8875

8906 8897 8888 8881 8875

8905 8896 8888 8880 8874

8904 8895 8887 8880 8874

8903 8894 8886 8879 8873

1.40 1 2

0 40 1 2 3 4

8873 8868 8864 8860 8858

8872 8867 8863 8860 8858

8872 8867 8863 8860 8858

8871 8866 8863 8860 8858

8871 8866 8862 8859 8857

8870 8865 8862 8859 8857

8870 8865 8862 8859 8857

8869 8865 8661 8859 8857

8869 8864 8861 8858 8857

8868 8864 8861 8858 8857

5 6 7 b 9

8857 8856 8856 8857 8859

8857 8856 8856 8858 8860

8856 8856 8856 8858 8860

8856 8856 8857 8858 8860

8856 8856 8857 8858 8860

8856 8856 8857 8858 8861

8856 8856 8857 8859 8861

8856 8856 8657 8859 8861

8856 8856 8857 8859 8862

8856 8856 8857 8859 8862

8862

8863

8863

8863

8864

8864

8864

8865

8865

8866

9

o.so

1.50

I'd)

-Jo

«•---■<«

For higher values of argument

rd)

=.

(i

Example:

-

\>r(x

r(4.7)

-!)-(*-

-

(3.7)1

-

3.7

(x

i)(i

-

X

2.7

-

2)r(z X

/o

1)!

-

2)

-

1.7 X 0.9086

x\

-

<»«- - rd + ») rd + »)

- x(.x -

= 15.43.

I)!

i(x

-

l)(i

-

2)!

GENERAL

1-118


Table 14.

DATA

[SEC.

Gamma and Factorial Functions: T(x) = y\

(Continued)

0

1

2

3

4

5

6

7

8

9

0.8862 8866 8870 8876 8882

8863 8866 8871 8876 8882

8863 8867 8871 8877 8883

8863 8867 8872 8877 8884

8864 8868 8872 8878 8884

8864 8868 8873 8879 8885

8864 8869 8873 8879 8886

8865 8869 8874 8880 8887

8865 8869 8875 8880 8887

8866 8870 8875 8881 8888

5 6 7 8 9

8889 8896 8905 8914 8924

8889 8897 8906 8915 8925

8890 8898 8907 8916 8926

8891 8899 8908 8917 8927

8892 8900 8909 8918 8929

8892 8901 8909 8919 8930

8893 8901 8910 8920 8931

8894 8902 8911 8921 8932

8895 8903 8912 8922 8933

8896 8904 8913 8923 8934

1.60 1 2 3 4

60 1 2 3 4

8935 8947 8959 8972 8986

8936 8948 8961 8974 8988

8937 8949 8962 8975 8989

8939 8950 8963 8977 8991

8940 8952 8964 8978 8992

8941 8953 8966 8979 8994

8942 8954 8967 8981 8995

8943 8955 8968 8982 8997

8944 8957 8970 8984 8998

8946 8958 8971 8985 9000

5 6 7 8 9

5 6 7 8 9

9001 9017 9033 9050 9068

9003 9018 9035 9052 9070

9004 9020 9036 9054 9071

9006 9021 9038 9055 9073

9007 9023 9040 9057 9075

9009 9025 9041 9059 9077

9010 9026 9043 9061 9079

9012 9028 9045 9062 9081

9014 9030 9047 9064 9083

9015 9031 9048 9066 9084

1.70 1 2 3 4

70 1 2 3 4

9086 9106 9126 9147 9168

9088 9108 9128 9149 9170

9090 9110 9130 9151 9173

9092 9112 9132 9153 9175

9094 9114 9134 9155 9177

9096 9116 9136 9157 9179

9098 9118 9138 9160 9182

9100 9120 9140 9162 9184

9102 9122 9142 9164 9186

9104 9124 9145 9166 9188

5 6 7 8 1

5 6 7 8 9

9191 9214 9238 9262 9288

9193 9216 9240 9265 9290

9195 9218 9242 9267 9293

9197 9221 9245 9270 9295

9200 9223 9247 9272 9298

9202 9226 9250 9275 9301

9204 9228 9252 9277 9303

9207 9230 9255 9280 9306

9209 9233 9257 9283 9309

9211 9235 9260 9285 9311

1.80 1

80 1 2 3 4

9314 9341 9368 9397 9426

9316 9343 9371 9400 9429

9319 9346 9374 9403 9432

9322 9349 9377 9406 9435

9325 9352 9380 9408 9438

9327 9355 9383 9411 9441

9330 9357 9385 9414 9444

9333 9360 9388 9417 9447

9335 9363 9291 9420 9450

9338 9366 9394 9423 9453

5 6 7 8 9

9456 9487 9518 9551 9584

9459 9490 9522 9554 9587

9462 9493 9525 9557 9591

9465 9496 9528 9561 9594

9468 9499 9531 9564 9597

9471 9503 9534 9567 9601

9474 9506 9538 9570 9604

9478 9509 9541 9574 9607

9481 9512 9544 9577 9611

9484 9515 9547 9580 9614

90 1 2 3 4

9618 9652 9688 9724 9761

9621 9656 9691 9728 9765

9625 9659 9695 9731 9768

9628 9663 9699 9735 9772

9631 9666 9702 9739 9776

9635 9670 9706 9742 9780

9638 9673 9709 9746 9784

9642 9677 9713 9750 9787

9645 9681 9717 9754 9791

9649 9684 9720 9757 9795

5 6 7 8 9

9799 9837 9877 9917 9958

9803 9841 9881 9921 9962

9806 9845 9885 9925 9966

9810 9849 9889 9929 9971

9814 9853 9893 9933 9975

9818 9857 9897 9938 9979

9822 9861 9901 9942 9983

9826 9865 9905 9946 9987

9830 9869 9909 9950 9992

9834 9873 9913 9954 9996

1. 0000

0004

0008

0013

0017

0021

0026

0030

0034

0038

X

V

1.50 1 2 3 4

0.50 1 2 3 4

s 6 7 8 9

1.90

2.00

1.00

Minimum value of the function: I'( 1.461632)

-

(0.461632)!

= 0.885603.

1

tables

mathematical

Sec. 1-2]

1-119

Exponential and Related Integrals

Table 15.

The Exponential Integral*. Functions usually used in combination are ( 1) the exponential integrals z or (2) Eo(z), E\(z), and Bt(x),* Ei(x) -Ei(-z). In all cases x ja positiveuse the formulas or series Small Valuea of the Argument. Where interpolation is unsatisfactory, solutions given in the tables. Large Values of the Argument for Ei(x) =■ Bi( — x), a. For 5 < z < 16 sec special tables following main tables. b. For 5 < x < 40. Ei(z) -Jf*(-«) F\e'*/z and Ei z Ft*?*/*, where Fx and Ft Are given below and e~* and e* are read from Table 6.

-

Ei(—x) and Ei

-

-

-

J F,

1.354

■♦I

i

For

c.

15

10

0.8516

Large

30

35

40

0.9408

0.9549

0.9627

0.9687

0.9729

0.9753

1. 1316

1.0781

1.0360

1.0440

1.0358

1.0305

1.0264

1

large values of x,

Fi

and

I!

+

X

Ft

can be computed

-

2!

3!

x'

X*

- -

Values of the Argument for Eotx) •» «-»/« and Ei(x)

-

from the scmioonvergent

F,

+

-

and

£1(2).

-

«-«/«

EoM

I1 X Use

+

- 2'

X*

the

+

3'

X*

series

+

formulas

given

In the tables,

1

2

3

4

5

6

7

8

9

m 9.048 4.094 2.469 1.676

99.00 8. 144 3.860 2.366 1.619

49.01 7.391 3.648 2.269 1.564

32.35 6 755 3.454 2. 179 1.513

24.02 6.210 3.278 2.093 1.464

19.02 5.738 3. 115 2.013 1.417

15.70 5.326 2.966 1.938 1.372

13.32 4.963 2.827 1.867 1.330

11.54 4.640 2.699 1.800 1.289

10. 15 4.352 2.580 1.736 1.250

1.213 0.9147 7094 5617 4517

1. 177 8907 6925 5492 4423

1. 143 8677 6760 5371 4332

1.111 8454 6601 5254 4243

1.079 8239 6447 5139 4156

1.049 8031 6298 5028 4071

1.020 7831 6154 4920 3988

•9921 7637 6013 4816 3908

•9653 7450 5877 4713 3830

•9395 7269 5745 4614 3753

3679 3026 2510 2096 1761

3606 2969 2464 2060 1732

3535 2913 2420 2024 1722

3466 2859 2376 1989 1673

3399 2805 2334 1954 1645

3333 2753 2292 1920 1618

3268 2702 2251 1887 1591

3206 2653 2211 1855 1564

3144 2604 2172 1823 1338

3085 2556 2134 1792 1513

1488 1262 1075 0.09183 7872

1463 1242 1058 9042 7753

1439 1222 1041 8903 7636

1415 1202 1025 8766 7521

1392 1183 1009 8631 7407

1369 1164 •9930 8500 7296

1347 1145 •9775 8370 7187

1325 1127 •9623 8242 7079

1304 1109 •9474 8116 6973

1283 1092 •9327 7993 6869

0.0 1.0 2.0 3.0

* 0

n 0.3679 10-' X 0.6767 10"' X 1.660 IO-« X 0. 4579

9.048 3026 5831 1.453 4042

4.094 2510 5037 1.274 3570

2.469 2096 4359 1. 118 3155

1.676 1761 3780 0.9816 2790

1.213 1488 3283 8628 2469

•9147 1262 2857 7590 2185

•7094 1075 2489 6682 1935

•5617 •9183 2172 5887 1714

•4517 •7872 1897 5190 1520

5 f. 7 ft 1

10-< X 1.348 10-' X 0.4131 10-' X 1.303 IO-« X 0 4193 10 ' X 1.371

1. 195 3677 1. 1621 3747 1.227

1.061 3273 1.037 3349 1.098

0.9418 2915 0.9254 2994 0.9831

8364 2596 8260 2677 8800

7431 2313 7375 2394 7879

6603 2061 6585 2141 7055

5870 1837 5881 1915 6318

5220 1638 5253 1713 5658

4643 1461 4693 1533 5068

0


25

20

0.9156

Ft

i.e.. /?a(x)

-

0 0 1 2 3 «

j 7 8 » 1.0 2

J

4

j | 7 ft »

0 0 0 0 0

• The functions are represented %nd the exponential

integrals

by the general formula E*(x) =

can be expressed in various

alternative

jj

«~*"a~" du.

forms.

These

functions

GENERAL

1-120 Table 16.

0 go

1

{Continued)

Si(-z)

1

2

3

4

5

6

7

8

9

1.823 1.223 0.9057 7024

4.038 1.737 1.183 8815 6859

3.355 1.660 1. 145 8583 6700

2.959 1.589 1 110 8361 6546

2.681 1.524 1.076 8147 6397

2.468

1 2 3 4

1.464 1.044 7942 6253

2.295 1.409 1.014 7745 6114

2.151 1.358 0.9849 7554 5979

2.027 1.310 9573 7371 5848

1.919 1.265 9309 7194 5721

5 6 7 8 9

5598 4544 3738 3106 2602

5478 4454 3668 3050 2557

5362 4366 3599 2996 2513

5250 4280 3532 2943 2470

5140 4197 3467 2891 2429

5034 4115 3403 2840 2387

4930 4036 3341 2790 2347

4830 3959 3280 2742 2308

4732 3883 3221 2694 2269

4636 3810 3163 2647 2231

1.0 1 2 3 4

2194 I860 1584 1355 1162

2157 1830 1559 1334 1145

2122 1801 1535 1313 1128

2087 1772 1511 1293 1111

2052 1743 1487 1274 1094

2019 1716 1464 1254 1078

1986 1688 1441 1235 1062

1953 1662 1419 1216 1046

1922 1635 1397 1198 1030

1890 1609 1376 1160 1015

5 6 7 8 9

1000 0.08631 7465 6471 5620

•9854 8506 7359 6380 5542

•9709 8383 7254 6290 5465

•9567 8261 7151 6202 5390

•9426 8142 7049 6115 5315

•9288 8025 6949 6029 5241

• 9152 7909 6850 5945 5169

•9019 7796 6753 5862 5098

•8887 7684 6658 5780 5027

•8758 7574 6564 5700 4958

2.0 1 2 3 4

4890 4261 3719 3250 2844

4823 4204 3669 3207 2806

4757 4147 3620 3164 2769

4692 4090 3571 3122 2733

4627 4035 3523 3081 2697

4564 3980 3476 3040 2662

4502 3927 3430 3000 2627

4440 3874 3384 2960 2592

4380 3821 3339 2921 2558

4320 3770 3294 2882 2525

5 6 7 8 9

2491 2185 1918 1686 1482

2459 2157 1893 1664 1464

2427 2129 1869 1643 1445

2395 2101 1845 1622 1427

2364 2074 1821 1601 1409

2333 2047 1798 1581 1391

2303 2021 1775 1560 1373

2273 1994 1752 1540 1356

2243 1969 1730 1521 1338

2214 1943 1707 1502 1322

1305 1149 1 2 1013 3 0.008939 4 7891

1288 1135 1001 8828 7793

1272 1121 •9882 8718 7697

1256 1107 •9758 8610 7602

1240 1093 •9637 8503 7508

1225 1079 •9517 8398 7416

1209 1066 •9398 8294 7324

1194 1052 •9281 8191 7234

1179 1039 •9166 8090 7145

1164 1026 •9052 7990 7057

5 6 7 8 9

6970 6160 5448 4820 4267

6884 6085 5381 4762 4216

6800 6011 5316 4704 4165

6716 5937 5251 4647 4114

6634 5864 5187 4591 4065

6552 5793 5124 4535 4016

6472 5722 5062 4480 3967

6393 5652 5000 4426 3919

6314 5583 4939 4372 3872

6237 5515 487) 4319 3825

4.0 1 2 3 4

3779 3349 2969 2633 2336

3734 3309 2933 2602 2308

3689 3269 2898 2571 2281

3645 3230 2664 2540 2254

3601 3191 2829 2510 2227

3557 3153 2796 2480 2201

3515 3115 2762 2450 2175

3472 3078 2729 2421 2149

3431 3041 2697 2393 2123

3390 3005 2665 2364 2098

5 6 7 8 9

2073 1841 1635 1453 1291

2049 1819 1616 1436 1276

2025 1798 1597 1419 1261

2001 1777 1578 1402 1247

1977 1756 1560 1386 1232

1954 1735 1541 1370 1218

1931 1715 1523 1354 1204

1908 1694 1505 1338 1189

1885 1674 1488 1322 1176

1863 1655 1470 1307 1162

5.0

1148

1135

1122

1109

1096

1083

1070

1058

1045

1033

1-112-2!

3- 3!

0.0


[Sec.

and Related Integrals.

Exponential

jBiCx) X

DATA

3.0

i'i(j-)

where In

/; «-'»u-' fz

du

-

—

= 0.5772 + In z

Ei(-z)

-/:

t-"u
mathematical tables

Sec. 1-2] Table

Exponential

16.

1-121

and Related Integrals.

(Continued)

Wix = Ei(x)

I

0

1

2

3

4

5

6

7

8

9

4.018

3.315

2.899

1.419 7042 2147 1783

1.329 6485 1721 2143

2.601 1.244 5947 1304 2498

2.368

1.517 7619 2582 1418

2.175 1.089 4919 0493 3195

2.011 1.017 4427 0098 3537

1.867 .9491 3949 *0290 3876

1.739 .8841 3482 •0672 4211

4870 8002 1.094 1.375 1.650

5195 8302 1.122 1.403 1.677

5517 8601 1.151 1.431 1.705

5836 8898 1.179 1.458 1.732

6153 9194 1.207

6467 9488 1.236 1.513 1.786

6778 9780 1.264 1.541 1.814

7087 1.007 1.292 1.568 1.841

7394 1.036 1.320 1.595 1.868

1. 895 2. 167

1.922 2. 195 2.470

1.949

2.222

1.977 2.249

2.004 2.277

2.058 2.332

2.086

2.750 3.036

2.778 3.065

2.921

2.113 2.387 2.665 2.949

2.140 2.414

2.721

2.031 2.304 2.581 2.863

3.212

3.242

3.271

3.544 3.857 4.183

« 0 0 — 1 1.623 2 -0.8218 3 .3027 4 + .1048

-

-

5 6 7 8 9


1.0 1 2 3 4

.4542

.7699 1.065 1.347 1.623

2.442 3.007

5 6 7 8 9

3.301 3.605 3.921 4.250

2.0 1 2 3 4

4.954 5.333 5.733

5

t

7 8 9

4.59'

3.331 3.636

7.074 7.576 8. 110

8.679 9 286

2.525 2.806 3.094

3.361

3.391

3.953 4.284 4.629

3.667 3.986 4.317 4.664

3.699

4.991

5.028

5.066

5.411

5.451 5.857

5.372

6. 154 6 601

2.498

5.774 6. 198

6.647 7.123 7.628 8. 166

4.018 4.351

4.700

5.815 6.242

6.286 6.740

6.693

7.172

7.221 7.733

2.553 2.835

1. 164 5425 0894 2849

I486 1.759

2.609

2.892 3.183

2.359

2.637

3.124

3.153

3.422 3.730

3.452 3.762 4.084 4.420 4.772

3.482 3.793 4.117 4.454

3.513 3.825 4.150 4.489

4.808

5.141

4.051

4.386 4.736 5.104 5.490 5.899 6.330 6.787 7.271

2.693

2.978 3.574 3.889 4.216 4.559

4.844

4.524 4.881

4.917

5.179 5.570

5.217 5.611

5.256 5.651

5.941

5.9B3

6.374 6.834

6.419 6.881

6.025 6.464 6.929

6.068 6.509 6.977

5.692

7.321 7.839

7.372

7.422

7.893

7.947

7.473 8.001 8.563

5.530

5.294 6.

Ill

6.555 7.025 7.524

8.738 9.349

7.798 9.412

8.277 8.857 9.476

7.786 8.334 8.917 9.540

10.07 10.77 11.52 12.33 13.19

10.14 10.84 11.60 12.41 13.28

10.21 10.92 11.68 12.49 13.37

10.27 10.99 11.76 12.58 13.46

10.34 11.06 11.84 12.66 13.55

10.41 11.14 11.92 12.75 13.64

10.48 11.22 12.00 12.84 13.74

10.55 11.29 12.08 12.92 13.83

7.680 8.221

8.055

8.390 8.978 9.605

8.447 9.039 9.670

8.505 9.735

9.801

9.224 9.867

9. 100

9.162

8.621

3 0 1 2 3 4

9.934 10.63 11.37 12. 16 13.01

10.00 10.70 11.44 12.24 13.10

5 6 7 8 9

13.93 14.91 15.96 17. 09 18.32

14.02 15.01 16.07 17.21 18.44

14.12 15.11 16.18 17.33 18.57

14.21 15.21 16.29 17.45 18.70

14.31 15.32 16.40 17.57 18.83

14.41 15.42 16.52 17.69 18.96

14.51 15.53 16.63 17.82 19.09

14.60 15.64 16.75 17.94 19.23

14.70 15.74 16.86 18.06 19.36

14.80 15.85 16.98 18.19 19.49

4.0 1 2 3 4

19.63

19.77 21.20 22.74 24.40 26.19

19.91 21.35

20.05 21.50

20.19 21.65

20.33

21.05 22.58 24.23

23.06

23.22

22.42

24.57

24.75 26.57

24.92 26.76

23.39 25.10 26.95

20.61 22.11 23.72 25.46

20.76 22.26

22.90

20.47 21.95 23.55

23.89 25.64 27.54

24.06 25.82 27.73

29.80

39.31

29.58 31.80 34.20 36.79 39.60

39.89

42.32

42.64

42.96

5 6 7 H 9 5.0

26.01 27 93 30.01 32 26

32.50

34.70 37.33

34 95 37.61

32.74 35.21 37.88

40. 19

40.48

40.78

where In yx

-

28.13 30.23

26.38 28.34

30.45

Ei

0.5772

+ In

i

x

"

28.54 30.67 32.97 35.47 38. 16

41.09 In yx + ■

21.80

25.28 27.15

27.34 29.37 31.57 33.95 36.52

28.75

28.95

29.16

30.89

31.12

33.21

33.46

35.73

35.99

31.34 33.70

38.45

38.73

39.02

41.39

41.70

42.01

II

+

x* -=—

2- 2!

z> + -=—

3-3!

36.25

20.90

32.03

34.45 37.06

GENERAL DATA

1-122

Exponential and Related Integrals.

Table 16.

Bix =

5 6 7 8 9 10

II

12 13 14

1

1

3

10"' 10-' I0"< IO-<

X 1148 X 0.3601 X 1.155 X 0.3767 10-' X 1.245

1.021 3211 1.032 3370 1. 115

9086 2864 9219 3015 9988

8086 2555 8239 2699 8948

10-' X 0 4157 1.400 10"' X 10-' X 0. 4751 1.622 10-' X 10-' X 0 5566

3727 1.256 4266 1. 457 5002

3342 1. 127 3830 1.309 4500

2997 1.012 3440 1. 176 4042

Six


-Ei(-x).

0

X

[Sec.

1

(Continued)

5 < x < 16 4 7198 2279 7364 2415 8018

5

7

6

8

9

6409 2034 6583 2162 7185

5708 1816 5886 1936 6439

5085 1621 5263 1733 5771

4532 1448 4707 1552 5173

4039 1293 4210 1390 4637

2687 2410 8149 0.9080 3089 2774 1.057 0.9495 3633 3266

2162 7315 2491 8532 2936

1939 6566 2238 7667 2640

1740 5894 2010 6890 2373

1561 5291 1805 6193 2134

= EHx). 5 < x < 16

S 6 7 8 9

10' I0» 10' 10' I0«

X X X X X

0.4019 0.8599 0. 1915 0.4404 0. 1038

4328 9300 2079 4793 1132

4662 1.006 2257 5218 1235

5026 1.089 2451 5682 1347

5419 1. 179 2663 6189 1471

5847 1.277 2894 6743 1605

6310 1.384 3146 7347 1752

6813 1. 501 3420 8007 1913

7360 1.627 3720 8729 2089

7954 1.765 4047 9517 2282

10 11 12 13 14

10* I0< I0» I01 10'

X X X X X

0. 2492 0.6071 0. 1496 0.3720 0.9319

2723 6641 1638 4076 1.022

2975 7265 1794 4467 1. 121

3251 7949 1964 4896 1.229

3553 8698 2151 5367 1.348

3884 9518 2357 5883 1.479

4246 1.042 2581 6449 1.622

4642 1. 140 2828 7070 1.779

5076 1.248 3098 7751 1.952

5551 1.366 3395 8499 2. 142

*.C>

1 2 3 4

i

6 7 S 9

xEdx)

= x[E„(x)

-

£,(*)]

2

3

4

5

6

7

8

9

0.7225 5742 4691 3894

0.9497 7048 5622 4602 3824

9131 6878 5505 4515 3756

8817 6715 5393 4430 3690

8535 6560 5283 4348 3626

8278 6410 5177 4267 3562

8040 6267 5074 4189 3500

7818 6128 4974 4112 3440

7610 5995 4877 4038 3381

7412 5866 4783 3963 3325

3266 2762 2349 2009 1724

3211 2717 2312 1978 1698

3157 2673 2276 1948 1673

3104 2630 2240 1918 1648

3052 2587 2205 1889 1623

3001 2546 2171 I860 1599

2951 2505 2137 1832 1576

2902 2465 2104 1804 1552

2855 2426 2072 1777 1530

2808 2387 2040 1750 1507

1485 1283

1442 1246 1080 9378 8160

1421 1228 1065 9247 8048

1400 1211 1050 91 19 7938

1380 1193 1035 8993 7829

1360 1176 1020 8868 7722

1340 1160 1006 8746 7617

1321 1143 •9920 8625 7513

1302 1127 ♦9781 8506 741 1

i.ooo

0.0

-

/

1

0

X

-

1.0 1 2 3 4

0.09645 8389

llll

1463 1264 1095 9510 8274

3 6 7 8 9

7310 6380 5577 4882 4278

7211 6295 5503 4817 4222

71 13 6210 5430 4754 4167

7017 6127 5358 4691 41 13

6922 6045 5287 4630 4059

6828 5964 5217 4569 4007

6736 5884 5148 4509 3955

6645 5806 5080 4450 3903

6555 5729 5013 4392 3852

6467 5652 4947 4335 3803

1.000 1.485 10' X 0.3753 10"' X 1.064 10' X 0.3198

0.7225 1.283 3297 0.9417 2842

5742 1. 2898 8337 2527

4691 0.9645 2550 7384 2247

3894 8389 2246 6544 1999

3266 7310 1980 5802 1779

2762 6380 1746 5146 1583

2349 5577 1541 4567 1410

2009 4882 1362 4054 1255

1724 4278 1203 3600 1118

10"' X 0. 9965 10 ' X 0.3183 IO-< X 1.035 10 < X 0.3414 10-' X 1.138

8881 7917 2842 2539 0.9259 8283 3057 2738 1.021 0.9149

7060 2268 7411 2453 8203

6296 2027 6632 2198 7356

5617 1812 5935 1969 6597

5012 1619 5313 1764 5916

4473 1448 4756 1581 5306

3992 1294 4258 1417 4760

3564 1157 3812 1270 4270

0 1 2 3 4 S 6 7 8 9

10-' X

Ill

Table 16.

P'M

P*(.z)

0 001-0. sooo 0 0000 0. 3750 0.0000 0 4998 -0150 3746 0187 0 -4994 0300 3735 0374 0 -4986 -0449 3716 0560 M 4976 0598 3690 0744 |

-

-

- 4878

10 II 12 13 14

-4850 -4818 1617 -4784 -1757 -4746 1895 -4706 2031

3379 1788 3303 1947 • 3219 '2101 2248 31.29 2389 3032

IS 16 17 18 19

--2I«6 - 4566 - %298 2427 -4514 - 2554 2679 -445(6

2928 2819 2703 2581 2453

2523 2650 2769 2880 2982

2320 2181 2037 1889 1735

3075 3159 3234 3299 3353

1577 1415 1249 1079 0906 0729 0550 0369 0185

45 46 47 48 49

4274

4206 -3146 //--4136 - 3254

- 3740 - 3825

3359 -4062 -3986 - 3461 3558 -3906 -3824 -3651

- 3738

- 3650 - 3558 - 3905 -3464 - 3981 - 3366 - 4052 - 3266 -4117

-0000

-3162 -3056

-0375

- 4178 -0187 - 4234

- 2946 -4284 - 0564 - 2834 - 4328 - 0753 -2718 - 4367 -0942 - 2600 -4400 - 130 -2478 - 4427 - 1317 - 2354 - 4448 - 1504 - 2226 -4462 - 1688 - 2096 - 4470 - 1870 - 4472 - 2050 - 1962 1826 - 4467 - 2226 - 1686 - 4454 - 2399 -1544 -4435 - 2568

-

1398 -4409

- 2732

-2606 -2488 -2360

- 0.4375 - 0. 2891

-0944 -0786 -0626 -0462

- 0296 -0126 + 0046 0222

- 4334 -3044 -3191 - 3332 --4285 4228 -4163

-3465

-4091 -4010

-3590

- 3920 - 3822 -3716

- 3707

-3815 -3914 -4002

+ 0.0898 + 0.3232 3166 0673 0441 3080 2975 + 0204 2851 -0037

-0282

2708 2546 2366 2168 1953

--0529 0779

- 1028 - 1278 -

-3600 -3475

-4080 -4146 -4200

- 1526 1772

-2884 -2713

-4284

-2705 -2919 -3122 -3313

1721 1473 1211 0935 0646

1913

60 61 62 63 64

0400 0582 0766 0954 1144

- 1746 - 1572 - 1389 - 1201 1006

65 66 67 68 69

1338 1534 1734 1936 2142

-0806 -0601 -0394 -0183 0029

70 71 72 73 74

2350 2562 2776 2994 3214

3397 3431 3453 3465 3465

0243 0456 0669 0879 1087

75 76 77 78 79

3438 3664 3894 4126 4362

3454 3431 3397 3351 3294

1292 1492 1686 1873 2053

80 81 82 83 84

4600 4842 5086 5334 5584

0800 1136 1484 1845 2218

- 2330 2021

3225 3144 3051 2948 2833

2225 2388 2540 2681 2810

85 86 87 88 89

5838 6094 6354 6616 6882

2603 3001 3413 3837 4274

-4030 2857 - 0053 - 2484 -3872 - 2056 - 3638 + 0431

2706 2569 2421 2263 2095

2926 3029 31 18 3191 3249

90 91 92 93 94

7150 7422 7696 7974 8254

4725 5189 5667 6159 6665

2079 2698 3352 4044 4773

1917 1730 1534 1330 1118

3290 3314 3321 3310 3280

95 96 97 98 99

8538 8824 9114 9406 9702

7184 7718 8267 8830 9407

5541 6349 7198 8089 9022

3727 4796 5954 7204 8552

1875 3151 4590 6204 8003

1.00

. 0000

. 0000

. 0000

0000

. 0000

- 2220

- 2071

-

Pi(x)

■»x for all values of z

- 3342 - 4242 -2014 -2251 -3199 - 3046 - 4270 - 2482 - 2531 - 2339

-2137

- 4284 - 4268

-4236 -4187

- 1925 -4121 - 1702 -4036 - 1469 - 3933 - 1225 -3810

0347 + 0038

- 3490

-0278 -0601 -0926

-2214 -2518

- 1253 -1578 - 1899

-0969

-3666

-3652 -3796 -3922 -4026 -4107

-0703 -0426 -0137

-3501 -3314 -3104 -2871 -2613

-4164 -4193 -4193 -4162 -4097

-2808 -3081

- 1685

-3995 -3855 -3674

-3918 -4041 -4119 -4147 -4120

+ 0164 0476

-

1321

-0928

-0506

0947 1496

- 3449

-3177

-

-

1570 1023

-041

026B 1017 1842 2744

- 3333 -3559 -3756

-3322 -2916

- 1802 - 1077 -2412

- 0229 +

40 41 42 43 44

.-

- 2800 - 2918 - 3034

-2713

55 , 56 57 58 59

1250 1098

1

-jMOO

/4338

--2962 - 2891 2808

-

P>M

1

35 36 37 38 39

-

0927 1106 1283 1455 1624

-0.

+

30 31 32 33 34

- 1332 - 1475

3657 3616 3567 3512 3449

1

28 29

-4904 -1187

-4662 -4616

11/

20 21 22 2! 24

- 4946 -0747 -0895 -4926 - 1041

-3099 -3066 -3021

0 50 51 52 53 54

1 .

-4962

05 Oft 07 OS 0'!

i


-

-0.3125 -3118

1

-

P.(x)

X

1

Pid)

1-123

Legendre Polynomials

1

■

tables

mathematical

Sec. 1-2]

0751

1-124


I

0

1

Probability

[SEC.

Function or Error Integral

:

1

erf x

2

3

4

5

6

7

•

9

| 0 0000 0113 0226 0338 0451

0011 0124 0237 0350 0462

0023 0135 0248 0361 0474

0034 0147 0260 0372 0485

0045 0158 0271 0384 0496

0056 0169 0282 0395 0507

0068 0181 0293 0406 0519

0079 0192 0305 0417 0530

0090 0203 0316 0429 0541

0102 0214 0327 0440 0553

9

0564 0676 0789 0901 1013

0575 0688 0800 0912 1024

0586 0699 0811 0923 1035

0598 0710 0822 0934 1046

0609 0721 0834 0946 1058

0620 0732 0845 0957 1069

0631 0744 0856 0968 1080

0643 0755 0867 0979 1091

0654 0766 0878 0990 1102

0665 0777 0890 1002 1113

10 1 2 3 4

1125 1236 1348 1459 1569

1136 1247 1359 1470 1581

1147 1259 1370 1481 1592

1158 1270 1381 1492 1603

1169 1281 1392 1503 1614

1180 1292 1403 1514 1625

1192 1303 14(4 1525 1636

1203 1314 1425 1536 1647

1214 1325 1436 1547 1658

1225 1336 1448 1558 1669

6 7 8 9

1680 1790 1900 2009 2118

1691 1801 1911 2020 2129

1702 1812 1922 2031 2140

1713 1823 1933 2042 2151

1724 1834 1944 2053 2162

1735 1845 1955 2064 2173

1746 1856 1966 2075 2184

175? 1867 1977 2086 2194

1768 1878 1988 2097 2V05

1779 1889 1998 2108 2216

20 1 2 } 4

2227 2335 2443 2550 2657

2238 2346 2454 2561 2668

2249 2357 2464 2572 2678

2260 2368 2475 2582 2689

2270 2378 2486 2593 2700

2281 2389 2497 2604 2710

2292 2400 2507 2614 2721

2303 2411 2518 2625 2731

2314 2421 2529 2636 2742

2324 2432 2540 i 2646

s 6 7 9

2763 2869 2974 3079 3183

2774 2880 2985 3089 3193

2784 2890 2995 3100 3204

2795 2901 3006 3110 3214

2806 2911 3016 3120 3224

2816 2922 3027 3131 3235

2827 2932 3037 3141 3245

2837 2943 3047 3152 3255

2848 2953 3058 3162 3266

29164 30(68 3172 3276

10 1 2 3 4

3286 3389 3491 3593 3694

3297 3399 3501 3603 3704

3307 3410 3512 3613 3714

3317 3420 3522 3623 3724

3327 3430 3532 3633 3734

3338 3440 3542 3643 3744

3348 3450 3552 3653 3754

3358 3461 3562 3663 3764

3369 3471 3573 3674 3774

3379 3481 3583 3684 3784

5 6 7 8 9

3794 3893 3992 4090 4187

3804 3903 4002 4100 4197

3814 3913 4012 4110 4207

3824 3923 4022 4119 4216

3834 3933 4031 4129 4226

3844 3943 4041 4139 4236

3854 3953 4051 4149 4245

3864 3963 4061 4158 4255

3873 3972 4071 4168 4265

3883 3982 4080 4178 4274

40 2 3 4

4284 4380 4475 4569 4662

4294 4389 4484 4578 4672

4303 4399 4494 4588 4681

4313 4408 4503 4597 4690

4322 4418 4512 4606 4699

4332 4427 4522 4616 4709

4341 4437 4531 4625 4718

4351 4446 4541 4634 4727

4361 4456 4550 4644 4736

4370 4465 4359 4653 4746

s 6 7 8 9

4755 4847 4937 5027 5117

4764 4856 4946 5036 5126

4773 4865 4956 5045 5134

4782 4874 4965 5054 5143

4792 4883 4974 5063 5152

4801 4892 4983 5072 5161

4810 4901 4992 5081 5170

4819 4910 5001 5090 5179

4828 4919 5010 5099 5187

4837 4928 5019 5106 5196

50

5205

5214

5223

5231

5240

5249

5258

5266

5275

5284

0 00 1 2

J

4

i

t 7

t


j

erf

i-

11(7)

- — [x,-"dl

\

\

\2753

M58

mathematical tables

Sec. 1-2]

Probability Function or Error Integral

Table 17.

I 0 50 1


60

70

80

90

1.00

fi

1-125 :

erf x.

(Continued)

0

1

2

3

4

5

6

7

8

9

0.5205 5292 5379 5465 5549

5214 5301 5388 5473 5558

5223 5310 5396 5482 5566

5231 5318 5405 5490 5575

5240 5327 5413 5499 5583

5249 5336 5422 5507 5591

5258 5344 5430 5516 5600

5266 5353 5439 5524 5608

5275 5362 5448 5533 5617

5284 5370 5456 5541 5625

5633 5716 5798 5879 5959

5642 5724 5806 5887 5967

5650 5733 5814 5895 5975

5658 5741 5823 5903 5983

5667 5749 5831 5911 5991

5675 5757 5839 5919 5999

5683 5765 5847 5927 6007

5691 5774 5855 5935 6015

5700 5782 5863 5943 6023

5708 5790 5871 5951 6031

6039 6117 6194 6270 6346

6046 6125 6202 6278 6353

6054 6132 6209 6286 6361

6062 6140 6217 6293 6368

6070 6148 6225 6301 6376

6078 6156 6232 6308 6383

6086 6163 6240 6316 6391

6093 6171 6248 6323 6398

6101 6179 6255 6331 6405

6109 6186 6263 6338 6413

6420 6494 6566 6638 6708

6428 6501 6573 6645 6715

6435 6508 6581 6652 6722

6442 6516 6588 6659 6729

6450 6523 6595 6666 6736

6457 6530 6602 6673 6743

6464 6537 6609 6680 6750

6472 6545 6616 6687 6757

6479 6552 6624 6694 6764

6486 6559 6631 6701 6771

6778 6847 6914 6981 7047

6785 6853 6921 6988 7053

6792 6860 6928 6994 7060

6799 6867 6934 7001 7066

6806 6874 6941 7007 7073

6812 6881 6948 7014 7079

6819 6887 6954 7021 7086

6826 6894 6961 7027 7092

6833 6901 6968 7034 7099

6840 6908 6974 7040 7105

7112 7175 7238 7300 7361

7118 7182 7244 7306 7367

7124 7188 7251 7312 7373

7131 7194 7257 7318 7379

7137 7201 7263 7325 7385

7144 7207 7269 7331 7391

7150 7213 7275 7337 7397

7156 7219 7282 7343 7403

7163 7226 7288 7349 7409

7169 7232 7294 7355 7415

7421 7480 7538 7595 7651

7427 7486 7544 7601 7657

7433 7492 7550 7607 7663

7439 7498 7555 7612 7668

7445 7503 7561 7618 7674

7451 7509 7567 7623 7679

7457 7515 7572 7629 7685

7462 7521 7578 7635 7690

7468 7527 7584 7640 7696

7474 7532 7590 7646 7701

7707 7761 7814 7867 79)8

7712 7766 7820 7872 7924

7718 7772 7825 7877 7929

7723 7777 7830 7882 7934

7729 7782 7835 7888 7939

7734 7788 7841 7893 7944

7739 7793 7846 7898 7949

7745 7798 7851 7903 7954

7750 7804 7856 7908 7959

7756 7809 7862 7913 7964

7969 8019 8068 8116 8163

7974 8024 8073 8120 8167

7979 8029 8077 8125 8172

7984 8034 8082 8130 8177

7989 8038 8087 8135 8181

7994 8043 8092 8139 8186

7999 8048 8097 8144 8191

8004 8053 8101 8149 8195

8009 8058 8106 8153 8200

8014 8063 8111 8158 8204

8209 8254 8299 8342 8385

8213 8259 8303 8347 8389

8218 8263 8307 8351 8394

8223 8268 8312 8355 8398

8227 8272 8316 8360 8402

8232 8277 8321 8364 8406

8236 8281 8325 8368 8410

8241 8285 8329 8372 8415

8245 8290 8334 8377 8419

8250 8294 8338 8381 8423

8427

8431

8435

8439

8444

8448

8452

8456

8460

8464

For larger values

'

-*(-

of x see next page.

r

1-126

GENERAL DATA :

erf x.

0

1

2

3

4

J

i

7

8

9

1.0 1 2 } 4

0.8427 8802 9103 9340 9523

8468 8835 9130 9361 9539

8508 8868 9155 9381 9554

8548 8900 9181 9400 9569

8586 8931 9205 9419 9583

8624 8961 9229 9438 9597

8661 8991 9252 9456 9611

8698 9020 9275 9473 9624

8733 9048 9297 9490 9637

8768 9076 9319 9507 9649

s 6 7 8 9

9661 9763 9838 9891 9928

9673 9772 9844 9895 9931

9684 9780 9850 9899 9934

9695 9788 9856 9903 9937

9706 9796 9861 9907 9939

9716 9804 9867 991 1 9942

9726 981 1 9872 9915 9944

9736 9818 9877 9918 9947

9745 9825 9882 9922 9949

9755 9832 9886 9925 9951

532 702 814 886

552 715 822 891 35

572 728 831 897 38

591 741 839 902 41

609 753 8.46 906 44

626 764 854 91 1 47

642 775 861 916 50

658 785 867 920 52

673 795 874 924 55

688 805 880 928 57

59 76 87 25 59

61 78 87 29 61

63 79 88 33 64

65 80 89 37 66

67 8T 89 41 68

69 82 90 44 70

71 83 91 48 72

72 84 91 51 73

74 85 92 54 75

75 86 92 56 77

For larger values of x,

n

0

1

, 2 3 4 5 6 7 8

, 1 I 1 1 1 1 1

2 3 4 5 6 7

1 3 6 10 15 21 28

9

1

9

1

10

3

n , + • • ■1

)

5

-

(n

6

-

r)!r!

7

9

8

1 + 6r + 15*' + 20j:' + 15j-« + 6i« + *« 84

1

126

84

36

45

20

210

252

210

120

45

10

1 1

126

1

1 8

9

1 7 28

|36

1 6 21 56

84

is

10

the sum of the number above

and the number

to the left of that number

The table can be extended indefinitely in this way.

24 4. 166667 X io->

Factorials and Their Reciprocals

120 8.333333 X 10-'

9

10

720

5.040

40.320

362.880

3.628.800

1.388889 10-'

1.984127 X 10-'

2.480159

2.755732

2.755732

»

X 10-'

4

Exampl e; (1 + *)' •C. Exampi 1 1 4 5 10 15 20 35 35 70 56

4

3

2 2 5

6

1 1 1

0.

1

t 1 x<

666667

C2»»)«

of rth term of (I + *)•»: "CV =

Table 19.

xl

1-3-5

X 10-'

X 10'

X

+

Each number in the table 84 = 120). (see box: 36

1-3 (2*")*

Coefficients of the Binomial Expansion

2

,

.

1

18.

Coefficient

10

I

I I

x\^\

8

Table

/,

'

— Ii

7

i

5

, erf

it

0.9999

6

Jl

0.999

X

} 6 7 8 9

0.99 0.99 0.99

1

(Continued)

%

2.0 1 2 3 4


Function or Error Integral

Probability

Table 17.

[SEC.

I0

mathematical tables

Sec. 1-2] 2

1-127

bibliography of english-language tables

Tables of most functions are available in considerable multiformity of range, interval, and number of figures. The following references are restricted to tables of not more than ten-figure accuracy, except where a table of higher accuracy offers some compensating advantage for the cumbcrsomeness associated with extra figures Many isolated standard tables are contained unwanted for most engineering work. in the literature of scientific societies, especially prior to 1930, but these are cited only There are where no equivalent table can be found in newer collections of tables. many good collections of four- to six-figure tables of elementary functions, including those in standard handbooks. For functions other than those listed and for tables of higher accuracy, see references under Bibliographies.


2.1

Bibliographies

Fletcher, A., J. C. P. Miller, and L. Rosenhead: "An Index of Mathematical Tables," Scientific Computing Service Limited, London, and McGraw-Hill Book Company, Inc., New York, 1946. A complete guide to the contents of all important tables. Over 2,000 references. M athematical Tables and Other Aids to Computation, National Academy of Science, National Research Council, Washington, D.C. (British Agents: Scientific Computing Service, A journal issued quarterly since 1943 (Report 1 published 1941), including London). information on new tallies, errors in tables, and specialized bibliographies (e.g., No. 7, II. Bateman and R. C. Archibald, A Guide to Tables of Bessel Functions, July, 1944). Bibliographies of Particular Functions. Most major tables of particular functions give pertinent bibliographies. 2.2

Four- and Five-figure

Tables

J. B.: "Five-figure Co., London, 1949.

Tables of Mathematical Functions," 2d ed., Edward Arnold & hyperbolic, Bessel, Legendre poly In x, circular, exponential, nomials, r, error, and other higher functions and elementary functions. Dwight, H. B.: "Mathematical Tables of Elementary and Some Higher Mathematical Functions," McGraw-Hill Book Company, Inc., New York, 1941. In x, circular, exponential, hyperbolic, Legendre polynomials; I\ error, other higher functions; and elementary functions. Flfigge, W. : "Four-place Tables of Transcendental Functions," McGraw-Hill Book Com Circular, expo pany, Inc., New York, and Pergamon Press, Ltd., London, 1954. nential, hyperbolic, Bessel, I\ error, and other higher functions. Jahnke, E., and F. Emde: "Tables of Functions with Formulae and Curves," Dover Publications, New York, 1945. In x, exponential, Bessel, Legendre polynomials, T, error, sin x/x, tan x/x, and many other higher functions. Dale.

2.3

Tables of Six or More Figures

Allen, Edward S. : "Six Place Tables," McGraw-Hill Book Company, Inc., New York, hyperbolic, and other elementary 1947. In x, circular functions, and exponential,

functions. P.: "Tables of Squares, Cubes, Square Roots, Cube Roots and Reciprocals of all Integer Numbers up to 12,500," E. and F. N. Spon, Ltd., London, and Chemical Publishing Company, Inc., New York, 1941. Fourth edition edited by L. J. Comrie. (First edition 1814.) Contains also reciprocals of square roots, etc. Becker, G. F., and C. E. Van Orstrand: "Smithsonian Mathematical Tables: Hyperbolic Functions," Smithsonian Institute, Washington, D.C, 1909, and reprints. «*, e~',

Barlow,

gd x.

"British

The vol Association for the Advancement of Science Mathematical Tables." umes of this series are published by the Cambridge University Press for the Royal Society. (The first editions of the first five volumes were published 1931 to 1935 in Many other tables have been published in London by the British Association.) British Association Reports since 1873. The Association has also issued interim Part Volumes and Auxiliary Tables, the latter formerly called Cards. The Committee is now reconstituted as the Royal

GENERAL DATA

1-128

Society Mathematical Tables Committee, and vol.

[SEC. 10 is the last

1

of the original scries.

Part Vol. A, "Legendre Polynomials," prepared by L. J. Comrie, 1946. Vol. 1, "Circular and Hyperbolic Functions, Exponential and Sine and Cosine Integrals, Factorial Function and Allied Functions, Hermitian Probability Functions," 3d ed., 1951. Vol. 6, "Bessel Functions, Pt. 1. Functions of Orders of Zero and Unity," 1937. Particularly convenient for reactor calculations. Vol. 10, "Bessel Functions, Pt. 2. Functions of Positive Integral Order," 1952. Cambi, E.: "Eleven- and Fifteen-place Tables of Bessel Functions of the First Kind, to All Significant Orders," Dover Publications, New York, 1948. First edition by Andrew Bell published as seven"Chambers's Mathematical Tables." Subsequent editions by James Pryde. Current edition, sixfigure tables in 1844. figure tables by L. J. Comrie, which see. Comrie, L. J.: "Chambers's Six Figure Mathematical Tables," W. and R. Chambers, Ltd., Edinburgh, and D. Van Nostrand Company, Inc., Princeton, N.J., 1949.


Vol. 1, "Logarithmic Values." log x, antilogarithms, logarithms of functions. Vol. 2, " Natural Values." In x, circular, exponential, hyperbolic, T, error, ele mentary functions. Comrie, L.J.:" Chambers's Shorter Six-figure Mathematical Tables," W. and R. Chambers^ Ltd., Edinburgh, and Chemical Publishing Company, Inc., New York, 1954. In x. circular, exponential, hyperbolic, and other elementary functions. Davis, H. T.: "Tables of the Higher Mathematical Functions," vol. 1, Principia Press, Bloomington, Ind., 1933. Harvard University: "Annals of the Computational Laboratory," Harvard University Press, Cambridge, Mass. A series of tables (some highly specialized) started in 1945 and successively under sponsorship of the Bureau of Ships, the Bureau of Ordnance, and joint sponsorship of the United States Air Force and the United States Atomic Energy Commission. Vol. 3, "Tables of the Bessel Functions of the First Kind of Orders Zero and One," 1947. Vols. 4-14, "Tables of Bessel Functions of the First Kind for Orders 2 to 135," 1947-1951.

Vol. 20, "Tables of Inverse Hyperbolic Functions," 1949.

-

sin
and of Its First Eleven Derivatives," 1949. Vol. 22, "Tables of the Function " Vol. 23, Tables of the Error Function and of Its First Twenty Derivatives," 1952.

Milne-Thomson, L. M.: "Standard Table of Square Roots," George Bell & Sons, Ltd., London, 1929. National Bureau of Standards (U.S. Department of Commerce): A major series of tables Volumes published from 1939 to all under sponsorship of NBS, initiated in 1938. March, 1943, were prepared under the U.S. Works Progress Administration (WPA) by the Work Projects Administration for the City of New York, and published by the Bureau of Standards. Since March, 1943, the work has been continued by the Mathematical Tables Project of NBS, which became the Computation Laboratory on July 1, 1947. Volumes are published by Columbia University Press or by the Bureau of Standards. In consequence of this history, there is some inconsistency in early catalogues and The early volumes were issued without numbers, references to tables of this aeries. but all volumes are now listed* numerically by NBS in three series: Mathematical Tables Series (MT) Applied Mathematics Series (AMS) Columbia University Series (CUP) The plan is to issue future tables, including all reissues, in the AMS series. Volumes of the MT and AMS series arc available from the Superintendent of Documents, Government Printing Office, Washington, D.C. Volumes of the CUP series arc available from Columbia University Press, Morningsidc Heights, New York. The following selected references are listed in the order of publication of first editions. * List of Publications LP 17 (Revised), "Publications of the Applied Mathematics Division," Department of Commerce, National Bureau of Standards, Washington, D.C. July. 1957.

U.S.

mathematical tables

Sec. 1-2]

1-129

14. "Tables of the Exponential Function «*," 1951. Reissue of MT 2, 1939; 2d ed. 1947. For an extension of this table see AMS 46, 1955. AMS 36. "Tables of Circular and Hyperbolic Sines and Cosines for Radian Arguments," 1953. Reissue of MT 3, 1939; 2d ed. 1949. For an extension of this table see AMS 45, 1955. AMS 43. "Tables of Sines and Cosines for Radian Arguments," 1955. Reissue of MT 4, 1940. MT 5. "Tables of Sine, Cosine, and Exponential Integrals," vol. 1, 1940. MT 6. "Tables of Sine, Cosine, and Exponential Integrals," vol. 2, 1940. MT 7. "Tables of Natural Logarithms," vol. 1 [In x = 1(1)49,999], 1941. 50,000(1)99,999), MT 9. "Table of Natural Logarithms," vol. 2 [In x 1941. AMS 31. "Table of Natural Logarithms for Arguments between Zero and Five to Sixteen Decimal Places," 1953. Reissue of MT 10, 1941. AMS 53. "Table of Natural Logarithms for Arguments between Five and Ten to Sixteen Decimal Places," 1958. Reissue of MT 12, 1941. AMS 41. "Tables of the Error Function and Its Derivative," 1954. Reissue of MT 8, "Tables of Probability Functions," vol. 1, 1941. AMS 26. "Tables of Arc tan x," 1953. Reissue of MT 16. CUP 3. "Tables of Circular and Hyperbolic Tangents and Cotangents for Radian Arguments," 1943, second printing, 1947. CUP 5. "Table of Arc sin x," 1945. AMS 25. "Tables of Bessel Functions Fo(x), Yi(x), A'o(x), Ki{x), 0 S £ 1," 1952. Reissue of AMS 1. AMS 45. " Table of Hyperbolic Sines and Cosines, x = 2 to x — 10," 1955. An extension of AMS 36. AMS 46. "Tables of the Descending Exponential, x — 2.6 to x — 10," 1955. An extension of AMS 14.

AMS

-


I

National

Research

E„(x)

=

/

"

Council

of

Canada, Division

e-'uu-"du," MT-1, No.

of Atomic

Energy: "The Functions

1547, 1946.

Royal Society Mathematical Tables Committee (see British Association). Thompson, A. J.: "Logarithmetica Britannica being a Standard Table of Logarithms to Twenty Decimal Places," published as "Tracts for Computers," Nos. 11, 14, 16-22, Cambridge University Press, London, 1924—1952. Also collected as Vols. 1 and 2, 1952. "Issued by the Department of Statistics, University College, London, to commemorate the tercentenary of the 14 figure Arithmetica Logarithmica published by Henry Briggs in 1624." Thompson, A. J.: "Table of the Coefficients of Everett's Central-Difference Interpolation Formula," 2d ed., Tracts for Computers No. 5., Cambridge University Press, London, 1943. 2.4

Cross Reference by Functions

In

the first column, dec. means decimal places and fig. means significant figures. In the second column, the notation 0(0.001)1(0.01)10 means that in the range 0 to 1 of the argument, the function is given at intervals of the argument of 0.001 and in the range 1 to 10 at intervals of 0.01. The intention is to describe the general character of a table rather than to define it precisely. For example, the number of decimal places or significant figures may be changed within a table to meet special conditions over a limited range of the argu ment. Also, a table described as in the above illustration may actually consist of two tables, 0(0.001)1 and 0(0.01)10, with an overlap in the range 0 to 1. No attempt has been made to reflect such detail. Particulars of the references given in the third column will be found in the preceding under Tables of Six or More Figures. The following alphabetical bibliography abbreviations are used:

BA

=

XBS NRC

British Association

= National Bureau of Standards = National Research Council of Canada

"Chambers" refers the two-volume

to the single-volume work and Chambers work, all by Comrie.

1

and Chambers 2 to

GENERAL DATA

1-130 Accuracy

Range and Interval

Squares,

Exacl I Exact

Square

roots

19 6 6 17

<

dec. fig. fig. fig. 6 fig. 6 dec. 6 dee. 6 dec. ,6 dec.

1

Reference

Reciprocals

1(1)1.000 1(1)1.000 1(1)3.400 1(1)12.500 1(1)1.000 1(1)1.000 1(1)1.000 1(1)12.500 1(1)1.000 1(1)1.000(10)10.000 1(1)1.000(10)10.000 100(0.1)1.000(1)10.000 1(1)12.500(10)125.000

C"H

Reciprocals

Roots,

Cubes, Square

1 Exact

S-uare..

[Sec.

Allen Chambers Chambers

2

Barlow Allen

Chambers

Chambers 2 Barlow Allen Chambers Chambers 2 Milne- Thomson

Barlow

Factorials


6 7 7 8

fig. fig. fig. fig.

1(1)100 1(1) 1.000 1(1)1.000 1(1) 100 log x (reduced

to range

In

6 dec. 6 dec. 6 dee. 16 dec. 16 dec.

Barlow

2

1-10)

1(0.001) 10 1(0.001)10 1(0.0001)10 1(0.00001)1.1(0.01)10 1(0.0001)10

6 dec. 6 dec. 6 dec. 8 dec 20 dec.

Chambers

Allen Chambers Chambers Chambers Thompson

I I

t

0.01(0.001)10 1(0.001)10 1(0.001)10 1(1)100.000 0.0001(0.0001) 10

Allen Chambers Chambers

2 7 and 9 NBS-AMS 31 and 53

NBS-MT

sin x and cos x (rn
6 dee. 6 dec. 8 dec. 9 dec. 10 dec.

0(0.001)1.6(0.1)10(1)54 0(0.001) l.6{0. 1)50(1) 100( 100) 5.000( 1.000) 10.000 0(0.001)25(1)100 0(0.0001)2(0.1)10 0(0.1)10(1)100

11 dec. 15 dec.

0(0.001)1.6 0(0.1)50

8 dec. or 8 fig. 10 dec.

NBS-AMS NBS-AMS

Becker

and Van

1

argument)

0(0.001) 1.6 0(0.001)1.6 also cot 0(0.0001)0.1 0(0.0001)2 0(0.1)10

i

2 43 36

Orstrand BA 1

BA

tan x and cot x (radian 6 dec. or 6 rig. 6 dec. or 6 fig.

Chambers Chambers

Chambers Chambers Chambers

NBS-CUP NBS-CUP

2 2 3 3

Sec. 1-2]

MATHEMATICAL

TABLES

Range and Interval

6 6 6 6

dec. dec. or 7 fig. dec. or 7 fin. dec. or 8-9 fig.

9 fig. 18 dec. 15 dec. 12 dec. or 19 fig.


6 6 6 7

dec. dec. dec. dec.

0(0.001)1(0.002)2(0.005)3(0.01)6 0(0.001)3(0.01)10(0.1)50 0(0.001)6(0.01)16(0.1)50(1)100 0(0.001)3(0.01)15(1)50 0(0.01)0.5(0.5)20 0(0.000001)0.0001(0.0001)1 1(0.0001)2.5(0.001)5 5(0.01)10(1)100

0(0.001)1(0.002)2(0.005)3(0.01)6 0(0.001)3(0.01)7.5(0.1) l2.5(various)» 0(0.001)3(0.01)7.5(0.1) l2.5(various)» 0(0.001)3(0.01)15

9 fig.

0(1)50

8-9 fin.

0(0.01)0.5(0.5)20 0(0.00000 1)0.000 1(0.000 1) 2.5 1(1)100 2.5(0.001)10

18 dec. 19 dec. 20 dec.

1-131 Reference

Allen Chambers Chambers 2 Becker and Van Orstrand

BA 6 NBS-AM8 NBS-AMS NBS-AMS

14 14 14

Allen Chambers Chambers 2 Becker and Van Orstrand Becker and Van Orstrand

BA 6 NBS-AMS NBS-AMS NBS-AMS

14 14 46

sinh x and cosh x

6 6 6 9 9

dec. dec. or 7 fig. dec. or 7 fig. dec. fig.

0(0.001)1(0.002)2(0.005)3(0.01)6 0(0.001)3(0.01)5 0(0.001)6 0(0.0001)2(0.1)10 2(0.001)10

Allen Chambers Chambers 2

NBS-AMS NBS-AMS

36 45

tanh x

6 dec. 6 dec. 6 dec. 8 fig. 10 dec.

0(0.001)1(0.002)2(0.005)3(0.01)6 0(0.00 1)3(0.0 l)6(various) oo 0(0.00 1)3(0.0 l)6(various)<» 0(0.0001)2 0(0.1)10

Allen Chambers Chambers 2

NBS-CUP NBS-CUP

3 3

coth z

6 dec. or 6 fig. 6 dec. or 6 fig. 8 dec. or 8 fig. 10 dec.

0(0.00 1) 3(0.0 1) 6(various) » 0(0.000 1)0. 1(0.00 1)3(0.0 1)6(variuus)» 0(0.0001)2 0(0.1)10

Chambers Chambers 2 NBS-CU1> 3

NBS-CUP

3

1-132

GENERAL Accuracy

Range

DATA

[Sec.

and Interval

Reference

I averse circular and hyperbolic functions

sin-1 x 6 dec. 6 dee. 12 dec.

0(0.01)0.9(0.001)1 0(0.001)0.99(0.0001)0.999(0.00001)1 0(0.0001)0.989(0.00001)1

Chambers Chambers 2

6 dec. 6 dec.

0(0.01)0.9(0.001)1 0(0.001)1

Chambers Chambers 2

6 dec. 6 dec. 12 dec.

0(0.01)3(0.1)13(1)100 0(0.001)2(0.01)10(0.1)35(1)80 0(0.001)7(0.01)50(0.1)300(1)2.000(10)10.000

Chambers Chambers 2

0(0.01)3(0.1)13(1)100 0(0.001)2(0.01)10(0.1)35(1)80

Chambers Chambers 2

0(0.01)3(0.1)13(1)100 0(0.001)2(0.01)10(0.1)35(1)80

Chambers Chambers 2

1(0.001)1.1(0.01)3(0.1)13(1)100 1(0.0000 1) 1.00 1(0.000 1) 1.0 1(0.00 1)2(0.01) 10(0.1)35(1)80

Chambers Chambers 2

0(0.01)0.9(0.001)1 0(0.001)0.99(0.0001)1

Chambers Chambers 2

1(0.001)1.1(0.01)3(0.1)13(1)100 1(0.0001)1.01(0.001)2(0.01)10(0.1)35(1)80

Chambers Chambers 2

cot 6 dec. 6 dec.

' X

*

6 dec. 6 dec. cosh-1 x 6 dec. 6 dec.


tanh-1 x 6 dec. 6 dec. coth-1

NBS-CUP

NBS-AMS

5

26

x

6 dec. 6 dec.

Besael functions of ttie first kind Jo(x) and

10 dec. 10 dec. 11 dec. 18 dec.

Ji(x) HA 6

0(0.001)16(0.01)25 0(0.01)10 0(0.001)0.5(0.01)10.5 0(0.001)25(0.01)100 Bessel functions of the second

8 fig. 8 fig. 10 dec.

NBS-CUP

2

Cambi Harvard

kind* Vo(-r) and Yt(x) BA 6

0(0.01)25 0(0.0001)0.05(0.001)1 0(0.01)10

NBS-AMS NBS-CUP

25 12

Modified bessel functions of the first kind Ioix) and /i(x) 9 Kg. 10 dec.

e-'Io(z) and e~'h(x) 8 dec.

0(0.001)5 0(0.01)10

BA 6

5(0.01)10(0.020

BA

* Weber's functions Yuix) and Vi(x) are also called Nt>(x) and

NBS-CUP 6

Ni(x) after Neumann.

2

1

Sec. 1-2]

MATHEMATICAL TABLES

Accuracy

1-133

Range and Interval

Reference

Modified bessel (unctions of the second kind Ko(i) and Ki(jc) 7 fig. fig.

8-9

8 dec.

e'Kt(z)

and

e'Ki(z)

Gamma

T(j), 6 dec. I 10 dec. log T(l), 10 dec.

NBS-AMS

0(0.0001)0.033(0.001)1 0(0.01)5

BA 6

5(0.01)10(0.1)20

BA 6

and factorial functions

Chambers Davis 1 Davis 1 Davis 1 BA 1

1(0.001)2 1(0.0001)1.1(0.001)2 1(0.0001)1.1(0.001)2 1(0.001)2(0.001)11(0.1)101 0(0.01)1

log T(x). 12 dec. *!, 12 dec.

Exponential and related integrals

Ei(z)

Alternative nomenclature: B

-EH-x)

BHx) BHx)

2?
E,(x)

-

ft i

7dee.,

5 fig.

i

m

NRC NBS-MT NBS-MT from BA NRC NBS-MT NBS-MT from BA NRC

0(0.001)10(0.1)15 0(0.01)2(0.1)10 Lcgendro

To n

-

2

Kid)

0(0.0001)2(0.1)10

7 dec, 5 fig.

5 6 (in part 1) 5 6 (in part 1)

polynomials Pud) 12

erf x = //(x)

6 dec. colog (1 — erf x), 3 dec. 15 dec. (1 — erf x). 8 fig.

Hi

0(0.01)2(0.1)10

10-11 fig.

7 dec.

— — Ei( — x)

0(0.01)2(0.1)10 0(0.0001)2(0.1) 10 0(0.001)10(0.1)15

-EH-x) -EH-x)

B,(x) =

25

=

BA, Part Vol. A

0(0.01)1

-L

V*

fX

e~<'dt

JO Chambers 2 Chambers 2

0(0.01)4 3(0.1)8 0(0.000 1) 1(0.00 1) 5. 6( various) 5.946 4(0.01)10

Chambers 2 and NBS-AMS 41 also give the derivative or ordinate

ff'(x) =

NBS-AMS NBS-AMS

—-

Vt

41 41

e tldt.

Chambers 2, NBS-AMS 23, and Harvard 23 give other forms of the probability integral and its ordinate.

1-3

UNITS AND CONVERSION

FACTORS

BY Harold Etherington Unit systems and conversion factors are given for quantities that may plausibly For other quantities see references.

be used in nuclear engineering. 1

DIMENSION AND UNIT SYSTEMS


1.1

Dimensions, Units, and Systems

1.11 Dimensions and Units. A dimension is a general expression for a particular A unit is a reference standard kind of physical quantity, e.g., length, time, mass. used to measure a physical quantity; e.g., one centimeter, one inch, and one foot are alternative units of the dimension length. These Some dimensions and units are chosen arbitrarily, as in the above examples. are called fundamental dimensions and fundamental units. Some dimensions and units are defined and measured in terms of others; e.g., areas and volumes are defined and measured in terms of length, and density is defined and measured in terms of mass Some dimensions and units are related to others by physical laws; per unit volume. for example, Newton's second law of motion provides a means of expressing force in the dimensions and units of length, time, and mass. Dimensions and units that are expressed (by definition or by physical law) in terms of fundamental quantities are called derived dimensions and units. 1.12 A system in which all the dimensions can be expressed Dimension Systems. in terms of the smallest possible number of fundamental dimensions (e.g., length, time, mass) is called a dimension system. 1.13 Unit Systems. A unit system consists of fundamental units (e.g., foot, second, pound-mass) corresponding to the fundamental dimensions of the system (e.g., length, time, mass) together with the derived units necessary to satisfy all the requirements of the system. The units are said to be consistent, self-consistent, or systematic if all physical relations within the system can be expressed without intro The principal metric and English systems of units are ducing numerical factors.* given in Tables 1 and 2, respectively. 1.14 Absolute Systems. Historically, units of length, time, and mass were most easily established as standards. Systems employing these dimensions and units as In a literal sense they are no fundamental quantities are called absolute systems. more absolute than any other arbitrarily chosen system. 1.15 It is sometimes convenient to use Nonsystematic Dimensions and Units. sets of dimensions and units that do not conform to the requirements of a true system; i.e., the number of dimensions and units may be greater or less than required, or the units may be made nonsystematic by mixing units from different systems. A greater than necessary number of units is frequently used in thermodynamics. Thermal conductivity may have the dimensions and units of length, time, temperature, and heat, e.g., Btu/(hr)(ft)(°F), and specific heat may have the units mass, temper ature, and heat, e.g., Btu/(lbj/)(°F); if these quantities both enter into a physical * The factor 4* enters into some of the physical relations of electrical and is accepted as not destroying the consistency of the system.

factor

1-134

systems.

This is a geometric

Quantity

symbol

. • .

L

u

T

F

/

M«L«T-.,» L-'T^

Statvolt Statohm

. . .

E

. .

R

Abvolt Abohm

Abampere

Abcoulomb

Abhenry /centimeter

No name

Volt Ohm

Amperet MLT-'I-' ML»T-"I-«

Coulomb?

IT

M MLT> MLT-i ML-'T-i M-"L-'T«I» MLI-T-'

Meter Second Kilogram Newton Joule No name Farad/meter Henry/meter

Unit

(Giorgi) system

mks

Dimensions, MLTI

The

The table gives selected units of the mechanical and electrical absolute systems. Additional derived units are given in the following conversion tables. Fundamental dimensions and units are shown in boldface. Consistent thermodynamic systems of unite may be obtained by adding the degree centigrade as unit to the cgs and mks systems. fundamental = IX newton = 10s dynes. joule/meter on Weights and Measures, International The coulomb was generally used in the United States as the fundamental unit, but the Tenth General Conference The coulomb thus denned as one Bureau of Weights and Measures, 1955. adopted the meter, kilogram, second, ampere, and degree Kelvin as systematic units. ampere-second instead of the ampere being defined as one coulomb per second.

M^L^T-igL"'T£-'

m^t-.,.-^

Statampere

mWl^tV4

M*L»V*

L-«T«
Statcoulomb

Staff arad /centimeter No name

aesuC L-iTie-i

M MLT-> ML«T-»

(aemu)

Unit

Centimeter Second Gram Dyne Erg

electromagnetic system

Units*

Dimensions, MLTV

Absolute

and

MK-LMT-,£W

Dyne Erg

M MLT"' ML>T-

a

(aeau)

Unit

Centimeter Second Gram

electrostatic system

of Dimensions

I

Potential, Resistance,

0

Current,

LT Dyne Erg Poise

Dimensions, MLTg

LT

M MLT-' ML*T-> ML-'T-i

Centimeter Second Gram

Unit

Absolute

Systems

LT

Charge,

Dimensions, MLT

cgs

Metric

LT

. ..

Length, Time, IVlass, M Force, Work, energy, £'. Viscosity, S. Permittivity, mPermeability,

and

system

Mechanical

Table


1.

t a

is

1

*

It X

1-136

GENERAL DATA

[Sec.

1

equation, five dimensions (mass, length, time, temperature, and heat) are represented, whereas four fundamental dimensions are sufficient (see later). In such cases, it is necessary to introduce a dimensional conversion factor in physical equations. The dimensions of this factor must satisfy the physical relation among the quantities, and the magnitude depends on the choice of units. It is desirable to choose the magnitude of a redundant unit so that the conversion factor is numerically equal to unity. In a limited range of application, it is permissible to use a smaller number of funda mental dimensions than required for a complete system. Such a set of quantities is said to be incomplete. Dimensionally consistent units from different unit systems (e.g., inches in place of feet in a system otherwise based on the foot, second, and pound) can be used, but a dimensionless factor mast be introduced in physical equations of such hybrid systems.


1.2

Mechanical Systems

Three fundamental dimensions are necessary and sufficient to express all the physical quantities of a mechanical dimension system. Length and time are invariably chosen for two of the dimensions; the third is usually either mass or force. These alternative systems are called the absolute, or MLT, and the gravitational, or FLT The principal metric systems are absolute systems, in the systems, respectively.* sense that length, mass, and time are the fundamental dimensions. The usual engi neering (English) unit system is a gravitational system, with length, force, and time as the fundamental dimensions. 1.21 The CGS (Centimeter-Gram-Second) System. This system is generally Derived consistent units are given in Table 1. used by physicists. Table 2.

Quantity and symbol

Mechanics and gas flow (gravitational system)

Dimensions,

FLT

Length, L Time. T Mass, M Force, F Work, W

L T

Flr'T' F FL

Wlf* FL-
English Systems of Dimensions and Units*

FL'T

Unit

Heat flow (nonsystenutic)

Dimensions, MLT OH

L

Foot Second Slug Pound Foot-|»und

T

M

Unit

Foot Hour Pound

Lbr/ft'

Slug/ft' Slug/(ft'i
Ther.nal stress (nonsystcmatic)

Dimensions,

T

L

Inch Hour

F

Pound

FL ML"> Ml, -1 i 6

HT-'L"'»->

UT-'L-»»-'

Btu, (hr)(ft>)(°F)

H HM-'IT'

ML'T"

I

Lbu/ft' Lb*/(ft>)(hr) °F Btu Btu, (lb«)(°F) Btu (!ir)(ft)(°F)

Lb*,(hr)(ft)

Unit

LT0H

0

H HT -'L-'» '

Lbr/in.' JF Btu lltu. (hr)(in.)(°Fl

• Fundamental dimensionsand units are s:town in boldface.

The MKS (Meter-Kilogram-Second) System. f This system is little used 1.22 Its in English-speaking countries by reactor engineers as a mechanical system. advantages are discussed in the article on electrical units. The foot, second, and pound-force are the fundamental Engineering Units. 1.23 The derived unit of mass is the slug, units of the gravitational system (Table 2). * The absolute system is also called the dynamical or physical system, and the gravitational system MLT and FLT are also written LTM and LTF. respectively, is sometimes called the technical system. t Sometimes called, after the founder of the system, the mks-Giorgi system.

Sec. 1-3]


1-137

FACTORS

The alternative absolute system (foot, second, poundequal to 32.17 pounds-mass. mass, with the poundal as the derived unit of force) is little used. In the practical treatment of fluid flow, the pound-mass, poundHydrodynamics. force, foot, and second are commonly used as a nonsystematic set of four units. It is therefore necessary to introduce a dimensional constant into Newton's second law of motion, force = mass X acceleration /gc or g, = mass X acceleration /force. The magnitude of gc depends on the units used, and, by definition of the pound-force, I, 32.1740 (lbjfXftVabfKsec2).

-

The ratio g/gc, which occurs frequently in hydrodynamic theory, is numerically approximately* equal to unity and has the dimensions lbr/lbjr. Thermodynamic

1.3

Systems

Four fundamental dimensions are required. Mass, time, and length are usually with temperature (the MLTfl system) or heat (the MLTH system) as the


chosen,

fourth dimension. 1.31 The CGS System. The cgs system of units is extended by addition of the Heat is a derived unit, expressed as ergs degree centigrade as the fourth unit. (dyne-centimeters). 1.32 Metric Heat-flow Units. For problems not involving interconversion between heat and work, a set of five fundamental units —centimeter, gram, second, Where rate of heat generation degree centigrade, and calorie — is commonly used. is expressed as watts per cubic centimeter and heat flow as watts per square centi meter, it is then simplest to change watts to calories per second (1 cal/sec = 4.18684 watts). This set of units becomes systematic if the gram is replaced by a derived unit of mass equal to 4.184 X 10' g (used in the caloric unit syxtem — 4.184 joules is the equiva lent of the "thermodynamic calorie" proposed for this system). 1.33 The MKS system is also extended to provide a thermodynamic system by addition of the degree centigrade. The usual thermodynamic set of dimensions (Table 2) con English Units. 1.34 tains more than the four fundamental dimensions that would constitute a true dimen sional system. The units are, however, consistent and are in general use for calcu lating heat flow and heat balances when there is no significant conversion from work to heat or vice versa. Thermal Stress Units. Thermal stress units (Table 2) also constitute a 1.35 special-purpose but self-consistent set of English units. Dimensionless Groups. 1.36 Data for calculating dimcnsionless groups are given in this handbook in English thermodynamic units. The quantities entering into the dimcnsionless groups are such that all units cancel, even though all five fundamental units are involved.

For example, Prandtl number All

five

units cancel.

-r-

has

the

units

...

L

X

Reynolds number, even when used

in connection with fluid-flow problems, is calculated in terms of thermodynamic units. This avoids the confusion and unnecessary duplication of viscosity data in two different systems of units. 1.4

Electrical Systems

Four fundamental dimensions are required. Usually the three dimensions of a mechanical system are complemented by a fourth dimension which may be chosen from among electric charge, current, voltage, resistance, energy, power, permittivity, and magnetic permeability (Table 1). Electrical units are based on the absolute metric mechanical systems of units. •The gravitational acceleration ffmis invariant.

g varies slightly with latitude and altitude (see Art. 2.14), whereat-


1-138

GENEHAL DATA

[Sec.

1

Classical development of electromagnetic and electrostatic theory led to two distinct and incomplete sets of cgs units. Many systems have been devised to unify these sets of units; the most successful is the mks system, which also yields units identical with the practical units of electrical engineering. The statcoulomb is defined as the charge 1.41 Absolute Electrostatic System. that exerts a force of one dyne on an equal charge one centimeter distant in free The statcoulomb is thus a derived unit, the centimeter and dyne being funda space. mental units. The permittivity of the dielectric appears as a divisor in Coulomb's law, and this quantity is therefore usually selected as a third dimension, the permit tivity of vacuum (or, for practical purposes, air) being taken as the unit of permit tivity. This incomplete set of three units is limited to static conditions; to accom modate change with time, the units are replaced by the three absolute mechanical The dyne (as units (centimeter, gram, and second) and the unit of permittivity. The units are called the cgs well as the statcoulomb) thus becomes a derived unit. absolute electrostatic units (aesu). In the aesu system, permittivity is numerically equal to the dielectric constant The aesu system is frequently used as an incomplete system, with (a pure numeric). the permittivity of space taken as the dimensionless constant unity. 1.42 Absolute Electromagnetic System. The abampere can be defined as that current which exerts a force of one dyne per centimeter of conductor length on an There are other equiva equal parallel current one centimeter distant in free space. lent definitions. As in the absolute electrostatic system, these two units (dyne and From Ampere's law, centimeter) are replaced by the centimeter, gram, and second. permeability is added as the fourth dimension, with the permeability of vacuum taken as the fundamental unit. The dyne, abampere, abcoulomb, etc., are con The units of this system are called cgs absolute electromagnetic sistent derived units. units (aemu). Like the aesu system, this system is frequently used as a simple cgs incomplete system, with permeability of free space assumed to be the dimensionless constant unity. The statcoulomb is greater than the 1.43 Symmetric Systems and Units. abcoulomb by a factor numerically equal to the speed of light. The difference in the units arises from the independent methods of defining charge; the magnitude of the factor is inherent in the nature of electromagnetic radiation. For problems involving both electrostatic and electromagnetic phenomena, a union between the absolute electrostatic and electromagnetic systems is necessary. Either system can be suppressed and the other extended to include all necessary units, but many of the derived units are inconveniently large or small. Alternatively the systems may be mixed, giving nonsystematic units with some power of c (the velocity of light) entering into many of the relations and definitions. Like the aes and aem systems, this system is also frequently used as an incomplete system with the dimensions of permittivity and permeability taken as zero and the magnitude as unity. 1.44 The practical electrical units are derived from Practical Absolute Units. the aemu system by applying powers of 10 as multipliers to yield units of a con venient magnitude: 1 ampere ("absolute practical ampere") = 1 X 10"' abampere, 1 volt = 1 X 10s abvolts, 1 ohm = 1 X 109 abohms, etc. Incomplete sets of practi cal absolute units (coulomb, ampere, volt, ohm, henry, farad, joule, and watt) are used in limited fields. 1.46 The MKS System of Units. The mks (or Giorgi) system provides a solution to the unsat isfactory coexistence of t wo absolute cgs systems, together with numerous compromise systems and an incomplete practical set of units. The units of the extended mks system are the meter, kilogram, second, and ampere (practical) or some other consistent electrical unit. Derived units include the coulomb, volt, ohm, watt, farad, and henry, all identical with the practical units as conventionally defined. Other derived units are so defined that proportionality constants are all eliminated with the exception of powers of 4jt inherent in spherical geometry. The derived unit of force is the newton, equal to 1 joule/meter or 106 dynes. The system is directly applicable to electrostatic and electromagnetic phe

Sec. 1-3]


1-139

FACTORS

nomena, alone or in combination. In this system the permittivity of vacuum is 1.11 X 10~10 inks unit, and the permeability of vacuum is 10~7 mks unit.* 10ll/cs 1.6

Comprehensive or Universal Systems

Five fundamental dimensions arc required to express all possible relations of mechanics, thermodynamics, and electricity. Length, mass, time, temperature, and quantity of electricity are suitable fundamental dimensions for such a system. The mks mechanical units, extended by addition of the degree centigrade and the ampere (or an alternative electrical unit), provide the best all-purpose unit system. 2

FUNDAMENTAL STANDARDS

AND EXACT EQUIVALENTS

Precise measurement has revealed very small errors in previously established equivalents of units. The errors have no engineering significance, but the conversion tables of this section are based on the following officially accepted equivalents.


2.1

Mechanical

Standards

Length, t'1 The primary standard of length is the meter as measured at 0°C 2.11 under prescribed conditions on the international prototype meter, a platinum-iridium bar kept at the International Bureau of Weights and Measures at Sevres, France. Similar secondary standards are held by countries contributing to the Bureau, and in 1927 the standard was provisionally given permanence and reproducibility by expressing it as 1,553,164.13 times the wavelength of the red line of the cadmium Other wavelengths are under consideration as spectrum (0,438.4696 X 10~7 mm). a basis for a new definition of the meter. 1

cm = 0.3937 in. (exactly)

J

2.12 Mass.1 The primary st andard of mass is the international prototype kilogram, standard kept at the International Bureau of Weights and Meas a platinum-iridium The kilogram was intended to ures, with secondary standards in other countries. equal the mass of 1,000 cm3 of water at the temperature of maximum density (3.98°C) and 760 mm of mercury, but the standard is actually equal to the mass of 1,000.028 cm* (approximately) of water under these conditions. 1 1

lb avoirdupois British lb

= 453.5924277 g (legal exact equivalent) = 453.59243 g (exactly)

The fundamental unit of time, one second, is 1/86,400 of a mean 2.13 Time. The mean calendar year, based on the standard procedure for selecting solar day.§ * Unrationalized system. The values for air are substantially the same as for vacuum, Superscript numbers refer to References at end of subsection. The Imperial yard is defined as X The legal equivalent is defined as 1 yd/1 meter = 3.600/3,937. Therefore 1 U.S. in. = 2.5400051 cm (approximately) and 0.9143992 meter, exactly.

t

1 British in. = 2.5399978 cm (approximately)

In both countries the industrially accepted equivalent, and the British Standards Institution, is

adopted by the American

Standards Association

1 in. = 2.54 cm (exactly)

This simplified equivalent is becoming increasingly accepted by other member nations of the Inter national Organization for Standardisation. Derived units are presumed to be based on the legal equivalents. } The solar day as measured by transit of the sun across a meridian is out of phase with the mean The variation throughout the year iB about half an hour. solar day as measured by the chronometer. The mean The mean solar or tropical year is 305.24220 mean solar days, or 305 days 5 hr 48 min 40 sec. See Ref. 2. It is expected that the calendar year is thus 26 sec longer than the mean solar year. International Bureau of Weight* and Measures will, in the near future, define the unit of time more precisely.

GENERAL DATA

1-140

[SEC.

1

leap year (years in which the date is divisible by four, except exact centuries for which the year date must be divisible by 400) is 365.24250 days or 3. 1 556952 X 10T sec. 2.14 Force. The kilogram-force and the pound-force are the forces exerted by gravity on the standard kilogram-mass and the standard pound-mass, at a location Standard gravity, or standard acceleration, of gratrity, is defined of standard gravity.* as go = 980.665 cm/secs equivalent to 32.1740 ft/sec1. 2.2 2.21 Temperature. is defined by assigning

Thermodynamic Standards

The Kelvin or thermodynamic scale, selected as fundamental, the temperature 273.16°K (exactly) to the triple point of

water, f The thermodynamic centigrade, scale conforms to the Kelvin scale, but. with the triple point of water taken as zero and the absolute zero as — 273.16°C. Since the definition of thermodynamic scales does not provide a readily measurable standard, the thermodynamic centigrade scale is represented by the international centigrade and specified scale, standardized by a platinum resistance thermometer calibration points. The Rankine and Fahrenheit scales are defined in terms of the Kelvin and thermo dynamic centigrade scales, respectively. The absolute zero is — 459.69°K on the Fahrenheit scale. Work, Energy, Heat. Nearly all engineering standards of energy are now 2.22 referred to the absolute joule.


107 ergs = 1

joule (abs)

= 1 watt-sec

(abs)

The foot-pound is related to the erg through the length and force equivalents. Heat units are defined arbitrarily but are expressed in joules. Heat. The IT calorie (International Steam Table calorie) and the corre 2.23 sponding Btu have been adopted as standards in these conversion tables, p* The following equivalents are approximate. 1 cal 1 1 chu

Btu

(centigrade

= 0.00396832 IT Btu joules heat unit) or pcu (pound centigrade unit or pound calorie) = 1.8 Btu = 453.592 IT cal = 1,899.13 abs joules

=1

IT

cal

= 251.996

= 4.18684 abs joules cal = 1055.07 abs

IT

The absolute joule is the fundamental unit of heat (adopted by the Ninth General Conference on Weights and Measures, Paris and Sevres, 1948). The advantages of the joule as a unit of heat are its expression in absolute mechanical units, its easy representation and measurement in electrical units, and its independence of the heat capacity of water and of the variation of heat capacity with temperature. These units were originally defined in There is no legal definition of the calorie or Btu. terms of the heat capacity of unit mass of water per degree of temperature rise at some specified temperature or temperature range. The International Steam Table Conference * Force, defined in terms of standard gravity, is independent of geodesy. Weight, however, varies with gravity, and if force is measured by comparison with the weight of a standard mass. Force

= weight

X —

where g is the local gravitational acceleration. The ratio g/go ranges from 0.9974 at the equator to 1.0026 at the poles and decreases by 0.000314 for each 1,000 meters of altitude (0.000090 for each1.000ft of altitude). f Tenth General Conference of the International Bureau of Weights and Measures, 1954. Originally, the Kelvin scale was chosen to retain, exactly, the historic 100°C difference between the meltingpoint of ice and the boiling point of water, both at standard atmospheric pressure. The melting point of ice at atmospheric pressure is approximately 273. 15°K on the newly defined Kelvin scale, and theboiling point is also only approximately defined. X The author is indebted to Donald D. Wagman, in charge of data on chemical thermodynamic properties, Thermochemistry Section, Division of Chemistry, National Bureau of Standards, for guidance in this description of heat units.

Sec.


1-3]

1-141

FACTORS

of 1929 defined the IT calorie as Jfjgo international watt-hour, which is equivalent to 4.18684 abs joules.* The Btu as defined above is derived from the relation 1 Btu/(lb)(°F) = 1 IT cal/(fr) (°C). Another calorie, the Ihermochemical calorie, defined as 4.18400 abs joules, is much used in chemical thermodynamics and thermochemistry. Table 3 gives approximate equivalents of heat units that have been widely used — all except the 20°C equivalent are by Griffith.3"

Table

3.

Various Calories and British Thermal Units Btu

Calories

Designation

4°C

cal

Designation

Joule equivalent

IT

Btu

equivalent

4.2040

1.0041

39°F

1059.52

1. 0042

60°F (60-6l°F)

1054.54

0.9995

equivalent

1S°C

4. 1855

0.9997

2CPC

(I9.5-20.5°C)

4. 1816

0.9987

iO-IOO°C)

4. 1897

1.0007

Mean (32-21 2°F)

1055. 79

1 .0007

Thermochcinical

4 184*

0. 9993

Thermochemical

1054.35

0. 9993

Mean


IT

Joule equivalent

* Exactly, by definition. 2.3

Electrical Standards4

The

practical electrical units (ampere, volt, ohm, etc.) were defined, more than a ago, from the cgs electromagnetic units by applying the following powers of 1 amp ("absolute practical ampere") — 1 X 10_1 abamp (cgs); 10 as multipliers: The International Congress 1 volt = 1 X 108 abvolts; 1 ohm = 1 X 10* abohms. of 1 893 retained these definitions but also set up measurement standards as alternative definitions —a mercury column for the ohm, a silver voltameter for the ampere, and cell for the volt. a standard These units, called international units, were legalized by act of Congress in 1894. Discrepancies became apparent between the theoretical definitions and the measure ment standards and also among the three measurement standards, one of which was In 1908 the standard of voltage was dropped, but instead of the other redundant. two inaccurate measurement standards being corrected, the unfortunate decision was made to retain these as fundamental units of the International System and to abandon the basic relation to the cgs system. Later, standard wire resistors and standard cells became the accepted international reference standards, these still retaining the original deviation from the absolute system and also showing small dis crepancies among the responsible laboratories of different countries. This unsatisfactory condition was rectified, effective Jan. 1, 1948, by abandoning the international units in favor of the absolute (practical) units. f Calibrated wire resistors and standard cells continue to be the practical measurement standards. Revised equivalents were established as a preliminary to abandonment of the international units.

century

1

hence

mean int ohm = 1.00049 abs ohm 1 int volt = 1.00034 abs volts 1 mean int joule = 1.00019 abs joules (approx)

* A figure of 4.18674 has been widely used in the United States, but this is based on the former United electrical joule, which deviated slightly from the international mean value. States "international" See Electrical Standards below and Ref. 3. t The International Commission on Weights and Measures denned the units in the inks system Legalization by the United States Congress (Jan. 13, 1949) was in terms of the cgs (Oct. 29, 1946). system. The resulting units are, of course, identical.

GENERAL DATA

1-142

[SEC.

1

The international standards formerly used in the United States (and in other countries) differed slightly from the international mean standards; the conversion factors based on National Bureau of Standards measurements are: 1 1 1 1

int ohm = 1.000495 abs ohms 1.00033 abs volts int volt int amp = 0.999835 abs amp int coulomb = 0.999835 abs coulomb

-

1 1 1 1

int henry

= 1.000495 abs henrys int farad = 0.999505 abs farad int watt = 1.000165 abs watts int joule = 1.000165 abs joules

These conversion factors apply only to measurements referred to the United States former international standards. Special Units and Standards

2.4

Nonsystematic special units are denned in context with the conversion tables of dimensionally equivalent units. The units and conversion factors of nuclear physics are given in Art. 4 of this section.

CONVERSION FACTORS


8

The conversion tables appear in the following sequence: mechanical, heat, electrical, In each category, the units generally regarded as fundamental and miscellaneous.* Within each table, metric units are given at the left in order of are given first. ascending magnitude; units of conventional engineering follow in convenient groups also in order of increasing magnitude. All factors are accurate to four significant figures. Quantities given to less than four significant figures are exact equivalents. 3.1

Use of Conversion Tables

To convert a quantity given in one unit to the equivalent expressed in another unit, locate 1 under the given unit and in the same horizontal row find the multiplier to convert to the desired unit. Example: Suppose a volume of 25 ft3 is to be expressed in cubic centimeters. The volume-conversion table (Table 6), fifth horizontal row, shows that 1 ft' is equal to A volume of 25 ft' therefore equals 25 X 2.832 X 10' = 7.080 X 10' cm'. 2.832 X 10" cm1. 3.2

Example ot Derivation

of Conversion Factors

Suppose it is desired to derive the conversion factor foot-pounds per minute. 1 kw — ft-lbf/min 1 kw X 1 min *

/ for

converting kilowatts

to

/

=

1

ft

X

1

lbF

It is necessary to develop the units of this ratio to units for which conversion factors are known. It is assumed for the purpose of this example that equivalents of the kilowatt in foot-pound-minute units are not known. 1

kw = 1,000 watts = 1,000 joules/sec = 10'° ergs /sec = 10'" dyne-cm /sec 1 dyne 10"> X dyne-cm ;/sec X 60 sec 60 — = f = V 1010 A X 1 lb,30.48 cm X 1 lb* 30.48

*

'

Known conversion factors can be substituted version table, 1 lbr = 4.448 X 10' dynes.

/

=

* Atomic conversion

^KTo 30.48

X

10'°

X ;

at any time, and from the force con

—

, i Q 1/ »»n 4.448 X 105 dynes

tables are given in Art. 4.

"

4425

x

10i

Sec. 1-3]

AND CONVERSION

UNITS

1-143

FACTORS

If as

it is desired to develop all ratios to fundamental units, 1 dync/1 Ihr is evaluated follows: In any consistent units, force = mass X acceleration; i.e.,

1 dyne

= 1 g 1

X dyne 1 \bF

1 lb^ =

1 em/sec2

X

1 g

32.17 lb.,,

1 slug

X

1 ft/sec2

1 cm /sec2

X

1 g

1 ft/sec2

X

32.17

= 32.17

X

lb* X

1

ft/sec2

1 cm

453.6 g

X

30.48 cm

1

X

4.448

10*

Table Centimeters

1 100 2 540 30. 48

Meters

0.01 1 0. 02540 0. 3048

Inches

Feet

0.3937 39.37 I 12

0.03281 3.281 0.08333 1

Metric Measures

=■I cm = I dm = I meter = I km 10-' meter 1 X 10"' mm = 1 1 micromillimeter m I x 10-« mm 10-' u 10-« (i = 1 X lO"1 mm I X 10-' cm = I X 10-< u = 0.1 imi I X 10"' A = I X 10-" cm


10 ram 10 cm 10 dm I.OOO meters I a = I X 1 iiim =

X

4.

Length U.S. Customary

Measures 12 in. = I ft 3 ft = I yd 5.280 ft = 1.760 yards = I mile I mil = 0.001 in.

Nautical Measures 6 ft I fathom

-

120 fathoms = I cable length 1 international nautical mile (adopted July I, 1954, by Departments of Defense and Commerce) = 1,852 meters = 6.076.10333 ft (appro*.) I U.S. nautical mile = I geographical mile = 1 min of longitude at equator — 6,080.20 ft I British nautical mile = 6,080 ft

-

I light year

Astronomical Measures 5.879 X 10" miles = 9.461

-

Iw-IX I A =

I X-ray

unit

(XII)

=

Table Square centi meters

1 X 10« 6.452 929.0

Square meters

1 X 10~« 1 6. 452 X 10 '

0.09290

Square inches

0. 1550 1.550 1 144

6.

X 10" cm

Area

Square feet

0.001076 10.76 0.006944 1

Electrical Unit I cir mil (area of circle 0.001 in. in diameter)

U.S. Customary Measures 144 in.' 1 ft' 9 ft' = 1 yd' 4.840 yd' = 1 acre 640 acres = 1 mile'

-

= 0.7854 X 10-8 in.s 5.067 X 10 ■cm^

-

Metric Measures I ha = 10.000 meters2 = 2.471 acrea I acre = 0.4047 ha

1-144

GENERAL DATA Table

Cubic centimeters

i

Liters

0 001000

1000 1 X I0« 16 39 2.832 X 10' 7.646 X I0»

1 1.000 0 01639 28 32 764.5

[SEC.

Volume

6.

Cubic meters

Cubic inches

Cubic feet

1 x io-«

0 06102

3.531 X IO-«

0 001000

61.03*

0 03532*

1

6.I02X I0<

1.639 X 10 >

1

0 02832

1.728

0.7646

4 666 X 10'

3785

3.785

0.003785

231

4546

4.546

0.004546

277.4

1

Cubic yards

0.001308 1 308

1

0.03704

27

1

0 1337 j

0 2642 264 2

5.787 X 10 < 2. 143X I0">

0.1605

Imperial Kalians

1 308 X 10 • 2 642 X 10-< 2 200 X 10 <

35 31

I

U.S. gallons (liquid)

0 2200 220 0

0.004329

0 003605

7.481

6.229

202.0

168 2

0.004951

1

0.8327

0.005946

1.201

I

1


Fluid Measure* Fundamental Standards of Fluid Capacity. The metric, United States, and British units are defined independently. One liter is the v -lume of one kilogram of water at 3.98°C and 760 mm of mercury. This is equal to 1.000028 cubic decimeters, f the deviation from one cubic decimeter being the difference between the intended and actual mass of the standard kilogram. I U.S. gal — 231 in.3 exactly

(definition)

One Imperial gallon is the volume occupied by 10 pounds avoirdupois 30 in. of mercury. This volume is 277.420 U.S. cubic inches.

of distilled water at 62°F and

Metric Fluid Measures I liter 100 cl

1.000 ml I ml

-

-

-

I cm1 (approx.,

see above)

U.S. and British Fluid Measures 4 gills (gi) =- 1 liq pt 2 liq pt — 1 liq qt 4 liq qt = 1 gal

among the gill, pint, quart, and gallon are the same in United States and Note: The relationships British systems; the magnitudes of corresponding units are in the same ratio as the United States and the Imperial gallon given in the table. (approximately). One U.S. gallon of water at 62°F weighs 8.336 pounds avoirdupois 10 pounds avoir One Imperial (British and Canadian) gallon of water at 62°F weighs, by definition, dupois (exactly). 16 U.S. fl oz I U.S. liq pt (I fl oz of water weighs 1.042 oz avdp at 62°F) 160 British fl oz = I Imperial gal ( I fl oz of water wciglis 1 oz avdp at 62°F)

-

1 U.S. bu I British bu

-

Dry Measures 2.150.42 in." 2.219.34 in.1 (U.S.)

-

8 Imperial gait

Shipping Measures Internal capacity: 100 ft1 =■1 register ton Cargo: 40 ft' — 1 U.S. shipping ton 42 ft3 — 1 British shipping ton One board foot before dressing. multiplied by the the dimensions of

Lumber is the volume of one square foot of board one inch thick ( 144 cubic inches), measured The feet board measure (fbm), or board feet of flat lumber, is the area in square fe«t thickness in inches. The board foot measurement of dressed stock is calculated from the rough lumber required; thicknesses less than one inch are taken as one inch.

between the liter and 1.000 cm' is insignificant but may lead to a fourth-figure in rounding off if the fifth figure is very close to 5. t Units of Weight and Measure (United States Customary and Metric), Definitions and Tables nf Equivalents, Natl. Bur. Standards Misc. Publ. 214. July I. 1955. The United States gallon is legally only {The Imperial gallon is both a dry and a liquid measure. a liquid measure — one-eighth of a United States bushel is sometimes called a "dry gallon" (equal to 1.16365 U.S. gal).

•The difference

difference

units and conversion factors

Sec. 1-3]

Table

7.

Mass

Grams

Kilograms

Pounds (avdp)

Short tons

Long tons

1 1000 453.6

0.001 1 0.4536 907.2 1016

0.002205 2.205 1 2000 2240

1. 102 X 10-« 0.001102 0.0005 1 1. 12

9.842 X 10"' 9.842 X 10-' 4.464 X I0< 0.8929 1

9.072 X 10' 1.016 X 10'

Metric Wright* 1g 1.000.000 wg (or t) 1.000 mg 1.000 g = 1 kg 1.000 kg "* I metric ton or tonne atone*: I carat = I metric carat =» 200 mg

-

For precious

1-145

-

Avoirdupois Weights 16 oz (os avdp) 100 lb 2.000 lb 112 lb 2.240 lb Engineer's unit: I slug (or geepound) =

=>7.000 grains =» I short cwt = I short ton I long cwt 1 long ton 32.174 lb

"

I lb (lb avdp)

-

I lb (lb troy)

--

Troy Wrights*

12 oz (oz troy)

5760 grains

* The grain is the same as the avoirdupois measure; I pound troy — 5.760/7.000 pound avoirdupois, troy 1.097 avoirdupois ounce. (5.760 X l6)/(7.000 X 12) "Pound" and "ounce" always mean the avoirdupois measures unless otherwise stated or implied by context. The apothecary's pound and ounce are the same as the corresponding troy masses.

-


1 ounce

-

Table 8.

Time

solar (or tropical) year = 365.24 days = 8,766 hr 5.259 X I0» min = 3.156 X 10' sec 1 day = 24 hr = 1.440 min = 8.640 X I0« sec

-

Table 9. Dynes

Newtona

Grams

(joules/'meter)

Force

Poundals

Pounds

Torque F°r equivalents of the dyne-centimeter, kilogram

(force)-meter,

1

X

10'

980.7 1.383 X 10' 4.448 X 10'

1 X 10 > 1 0.009807 0. 1383 4.448

0.001020 102.0 1 14. 10 453.6

Table 10.

7.233 X I0» 7.233 0.07093 1 32. 17

and foot-pound, use 2.248 X 10 • Table 15 (Energy, Work, 0.2248 faeand Heat) conversion 0.002205 tors. For quantities mens0.03108 ured in inch-pounds, first 1 divide by 12 to convert to foot-pounds.

Angular Measures Angular Velocity

Plane Angle Seconds

Minutes

Right angles or quadrants

Degrees

0.01667 2.778 X 10 • 3.086 X IO-« 1 0 01667 1.852 X 10 1 60 1 1 0.01 111 3600 60 5400 1 90 3 24 X 10* 4 1 296 X 10* 2 16 X I0< 360 3438 57 30 0 6366 2 063 X I05 » radians

-

Revolutions or circum ferences

Radians

7.716 X 10-' 4.848 X IO-« 4.630 X 10 1 2.909 X 10 ' 0 002778 0.01745 0.25 1.571 6 283 1 0 1592 1

180°

I". . S-

newton-

metcr,

= 0.0174533radians 180 = 100centesimal minutes 1grade 100grades — 1right angle

Revolu tions per second

1 0 01667 0 1592

Revolu tions per minute

JVidians per second

60 1 9 549

6.283 0.1047 1

Solid Angle* I sphere (or steregon) = 4* (or 12.5664) steradiana = 8 spherical right an gles. A ste radian is the solid angle subtended at the center of a sphere of ra dius r by an area r1 of the spherical surface.

GENERAL DATA

1-146 Table 11. Centi

Meters

meters per BCOOIIfi

1 100 30. 48 0.5080 44.70

[Sec.

Velocity

Table 12. Cubic

Feet per second

Feet per minute

Miles

per second

0.01 1 0.3048 0.005080 0.4470

0.03281 3.281 1 0.01667 1.467

1.969 196.9 60 1 88

0.02237 2.237 0. 6818 0.01 136 1

centi meters per second

per hour

1 472.0 63.09 75.77

1

Flow

Cubic

U.S.

Imperial

feet per minute

gallons per minute

gallons per minute

0 01585 7. 481 1 1.201

0.01320 6.229 0.8327

0.0021 19 1 0. 1337 0. 1605

For other conversions involving no change of unit use volume-conversion table (Table 6). I U.S. gpm = 8.02lp lb/hr = = density, lb/ft3, p' = 500. 7p' lb/hr where p

Nautical Velocity I knot (U.S.) = I U.S. nautical mile/hr = 1.152 statute miles/ hr = 1.689 fps = 51.48 cm/sec l or other nautical miles see Length convention factors

time

density,

g/cm*.

Mass Velocity mass per unit of time

Mass velocity

cross-sectional

= velocity X density

area of stream

The units are usually pound-mass, foot, hour (occasionally, pound-mass,


Table 13. Grams per cubic centi meter

Kilograms per cubic meter

Pounds per cubic inch

Density

rounds

Pounds

per cubic foot

U.S.

per

Degrees

API

(for petroleum

Degrees Baume

Sprcific

Density

Pounds per gallon

10.02 0.01002 277.4 0. 1605 1.201 1

141.5 sp gr 60°F/60°F

(for other liquids)

Liquids heavier than water: Degrees Banine. (all liquids) = 145 —

140 sp gr 60°F/60°F

131.5 1)0

145 sp gr 60°F/60°F

Mass per Unit Length 1 g/cm = 0.005600 lb/in. = 0.06720 I lb/ in. = 178.6 g/cm 1 lb/ft = 14.88 g/cm

Mass per Unit Area

I g/cm* I psf

-

2.048 paf 0.4882 g/cm*

Specific

Gravity.

of a substance at a specified tem perature. Specific gravity is the ratio of the weight of a substance at temperature to the a specified weight of an equal volume of a reference substance (usually water), In also at a specified temperature. notations such as 20°C/4°C or 20/4C, the first temperature refers to the material and the second to If the the reference substance. is (as in the reference temperature 4°C, specific gravity is example) numerically equal to density in grams per milliliter. The dimen is ML-*; specific sion of density gravity is a numeric.

Gravity of Liquid* by Hydrometer

products)

and

Density is the mass per unit volume

Imperial

gallon

62.43 8.345 I 1000 0.03613 0.001 1 3.613 X 10' 0.06243 0.008345 1 1728 2.768 X I0< 231 27.68 16.02 5.787 X 10' 1 0. 1337 0.01602] 1 0. 1196 119.8 0.004329 7.481 0.8327 0.099781 99.78 0.003605 6.229

Liquids lighter than water:

foot, second).

lb/ft

Table 14. Baryes or dynes per square centi meter 1 1.807 X 10! 10 6.895 X I0< 478.8 1.333 X 10« 3.386 X I0< 2488 2.986 X 10* 1.013 X I0«

factors

units and conversion

Sec. 1-3]

Pressure

Kilo grams per square centi meter

~T7o2
1.020

x 10'

Newtons per Bquare meter 0. 1 9.807 X 10' 1

Centi

Pounds per square inch

Pounds per Bquare foot

1.450

0.002089

X 10' 14 22

meters of mercury at 0°C

1.450 io-< 1

73.56

0.02089

9.869

2.036

27.71

2.309

0.03591

0.01414

0. 1924

0.01603

0.3937

5.358

0.4465

0.01316

0. 1934

27.85

0 03453

3386

0.4912

70.73

2.540

0 002538 0.03045

248.8 2986

0.03609 0.4331

5. 197 62.37

0. 1866 2.230

1

2116

3.349

5. 171

1333

14.70

4.018

0.9678

X 10'

0.006944

1.013

2.953

32.84

9.869_

X 10-'

X 10'

47 88

X 10'

394. 1

3.349

X 10 '

Atmos pheres*

X 10'

4.882 io-< 0.01360

1

4.018

x io«

Feet of water at 60° F

X io-<

6895

1.033

28.96

7.501

144

Inches of water at 60° F

2.953" X 10 >

7.501

0.07031

x

Inches of mercury at 0"O (32°F)

X 10'

2048

x

1-147

76

x io-« 0.06805 4.725

X 10'

1

13.61

1. 134

0.03342

0.07348 0.8818

. 1 12

0.08333 1

0.0024S6 0.02947

29.92

407.2

33.93

1


Units of Atmospheric

Pressure I bnrye — I dyne/em* = I bar 10' dynes/cm1 (This is the internationally accepted meaning; but 1 bar is also sometimes I dyne/cm9.) or

used as a synonym

of 1 barye

•The standard atmosphere is based on 76 cm (= 29.92 in.) of mercury at 0°C, at a location of standard gravity and iB defined by the Tenth General Conference on Weights and Measures ( 1954) as 1.013.250dynes/cm* or 101.325 newtons/ meter'. Hence I atm equals 1.01325 bars or 1.03323 kg force per square centimeter. (In power plant practice, steam condenser vacuum is referred to a 30-in. barometer instead of 29.92 in.)

Table 16. Kilo

Ergs (dynecenti

(force)-

meters)

metera

gram

Energy, Work, Heat

Joules (newtonmeters,

watt

Kilo-

Kilo-

Calories

watthourB

calories

. Foot pounds

Horse powerhours

British thermal units

Centi grade heat units

seconds) 1 9.807 X 10' 1 X 10' 3.6 X 10" 4.187 X 10' 4. 187 X 10" 1.356 X 10' 2.685 X 10" 1.055 X 10" 1.899 x 10"

1.020

1

X I0 »

X io-'

1

9.807

2.778

X 10-"

2 342

2.724

2.388

2.388

x io-»

X 10"" 0.002342

7.376

X 10 • 7.233

1

3 671 X 10» 0. 4269

3.6 X I0<

426.9

4187

0. 1383

1.356

2. 737 io» 107.6

X 10'

193.7

1899

x

4. 187

2.685 1055

0.2388

2.778

2.388

X 10'

X 10-' 1 1. 163

8.598 X 10' 1

0.001 163

1000

X 10 • 3.766

x io-'

859.8 0.001

0.7376 2.655 X I0« 3.088

1

0.3238

3088 1

3.238

X 10'

2.931

0.2520

778.2

5.275

453.6

0.4536

1401

X 10' X I0 «

641.2

3.725

1.98

9.478

5.266

X 10""

X 10""

0.009295

0.005164

9.478

5.266

X 10-'

X I0 «

X 10<

1.341

3412

1896

1.560

0.003968

0 002205

3.968

2.205

X 10 • 0.001560 5.051 1

0.001285 2544

7. 139

X io-< 1414

X I0«

Energy. Work, Heat I liter-atin = 24.20 cal 101.3 joules I (ft') (atm) 2.719 Btu I (ft1) (pai) = 0.1850 Btu

-

3.653

X IO"'

6.412 X 10' 252.0

0.7457

10"

X io-«

X 10-' 0. 1020

3.725

X

-

3.930

1

7.074

1.8

X 10 « X io-«

= 0.09604 Btu

0.5556 1

GENERAL DATA

1-148

Table 16. Kilo Ergs per second

1

gram

Watts

(force)meters per second

(joules

1.020

per--

Kilo watts'

second)

1

1

Calories per second

2.388

X 10 s

x io-«

X 10'

X 10-'°

9.807 X 10'

I

9.807

0.009807

2.342

1 X 10'

0. 1020

0.001

0. 2388

1 X 10"

102.0

1000

4. 187 X 10'

0.4269

4. 187

0.004187

4. 187

426.9

4187

4.187

1

238.8

1

[SEC.

Power

Kilo-

Foot

Foot

calories per second

pounds per second

pounds per minute

2.388

7.376

x io-» X

10"

7.233

2.388

0.7376

44.25

0. 2388

737.6

0.001

3.088

X I0«

1000

1

3088


434.0

t

Btu

(IT)

Horse power

per hour

1.341

3.412

X 10-" 0.01315

1.356

0.001356

2.260 X 10'

0.002304

0.02260

7.457 X I0«

76.04

745.7

2.931 X I0«

0.02989

0.2931

5.275 X 10«

0.05379

0.3238

3.238

1

0.5275

2.260

0.7457

2.931

x io« 5.275

X 10-«

0.005397

5.397

0.01667

0.07000

0. 1260

550

7.0000

0.2162

X 10' 1.260

X I0«

X 10'

33.46

18.59

4.425 X 10'

1.341

3412

1896

185.3

0.005615

14.29

1.853

5.615

1.429

60

1

3.3 X 10' 12.97

7.937

7937

X 10' 0.001818

3.030

X 10-'

0. 1781

1.896

X 10-'

1.896

X 10« 178. 1

per

hour

3.412

X 10-' X 10'

Chu

0.001341

X 10' 0. 1383

1.356

4.425

X 10'

2.342

X I0»

X 10" X 10'

1

4.626

2.570

0.07710 0.04284

1

2544

1

3.930

1414

0.5556

X 10-' 0.3891

23.35

7.074

1.8

X 10-'

Horsepower = 33.000 ft-lb/min = 1.980.000 ft-lb/hr 1 hp = 550 ft-lb/sec (definition) = 1.0004 hp I electric hp = 746 watts (definition) I metric hp = 75 kg (force) -meters/sec (definition) 735.499 watts = 0.9863 hp I boiler hp = 34.5 lb water evaporated per hour from and at 2I2°F = 33.472 Btu/hr = 13.155 hp (heat equivalent)

-

745.70 »atu»

-

Refrigeration = 12.000 Btu/hr = 200 Btu/min = 4.716 hp = 840.0 cal/sec = 0.9055 British ton of refrigeration 1 ton of refrigeration (British) = 321.200 Btu/day = 13.385 Btu/hr = 223.1 Btu/min = 5.260 hp = 936.9 kcal/sec = I.I 15 U.S. tons of refrigeration equal to the latent heat of fusion of one short ton and one long The units arc approximately respectively, of ice per 24 hr. 1 ton of refrigeration

(U.S.) = 288.000 Btu/day

ton.

Sec.

1-3]


1-149

FACTORS

Table

17. Viscosity Absolute Viscosity, u

Centipoises

Poises

1 100 1000 1488 4.788 X I0« 0.4134

0.01 1 10 14.88 478.8 0.004134

KiloKrani-mass

Pound-mass

(»ec)(nioter)

Pound-force (sec) (fti)

(see) (ft)

0.001 0. 1 1 1.488 47.88 4. 134 X I0<

6.720 X 10-' 0.06720 0.6720 1 32.17 2.778 X 10"«

2.089 X 0.002089 0.02089 0.03108 1 8.634 X

Alternative Dimensional System* The units of absolute viscosity are expressed alternatively in absolute equivalent gravitational units (dimensions FT/L1).

10'

I0«

units (dimensions

Pound-mass

(hr)(ft)

2.419 241.9 2419 3600 1. 158 X 10' 1

M/TL)

or in

1 dyne-sec /cm* 1 poise ™ 1 g /(sec) (cm) 1 lbM/(sec)(ft) I poundul-*ec/ftJ 1 slug/(sec)(ft) I lbF-sec/ft1 ' kgM/(sec)(metcr) = I newton-sec /meter*

-


All tabulated viscosity data in this handbook are expressed in pound-mass per hour per foot units Laboratory data are expressed formally in centipoiaes, a metric unit of convenient size Hast column). for many liquids; water has a viscosity of I centipoise at a temperature of 20. 20°C (68.4°F). Kinematic Viscosity, v The kinematic viscosity is the absolute viscosity divided by the density cgs unit of viscosity is the stoke. I stoke = I cmVsec

= 0.1550 in.Vsec = 3.875

of the fluid: » ** u/o.

The

ftVhr

Viscosity can be measured in the laboratory by the time in seconds for free discharge of a given volume of fluid through the capillary of an efflux viscometer (viscosimeter). The kinematic viscosity in stokes can be calculated from the efflux time t by the empirical formula v = At —

B/t

cm1 /sec

on the Saybolt Universal scale (United States) and where A and B are 0.0022 and 1.8, respectively, 0.0026 and 1.72, respectively, on the Redwood scale (British). The viscosity of reactor coolants is below the useful range of these and similar viscosity scales.

GENERAL DATA

1-150

Table 18.

[Sec.

1

Temperature

Centigrade

to Fahrenheit (Values are exact)

°c

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400

32 212 392 572 752

50 230 410 590 770

68 248 428 608 788

86 266 446 626 806

104 284 464 644 824

122 302 482 662 842

140 320 500 680 860

158 338 518 698 878

176 356 536 716 896

194 374 554 734 914

500 600 700 800 900

932 1112 1292 1472 1652

950 1130 1310 1490 1670

968 1148 1328 1508 1688

986 1166 1346 1526 1706

1004 1184 1364 1544 1724

1022 1202 1382 1562 1742

1040 1220 1400 1580 1760

1058 1238 1418 1598 1778

1076 1256 1436 1616 1796

1094 1274 1454 1634 1814

1000 1100 1200 1300 1400

1832 2012 2191 2372 2552

1850 2030 2210 2390 2570

1868 2048 2228 2408 2588

1886 2066 2246 2426 2606

1904 2084 2264 2444 2624

1922 2102 2282 2462 2642

1940 2120 2300 2480 2660

1958 2138 2318 2498 2678

1976 2156 2336 2516 2696

1994 2174 2354 2534 2714

Interpolation Table °C

I

2

1

3

4

°F | 1.8 3.6 5.4 7.2 Example: I054°C =- 1922 + 7.2 I929.2°F.

-

5

6

1

7

8

9

9.0

10.8

|

12.6

14.4

16.2

to Centigrade to nearest

Fahrenheit


(Centigrade temperatures °F 0 100 200 300 400

-

0 17.8

-

10 12.2

37.8 93.3

43.3 98.9

148.9 204.4

154.4 210.0

500 600 700 800 900

260.0 315.6 371.1 426.7 482.2

265.6 321.1

1000 1100 1200 1300 1400

537.8 593.3 648.9 704.4 760.0

543.3 598.9

1500 1600 1700 1800 1900

815.6 871. 1 926.7 982.2 1037.8

821.1

2000 2100 2200 2300 2400 2500

376.7 432.2 487.8

20

30

-6.7

-1.1

48.9

710.0 765.6

4.4

54.4

104.4 160.0 215.6

110. 0 165.6 221.1

271.1

276.7 332.2 387.8 443.3 498.9

326.7 382.2 437.8 493.3 548.9

654.4

40

60

10.

15.6 71.1 126.7 182.2 237.8

65.6

60. 115.6 171. 1 226.7

121.1 176.7 232.2

282.2

287.8 343.3 398.9 454.4 510.0

337.8 393.3 448.9 504.4

604.4

554.4 610.0

615.6

565.6 621.1

660 . 0 715.6 771.1

721.1 776.7

671. 1 726.7 782.2

293.3

348.9 404.4 460.0 515.6

80

90

21.1

76.7

26.7 82.2

32.2 87.8

132.2 187.8 243.3

137.8 193.3 248.9

143.3 198.9 254.4

298.9 354.4 410.0

304.4

310.0 365.6

465.6 521.1

360.0 415.6 471.1 526.7 582.2

587.8 643.3

676.7

637.8

737.8 793.3

743.3 798.9

693.3 748.9 804.4

698.9

732.2 787.8

754.4 810.0

837.8 893.3

843.3 898.9

954.4

854.4 910.0 965.6

860.0 915.6

865.6

948.9

848.9 904.4 960.0

1043.3

1004.4 1060.0

1010. 0 1065.6

1015.6 1071. 1

1093.3 1148.9 1204.4 1260.0 1315.6

1098.9 1154.4 1210.0 1265.6 1321. 1

1104.4 1160.0 1215.6 1271. 1 1326.7

1110.0 1165.6 1221. 1 1276.7 1332.2

1115. 6 1171. 1 1226.7 1282.2 1337.8

1121. 1 1176.7 1232.2 1287.8 1343.3

1371.1

1376.7

1382.2

1387.8

1393.3

1398.9

1021. 1 1076.7

971. 1 1026.7 1082.2

921.1 976.7 1032.2 1087.8

1126.7 1182.2 1237.8 1293.3 1348.9

1132.2 1187.8 1243.3 1298.9 1354.4

1137.8 1193.3 1248.9 1304.4 1360.0

1143.3 1198.9 1254.4 1310.0 1365.6

1404.4

1410.0

1415.6

1421. 1

Interpolation Table 1

I

2

4

3

°C 1.7 0.6 1 I.I 1 2.2 Example; 1054°F =■565.6 + 2.2 567.8°C.

-

Centigrade to Fahrenheit: Fahrenheit to Centigrade: Degrees Kelvin: Degrees Kankine:

5

6

2.8

3.3

Conversion Formulas

= 9i'c + 52 — 32) tc = %{t¥ *K ■ tc + 273.16 (R = t¥ + 459.69 (R HtK

(f

-

532.2

576.7

832.2 887.8 943.3 998.9 1054.4

"F

421.1 476.7

632.2 687.8

571.1

826.7 882.2 937.8 993.3 1048.9

876.7 932.2 987.8

70

626.7 682.2

560.0

665.6

0. 1°C)

50

1

7

8

9

3.9

4.4

5.0

units and conversion factors

Sec. 1-3]

Table 19.

1-151

Specific Heat

-

-

= I kcal/(kg)(°C) 1 Btu/(lb)(°F) I ohu/(lb)(°C) = 4.187 joules/(g)(°C) Specific heat was originally denned as the heat capacity of a substance relative to that of an equal mass of water at a specified temperature. The property was therefore a diinensionless ratio. The term is now generally used in the sense of heat capacity per unit mass per degree of temperature rise. 1 cal/(g)(°C)

Table

Thermal

21 .

Table 20. Heat Capacity

Conductivity

Cal

Watts

Btu

Btu

Btu

Calorie

Joule

Btu

(»tc)(cm)(°C)

(em)CC)

(hr)(ft)(°F)

(hr)(ft')(°F/in.)

per gram

per gram

per pound

241.9 57.78 12 1 0.08333

2903 693.3 144 12 1

1 0.2388 0.5556

4. 187 1 2.326

1.8 0.4299 1

0 0 0 3

1 2388 04961 004134 445 X 10-'

20.02

4. 187

4.782 1 0 08333 0.006944

0.2077 0.01731 0 001441

1 kcal/kg


-

-

I cal/K = I elm /lb

1 joule/(sec)(cm)(0C) 1 watt/(cm)(°C) 1 joule/Ojec)(m)(°C) ! watt/(m)(°C) 0.01 watt/(cm)(°C) Thermal conductivity is frequently stated as heat units per unit time, per unit area, per unit of teml*;r;ture difference per unit thickness, written, for example, Btu/(hr)(ft,)(°F/ft). For consistent The inconsistent units shown in the last column units the designation can be simplified as in the table. are commonly used in practical problems involving heat conduction through plates, walls, insulation, etc. Conversion factors for the chu used in conjunction with the degree centigrade are the same as for the Btu used in conjunction with the degree Fahrenheit; e.g., the Btu/(hr)(ft)(°F) column serves also

-

forchu/(hr)(ft)(°C).

Table 22.

Heat Flux and Conductance

Heat Flux

Thermal Conductance

Cal

Watts

Btu

Cal

Watts

Btu

(sec) (cm1)

cm*

(hr)(ft»)

(hr)(ft<)

(sec)(cm')(°C)

(cm>)(°C)

(hr)(lt«)(°F)

1

4. 187

1.327 X 10' 3170 1

7373 1761 0.5556 1

0.2388 1.356 X I0«

0.2388 7 535 X 10 "■ 3 155 X 10' 1 356 X 10 -« 5.678 X 10-'

Chu

1.8

- IX

I watt/cm1 I joule /(sec) f cm1) I joule/CsecXmeter1) — I watt/meter* 10"* watt /cm'

Table

1

4. 187 1 5 678 X I0~«

I joule/(sec)(cm')(°C) I joulo/(sec)(meter,)(°C)

= 1 watt/(cm>)(°C) = I watt/(meter,)(0C) I X 10< watt /(cm') (°C) I Btu/(hr)(ft>K°F)

-

1 ohu/(hr)(it»)(°C)

Internal Heat Generation and Power Density

23. Watts

Cal

cm1

(scc)(cinl)

1 4 187 0.01788 1 035 X I0»

0.2388 1 0.004272 2 472 X 10 •

I watt/cm" I chu

-

Btu (hr)(in.<)

55.91 234. 1 1 5 787 X 10"«

1 Mw /meter1 1.8 Btu

-

7373 1761 1

Btu (hr)(ft»)

9.662 X 10' 4.045 X 10' 1728 1

1.000 kw/liter

1-152


Absolute (cgs) electrostatic, esu M KS or practical, prau Absolut* (ogs) electromagnetic,emu

Potential differenceand electromagnetic force

Electric current

Capacitance

Resistance

I0~'cstatamp 1amp

!0"c-i statvolt 1 volt

lO'c-'statohm 1 ohm

10 -V Statfarads 1 farad

10"" abamp

10-abvolts

10"abohms

10 ■abfarad

Magnetic field intensity

Magneto motive force

Magnetic flux

Magnetic flux density

1 praoersted 10"' oersted

1 pragilbert 10"' gilbert

1 weber 108maxwells (or lines)

1 weber/meter* 10*gauss

Absolute (cgs) electrostatic, esu. 10-"cstatcoulombs MKS or practical, prau 1 coulomb Absolute (cgs) clectroniaiinetir, 10-i abroulomb

System

1

Electrical Units

Quantity of electricity

System

[SEC.

Inductance

I0*c~sBtathenry 1 henry I0» abhenrys

c = 2.99793X I010,a numeric equal in magnitude to the speed of light in centimeters per second. Approximately,


c

- 2.998 X

lOio, ct = 8.988 X 10», tT»

- 0.3336X IO-",

c~*

-

1.113X 10-"

Example*: 1. 500amp = 500 X 10-' abamp = 50 abamp. 2. I X 10»statcoulomhs (1 X I0»)/(I0"« X 2.998 X 10") coulombs

-

1 gilbert «■

- 3.336 X

10"* coulomb.

ampere turns

1 oersted — 1 gilbert/cm I gauss = I maxwell (or line) /cm*

Table 25. Grama per

liter

1 0.001 0.01712 0.01425

Parts per million*

1000 1 17. 12 14. 25

Concentration

Grain* per U.S. gallon

Grains per Imperial gallon

58.42 0.05842 1 0.8327

70. 15 0.07015 1.201 1

mg

mg

(em1) (month)

mg

(dm:)(mnntli)

(dm»)(
1 0.01 0.3042 211 7p 0.2117s 2. 540p

100 1 30. 42 2. 117 X 10V 21. 17P 254. Op

3 288 0.03288 1 695. 9p 0.6959p

1 month = Ka mean solar year. * p = density of material, g/cm1.

8.34l„

-

-

* Conversion based on parta by weight in water of density

Table 26.

Normal solution: One gram equivalent weight divided by hydrogen (molecular equivalent) per liter. Gold content of allay: The carat is the number of parts by weight of gold per 24 parts of alloy. 1 carat >i« (any mass 41.6667 mg/g. units)

I g/cm!.

Corrosion* In. /year

0. 004724/p 4.724 x 10 »/p 0.001437/p 1 0 001 0 012

Mi la /month

0. 3937/p 0.003937/p 0. 1198/p 83.33 0.08333 1

Mils/year

4. 724/p

0.04724/p 1.437/p 1000 1 12

Sec. 1-3] 4


ATOMIC UNITS, CONSTANTS, 4.1

1-153

FACTORS

AND CONVERSION

FACTORS

Units

The units of classical

physics (the cgs units and the supplementary units of the electrical systems) are generally used in theoretical physics. Since these classical units are inconveniently large for practical calculation on an atomic scale, convenient nonsystematic units are defined for limited use. 4.11 Atomic Mass Unit, amu. The amu, expressed in the physical scale, is defined as one-sixteenth of the mass of an O" atom. absolute

1

amu = (1.659790

+ 0.000044)

X

10~» g

this is the reciprocal of Avogadro's number on the physical scale. Electron Volt, ev. The electron volt is the kinetic energy acquired by a particle of unit charge (the charge of one electron) in falling through a potential difference of one (practical) volt. Numerically, 4.12

1

Barn.

4.13

This unit of


1 4.14

Other Atomic Units. handbook.

= 1,000,000 ev

area is a measure 1

of the

Mev

barn millibarn

of nuclear cross section.

= 10~" cm2 = 0.001 barn

Other special units are defined in the various sections 4.2

Physical Constants

Several important dimensional constants are introduced to express laws of atomic nuclear physics. The values in Table 27 are from the compilation by Du Mond, et al.* The compilation by Birge* was in general use until Cohen,

and

recently.

4.3

Masses and Energy of Particles

Physical and Chemical Scales. In the physical scale, the atomic weight of as 16 exactly, whereas in the chemical scale, the atomic weight of oxygen in its natural isotopic composition is taken as 16 exactly. Atomic masses and derived constants may be expressed in either scale. Birge' gives 1.000272 ± 0.000005 as the ratio of atomic weight in the physical scale to atomic weight in the chemical scale, and this value is adopted by Cohen et al. Birgc's ratio is based on an isotopic composition Ol,:0":0H = (506 ± 10):1:(0.204 ± 0.008), but Nier7 and others have since shown that the isotopic concentration of natural oxygen varies with the source to an extent that causes the calculated ratio of the physical to the chemical scale to In the absence of an assigned standard range at least from 1.000268 to 1.000278. composition, the ratio therefore includes an inherent uncertainty, not associated with precision of measurement. Constants expressed per gram-molecule are all smaller, by the given ratio, in the chemical scale than in the physical scale. This applies to atomic weights, Avogadro's molar volume, the Faraday, and the universal gas constant. Atomic number, weights of the elements in their natural isotopic abundance are expressed in the chemical scale; hence the chemical scale generally applies in engineering calculations Particle and isotopic masses are expressed involving the above-stated quantities. in terms of amu in the physical scale. 4.32 Particle Masses, Energy Equivalents, and Wavelengths. Mass-energy equivalents are given in Table 28, data for particles in Table 20, and wavelengths and energy relations in Table 30. The values are those of Cohen et al.* 4.31

0"

is taken

1-154


[SEC.

Physical Constants*

Constant Avogadro number and related constantst Avogadro number N:

Value

Chemical

scale

(6.02322 (6.02486

Chemical

scale

22414.6

Physical scale Molar volume of ideal gas at 1 atm:

J

± 0.00016) X 10" mole"' ± 0 00016) X 10" mole > ± 0. 6 cm'}

X 10" cm > X I0"'«erg/°C IO-»ev/°C 10-'

/;'

Physical scale 22420 7 ± 0.6 cm' Loschmidt number (molecules /cm" of ideal gas at I atm) (2 68719 ± 0 00010) Boltzmann constant and related constantst oonstant, BolUmann k (1.38044 + 0 00007) (8.6167 ± 0 0004) X (4.7871 ± 0.0002) X = Universal gas constant. Nk:

ev/°Ft

(8.31470 0.00034) X 10' erg /(mole) (°C)t 2781 . 70 + 0. ft-lb/(lb-mole)(°C)t 1545.39 06 ft-lb /(lb-mole) (°F)t (8.20575 0.00034) X 10"' liter-atm/(mole)(°C)t 1.98591 >0 0008 cal/(mole)CC)t 1.98591 >0. 0008 chu/(lb-mole)(°C)t 1.98591 >0. 0008 Htu /(lb-mole) (°F) (8.31696 0.00034) X 10' erg/(inole)(°C) ±

scale

Physical scale Electronic data: Faraday constant,

F:

R.

«»

-

( scales,

1.000272

X X

X X

1

X lO^'ergscc X lO"" cv sec

0.00023) 0.00007) 0.00004)

10"

erg sec

+ 0.000005t

constant: 2*'me'/ch<

0.012cm'

109737.309 109677 576

± +

Rydberg

to chemical

r

H/1t Ratio of physical

1 1

± ±

(6 62517 (4 13541 1.05443

-

X

...

h

Planck's constant: Planck's constant,

e/m.

G

ratio of the electron,

Gravitational constant,

h

F/N

± + +± ± ± ±

=

96495.7 ±1.1 coulombs /molet 9652. 19 0. emu/mole (2 89366 0.00003) I0'
e

Electronic charge,

(0.56687 (1.35393 (0 99831 (0.171 18 (7.5635 (0.289782

0.00010) X 10 erg/(cm>)(°K)<(sec) >0. 00024) X 10"" calAcm'lCKi^secOt >0. 00018) X I0"' chu/(ft')(°K)< (hr) >0 00003) X I0-" ntu/(ft')(°R)«(hr)t 10"" erg/(cm»)(°K)«t 0.0013) 0.000013) (cm)(°K)

0.012cm-'

constant:

Fixed nucleus, lm/h'

atom, Hydrogen electron) Velocity of light:

lix/h1

(p = reduced

(1.63836

00007) X

00007) X 10" erg-' cm

mass of (1.63748

I0i;erg-'

± 0

Schrodinger

+

Density constant. 4,/e Wein's displacement law \„,*XT

X

'

-

X

cr

±

constant,

+ 0

Surface

± ± ± ±

Stefan-Boltziriann and related constants:

cm"1

1

X

a

A

is

t

±

:

2

c

+ 0 0

10'° cm/sec (2.997930 000003) c* 0000181 X 0!° em'/scc't (8.987584 + * Except where otherwise indicated, all data are taken directly from tables by E. R, Cohen et al.. Analysis of Variance of the 195 Data on the Atomic Constants and New Adjustment. 1955. Rev. Mod. Phyt., »7 363-380 (1955). adopted The ratio of the physical to chemical scales as given by R. T. Tiirite (1.000272 0.000005) by Cohen et al. In this table the constants on the chemical scale have been calculated using the ratio 1.000272 exactly. The "mole" and "lb-mole" mean the gram-molecule and pound-molecule, respec tively, on the appropriate scale. Derived from tables by Cohen et al.. lor. cit. Value by R. T. Hirge. New Table of Values of the General Physical Constants, Rett. Mod. Phym., Also, R. T. Birge, The General Physical Constants as of August, 13: 233 (1941). 1941. Phyt. Sor. (London), Reptt. Progr. in Phyt., 8: 90 (1942). *J X


Chemical scale Physical scale

Charge-to-mass

J

± ± ± ± ± ± 0.

1 I

Chemical

1


Sec. 1-3]

4.4 1

Mev/sec

1

watt

= 1.60206 = 3.8264 = 6.24196

Power

X lOr" watt = 1.60206 X 10"" kw X 10"" cal/sec = 5.4678 X 10"" Btu/hr X 10" Mev/sec

Power density conversion factors are given in Table 23. equivalents are given in Art. 5 of Sec. 1-1.

Table 28. Amu

(1.073951

+ 0 000011) »

± 0.00016)

X I0»

670.354

± 0.019

Equivalents*

(1.659790

Ergs

± 0.000044)

X 10

1

(1.78252

"

(1.491750

(5.61000

± 0.00004) 1

± 0.00011)

(1.60206

(0 624196 ± 0.000012) X I0«

± 0.00003)

X 10'

(8.987584

X 10"

± 0.000043)

X 10-'

X 10-"

± 0.000018)

X 10"

(1. 112646 ± 0.000002)

1

X 10"

I cv = (1.60206 ± 0.00003) X 1 Mev 1.60206 X I0"" joule

-


Power, flux, and burnup

Grains

931. 141 ± 0. 010

x io

(6.02486

Mass-Energy

Mev

1

1-155

FACTORS

-

I0~"erg 4.4508 X I0"> kwhr

-

1.5188 X 10 = 3.8264 X I .• 2.147 X 10" cal = 8.518 X 10" Btu 8.988 X 10" joules = 2.496 X 10' kwhr lib 4.077 X 10" joules 1.132 X 10" kwhr = 9.737 X 10" cal = 3.863 X 10" Btu 1 ev/atom (or molecule) = 23047 cal/mole (chemical scale) I cal/mole (chemical scale) *« 4.3389 X 10"s ev/atom (or molecule)

-

-

-

" Btu 10" cal

•Equivalents are derived from the following values by Cohen et al.: c = (2.997930 ± 0.000003) X I0>»cm/sec, I amu = 931.141 ± 0.010. I g = (6.02486 ± 0.00016) X 10" amu. 1 ev (1.60206 ± 0.00003) X 10"" erg. The heat-conversion factor is taken as 4.18684 joules /cal, and the molar volume on the chemical scale as 2241 4.6 cm3.

-

Table 29. Particle

Amu

Rest Mass of Particles* Mev

(5.48763 ± 0.00006) X 10"' 0.510976 939.505 1.008982 ± 0.000003 1.007593 + 0.000003 938.211 Proton 938 722 1.008142 ± 0.000003 Hydrogen atom. . .

+0.000007 ± 0 010 + 0 010 + 0. 010+

Grams

(9. 1083 + 0.0003) X (1.67470 ± 0.00004) (1.67239 + 0 00004) ( 1.67330 ± 0.00004)

10"" X 10" X 10 " X 10 !'t

Hydrogen mass /proton mass = 1.000544613 ± 0.000000006 Proton mass /electron mass = 1836.12 + 0.02 Reduced mass of electron in hydrogen atom = (9.1034 ± 0.0003) X I0~3B k * Except where otherwise indicated, all data are taken directly from tables by E. R. Cohen et al., Analysis of Variance of the 1952 Data on the Atomic Constants and a New Adjustment. 1955. Reva. Mod. ny$., 27: 365-380 (1955). t Calculated from data by Cohen et al., loc. cit.

GENERAL DATA

1-156 Table 30.

[SEC.

1

Wavelength and Energy Relations*

Property

Value

Photons:

-

Wavelength, X. cm Wave number, f

(12397.67 ± 0.22) X \0'>/B (8066.03 ± 0. \ 4)E

1A, cm"' Electrons: Compton wavelength, h/me, cm de Broglie wavelength, h/mv, cmt Neutrons: Compton wavelength. h/.\l„c. cm de Broglie wavelength, h/M.t: omt

(24.2626 ± 0.0002) X 10-11 (1 . 226378 ± 0.000010) X

lO'/JS"

(13.1959 (3.95603

± 0.0002) X 10"" ± 0.00005) X 10-'/»

(2.86005 ± 0.00004) X \0'/EH Velocity, v, om/sect (1.38320 ± 0. 00003)£^ X 10«J Velocity of 0.025-ev neutron, meters/sec 2187.036 ± 0.012 Energy, B, evt (5 22671 ± 0. 000006)v' X 10"t Energy of 2.200-meter /see neutron, ev 0.0252973 + 0.0000003 kT temperature for 2.200-m/sec neutrons, °C 20.426 ± 0.022J v = velocity, cm /sec E — energy, ev * Except where otherwise indicated, all data are taken directly from tables by E. R. Cohen at al.. Analysis of Variance of the 1952 Data on the Atomic Constants and a New Adjustment, 1955. Reve. Mod.

Phye., 17: 363-380

t

t

(1955).

Quantities involving velocity are for the nonrelativistic range. Calculated from data by Cohen ct al., lac, cit.

REFERENCES Units of Weight and Measure (United States Customary and Metric), Definitions and Tables of Equivalents, Natl. Bur. Standards Misc. Publ. 214, July 1, 1955. 2. Standard Time throughout the World, Noll. Bur. Standards Circ. 496, Aug. 1, 1950. 3a. Griffith, Eier: "The Heat Unit," Institute of Mechanical Engineers, London, 1951. 6. Stimson, H. F.: Heat Units and Temperature Scales for Calorimetry, Am. J. Phys.,


1.

23: 614-622

4. 5.

6a.

(1955).

Establishment and Maintenance of the Electrical Units, Natl. Bur. Standards Circ. 475, June 30, 1949. Cohen, E. R., J. W. M. Du Mond, T. W. Layton, and J. S. Rollett: Analysis of Variance of the 1952 Data on the Atomic Constants and a New Adjustment, 1955, Revs. Mod. Phys., 87: 363-380 (1955). Birge, R. T.: A New Table of Values of the General Physical Constants, Revs. Mod. Phys.,

13: 233 (1941). R. T.: The General Physical Constants as of August, 1941, Phys. Soc. (London), Repts. Progr. in Phys., 8 : 90 (1942). of the Relative Abundances of the Isotopes of Carbon, 7. Nier, A. O. : A Redetermination Nitrogen, Oxygen, Argon, and Potassium, Phys. Rev., 77: 792 (1950). b. Birge,

BIBLIOGRAPHY CF-51-8-10: "Manual of Pile Engineering," AEC Technical Information Service, Oak Ridge, Tenn., Dec. 28, 1951. Cohen, E. R., K. M. Crowe, J. W. M. Du Mond: " The Fundamental Constants of Physics," Interscience Publishers, Inc., New York, 1957. Fundamentals," 2d ed.. Sec. 1, pp. Eshbach, O. W. (ed.): "Handbook of Engineering 148-166 and Sec. 3, pp. 01-35, John Wiley & Sons, Inc., New York, 1952. International Bureau of Weights and Measures: Reports of General Conferences. National Bureau of Standards: Various Circulars and Miscellaneous Publications. Zimmerman, O. T., and I. La vine: "Conversion Factors and Tables," 2d ed., Industrial Research Services, Inc.. Dover, New Hampshire, 1955.

SECTION

2

NUCLEAR DATA BY

SOODAK, B.S., M.A., Ph.D., Assistant College of New York.

HARRY

Professor of Physics,

The

City

CONTENTS 1 Fission- process Data 2 Cross Sections of Fissionable and Related


Heavy

Atoms

3 Moderator Data Cross Sections 4 Fission-product 5 Thermal Cross Sections

PAGE

2-2 2—5

2-8 2-10 2-12

6 7 8 9

Integral Data Fast-neutron Data Radiations and Their Ranges Atomic Weights

Resonance

References

2-1

PAGE

2-24 2-27 2-34 2-35 2-36

NUCLEAR DATA BY Harry Soodak FISSION-PROCESS DATA

1

1.1

Energy from Fission*-1'8


The energy from slow-neutron fission of U"5 is distributed as listed in Table 1. The large amount, 168 Mev, of fission-product kinetic energy is converted into heat The 7 Mev of 0 within the very short range of the fission products (see Table 35). The 18 Mev plus the capture energy is also converted to heat near the point of fission. 7 energy amounts to about 10 per cent of the total energy and is a source of heat in all parts of the reactor. The 11 Mev of neutrino energy escapes from the reactor and is not a source of heat. Table Space

1.

Energy from Slow-neutron

U'" Prompt plus

Prompt energy, Mev

distribution

Fission of

Delayed

energy,

Mev

of heat

Localized heat . . Fission-product kinetic energy, 168*

Fission-product decay fin, 7{

Neutron kinetic energy ( 5) pluH prompt fission ys (7.5)t = 12.5

Fission-product decay 79, 6J

Dispersed

heat. .

plus capture

Total heat

delayed

energy,

Mev

175 18 plus capture

7a

193 plus capture

7a

7s

180 plus capture

ya

13

No heat

Neutrinos from Q decay, 1 1H

i

• E.

J. Whitchouse, Progr. Nuclear Phy*., : I 20 (1952). t R. Gambia, Ph.D. Thesis, University of Texas, June, 1955. t Old, C. C, and J. W. Weil, TID-65, 1948. 1 K. Way and E. P. Wiener, Phya. Rev., 73: 1318 (1948). 1.2

Prompt-neutron

Spectrum1

Of the total number v of neutrons emerging from a fission process, the large fraction — 0) is emitted promptly, and the small fract ion 0 is the delayed neutrons emitted by excited nuclei formed in the jS decay of fission-product atoms. Values for the total neutron yield v are given in Art. 1.3. The c(l — 0) prompt neutrons emitted from a slow-neutron fission of U,si are born with an energy spectrum that is fairly well fitted by the theoretical expression (1

N(E) where

N(E) dE

=

{-J

e-" sinh (2E)H

is the fraction of neutrons emitted in the energy range between

* Superscript numbers

refer to References

at end of section.

2-2

(1)

E

and

Art.

(E + dE) and

E

mum at

E

is in Mev.

= 0.72.

The average neutron energy is 2.0 Mev.

Table

functions of E. The prompt-neutron that for U"s.

2-3

DATA

FISSION-PROCESS

1]

15

N{E) is a maxi

f* N(E) dE, and jg

of Sec. 7-3 lists N(E),

N(E) dE

Yields3

Delayed- and Prompt-neutron

*

1.3

within experimental error, the same as 5

spectrum for Pu23'

is,

as

(3

is

is

2

The delayed neutrons with significant yields fall into six groups. The neutrons of each group are emitted exponentially from the time of fission with a certain halflife. The half-lives are listed in Table along with the mean energy of neutrons in Also listed the precursor of each group. The delayed neutron each group. emitted following decay of the precursor atom. Delayed-neutron

2.

Littler

Half-life, sec

54 22

Precursor

Br"'

250 500 400 500 400

5.8 0.46 0. 16

im

Br'»»-»i>t

(I'")t

(Sb'» or As»')t

(Li')t

compiled from the various data presented in R. A. Charpie, J. Horowitz. D. J. Hughes, (eds.), "Physics and Mathematics," McGraw-Hill Book Company, Inc.. Now York,

t

Data in parentheses are uncertain. Li9 fragment in a small fraction of the fissions.

Tables

Mean energy, kev

is

J.

is

• This table

and D. 1956.

presumed

to be formed

as the long-range

low-mass

4

2

of Sec. 8-1 present data on relative yields of the various delay groups. to An important reason for this that the seen that the half-lives are not all alike. observed emission of delayed neutrons can be analyzed into sums of exponentials in more than one way. The relative yield of neutrons emitted in group 0%/f), where the number of delayed neutrons emitted in group per fission and vff the total rfii number of delayed neutrons emitted per fission. is

is

i

is

i

is

It is

Delayed Photoneutrons4

1.4

y

Delayed neutrons may also arise from photoneutron production by the delayed Because of the lack of high-energy 7s in fission-product radiations, delayed rays. photoneutrons are of significance only for reactors containing deuterium atoms or

Half -life and Maximum Yield

of Delayed Photoneutrons D20-U!3S Reactor*

2.5 41 144 462 1.620 5.940 15.840 190.800 1.105.000 * A. Lundby and N. Holt. Nucleonics, 12(1):

Yield. 78 24

10

•

Half-life, tec

B. 4

in

a

Table 3.

4.0 2.5 2.8 0.39 0. 12 0.05 24 (January, 1954).

chap.

1.

See Kef.

3.

if

is

3.

is

is

For these atoms, the photoneutron threshold energies are 2.22 and beryllium atoms. The half-life of each delayed photoneutron group 1.67 Mev, respectively. the the precursor of the excited atom emitting the 5-decay half-life of the atom that y ray. The half-lives and maximum yields of delayed photoneutrons in a DzO-UJ3S reactor are presented in Table The maximum yield that obtained all the *


6 5 4 3 2 1

Group index

Half-lives, Energies, and Precursors*

2. 1

Table

2-4

NUCLEAR DATA

[SEC. 2

Absorption or fission-product 7 rays were born inside an infinite block of DjO. degradation of the y ray by fuel or other material and leakage of the y ray out of the important neutron region of the reactor would both reduce the yield below the maximum. Prompt Gammas

1.5

A fission in U"6 results in the emission of 7.5 Mev of prompt 7-ray energy. The energy spectrum of the individual y rays is roughly a decreasing exponential with 7-ray energy. The average 7-ray energy is 0.9 Mev, and thus somewhat over 8 7s are emitted (on the average). Delayed Gammas and Betas''8

1.6

About 6 Mev of 7-ray energy is emitted by the fission products as they 0 decay The rate of release of this 6 Mev diminishes with time and for not toward stability. too short times roughly behaves as

r

= 1.3r12

for

Mev/(sec) (fission)

t>

10 sec

(2)

»

= 0.032P«[r°

-

(t

+

for

t

Py

■•]

where I is the time in seconds after the occurrence of the fission and T is the 7 energy emitted in Mev per second per fission. This formula leads to > 10 sec f0

t

7

is

X

10uPje[r0-2

-

+ to)-"-*]

Mev /sec

for

I

-

(t

Py

2

If

is

is

a

is

the power of a reactor that has been running steadily for time sec, at where Pr the delay power at the time sec after which time the reactor shut down, and P-, shutdown. Pr expressed in watts and Py in Mev per second, then > 10 sec

(4)

is

7

0

I

0

3

t

7

7

is

somewhat higher than the value of Py The rate of energy during the first 10 sec rays peaks at about 0.8 Mev at — 10 sec. The spectrum of these after-shutdown and extends up to about Mev. For fission-product > 10 sec rays, the energy release for roughly the same as energy. that for rays. As a result, Eqs. (2), (3), and (4) apply also to the delayed The Fission Products'

1.7

ll

a

is

In 99.8 per cent of all fissions, the products are two heavy nuclei (fragments). In light particle of long range also emitted. the remaining 0.2 per cent of all fissions, Fission -product Distribution*

Atom or

I.

t

atoms

Rare earths Rare (rases Zr Mo Cs Ru Sr Te Ba

Fraction of total number of atoms after a burnup of 10 per cent at a thermal flux of I0U, per cent 24. 16.2 15.0 12.5 4

Table 4.

6.4 5.4 4.4 4.0 3.2 0.8

Br

92.3 7. 7

Subtotal Other element.*

is

*

J.

ToO Total W. L. Robb, J. B. Sampson, .1. H. Stehn, and K. Davidson, Nucleonics, 13 (12): 31 (1955). Nd, 9.9 per cent; Ce, 6.0 per cent; La, Distribution of specific ran? earths at 10 per cent burnup 3.1 per cent; Pr, 3.0 per cent; Pm, l.f per cent; Sm, 0.7 per cent; Eu. 0.5 per cent; Gd, 0.1 percent, See also Table of Sec. for a total of 24.4 per cent. 1 1.

4

t


(3)

Art.

CROSS SECTIONS

2]

These light particles are presumed to be energetic up to 28

2-5

OF FISSIONABLE ATOMS

a particles with energies ranging

Mev.

The main fission products fall into two groups, the heavy group and the light The light group peaks at a mass number near 95, and t he heavy group peaks

group.

near 140. The yield at the peaks is G to 7 per cent and is lower by a factor of several hundred at mass number ~120 (corresponding to symmetrical fission) and at mass numbers 80 and 155. The kinetic energy of the light fragment peaks near 90 Mev. The heavy-fragment energy peaks near 60 Mev. The average total kinetic energy of both fragments is 168 Mev. Table 4 lists the atomic composition of fission products in a reactor that has had The composition 10 per cent of its U"5 content burned up at a thermal flux of 10u. changes only slightly with further burnup. 2

OF FISSIONABLE AND RELATED HEAVY ATOMS

CROSS SECTIONS

2.1

Spontaneous Fission14

Fission is an exothermic reaction. As a result it Table 5 lists rates for some heavy atoms.

Thermal Cross Sections*18

2.2


can and does occur spontaneously.

Table 6 lists thermal-neutron data for fissionable and fertile materials. Data not taken from "Neutron Cross Sections" (BNL-325) are from Ref. 3, Chap. 1. The notation used is defined in Art. 5, Thermal Cross Sections. The numbers in paren numbers defined in Art. 5. The theses preceded by a multiplication sign are the

/

Spontaneous Fission Rates of Some Materials*

Table 5.

Material Th"» Us"

Number

U"<

gm U»"

rjm Np"'

Pu>" Pu>"

Pu»" Pu>«

Pu»«

of fileiont/(g)(tec) 4. 1 X 10' 8. 2 X I0>

» Studier and Huiienga, Phyt. Rev.. »6: 546 (1954).

Table 6. Atom

Thermal Data for Fissionable and Fertile Materials* ff„i,(2.200)," barns

Th»" U(nat) U»»

U"« rjiaa

Pu»»

.7,(2.200),

V

a(2.200)

,((/>) at 20°C

2 47

0.837

1.33

0.977) 524

2.55

0. 132

2. 29

590

2.47

0. 183

2.09

2.91

0.416

2.02

V^barns^

7.0 7.68

(X0.99) 593

(X

4.18

■

(X0.996)

(XI. Oil)

(X

(X0.977)

698

0.974) 2 75 1032

(XI. 073)

0 729

(X

».((*). barns

12.5 8.3

10 8.3

1.056)

for USM, Um, Pw,M are taken from page I of R. A. Charpie, J. Horowitz, D. J. Hughes, Littler (eds.), "Physics and Mathematics," chap. I by J. A. Harvey and J. E. Sanders, McGraw-Hill Book Company, Inc., New York, 1956, and differ somewhat from the BNL-325 values •

and of

The D.

Table

values

J.

23.

* Sec Ref. 3, chap. 1.

2-6

NUCLEAR DATA

[Sec. 2

neutron yield per fission v is independent of neutron energy in the thermal region. The cross-section ratios do, however, change (according to the As a numbers). consequence, the neutron yield per absorption ij is a function of neutron energy. For 2,200-meter/sec (0.0253-ev) neutrons, the ij values may be calculated by dividing » For a Maxwell distribution at temperature ~20°C, the values are by 1 + a(2,200). different and are listed under i)(th). For U, U233, and U236 the value of r)(lh) is insensi tive to neutron temperature change in the thermal region. For Pu23', it is found that Thus for Pu*" ri(lh) decreases by 0.0007 per centigrade-degree rise.

/

dr,(th)

dT 2.3

0.0007/°C

(5)

Cross Sections at Various Energies*

Table 7 presents for U"5 the average value of ofE^ (the product of fission cross By average value section times square root of neutron energy) for several energies. is meant the average over resonances in the neighborhood of the listed energy. For a pure l/v cross section, orE^ would be independent of energy. The table exhibits the departure of the average fission cross section from the l/v behavior and shows the cross-section "window" in the energy range of a few electron volts. Table 7.

Average Fission Cross Section of

U23S as Function Average value of

of Neutron Energy*


Energy E

or energy range barn y/ev 0 97 1. 5-6 cv 21 7. 8-10. 8 ev 240 70 ev 240 100 ev 245 200 ev 275 270 400 ev 275 700 ev 1 kev 230 1. 25 kev 320 1. 75 kev 240 2. 2 kev 340 3 kev 305 5 kev 340 10 kev 370 20 kev 400 470 50 kev 100 kev 570 200 kev 670 * R. A. Charpie, J. Horowitz, D. .!. Hughes, and D J. Littler (eds.), "Physics Book Company. Inc., New York, 1956. chap. 3 by H. A. Bethe, McGraw-Hill

Table 8.

Capture-to-fission

Ratio for U2" and

Pu239

and Mathematics,'

for Various Neutron Spectra*

Capture-to-fission ratio for Spectrum of median fission energy, cv

TJjii

30 100 1.200 15.000

0.52 0.47 0.41

Pu«»

0.65 0.81

0.60 0.45

J.

* R. A. Charpie, J. Horowitz, D. J. Hughes, and D. Littler (eds.), "Physics and Mathematics," II, pp. 378-379. by II. Hurwiu, ,Ir., and R. Ehrlich, McGraw-Hill Book Company, Inc., New York, 1956.

chap.

Table 8 lists values of the capture-to-fission ratio a = ajar for U235 and Pu23* for These values were obtained by measuring the various spectra of incident neutrons. •See Ref. 3, chaps. 3 and 11.

Art.

2-7

CROSS SECTIONS OF FISSIONABLE ATOMS

2]

number of fissions and the number of captures that occurred in a sample placed in the

The incident-neutron spectrum was controlled by using various Hanford reactor. the sample. Measurements of 7; for U"3, U"5, and Pu2" at neutron energies of 30 and 900 kev are presented in Table 9. The neutrons used in these measurements were photoneutrons. shields around

Cross Sections for Fast Neutrons*

2.4

Cross sections averaged over the neutron spectrum at the center of some fast Also given are the calculated cross sections averaged reactors are listed in Table 10. The fast reactors and the neutron spectrum are described over the fission spectrum. in Art. 7. Transport cross sections are given in Table 32, Art. 7. It may be noted from Table 10 that the value of a for U"5 in the central spectrum of EBR I is

0.15 1.32

0.114

Using the value v = 2.47 + 0.1 = 2.57 leads to 17 = v/(l + a) = 2.31, which agrees with the 900-kev value of Table 9. For Pu"9, however, use of r 3.01 2.91 + 0.1 and a = 0.11/1.87 = 0.059 leads to tj = 2.84 as opposed to the 2.52 of Table 9.

-

-

Cross sections averaged over the equilibrium neutron spectrum of natural Table 31 characterizes this spectrum. given in Table 11.

uranium

are


Table 9.

Values of

n

Materials at

for the Fissionable

Value of v for neutrona

30 and 900

Kev*

of energy

Atom

U"'

900 kev

2.25

2.60 2.28 2.52

1.86 2.01

Pu>» * M. S. Kozodacv,

30 kev

Proc. Intern. Con}. 4: 352 (1955).

Fission and Capture Cross Sections

Table 10.

Cross section, Atom and reaction

Fission in:

Tb"» U«" U"» U«" Np»"

EBR I

Godiva

at the Centers of Fast Reactor Cores*

barns,

at center

Zephyr I

of

Zephyr

Cross section for a fission spectrum,

II

barns

0.04

0.05

0.06

0.075

1.32 0. 152

2.36 1.46 0. 18

2. 19 1.36 0.21

1.94 1.28

0.8

l.87t

0.9 1.87t

1.3

0.79f

0.20 0.9

Au"" U"»

0.25f

0. 12

0. 174

0. 146

0. 137

lit

0. 10

0. 133

Pu»"

0. 15 0. I32t 0.

0. 130

0.096

Pu>"

Pu'"

l.87t

l.87t

0.28 1. 18 1.89

Radiative capture in: rjiM

* R. A. Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Phyaica and Mathematics," chap. 9. p. 289. by J. Codd, L. R. Shepherd, and J. II. Tait, McGraw-Hill Book Company, Inc., New York, 1956. t These values are calculated by using known cross sections aa functions of energy and averaging over the measured central spectrum of EBR I. X This cross section was obtained at the core boundary of EBR I. •See Ref. 3, chap. 9.

2-8

NUCLEAR DATA

Table

11.

[Sec. 2

Average Cross Sections in Natural Uranium Equilibrium Average cross section, barns

Atom and reaction

Fission in: U>"

Spectrum*' t

2.8

U"'

1. 8 0.01 0. 16 1. 80 0. 25

U"" Np'» Pu"« Pu»" Radiative capture in: Au1"

0.43 C»" 0.21 * R. A. Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Physics and Mathematics." chap. 9. p. 291. by J. Codd. L. R. Shepherd, and J. H. Tait. McGraw-Hill Book Company. Inc., New York. 1956. t Neutrons in spectral equilibrium in natural uranium. Such a spectrum is found some distance inside a block of uranium at a depth sufficient for the source neutrons to have lived several generation times in the uranium. Table 35 describes such a spectrum.


2.5

Fission "Thresholds""

"

Fission, being an exothermic process, occurs spontaneously and is not a true threshold process. The probability per unit time that a nucleus splits is quite small, as shown by the spontaneous fission rates of Table 5. If the nucleus is given sufficient excitation energy, however, the fission activation energy barrier is overcome and the nucleus splits promptly. To within a few per cent, 5.25 Mev of -y-ray energy is sufficient to cause measurable photon-induced fission in various nuclei ranging from Th to Pu. This is the photofission "threshold." For the thermally fissionable nuclei, the binding energy of an incident neutron is sufficient to overcome the activa tion energy barrier. For the nonthermally fissionable nuclei, the binding energy must be supplemented by kinetic energy of the incident neutron in order to overcome the barrier. As a result, the fission cross section of these nuclei is essentially zero up to some energy and then rises rapidly to a first plateau, at which point the barrier is Table 12 lists the values of the fission cross section at the first essentially overcome. plateau that occurs near 2 Mev. Also listed are the energies E\<, and E\\a at which the cross section is one-half and one-tenth of the plateau value. Table 12.

Fission Cross Sections of Nonthermally Atom

«y(~2 Mev), barnst

Th«»

By,. Mevt

U"«

15 0.8

U"<

0 54

U"<

Nuclei*

Mev",

1. 25 0.46 0. 36

0.8 0 62 1. 1 1.45 0 6

14

Np>"

BMo.

1 4

0. 11 1.0

Pa"'

Fissionable

0.8 1 25 0 37

* Values from the curves of BNL-325. This value is reached at about t This is the value of the cross section at the first plateau. is the energy at which the fission cross section drops to half the plateau value, %

t

B^

to one-tenth

'a the energy corresponding

3

3.1

of the plateau

2

Mev.

value.

MODERATOR DATA

Nuclear Properties of Standard Moderators*-16-18

Table 13 presents some important nuclear properties of the usually considered moderators. The number N of molecules per unit volume is calculated from the formula

♦ See Ref.

1, chap. 1.5.

N-%

(6)

Art.

moderator data

3] Table 13.

2-9

Nuclear Properties of Standard Moderators

Moderator

Nominal density p g/cm'

Number of molecules per unit volume A', 10 "/cm'

Macroscopic absorption cross section AV.i. (2.200), cm-1

Slowingdown power N(t. (epi), cm-1

H*0 DrO M) (0 16% H.O) Be BeO C

1.00 1.10 1.10 1.84 2.96 1.60

0 0334 0 0334 0.0334 0.1229 0.0713 0 0803

0.0220 0 000037 0 000072 0 00123 0 000727 0 000386"

1 38 0.180 0.180 0.156 0 123 0.0595

U

Diffusion coefficient = L»AV„s. (2.200), cm

Thermal- Age from diffusioD fission to length thermal T, U cm cm*

2.85t 170 115 21 27 52

0.179 1 07 0.952 0.542 0.530 1.04

31 125 125 97 105. 155 365

'This value ia based on v«s« (2,200) = 4,8 millibams rather than the Table 19 value of 3.2 millibarns. The higher raiseis mentioned in Ref. 1. chap. 1.5. The actual value of
-

N

ia

number,

p

Avogadro's

is

is

The macroscopic absorption

is

is

is

if

is

r

is

is

is

is

in

a

is

£

is

is

density.

3.2

Metal-Water Mixturest

= — D&ptk where jtk chap. 1.5. 1.

in

t.See Ref.

is

is

The age from fission to thermal in mixtures of water with aluminum and water with zirconium given in Table 14 of Sec. 6-2. *


M

the molecular weight, and the moderator cross section for neutrons of velocity 2,200 the sum of the absorption meters/sec is given by Afo-Oi,.(2,200), where o-0(,,(2,200) cross sections of the atoms composing the molecule, as given in Table 18. The given by N£a,(epi), where {o-.(ept') the sum over the atoms slowing-down power composing the molecule of the products times ' given by Ii'AToVu (2,200) and value to use in Fick's law* absorption cross sections used in the calculations are If, however, the Maxwell average cross sections
the current density

and ytk the flux of thermal

neutrons.

2-10

NUCLEAR DATA

[SEC. 2

Collisions and Time to Slow Down

3.3

According to the continuous-slowing-down theory, the number of collisions required to slow down a neutron from the energy Ea to the energy E is given by

fE"

IdE

Eo

1 ,

which is equal to 18/{ when E„ = 2 Mev and E = 0.0253 ev. Thus, 18/( may be called the number of collisions required to thermalize and is listed in Table 14. Table

Number of Collisions and Time Required to Thermalize

14.

Moderator

Time, 10 ' sec

Number of collisions

HiO

6.6

19 35 87 103 113

D,0 Be BeO

C

31 58 74 153

The time required to slow down from the energy

l


[E>

dE

_

E<>to the

/l _

2

energy

E

is given by

1\

For the energy values chosen above, l/i>0 is negligible where v is the neutron speed. The resulting time may be called the slowing-down time and v ■» 2,200 meters /sec. and is also listed in Table 14. 4

FISSION-PRODUCT CROSS SECTIONS

For absorption of thermal neutrons, groups,

Xe136,

the fission products are divided into three fission products. Sm14', and the remainder, or "low-cross-section,"

Xenon10''

4.1

The first group, Tc>36

Xe'",

is part of the fission-product chain

2 min ->

I us

6.7 hr ►

Xeus

9.2 hr

>

Cs1"

2X10«yeara »

Ba1"

For fission in Um,

The yield the Xe13e atom is formed in 6 per cent of all fissions. for direct formation is 0.3 per cent. In the remainder of the 6 per cent, the Xe"* is formed by decay of I'". The Xe136 yield in Pu33* fission is about the same as in The thermal-neutron (Maxwell average) absorption cross section of Xe13' is TJ13S listed in Table 15 as a function of neutron temperature. The two possible sets of values that are listed are 13 per cent apart. Picking a cross-sectional value of a = 2.3 mcgabarns and taking the yield y = 0.06 result in the product
The atom

Sm1<9 is

Samarium13'"

the stable isotope of the fission-product chain 1.7 hr Nd>«»

>

Pm'»

47 hr »

Sm1"

and is formed in 1.3 per cent of fissions in U135. The thermal-absorption in the neighborhood of room temperature is 6.6 X 10* barns. * See Ref. 3. chap. 5. discussion

t For further

of Xe1,& yield and cross section see Table 26 of Sec. 1-1.

cross section

Art.

fission-product cross sections

4] Table

2-11

Maxwell Average Absorption Cross Section of Xem*

16.

Cross-section

possibility!

Neutron temperature,

°K

A, megabarns

B, megabarns

2 24 2.05 1.85 1.67 1.48 1.34 1.21 1.08

300 400 500 600 700 800 900 1000

2.50 2.30 2.08 1. 88 1.70 1.54 1.38 1.23

• R. A. Charpie. J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Physics and Mathematics," chap. 5. p. 177. Fig. 9. by S. Bernstein and E. C. Smith, MoGraw-Hill Book Company, Inc., 1956. Possibility A refers to the case that the Xe115 reso t Tbe spin of Xe"1 is presumed to be / — — M m »» — Possibility B refers to the other oase, nance is due to + yi m 2.

J

J

Um Fission-product

Table 16.

Atom


Xe'».

-

M -

8m'" Low-crons-section group.

Table

Estimated Fission-product

17.

Neutron

Cross Sections for Thermal Neutrons

Yield y and absorption cross section or«l« V = 6% 2.37 0.0018 (7" » neutron temperature in °K 300 < T < 1,000 1.3% V 6.6 X I0< barns (2.200) 100% V 1-80 barns at 0 % burnup , , \ = 65 barns at 10% burnup for 10" flux

energy cv

E,

Average

10* 10" 10' I0> 10*

Um

-

-

"

Cross Sections as Function of Neutron Energy*

fission cross section ay, barns

Fission-product absorption

cross section per fission, barns

23

15 2.8 0.49 0. 10

8.S 3.7 1.7 1.3

0.04f

* R. A.

chap. 1956.

Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Physics and Mathematics," 11. p. 376. by H. Hurwitz, Jr., and It. Ehrlich, McGraw-Hill Book Company, Inc., New York,

t This

value is about half of that estimated

4.3

on the basis of the

t-Mev cross sections of Table 25.

Low -cross-section Fission Products"•,1•"

The "low-cross-section" fission products can be regarded as if they were a single The cross section of this product fission product formed with a yield of 100 per cent. for U1" has been estimated to be 80 barns at 2,200 meters/sec. The composition of these low-cross-section fission products changes with time in the reactor because of the greater neutron absorption in the higher-cross-section atoms. As a result, the average cross section diminishes with time. At a neutron flux of 10'4, the cross section has dropped to 65 barns after a burnup of 10 per cent of the U2'5. The above results are summarized in Table 16, where the Xe cross section is taken at about the average of the possibilities of Table 15. 4.4

Table

17 presents

mates of resonance

Intermediate and Fast Cross Sections

fission-product cross sections parameter distributions.

at various energies based on esti

NUCLEAR DATA

2-12 6

[Sec.

2

THERMAL CROSS SECTIONS Notation and Values"

5.1

All cross Tables 18 and 19 present slow-neutron data for isotopes and elements. in these tables arc those of "Neutron Cross Sections," BNL-325. Table 16, however, suggests the use of a different value for the Sni1" absorption cross section. Tabic 6 lists different values for thermal data of the fissionable atoms. Also, Table 16 gives a different value for the thermal cross section of Xel". The subscripts describing the cross section a have the following meaning:

sections

A quantity denoted by 0(2,200) means the value of Q for neutrons of speed 2,200 A quantity denoted by meters/sec, corresponding to a kinetic energy of 0.0253 ev. If a Q(th) means the value of Q for thermal neutrons at a particular temperature. thermal value is listed without a corresponding temperature, it may be assumed that the temperature of the material in which the measurement was made was not far from The scattering cross section is listed for both thermal and room temperature. 'epithermal" neutrons. The epithermal cross section
a(th) tr,(epi)

= value for 2,200-meter

/sec neutrons value for thermal neutrons = epithermal or free-atom scattering cross section =

The quantity £ is defined as the average loss in the natural logarithm of the neutron For neutrons slowing down through the reso energy due to a scattering collision. nance region, £ is given by £

- i + Yzrr

r =

where

-

ln r

[j^)IV (M

^ (10,

and M is the atomic weight of the atom. The values of £ in Tabic 18 arc calculated from these equations. An approximate expression accurate to 1 per cent for M > 10 is

*mWTH

(u)

coil

=

T^j

The quantity cos 6 is the average value of the cosine of the angle through which the neutron is scattered. For neutrons slowing down through the resonance region, cos 6 is given by (12)

For thermal neutrons, which the formula used to obtain the values in Table 18. Eq. (12) may also be used except for small M and except for materials in which the effects of binding between the atom and its neighbors are important. is


abs = absorption by any process including fission » = scattering F = fission act = activation c = capture, radiative capture, absorption by the ny process

Art.

THERMAL

5] Table 18.

Element

iH iD •II.-

iLi .Be >B
.0

iF itNe

uNa

uMl 11AI ••Si

I.P

i<8

itCI ,»A

,.K


uCa tiSc

nTl nV ..Cr :.Mn nFe

■,Co

»Ni »Cu isZn i:Ga wGe ■>Aa i«Se

Atomic weight

4.003 6.940 9.02 10.82 12.01 14.008 16.000

26 97

28.06 30.98 32.06 35.457 39.944 39.096 40 08 45 10 47 90 50 95

52.01 54.93 55.85 58.94 58.69 63.57 65.38 69.72 72 60 74.91 78 96

i&

85 48 87 63 88 92

«Zr

uNb uMo oTe •iRu

uRh «Pd

(XI0

»)

91.22 92.91 95.95

0.534 1.84 2.535 1.60

0.0463 0. 1229 0. 1411 0.0803

102.91

0.9031 0.9255 0 9379 0 9440 0.9520

0.9580 0.9646 0.9667

0.9708

1.741 2.699

0.0431 0.0603

0.087 0.0730

0.0455

0.9724 0.9751 0.9760 0.9783

0.0376

0.9790

0.0616 0.0558

2.42 2.34 2.0

0.0519

0.9810

0.9832 0.9828 0.9832

0.87

0.0134

1.54 2 5 4 5

0.0231 0 0334

0.0566

0.9860

5.96 6.92

0.0705 0.0801 0.0789 0.0847 0.0890 0.0913 0 0842

0.9868 0.9871

7 2 7.85 8.71

8.9 8 89 7. 19

5.9 5.46 5 73

4.5 3.12

0.0662 0 0510 0 0453 0 0461 0 0343 0.0235

1 532

0.0108

2.6 3.8 6.44

0.0179

0.0257

0.9851

0.9878 0.9880 0.9886 0.9885 0.9894 0.9897

0.9904 0.9907 0.9910 0.9915 0.9916 0.9920 0.9921 0.9923 0.9924 0.9926

1.88

0.00019 0.009 0.505 0.063

0.230 0. 13 0. 19

0.49 31.6 0.62 1.97

0.43 24 0

5.6 4 98

2.9 37.0 4.6 3.62 1.06 2.77 2.35 4. 1 11.8

6.6

0.70 1.16 1.28 0. 180 1. 1 2 5 100 150

0.9938 0.9940

0.0186 0.0178

62 2,550

0.0175

iiSn i.Sb ■iTe

118 7 121 76 127.61

7.28 7.3

0.0382 0.0370 0.0331 0.0295

0.9941 0.9943 0.9945

0.0234

0.9947

6.691

126.92

6.25 4.94

131.3 132.91 137 36

1.873 3 78

6.15 6 9 6.475

6.96

0.0586

0.0085 0.0166 0.0267 0.0297 0.0277 0 0291

0.9947 0.9949 0.9949 0.9951

0.9952 0.9952 0.9952 0.9953 0.9954

0.0169 0.0165 0.0157 0.0158 0 0153 0 0151 0 0146 0.0145 0 0143

0.0142 0.0139 0.0137

2.46 8.0

(XI.

38 (gas) 7

0.8 1.4 7 4

4.8 10

4.2 3.9

4.0

3)

190

0.60 5.5 4 5 6 7 35

29.0 1. 17

8.9 0 70 11.2 46 60

20.4

3.4 0.83 0.9 6.

II

3.7 4.66 9.9 3.75 3.6 2.6 3. 1

3 6 1.4 1.7 5 1. 1 16 1.5

3.4

3 24 4 5

3.0

15

2 3

1.4

2.2 3.4 1. 1 0 68 2. 1

4.2 4.9 3.9

II

1.9 11.4

7 17.5

17.4

7.2 3.6

7.7 4.0

4 3 6

7.3

II

6

28

0.0197 0 0195 0 0188

0.0732 0.0686

((A), barns

3.0

13.2 2 53

0.0202

c

2 4

<2.8

0 9934 0.9935 0.9937

114.76

•iPm

0.0032

0.9932

uln

144 27 147

0.0225

755

0 0723

0.0463

140.92

0.0239 0.0234 0 0228

71.0 0.010

12.2 12.5 12. 16 10.5

8.65

»Pt »Nd

0.0250

0.00046 0 0070

0.9930

112.41

140. 13

0 0497 0 0507 0 0495 0 0441 0 0415 0 0391 0.0383 0.0363 0 0357 0.0338 0.0340 0.0314 0.0305 0.0287 0.0275 0 0267 0.0253

0.332

0.0640

0.9928

■ ■Cd

138.92

0.0702 0.0637

»„», (2.200), barns

0 0220 0.0216 0 0209

8 4 10.2

0.0425 0.0545

106.7 107.880

i»Cs I'-i »:La isCe

1.0 0.7261 0.4281 0.2643 0.2078 0. 1756 0. 1589 0. 1373 0. 1209 0. 1025 0.0967 0.0852

0.0254

«Ar

ul nXe

£

0.9712

99

101.7

1 -- COS S

0.333 0.666 0.832

20. 183

22.997 24.32

Properties of the Elements*

Atoms per cm3

19.00

79.916 83.7

»Y

Slow-neutron

1.008 2. 0147

«Br . Kr .-Rl,

Nominal density, g/em'

2-13

CROSS SECTIONS

5.8

8.7 7.8 7.8 6.0

7.2 12 10 3 8 5 7 6 5 3 6 6 7

2.2 4 4 3 5

3.6 4.3 20 ± 10 8 15 ± 5 9 ± 6

5 4

9.8 6.2 6.5 6.0 6 S 5 5 4 7

6.4

4.8 4. 1

4.4 3.7 6.9 5.9 9.2 2.8 3.9 16

2-14

NUCLEAR Table 18.

Element

Nominal density. K/cm'

Atomic weight

[Sec. 2

Properties of the Elements.*

Atoms per cm1

(XI0 -")

1 — cos 6

i

150. 43

7.7

0.0308

0.9955

0.0134

5,500

eiEu

152.0

5.24

0.0208

0.9956

0.0132

4,600

«Gd

156.9

7.94

0 0305

0.9957

0.0128

..Tb

159.2 162.46 164.94 167.2 169.4 173.04 174.99 178.6 180.88 183.92 186.31 190.2 193. 1 195.23 197.2 200.61

8.33

0.0315 0.0317

0.9958

0.0126

0.9959 0.9959

0.0124

«iHo I l-'.r saTm

ioYb 7iLu

7.Hf

jiTa

7.W TtRe 7«Oa 77lr 7«Pt 7«Au •oHg

.iTl

t.U

204.39 207.21 209.00 210 211 222 223 226.05 227 232. 12 233. II 238.07

9lNP'" ,.Pu'»

239.13 239. 13

..Pb

uBi

uPo ■lAt boRq »7Fr

..Ra"«

iiAc"' •oTh

•iPa"'

* Cross sections

8.56 8.76 9.16 9.34 7.01

9.74 11.4 16.6 19.3 20 22.48

22.42

0.0320 0.0330

0.9960

0.0332

0.9960 0.9961

0.0244 0.0335 0.0384 0.0553 0 0632 0.0647

0.0712 0.0699

21.37 19.3 13.546

0.0659 0.0589 0.0407

11.86 11.347

9.8

0 0282

0.9962 0.9962 0.9963 0.9963

0.0122 0.0120 0.0119 0.0116 0.0115 0.0113 0.01 11 0.0109 0.0108

0.9964 0.9965

0.0106

0.9965 0. 9966

0.0104 0.0103

0.9966 0.9966

0.0102

0.0349

0.9967

0.0330

0.9968 0.9968 0.9968 0.9968

0. 0098 0.0097

0.9970 0.9970

0.0100

0.0096 0.0096

0.9970 0.9970

11.5

0.0298

0.9971 0.9971

0.0089 0.0089 0.0087 0.0087

0.9972

0.0084

19.6

0.0494

0.9972 0.9972

». («P0, barns

5) 8

(X.95)

46.000

(X.85)

44 1. 100 64 166 128 36 108 105 21.3 19.2 84 14.7 430 8. 1

98.8 380

(X95) 3.3

0. 170

0.032 7.0

100 15 12 8 5 5 14

6

5.6

II

IS

10

12

9.3 20 14

II

9

10 11.3 9. 28

0.0090

0.0133

0.0473

(XI.

barns

0.0095 0.0091

5

18.7

(Continued)

«Sm

6oDy


Slow-neutron

DATA

0.0084 0 0084

20 510

7.45 40

7.68

12.5

12.5

8.3

(X.99)

60 1.025

(XI.

9.6 075)

from BNL-325. 6.2

2,200

Meters/sec and Thermal Cross Sections

To calculate the thermal cross section for absorption or for fission, it is necessary to know the spectrum of the thermal neutrons and the cross-sectional variation with neutron energy through the range of thermal energies. The thermal cross section o(th) is an average value defined such that the product of a(lh) [or a macroscopic cross section based on a(th)\ times the total flux of thermal neutrons results in the correct value for the reaction rate. If the cross section varies inversely with the neutron speed (is proportional to \/v), and if the thermal spectrum is that of a Maxwell distribution at the absolute tem perature T„, then a(th)

=

^
(13)

where a(kT„) is the value of the cross section for neutrons of kinetic energy kT„ and k is the Boltzmann constant, (the universal gas constant R per mole divided by AvoThe kinetic energy of a neutron of speed 2,200 meters/sec is gadro's number).

Art.

thermal cross sections

5]

Table

2-15

2,200 Reaction Cross Sections by Isotope and Element*

19.

Reaction cr »s sections. 2.200 meters/sec Element

Isotope,

%,

TyJ

•H

332 ± 2 millibarns

H"(-~I00) H=(0.OI5)

0. 46 ± 0.10 millibarn

He'(0.000l3)

np 5.400 ± 300 0 71.0 ± 1.0 (m» 945)

I2.4years,

0.57 ± 0.01 millibarn

■He He«(~IOO) •Li

.Ho

Li«(7.52) Li'(92.48) Be'(54d)

iB

Be»(IOO) (18.5 per cent B">)

B'«(I8.8)

B"(8I.2) •C

C'«(98.89)

C'»(l.ll) jN

C"(5570 years)


N»(99.63)

iO

.F

iiNe

N'»(0.37) 0'«(99.79) O"(0.037) O"(0.204)

F"(I00)

np 51.000 ± 6.000 no < 1 10 ± 1 millibarns 755 ± 2 (na 4.010) np <0. 2

3.2 ± 0.2 millibarns 0.5 ± 0.2 millibarn <200 1.88 ± 0.05 np 1.75 ± 0.05 ny 0.08 + 0.02 <0. 2 millibarn na 0.25

± 0.15

< 10 millibarns

<2.8

0 23 ± 8 millibarns 0.85 sec, 33 ± 5 millibarns

2.7 X 10« years, 9 ± 3 millibarns 0.45

± 0.20

0.03 sec, <50 millibarns

3.3 ± 0.2 millibarns 5,570 years, 0.9 ± 0.3 millibarn 2. 4 sec,
7 . 4 sec, 24 ± 8 microbarns 5.570 years C», 0.5 ± 0. 1 29 sec, 0.21 ± 0.04 millibarn 11 sec, 9 ± 2 millibarns

Ne*>(90.92)

11N1

Ne»(0.26) Ne"(8.82) Na»( 100)

Mg>'(78.60) M«»(I0. 1 1) Uk'H 11. 29) i>Al

A1"(I00)

i.P

Si»(92.27) Si»(4.68) Si"(3.05) P»'(IO0)

i«Si

■*S

S"(95.0I8) cairn

75m

S'«(4.2I$) S"{0.0I7) 11CI

Cl»(75.4)

I.A

,.K

Cl»(3 .08 X I0> years) Cl"(24.6)

505 ± 10 millibarns 63 ± 4 millibarns 33 ± 10 millibarns 270 ± 90 milliharns 60 ± 60 millibarns 230 ± 5 millibarns 0. 13 ± 0.03 80 ± 30 millibarns 0.27 ± 0.09

0.4 ±0.4

0. 19 ± 0.03 0.49 ± 0.02 na 1. 8 ± 1.0 millibarns np 125 ± 100 millibarns no < 8 millibarns

31.6 ± 1.0 np 0.30 ± 0. 10 tut < 0.05 millibarns

0.62 ± 0.04

A»(0.37) A«(0.063) A "(99 . 600)

A"(

109 min)

K"(93.08) K»(0.012)

K"(6.9I)

1.97 ± 0.06 1.87 ± 0. 15 70 ± 20 np
40 sec, 36 ± 15 millibarns 15.0 hr, 0.53 ± 0.02

9.5 min. 26 + 2 millibarns 2.30 min, 0.21

± 0.02

2.62 hr, 110 ± 10 millibarns 14.3 days, 0. 191 ± 0.010

25. 1 days

P",

2.3 ± 1.0 millibarns

87 days, 0.26 ± 0.05 5.0 min, 0. 14 ± 0.04 3 .08 X 10' years. 30 ± 20 87 days S", 0. 17 ± 0.04 90 ± 30 37.5 min, 0.56 ± 0. 12 35 days. 6 ± 2 265 years, 0.8 ± 0.2 109 min, 0. 53 ± 0.02 >3.5 years, >60 millibarns 1.3 X 10" years.

3 ± 2{

12.5 hr. 1.15 ± 0.11

2-16

NUCLEAR DATA

Table 19. Element

2,200 Reaction Cross Sections by Isotope and Element.* Isotope,

%,

Ca«(96.97) Ca«(0.64) Ca«(0. 145) Ca«(2.06) Ca»(0.0033)

Ca"(0.

«Ti

185)

So«(IOO)

Ti«'(7.95)

Ti"(7.75) Ti«>(73.45)

Ti«(5.5l) ..V

Ti«(5.34) V»°(0.24)

nCr

iiMd


i.Fe

\"(99.76)

Cr'»(4.3l) Cr»(83.76) C'r"(9.55) Cr"(2.38) Mn»(100) Fe»(5.84)

Fe"(9l.68) Fe»(2. 17)

Fe"(0.3l) 17C0

Reaction

TyJ

loCa

«8o

Co»(IOO)

min) Co«»(5. 28 years)

Ni"(67. 76) Ni«°(26. 16) Ni"(1 . 25)

Ni"(3.66)

Ni"(l.

iiCu

24.0

5.6 ± 0.4 0.6 ± 0.2 1.6 ± 0.3 8.0 ± 0.6 1.8 ± 0.5 <0.2 4.98 ± 0.02 250 ± 200

2.9 ± 0.2

16.3 + 1.3 17.5 ± 1.4

<0.3

13.2 ±0.4 2.53 ± 0.06 na <5 millibarns 2.2 ± 0.2 2.6 ± 0.2 2.4 ± 0.2 2.5 ± 2.0 no < 1. 5 millibarne 37.0 ± 1.5

4.6 ±0.2 4.2 ± 0.3 2.5 ± 0.2 1.9 ± 1.0 15 ± 2

3.62

no 15 ± 10 microbarns

Zn"(27.8l)

na < 20 microbarns na 6 ± 4 microbarns na < 20 microbarns

Zn'°(0.62) iiGa nGe

± 0.04

Zn«<(48.89)

Zn«»(18. 56)

5.8 min, 0. 14 ± 0.03 3. 76 min, 4.5 ± 0.9 27. 8 days,

0.73 ± 0.06

4.3 ± 0.3 2. 11 ± 0.17 1.06 ± 0.05

Zn"(4. 1 1)

152 days, 0.63 ± 0. 12 4. 8 days, 0.25 ± 0. 10 8.5 min, ±0.1 20 sec, 10 ± 4 85 days, 12 ± 6 20 sec + 85 days, 22 ± 2 (20 sec-. 85 days)

I.I

± 1.0

Cu"(69. 1) Cu«(30.9)

■oZn

(Continued)

2.200 meters see

40 ± 3

16)

Ni"(2.57hr)

cr<)»ssections,

0.43 ± 0.02 0.22 ± 0.04

Co»"»(l0.4

iiNi

[SEC. 2

2.77

± 0. 12

Ga«(60.2) Ga»(39.8)

2.0 ± 0.2 4.9 ± 0.4

Gp'»(20.55) Gp"(27.37) Ge"(7.67) r.o»(36.74) Cic"(7.67)

3.3 + 0 3 0.94 ± 0.09

2.35

13.5

± 1.4

3.6 min. 0.37 ± 0.04 2.58hr. 13.4 ±0.3 2.96 years, 2.2 ± 0.5 46 days, 0.9 ± 0.2 10.4 min, 16 ± 3 5. 28 years, 20 ± 3 10.4 min + 5. 28 years, 36.0 ± 1.5 (99.7% of 10.4 min -> 5. 28 years) 1.75 hr, 100 ± 50 1. 75 hr, 6 ± 2

2.56 hr. 1.6 ± 0.2 56 hr, 6 ± 3 12.8 hr, 3.9 ± 0.8 5. 14 min, 1.8 ± 0.4 250 days, 0.5 ± 0.1 12. 8 hr Cu«< < 10 microbarns

13.8 hr. 97 ± 10 millibarns 52 min, 1.0 ± 0.2 2.2 min, 85 ± 20 millibarns 20. 2 min, 1.4 ± 0.3 14.2 hr, 5.0 ± 1.0

± 0 20

13.7 ±

I.I

0.60 ± 0.06 0.35 ± 0.07

11.3 days,

3 ± 1

82 min, 0 .45 ± 0.08 57 sec, 30 ± 20 millibarns 12 hr, 0.2 ± 0. 1 <~50'; of 57 sec -« 12 hr)

Art.


5]

Table

19.

2,200 Reaction Cross Sections by Isotope and Element.* Reaction cross sections,

Element

2-17

Isotope.

%.

2.200 meters/sec

TyJ »„«•!

OmhtX 11A1 i«Se

Se»(49.82)

4.1 ± 0.2 11.8 ± 0.4 48 ± 7 82 ± 7 40 ± 4 0.4 ± 0.4 0.59 ± 0.06

Se«(9. 19)

2.0 ± 1.4

Ai"(IOO) Se'<(0.87)

Se»(9.02) Se"(7.58) Se>«(23.52)

6.6 • 0.5

uBr

: Kr

Br»(50.52)

10.4 ± 1.0

Br"(49.48)

2.6 + 0.4

28+5

Kr'«(0.35) Kr'°l2.27)

Kr"(56.90)

95 ± 15 45 ± 15 205 ± 10 <2

Kr«(9.4years)

Kr»»(l7.37)

<2

KrS'OI.56)


Kr"( 11. 55)

nRb

Kr"(77

min)

Rb"(72.

15)

0.70 ± 0.07

Rb';(27.85)

Rb"(l7.8

ruin)

i»Sr

»Y «Zr

Sr"(0.56) Sr"(9.86) Sr»'(7.02) Sr"(82. 56) Sr«(53 days) Sr»°(l9.9 years)

«Mo

180+4 millibarns 0.10 + 0.07

1. 23)

± 0.08

0.25 + 0.12 0.08 + 0.06

Zr"(2.80)

0.1

Nb»(2.2 X

I.I

Mo">°(9.62) Tc"(2. 1 X I0> years)

Ru»(l2.8) Ru'«(l2.7) Ru'"'(l7.0) Rui«(31.3) Ru'"(l8.3)

± 0.1 ± 0.1

I0< years)

Mo"(l5.86) Mo"(9. 12) Mo»( 15 . 70)

Ru"«(5.7) Ru»H2.2)

123 days, 26 ± 6 18 sec, 7 ± 3

min, 30 ± 10 millibarns min, 0.5 ± 0.1 sec, 50 25 millibarns min, 4 ± 2 millibarns (Order of isomers unknown) 57 17 67 25

i

4.6 hr, 2.9 ± 0.5 18 min, 8.5 ± 1.4 35.9 hr, 3.5 ± 0.5 34.5 hr, 2.0 ± 0.5

4.4 hr. 0.10 ± 0.03 9.4 years, 60 ± 20 millibarns (23^ of 4. 4 hr -» 9.4 years) 77 min, 60 ± 20 millibarns 2.8 hr, <600 19.5 days, 0.72 ± 0.15 17.8 min, 0. 12 ± 0.03 15.4 min, <200 1.0 ± 0.3 1.3 ± 0.4

53 days, 5 + 1 millibarns 19.9 years, < 90 9.7 hr, 1.0 ± 0.6 64 hr, 1.26 ± 0.08 6 1 days, < 7

1.52 ± 0. 12

Zr»(l7. II) Nb"(IOO)

27 hr. 4.2 ± 0.8

65 days, 2.80 hr,

Zr"(5l.46)

Mo"(16.50) Mo»'(9.45) Mo»»(23.75)

uTc uRu

± 0.06

1.28

Zr"(17.40)

uNb

1.16 <3

V"(I00)

Y'»(63 hr)

Zr"(l

(Continued)

2.5 ± 0.2 <0.3 13.4 ± 1.3 1.2 ± 0.6 2.1 ± 0.7 0.4 ± 0.4 0.5 ± 0.5 100 ± 25 2.46 ± 0.12

63 days. 0.09 ± 0.03 17.0 hr, 0. 10 ± 0.05 6.6 min, 1.0 ± 0.5 36 days. 15 ± 4

6.9 hr, <6 millibarns

67 hr, 0.45 + 0.10 14.3 min, 0.20 ± 0.05

2. 8 days,

10+4 millibarns

41 days, 1.2 ± 0.3 4.5 hr. 0 7 ± 0.2

2-18

NUCLEAR DATA

Table 19.

2,200 Reaction Cross Sections by Isotope and Element.* Reaction

Element

..Rh

Isotope,

%,

Rh'"(100)

8.0 ± 1.5 Pd'«(0.8) Pd"»(9.3) Pd'»!(22.6) Pd'»«(27.

2.200 meters/sec

4.5 min. 12+2 44 sec, 140 + 30 17.0 days, 4.8 ± 1.5

13.6 hr, 12 ± 3 22 min. 0.3 ± 0.1 62 ± 2

30+2

2.3 min, 44 ± 9 270 days, 2.8 ± 0 5 24. 2 sec. 110 ± 20

84 ± 7 2.550 ± 100

Cd'»«(l . 22) Cd"»;0 87) Cd»°(l2.39)

(not

X 1.3)

Cd>"(24.07) Cd"MI2.26) Cd"<(28.86)

190 ± 10 In>»(4. 23) In»>(95.77)

0.60 ± 0. 10 Sn"'(0.95) Snm(0.65) Sn»M0.34)

Sn'»(4.7l) Sn"'<(5.98)

5.5 ± 1.0 5.7 ± 0.5 3.9 ± 0.3

4.5 ± 0.2

itTe

Te'"(6.99) Tc>"(!8. 71)

70 ± 70 2.7 ± 0.9 390 ± 30 6.5 ± 1.2 1.50 ± 0. 15 0.8 ± 0.2

Te'»(3l.79)

0.3 ± 0.3

Te'»«(34.49)

0.5 ± 0.3

Tr>»(0.0S9) Te»»(2.46)

Te'"(0. 87) Tc"«(4.6l)

49 days, 56 ± 12 72 sec, 2. 0 + 0.6 54. 1 min, 145 ± 15 13 sec. 52 ± 6 1.3 ± 0.3

112 days.

250 days,

Sn"«(24.0l) Sn"«(8. 58) Sn»»(32.97)

Sb'"(57.25) Sb'»(42.75)

43 days, 0. 14 + 0.03 53 hr, 1.1 ±0.3 2.9 hr, 1.5 ± 0.3

6 ± 2 millibarns

14.5 days,

Sn»«(14.24) Sn>>'(7.57)

D1Sb

30 ± 15 millibarns

5. 1 years, (20.800) (not 1/p, X 1.3)

Cd"»(7.58)

loSn

6.7 hr, 1.0 ± 0.5 49 min, 0. 2 ± 0.1

Cd'"(l2.75)


(Continued)

1)

Pd'°»(26.7) Pd»°(13.5) Ak'»'(5I.35) Ag'»(48.65)

iiId

rr jss sections.

r^t 150 ± 7

uPd

4lAg

[SEC. 2

10 ± 6 millibarns

> 400 days, 1 ± 1 millibarn 27.5 hr. 0. 14 ± 0.03 130 days. 1.0 ± 0. 5 millibarns 40 min, 0 .16 ± 0. 04 (Order of isomers unknown) 10 min, 0. 2 ± 0. 1 10 days, 4+2 millibarns (None of 10 min -» 10 days) 2. 8 days, 6.8 ± 1.5 21 min, 30 ± 15 millibarns 1.3 min, 30 ± 15 millibarnfl 60 days, 2.5 ± 0.5 {% of 21 min and 1.3 min -» 60 daya

unknown) 110 days, 58 days,

T.I ± 0.5 5 ± 3

110 days, 90 ± 20 millibarns 9.3 hr. 0.8 ± 0.2 33 days, 15 ± 5 millibarns 72 min, 0. 13 ± 0.03 30 hr, <8 millibarns 25 min, 0. 22 ± 0.05

Art.


5]

Table 19.

2,200 Reaction Cross Sections by Isotope and

2-19 Element*

Reaction cross sections. Element

Isotope,

%.

2.200 meters/sec

Titt r—1

Safel

ul *iXe

6.7 - 0.6

I'i'ilOO) I»»(l.7 X

10' years) 1 days)

I'"(8.

(Continued)

25.0 min, 5.5 ± 0.5 12.6 hr.

II

± 4

2.3 hr. 200 ± 150}

35 ± 5

Xe»<(0.096) Xe'"(0.090>

Xe'"(

1. 92) Xe>»(26.44)

Xc'»(4.08) Xe»"(2l. 18) Xe»>(26.89) Xe»'(10.44)

Xe'»r9.l3 »Cs

hr)

Xe>»(8.87) r»i»(IOO)

<5 45 ± 15 <5 120 i 15 <5 <5 2.72 ± 0.11 X 10" (not l/», X 1. 16) <5 29.0 ± 1.5

5. 3 days, 0.2 ± 0.1 9. 13 hr, 0.2 ± 0. 1

3.9 min, 0. 15 ± 0.06 3.2 hr, 17 ± 4 niillibarnN 2. 3 years,

26 ± 5 of 3.2 hr— 2.3 yea r») 13.7 days. 15 ± 8 33 min. <2

(-100',


uBi

(b'«(2.6 X 10" years) Cs'"(33 years) Ha""(0. 101) Ha'"(0.097) Ba'»(2.42) Ba"»(6.59)

Ba'"(7.8l) Ba»'(ll.32) iiLa

Ba"»(7l.66) Ba'»(S5 min) La">(0 089)

1.17 ± 0.10

12.0 days, 10. 1 ± 1.0 7 . 2 years. 7 ± 2 2 ± 2 5.6 ± 0.9 0.4 ± 0.4 4.9 ± 0.4 0.68 ± 0. 10

8.9 ± 0.3

I.ai"(99.9ll) nCe

*

I.a'«»(40

hr)

(>'"(0.

19)

25 ± 25

O"<(0.26)

9 ± 6

Ce>«(88.48)

0.63 - 0.06 1.0 ± 0.2' 11.2 ■_0.6

II.

f. •«;>«( 07) Pr'"< UM) Pr'«»(l9.2 hr)

■«Nd

40 hr, 8.4 + 1.7 3.7hr, 3.1 ± 1.0

34.5 hr. 0.6 ± 0.2 8.7 hr, 6.3 ± 1.5 55 sec, 7 ± 5 millibarns 140 days. 0.6 ± 0.3 32 days, 0.31 ± 0. 10 32 hr. 0.95 ± 0.05 19.2 hr. 10+3 13 7 days. 18 ± 3

46 ± 2

Nd'««(27.l3)

Nd""(l2.20) Nd'"(23.87) Nd'«s(8.30)

Nd'«(l7. ii Pm

0.70 ± 0.08

85 min. 0.5 ± 0. 1 12.8 days. 4 ± 1

18)

Nd>"(5.72) Nd>"(5.60) Pni1,,(2. 5 years)

itSm

Sm,M(3. 16) .Sin'«(l5.07) .Sm'«"(ll.27) ^m>«(l2.84) Sml»(7.47) Sm'»'(73 years) .Sm'"(26.63) Sni""(22 53) ■iEu

Eu">(47.77)

Eu'"(l3

years)

Eu'"C52.23)

18+2

280

■ 20

4.5 ± 0.5 52 ± 4

9.2 ± 0.8 3.2 ± 1.0 2.8 ± 1.5

11.3 days,

+ 0.6

1.8

1.8 hr, 3.7 ± 1.2 3.3 days.

60 ± 20

400 days.

<2

5.500 ± 200 (not l/i', X 1.5)

66.000

± 3.000}

10.000 ± 2.000}

4.600 ± 400 (not l/i>. X 0.95) 9.000 ± 3.000} 5.500 ± 1.500} 420 -•- 100}

47 hr, 140 ± 40 24 min, 5.5 ±

I.I

9.2 hr, 1.400 ± 3001

2-20

[Sec. 2

NUCLEAR DATA

Table 19.

2,200 Reaction Cross Sections by Isotope and Element.* Reaction cr
Element

2.200 meters/sec

%,

Isotope,

ffdb.t

Eu'"(l6 ,.Gd

years)

Eu'"(l . 7 years)

1.500 ± 400$ 14.000 ± 4.000 J 46.000 ± 2.000 (not I/d, X 0.85)

Gd>"(0.20)

Gd"'(2. 15) Gd»'( 14.73)

uTb s«Dy

(Continued)

Gd'"(20.47) Gd"7(l5.68) Gd"«(24.87) Gd'«°(2l.90) Tb'»(100) Dy>"(0.052) Dy'«(0.090)

»«.«!

230 days,

70.000 ± 20.0008 160.000 ± 60.000 1

18.0 hr. 4 ± 2

3.6 min. 0.8 ± 0.3 44 ± 4 1.100 ± 150

73 .lays,

>22

Dy>*>(2.298)

Dy"»(l8.88) Dy'«(25.53) Dy'«(24.97)

Dy'"(28.

18)


Dyi«'(l39 BTHO

„Er

min)

Ho'"

Er'"(0. 136) Er'"(l.56) Er'"(33.4)

64 ± 3 166 ± 16

Er""(22.9) Er"»(27. 1)

s»Tni >.Yb

Er"°(l4.9) Tm'"(l00)

1.3 min. 510 ± 20 139 min. 2.100 ± 300§ (1.3 min — 139 min) 82 hr. 5.000 ± 2.000& 27.3 hr, 60 ± 12

9.4 days. 2.0 ± 0.4 2.5 sec + 7.5 hr, 9 ± 2 128 ± 4 36 ± 4

Yb'«"(0. 140) Yb"°(3.03)

129 days,

32 days.

130 ± 30 1 1.000 ± 3.000§

Yb"'(l4.3l)

Yb'»(2l.82) Yb'"(16. 13) iiLu

Yb"<(31.84) Yb»«(12.73)

101 hr, 60 ± 40 l.8hr, 5.5 + 1.0 108 ± 5

Lu"»(97.40)

nHf

Lu'"(2.60) Hf"'(0. Hf"«<5.

18) 15)

Hf'"(18.39) Hf>»(27.08)

Hf>»(l3.78) 7,Ta

t.W

Hf'»»(35.44)

Ta">(l00)

Ta'«(lll

6. 8 days, 105 ± 5 1.500 ± 1.000 15 ± 15 380 ± 30 75 ± 10 65 ± 15 13 ± 5 21.3 ± 1.0

days)

Wi»o(0. 14) W>»»(26.4)

W>"(I4.4) W"<(30.6) W"»(28.4) Wi»'(24 hr) Re>"(37.07) Rc>"'(62.93)

3.7 hr, 35 ± 15

19.2 60 ± 19 ± 11 +

± 1.0 60 2 1 2.0 ± 0.3 34 ± 3 84 ± 4 100 ± 8 63 + 5

4,000 ± 800

46 days, 10 ± 3 16.4 min, 30 ± 10 millibarna days. 19 ± 7 (~95<-; of 16.4 min -> days) 5.5 days, 30.000 ± 15.000

III

140 days,

III

10 ± 10

73 davs, 2. 1 ± 0.6 24 hr, 34 ± 7 65 days. 90 ± 40 92 hr, 100 ± 20 18 hr, 75 ± 15

Art. Table


5] 19.

2-21

2,200 Reaction Cross Sections by Isotope and Element.* Reaction Isotope,

%,

cross sections,

(Continued)

2.200 mcters/seo

T^t

14.7 ± 0.7

..Ob

97 days,

Os""(O.OI8) 0»'«( 1. 59) Os'«( 1. 64)

<200

Osi"(l3.3)

Os>»( 16. 1) Os"">(26.4)

..Ir

raPt

16.0 days, 8 ± 3 31 hr, 1.6 ± 0.4 700 days, 60 ± 20

Os'"(4I.O) Os'"(3lhr)

430 ± 20 1.4 min, 74 days, (1.4 min 19.0 hr,

Ir">(38.5) Ir»>(6I.S)

8. 1 ± 0.4 I50± 150 8 ± 8 1.2 ± 0.9 27 ± 2 0.7 ± 0.7

Pt'»(0.012)

Pti"(0.78) Pt>"(32.8) Pt>"(33.7)


Pt>«(25.4)

r,Au

98.8

■

•

■Hi

Hg""(0. 146) Hg'»'(l0.02)

Hg'»(l6.84)

«Fb

oBi

18 hr, 80 ± 1.0 31 min, 3.9 + 0.8 2. 7 days, 96 ± 10 3. 15 days, 26.000 ± 1,200

± 0.3

380 ± 20 (not l/p, X 0.95) 3.100 ± 1.0001 2.500 + 800S

Hg»»(23. 13) Hg*»(l3.22) Hg»»(29.80)

uTl

4. 3 days, 90 ± 40

4.0±0.5

Pt»»(7.2) Au'»'(l00) Au'"(2.7days)

<60l <60J <60J <60j 3.3 ± 0.5 11.0 ± 0.9 0.77 + 0.08 170+10 millibarns 0.8 ± 0.6 25 ± 5 millibarns 0.70 ± 0.03 <30 millibarns

Hg*><(6.85)

TT«(29.50) Tl»»(70.50)

Pb»'(l48)

Pb«"(23.6) Pb»'(22.6) Pb«"(52. 3) Bi»»(I00)

32 ± 2 mUlibarns

Reaction

Ele

ment

«Rn

tiAc BTh

Isotope,

%,

260 ± 100 700 ± 200 —» 74 days) 130 + 30

47 days. 3.8 ± 0.8 5.5 min, 0.43 ± 0. 10

2.7 years. 8 ± 3 4.2 min, 0. 10 ± 0.03

3.2 hr, 0.6 + 0.2 millibarn 5.0 days, 19 ± 2 millibarns

cross sections,

2.200 meters/sec

Tw of

rati 30 Bee, <0.2J 11.2 days, 0.72 ± 0.07| 3.64 dBys, 130 ± 20} I4.8days, 12.0 ± 0.5| 41.2 min, 20 ± 3J

Rn«»(54 sec)

Rn'"(3.83

days) Ra">( 11.2 days) Ra»«(3.64 days) Ra«»( 1.620 years)

Ra""(6.7

years) Ac«'(22 years) Th'"* IB. 6 days) Th"«( 1. 90 years) Th"«(7.3 X I0» years) Th»°(8.0 X 10* years)

Th«»(l00) (1.39 X 10" years)

510 ± 40

<10 min, 36 ± 5J 6. 13 hr, 520 ± 50

7.3 X I0» years, 123 ± I5| 27 ± 2

7.45± 0.15

25.6 hr, 35 ± 10} 23.3 min, 7.34 ± 0.15

<100

<0. 1 milli barn <2

<2

1.500 ± 1.000

<0.3

45 ± II < 1 millibarn <0. 2 milli barn

2-22

NUCLEAR DATA

Table 19.

[SEC. 2

2,200 Reaction Cross Sections by Isotope and Element.* Reaction Isotope,

2,200 meters/sec

cross sections,

%, T,

Th">(23.5 min) Th»«(24. I days)

24. I days, 1.400 ± 200} <10 min, 1.8 ± 0.55

Pa»">(17.3 days) Pa"'(3.4 X I0« years)

1.32 days,

(27. 4 days)

Pa'»(UXi) (1.18

(UZ) (6.7 hr)

200 ± 155

27. 4 days. 760 ± 1005 1.18 min, 40 ± 6 6.7 hr, 26 ± 4

Pa»"( 1. 32 days)

Pa'"

min) 7.68 (not

± 0.07 X 0.95)

U"»(20.8days)

U"'(4.3

days)


U"«(73 years) U'"(l .62 X I0l years) U"<(0.0057) (2.52 X I0> years) U»»'(0.714) (7.1 X 10>years) U'"(2.40 X 10' years)

U'"(99.3) (4.50 X I0» years) U'"(23. 5 min) Np'"(4. 4 days)

Np«"(22 hr) Np«"(2.2 X 10' years)

588 ± 7 92 ± 7

I. 62 X 10' years. 300 ± 2005 2.52 X 10' years, 52 ± 2 7. 1 X 10' years, 72 + 105

694 ± I0||

2.40 X 10' years,

6 ± 2 2.73 ± 0.03

6.7 days. 9 ± 25 23.5 min,

2.76

107 ± 5J

± 0.09

17 hr. 22 ± 55

170 ± 20

2. 10 days,

169 ± 6

Np"«(2. 10 days)

Np'»(2.3

days)

Pu"'(89.6 years) Pu'"(2.44 X I0< years)

1,025 + I3*»

Pu"'(l3.2 years) Pu'«(3.8 X 10' years)

64 ± 15j

Pu"»(6.6

X 10" years)

Pu«"(4.98 hr) Pu"<(7.6 X 10' years) Pu"«(l0 hr) Am'"(470 years) Am««m(15.8

Cm'"(l9.2years)

Cm'"(>500 years) Cm'"(2.000 years) Bk»'(290 days) Cf"»(470 years)

CP"

years)

Cf>'(2. 2 years)

CP"

E>" (20 days)

E'"

(See footnotes

620 ± 30

7.3 min, 35 ± 105 60 min, 25 ± 155 2.44 X 10' years, 403 ± 105 6.6 X 10' years, 315 ± 16»* 13. 2 years, 250 ± 40 ~5 X 10' years, 380 ± 505 4.98 hr, 9 ± 2 7 .6 X 10' years, 170 ± 90} II hr, 1.9 ± 0.35 I I. 2 days, 260 ± 1505 15.8 hr, 700 ± 2005 500 years, <50j (20% of 15.8 hr — 500 years)

hr)

Am'"(500 years) Am"'(8.0 X 10' years) Cm'"(162. 5 days) Cm»"(35. 1 years)

Cf"°(l0

(Continued)

on page 2-23.)

8.000 ± 1.0005

500 ± 300 |

900 ± 400 {

26 min, 74 ± 4 20 ± 105 19.2 years, 490 ± 70 >500 years, 15 ± 105 2,000 years, 200 ± 1005 15 ± 105 500 ± 2005 270 ± 1005 1.500 ± 1.0005 2.2 years, 3.000 ± 2.0005 18 days, 30 ± 105 <25 38 hr, 300 ± 150}

15 ± 25

<0.65

582 ± 10]

<0.5 millibarn 15 ± 35 900 ± 300 2.800 ± 800 19 ± 3 mUlibarns 1.600 ± 100 <3 18 ± 2 738 ± <4 950 ± 50

9"

3.2 ± 0.2 2.500 ± 1.000 4,600 ± 300 <50 millibarns <55

1.800 ± 300

600 ± 400}

Art.


5]

0.0253 ev and is the value of kTn for as a{th)

T.

= 293.6°K.

2-23

Thus, Eq. (13) may be rewritten

=^5(^)^.(2,200)

(14)

where Tn is the neutron temperature on the Kelvin scale. All listed absorption and fission cross sections except those specially marked follow the 1/v law in the thermal region. The cross sections that do not follow the l/v law closely are so marked by placing underneath them the symbol (X/), where the num ber following the multiplication sign is the "/ number." For these cross sections, the number is denned such that the thermal value in a Maxwell distribution at Kelvin temperature Tn is given by

/

/

/

for not too high temperatures. For high temperatures, sectional curves of BNL-325 should be consulted.

In

used

computing ratios of absorption or fission cross sections, if the cross sections are l/v. In general °w(th)


*™(lh)

=

/">o-<»(2,200) (2,200)

the

/

curves and cross-

the 2,200 values may be

U '

where superscripts (1) and (2) refer to the different cross sections and where it is understood that = 1 for a l/v cross section. In problems concerning diffusion of thermal neutrons, the correct cross sections to use are the thermal ones rather than the 2,200 values. Use of thermal values is con sistent with the use of thermal scattering cross sections and measured diffusion In problems where measured values of diffusion lengths are used, the 2,200 lengths. absorption cross sections may be used with only slight errors arising if they are multi value and if the diffusion coefficient D in Fick's law is chosen to be plied by their consistent with the absorption cross section jaau (2,200) and with the measured value of the diffusion length. The resulting values of D and of quantities (such as transport cross section) derived from D are not then consistent with scattering data and should Also, if the diffusion coefficient D is not be used together with scattering data. obtained directly from scattering data, then the use of 2,200 cross sections is incorrect and thermal cross sections should be used.

/

/

Footnotes to Table 19. • From BNL-325 and suppl. I. t Stable iaotopea are listed with their per cent (by atom) abundances in parentheses, whereas unstable isotopes are listed with their half-lives in parentheses. X Except where marked np (for a proton release reaction) or na (for an a release reaction), all absorp For the higher atomic tion cross sections up to uBi are due to radiative capture (n-y) reactions. numbers, all absorption cross sections are due to ny or fission reactions or both. •| The activation cross sections are the cross sections for the process that leads to the listed activity. Thus, for example, 0.S7 millibarn is the cross section of H' for formation of a radioactive atom of halfAll activation cross sections are due to ny reactions, except where the radioactive life 12.4 years. product is listed. In some atoms, more than one activity is induced by neutron absorption. Thus for CI", a 3.08 X l0*-year activity is caused by an ny reaction with a cross section of 30 ± 20 barns, and In an 87-day activity (due to S") is caused by an np reaction with a cross section of 0.17 ± 0.04 barn. The 10.4-min the case of Co" two activities are induced by the ny reaction, both being due to CoM. activity is that of an excited state of Co*0 and is formed with a cross section of 16 ± 3 barns. The 5.28-year activity is that of the ground state of CoM and is formed with a cross section of 20 ± 3 barns. Finally, The cross section for formation of either one of these activities is given as 36.0 ± 1.5 barns. it is stated that 99.7 per cent of the decays of the 10.4-min activity results in the formation of the 5.28-year activity. Other cases of isomeric activities (activities due to different states of the same nucleus) are treated in the same way. When the order of decay of the isomeric activities is unknown, it is so stated. f All cross sections so marked are values for thermal neutrons rather than 2,200 values. J Not l/v, X 0.981 ± 0.005. Not l/», X 1.075 ± 0.010.

»

2-24

NUCLEAR DATA

[SEC. 2

When a reaction rate is computed as the product of cross section times flux, same (thermal or 2,200) cross section should be used as was used in obtaining the value. If the flux value is calculated by dividing a measured reaction rate by 2,200 cross section, then this same 2,200 cross section should be used to obtain

the flux the the

reaction rate. Neutron Temperature"-"

6.3

Because of neutron loss by absorption or leakage, the neutrons are not in a Maxwellian distribution at the temperature T of the material. Because of stronger absorption at lower energies, the spectrum is shifted toward higher energies and the effective temperature T„ of the neutrons is higher than the material temperature. A . . relationship that may be used is

T

= 1 + o.9Af

(17)

am

6

INTEGRAL DATA

RESONANCE

Homogeneous Mixture*'"

6.1

For a homogeneous mixture of absorbing material and a weakly absorbing diluent material, the Wigner formula states that the probability p(i?o, Ec) of a neutron escaping capture by the absorbing material while slowing down from the energy E0 to the energy Ec is given by In p(£„, Ec) = .

.

Wnete

a„

=

an =

Nd

~

£ = £d

=

For

_ ~

-

A

1

(18)

^

fE°

h.

+
+

rj dEE

absorption cross section per atom or molecule of absorbing material resonant scattering cross section per atom or molecule of absorbing material nonresonant scattering cross section per atom or molecule of the absorbing material [ °° , the absorption integral A becomes A~ =

fB»

)b.

, =

dE a'h'-E

(20)

See Ref. 3, ohop. 6.

is

5

0 .

a

is

Values of A„ are listed in Table 20 for the usual experimental arrangement in which surrounded by thin layer of cadmium in order to eliminate capture in the sample by very slow and by thermal neutrons. For such cases, the value of Ec, the cadmium cutoff energy, depends on the thickness of cadmium used and on the ev. The energy E0 energy variation of
*


where M is the atomic weight of the important moderating atom (H in HjO, D in D2O), a,(th) is the thermal scattering cross section of the important moderating atom, and 25, or for sufficiently absorbing media, Mo^lo, > 0.5.

Art.

6] Table 20.

INTEGRAL

RESONANCE

Infinite Dilution Resonance Absorption Integrals (Barns) from Cadmium Cutoff (~0.5 ev)* Measured

Calculated!

l/i

0. 12

31 0.25 0.027 0.095 0.06 0.084

0.22

Activation

-0.24 ~0. 16 ~0 . 072

14.3

0.89 2.0 10.6

2.5 2.2

~10.7 ~2. 2

5.9

-11.8

90

16.3

49.3

2.0 0.8 0.8 20 170 62

1.6 1.9

0.92 0.48 0.62 1.38 1.8

4.8

Absorption

28 0.27

0.9 0. 18

0.5 <2

0.6

I.I

12

I.I

by

12

1.3


2-25

DATA

2

3.0 1.9 10.8 2. I. 2.3

3.7 4.4 2.2

4. 3.3

9.2 15 36.8 147

0.53

33 16

0.62

0.91

0.47

3.87

3

1.060

1.1 66

656

3.6 82 1.206 1.752

27 13 37 26 87 0. 26

3.0

I.I

150

2.0 2.9 0.53 4 0

4.9 61 1.400 2.778 555 180 1.630 310 7.680 976 1.390

51 4. 9.

IS

44 33 420 37 3. 45

8.3 13 375 23 >650

74 1.160 1.050 2.640 4.3 115 162 -138 140

11.3

36 130 7.5 II. 5.5 7 40

> 1.750 950

'~ 1.750 21. 590 355 1,160 305 3.500 1.370

720 2.800

170

69 1.558

2-26

NUCLEAR DATA

[Sec. 2

Infinite Dilution Resonance Absorption Integrals from Cadmium Cutoff (~0.6 ev).* (Continued)

Table 20.

(Barns)

Measured by

Calculated t Atom

i/v

Resonance

wHg siTl'O'

.lU«"

HU">

«Pu»"t

83 650 >260 290

234 255

1.2

Absorption

31. 73

4.8 0.04 0.08 0.014 3.2

iiTI»» uPb uBi toTh*»

Activation

129

0.5 0. 1

0.5

69.8 271 282; 297

250H


* R. A. Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Physics and Mathematics," chap. 6, p. 186, by R. L. Macklin and H. S. Pomerance, McGraw-Hill Book Company, Inc., New York, 1956. t The calculations of the resonance integrals separate the value into two components, a pure reso In this table, the l/p contribution is taken as 0.44 times nance contribution and a l/p contribution. the cross section at 2,200 meters/sec. X These are values of the resonance fission integral. The cutoff •I This value is estimated graphically from a curve of fission cross section vs. energy. energy here is 2.0 ev.

energy and may be regarded as fission spectrum energy. The value of Aa is inde pendent of the temperature of the mixture. For finite dilution, the absorption integral A is a function of Nd
cross section

(barns)

present

per absorbing

atom of

Absorbing atom

Th«» * L. Dresner,

20

80

200

2.000

12.5 11.5

25 21

38 31

99 66

Nuclear Sci. Eno., 1: 68 (1956). 6.2

Lattice Arrangement*

For a lattice cell consisting of a lump of material surrounded by a weakly absorbing moderating material, Eq. (17) is replaced by In p(£„, B.)

Nuiwu +

A„ NdZdo-d

where it is here assumed that the lump may contain absorbing material and * See Ref. 1, chap.

1.5, and Ref. 3, chap.

6.

(21)

+ fr*

diluent

material

as well

as

Art. "u

FAST-NEUTRON

7]

= epithermal

2-27

DATA

scattering cross section c,(epi) per atom or molecule of moderating

material

in = average loss in In £ per scattering with moderating material Nm = ratio of number of moderating atoms or molecules in cell to number of absorb ing atoms or molecules = "effective" absorption integral A.ff Equation (21) is valid only when the energy average flux of resonance neutrons is con stant across the lattice cell. Any spatial variation (such as the spatial depression caused by absorption in the lump) must be corrected for. According to a modification of the Wigner formula, the effective absorption integral is given by 1.//

-A[

1

+

n/A (M/S) +

A +

Xo-

(M/S) +

(22)

X„

A is the absorption integral of the lump material and is given by Eq. (19), M/S ratio of lump mass to lump surface area, n is an integral involving the square of the integrand of Eq. (19) and is such that for large lumps (large M/S) the surface absorption Atff — A is given correctly, and X0 is a parameter chosen to allow A,// to where

is the

approach A* as the lumps diminish Thus, X0 is given by


X« =

toward

atomic size

*/A

(AJA)

-

(M/S

approaches

zero).

(23)

1

Table 22 lists values of A, ii/A, and X0 for lumps of undiluted U metal and Th'" For lumps of diluted U or Th (e.g., oxides of the metals), A may be obtained metal. Table 23 lists values of n/A for diluted U as a from Table 21 and Am from Table 20. function of Ndcd + «■»., the scattering cross section present per atom of U. Table

22.

Parameters for the Effective Absorption Integral from Cadmium Cutoff in Lumps of Ths» and U"** Material

A, barns

nl A, g/cm'

Xo, g/om*

8.37

0.55 2.95

o.ot

Th"" metal U metal

10.2

0.

II

J.

" P.. A. Charpie, J. Horowitz, D. Hughes, and D. J. Littler (eds.), "Physics and Mathematics," chap. 6, p. 104. by R. L. Macklin and H. S. Pomerance, McGraw-Hill Book Company, Inc. New York, 1956.

Table

23.

Surface Absorption Integral p/A in Homogeneous Containing Uranium* Scattering croee lection per U atom,

Surface absorption integral p/A,

tarns

g/cm'

8. 3 (pure U) 20 40 60

2.95

1.8 1.0

0.5 * From U.S. Atomic Energy Commission, " Reactor Handbook 3645. McGraw-Hill Book Company, Inc., New York, 1955. 7

Mixtures

vol. I. p. 521. Fig. 1.5.20, AECD-

FAST -NEUTRON DATA

Cross sections of fissionable and related heavy atoms are given in Tables 7 to 12. Cross sections of fission products are given in Table 17.

NUCLEAR DATA

2-28 7.1

[Sec. 2

Absorption Cross Sections*

Table 24 lists for some materials the ny cross sections for Sb-Be photoneutrons of effective energy 30 kev. Table 24.

Radiative Capture Cross Sections for Sb-Be Neutrons of Effective Energy 30 Kev* Crou

tectum, f barns 0.0016 0.059 0.078 0.065 0.033 1.2 0.57 0.20 1.4 2. 1 1.0

Atom

itAl" »V« »Mn" »Cu" ioZn"

iiAs" uMo'" uRh'"

»

«Pd">« 47Ag'0' nSb'«>

2.8 2.2

7»Pt>M

0. 28

I.S


* V. Hummel and B. Hamermesh, Phyt. Rev., 81: 67 (1951). t The quoted errors are ± about 20 per cent.

Table 25 lists the ny cross sections for fission neutrons of listed effective energy Mev. It is seen that the behavior of the 1-Mev capture cross section is a general increase toward heavier atoms, accompanied by sharp drops at the magic neutron numbers 50, 82, and 126. Table 26 lists cross sections averaged over the fission spectrum for threshold np and not reactions in various materials. 1

7.2

Transport Cross Sections

A cross section often used in multivelocity diffusion cross section a,„ defined by "u = "tot — «"««cos 8, where

calculations is the transport (24)

o> = transport cross section a„ = elastic scattering cross section

cos 8, = average value of the cosine of the angle of scattering in elastic scattering ffioi = o-„t, + a„ + "Vn« = total cross section oin, = inelastic scattering cross section

In one-group calculations, use is made of the one-group transport cross section a',r, defined by o'lr = Oft — "tt COS 6, — (7i„, cos 0,-„

where cos

0
(25)

average value of the cosine of the angle of scattering in inelastic that cos 6\„ = 0, in which case the two transport

It is often assumed scattering. cross sections are equal.

trtr —

a'„

when cos

0,„

— 0

(26)

The two cross sections are always equal for lower energy neutrons where inelastic scattering does not occur. Thus, when o-,,, = 0 "tr = a'tr — ftct — a, COS 8 =
-

* See Ref. 3, chap. 9, and Ref. 29, chap. 4.

Art.

FAST-NEUTRON DATA

7] Table 25.

Atom

nNa"

nH|» iiAl" i«Si"

iiC\"

i«A'»

nK" ioCa«

uV" «Mn« «iCo» i«Ni"

JlCu" iiCu" i.Zn»»

i.Ga" iiAs" i»Br" >iBr«


i.Kr" i.Kr"

itRu" itRu" iiSr" »Y» .,Nb" uMo" ,iMo'» .iRu'" .iRu"< «.Rh"»

.rAg'" «.In"»

ioSn'" ioSn">

ioSn'"

nSb"«>

ill"'

nXe>»

itBai" i7La'» i«Ce'" iiCe'«

••Pr'" «Nd'<« ■oNd""

;iLu"»

tiLu"' iiTa"" nW'» «Re>»

«Re'" nPt1"

«Au'" »Hg"< ■tPb«M

nBi"' •

t

2-29

Radiative Capture Cross Section for Fast Neutrons of Effective Energy 1 Mev* Number of neutrons

12

M 14 16 20 22 22 28 28 30 32 36 34 36 38 38 42 44 46 48 50 48 SO 50 30 52 56 58 58 60 58 60 62 66 70 72 74 70 74 82 82 82 82 84 82 86 88 104 105 108 112

no

112 120 118 124 126 126

in nuclide

Cross section, millibarns

0.26 0.6 0.37 1. 1

0.74 0.93

2.9 1.9 1.8

3.82

II.

0 5. 1 11.4

6.0 23.2 20.9 22.5 42.5 19

2.4

23. 1 1.8 2. 1

7.0 41 10.4 12.3 30 31 109 85 178 223

>I4

>I2

19 90 105 1.0 2.3

5.0 5.4 4.2 11.0 40 80 158 330 142 71 180 165 64 120 102

2.0 3.4

D. J. Hughes, R. L. Garth, and J. S. Levin, Phy: Rev., 91: 1423 (1953). Values are given ± 50 per cent.

2-30

NUCLEAR

Table 26.

Cross Sections of Threshold

[Sec. 2

DATA

Reactions for Fission Spectrum Neutrons* Croat tection, 10

Reaction

iBe'di.ajHe* .B>'(n,a)Li'

millibarnit

0.085 0.014 0.5 4.5

iF"(n,p)0" nNa"(n,p)Ne" nNaM(n,«)F» iiAl"'(n,p)Mg" i«Al"(n,a)Na» uMg»(n,p)Na" uMg»(n.p)Na» MSi»(n,p)Al" i.Si»(n,p)Al» iiP"(n,p)Si» i«P"(n.a)Al» »S"(n.p)P" nS»(n,a)Si"

0 7

0.4 2.8 0.6 1.0

2.0 4 2 7 19 1.43 30

3.0

i7Cl»(n,p)S»

16

i:Cl"(n,a)P" i7Cl"(n,p)S» l,V"(n,a)Sc"

3.0 0.24 0.08 "Pile Neutron Research," Table 4-1, p. 100, Addison-Wesley Publishing Company,

* D. J. Hughes, Reading, Mass., 1953.

Table 27.

Transport


Element

Cross Sections Cross section, barns

«Ti

2.1 2.1

•iCo

47Ag
3.0 2.5 2.8 3.3 3.5 4.8 4.6 4.2 4.6 4.4 4.4

win

4.1

i.Fe t.Ni »Cu icZn i«Se

iiSr nZr

.iNb «Mo

Neutrons of Energy

Element

1

Mev*

Cross section, barns

ioSn ■.8b

•iTe isBa iiCe 7lHf 7iTa 74W 7tAu toHg 11Pb

*.[

tiBi

4. 1

i.Th

5.3

* M. Walt and H. H. Barschall, Phyt. Rev.. 93: 1062 (1954).

Equation (27) applies to slow and intermediate energy neutrons, with a, being the ordinary elastic scattering cross section. The equation is sometimes written as — cos 6), which is valid if a^, is small. otr = o-,(\ Table 27 lists values of the transport cross section o-,r for 1-Mev neutrons. Table 28 lists values of the one-group transport cross section a'tr for neutrons having the energy spectrum in Godiva. 7.3

Fast Reactors

Tables 10 and 28 present average, or one-group, cross sections appropriate to the neutron spectrum in the listed reactors. The reactors include EBR I, Godiva, Zephyr I, and Zephyr II. The compositions of the cores of these reactors are given in Table 29. Godiva is an unreflected reactor. The others have reflectors of natural uranium. A rough comparison of the neutron spectra in Godiva, EBR I, and the fission spectrum is presented in Table 30, which pertains to the spectra at the center of the listed reactors. The spectrum at the core boundary of Godiva is slightly more energetic than the central spectrum because neutrons near the boundary have made fewer collisions. In EBR I, the spectrum at a point near the core boundary was

Art.

FAST-NEUTRON

7] Table

28.

2-31

DATA

One-group Transport Cross Section Transport

a'iT

in Godiva Spectrum*

cross section barns

Atom Be B C

1.1 2.2 2.2 2.5 4.5

Fe

An Bi Th»»

5.1 5.1

U»" U"s

6.0t 6.5 5.6

U>"

7 6 Pu»» * R. A. Charpie, J. Horowitz, D. J. Hitches, and D. J. Littler (cds.), chap. 9. Tables 7 and 8. by J. Codd, L. R. Shepherd, and J. H. Tait. Inc., New York, 1956. fThis number is from Zephyr II.

Table 29.

"Physics and Mathematics,"

McGraw-Hill Book Company,

Composition of Fast Reactor Cores*

Fraction of volume occupied

by material in core of


Material Godiva

EBR I

1 0 0

0.55 0.25 0.20

u»»t NaK Steel

Zephyr I

Pu U(nat)

Ni

Cu Fe

Air

Zephyr

0.48 0.33 0.10 0.02

II

0.52 0 0.11

0.02 0.06 0.29

0

0.07

* R. A. Charpie. J. Horowitz, D. J. Hughes, and D. J. Littler (eds.), "Physics and Mathematics," chap. 9. p. 287. by J. Codd, L. R. Shepherd, and J. H. Tait, McGraw-Hill Book Company, Inc., New York. 1956. t This is uranium enriched to above 90 per cent in U"1.

Table

30.

Comparison of Neutron Spectra in Fast Reactors Fraction of neutron density in the group for

Energy group, Mev

0-0.4 0.4-1.4 >1.4

Godiva center*

EBR I centert

Fission spectrum

0.21 0.41

0.31

0.10 0.34 0.56

0.38

0.37 0.32

•

R. E. Peterson and G. A. Newby, Nuclear Set. Eno., 1:112 (1950). t H. V. Lichtenberger, F. W. Thalgott, F. W. Kato, and M. Novick, Proc. Intern. Con/. 3: 345, Geneva (1955). found to be slightly degraded in comparison with the central spectrum as a result of inelastic scattering in the uranium reflector. The spectrum deep inside the uranium reflector is that for which the cross sections of Table 11 apply. An approximate spectrum of neutrons in spectral equilibrium in uranium is presented in Table 31. This neutron flux spectrum is fairly uniform in the

2-32

NUCLEAR DATA

[Sec. 2

range 0 to 0.2 Mev with a peak at about 0.1 Mev. For higher energy neutrons, it At 1 Mev, the flux per unit energy range is drops approximately exponentially. about 10 per cent of the peak value. Natural Uranium Equilibrium Spectrum*' f

Table 31.

Energy group, Mev

0-0.2

Fraction of

neutron flux in energy group

0.35 0.55 1-2 0.08 0.02 >2 * From R. A. Charpie, J. Horowitz, D. J. Hughes, and D. J. Littler (eda.), " Physics and Mathematics," chap. 9, p. 290, Fig. I J, by J. Codd, L. R. Shepherd, and J. H. Tait, McGraw-Hill Book Company, Inc., New York, 1956. t This is the spectrum measured at a depth of about 20 cm inside the uranium blanket surrounding the core of Zephyr. It is claimed that this spectrum is only slightly less degraded (from the original fast neutrons leaking out of the Zephyr core) than that which would obtain at a larger depth inside a larger uranium block. See Table 15 for cross sections averaged over this spectrum. 0.2-1

7.4

Inelastic and Nonelastic Cross Sections

The nonelastic cross section an, is defined by


0~ne — Olol — 0~$t ™ 0~abt

T ff»n»

(28)

Values of an, for various materials at several neutron energies are presented in Table If the absorption cross section is small, then a„, = e^,,.

32.

Table 32.

Nonelastic Cross Section for Fast Neutrons at Several Energies* Cross sectiont (barns) at Element 2.5 Mev

.C

nNa nMg

0.5 0.8

11AI

1.0

uP isS

irCl «oCa

»iCr wFe »rCo

isNi

iiCu

4. 1 Mev

0.8

0.6 0.8 0.8

0.7 0.5 0.6 0.4 1.4 1.2 1.4

0.8 1.6

1.5

2.2

1.7 1.6

1.4 1.35 1.4 1.4

1.4 1.7

41M0

1.9 1.9 2. 1

.«Cd

2.2

2.3

2.3

itSn .iSb uTe

1.7

1.7 2. 1

2. 1 2. 1

uBl

0.6

nPb

1.7

M. V. Pasechnik, Proe. Intern. Conf. t Quoted errors range from 5 to 50 per

1.8 1.8 1.9

2.0 1.9

2 0

0.8 2.6 2.6

1.7 1. 1

1.3

2.2

••I ••Ba 7.W -..11k

0.6

1.6

i.Sc

2.0 2.0

14 Mev

0.9

loZn

«Ag

*

3.3 Mev

2.2 1.0

V4

0.6

1.7 1. 1

2 4

2.4 2.4

2. Paper 3 (1956). The majority are in the range cent.

10 to 15 per cent.

Art.

FAST-NEUTRON

7]

Table

2-33

DATA

33 presents the (inelastic) cross section for scattering of neutrons above the

energy E to below the energy E for various materials and several E values. If the neutrons of energy greater than E are treated as one group, the listed cross section is then the group average cross section for scattering out of the group (to below E). The listed cross sections apply to the case in which the neutrons in the group have the fission spectrum. Table 34 breaks up this one-group cross section for E = 1.4 Mev into two parts. The cross section for scattering into the energy range 0.4 to 1.4 Mev is listed along with that for scattering to below 0.4 Mev. The sum of these two is the net group cross section of Table 33. Table 33.

Inelastic

Scattering of Fission Neutrons*

Cross section, t barns for scattering from above to below


Element 0. 7 Mev

1. 4 Mev

5 Mev

iiAl

0.09 0. 19

0.32 0.56 0.58 0.69

0.76

»«Ti

..V »Fe iiNi itCu

0.22 0.28 0.28 0.30

■oZn

0.31

wZr «7Ag «.Cd io8n 7.W 7»Au

uPb

uBi

1.20 1.31 1.41

0.71

0.90 0.96 0.96

0.30 0.84

1.57 1.99 2. 14 2.01

1.66 1.52 1. 12 2.23

0 66

0.37 1 08 1.00 0.21

0.20

2.04

2.72 2.68

0.71

2.21

0.73

2.35

A. Bethe, J. R. Beyster, and R. E. Carter, LA-1429. December, 1955. Quoted errors are about 15 per cent for the 0.7 Mev cross sections, 7 per cent for the 1.4- and 5 per cent for the 5-Mev cross sections. • H.

t

Table

34.

Inelastic Transfer Cross

Section for Fast Neutrons*

Transfer cross section,

f

barns

Atom

iiAl

.iTi nV

itFe

IiNi iiCu loZn .oZr

«Ag 4iCd toSn r«W 7§Au •«Pb •iBi

J.

»li

ffn

0.27 0.48 0.49 0.56

0.05 0.08 0.09

0.61

0. 13 0. 10

0.78 0.82 0.80

0.12 0.13

1.19

0.47 0.60 0.25 0.55 0.53

0.92 0.87 1.68 1.51

0.60 0.61

0. 16

0.

II

0.12

R. Beyster, and R. E. Carter, LA-1429. December, 1955. **§ is the atomic cross section for scattering neutrons (by inelastic scattering) from group group ;. Group I is 0 to 0.4 Mev; group 2 is 0.4 to 1.4 Mev; group 3 is greater than 1.4 Mev. • H. A. Betbe,

t

i

to

2-34

NUCLEAR DATA 8

[Sec. 2

RADIATIONS AND THEIR RANGES 8.1

Charged

Particles"-"

Charged particles at velocities usually encountered in reactors and related facilities lose most of their energy by ionizing the atoms of the material they are traversing. On the average, the particles lose 33 ev of kinetic energy per ion pair produced. The range of electrons in aluminum is given to within 5 per cent by the formula

„.

.

«3(mg/cm«)

J |

= 412£» -m-o.oin m* 530£ 106

_

_

for 0.01 2 5

.„„,

(29)

where i?fl(mg/cms) is the range in milligrams per square centimeter of aluminum of an The range in other materials is roughly the same electron of kinetic energy E Mev. in milligrams per square centimeter. Table 36.

Maximum

Range of Fission Products* Range, mg /cm1

Material

Air Al

3.0 3.7 5.2

Cu


* E. Segr* and C. Wiegand,

U

12.6

Phyt. Rev., 70: 808 (1946).

The range of a particles in air at standard conditions is given to within by the formula . . . / = 0.56S for E < 4 „ . Ka(cm of air) _ I UE _ 2 62 for 4 < £ < 8

j

10

per cent /om (30)

In other materials, the range of a particles where E is the a-particle energy in Mev. is given to within 15 per cent by the formula fi„(mg/cm')

-

0.56A^ft„(cm of air)

(31)

where A is the mass number of the material. For protons, an approximate formula is

RP(E)

« Ra(4E)

(32)

which states that the range of a proton of energy E is approximately that of an a of energy 4E. The maximum range of fission products in some materials is given in Table 35. 8.2

Neutrons*""

In contrast to charged particles, which have a definable range, neutrons and y rays In this exponential approximation, use are stopped approximately exponentially. is made of the relaxation length, which is the thickness of material that, owing to absorption, causes a drop in intensity by a factor of e. For thermal neutrons, the relaxation length is equal to the thermal-diffusion length, Thus, thermal neutrons can which can be made quite small in absorbing materials. be shielded out in a fraction of an inch of boral. Neutrons of energy up to about 1 Mev may be readily stopped by moderating them to low energies where they can be absorbed easily. The relaxation length in such cases is about equal to the slowingdown length (square root of the age of the neutrons), which can be made as small as a few centimeters by use of hydrogen atoms. The very energetic neutrons, several Mev and up, are the most difficult to shield against. Table 36 lists illustrative values of the relaxation lengths for these neutrons in various materials. Final absorption of the neutrons is usually accompanied by the emission of y rays. The use of boron results in a comparatively soft y ray of 0.5 Mev. Lithium, however, absorbs neutrons with essentially no production of y rays. Table 9 of Sec. 7-3 • 8ee Ref. 8. chap. 2.

Abt.

WEIGHTS

ATOMIC

9]

presents data on the production of y rays by thermal-neutron capture.

Gammas also

result from inelastic scattering of neutrons. Table 4 of Sec. 7-2 lists the values of the neutron flux, at various neutron that results in a dose rate of 7.5 mrem/hr. Shielding of neutrons and radiation dose are treated in detail in Sec 7-3. Gamma Rays"

8.3

energies,

"

For y rays in the energy range 0.1 to 10 Mev, a flux of about 600 Mev/(cm')(sec) is required to produce a dose rate of 1 mrem/hr. -»

~600 Mev/(cm»)(sec)

for photons in the energy range 0.1 to 10 1 mrem/hr diminishes for softer photons. The reason for this is the high (cm,)(sec). which causes the y energy to be absorbed in

1

mrem/hr

(33)

The energy flux that results in At 0.01 Mev, the flux is 3.5 Mev/ photoelectric cross section for soft ys, short distances. For the same reason, however, the soft 7s can be shielded out very easily. Table 36 lists illustrative values of relaxation lengths in various materials for y rays coming out of a reactor. Detailed information concerning -y-ray shielding is presented in Sec. 7-3.


Table

36.

Mev.

Illustrative Values of Relaxation Lengths in Various Materials for Fast Neutrons and for Gamma Rays*

Density,

Material

Relaxation length, cm

g/om*

Fast neutrons

1.0 1.62 1.85 2.3 2 3 2. 7 3. 5 4. 3

Beryllium oxide

Gamma rays

~I0 ~ 9

~ 9 ~ 9 11 ~10

8.0 6

7.8

~ 6

11.3

9

~

J

30 19 18 14 15 13 10 8

3.7 2.5

* S. Gladstone, *' Principles of Nuclear Reactor Engineering, " Table 10.3, p. 609. D. Van Nostrand Company. Inc., Princeton, N.J., 1955. t Portland concrete is a mixture of Portland cement, gravel, and sand. t Barytes concrete is a mixture of Portland cement and BaSO< aggregate. 4 Brookhaven concrete is a mixture of Portland cement and iron aggregate.

ATOMIC WEIGHTS'

9

The atomic of the atomic

0"

"

mass or weight scale is a scale of the masses of neutral atoms in terms The amu is defined such that the mass of the mass unit, the amu.

atom is exactly 16 amu. The conversion factors between the amu and the gram and between the amu and its energy equivalent (through E = mc1) are 1 g =

N

1

N

amu

- 6.025

X 10"

amu = 931 Mev

- Avogadro's

number

(34) (35) (36)

Reference 37 gives a list of the atomic masses of the nuclides. is defined as

The packing fraction

/

/-^

(37)

2-36

NUCLEAR DATA

where M is the atom mass in amu and A is the mass number. per particle BE /A can be obtained from through the relation

/

[Sec.

2

The binding energy

(38)

where Z is the atomic number. The binding energy of an additional neutron, when added to the incident kinetic energy, gives the energy of excitation of the compound nucleus formed by neutron capture.

REFERENCES 1. 2. 3.

4. 5. 6.


7. 8. 9. 10. 11.

U.S. Atomic Energy Commission: "Reactor Handbook," vol. 1, chap. 2, AECD-3645, McGraw-Hill Book Company, Inc., New York, 1955. Gamble, R.: Ph.D. Thesis, University of Texas, June, 1955. Charpie, R. A., J. Horowitz, D. J. Hughes, and D. J. Littler (eds.): "Physics and Mathematics," chap. 7, Progress in Nuclear Energy, ser. 1, vol. 1, McGraw-Hill Book Company, Inc., New York, 1956. Lundby, A., and N. Holt: Nucleonics, 12 (1): 24 (January, 1954). Way, K., and E. P. Wigner: Phys. Rev., 73: 1318 (1948). Rockwell, III, Theodore (ed.): "Reactor Shielding Design Manual," chap. 3, U.S. Atomic Energy Commission, TID-7004, March, 1956. Moteff, J.: Nucleonics, 13 (5): 28 (May, 1955). Clark, F. H.: Report NDA-27-39, Dec. 30, 1954. Whitehouse, E. J.: Progr. Nuclear Phys., 2: 120 (1952). Steinberg, E. P., and L. E. Glendenin: Proc. Intern. Conf. 7: 3 (1956). Robb, W. L., J. B. Sampson, J. R. Stehn, and J. K. Davidson: Nucleonics, 13 (12): 31

(1955). Studier, 13. Hughes,

M. H, and J. R Huizenga: Phys. Rev., 96: 546 (1954). D. J., and J. A. Harvey: "Neutron Cross Sections," BNL-325. 14. Koch, \V.: Phys. Rev., 77: 329 (1950). 15. von Dardel, G. F., and N. G. S. Sjdstrand: Phys. Rev., 96: 1245 (1954). 16. Scott, F. R., D. B. Tomson, and W. Wright: Phys. Rev., 96 : 583 (1954). R., G. S. Mani, P. K. Ivengar, and B. V. Joshi: Proc. Intern. Conf. 5: 17. Ramanna, 24, Trombay, India (1956). 18. Geraseva, L. A., A. V. Kamayev, A. K. Krasin, and I. G. Morosov: Proc. Intern. Conf. 5: 13, Trombay, India (1956). 19. Antonov, A. V., A. I. Isakoff, I. D. Murin, B. A. Neupocoyev, I. M. Frank, F. L. Shapiro, and I. V. Shtranich, Proc. Intern. Conf. 5: 3 (1956). 20. Melaika, E. A., M. J. Parker, J. A. Petruska, and R. H. Tomlinson: Can. J. Chem., 12.

33: 830 (1955).

21. 22. 23.

Hurwitr, Jr., H.: Proc. Intern. Conf. 4: 328 (1956). Deutsch, R. W.: Nucleonics, 14 (9): 89 (1956). Hughes, D. J.: "Pile Neutron Research," chap. 3, Addison- Wesley Publishing Com

pany, Reading, Mass., 1953. 24. Coveyou, R. R., R. R. Bate, and R. K. Osborne: /. Nuclear Energy, 2: 153 (1956). 25. Cohen, E. R.: Proc. Intern. Conf. 5: 405 (1956). 26. Dresner, L.: Nuclear Sci. Eng., 1: 68 (1956). 27. Walt, M., and H. H. Barschall: Phys. Rev., 93: 1062 (1954). 28. Peterson, R. E., and G. A. Newby: Nuclear Sci. Eng., 1: 112 (1956). 29. Pasechnik, M. V.: Proc. Intern. Conf. 2: 3 (1956). 30. Bethe, H. A., J. R. Beyster, and R. E. Carter: LA-1429, December, 1955. 31. Katz, L, and A. S. Penfold: Revs. Mod. Phys., 24: 1 (1952). 32. Aroux, W. A., B. G. Hoffman, and F. C. Williams: AECU-663. 33. Segre, E, and C. Wiegand: Phys. Rev., 70: 808 (1946). 34. Coryell, C. D., and N. Sugarman: "Radiochemical Studies: The Fission Products,"

Book Company, Inc., 1951. " Principles of Nuclear Reactor Engineering," chap. 10, D. Van Nostrand Company, Inc., Princeton, N.J., 1955. 36. Mittelman, P. S., and R. A. Liedtke: Nucleonics, 13 (5): 50 (1955). McGraw-Hill

35. Glasstone,

37. Physica,

S.:

21 : 367-424

(1955).

SECTION

3

MATHEMATICS BY

ALSTON

S.

HOUSEHOLDER, Ph.D.,

Head of Mathematics

National Laboratory. WARD CONRAD SANGREN,

Panel, Oak Ridge


A.B., M.A., Ph.D., Chief of Computing, General Atomic, General Dynamics Corporation; formerly Chief of Computing and Mathematics, Curtiss- Wright Research; Assistant Chief of Mathematics Panel, Oak Ridge National Laboratory.

CONTENTS ALGEBRA AND GEOMETRY

3-1

BY

ALSTON

S.

HOUSEHOLDER

Algebra of Scalars, Vectors, and Matrices Trigonometry and Complex Numbers . . . Loci: Curves and Surfaces Algebraic Equations Probability and Statistics References and Notes

1 The 2 3 4 5

3-2

BY

PAOE

3 Series and Expansions of Functions 4 Differential Equations 5 Other Topics

Bibliography

3-2 3-15

3-3

3-24

3-35 3-55 3-62

3-05 3-106 3-125 3-140

OF HIGH-SPEED MACHINERY

BY WARD CONRAD SANGREN AND ALSTON

8.

HOUSEHOLDER

1 Digital Computing Machinery 2 Analogue Computing Machinery

ANALYSIS

Bibliography

WARD CONRAD SANGREN

1 Differential and Integral Calculus 2 Function Theory

PRINCIPLES COMPUTING

PAOK

3-64 3-75

3-1

3-142

3-146 3-148

3-1

ALGEBRA AND GEOMETRY BY Alston

S. Householder

Generally speaking, algebra and geometry are distinguished from analysis in that Nevertheless most practical methods for they do not involve the notion of limit. solving equations do involve limiting processes, even though the definition of the solution does not. Hence some familiarity with limits will be presupposed in certain portions of the present section. 1

THE ALGEBRA

OF SCALARS, VECTORS,

AND MATRICES


This article will begin with the basic concepts and operations of algebra and of analytic geometry. 1.1

Sum, Products, and Powers

The real numbers include the integers (the null element 0; the positive integers 1, 2, — 2, — 3, . . .); the rationals, which ; and the negative integers —1, include the integers as well as any number expressible as a fraction whose numerator 3, 4,

...

and denominator are both integers; and the irrationals, such as r, y/2, which are not so expressible. 1.11 The basic arithmetic operations are the two summative operations of addition and subtraction and the two multiplicative operations of multiplication and division. These operations can be further classified as direct (addition and multiplication) and Any two real numbers can be combined by any indirect (subtraction and division). of the four basic operations, and the result is again a real number with one exception: Division by zero is not allowed. The indirect operations are related to the direct operations as follows: The differ ence a — 6 is defined to be that number x which satisfies the equation 0 = 6+1; the quotient a/6 is defined to be that number y which satisfies a = by. Since Qy = 0 whatever y may be, it follows that when 6=0 and a = 0, then y could be anything, and when 6=0 and aÔ, there is no y that could satisfy the equation a = by. Signs of aggregation are parentheses ( ), brackets [ ], braces j }, and, less com monly, the vinculum or horizontal bar. They have the force of punctuation marks and signify that the enclosed operations are to be performed before any further combination is effected. Thus 6/(2 + 1) = 6/3 = 2, the division being withheld until the addition set off by the parentheses has been performed. Where signs of aggregation do not intervene, the following conventions are always understood to hold: 1. Multiplicative operations are performed before summative operations. 2. Multiplicative operations are to be performed in the order in which they occur. 3. Summative operations are to be performed in the order in which they occur. Thus, 6/2 + 1=3 + 1=4; 1+6/2 = 1+3=4; 6/2-3=3-3=9, but 6/(2 • 3) = 6/6 = 1; 6/2/3 = 3/3 = 1. The commutative laws and the distributive law are satisfied by all real numbers as follows: Commutative law: a + 6 = 6 + o. 06 = 6a. Distributive law: 0(6 + c) = 06 + ac.

3-2

3-3

ALGEBRA AND GEOMETRY

Sec. 3-1]

A variant of the distributive law

is

(6

c)/a

±

- b/a

c/a

±

and this constitutes the rule for summing fractions with a common denominator. Other rules for combining fractions are as follows: a

c

b'

d

a/b eld a

c

b+d

"

ac bd ad

"

be

ad + be

=

bd


and the cancellation law is ac

a

bc~

b

A ratio is a quotient; a proportion is a statement that expresses the equality of two Thus the ratio of a to 6 is commonly written o:6 and is simply the quotient ratios. The colon is therefore a particular sign of division. Four numbers a, b, c, and d a/b. This is sometimes written are said to be in proportion if a:b — c:d, or a/b = c/d. a:b: :c:d, the double colon being thus a form of the equality sign. A number c is said to be the third proportional to a and b in case a/b = b/c, and then b is said to be the mean proportion between a and e. If a/b = c/d, then it is also true that b/a -- d/c

(o ± b)/b = (c ±

a/c

d)/d

- -

= b/d

(a +

ad «= be

6)/(o

6)

(C

More generally, if a\/bi = Oj/ftj = 03/63 =

then

(moi + n,as +

n»a3

■■

+

provided only that the numbers ni,

• • • —

-)/(n,bi + nj&j +

7i2, n3,

ni&i + nibt +

. .

n>b3

. are such

+

• • •

+ rf)/(c

- d)

f

+•••)■"'

n3&3

that

^0

When two or more quantities are to be added, each is called a term; when they The product of n equal factors is called each is called a factor.

are to be multiplied, a power:

a ■a ■ ■ ■ • — a"

the number a is called the 6ose and n the exponent of the power. bined according to the following rules: an • am = an+m a»b»

aH/am ™ an~n = \/am~n = (ab)n a"/6" = (a/b)"

Powers are com

(a*)m — anm

When n = m, the second of the above relations is to be regarded as providing a m, the definition: a" = 1 except when a = 0, and 0° is meaningless. When n second relation also defines the use to be made of negative exponents: a~~» = 1/a". The operation inverse to that of taking a power is that of extracting a root. Thus if xn = a

then x

=

\/ a

Note that by definition, provided it is further true that a and x are of like sign. when n is even and a negative, no such (real) x exists (see the discussion of complex

3-4

MATHEMATICS

[Sec. 3

Also When such an x does exist, it is called the nth root of a. numbers in Art. 2.1). When n is even and o > 0, there is always an o is the base or radicand, n the index. x > 0 satisfying the relation x" = o, and in that case it is also true that (— x)" = a. Although —x is also an nth root of a, nevertheless the radical designates x and never —as.

The following relations govern the use of radicals: m f~ V"/ V a \/a \/b

= =

nt

nm /~

V«*

V»

n/my—

=

yfa/y/b

\/ab

. = am/"

Va =

\V a/b

These and similar relations are readily established by using fractional exponents; for

V

tya = [(a)1/m]1/n = a"*""' = "\Va. In the second relation above, the example, fraction n/m is not to be used as index of a radical unless n/m is in fact an integer. However, the form am,n is always permissible, and the equation is to be taken as It can be verified that all defining the expression when the exponent is fractional. previous laws of exponents are valid even with fractional exponents. A natural extension permits the use of even irrational exponents. As special cases of the above laws of radicals, the relations


y/aPb = a

y/ a/b"

y/b

=

yfa/b

give examples of possible "simplification" of radicals. When attention is fixed upon the relation between the power and the exponent, the There are two frequently used base being fixed, the exponent is called a logarithm. systems of logarithms, the natural, or Napierian, with base e = 2.718 . . . , and The natural logarithms are useful in analysis the common, or Briggsian, with base 10. and will be discussed there and in Art. 2. The common logarithms are useful in The following identities hold for any (fixed) base: computation. log (NM) = log N + log log log (N/M) = log N = m log N log Nm

-

log

For

common logarithms

it

ty N

is true

= (log

M M

N)/m

in particular that

log (n

X

10m)

= m

+ log n

is not affected by the location of the decimal point in the number, and this accounts for the utility of 10 as a base. Let N = n • 10™, where m is an integer, positive or negative, and 1 < n < 10. Then, since log 1 = 0 and log 10 = 1, it follows that 0 < log n < 1. Hence log N is the sum of an integer m, called the characteristic of the logarithm, and a pure decimal log n, called the mantissa of the logarithm. The mantissa is always nonnegative, and its value is independent of the location of the decimal point in N . The characteristic may be positive, negative, or zero, and its value is determined solely by the location of the decimal point in N. Note that

This implies that the decimal part of the (common) logarithm

log AT-«

-

-(m

+

1)

+

(1

- log n)

Hence, unless log n = 0, then 1 — log n is the mantissa of log N~l and — (m + 1) is its characteristic. 1.12 Bases of Enumeration. The statements just made rest upon the fact that moving the decimal point corresponds to multiplying or dividing by a power of 10, and this, in turn, is due to the use of the decimal base in our common system of enumeration. That is to say, any number is represented in the form • • ■

where

each

+ dt

■10s

+ di

■10

+

do

+ d_i

d is one of the 10 integers 0,

1,

• 10"1 2,

+

. . . , 9.

10"' +

■ ■■

Any number whatever

3-5


Sec. 3-1]

(instead of 10) could be used as base, and in recent years the base 2 (or sometimes 23 or 24) is used for many automatic computers, for scalars, and in other special situ ations. In the binary system a number is represented in the form • ■•

+

a2

• 2»

+

an

■2

+

a„

-f

• 2"1

• • •

+

where each a is either 0 or 1. It is sometimes desirable to convert from decimal to some other specified representation or back again. If the numbers are integers, let the two representations be JV = d0 + di ■10 + dt ■10l + • • • = 6o + b,/3 + 62/3s +

- I:

where 0 represents the other base and each ft

• • •

of the values 0,

6 has one

M'

+

1,

Given the decimal representation: Divide N by 0, and the remainder is the quotient by 0, and the remainder is 6i ; continue, obtaining 62, 63, . . . Given the representation in the base 0: Form in sequence &'„_,

b',-i

Then

b'o

D

=

...

,

divide in turn.

b0; ,

b,P + 6,_,

= &'»-t0 + = 6\_20 +

= N in decimal form. numbers are not integers, let the nonintegral parts have the representations

d_,

• 10-»

+ d_,

•

10"' +

• ■•

- 6_i

•

/3"1

+ 6_j

•

+

0"»

• • •

+

Given the decimal representation: Multiply by 0, and the integral part of the product will be 6_i ; multiply the nonintegral part of the product by 0, and the integral . part of this product will be 6-2; Given the representation in the base 0: Form in sequence

...

b'-p+t = 6'-,1+i/0 + 6-M+2 = b'-n+i/p +

6'_M+>

Then D

It

=

is presupposed that the computations themselves are performed decimally. Analogous rules can be formulated when the other base is to be used for computation. 1.13 Polynomial products can be formed by repeated application of the associative law, but the following are of frequent occurrence:

+

-

x» x4

xi

xy + t/s)(x + y) = x» (x» x*y* — xy3 + y*)(x + y) = x'

+ y' + y*

+

x»-'i/ +

*"~V

(3)

- x'

+

y*

are cases of the binomial series that can be written

+

(x + y)n

bc)xy -f- cdy*

y* y* y*

(x ± y)* = xs ± 2xy + y* (x ± y)* = x' ± 3x*y + 3xy* ±y3 (x ± y)* = x* ± 4x'y + 6xV ± 4xy>

The identities in this last group as follows:

+

y1

*"~V

•

•

(x* — x*y

xV

-

acx* + (ad x1

•

+

= = = = =

+

(z*

-

(ax + by) (cx + dy) y) (x + i/)(x y) (xl + xy + y')(x y) (x» + x*y + xy2 + ys)(x x'y + y) + xy' + y*)(x

(")


If the

=

2,

3-6

MATHEMATICS n(n

where

-

• • •

r(r 1)

1)

-r+

(n

3

[Sec. 1)

■ • • •2 • 1

When n is a positive integer, then

the series represents y/2). one way of extracting roots. Another to use logarithms available. Thus from a table of common logarithms one finds log

it

<

1,

y

a

1,

\y\

When n is not a (see also Art. 5), and the polynomial has exactly n + 1 terms. positive integer, there are infinitely many terms defined. If x — 1 and can be shown that the infinite series converges (cf. Art. 3.6 of Sec. 3-2) and provides = n = method of computing the quantity represented on the left (thus for x =

=

is

table

a

if 2

is

This

is

— J>£,

0.30103.

.

Un,

If

r

is

a

if

is

■ ■

a + ar + ar* + ■

+ ar»_1 =

is

geometric progression

o(l

= n[a

+

- r")/(l -

(n

is

- l)d]

- l)d/2]

r)

+

(n

[a

+

and the sum of n terms of

2d) + ■ ■ ■ +

a

+

+

(a

a

d)

If

is

if

d

+

is d is

+ (a

is

each of which, after the first, obtainable from the preceding by fixed rule. arithmetic; where fixed, the progression Un = Un-i u„ =■Un-\r where fixed, the progression geometric; Un = vn~l where the vs form an arithmetic pro harmonic. gression, then the progression (of the us) ui = a, the sum of n terms of an arithmetic progression

sequence

is

+

■

+

A +

+

is

of

is a a

is

a

a

is

series

is

a succession of numbers Ui formed according to some rule, and a formed by summing these numbers. Simple formulas for sum (as in the arithmetic and geometric progressions) are available in only few special cases. If the nth term u„ polynomial in n with constant coefficients, then the sum of the first n terms also The method undeter polynomial of degree one greater. mined coefficients can be used to obtain the sum. This best illustrated by an • • cubic in n. example. Let Un be quadratic in n. Then «„ = ui ut u„ Hence let Bn + Cn1 + Dn> s„ = Un = a + bn + cn1

A

SD = 2a + 36

27

D

= 3a

+

66

+

+

+

-f

AC 9C

+

A+B+C+D=a+b+c

+ 2B A + SB

A

=

0

A

3,

2,

1,

0,

c

C,

For deter where a, and D are to be determined. and are known and A, B, mining these four coefficients one can assign to n any four convenient values (including one obtains even 0). On taking n = b,

5c

+

14e

see Art. 4. The method of undetermined coefficients can be applied in other situations but only when the form of the result is known or can be conjectured.

For methods of solving such equations

1.2

The Algebra of Vectors

is

This may be applied in situations where spatial orientation and localization become significant. vector exemplified by a directed line from one point (its origin) to A


a

. .

Ml, Vt, U

is

A

is

Then log \/2 half this, or 0.15052. Hence one has only to return to the table to find what number has 0.15052 as its logarithm. For still other methods see Art. 4. 1.14 of Progressions, Sequences, and Series. succession progression numbers


Sec. 3-1]

point (its terminus). If a is the vector with origin 0 and terminus P (Fig. 1), vector with origin P and terminus 0 is — a. If b is the vector with origin P and terminus Q, then a + b is the vector with origin 0 and terminus Q. If P, Q, and R are in a straight line, and if the segment PR «= aPQ, then the vector with origin P and terminus R is orb. If the line ST is parallel to the line PQ and the segments are equal, ST = PQ, then the vector with origin S and terminus T exemplifies (or is) the same vector b. A vector may therefore be thought of as freely movable from place to place but fixed in length and direction. (In more rigorous phraseology the vector b is the class of all segments parallel and equal to the segment PQ and similarly another

then the

oriented.) Examples of vectors are finite motions, velocities, accelerations, and forces. On the other hand masses, temperatures, and energies are nonvectorial, or scalar, quanti ties. When vectors are being discussed, it will be convenient to represent them by


Fia. boldface,

1.

letters and to represent scalars by lower-case Greek

lower-case Roman

letters.

The null vector 0 satisfies with any vector a the identities

0a=0

a + 0 = a Hereafter the null vector will be indicated holds

for addition and for multiplication a

Subtraction

+ b

= b + a

is defined as follows:

There are two distributive <*(a

laws:

as a simple zero. by scalars:

a + (b

+ b) = aa + ab

The

commutative

lav)

a& = aa

- a)

= b

(a + 0)a = aa +

/3a

If

two vectors are parallel, one can be expressed as a scalar multiple of the other are said to be linearly dependent. In general, vectors a and b are linearly dependent in case scalars a and f) exist, not both being zero, such that aa + /3b = 0. If this equation is satisfied only by a = f? = 0, then a and b are linearly independent. It should be observed that if one vector, say b, is null, the two vectors are necessarily linearly dependent whatever a might be, since one has only to take a = 0 and /3 arbitrary but ^ 0. Let ei and e* be any two linearly independent vectors, and let x be any vector in the plane of these vectors. Let these be drawn from a common origin O, and let and they

terminate at Bi, Et, and X, respectively (Fig. 2). From X draw PX parallel 0Et and intersecting OEi (extended if necessary) at P. Then 0 is the origin

them to sad

P

scalar

the terminus of a scalar multiple of et; P is the origin and multiple of es. Let these be { iBj and £2e2. Then

X

the terminus of a

x = £tei + hei Thus x is resolved into components along ei and es, or it is expressed as a linear combi nation of ei and ei. Its components are i-iei and f2e2; the scalars £i and {2 are its

3-8

MATHEMATICS

[Sec. 3

in the vectorial coordinate system (d, e2). Any other point 0' could have This would have led to different instead of 0 as the common origin. points E'i, E't, X', and P' but to the same scalars fi, {j. When a vectorial coordinate system such as (ei, e») is associated with a particular Associated point 0 as origin, the system forms a ■point-coordinate system (0; ei, ej). The with each vector x drawn from 0 as origin is a particular point X, its terminus. coordinates fi, £2 of x in the vectorial coordinate system (ei, es) are said to be the coordinates of the point X in the point-coordinate system (0; d, e»). Any vector not in (or parallel to) the same plane with ei and ei is not expressible If e> is such a vector, then ti, et, and es are as a linear combination of Ci and e2. linearly independent. Any vector in the same 3-space is expressible as a linear Hence they form a vectorial coordinate system combination of these three vectors. for the 3-space, and, when associated with any point 0 as origin, the point and the vectors form a point-coordinate system for the 3-space. In general, n vectors ai, as, . . . , a. are linearly dependent if there are n scalars ai, at, . . . , a„, not all zero, such that coordinates been used

"

2a, Ei

aiai +

ai&2

• • •

+

+

or„a„

= 0

If

the equations can be satisfied only by taking ai = orj = • • • = a„ = 0, then the vectors are linearly independent. A space of vectors is of dimensionality n in case there is a set of n linearly independent vectors, whereas any n + 1 vectors are linearly dependent. In an n-dimensional space let ei, e2, . . . , e„ be linearly independent. Then if x is any vector, x, t\, e«, . . . , e„ are linearly dependent and, in fact, one can Generated for wjivans (University of Florida) on 2015-09-23 02:46 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

express x = 2fce;

Thus

one may speak of the vectorial coordinate system (ej). When these vectors are associated with an origin O, the vectors and the point form a point-coordinate system (0; e<) as in the cases of two and three dimensions. If the vector x is called a geometric vector, the set of its coordinates ({i, fa £„) in any coordinate system may be called an arithmetic vector, and each coordinate & is an element of the arithmetic vector. For purposes of notation it is convenient to distinguish the column vector, where the d are written in a column, from the row When the column vector is intended, we vector, where they are written along a line. shall enclose the elements between braces { fi, . . . , £„ J for convenience of printing. The column vector will be designated x; the row vector will be called the transpose of the column vector and designated xT. Either arithmetic vector x or xT represents the geometric vector x in the given coordinate system (e,), but a change in coordinate system will in general change the representation (see Art. 1.3). If y = 27j,ei is any geometric vector, represented in the coordinate system e< by the

arithmetic vector y

«■

jm,

. . ,

.

x is represented

t/»J, then the sum

+ y

= 2(fc

+ m)ti

by the arithmetic vector X

the product

+

V

=

[fl

+

11,

. . . , fn

+

IJ»)

ay = 2(ai),)ei vector ay = (cnti,

by the arithmetic . . . , avn). null vector, and the null arithmetic vector is (0, . 1.3

Transformations

In particular, if a . .

= 0, ay is the

,0).

and the Algebra of Matrices

In the point-coordinate system (O; e;), let the point 0' correspond to the vector d If X corresponds to x in the same system and to x' in the system (O'; e,), (Fig. 3). then x = d + x'. In terms of arithmetic vectors and coordinates, x = d + x'

& = «(

+

£'<

3-9


Sec. 3-1]

These equations relate the coordinates of a given point before and after a translation of axes (changed origin but fixed vectorial system). To be compared with the coordinate change caused by changing the coordinate system is the coordinate change brought about by

moving the point but keeping the coordinate system If the point X undergoes a translation, rep fixed. resented by the geometric vector d, to the position X', then x' = d + x, so that x'

= d

+x

«'(

= *<

+ ii

Here it is understood that the coordinate system remains fixed as (0; e<). Let e'i, . . . , e'» be any n linearly independent vectors in the same n-dimensional space. Then any 0' Fio. 3. Translation of axes. vector x = 2{ie< is also expressible in the form z = 2£',-e'i. But any e',- is expressible as a linear combination of the e,, and likewise any e< is expressible as a linear combination of the t'j. Hence for some ttj and «'/», e', = Xtete't,

2,e',t,i

e< =

(1)

Hence

SiXten'tii'i

h

= 2,«'*,«'y

t'i

follows that

=

From these relations arise the rules of matrix algebra. written in a square array as follows:

/til

• •

«12

(2)

Z<«i(2)

Let the numbers

t,-,- be

«ln\

is

is

is

a

the first subscript being constant along Then the row, the second along a column. the product of the array, regarded as a matrix E, by the column arithmetic vector x column arithmetic vector x' according to Eqs. (2). then If E' the matrix of the the product of E' by z' Hence x.

E'x'

x =

x' = Ex

(4)

multiplication rules

e

e' = eE'

,

.

,

e'

Let e stand for the arr,ay (ei, . . . e„) and for the array (e'i, . . e'„). That is, these are row vectors whose elements are geometric vectors. Then, by the same e'E

=

(5)

matrix can be thought of as made up of column vectors or of row vectors. To multiply matrices one multiplies the first matrix by each column of the second or else each row of the first matrix by the second matrix. Thus the matrix whose . columns are . =

-

(/.,/.

(E/„ Eft,

is

U)

.

.

,

EF

then

.

F

:

.

if

F

A

Efm)

Matrices

need not be equal.

- (A B) +C = (AB)C - aC + PC +

+ (B + C) A(BC) + 0)C (a

-

+ + A

B

A

B

= + + AC A{B + O = AB aB aC a(B C) But note that in general AB and BA

A

it

are added by adding corresponding elements, as with the addition of vectors. scalar multiplies matrix when multiplies every element. The following commutative, associative, and distributive laws are satisfied: a

A

+


It

x = Zie,£, = ZiZ,e' = Eye',*', =

3-10

MATHEMATICS

Since

e'E

e =

it follows that the

=

=

(eE')E

effect of the product matrix

Hence

E'E

the identity matrix, and

E'

[Sec. 3

e(E'E)

E'E

upon e is to leave it unchanged.

/

=

is said to be the reciprocal or inverse of

E'

- E' - E~l

E:

Both designations F' and E~' are in use, and both will be used here, according to convenience. It is also true that

E

EE' =7

- E"

The effect of changing the vectorial coordinate system is simply expressible now in terms of matrices and arithmetic vectors: Let

eV

x = ex =

be a geometric vector represented by the arithmetic vector x in coordinate system (e)

and x' in coordinate system (e').

Let

e =


relate the two systems.

e'E

e' = eE>

Then the representations are related by x' = Ex

- E'x'

x

This gives the result of changing the vectorial coordinate system. Suppose, on the other hand, that the coordinate system remains fixed but every point in space is dis placed in some fashion. The result is a point transformation, and it is linear provided there exists a set of vectors (f) such that any vector x = ex is transformed into x* = fx. That is to say, the arithmetic vector x that represents x in the system (e) represents x* in the system (f). In particular e< is transformed into fi for each * «. 1, 2, Let f = eF . Then x* = ix = eFx = ex*

-

....

Hence x* is represented in the system (e) by the vector x* = Fx

Now let the of (e).

same transformation

Then x — e'x' x* = eFx =

be referred to the coordinate system (e') instead

x'

Ex

—

e'EFX

=

e'EFE'x'

= e'x'*

Hence the arithmetic vectors x' and x'* that represent x and its transform x* in the system e' are related by

x'*

=

EFE'x'

The matrices F and EFE' are said to be similar. To return to the system (e), if F is any matrix, then F defines a linear vector trans formation if the vector x* = eFx = ex* is made to correspond to any vector x = ex. Thus, with respect to a given coordinate system, every linear vector transformation is specified by a matrix and every matrix defines a linear transformation. It was not required that the system f = eF of vectors into which the coordinate vectors transform should be itself a coordinate system, i.e., that the vectors fi be linearly independent. If the vectors f< are linearly independent, then the vectors e< can be expressed as linear combinations of the U, then the matrix F' exists, and e

- fF'

3-11


Sec. 3-1]

In that event any vector x* is the transform of one and only one vector z, where, in In this event the matrix F and the fact, if x* = ex*, then x = ex with x = F'x*. transformation are said to be nonsingular or of rank n. If the vectors U are linearly dependent, then there exists an arithmetic vector x ^ 0 such that fx = 0. Since f = eF, eFx = 0. But the vectors ei are linearly inde pendent, whence Fx = 0. In this event the columns of F are said to be linearly dependent, F is said to be singular, and its rank less than n. If Among the vectors f< there is a maximal set of linearly independent vectors. there are r vectors in this set, then every vector in the set (f ) is expressible as a linear combination of these r vectors and every column of the matrix F is expressible as a linear combination of the same r columns of F. The set (f) and the matrix F are both The transformation transforms every vector x in the space into said to be of rank r. a vector x* which lies in the r-dimensional subspace of linear combinations of these r independent vectors of the set (f). Multivectors

1.4

and Determinants

1.41 Bivectors and Trivectors. If the vectors a and b are drawn from the same origin, they can be regarded as forming two sides of a parallelogram or bivector, which will be designated [a,b]. Interchanging the order of the vectors will be taken to reverse the sign of t ~y\ T — C°» b J A the bivector, just as interchanging origin and termi1 nus of a vector reverses its sign. This accounts for x~& \ the first of the following relations: V° \

y'


\

[a,b] = -[b,al = [a,«b] = a]a,b] [a, b + c] = [a,b] + [a,cj

(6) (7) (8)

!<*a,b]

Flo

r

K"B-J 4

A

bivector.

The second of these is easily verified. The third provides a rule of composition con sistent with that for vectors. From these relations one proves readily that [a,a] = 0 [a,b] = [a + Xb, b]

where

X

a =

aiei +

a2e2

[a, b

+

Xa]

Moreover, if

is an arbitrary scalar (see Fig. 4).

then

-

= eo b = 0iei + /32ej = eft = [a,b] |o6|[ei,e2]

where I I

°" at

01 I

Pi

|

_

a

a

_

a* I

I W1 1 Pi

Pa I

is called the determinant of the arithmetic vectors o and 6. If the vectors a, b, and c are drawn from the same origin, they can be regarded as forming three edges of a parallelepiped or trivector, which will be designated [a,b,c]. Interchanging any pair of the vectors will be taken to reverse the sign of the trivector. This and the other two relations given below form a natural generalization of the case for bivectors: [a,b,c] [Xa,b,cj [a, b, ci + cs]

One verifies readily that [a,b,c]

Moreover if

then

= [a

+

= -[b,a,c] [c,b,a] [a,c,b] = [a,Xb,c] = [a,b,Xc] = X[a,b,c] [a,b,ci] + [a,b,Ci]

=

■• ■

-

-

[a,a,c] = [a,b,a] = [a,b,b] 0 Xb, b, c] = [a + Xc, b, c] = [a, b a = aiei + cuaes + asej = ea b = 0iei + foej + 0»e3 = eb c = 7iei + 7ie2 + yie» = ec [a,b,c] = |a6c|[ei,ej,e,]

+

Xc, c] =

(9) (10) (11)

3-12

MATHEMATICS

where \abc\

at

=

=

ft

at

ft ft

«i

at

5.

ft

7i

71

en

71

yt yt

ft °l ft

—

«3

ft

yt

7s

ft

71

71

ft

71

P»

73

<*i

at

7J

71

„

or?

7.

ft

[Sec.

-ft

ft ft +

71

ai

ft

3

71 71

oct

ft

The two-by-two determinants are called minors. In general, when any row and any column of a determinant are deleted, the resulting determinant is a minor of the The first line of this continued equation defines the value of the determi original. nant by giving an expansion. Note that by virtue of Eq. (9) other expansions are possible, for example, |aic| =

—

|6oc|

OTl

= —

_P

ai as

I

VI ai

yi

as

71

'

+L

«, at

a ft

- , 73 !

or,

a ft «i

71 71 I

In the complete expansion

more easily illustrated in particular than stated in general.

each term contains exactly one element from each column and exactly one from each row. For a fourth-order determinant one can write |aocd|

aiAi ftBi

=

a,A, + ftB, +

where the quantities Ai, Bit Ci,

ft ft ft At

Z>
71

*2

73

4s

74

*4

Bi

=

-

at at at

*2

71

7i

St

74

6t

*4

atBt

ftB,

a%B% -|- atJBt = «,£>2 = 7iC2

0 0

aiA,

+

a\B\

+ (3tA, + (3,At + t3,A, = 0 + +

/Mi

Likewise

respectively, the cofaclors of ai, ft, yi, Si, and

it follows that

-f-

= 0,

+«ii>i

71

+

|66crf|

71C1

71

=

74

Since

+ ajAt + ajAj + a>At + PtBt + |3jJ5i + ftB<

= =

Properties (9) to (11) for multivectors apply directly to determinants. third-order determinant, one has \a,b,e\

"

-\b,a,c\

=

•

ple, for

For exam

a

• ■

1.43 Products of Determinants of the Same Order. In general, let the n vectors t\i, a2, . . . a„ in n-space be expressed in terms of the vectors e< by the relations ,


The next line in the above equation asserts that the transposed determinant has the same value and permits still other expansions. According to Eq. (9), an interchange of columns in a determinant leads to a reversal of sign. If one first transposes, then interchanges columns, then transposes again, the result is an interchange of rows in the Hence an interchange of a pair of rows leads to a reversal of sign. original. 1.42 Multivectors. One can readily carry on to four-vectors, five-vectors, . . . , and for each case write down the basic relations (9), (10), and (11) and obtain the determinant. The rules for manipulating and expanding determinants are much

algebra and geometry

Sec. 3-1]

3-13

or, in matrix form,

•

Then the multivectors

are related by [»i

the determinant

where

•

eA

a =

= [ei,e,, . . . ,e„]|A|

a»]

\A\ = |oi,Oj,

fA'

a =

A'

. . . ,a„] =

[a,,a,, [e,,e,,

. . . ,On\

fE

e =

Let so that Then

-

. . . ,e„]

-

[f,,f,, [f , ,f,,

EA .

,U]A'\

. .

. . . ,f„]

E\

By comparing these identities one finds

|A'| = \EA\ = \E\

■

\A\

-

-

|/|

1

E

1

|/|

Thus the determinant of the product of two matrices is the product of their determi nants. In particular, since = = A~l, therefore when |A~»A| =

is,

Hence

=

|A-'|-|A|

|A|-< is

a

it

the determinant of the inverse of matrix (when the reciprocal of exists) determinant of the matrix. 1.14 Linear Dependence and the Solution of Equations. Let the m vectors = eai = . . . m) be such that the multivector

that

—

[y

. .

.

[y,a,,

•

• •

+

am£m

{2a2 — .

■

• — £».a,»,ai,

,a„]

= ii[ai,a2,

,a„] = {,[a,,a2

a„]

and likewise

a„l = {ataâj,

[ai,y

Suppose

Then

. . . ,am].

- [fiai.a,,

Thus

+

•

[y,alp . . . ,a„] =

= a!?,

.

[y,as,

be linearly dependent.

.

and form the multivector

y

vectors a; and y

+

1

and let the m

^

am]

0

,

1,

(»

li

[ai,a,

. . .

... . .

.

,a«,]

,aj

,a„]

if

.

if

. .

is

y

is

is

Each of the multivectors on the left a scalar multiple of that on the right, and to evaluate this scalar multiple one can take any minor on the left and the corresponding minor on the right. This Cramer's rule for solving linear systems. The arbitrary vector resolved along the vectors ai am, and these must therefore be linearly independent. In fact, any m vectors ai, . ,an are linearly dependent and only the multivector [ai.aj, . . ,am] vanishes. 1.5

Metrics is a

is

B

A

If

and perpendiculars are drawn from points to given line, intersecting this A' and B', respectively, then the segment A'B' the orthogonal projection of AB on the given line. The orthogonal projection of a broken line made up of seg ments AB, BC KL simply A'U, the same as the projection of the straightline at

line segment AL. = ex and Let the

Therefore

y

= ey be two vectors. The scalar product of these vectors xy product of the length of either vector into the projection of the other upon it.

x

is


the

xy = xZew = J(ie,)i?i = ZSf,e,e,)j,-

3-14 Let

MATHEMATICS

[SEC. 3

7,7 = e.ej, and let G = (7,,) be the matrix of the 7,7.

Then

xy = xTGy = yTGx

(12)

Each element 7,, of G is the scalar product of two of the coordinate vectors e* and e,, and in particular, for i = j, yu is the square of the length of e,-. The matrix G is said to represent the metric of the space. If each e; is of unit length and orthogonal to all other e,-, the vectors e,- are said to form an orthonormal basis. In this event G = /, xy = xTy = yTx. In general the matrix G can be written

G Let

= eTe

(13)

IE

e =

(14)

Then

represent a coordinate change.

ETHE

G = ETiTtE =

H

where

Ff

=

(15) (16)

represents the metric corresponding to the base f. If f and e are both orthonormal, then O =

H

=

I

1 =

ETE

=

EET

-

matrix.

orthogonal

\E\ = ±1

1

|/|

\E\* =

if

0,

G is

1.

A

x it is

G is is

0,

G is

-

e*

is

to say,

if

is

/

e*V

That

the ith vector of the system («'), then

(e) and (e') are said to be reciprocal. = ey = e'y', follows that

Let

e' = efi'.

Then

it

y

x

The coordinate systems = ex = e'x',

y'

y

is

x

is

If

is

E

G

is

0

if

if

G

if

a

is

G,

is

and the determinant of an orthogonal matrix either +1 or — xx = xTGx Since, for any metric squared length, essentially positive and, = 0. the e,- are linearly independent, can vanish only when matrix sym metric GT = G; a symmetric matrix for any x, xTGx > positive semidefinite Any matrix expressible and xTGx = then only when x = -positive definite. = ETE in the form positive semidefinite, and conversely, any positive semidefinite matrix can be so expressed. If nonsingular, then posi (and hence G) tive definite. 1.61 Covariant and Contravariant Representations. x = ex, the arithmetic vector x sometimes said to be the contravariant representation of in the system (e) =■Gy, said to be the covariant representation. If = ey, and the vector x' = Gx then xy = x'Ty = xTy' = y'Tx = yTx'. Let the system (e') satisfy

if

xy

—

xree'y'

—

xTIy'

= xTy' x

is

-

j

[a,b][x,y]

ax

bj

ay

^

.....

I

is

x

y'

and y. Hence x' and are the covariant representations of and for any vectors contra Hence covariant with respect to a system (e) representation that y. variant with respect to the reciprocal system (e'). The product of two bivectors [a,b] and [x,y] will be defined as the determinant a


Then the matrix E, whose transpose is its own reciprocal, is called an Since \ET\ = \E\, it follows that

In general, the product of two multivectors of order m will be the determinant order m whose elements are the scalar products of the vectors taken in pairs.

of

3-15


Sec. 3-1]

The magnitude of a vector a is its length and is designated and Hall* = aa.

Clearly

||a||.

||a||

> 0,

The magnitude of

l![a,b]|[.

Clearly

a bivector [ab] is the area of the parallelogram and is designated ||[a,b]|| > 0, and one can show that ||[a,b]||'

= [a,b][a,b]

In like manner the product of a multivector by itself is the square of its own magnitude. = 1, and let 8 represent the angle from a to b. Choose an orthoLet |[a|| = [|b||

normal system (e) in the plane of a and b. l|e,|| = ||e,|| Let a =- eo, b = e6.

Then

-

Then by definition sin 8 = \ab\

sin

= 1

||[e„e,]|[ 8

and cos

8 are

"

cos 8 = aTb

the quantities (see Art. 2)

ab

More generally, if
- IWI


\xy\

• llyll

sin

xTv = IMI

*•

• IIj/II cos *>

The bivector [x,y] may be regarded as a kind of product of the vectors x and y. This is called the outer product, and the scalar product xy is sometimes called the inner product of x and y. Neither the outer product nor the inner product is a vector. Ip 3-space, but only in 3-space, one can define the cross (or vector) product

z=xXy as

follows:

If

=

It follows that if w =

-y

X

x

the system (e) is orthonormal, then

fi

and

=

£»

H

£•

v>

/. {'

_

£s

11

£1

vt

j.

s'

{1

iji

£.

t»

xz = yz *» 0

etc is any vector in the same 3-space,

then

zw = \xyw\ 2

TRIGONOMETRY AND COMPLEX NUMBERS 2.1

The equation x* +

=0

1

Complex Numbers and Vectors cannot be satisfied by any real number, but if one defines t =

will be satisfied by either i or by — i. Numbers of the form x + iy, where x and y are both real, are called complex num bers. If one requires such numbers to satisfy all the ordinary rules of algebra, it can be shown that every algebraic equation of degree n has exactly n roots of this form. If x = 0, a number is a pure imaginary. From the definition it follows that the equation

J*" =

1

=

J

j«»+S =

_1

j4*+3

=

_j

(17)

where n is any integer, positive or negative. This suggests a geometric interpretation of complex numbers as follows: With any origin O, a horizontal (real) axis and a vertical (imaginary) axis represent x + iy by the terminus of the vector (x,y) or by the vector itself. Then the sum of x + iy and u + iv is represented by the vector sum of the vectors that represent these numbers. The product i(x + iy) is obtained by rotating the vector x + iy counterclockwise through 90°. The vectors u(x + iy)

3-16

MATHEMATICS

[Sec. 3

sine

v'eose, AW-e) ;

.

»in(-e)l

i

o

Fio.

u

A

geometric plex numbers. 5.

interpretation of com-

Fio.

6.

and vi(x + iy) will be obtained by stretching the initial and rotated vectors by factors u and v, respectively, and the final product

-

-f

vy)

i(vx + uy)

is the sum of the stretched vectors. Geometrically (Fig. 5) the construction is as follows: Let U, P, and Q be the points corresponding to 1 + 0 • i, x + iy, and w + iv, and let W correspond to the product. Then the triangle OQW is similar to the triangle OUP [cf. Eq. (21)].

Let

r = Vac' +

V1

> 0

represent the length of the vector x + iy, and let 0 represent the angle measured in a counterclockwise direction from the real axis to the vector. Then one defines the quantities cos 0 and sin 0 by sin

y/r

0 =

cos 0 =

x/r

These definitions are equivalent to those given in Art. 1.5. These are functions of the angle 0 alone, their values being fixed by 0 and independent of r. Clearly sin 0° = 0 cos 0° = 1

sin 180° =0 cos 180° =

sin 90°

and, in general, for any

0

-1

By taking r

= 1 and applying

< sin

0 < 1,

1 < cos 0

<

(18)

1

the Pythagorean theorem one has

sin' for any

1 sin 270° cos 270° = 0

-1

cos 90°

0

+

cos* 9 = 1

(19)

0.

Any complex number can

be written in either of the equivalent

- r(cos

x + iy

0

+

i

sin

forms (20)

0)

called, respectively, the rectangular and the polar forms. One calls r the radius vector 0 the amplitude or argument, x the real part, and iy the imaginary part of the number. From geometry it is clear that in multiplying two complex numbers the modulus of the product is the product of the moduli and the amplitude of the product is the sum of the amplitudes of the separate factors (cf. Fig. 5): sin if + t'(sin

the final form being given by direct multiplication.

sin

(0

rs[cos

sin

i

—

—

cos if

+ ?)] + cos

0

cos

+

i sin

0

= rs[cos

+

sin 0)8(cos

*>)

i

(9

+

0

0

if

r(cos

or modulus,

0


(u + iv)(x + iy) = (ux

sin

*>)]

(21)


Sec. 3-1]

3-17

of results follow from this identity. Let r = s = 1, and take succes The results can be summarized in the following equal to 90, 180, 270, 360°.

A number sively table:

sin cos

90° + 9

180° + a

270° + e

360° + 9

cos 9 — sin 9

— sin 9 — cos 9

— cos 9 sin 9

sin 9 oos 9

(cos 9 + i sin 9) (cos 9 — i sin 9) = 1 (cos 9 + i sin 9)[cos ( — 9) + » sin ( — 9)] = 1

Since and

that

it follows

cos ( — 9) = cos 9

sin

(-9)

= — sin 9

(22)

This is illustrated by Fig. 6. Hence the above table can be supplemented:


90°

s =

1

9

180°

cos 9 sin 9

sin cos

Still with r =

-

-

9

270°

-

9

360°

— cos 9 — sin 9

sin 9 — cos 9

-

9

— sin 9 cos 9

in Eq. (21), on comparing the second and final forms one has sin (if + 9) = sin + 9) = cos
(23)

On referring to Eq. (22) one has also

H

9 =

sin (y> — 9) = sin ^ cos 9 — cos cos 9 + sin
n

sin 29 = 2 sin 9 cos 9 cos 29 = cos' 9 — sin1 9 = 2 cos'

If

9 in

(24)

then

9—1 =1—2

sin* i

(25)

Eq. (25) is replaced by 9/2, one obtains sin (9/2)

=

±

- cos

V(l

9)/2

cos (9/2) = ±

+

cos 9)/2

(26)

Given any complex number z —

x + ty

™

r(cos

9

+ i sin

9)

i sin

9)

the conjugate complex number is

I

= x —

iy

= r(eos 9 —

Note that the product of a number by its conjugate is the square of its modulus, sum of a number and its conjugate is twice the real part, z + z = 2x. The sum, difference, product, or quotient of two conjugates is the conjugate of the sum, difference, product, or quotient of the numbers themselves. zz = r»; the

MATHEMATICS

3-18

The Trigonometric

2.2

Along with the functions sin

9

and cos 6 =

tan cot

Functions; 9

Basic Identities

already defined, one can form four others:

y/x

9 /cos 8 =

sin

9/sin 8 = x/y = 1/cos 9 = r/x = 1/csc 8 = r/y

9 — cos

sec 9 esc 8

where, as usual,

[Sec. 3

* + iy

(27)

= r(cos 8 -f- t sin 9)

These satisfy the following identities: + tans + cot*

1 1

8 = sec5

8

0

9

- csc«

(28)

The first is obtained from the complex number 1

+

8 = sec 9(cos

tan

*

8

+

i

sin

9)

the second from the complex number


cot

8

+

i

= esc 9(cos 8

+

i

sin

9)

Complete reduction formulas for all the functions can be obtained by applying those The following table summarizes: derived for sin 9 and cos 9 and the above definitions.

-8 sin COB tan

cot sec CSC

The quadrant of an angle

- sin 6 cos B — tan 8 — cot 8 sec 8 — esc 8

9 is

90° ± 8

180° ± 8

cos 8 +■sin 9 T cot 8 ¥ tan 0 T esc 9 sec 0

+ sin 9 — COB 0 ± tan 8 ± cot 0 — sec 8 + esc 8

9'

— cos ± sin + cot + tan ± esc — sec

8 0 8 8 8 0

defined by first expressing it in the form 8 = 8'

where 0 < of 9 is

270° ± (9

+n-

360°

< 360° and n is any integer, positive or negative.

I II III IV

if if if

0 < 9' 90° < 9' 180° < 9' 270° < 9'

if

Then the quadrant

< 90° < 180° < 270° < 360°

The algebraic sign of each function is constant throughout any quadrant according to the following table:

sin cos

tan oot sec 080

I

II

+ + + + + +

+

— — — —

+

III

IV

—

+ — —

_ + +

— —

+

Sec. 3-1]


3-19

Since adding or subtracting integral multiples of 360° has no effect upon the value of any of the trigonometric functions, these functions are said to be periodic with period Actually the tangent and cotangent have periods 180°. 360°. Exact values of the functions of certain particular angles can be obtained by elemen tary methods (in an isosceles right triangle each acute angle is 45°; in an equilateral triangle any altitude forms congruent right triangles) :

sin

0

cos

1

tan cot

so

0

30°

45°

60°

H

\,Vl

Vi/2

■

120°

1

V3/2

«

Vi/2

I/V2

VI

1

Vi

00

1

1/V3

0

2

00

2/Vj"

1

V2

1

sec

90°

Vi

2

-Vi

\/V2

H

-1

-1

-1/V5

-2 2/Vi

V2

00

the reduction

(29) (30) (31)

9)

(1

9)

-

+ + +

[(8

-1

2

(32)

-

9

9)
(33)

9

(9

2

cos )/2] —2 sin [(9 + *>)/2] sin [(9 »>)/2] sec sec v tan
cot ± ±

9

tan cot

9

+

cos

9

9

9

2

9

2

+

2

-

0 30

-21 VI

III and IV by applying

tan (6 ± = *>)/2] — sin

-VI

1

0

is

a

is

unit circle intercepted The radian measure of an angle the length of the arc of by the angle when the center coincides with the vertex. Since the complete circum 2r, the radian and degree measures are related by ference radians = 180° of length

s,

is

if

the intercepted arc

(34)

the radian measure

9

is

For a circle of any radius of the angle

r,

t

=

s/r

(35)

e

...

(ri)

exp

9

9

sin

= cos sin = [exp (t'9) + exp (-»9)]/2 = — i[exp (t'9) — exp (-i'9)l/2 9

t

9

exp (i9) cos

1

=

follows that +

it

ewi

from which

= —

i

is

a

is

is

a pure number. Hereafter, unless otherwise specified, angles in the formulae and this will be understood as given in radian measure. the sum of the The rule that the amplitude of product of two complex numbers the same as the rule for combining logarithms amplitudes of the factors [Eq. (21)] = 2.718 to Hence one can define the real number (Art. 1.1).

(36)

is a

e

convenient logarithmic base, and logarithms For theoretical purposes are known as natural logarithms and abbreviated In.

to the base

exp

+ i9) = exp

[p

iy

sin

(p

+

can be written 1

= r(cos

*

x +

9

complex number x

iy

Any

+

e


180°

-VI

- \/Vi

The table can be extended through the quadrants formulas. Other common multiple angle formulas are

150°

-I/V2 -Vi/2

-H

0

135°

9

0°

+ i(9 + 2«r)]

(37)

3-20

MATHEMATICS

[SEC.

where p =• In r is the real natural logarithm of the positive real number r. also say that In (x + iy) = p + i(0 + 2nr)

3

One may (38)

where n is any integer, positive or negative. The natural logarithm is defined, there fore, only up to integral multiples of 2xt. Inverse trigonometric functions are also multiple-valued. By definition if fen represents any of the six trigonometric functions, then 9 =

Thus arcsin

}£

if

arcfen u

u = fen 8

may have any of the values

r/6 5t/6

± ±

2m 2nr

i.e., 30° ± n • 360° i.e., 150° ± n • 360°

and arccos }£ may have any of the values ±

\o

± 2nx

one defines the principal value of an inverse function, A common con designated by a capital letter, to be just one of the possible values. vention is as follows: — jt/2 < arcsin u < ir/2 0 < arccos u < -r — t/2 < arctan u < t/2 ,„Q. (M> < arccot u < r/2 0 < arcsec u < x


To remove this ambiguity

-t/2 —t/2

< arccsc

u <

r/2

the symbols sin-1 u, cos"1 u, . . . are some In place of arcsin u, arccos u times used. The hyperbolic functions are defined by applying real exponents to the exponential:

,.-.

x — exp (— x)j/2 x + exp (— x)]/2 sinh x/cosh x cosh x/sinh x 1/cosh x = 1/sinh x

sinh x — [exp cosh x =■ [exp tanh x = coth x = sech x =csch

They

are

related by identities

i

analogous

identities:

/..■>

to those satisfied by the trigonometric

-

cosh1 x — sinh* x = tanh* x = 1 coth* x — 1 = sinh (2x) = 2 sinh x

1

sech' x csch* x cosh x cosh (2x) = 2 cosh* x — 1 = 2 sinh* x + 1 — cosh* x + sinh* x sinh (x/2)

= -y/(rosh x

cosh (x/2)

=

The circular (or simple trigonometric)

follows:

sin 0 = — i sinh (i6) cos 9 = cosh (i$) tan 9 = — i tanh (id) cot 9 = i coth (ifl)

V(cosh

-

x +

(42)

,,„.

(

l)/2 l)/2

(44>

and the hyperbolic sinh x =

—

i

i}

sin

functions are related as (t'x)

cosh x = cos (jj) tanh x = — i tan (uc) coth x = i cot (ix)

,,-..


Sec. 3-1]

3-21

The Solution of Triangles

2.3

Of the six elements in any triangle (three sides and three angles), given any three of which at least one is a side, the other three can be computed by means of tables of trigonometric functions. In case the triangle is a right triangle, let the two legs be x and y, the hypotenuse r. Then the acute angle adjacent to the side x is 8, where

+ iy

x

The other angle is the complement of sin

8

-

8)

Of the four formulas

8.

y/r - x/r

cos 8

+ i sin

= r(cos 8

x1

tan 8 = y/x + y1 = r'

any given pair of the elements x, y, r, and 8 will be found in exactly two, and each of these can be used to give one of the others. If and y are given, the Pythagorean theorem may be incon venient to use for finding r. An alternative is first to find 8 and then use one of the two formulas

i


r = y/sin

8 =

x/cos

8

An isosceles triangle can be divided into two congruent For an arbitrary scalene triangle, let the right triangles. angles a, 0, and y be, respectively, opposite the sides a, b, and c Let R be the radius of the circumscribed circle (Fig. 7). Then the law of sines is The law of

The

cosines is

2R = a/sin a = 6/sin 0 = c/sin y

-

a* = 6' + c* — 26c cos a 6" c« + a* 2ca cos 0 c* =» a* + 6* — 2ab cos y

-

fam of tangents is (6

(c (a

Let r

- c)/tan - 7)/2] - a)/tan - 6)/tan [(a - a)/2] 0)/2] -

= (6 + c)/tan

[(f)

[(7

(c

(a

[(0 + 7)/2] + o)/tan [(7 + a)/2] + 6)/tan [(a + 0)/2]

be the radius of the inscribed circle, and let s = (a

be half the perimeter of the triangle.

+

6

+ c)/2

(49)

Then

- o)(s - 6)(s - c)/*]M r = - 6)(g - c)/(&c)]V* sin («/2) = cos (a/2) = [s(s - a)/(6c)]tt [(«

and

(48)

(50)

[(«

tan (a/2) = r/(s

with

— a)

(51)

corresponding formulas for the functions of 0/2 and y/2.

While space does not permit giving the derivation of all these formulas, it may noted that some are direct interpretations of vector formulas. Thus let c = a — b cc = aa — 2ab + bb

Then which has the interpretation ||C||«

where y

be

=

||*||«

-

is the angle between a and b.

2||a||

•

||b|| cos

This is

y +

|tb||s

one of the forms of the law of cosines.

3-22

MATHEMATICS

[SEC. 3 Then

= ab sin

-

c]

—

c, b

[a

=

y

b, a

sin a = ca sin

[c

6c

which proves that

—

b)

—

(3

=

—

a]

— a,

c

[b

.

||c

Again let a, b, and c be vectors to the vertices A, B, and C of the triangle. — b||, . . . One verifies that a =

= 180°

y

+

0

a + a, then find

2R

=

0

b

= 2R sin

o/sin a c

and thence

= 2R sin

y

the given side

is

If

a

1.

is

This equivalent to the usual statement of the law of sines. For solving a triangle the following cases are to be distinguished: The third angle can be found from side. Given two angles and

2. Given two sides and the included angle. Method a. Use the appropriate form of the law of cosines to find the c,

7)/2]

y)l2 one

can find

Let

(a/2)

Thence use the law of sines

and y.

these be a,

and a.

From the law

= (6/a) sin a

and thence by another application of the law of If There are possible complications, however. sin a > a no solution.

If

b

sin a < a <

is

impossible and there 6

the triangle

is

b

sines.

c)] cot.

c

and a, one finds

y

0

Knowing both

- c)/(b

nonincluded angle. sin

(a/2)

+

7)/2] = [(6

0

-

0

a

(0

— Knowing >)/2 and to find a. 3. Given two sides and of sines

(0

tan \(H

+

Hence the law of tangents yields

- a/2

= 90° = cot

third side,

Then

6,

+ y)/2

+

tan [(0

cos a = (6s

+ cl

(3'

6. a

0,

8

is a

possible There two solutions are possible (Fig. 8). = 180° — (3, and possible obtuse angle acute angle either of which may be the angle opposite side One may find a from the law of cosines: 4. Given three sides.

j,

- al)/(2bc)

Correspond

is

0

or one may use the formula for sin (a/2), for cos (a/2), or for tan (a/2). or 7. ing formulas can be used for The area of triangle a

Area = (,H)bc sin a = rs = [*(» 2.4

- a)(s

—

b)(s

—

c)]>4

De Moivre's Theorem and Other Identities

i

sin $)n = cos (n0)

+

i

(cos

+

Repeated application of Eq. (21) gives 6


(0

b

b.

then the law of sines to obtain another angle. and the angle a. Suppose the sides are Method

sin (n0)

(52)


Sec. 3-1]

If

9

3-23

is replaced by 9/n and the nth root extracted on both sides, it follows that (cos 9 +

i

i

sin 9),A* = cos (9/n) +

sin (6/n)

However, if p is any integer, cos 8 ** cos (9 + 2pr)

Hence

i

(cos 9 +

9 =

sin

sin 9)"» = cos (9/n + 2px/n)

sin

i

+

(9

+ 2px)

sin (9/n + 2pir/n)

(53)

where p is an arbitrary integer. This is a special case of Dc Moivre's theorem. Exactly n distinct values are obtainable on the right by taking p = 0, 1, 2, . . . ,

*

— 1.

When 9=0, Eq. (53) defines the n roots of unity. These roots include 1 in all cases, — 1 when n is even, but no others are real. The root w denned by taking p = 1 In fact, if u = ui and is a primitive root, all roots being powers of this one. up = cos (2xp/n)

then clearly

and since

ap —

«i»

X' -

&> ^ 1,

— 1

=

it must

(x

Any

»' sin

= exp (2pxt°/n)

(2xp/n)

(54)

up satisfies

- l)(l"-'

+

I""'

• • •

+

+

X

+ I)

=-

0

be true that

w.-i + „»-* + «„_i + w„_j +

or Generated for wjivans (University of Florida) on 2015-09-23 02:46 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

v*.

+

. . . • • ■

Expansion of the left member of Eq. in terms of sin 9 and cos 9:

1 = 0

+ u + + m +

1

-0

(55)

(52) permits the expression of sin (n9) and

cos (n8)

sin (nd) = n sin cos (n9)

It

- cos"

9 cos"-1

9 —

sin'

9 —

sm* 9 cos"

J

9

cos""*

9

+

• • •

(56)

+

9

• - •

is possible also to express sin (n9) /sin 9 and cos (n9), as polynomials in cos a recursion. Note, first, that

9 alone

For this we deduce

cos 9 = cos 8 2 cos2

Now by Eq.

(32) we have

cos (29)

-

9-1

cos (n9 + 9) + cos (n9 — 9) = 2 cos 9 cos (n9)

Hence when cos (n9) and cos [(n cos [(n + 1)9]. With n — 3, 4,

—

...

1)9] have been so expressed, we obtain sequentially

one

can obtain

cos (39) = 4 cos' 9 — 3 cos 9 cos (49) = 8 cos4 cos»

9-8

9-1

Next, and by Eq. Hence

sin 9 — Bin 9 sin (29) = 2 sin (32) Bin (n9 + 8)

+ sin

(n9

—

9)

8 cos 9

= 2 sin (n9) cos 9

9-1)

Bin (39) = sin 9 (4 cos1 sin (49) = sin 9 (8 cos" 9

-

4 cos 9)

When expressions involving trigonometric functions arc integrated analytically The formulas 1.2), it is desirable, when possible, to replace products by sums.

(Art.

3-24

cos

—

(0

4- cos

+ sin

— —

)

+

+ +

[SEC. 3 (9 (9

— cos (0

+

*>)

sin

= sin cos cos

(9 (0

cos

0 0

sin sin

9

2 2

2

MATHEMATICS #>)

(57)

+

-

40)

0

n even

40)

20) +

-

■■•

cos (n0

-

n odd

40)

+

(58)

• ■

•

-

sin (n0

Parametric and Nonparametric Representation system (0; e),

the coordinates

of the vector x = ex

£<

x

are unrestricted, then the vector can represent any point in the space. are required to satisfy some equation

,«.) =

the

(59)

0

• • ■

¥>(fc,fc,

But

if

Given a point-coordinate

is is

is

,f.V({i

■■

•

£0

{,*

=

-

- ««,

(|i

-

follows that the curve I,1

=

?2'

+

£«)

0

Thus, in the plane, since

0

this event the surface consists, in fact, of two surfaces
In

,£.) =

0

possible.

is

it

3,

|,-

is

A

*>(«■,{., - - -

it

2,

if

x

only those vectors which terminate on some hypersurface will be allowed and Eq. In fact, all coordinates but are (59) will be the equation of this hypersurface. must be such that Eq. (59) satisfied. When n = the "hyper specified, then surface" an ordinary curve; when n = an ordinary surface. Certain simple geometric properties of the surface can sometimes be deduced directly from Eq. (59). one for which the factorization degenerate surface

+

2{a

=

= 0.

£i

0

£2

+

0 is

£i

£i«

=

3

£i

—

=

with respect to the hyperplane present in case Thus the curve without affecting the equation.

Symmetry

by

£,

=

0

-

£j

£1

consists actually of two straight lines, can be replaced

-

0

£2J =

1

is

£<-£/-•■• +

symmetric with respect to the origin

£1

£i* +

=

£2

Thus the curve

=

0.

...

{i

is

When the simultaneous replacement of symmetric with respect to the line by — £,, respectively, leaves the equation unaltered, the sur £/, face symmetric with respect to the linear space defined by £i,

is


• ■•

LOCI: CURVES AND SURFACES

if

3.1

0

p

-

-

cos (n0

- 20)

(n0

= cos (ra0) + n cos (n0

3

0

2"-1 cos"

20)

(")

- n sin

= sin (n0)

-

(n0

(^)

sin»

- n cos

+

_(l)(»-i)/«2»-i

= cos (n0)

0

sin"

0

_(l)«/i2»-i

a

9

0

9.

can be applied repetitively to break up products of sines and cosines. They also and cos permit decomposing powers of sin This can be done recursively [i.e., < n, one can form cos as sum of terms in cos (p9), having expressed cos" cos" and apply Eq. (57)]. Otherwise, one can apply the following formulas:


Sec. 3-1]

3-25

Asymptotic cylinders may reveal themselves when one solves Eq. (59) for some Suppose the solution has the form coordinate £, as a function of the others. =

«i/ft £,•

ii

that they

is

and are such where wi and ft arc independent of the particular variable constant, the equation Unless ft do not vanish simultaneously. 0

ft

=

a

-1=0

iiit

£2:

1/f.

VV

+4

=

4

h

=

±

2{i

-

{,«

=

a straight line

"cylindrical

in the solution for some as a assign values of these variables

0

it

fi

-

4{,» can be solved:

an asymptote; the second that

is

0

is

is

0

£i

£»

The first solution shows that the straight line = an asymptote (in the plane the straight line surface"). shown to be limited when, The extent of a surface possible to function of the other variables, one finds Thus the equation complex. that would render

fc

=

a

=

is

l/{.

±2

V*i*

-

1

=

«s

Zi

and for

&

is

is

£i

can be solved for

x

=

h

£i 1,

h

A

1,

1 £i

£i

1

1

fi

a

&

will yield real value of &; on the other hand, unless fj* > Any real value of < — > then would be complex. Hence no value of or else i.e., either will be found on the curve. on the range — < < It sometimes happens that symmetries, asymptotes, etc., are not apparent in the equation as given initially but that they can be revealed by suitable transformations This requires the selection of a vector of coordinates. and matrix such that on setting £i

+ Ax'

=

0

=

^

p

|;

{,-

will exhibit the features of interest. For surfaces the equation in the new variables of the second degree the theory of these transformations will be developed below (Art. 3.3). are required to satisfy two equations simultaneously, say If the coordinates

is

a

a

In the plane two the points, in general, lie on a surface of lower dimensionality. equations generally define a finite set of points; in 3-space they define a curve. In finite set of points; in 4-space they define curve. 3-space three equations define the common In n-apace, n — equations define a curve in general. This curve intersection of the n — hypersurfaces (in 3-space the intersection of two ordinary surfaces). Consider in 3-space the curve of intersection of the surfaces 1

1

=

ftto

Ei

=

ft(r)

=

Mr)

and

£»

£3,

(60)

and solve for

£i

In general one can select arbitrarily a &, say £j. The equations are then in the form

=

0

= *>«(«i,f.,{»)

£,

Vi(ti,M>)

£,


,

,

is

a

If

.

(n

— a surface in the l)-dimensional space of d, . . e,-_i, ej+i, . . . e». drawn in the direction e, through every point of this surface, one has line cylindrical surface in the original n-space, and this also has the equation ft = 0. This to say that as asymptotic to the surface of Eq. (59), which cylindrical surface becomes infinite through either positive values or negative values or both, the surface of Eq. (59) approaches the cylindrical surface. Thus the equation

defines

as functions of

(61)

3-26

MATHEMATICS

[SEC. 3

where, in this case, ^-s(t) = t. In general, when each fc is defined as a function of an independent variable r, one says the equations of the curve are in parametric form. Given equations in the form of Eq. (61), one could solve, say, for t as a function of Is and substitute into the other two equations, obtaining ft =

or, what comes to the same, £i

-

it

&>l({3)

= £»

o>,(£,)

-

= U>l(h)

*>,({,)

-

0

special case of Eq. (GO). Analogous steps are possible in n-space. Analogously a surface in a space of three or more dimensions can be represented by two parameters: ii = Mr,*) (62) ft

A curve, in a space of however many dimensions, is a one-dimensional manifold; s Either an r-dimensional manifold in an surface is a two-dimensional manifold. n-dimensional space, n > r, can be represented in parametric form: «.

n,

. . where, in particular, sented by requiring that n


*>,(«

Let

-

(63)

■»)

any r of the coordinates, or it can be repre be satisfied simultaneously:

. , rr may be — r equations {»)

3.2

= *<(n

-

• • •

¥>»--•(£,,

. . . ,{,,)

= 0

(64)

Lines, Planes, and Hyperplanes

The equation

a be any fixed vector and a a constant.

-a

ax

= 0

(65)

is satisfied by any vector z whose projection upon a is fixed and equal to a/||a|| (see Hence when z and a are drawn from a fixed origin O, x must terminate on Art. 1.5). Hence Eq. (65) is the a hyperplane orthogonal to a and at a distance a/||a|| from O. vectorial equation of this hyperplane. Given the point-coordinate system (0;e), let

Then if a — ea, x = ex, it follows that ax = aTGx. represent the metric. = a' Oa is the oo variant representation of a, Eq. (05) takes the form

-a

a'Tx

Hence

= 0

if

(66)

Conversely, given an equation in the form of Eq. (66) and a particular pointcoordinate system (0;e), let a be the geometric vector whose covariant representation is a': a = eG 'a', and let x = ex. Then Eq. (66) is equivalent to Eq. (65) and is Thus when Eq. (59) is linear as therefore the equation of the hyperplane as described. It should be observed in Eq. (66), the hypersurfacc is a hyperplane, and conversely. that Eq. (05) is independent of any coordinate system but Eq. (66) is not. In case the system (0;e) is orthonormal, a' = a and G = /. In this event

The vector

n = a/||a|| is a

unit vector in the direction

-

equation nTx where n = en, is equivalent tion of the hyperplane.

If

to Eq. (66).

—

If

v =

a/||a||, then

v = 0

This

nTy = 0

the (67)

is called the normal form of the

y = ey represents any point

nTx

a.

Y

equa

in the space, the equation

3-27


Sec. 3-1]

the hyperplane through Y parallel to the original hyperplane, and the between the two planes is nTy — v. This is also the perpendicular distance from the hyperplane of Hence, the distance S from a Eq. (67) to the point Y. hyperplane to a point Y is given by represents

distance

S =

nTy

-

v

(68)

positive direction is that of the normal n. given an orthonormal system, any unit vector n has coordinates (cos u, sin a) for some angle pIa g Hence m, this being the angle from d to n (Fig. 9). the normal form of the equation of a straight line in the plane can be written where the

In

the plane,

u +

£i cos The general

|j

sin a

(69)

equation (66) of the straight line can be written

-o

«i£i + ath

= 0

(70)

involving three parameters, an, at, and a; but in fact only two are significant, since one can multiply through the equation by any constant. Hence two geometric con ditions suffice to determine the line. Such a pair of conditions might be the unit normal n (determined by the angle u) and the normal distance v from the origin. They might be the intercepts ptei and p2ei where the line crosses the coordinate axes. H Pipi 0, then the equation is £i/pi + {j/pj = 1 This is known as the intercept form of the equation. The inclination of a line is the angle it makes with the axis of the

line is the tangent of this angle.

the slope is <• =

system is orthonormal. equation of the line is

(fi

the

and

(0; e,), and the slope and x" represent two points on the line,

If x'

- r»)/(t"i -

When the slope

when the

ii

—

when the slope n and any point

and the intercept

p2e2 are

known,

pi = 0

—

a are

p.

1'.)

known, the equation is

- at) - »(£i - ax)

({i

= 0

These are called the slope-intercept and the point-slope forms, respectively. Returning to a space of arbitrary dimensionality, let ai and a} represent any two points. If z represents a point on the line joining them, then x — ai and x — as are linearly dependent.

Hence there exist scalars at and as, not both zero, such that

ai(x

X =

or

- a,) + at(x -

a2)

= 0

(oia, + a2a5)/(ai + at)

(71)

x represents the centroid of the points a, and a2 weighted by ax and at (either or both "weights" may be negative); x represents the That the point which divides the line segment from a, to ai in the ratio at/ax. points are Ax, At, and X, then

if

is,

This has several interpretations:

equivalent to

Again,

or to

= ax/ (ax

at)

- r*, + - r)a, r(ai (1

Eq. (71)

x x

t then

at/ax

+

significant).

=

if

direction

= a2

+

'the

is

A\X /X Aj

is


— v = 0

aa)

(72) (73)

3-28

MATHEMATICS

[SEC. 3

either of which is a parametric form of the equation of the straight line through A i and Ai. Likewise x = a + tu (74)

A in the direction u. Given any coordinate system (0;e), not necessarily orthonormal, these equations That Eq. (71) becomes can be written directly in terms of the arithmetic vectors. is,

is a parametric form of the equation of the line through

+ atai) /(«i

at)

+

X = (ai
(75)

X

if

0

+

if

if

•

•

+

•

+ a,)

(76)

,

,

is

1

3.3

■■•

+ ar)

(77)

plane, since of the parameters on, possible, for example, to hold any one r

1)

—

it (r

the parametric, equation of the — are essential and

a„ only

a&r)/(ax + is

■■■

+

Conies, Quadrics, and Hyperquadrics

it

r

- 2(a -

p

+ Ut

be

vector

ApVUl

of

(79)

whose directions are given by the columns of

+

r

PUTAUt

P

a point in the plane through Substitution into Eq. (78) gives

-

pTAp

- 2aTp + a

=

0

U

x

quadric (78)

t

=

< n linearly independent columns, and let be any matrix of If represents any point P, then elements. p

r

Let

- 2a.Tx + a

0

xTAx

a

is

If Eq. (59) quadratic in all the variables, represents a hyperquadric in 3-space, conic in 2-space). Such an equation can be written in the form

(a

fixed.

«A)/(ai +

+

plane through Ah . . . Ar, and any point in this plane can A, with In fact, X the centroid of the points Au . . . a,. With any coordinate system (0;e), Eq. (76) equivalent to

r

is ,

.

. .

• • •

+

on

is

U.

(80)

lTAt

—

a' = a

where

Eq.

(80) has the form

a' = — aTp

0

case,

/,

In this

+

can be solved for p.

if U

a

is

a

is

t.

is

Hence the intersection of a linear quadratic in the elements of with hyperquadric again a hyperquadric (for brevity we shall omit the prefix and speak only of quadrics). If nonsingular, the equation Ap = (81)

This equation space

.A

;

t

t

it

is

it

A

is

is

P

is

Hence the point a center of symmetry (since replacing by — does not affect the bisects all chords through and called the center of the quadric equation). then the quadric a central quadric. If may or may not be possible singular, to satisfy Eq. (81). Consider first central quadrics.

It


—

= (cr,a,

X = (<*,a,

This

=

is

be so represented.

weights

- a,)

represented by a vector

(r

is

a point in the

«r(z

1)

point

x

a

Hence

• • ■

+

- a,)

ai(x

.

.

if

1)

(r

is

,

. .

,

z

x

. ,

r

and elsewhere each geometric vector can be replaced by the arithmetic vector that represents it. Given any distinct points A\, A%, . . A„ represented by a,, a.., . . ar, the — and only in the same plane with these points point X represented by — a, are linearly dependent, hence and only there the vectors x — at, . . . a, not all zero such that exist scalars a\, at, .

Sec. 3-1]

3-29


We may suppose the origin to have been moved to the center. the equation has the form xTAx + a = 0

Again the substitution

the columns of

U

(82)

of Eq. (79) gives

tTUTAUt + 2pTAUt + pTAp + a

If

Hence a = 0, and

= 0

(83)

are such that

pTAU

= 0

(84)

then P is the center of the quadric of Eq. (83). When the coordinate system (0;e) is orthonormal, Eq. (84) means that the columns of U are orthogonal to the vector pT A. They will be said to be conjugate to the vector p. There is a hyperplane of vectors orthogonal to a single vector pTA. Hence a matrix U with linearly independent columns and satisfying Eq. (84) could have as many as n — 1 columns. If P is on the quadric, pTAp + a = 0, and if U satisfies Eq. (84), then Eq. (83) becomes simply = 0

tTUTAUt and the hyperplane through Its equation is quadric.

P

whose

directions are given by U is tangent to the

pTA(x

- p)

= 0

or, since p is on the quadric,


pTAx + a

= 0

When the coordinate system is orthonormal and A =

/, Eq.

(82) takes the form

xTx + a = 0 which states that the squared length of x is — a. Hence for the surface to be real it is necessary that a < 0, in which event this is the equation of a (hyper)sphere (a circle in 2-space) whose center is the origin and squared radius —a. Returning to the more general equation (82), we can rephrase the property of conjugacy as follows: Two vectors u and v are mutually conjugate in case uTAv = vTAu = 0

(85)

If

u and v are conjugate, let the line through the center of the quadric in the direction u intersect the quadric at P. Then v is parallel to the tangent hyperplane at P. The same statement holds when u and v are interchanged. A vector v is not, in general, orthogonal to the vectors to which it is conjugate, but it will be if v and Av are linearly dependent: Av

- Xf

Vectors v exist satisfying Eq. (86) only if the matrix A is such that the determinant of this matrix vanishes: \A

-

—

\I is singular,

X/| = 0

This

(86)

i.e., only if

X

(87)

determinant expands into a polynomial in X of degree n, and hence Eq. (87) has n roots, not necessarily distinct. For each such root X there is a nonnull vector v satisfying Eq. (86). In fact, if X is a root of multiplicity n», then Eq. (86) will be found These can be chosen mutually orthogonal to have m linearly independent solutions. and of unit length. Any root X of Eq. (87) is called a proper value (or latent root or characteristic value or eigenvalue) of A, and a vector v satisfying Eq. (86) is called a proper vector (or latent rector or characteristic vector or eigenvector) belonging to X. It can be shown that In fact, let proper vectors belonging to distinct proper values are orthogonal.

Then But since

= itu X ^ ft Av = Xn vTAu = uTAv = iwTu = \uTv

Au

uTv = vTu, it ^ X; therefore uTv

=

vTu = 0.

3-30

MATHEMATICS

[SEC. 3

Let V represent the matrix of n orthonormal proper vectors:

VTV Thus V is an orthogonal matrix. form .A V one obtains

I

=

=

VVT

In multiplying A by each column of V in turn to

(\iv,,\tv2, . . . ,\,v„) = VA A = diag (Xi X„)

where

is formed of the proper values of A, each with its appropriate multiplicity.

A

or

=

Now let

AV

= VA

VTAV

VAVT

x = Vy

Then Rq. (82) becomes

Hence

= A

(88)

VTx = y

yTVTAVy

or, by virtue of Eq. (88),

+c

(89)

= 0

yTAy + a = 0

The substitution of Eq.

x

eVy eV

=

Z\,V

a =

0

in the especially simple form +

is

(90)

= =

In

of axes.

v

f„

0,

If

0.

A is

is

a

is

is

is

is

A

A

is

t,

if

Xi is > 0

if

±

— a/X, The quadric intersects the (0;f.) axis at the two points which are —a/X, > 0. If this holds for every real and only intersected and every axis the quadric an ellipsoid. There no restriction in supposing a < a < and the matrix then every If the inequality fails for positive definite. some Xi, the quadric hyperboloid (for methods of carrying out the reduction numerically see Art. 4.5). If the matrix singular, Eq. (81) cannot be satisfied in general but the reduction of to the form of Eq. (88) still possible, with one or more null elements in the made in Eq. (78), the result has the diagonal of A. If the substitution of Eq. (89) form 2bTy yTAy a = (91)

3.4

0

parabola; see Art. 3.4).

Loci and Coordinate Systems

the curve containing all points that satisfy a given geometric Thus circle with radius condition. and the locus of all points X(x) at a center ('(c) distance from C. Hence the vector x must satisfy the equation

(0;e),

if

system

x

—

ox

c)

- e)f(x - -

P»

= ec, the equation (x

c

In an orthonormal and

is

(x

c)

- c)(x - -

p'

p

is

p

a

locus in the plane

is

A

+

paraboloid (in the plane

a

a

surface

is

Such

a

-

X

a

1,

<

1,

<

if

a

1,

c

a

«

if

d

Ellipse.

F

10.

is

A

defined as the locus of points conic such that the distance of X from a fixed point fixed line (called the focus) and that from If the the conic in fixed ratio eccentricity). < the are directrix) (called (called = a hyperbola (Fig. 12). parabola (Fig. 11); an ellipse (Fig. 10); > Fio.

is


— ex

f

where

transformation

fj/

fact,

Equation

(90)

(89) represents an orthogonal


Sec. 3-1]

Fia.

11.

3-31

Fig.

Parabola.

12.

Hyperbola.

Let the directrix in normal form (Art. 3.2) be

nz = r and let

f represent the focus.

Then the conic is

- f)(x - f)


e«(x referred

= (nx

-

to an orthonormal system it is

- f)T(x - f)

For a parabola, < = 1,

„

it

-

= (nTx

(92)

»)*

,)*

(93)

is convenient to take

= {0,1 j

v =

Then Eq. (93) becomes

(I.

£iJ + or

-

-X/2

/=

< 0

JO.X/2}

X/2)8 = (f2 + X/2)* ti* = X£,

(94)

This is the form to which Eq. (90) reduces after a suitable change of origin when X» = 0, and it corresponds to a directrix parallel to the axis (0;ei). When Xi = 0, the directrix is parallel to the axis (0;e3) and the equation can be written

Any parabola can have its equation written in the form of Eq. (94) after a suitable rotation and translation of axes. Conversely any curve whose equation can be put into the form of Eq. (94) is a parabola. The parameter X is the length of the lotus rectum, which is the focal chord parallel to the directrix. It is also twice the distance of the focus from the directrix. The curve is symmetric with respect to the line through the focus perpendicular to the directrix [the axis (O;e0 in Eq. (90)], and this is therefore called the principal axis of the curve. The principal axis intersects the curve at a point midway between the focus and the directrix [the origin in Eq. (94)], and this point is called the vertex. The curve lies entirely in the half plane {a > 0. For the ellipse and hyperbola, if the directrix is taken parallel to either axis (0;ei) Eq. (93) will have a form like Eq. (78) with A already in diagonal form: A « A. Suppose the directrix parallel to the axis (0;ej). Then after a translation to the form of Eq. (82) with a = —1, the equation of an ellipse becomes

or (0;e2),

fiVoi' and that

+

of a hyperbola

faV-i*

= 1

- i«V«»s

«i

>

= 1

ai

> 0

(95)

(96)

3-32

MATHEMATICS

[Sec.

3

Both curves are symmetric with respect to both axes. Hence each has two foci and two directrices. The longest chord in the ellipse is of length 2on, called the major axis; en is the major semiaxis. The shortest central chord (diameter) is of length The points +aie, are the vertices; 2otj, called the minor axis;
£1/01

it/at

±

are asymptotes.

For the ellipse and for the hyperbola, alternative definitions are possible relating to the two foci instead of to a focus and associated directrix. The foci being f and F z and X being any point on the ellipse, the relation

\

F\X holds, and if

X

+

FtX

—

2a i

is any point on the hyperbola,

FiX

- FtX

±2a,

=

Coordinate systems of the form (0;e) hitherto discussed are called rectilinear or If X is any point Other types are possible and sometimes convenient. Cartesian. in the plane, given by x = ex, and if the system (0;e) is orthonormal, let

(97)

Then p and
P! =

p.

||

-

||s

p, =

-

p»,

if

p,||

(98)

0

;

^

+

Since cos cos
8

£1=0

8

a —

p

9

X.

cos #>

8

8

p p

p

p

£j

ii

one has

= sin = a sin ^> = cos = cos = sin

=
sin

(100)

^

0

are the spherical coordinates of

£ 3

p, ^,

{2 £j

Then

p«

c* = «i« + £•'

p

0,

p

then pi and pi are called bipolar coordinates of X. and to satisfy Spherical coordinates in 3-space are defined thus: Take a >

8

-

3.6

p«« =

£2»

arctan (£i/pi)

pi'

pt* • •

arctan (£j/p2)

• • •

+ pi*

= £1'

+

p,! = Si*

■

for the roverse transformation. Spherical coordinates in higher dimensions can be defined recursively:

Approximate Representation of Functions

is

;

Many useful functions, such as sin x and log x, are not easy to compute directly some experimental functions are known only to the extent of a limited number of measured values. In either case one often wishes to deduce from a set of tabulated values at least an approximation to some value not included in the table. This can be done in various ways when the function sufficiently smooth, and one of the is

by polynomial interpolation. commonest and generally most convenient ways One supposes that a polynomial can be found which agrees with the function at a sufficient number of known points, should be in fairly close agreement elsewhere, at least within the interval between the extreme points. it

if


P» = £i» + £»' — p cos if £i


Sec. 3-1]

3-33

Let fix) be the function to be represented (a scalar function of the scalar variable Let P(x) x), and suppose that the values f(xi) are given at each of a set of points Xj. of lowest possible degree such that be the polynomial

P(Xi) =/(*,)

(101)

Taking Fix) as an approximation to fix) is interpolation for each x,-. between the least and greatest of the x>, extrapolation otherwise.

when x lies

The case is simplest when the x, are equally spaced : Xi = xo + ih

The

where ft is some constant.

displacement

(102)

E

operator

is defined by

ft)

Efix) = f(x + « fix + E[Ef(x)] = Efix =

fix

2ft)

£>/(i)

=

fix +

uft)

-

fix

In

particular

ft)

=

by

8

can be defined symbolically

- - E-> 1

V

=

E -

1

A

The difference operators A, V, and

_ £-»

= EM

(105)

is

to be emphasized that these have no meaning except when applied to The last of these definitions, for example, has the interpretation that =

*/(*)

fix +

function.

- fix - ft/2)

ft/2)

p

is

These operators are called, respectively, the forward-, backward-, and central-difference defined by The central mean operator operators. = (#H

x =

This

(106)

A Aa

+

PM

A»

• • -

+

(107)

u =

wft

A'/(x„) +

—

• •

■

uAfixo)

Q

applied to /(x0), one has

fix,) + Xo

+

u&

+

=

fix)

+

(x

equation

+

1

=

(")

A)» =

1

+

is

thus operational

-

+

E>

(1

whence

+ £-H)/2

follows that

E

it

From the first of Eqs. (105)

If

Xo)/A

if

+ uA/(x0) +

+

A"/(x,

(109)

)

- fix,)

(")

Pix)

• • •

is

if

is

However, series terminates only u a nonnegative integer. one truncates the series after n + terms, the result a polynomial in x of degree n: 1

1,

is

. ,

1,

0,

i

= . . n. In particular, for n = one has the satisfying Eq. (101) for linear interpolation polynomial determined by xo and ij. This Newton's forwarduseful for not too large positive u. difference formula, and Since, likewise,

-

(1

E

is


(104)

any real number, positive, negative, or zero.

£-•/(*)

It

ft)

E*f(x)

6

where u

is

and by extension

(103) 2ft)

a

one can write

+

whence

+

Then

- V)-'

3-34

MATHEMATICS

[Sec. 3

E in powers of V and obtain Newton's backward-difference useful for numerically small negative u: one can expand

P(x)

-/(*,)

+ uV/(x») +

This polynomial

agrees

J

("

with /(x) at

- /(*.)

+

MM*/(X.)

|

M*

+

x0,

+■••+("

x_i,

x_„. Xi, x_i,

. .

with /(x) at

A polynomial which agrees difference formula

P(s)

V»/(x.)

X)

x%,

M*V(X„)

3,

x_s,

+

V"/(x„)

n"" J)

. ,

xo,

— l2) u(lZ* U '

«•/(*.) +

+

...

ul(ui WV

— 1»1 '

01

(110)

is the central-

4,

+,(u»-i»)(u.-2>)^/(x;)

formula

+

»V(X,)

.,,

(111)

When forward differences are to be used, it is convenient to tabulate these according to the following scheme: f(Xo)

Xo

x, Xj

/(*,)

Xi

f(li)

A!/(xo)

A/(Xo)

4f(x,) A/(x,)

/(x5)

AV(xo)

A'/(x.)


Central differences and means can be tabulated as follows:

X-t

f{x.t)

x_,

/(!_,)

m*/(x-,) a/(x_M)

*'/(x_,)

Xo

/(xo)

/.«/(xo)

*'/(x0)

x.

/(xi)

Xj

/(Xj)

if(*-H)

if(xu)

one forms the divided differences:

- /(x,)]/(x„ - X.)

= [/(Xo) = (/(Xo,X|)

/(x„,x,)

-/(Xi,Xj)]/(Xo

- [/(Xo,X,,X2)

/(Xo,Xi,X2)

/(x0,X,,X2,Xj)

«'/(xo)

«V(x«)

a«/(x,)

/.a/(x.) 8/(xM)

When the x, are not equally spaced,

It

«'/(x-n) /x8'/(xo)

- Xi)

-/(Xi,XS,X3)]/(Xo

- xt)

can be shown that the divided differences are symmetric in their arguments:

- /(X|,Xo) - /(Xo,Xj,Xi) - /(Xi,Xo,X2)

/(x0,Xi) /(Xo,Xi,Xj) Then the polynomial

P(x) =/(x0) +

(X

P(x) that

- Xo)/(Xo,Xi) +

(x

—

agrees with

+

(x

xo)(x

/(x)

- X0)(x —

Xi)

=

• • •

at x(>, x\, . . . , x„ is

■*■ X|)/(lo,Xi,Xi) ■• •

(x

—

+

• • •

x„_i)/(x0,xi,

. . .

,x„)

(112)

Explicit formation of the interpolation polynomial is not necessary and not desirable unless the same polynomial is to be used several or many times. If only one or two interpolations are to be made, Aitken's method is to be recommended. This is a J(xi), recursive procedure, each step being like a linear interpolation. Let P.(x) and let Pit(x) represent the linear interpolation to x based upon x,- and xy. Then

"

Piiix)

= [(x

- *.)P, - - x,)P.]/(x, (x

Now let P,,jt(x) represent the quadratic interpolation and xj. Then it turns out that P
= [(X

- Xi)Pih -

(x

Xi)

to x based upon the points

- Xj)Pit]/(x, - Xi)

x,-,

x,,


Sec. 3-1]

Likewise if Pi,u(x) represents the cubic interpolation to then

Pi mix)

= [Or

3-35

i based

upon xi, x,, x*, and x(,

- Xi)Pikl - - i,)P;*i]/(x, (x

Xi)

It is to be noted that each P.y ... is symmetric in all subscripts. f(x) were exactly a polynomial, say of degree n, then any selection of n + 1 points ii, Xj, Xk, . . • would give the same Put.. Ax). At each stage of Aitken's inter polation it is necessary to choose two different sets in order to take the next step of Whenever these two different sets yield going to a polynomial of higher degree. results that are in sufficiently good agreement, it is unnecessary to go farther. The error made in interpolating by a polynomial of degree n based upon xo, xt, . . . , i, can be shown to be not greater than

If

where

M\u(x)\/(n +1)1 — xi) • •

= (x — Xo)(x

w(x)

•

(x

—

x„)

/

and M is the maximum value of the (n + l)st derivative of on the interval that con tains all n + 2 points x, Xo, ii, . . . , x«. The polynomial w(x), and hence the error, ran become quite large outside the interval containing the n + 1 points Xo, X\, . . . , i„ whence extrapolation is generally unreliable. Within the interval w(x) oscillates, and when the points x< are uniformly spaced, or nearly so, the oscillations will be greatest near the ends and least toward the middle.


4

ALGEBRAIC

EQUATIONS

This article on algebraic equations will be concerned principally with techniques of numerical solution, and certain general principles upon which these techniques are based. In Art. 4.1 the treatment of linear equations will be discussed, most of the background having been laid down already in Arts. 1.2 to 1.5. The general theory of nonlinear algebraic equations will be outlined in Art. 4.2. In particular, one is some times interested in certain relations among the roots rather than in the roots them selves, and these relations can often be determined in a simple way from the coeffi cients. With the background given here, the strictly numerical techniques will be given in Art. 4.3, some of which apply to transcendental equations as well. A brief resume of methods applicable to systems of nonlinear equations will follow in Art. 4.4. The latent root problem was raised in Art. 3.3 and the numerical treatment will be developed in Art. 4.5. 4.1

Systems of Linear Equations

The problem of solving a system of linear equations is that of finding the elements of an arithmetic vector x to satisfy Ax

= fc

(113)

where A is a known matrix and k is a known arithmetic vector. Whatever may have been the original setting of the problem, one is always free to postulate an arbitrary coordinate system (0;e), orthonormal or not, and relate the equations to the system. Thus if eA » a » (ai,as, . . .) eA- = k Eqs. (113) can be interpreted as those satisfied by the coordinates of k in the resolution = ex and inter along the vectors ai, as, . . . of the columns of A. Or one can take pret each scalar equation (113) as the equation of a hyperplane whose common intersection x is required. In the special case when A is positive definite, one may suppose that the system e is such that ere = A. Then if x = ex, one is asking for the contravariant representation x of x (Art. 1.5) given its covariant representation k. Finally, if A is symmetric, whether or not it is positive definite, the equations can be related to Eq. (81), with x =• p and k = a, and interpreted as defining the center of a central quadric. Each of these interpretations suggests techniques for solving. Equations (113) axe said to be homogeneous when k = 0, nonhomogeneous when

I

3-36

MATHEMATICS

[Sec.

3

0. When A is square and nonsingular, there is a unique solution x for any k. Clearly if k = 0, x = 0, and this is said to be a trivial solution of a set of homogeneous equations. Consider the case k = 0; suppose that A has n rows and N columns, where n and N may or may not be equal; and let p be the rank of A. If p < N, the columns of A are linearly dependent and, in fact, there are N — p linearly independent vectors xi, xt, . . . , zy_p, each of which satisfies the homogeneous equation Ax = 0. Moreover, Since any linear combination of these xs also satisfies the homogeneous equations. p < n, one can pick out p rows of the matrix A that are linearly independent. These correspond to p of the equations, and any x that satisfies these p homogeneous equa tions will satisfy them all. From the resulting matrix one can pick out p linearly inde After a possible rearrangement these can be supposed to be the pendent columns. first p. The p equations can then be solved for £i, £i, . . . , in terms of the remain to give the general solution of the homogeneous equations. ing and only the rank of the augmented Nonhomogeneous equations have a solution matrix (A,k) the same as the rank of alone. If = n, this always the case, since the rank of a matrix cannot exceed the number of its rows. If the nonhomo geneous equations (113) have a solution x, and the homogeneous equations Ay — also have a solution y, then clearly

Ax

~

k

is

0

p

if

is

.4

if

if

{s

£/>

k

- A(x

+ \y)

+ \y also satisfies the nonhomogeneous system, where In experimental work often happens that the elements of

r

+

is

is

k

is

the vector

is

is

r

A

is

h

t

is

r

where the residual vector to be small in some sense. The simplest criterion hat rTr be minimized. In terms of orthonormal coordinates, this means that the vector and hence that projected orthogonally upon the space of the columns of is orthogonal to all columns of A: rTA = 0. Hence x determined by the normal equation

ATAx

=

ATk

(115)

= diag [ui,ci>i,

. .

$2

0

is

This the method of least squares. It may be that the measurements will not be regarded as of equal reliability but that measures of relative reliability can be assigned. Let ui > represent the weight assigned to the ith measurement, and let . ,«»]

0,

r

is

fi

Then instead of minimizing rTr one minimizes rTUr. Thus (being positive definite) taken to define an underlying metric, and will be taken orthogonal to the columns of with respect to the metric O. This means that rTQA = and the normal equation ATUAx = ATSik (116) is

A

/,

it

is

it

A

is

A

is

is

if

is

n

is a

= and both are spe Evidently Eq. (115) special case of Eq. (116) with cial cases of Kqs. (113) in which, indeed, the matrix positive definite or, at least, semidefinite. To return to Eqs. (113), square and theoretically singular, then a mathe If matical analysis known numerically only and required. square, will seldom turn out to be strictly singular, although may be so nearly singular that an accurate solution difficult (or even impossible). At any rate, the "solution" first arrived at, which will, in any case, be only an approximation because of rounding errors, will need to be improved by an iteration of some kind. Most of the common methods can be classified as direct and iterative. direct method would yield an exact solution after finite number of arithmetic steps, except for the presence of rounding errors. While such methods are themselves iterative in a

A

is


X

any scalar. are the results of experimental measurement and are therefore subject to experimental error. In such a case the system of Eqs. (113) often overdetermined, which to say that n > N, but one expects the system to be satisfied only approximately. Strictly one seeks a vector x satisfying Ax=h (114) it

so that x

Sec. 3-1]


3-37


character, the term will be reserved here for those methods whose individual steps are relatively simple but which define merely an infinite sequence of approximations approaching the true solution as a limit. Strictly, an iterative method is one t hat will improve any given approximate solution, although beyond a certain point rounding errors will contaminate the results and there will be no assurance that subsequent application of the method will, in fact, yield an improvement. A large class of direct methods are based upon Gaussian elimination and differ only If an ^ 0, multiply the first in the order in which the computations are carried out. In the resulting equation the equation by an/an and subtract from the second. coefficient of {i will be — OT21 (cni/oii)ofn = 0

Thereafter use this new equation in place of the second equation. The Next multiply the first equation by orsi/an and subtract from the third. coefficient of £i in the resulting equation will likewise be 0. Thereafter use the new equation in place of the original third. Proceeding thus one obtains n — 1 new from all of in equations, which the term fi is missing. The original first equation together with these n — 1 new equations constitute a system that is (apart from the inevitable rounding errors) entirely equivalent to the original system. Thus £i has been eliminated from the last n — 1 equations by means of the first. One could, of course, have used any equation other than the first, say the tth pro Rounding vided an 5^ 0, in order to eliminate £i from every equation but the tth. errors will be minimized by selecting that an known to the largest number of significant figures or, when these do not differ, selecting the largest. One could also have selected any other J to be eliminated. Having eliminated, say, {i from n — 1 of the equations, one selects one of these, using it to eliminate {j, say, from the remaining n — 2 equations; from these one is selected to eliminate |3, etc. The final result will be one equation that contains {„ but no other £; one that contains only {„_i and £„; one which contains only £„_j, f„_i, and It is now possible to solve for £„; substitute and solve for sub stitute both and solve for £„_.; . . . , eventually obtaining all the elements of the vector x. This is called back substitution. To organize the computation exactly in accordance with the description would necessitate rather more recording than is strictly necessary. One notes that assuming the equations to have been taken in order and the & eliminated in order, the result of the elimination is a set of equations in the form Wx = h

(117)

where IF is an upper triangular matrix, which is to say that all its nonnull elements aro on and above the main diagonal. Such matrices are readily inverted (precisely by the

method of back substitution). Moreover, the inverse is upper triangular, and the product of two upper triangular matrices is again upper triangular. An upper triangular matrix with units along the diagonal is unit upper triangular. The inverse of such a matrix is also unit upper triangular, and the product of two such are again of the same type. Analogous definitions and statements hold for (unit) lower tri angular matrices. Now in the reduction of the original system to the form of Eq. (117), each step involves the multiplication on the left by a unit lower triangular matrix of a simple form (only one off-diagonal element is nonnull at each elimination). The result of all these is again that the system has been multiplied on the left by a unit lower triangular matrix. This proves that in general, given any matrix A, it is possible to factorize it into a product of the form A = LW (118) where L is unit lower triangular and W is upper triangular (or where W is unit upper triangular and L lower triangular). The diagonal elements of L are all known (to be ones) and those above the diagonal are zeros; the elements of W below the diagonal are zeros. Hence in L and W together

3-38

MATHEMATICS

[Sec. 3

Crout's there are ns elements to be determined, and these fill out a square array. method and the Doolittte method amount to writing first the rectangular augmented matrix (A,k) and passing directly to a new rectangular array that can be written schematically

(L'\W,h)

L being placed where the null elements of W would go. The elements in this array are determined by relation (117) together with

the significant elements of

Lh

k =

and can be determined in sequence as follows: first row of W, first column of L, second row of W, second column of L, . . . , and each element of h can be found along with the corresponding row of W. Thus the first row of IT is precisely the first row of A ; the elements in the first column of L are determined from the relation X,i«n ~ an knowing

X21, one

can find the elements of the second row of W by +

XjlWl, the elements of the second


t > 2

U2;

L

column of

> 2

are found from

X.'iwu + Xi2Ci)u =

....

j

atj

=

i; > 3

an

To complete the solution the factorization must, of course, be followed by back substitution. Note that one could solve simultaneously any number of systems, each having the same matrix A of coefficients, by merely replacing A by a matrix, one In particular, if h is replaced by the identity /, then column for each of the systems. after the factorization L~' will appear instead of k and the back substitution will produce A~'. If D is a diagonal matrix whose elements are those of the diagonal of W, then D-*W where

U

is

unit upper triangular.

=

U

Hence A can be written

A

=

LDU

In particular, if A is symmetric, U = LT. The recording can be further reduced in that case in that only L and D (or D and U) need be written. If V = LD^, a sym metric matrix can be written A

=

VVT

and this is the Choleski method, or the square-rooting method. Unless A is positive definite, this method has the slight disadvantage of introducing imaginary quantities. The method of partitioning is rather different in character. Let A be partitioned in any manner

A

=

it1 \An t2) An)

with, however, An and An. both square matrices. if the inverse is similarly partitioned

VC»

one can verify that

Cn

=

C» Cu

=

-

(An

—

Suppose An-1 is known.

Then

Cn)

At\An~1A\tTl

-cJUnAu"

A.r'U,,

- A„C„)

(119)


Sec. 3-1]

3-39

is useful in special cases when a particular submatrix is easily inverted. 2X2 minor, one can border this to a 3 X 3, then border this to a 4 X 4, and so on, sequentially, where column vector, and A2! a scalar. at each stage An is a row vector, A large class of iterative methods are based upon a decomposition of A into the sum of two matrices: The method

One can. also

invert by progressive enlargement: having inverted the

i»a A

A, + At

=

(120)

the first is readily inverted, and defining the sequence x, by

of which

A\x,+i

r,

Let

= k

An,

—

k - Ax, - As, m

(121) (122)

It is essential that the vectors r, (or «,) become progressively smaller. If an iteration form of Eq. (121) converges, t hen it converges for any starting vector Xo whatever. In the method often called the (Gauss-) Seidel method Ai agrees with A on and below the main diagonal and is null elsewhere. The method is known to converge when A is positive definite. It also converges when each diagonal element exceeds in magni tude the sum of the absolute values of off-diagonal elements in the same row or again if it exceeds the sum of the absolute values of off-diagonal elements in the same column. Another common method is to take for Ai the diagonal elements of A. This also converges if each diagonal element exceeds in magnitude the sum of the magnitudes of the off-diagonal elements in the same row (or in the same column). In the Seidel method one takes the individual equations in rotation and solves the ith equation for ^ i), thus using the current approximations for the

- ... -

*.•

-

«,.«_,&_,(»+«"

(j

{ ,

J,-

- -

«iiti<'+«

o,,4+1eJ+,<»)

- ... -

«„„£„<" (123)

until the end of a complete cycle:

in the other method one retains the

whereas

= xi

«nti('+»

- o.iii'') -

• •

—

- fl«.<+i{ -

ow-ifa-i1'1

•••

-

«(,„{„<') (124)

Afc('>

-

-

A

is

£.

is

is

a

is

The method of relaxation differs from the Seidel method only in that instead of following strict rotation, that equation always selected which gives rise to the greatest a desirable procedure when the computation This by hand but change in the not feasible with automatic machinery. Often an iteration of the form of Eq. (121) [including Eqs. (123) and (124) when the equations are taken in strict rotation but not otherwise] can be made to converge more operator by rapidly by application of Aitken's i1 process as follows: Define the

A'fi'"' {< A'fi'"' = A£i - Afc'"'

=

A£i - Afc'"'

{j

Having obtained three consecutive sets of iterates £<»)

of forming

instead

{|C'+»

in the ordinary way, one forms £'<

-

I/'4-*

- [Afc]7A»fc<"

think of the {'< as being elements of new starting vector z'o. How process will sometimes convert divergent iteration to a convergent one. should not be applied for theirs* time until after a number of simple iterations

ever,

a

4*

it

The

a

One can

have been carried out.

The iterations of Eqs. (123) and (124) are especially useful when the equations are In such a case difference equations used to approximate differential equations. the matrix has a great many null elements and the equations are simple but many number. It sometimes happens that the equations can be grouped in such way that instead of solving each equation individually for the corresponding {i, one can a

A

the in


of the

3-40

MATHEMATICS

[Sec. 3

solve a group simultaneously for the corresponding set of A might have the form

A

/A0 An

= I

M«

A15

Ao A3!

{s.

For example, the matrix

Au A*i Ao

where the same small submatrix Ao occurs repeatedly. Then Ao can be inverted once and for all. The block method will also accelerate convergence. The 4' process can be applied equally well to the results of such an iteration. However, every application of the 4* process should be followed by a simple iteration to reduce rounding errors unless the simple iteration is divergent. Geometric considerations lead to other iterations useful in some cases. Let Eqs. If u is any (113) represent n hyperplanes whose common intersection is required. vector whatever, then uTAx = uTk (125) is also a hyperplane through the point. Let xo represent any point. Its orthogonal projection Xi upon the plane of Eq. (125) will be closer to x than is Xo unless xo is itself on Eq. (125). The projection can be found by solving to obtain that X for which Xi = Xo

+ \ATu


If

satisfies Eq. (125). The simplest choice for u is one of the coordinate vectors a. these are taken in rotation and a complete cycle is regarded as one iteration, the 4' process can be applied to any three consecutive iterates. Let Eqs. (113) represent the resolution of the vector k upon the columns of A. Let

represent any approximate resolution that yields a vector Axo deviating from k by the residual xo

ro = k — Axo

If u is any vector, project ro orthogonally upon the vector Au, adjoining the project ion to Axo. Unless r0 is itself orthogonal to Au, the new residual will certainly be shorter than ro. Analytically one takes Xi — Xo + Xu so that uTAT(rD XAu) = 0 — =» since XAu ro ri

-

is the new residual that is to be orthogonal to Au upon which the projection is made. Again the simplest choice for u is one of the ej, and if a complete cycle constitutes an iteration, the 4! process can be applied to three consecutive iterates. The last method can also be applied to the problem of least squares of Eq. (114), and the normal equations need not be formed explicitly. If each new vector Au is chosen, not arbitrarily, but orthogonal to all preceding, then the iteration terminates and the method is a direct method. When any direct method is used, one obtains a vector x0 that, in fact, will deviate more or less from the true solution x because of rounding errors. Hence when the computed vector x0 is substituted into the equations, a vector Ax0 will result that

deviates from k by

r0 =

where

k

—

Axo = Ax

— Ax0

=

— s0 = x x0

A(x

Hence Xo differs from the true solution x by a vector the equation A.s„ = r0

s0

—

xo) = Aso

that, though unknown,

satisfies (126)

having the same matrix as the original set. If the Crout method has been used or anv factorization method that gives A in the form of Eq. (118), then the major portion of the work required for solving Eq. (126) has already been done. Naturally one will

I


Sec. 3-1]

3-41

Nevertheless, not obtain the true solution of Eq. (126) but only an approximate one. if this solution is added to xo, the result will generally be an improvement. If Co is an approximation to A'1, then C, be a better one.

will often

- C0(2/ It

AC)

= (2/

-

CA)C

(127)

will be better provided the deviation matrix

I

=

R

- AC

small in some sense. A sufficient condition is that the sum of the magni elements of every row (or of every column) be less than 1. If this is true, be assured that C will be an improvement over Co and that, in general, any

is sufficiently tudes of the one can

Cr+,

=

-

C„(2/

AC,)

C

until the improvement becomes submerged in the improvement over errors. A complex system can always be Hitherto all matrices have been supposed real. written in the form (A + iB)(x + iy) =h+ik will be an rounding

When this is

multiplied out and real terms and imaginary terms equated, one has = h k

-

this is equivalent to the system of order 2n:

a)

(b

=

linear equations therefore reduce directly to the real case. Theory of Equations

4.2

In this article will be considered polynomials of degree

that

■• •

+ o„

+

•

±cn)

^

an

(128)

the roots of the algebraic equation

f(x)

=

(129)

0

and their zeros,

-

+

s»

• •

-

atx" + aii"~l + aix"-1 + cix"-1 ctx'~* ao(i»

™

is,

fix)

n > 0:

0

Complex

0

(fc)

Hence

algebra asserts that an algebraic equation has at least one fundamental theorem root, real or complex. Given this easy to prove that an equation of degree n has ezactly n roots, which may or may not be all distinct. The remainder theorem asserts that after dividing f(x) by x — any constant, the remainder f(r): m

r

is

is

7i

f(x)

(x

is

if r

it

of

The

-r)q(x) +/(r)

— r)q{x)

(130)

R

+

s

r is

if

if

is

is

r

is

r

{x

Indeed, the relation f(x) m must hold identically (i.e., for all x, in particular), which proves the theorem. The factor theorem asserts including x = that x — a factor otf(x) (the division and only a root of Eq. (129). exact) This a direct consequence of the remainder theorem. is

0,

if

(x

.

x.

are roots of Eq. (129) and there are no others.

(x

2.

■.

,

that X\, Xt,

(x

is

1.

is

is

and

(z

is a

of

If

— xClq\(x), where qi(x) root of Eq. (129), then f(x) a polynomial xi degree n — also a root of Eq. (129), and con Evidently every root of qi =0 versely, any root of Eq. (129) must be either xt, which causes x — Xi to vanish, or else — x2)
a


Ax — By Bx + A y

This proves

3-42

MATHEMATICS

[SBC. 3

the assertion made above that there are exactly n roots. whence

f(x)

- xi)(x - xi)

= a0(x

■ • •

(x

It

is clear that q„ =

- in)

o0,

(131)

The fact that an algebraic equation of

degree n can have no more than n roots can often useful forms: If a polynomial of degree n or less vanishes for n + 1 distinct values of x, it vanishes identically, and if two polynomials of degree n or less are equal for n + 1 distinct values of x, then they are identically equal. On multiplying out the right member of Eq. (131) and comparing coefficients of the various powers of z with those in Eqs. (128), one has be phrased in the following

-

y Xi = ci

I

I

i

XtXj

—

oi/ao

= c2 = aj/ao

(132)

ii.j) XiXjXk

= Cj = —at/ao

«j,t) The summations Thus for n 3,

are

extended over only the distinct combinations y

XiXj

- X\Xi

+

Ills

+

X\X%

of distinct xs.

These are called the elementary symmetric functions of the roots. If one substitutes x = y + h in Eqs. (128), the result can be written in the form

f(y +

h)

=f(y) +

hf'{y) +

hT(y)W

■■■

+

+ h"P»Ky)/nl

(133)

after collecting the several powers of h. In this expansion, f'(y), the coefficient of h\ is called the (first) derivative of/ evaluated at y. Likewise /" is the second derivative, /'" the third, . . . (cf. Art, 1.1 of Sec. 3-2). The derivative of a sum is the sum of the derivatives, the derivative of a constant is zero, and multiplying a function by any constant multiplies any of its derivatives by that constant. By applying the binomial theorem (Art. 1.1) to (y + h)' one obtains the formula for the derivative of a power: (y»)(.)

_

n(n

-

• • 1) • («

-

*

+

l)y-*

(134)

Also one verifies directly that

It

/(,/ + h)g(y +

follows that

+ =

- g(y)

(i

+ jHh derivative of the function

hg'(y) +

f(y)g(y) + h[f(y)g'(y)

•

+f'(y)g(y)]

W-fg'+fo

+

whence

h)

g(!l

then

the

(135)

• •

any polynomial,

derivative

k)

If g(i)

t'th

+

the jth derivative of an is

that itself.

is

[/(.)|0) =/(.+/) is,

■■■

(136)

is

Corresponding formulas for higher derivatives are obtainable from other terms in the above expansion. Let [xo,f(xo)\ represent a point on the curve, whose equation

U


-

-/(«.)


Sec. 3-1] The straight

3 43

line whose parametric equations are

ii

= xo + h

£2 =

/(xo) + mh

h, has slope m, passes through the point [xn,/(xo)], points corresponding to those values of h which satisfy

with variable curve at

/(*, +

f(x0) + mh = On applying

Eq.

If

itself.

h)

(133) and collecting terms this gives

- m] + A»/"(*o)/2!

0 = h[f\x0) The equation

and intersects the

+

. . .

(137)

is evidently satisfied by h = 0, which corresponds to the point [xo,/(x0)]

one takes m = /'(x0)

is a double root of the equation in h. The line is then said to be tangent If it is also true that/"(x0) = 0, t hen /'(x0) is the slope of the curve. root, and the point [xo,/(xo)l is said to be an inflection of the curve. Any root of Eq. (129) is given by Suppose x0 is a root of Eq. (129), and let m = 0. it + k, where h is a root of Eq. (137) with m = 0. If f'(xo) = 0, then h — 0 is a double root of Eq. (137) and xo = 0 a double root of Eq. (129). Hence Xo is a double mot of Eq. (129) if and only if then h = 0

to the curve, and h = 0 is a triple

0

root also of the first derived equation

fix)

=

(138)

0

double root of Eq. (129)

is a

a

is,

that

any convenient

7,- >

-/

7./i = 7i+l/.+l
/<+J the

—

remainder after =

/-+.

fi+i

(139)

Let

division.

m be the smallest

0

is

the quotient, for which

where q>+t index

1

/»-/'

let

a

and for

0

/.

is

if

1

-

is

a

is

More generally, a root, of each of the first m — root of multiplicity m of Eq. (129) derived equations. Also a root of multiplicity m of Eq. (129) a root of multiplicity m of Eq. (138). Hence all multiple roots of Eq. (129) are zeros of the highest common factor of f(x) there no such factor (other than a constant), then Eq. (129) has only and/'(x), and Consider the following algorithm: For uniformity let simple roots.

fi

if

of

is

A

,

/,

is

0,

is

a

V U

1-

j3

H

if

-)

...

,

/o

is

/„ the highest common factor of and /1. Many numerical methods require that the roots be isolated at the outset, i.e., that for any desired root some region be known to contain that and no other root of the equation. The sequence of Eq. (139), besides providing the highest common factor of /and /', permits the isolation of real roots by means of Sturm's theorem. When the functions are written in order, /0, fi, ft, and evaluated for any a, let Va represent the number of variations in sign in the sequence has two (thus the sequence Then variations, whereas +H — and each has but one). neither a nor are ignored in the sequence, the difference root of Eq. (129), and vanishing each root being counted once the number of roots between a and m — Vf regardless of multiplicity. Budan's theorem more easily applied but gives less information: In the sequence /", . . . the difference in the number of variations at a and may exceed the number of roots on the interval (roots being counted according to multiplicity) but Descartes' rule special case signs: The number of always by an even number. Then

/',


/(x„) = /'(x„) =

3-44

MATHEMATICS

[SEC. 3

sign variations in the coefficients may exceed (and is never less than) the number of positive real roots, but always by an even number. In Eq. (130) let q(x) = qox"-1 + g,x"-J + • • ■ + g„_i On multiplying

and comparing coefficients one finds 'la

= a0

?<

/(r)

+ qi-yr

— Oi

= a» + qn-ir

This justifies the algorithm known

as synthetic division: For dividing /(i) by x — r, the coefficients of the quotient and the final remainder are obtainable from the following scheme: o0

qo

Let f(V + r)

at

a2

• • •

q0r

qir

■ • ■ q„_ir

qi

92

= g(y)

• ■•

a„

[r

/(r)

^ b*y + bxy*

+•■•+&„

(140)

If Eq. (129) has roots xi, x xu, and if g(y) = 0 has roots jfi, j/2, . . . , yH, then are roots each suitably the Hence one says that to obtain Xi = v; + r. (if ordered) the equation g = 0, the roots of Eq. (129) are reduced by r (which could be a negative Now

number).

g(y) = g(x

- r) -

b„(x

- r)n + bi(x - r)""1 +

• ■■

+

b„

Hence 6„ is the remainder after dividing f(x) by x r; 6„_! is the remainder after dividing the quotient by x — r; . . . . Hence for reducing the roots of an equation one can extend the method of synthetic division:


—

Ol

ai

9or

9ir

9o

9i

92

9'o

9'.

9'»

9V

q'ir

■• ■ o„_i ■■• ■■■ • • •

9»-i ?'„_sr

a. 9»-ir b.

6.-!

In this table if r > 0 is such that the qs on any line together with the 6s on and above that line are all nonnegative, then r is an upper bound of the roots of Eq. (129). For this implies that every bi > 0, whence g > 0 for y > 0, whence > 0 for x > r. r is an upper bound to the roots of }(—x) = 0, then — r is a lower bound to the roots of Eq. (129). In case the coefficients a, of f(x) are all integers, if r is a rational root in its lowest terms p/q, then p divides a„ and q divides do. Thus all rational roots can be found by trial and divided out, and all multiple roots can be removed after the highest common factor of and /' is found. Since/' represents the slope of the graph off, it is evident geometrically that between must vanish at least once. This is Rolle's theorem (between distinct zeros of f(x), consecutive zeros of there must be an odd number of zeros of/', counting multiplicities). Between consecutive zeros of /' there may be one or no zero of /. An nth-order linear homogeneous difference equation is a relation of the form

/

/

/

If

f

oo{»+»

+ aii„+r-i +

• • ■

+ a„i, =0

oo

^ 0

(141)

connecting any n + 1 consecutive terms of a sequence £». If a sequence is such that any n + 1 consecutive terms satisfy Eq. (141), then the sequence is said to be a If J, and r\, are any two solutions of Eq. (141), solution of the difference equation. then a(, + f}y, = f„ is also a solution, where a and (3 are arbitrary fixed scalars. If Xt is a double Xi is any root of Eq. (129), then the sequence *<» satisfies Eq. (141). root of Eq. (129), then it can be verified that vxt' also satisfies Eq. (141); if Xi is a

If

3-45


Sec. 3-1]

l)!,'-'] satisfies Eq. (141); . . . . root, then also vhd' [or, equivalently, v{v there are always at least n distinct solutions of Eq. (141). Conversely it can be shown that given n distinct solutions {,(u, . . . , {»'"' of Eq. (141), any solution is as a linear combination of these. Equation (129) is the characteristic expressible tquation of Eq. (141). In case the terms £0, ti, . . . , £„-i of a solution are known, other terms of the sequence can be obtained recursively by applying Eq. (141) with v = 0, 1, 2, . . . Hence these initial values determine the solution uniquely. Of par consecutively. ticular interest is the solution —

triple

Hence

an = n

The

initial terms of the sequence a, can be shown to satisfy Newton's identities

=0

awn + Oi Oocrs

+

ai
floff.i

+

Oia2


aoff„

If

n

on ^ 0, =

+ +

= 0 = 0

2a2 ojff3

+ aia„_i +

+

• ■•

3oa

+ Hn-Kn

(143)

+ na» = 0

the difference equation can be reversed to obtain

. . .

(note that

n). If one sets h(v), a function of y, on the right of Eq. (141), the result is a nonhomo-

geneous

difference equation.

Given any particular solution

f,

of the nonhomogeneous

equation, if £„ is a solution of the homogeneous equation, then j", +
The Solution of Algebraic Equations

The two roots of a quadratic equation ax' + bx

+

c

=0

are given by the quadratic

formula x = The quantity

which occurs the roots are A = 0.

(-6

±

A =

V^-4oe)/(2a) 6* — 4ac

If A < 0, under the radical is called the discriminant of the equation. Otherwise they are real, and they coincide when complex and distinct.

exist for the roots of a cubic and also for the roots of a quartic equation. numerical computation they are of little practical value, and for equations of degree 5 or higher such formulas do not exist for the general case (clearly they do in However, there are special cases, as when factorization is known to be possible). various methods of successive approximation, which permit a determination as accurate as may be required (or as the accuracy of the coefficients will justify, in case these are only approximate). It will be supposed in this article that the equation to be solved has only simple roots. It has already been shown (Art. 4.2) that all multiple roots satisfy an equation of lower order which is obtainable from the equation and its first derived equation. As a practical matter, however, an equation can have even a single pair of equal roots only when its coefficients satisfy a certain algebraic identity. If the coefficients were selected at random, the probability is zero that such an identity would be satisfied. Formulas

For

3-46

MATHEMATICS

[Sec.

3

Other equalities among the roots imply other identities to be satisfied by the coeffi One can be assured that if the coefficients are given experimentally, then cients. unless they are required a priori to satisfy the conditions, they will not do so in fact. However, there may be roots that are nearly equal. A pair of nearly equal real roots will be nearly equal to a root of the derived equation (this will lie between them), and it may be advantageous to find first this root of the derived equation. For finding a real root, the method of false position is one of the simplest. Let the By graph equation be given by Eqs. (128) and (129), and let a be the root required. ing the function or otherwise, one can determine numbers io and xi such that the root a alone lies between x0 and xi. The root a can be supposed a simple root, since multiple roots can be found otherwise (Art. 4.2). Then/(x0) and/(xi) will be opposite in sign. The chord joining the points [xo,/(xo)l and [x\,f(xi)] will therefore cross the axis between

xo

and xi at

- Xl/(X0)]/[/(Xl) - /(xo)]

12 = \xo}(xi)

If /(xo) and /(x2) are opposite in sign, proceed with x% replacing xi ; otherwise proceed One obtains thus a sequence of values of x that approach a. with i2 replacing x0. Horner's method is equally simple. Suppose xa is an integer and Xt < a < x<>+ 1. Reduce the roots (Art. 4.2) by x0. In the new equation the desired root is a — xo By trial determine the largest number of the set 0.1, 0.2, 0.3, . . . , 0.9 that < 1. Let xi be the total amount does not exceed a — Xo, and reduce again by this amount. by which the roots have been reduced. Then the desired root in this equation is By trial find the largest number of the set 0.01, 0.02, 0.03, 0.09 a xi < 0.1. At that does not exceed a — Xi, and reduce by this amount, making a total of Xi. each step one finds an additional decimal in a. In practice only a few steps of this kind will need to be taken (one, two, or three) At some stage let z = a — Xi satisfy unless there is another root very close to a.


-

or

....

boz" + fciz"-' + • bn-iz = —b„ — b„_2Zs

• ■ —

bn = 0 • ■• — boz*

+

(144)

If 2 is very small by comparison with all other roots, then all terms on the right that contain z will be small by comparison with b„. Then, approximately, z =

-6»/6»_i

and this approximation, when substituted for z on the right, gives a still better approxi mation. This cycle can be repeated as many times as desired. This process of repeated substitution is one of a very large class of iterative methods. If the equation is written in the form x = *>(x)

(145)

which can be done in infinitely many ways, one can define a sequence by x„+i = ip(x,)

(146)

with a suitable initial term xo, and if the sequence converges, it converges necessarily to a root a of Eq. (146). When xo is sufficiently close to a, the sequence will converge provided !*>'(*) I <

i

throughout some interval containing a (for derivatives of functions other than poly nomials, see Art. 1.1 of Sec. 3-2). This means that the graph of has a sufficiently small slope throughout the interval. The optimal situation is that the curve shall be horizontal at x = a:

f

*>'(<*)

= 0

and this holds for Newton's method, which utilizes the iteration x„+i =

x,

- f(x,)/f'(xr)

(147)

Sec.

3-47


3-1]

Newton's method does not differ greatly from Horner's except that in the former one chooses xo to satisfy either zo < a < zo + 1 or else i0 > a > lo — 1, always so that neither /' nor /" vanishes between xo and a and so that/(io) and /"(xo) have the same sign. This means that if the curve is concave One then down, one starts below the axis; if concave up, then above the axis. diminishes the roots by xo, then by the nearest tenth, then the nearest hundredth, but never by enough so that the curve is crossed. Again the equation can eventually be put into the form of Eq. (144) when z has become small enough. = 0 are to be included among those of Eq. (145), then >(x) must be If the roots of of the form In practical application

method

/

-f(x)/g(x)

/ have

no zeros in common. Conversely, if
and

can use

it to

define

-

z,+i

(z, + o/z,)/2

Convergence of these and of many ot her sequences can be accelerated by the appli of Aiiken's S* process. This was described in Art. 4.1 for sequences of vectors and applies equally to sequences of scalars. It will, in fact, convert a divergent sequence to one that converges. A sequence of a different type is used in Bernoulli's method. Any solution {, of the difference equation (141) can be expressed as a linear combination of the powers of the


cation

roots

of Eq. (129):

£„ =

as already explained. Then if a, 5* 0, £„ =

r increases, Hence in the limit

and as

If

a,Xi' +

• • •

+

orsXi'

+ a»Z.'

the equation has a root of largest modulus, call this X\. ■■•

Xi'[a, + caixt/Xi)' +

+an(xjxl)']

all terms but the first within the bracket will decrease

-

hy

*,+i/£,— x,

to zero. (148)

If the initial values {0, (it ■ ■ • . in-i are chosen arbitrarily, there is zero probability 0/ selecting these by chance so that ai = 0. Nevertheless on might be so small as to retard the convergence seriously. The special choice £, = a, (Art. 4.2) gives all roots equal weight. For accelerating convergence, the t1 process should be applied to the sequence of quotients h, rather than to The method can be restated by saying that the roots of the sequence of equations I

form a sequence that approaches that of any other root. If

«'

I

-0

zi in the limit provided the modulus of

> 1**1 > then

1

it is also true that the roots of the «.

W

•

>

sequence 1

> |z.|

of quadratics

Xi exceeds

3-48

MATHEMATICS

[Sec.

3

form two sequences that approach Xi and Xj in the limit. Thus if Eq. (129) has a pair of conjugate complex roots of maximum modulus, then the sequence £,+i/£r will not approach a limit but these quadratics will define converging sequences. Again let >

|i.| Then the

sequences

W

>

M

>

W

>

formed from the three roots of

i,

£,-i

{,-.

£»+i

z.

Sr-1

t,+.

s»+l

i,+l

t.+i

1

I

i.

X

«,+ !

X

approach x,, x«, and i>, and likewise for the equations of higher degree similarly constructed. Lot

that are

£,(•> = 1

z,w

= z,


. Each of these determinants following manner:

i.

«»-!

Jr+l

{.

l,-l\

«,-! =

£,

{,-.

£r+l

«»

1

can be formed from determinants

of lower order in the

and the sequence of quotients of consecutive determinants have the products of the roots as their limits: i.x, A/2' = £,+,<»/$, - i.x,

-

In particular, XiXj,

if |xi I

>

|ij|

> |z,|

>

• ,

since A, approaches Xi and

A,<2)

approaches

fc,<»/A,-» xs Likewise

In principle, therefore, all roots can

be found by this process, although rounding errors increasingly difficult to control. The smaller roots, however, can be obtained independently. One has only to observe that the roots of become

a„x" + a„-ix"_l +

+

0

are the reciprocals of the roots of the original equation. Consequently the largest roots of this equation are the reciprocals of the smallest roots of the original equation. Bernoulli's method has the advantage that rounding errors do not accumulate within the same sequence (although they do build up as one passes from one sequence to a The method has the disadvantage of slow convergence. sequence of higher order). To Graeffe's method converges more rapidly but accumulates the rounding errors. apply this, one writes the equation in the form
+ a.x"

2

+

a3x"


Sec. 3-1]

3-49 The result

squares both sides of the equation, and re-collects all terms on the left. If one substitutes an equation containing only even powers of x.

is

i?

of a root of the original the result is an equation each of whose roots y,- is the square After v repetitions, the result is an equation whose roots are the pth powers equation. of the roots of the original equation, where p — 2'.

It

is most convenient to divide through the original equation by start with the form

a0

at the outset and

hence to

X" + After

»

fjl"-1 +

steps let the equation be

X"

- C,X"-'

Then, with p = 2»,

+

CtX--'

C,X"~» +

where all terms after the first within

Nil v is

- *l'(l

■■■

X|» + Xlp +

■■■

+ ex*'* +

■• ■

=0

■■■

+ XSISX," +

= Xi"X2"X,"

But

In this event, when

-

C, = x," + x2" + ■ • • & = X,"X;- + XfXt' +

d

■■■

+ Xj»/Xi» +

• •

■)

the parentheses become small, provided

> N»l >

Nil

>

sufficiently large, <"i =

Xl"

Likewise if

Xi'Xj' +

then where,

when

y

Nil > Nil Xi»Xi» + ■ •

> •

Nil > |x.| > • '• • x,"xtP(l + XjO/Xj" +

=

• •

-)

again, all terms after the first within the parentheses become small. is sufficiently large, Ct = xi^xi" C./C, = x+

Hence,

sufficiently large,

p

Nevertheless, when cos (pfl)

v,

0

+

is

-

if

it

•

(\ * p"

but C\ will oscillate because of the term in cos (pfl). also true that close to ±1, C, = 2p» cos (pfl) the equation and therefore, C,X X1 C\ = taken at suitable values of

■

+

-

■

!■> = exp (—i0) > |ij| > \xt\ > exp (ifl) Ci = 2p" cos (pfl) 4- x,» xt" + ■ ■ ■ p" + 2xa»p* cos (pfl) + • • ■ Ct p

=

the roots will tend toward x,» and x^.

is

Hence for

xi

*

Then

p

A

is,

Other roots can be found in like manner until the sequence of inequalities is broken = |xi+2|. by an equality; that |z<| > |xi+i|, but |x,+1| Whenever an equality occurs between moduli of roots, the pth root of the corre If the coefficients of the equation are sponding quotient C;+s/C»+i will not converge. real, complex roots occur only in conjugate pairs, and failure to converge will ordinarily be due to the presence of such a pair. complete analysis of all possible cases would be quite long, but suppose, for example, that

is


■ ■ ■ = c,x"~l

CiX"'4 +

3-50

MATHEMATICS

[SEC. 3

For solving transcendental equations Newton's method applies directly. Bernoulli's and Graeffe's methods also apply but with the following modification: If z = x~l, let viz) = z"/(z-') = 0 Then these methods can be regarded as methods for finding the roots of beginning with the smallest instead of the largest. Interpreted as methods of obtain ing the smallest roots, then they apply to transcendental equations viz) = 0.

Systems of Equations

4.4

Newton's method can be generalized to apply to systems of equations, but in order to describe the method it is necessary to make use of partial derivatives (Art. 1). Let the equations be 0 v>itx,h, ■ ■ ■ ,{.) *.({>,&,

If

we let x and

/ represent x =

• • •

,«.)

-0

the vectors

...,{» |

ifi.ij,

/

= \vi,v% . . .

,vA

these equations are, in vector form, = 0

The first partial derivatives of the which will be represented

with respect to the

U

fia)

form the Jacobian matrix,

=

Let a represent the solution vector, and let y<, = a from a. The Taylor expansion has the form = /(x„)

—

+/,(x„)2/, +

Xo represent

the deviation of

/

=

-

-hia

x0)

+

x<>

■• ■

where the terms omitted are of second degree or higher in the elements since a is the solution, fia) = 0, whence

v>

of y.

But

• • •

/

-

The solution

not possible

is

is

are solved for xi, then xi should be closer to a than x0. nonsingular, but in this event unless the matrix /*(xo)

is

frit

f

/*Xl =

is,

where the elements of and of /, are to be evaluated for x = x0. Suppose xo is close Then if these terms are to a so that the terms that have not been written are small. the equations neglected and the equations solved for x\ with xi replacing a, that

Xt — Xo — /,"'(xo)/(Xi)) to continue the iteration and write Xf = Xi

sequence

of high order, one would wish to limit the number of that converges, although more slowly, given by

is

if

the system

A

However, inversions.

-

- fx~lix<,)fixp)

x,,+i = xp where the same reciprocal matrix

/r'(ii)

Another method of solving the system The function solving linear systems. 2vix)

matrix

is

possible

used throughout.

is is a

It

is


fix)

generalization of the Seidel method for

=/r(x)/(x)


Sec. 3-1]

3-51

Let is essentially positive and has its minimum value of 0 at x = a. of partial derivatives of ip with respect to the £,-. Then the system
is equivalent detail,

system if

to the original

*>«,(ti,£«,

is nonsingular.

• • •

£,<0) to

These equations are, in

,£.) = 0

-0

,*.)

«.(«!,« One selects an approximation

the vector

= o

14

/,

the solution, and thereafter one solves in sequence

«>t,[£.(I>,S*(0W>,

• . • ,I»<0)1

#>f,[«i(,>.i«(,>,{it0).

• • • >«»(0)1

-0 -0

for {i(,) in terms of the assumed {j'0', . . . , |„, Thus the solution of the original system of nonlinear as many times as necessary. equations is approximated by solving a sequence of nonlinear equations each in a For solving each equation of the sequence one can use any of the single unknown. methods described in Art. 4.3. These methods also apply to transcendental systems. It has been presupposed here that the solutions to be found are real. For complex solutions, in place of the vector x write + iy, where x and y are real. When this substitution is made, the vector becomes


....

i

/

f(x + iy) = g(x,y) + ih(x,y) where g and h are real vectors, each depending upon the two real vectors x and y. Instead of the system f(x) = 0, one has now the system of order 2n: g(x,y) = 0 h(x,y) = 0

Either the Newton or the Scidel method can 4.6

For

be applied to this system.

Latent Roots and Vectors

a symmetric matrix A its characteristic equation \A

has arisen in Art. 3.3. If is not symmetric.

-\I\

=0

The equation has important applications also in

E

is nonsingular

A'

cases when A

and =

E-*AE

then A and A' are similar and represent the same Since coordinate systems (Art. 1.3). then

(149)

transformation

but in different

= A' -\I - \I\\I)E = \A' - \I\

E-'U

Hence the characteristic function

\A

?(X) = \A

- \l\

(150)

is the same for similar matrices, and hence so are the characteristic roots (or latent roots or proper values). A diagonal matrix A = diag (X„\2, . . . ,X») (151)

3-52

MATHEMATICS

has the characteristic function

v(\)

- X)(X5 - X)

= (X,

[SEC. 3

• • •

-

(X.

X)

is

A

is

- \iXi

ytTA = \njiT

(152)

- TiX""1

(-)"(*" 71

= trace

+

=

-

7jX"-«

.

• ■•

7„ = \A\

A

¥>(M

Then

j/,-

0

is,

both have nontrivial solutions (that and solutions a\ ^ are the latent vectors (or characteristic vectors or proper vectors) can take x,- = j/,. The characteristic polynomial can be written

^ 0).

If

A

An

is

X,-

Hence if A is known to be similar to A, the diagonal elements of A are necessarily the latent roots of A. — If any latent root of A, then the matrix singular and the equations X(/

Such solutions symmetric, one

±T.)

(153) (154) is

A

if

X

is a

X,

is

t

is

A

is

the sum and yn where trace the sum of the diagonal elements of A. Also 71 the sum of all principal minors the product of the latent roots of A. In general, 7, of order of A. by If ^(X) any polynomial in then ^(A) matrix obtained by replacing in ^(X). Whereas in general two matrices are not commutative,
theorem states that any matrix

-

It

u(A)+(A) satisfies its own characteristic

0

The Cayley-Hamillon equation:

0

is

(\/>

0

is

In general, possible to form matrices satisfying equations of degree less than n. the polynomial ^(X) of lowest degree with leading coefficient unity for which ^(A) = the minimum polynomial for A, and the may or may not be different from >) equation *(X) = 0,

is

0 0 1

0 0

AX

.

a

0,

/,

(157)

• ■

similar to A, there exists a nonsingular

- A-A

occur among the columns of normal form of the matrix. A

(156)

• •

(158)

X.

The matrix

A is

All latent vectors of

Since

A

of order then /1" = such that

X

0.

/1

matrix

v,

is

If

=

scalar and equal to a latent root X,,

• ■

- X./

"0 and

either

(155)

,Am)

+

A*

.

.

0 1 0

similar to A, where m < n and each Ai or else

(Ai.Aj,

is

= diag

is

A

a

X

is

if

it

is

1

is

0.

is

\j/

0,

a

if

is is

the minimum equation. Then any polynomial for which u(A) = u(X) ^(X) divisor of w(X). If x any vector and the minimum polynomial for A, certainly
/1


(A)

the Jordan


Sec. 3-1]

3-53

A is either diagonal or upper triangular, and its diagonal elements are roots X, of A. If w(X) is any polynomial, »(A) is also upper triangular and One proves inductively that for any integer *, diagonal elements are a>(X,).

The matrix the latent its

A'

=

XA'X-1

that

and hence

o,(A) =

X«(A)X-'

latent roots of a(A) are «(X,). The matrix A is always a diagonal matrix and the columns of X are all latent vectors of A unless the elements of A satisfy certain algebraic identities. Unless such relations or equivalent ones are assigned a priori and imposed, a matrix whose elements are fiven numerically as a result of measurements subject to error may be assumed to be diagonalizable. Only this case will be considered. Even real matrices may have complex latent roots and latent vectors. Hence it is A complex matrix A is said to be Hermilian necessary to consider complex matrices. in case it is equal to the conjugate of its transpose. The conjugate transpose will be Hence the

by an asterisk:

designated

A*

=

AT

=

/ if

is it is

real

i*i

is

if A

it

A

Ax x*Ax

then

= \x = Xx*x

X

is

it

But since

matrix are

p

A

is

is

is

x*Ax when Hermitian. real, and so Latent vectors associated with distinct latent roots of a Hermitian orthogonal. Ixst Ax = Xj Ay = ny ^ Then y*Ax - Xy*x x*Ay = iix*y But

is

UU*

Evidently in case reciprocal to its own conjugate transpose. unitary, orthogonal. Hermitian and Hermitian matrix has only real latent roots, for is,

that and

=

U

U*U

real, on taking the conjugate of the second equation one has

y*Ax

= ny*x 0.

One

1

0

if

This can hold only y*x = x*y = and, indeed, also y*Ax = x'Ay = Suppose a latent root Xi and an associated latent vector u\ have been found. can normalize Uj so that U\*U\ = a

unit vector orthogonal to Ui, u'z a unit vector orthogonal to both Hi and ii'„ a unit vector orthogonal to all preceding vectors. Then

Ui

= (wi,M'a, .

^'-(o1

,«'«)

a)

unitary and the matrix

. .

.

,

.

.

u'j be

is

Let u'j,

similar to A. Hence when one latent root and vector of a Hermitian matrix are known, one can obtain a Hermitian matrix of order n — whose latent roots are the Moreover, since the process can be repeated with A\, this remaining roots of A. shows that any Hermitian matrix call be diagonalized (cf. Art. 3.3). 1

is


Evidently a real matrix is Hermitian if and only if it is symmetric. In dealing with Hermitian matrices it is convenient to say that complex vectors x and y are orthogonal if x*y = 0. A complex matrix U is said to be unitary in case

3-54

MATHEMATICS

[SEC. 3

Given u\, a matrix Ui can be formed as follows: Let Ui be written in the form

«-(:) —

where u is a scalar and w a vector of dimension n

v

=

So,

\t»

-w* \ / — llWW*/

_

(1

1.

Then

_ -)/(1 _ -^

satisfies requirements. If AiVs = Xsf2, then

Hence if and

X»

i is a latent root and i)j its latent vector vector for At, then

X8

is a latent root of A

is an associated latent vector. Nothing has yet been said about finding a particular root Xi and the corresponding vector ui. Let v be any vector, and define the sequence v,

v0 = v

=

A'v

is known that A can be put into the form A =

where U is unitary and A is diagonal. v, =

U\U*

Hence

UA'U*v

Hence each element of v, is a linear combination of »th powers of the X,-. Hence as v increases (cf. Bernoulli's method, Art. 4.4), the ratio of corresponding elements in consecutive iterates approaches the largest latent root Xi and the vector v, approaches the latent vector u\ associated with \\. For the root, let
-

Then having formed v,

one can form

For

a,

= w

= VoTv,

sufficiently large the matrix

(a,

w,

—

0, P,

A

wo

Likewise let

'too

= WoTw,

\

a

v,

is,

which, in general, gives a better estimate of the root. Convergence can be accelerated by using the i* process on the vectors t>„. A quite different device can also be used effectively at times. Convergence is rapid when the Since A — id has the same latent vectors largest root is much larger than the others. as A and the roots Xi — /x, it is sometimes possible to adjust m so as to increase this ratio. If the largest root is a multiple root, it will have associated a characteristic subspace of dimensionality equal to the multiplicity of the root. On proceeding with the matrix At, one will obtain Xi again and a latent vector of At that will correspond to another latent vector of A, that one lying in this characteristic subspace. If the two largest roots are nearly equal but not quite, convergence will be slow. If so, let to be vector distinct from and form the similar sequence

v


It

Av,-\

•*


Sec. 3-1]

3-55

will be singular if the roots are strictly equal, and if not, the roots will satisfy the equation 1

Also if

ii and

0,

<*„

X

o«i

0»+i

X

Q£r+J

&y+2

= 0

xs are the associated latent vectors, then approximately

v, = SiXi'Xi W, = »7lXirXi

+ £i\t>xt + V2*l'Xl

from which equations Xi and xs can be found. Analogous conclusions can be drawn when three or more roots are nearly equal. For nonsymmetric matrices the case is somewhat more complicated because of the possibility of complex roots and because associated with any latent root X will be two latent vectors x and y (more, of course, for multiple roots), which satisfy

Ax

A*y

= Xx

=

\y

It will be assumed that the matrix can be diagonalized. Then it can be shown that if Xi and X» are distinct latent roots, Xi being the latent vector of A corresponding to Xi, y: that of A* corresponding to X2, then xt and yt are orthogonal:


In fact,

yt'x, one has

=

xi*yi

= 0

Ax, = X,xi yt*A yi*Ax, — X,!/2*xi =

whence

= X2!/2*

X^'i,

and the conclusion follows. If A has a latent root \, that is larger in magnitude than all others, then the sequence

u,

uo = u

=

and the sequence v, =

»o "■ v

Au,-i A*v,-i

will approach the corresponding proper vectors, and the ratio of corresponding Indeed, if both sequences u, and v, are elements of consecutive iterates will be Xi. being formed, and if a,

then the sequence

= Vo*u,

= vr*Uo

dy+i/a,

also approaches Xi. Suppose Xi and Xj are equal in modulus and exceed in modulus all other roots. for sufficiently large v one has approximately

u,

=

Then

tiXi'xi + hM'xi

v, = viXi'y, + ijjXj'i/s

Let a, be vectors.

as defined above, and let |8, be formed by means of iterates of two other

Then Xi and

X*

will satisfy 1

a.

(3,

X

Cty+l

X2

a,+i

0*+i 0r+i

= 0

and Xi, xt, y\, yi can be found from the above equations, together with the correspond ing ones using the other iterates. 6

PROBABILITY AND STATISTICS

The development to be given here of probability and statistics will be primarily algebraic in character, with indications, however, of how the theory is extended by the use of derivatives and integrals.

3-56

MATHEMATICS 6.1

[Sec. 3

Basic Principles

The abstract theory of probability is concerned with a space or set 5 (sample space or phase spare) of points Eh Eit . . . which may or may not be discrete. In case they are discrete, then with each point Ei is associated a positive real number p(E,) such that

Zp(£\)

= 1

(159)

the sum is taken over all points Ei in S. Then associated with A is the real number

Let A be a subset or region of 5.

where


p(A)

=

S

p(Ei)

(160)

where the sum is taken over all points Ei in A. The points Ei will represent elemen tary events, the regions A compound events, and the numbers p(Ei) and p(A) their Each Ei is (or represents) a possible outcome of an experiment; the probabilities. event A will be said to occur in case there occur any of the Ei that belong to A. The actual assignment of the probabilities p(Ei) is a physical problem, not mathematical. Thus, in tossing a single coin, one commonly assumes that the outcome will be one of but two possible events, heads, which we denote Ei, and tails, which we denote Et Then S (the possibility that the coin may stand on edge is generally neglected). consists of only two points, Ei and £2. Usually one supposes that p(Ei) = p(Et) and hence that each is H> but this assumption is by no means necessary, and, indeed, experience or observation may suggest the need for a different hypothesis. Suppose the coin is tossed twice. Let E\ and represent the two possible events on the second toss. Considered as a single experiment, there are four possible out Let these be comes.

E\

E"i

—

(Ei,E'i)

E" i

=

(Ei,E\)

E"i

=

(Ei,E'i)

E'\

**

(Ei,E't)

E"i is the event that heads occurs on the first toss and tails on the Then A consists of E"\, E"t, and be the event "at least one head."

Thus, for example, second.

Let A

have equal probability, but again mathe E"i. One commonly supposes that the One could assume that matical probability as such imposes no such requirement. two unlike throws are more probable than two like throws and hence that E"s and E"3 both have higher probability than either E"i or E'\. Such an assumption might require some revision of physical laws but would not affect the laws of probability. If A and B are two subsets Some new symbolism will be convenient in this section. of S, then AB is that subset whose points belong to both A and B, —A is the subset of all points of S not belonging to A, B — A is the subset of all points of B not belong B is the subset of all points belonging to either A or B or both. The ing to A, A assertion A = 0 signifies that there are no points in A [and hence that p(A) = 0); A signifies AB = 0 signifies that A and B have no points in common; A C B or B B — S signifies that, that all points of A belong also to B (hence p(A) < p(B)}; every point of S belongs to A or to B or to both; EitA signifies that the point Ei belongs to A. If AB =0, then

\J

A\J

U«)= p(A) + p(B) U«)= p(A) + p(B) - p(AB)

3

p(A

and in all cases

p(A

(161)

The conditional probability of A given B is

p(A|B)

- p(AB)/p(B)

(162)

provided p(B) ^ 0. In case p(A|B) = p(A), then A and B are said to be (statisti Thus, in the above example of two tosses of a coin, let A be JE" , cally) independent. and E"z (heads on the second throw), and for B take E" \ and E"t (heads on the first


If p(E"i) = for every *, then throw). B are statistically independent. But p(E"t)

=

=

say,

- X - p(B)

p(£",)

H

p{E'\)

p(A)

if,

\i

and

=

-

p(E",)

=

3-57 p(A\B).

Then A

%

Sec. 3-1]

H

= p(B), but p(AB) = p(£"i) = Hence whence p{A\B) = p(A) = and would not be independent, and whereas the a priori probability of heads on the second throw remains J^, the probability that heads would follow heads (granting this new assumption) only J>£. is

A

B

then

Equation (162) can be rewritten

p(A\B)p(B) in

the case

of

B

If

can be continued. one has

f(iU

C) = p(A) + p(B) + p(C)

one replaces

by

(164)

B"

B\J

—

C

If

p(A\BC)p{B\C)p{C)

- p(BC) - p(AC) - p(BC)

in Eq. (161)

+

=

and expands,

and the process

p(ABC) (165)

is

it

The extension Eq. (165)

to more complex cases can be made in like manner. necessary to note that

A(B These

\J C)

=

ABKJ AC

In

order to obtain

ABC

=

(AB)(AC)

relations are readily verified. ,

is

a

A

A

is

that of sampling from type of experiment often considered in probability theory population. population of n consists of n objects aj, o2, . . . a„ that are at least Let the event Ei represent the drawing of a,-. To say that conceptually distinct. the objects have equal chances of being drawn to say that all p{Ei) are equal and

hence

that

p{Ei) = n-' 1

A

i

r

may > sample of (other assumptions are possible and often true). If with replacement, then one may drawn either with or without replacement. (but need not) assume that before the second drawing conditions are restored to their former state. If so, one may let En represent the event of drawing oj on the first = trial, a,- on the second, with not excluded, and all have equal probability,

for every

if

j

i

be

then

p(Eit)

= n~*

-

+

1)

• • •

r

- n(n -

(»

*r

1)

a

is

of n

r

More generally, any particular sample of drawn with replacements from a population If the draw has probability n~r, granted the above assumptions of uniformity. ing without replacement, no o, can be drawn twice and a total of ■»

'P,

r!

is

distinct drawings are possible, account being taken of the order of the drawing (in stud The probability of a particular poker, for example, the order influences the betting). 1/n,. But the same final sample (e.g., the same hand in bridge, where drawing = 13, re = different ways. Hence there 52) could have been drawn in any one of

r

are

=

\n

- rj

n

\

n! Vz (A W _ - r!(n-r)!

(

M

r\

„c

'


= p{AB)

by B' = BC in Eq. (163), the formula can be extended to

p(ABC)

BKJ

(163)

independence,

p(A)p(B) one replaces

p(AB)

B

which becomes,

=

3-58

MATHEMATICS

[Sec. 3

distinct possible samples of r, no account being taken of order. are n, permutations of n things r at a time and I

1

One says that there

combinations of n things r at a time,

the permutations distinguishing order, the combinations not. If all combinations have the same probability of being drawn, then the probability of any particular one

"'/(?)■ Drawing

a sample of r from a population of n is equivalent to separating the n objects into two classes, one of r objects (those drawn), the other of n — r objects More generally, n objects can be divided into fc classes with rt (those not drawn). in the first, r2 in the second, . . . , r» in the fcth in

n!rj!

• • •

rt!

distinct ways, it being understood that

U +

r%

+

■■■

+

rk = n


If there are n Important in nuclear theory is the placement of objects in cells. objects and r cells, and if no cell can accommodate more than one object, then, if r r > n, there are ( I distinct ways of accommodating the n objects. If all arrange ments have equal probabilities,

the probability

of any particular

one is 1

/ ( r\>

it

If any cell can accommodate being assumed that the objects are indistinguishable. any number of objects, each object has r choices, independently of where the others have gone, whence there are r" arrangements and any one has a probability r~", although not all these are distinguishable if the objects themselves are not distinguish able (if two objects exchanged places one would not know it). The number of dis n

tinguishable arrangements is ( Bosc-Einstein

The

)>

and if these are equally probable (as in

statistics), the probability of any one is

1

/I

)•

example of independent events. Note that if independence is assumed (as it normally is) in tossing coins and dice, tossing a coin any number of times is equivalent to drawing from a population of 2 with replacement and tossing a die is equivalent to drawing from a population of 6 with replacement. In drawing without replacement, however, the outcomes of successive drawings are not independent. To return to the general formulas, Eq. (160) can be generalized also as follows: case of drawing with replacements provides a typical

p(A

If and if BiB,

U

=

B\C)

p(A\C) + p(B\C)

- p(AB\C)

B.WB!U--WBl=S = 0 for every

Hence

p(A)

From the identity

i

and

-

p(Bi\A)

\J

then also

(ABi)(ABj)

\J

= 0 for every t and

j and

\J

■■■ ABk = A ABt p(ABt) + p(ABi) + ■ ■ ■ + p(ABt) p(4|B,)p(B,) + ■ ■ • + p(A\Bk)p(Bk)

ABi

=

j,

(166)

=

piB^/piA)

=

p(A\Bi)p(Bi)/p(A)

one deduces Bayes's rule k

p(BAA) =

[p(A|B,)p(B,)l

p(A\B<)p(B [p(A|B,)p(B,)l (167)

l

(167)

3-59


Sec. 3-1]

When the points in the sample space are not discrete, then one assumes a probability The integral function whose integral over the entire sample space 5 is unity. over any subspace A is the probability p(A). The case can be reduced approximately to the discrete case as follows: Divide S into regions A such that each p(A) is small; from each such region select a point E and associate with E the probability p{A). density

Moments

6.2

Let a number x; be associated with each point Ei of a sample space S. This asso ciation defines a function X over the sample space in the sense that the function X will be said to take on the value xt when event Ei occurs. Such a function X is known as a random variable. For a given number x, let A represent the set of all points Ei = x. for which If is not included among the x,, the set A is vacuous, but in any

i

ii

event

p(X

= x) =

V

p(E<)

(168)

A is the

probability

that

X

will take on the value x.

p(X


otherwise the sum is taken over all Let p represent the vector whose

D,

If A

is vacuous, then

= x) = 0;

Ei in A. ith element

is

p(Ei), and let

= diag (zi,xs, . . . ,x„)

represent the diagonal matrix of the values x, assumed by the vector each of whose elements = 1, the rth moment of Mr =

X.

X

Then if

e

is (as usual)

is

eTDx'p

(169)

This is also called the expectation of X' and is sometimes denoted E(Xr). The first moment m = m is called the mean, and the zeroth moment mo = eTp = 1, since this is the sum of the p{Ei). For any X the rth moment about X is

a'r = er(D. = Mr Of

- \I)'p

- r\Mr-,

+

particular interest are the moments almul •v =

«r(D.

XS"-» +

■• •

(170)

the mean

- m/)'P

(171)

Evidently r, = 0. The second moment about the mean,
-

which expresses the variance in terms of the moments m and miSince the same formula can be written with n't and m'i replacing mi and mi, it follows that the second moment is least when taken about the mean. If Y is another random variable defined over the same space S, one can define its Let m<, m», »», o-„ represent the means and standard moments in the same fashion. deviations of the two random variables. Then the mean of the sum X + Y is the sum of the means: m*+, = eT(Dx + Dy)p m* + m, (173)

-

The variance of the sum is somewhat more complicated. of A' and Y by

First

define the covariance

3-60

MATHEMATICS

- n,I)(D, - ji„7)p - n,eTD„p - itveTDIp

= eT(Dx = eTDxDyp C» = eTDxD,p cxy

and the correlation

-

(Dx

+D,

+ ji*M»erP (174)

m«m»

r,„ by =

ffi»»r»,

Then since

[Sec. 3

- »A - m,/)"

- ,x,/)2

(D,

=

it follows that the variance of

X

c„

(175)

- mz/)(D» - nj)

+ 2(DX

+ (D,

- „,/)»

+ Y is given by = ax% + at* — 1a±ayrxy

When

rXy

Y

= 0, the variables Ar and

(176)

are said to be nncorrelated

or statistically inde

pendent, and then one has simply = a,* + a,*

ax+y'

rXy = 1, the variables are proportional, and in all cases 1 > rxy > —I. All the above definitions apply whether the probabilities p(Ei) are known a priori or empirically. By the latter is meant that in N trials one observes that the event Ei occurs ni times and then takes p(Ei) = ni/N. When the sample space S consists of infinitely many points Ei, whether discrete or not, the definitions have a natural

-

p{Y

t/)

extension but require the use of the notion of limits. Among the values assumed by the random variable Y, let y be a particular one, and let B represent the set of all Ei with which this value y is associated. Let e« represent the vector whose ?'th element is 1 if EaB but is otherwise 0. Then

- eBTp

p(B)

=

was said that

Y

and

is

if

is

1

are independent in case =

otherwise 0.

Then

p(AB)

=

p(AB)

= eABTp =

but

y)

5.1

= x,

EitAB

B

In Art.

it

p(X

represent the

p(A)

= eATp =

represent the vector whose i'th element

A

eAB

x)

p(X » Let

ex

is

if

1

is

A

Likewise let represent the set of all Ej for which X = x, and let EjiA but otherwise 0. Then vector whose j'th element

p(A)p(B) is

if

it

Y

e

Y

;/

B

.

A

A

At, . . are the sets If associated with all the distinct valuesxof X; Bi, Bt, . . . of Y; and the sets associated with all the distinct values every Aa and Bb arc independent. The proof can be made independent, then the variables X and by expressing as the sum of all the vectors e^s in the expression eTDxDyp, which can then be shown to reduce to n*iiy, whence follows that A' and are independent. i,

5.3

Distributions

q

p, pi

«■ p,

and
-

p* =

p(l

-

p)

— m

p

Hence

S)

-

■>■

(J

1

p

1

Consider an experiment with but two possible outcomes, "success" or "failure." — p, respectively. = these have probability and If E\ denotes success and = = Then let and 0. failure, Ex x\ xt

Let

p

= pq

Now let the experiment be repeated n times. Altogether there are 2" possible distinct outcomes for such an experiment. Let Xi have the value in case of success 1


When


Sec. 3-1] on the ith p(Xi = 1)

3-61

trial, 0 otherwise, irrespective of outcomes on the remaining trials. = p, and

X

=

ZX,

variable whose value is equal to the total number of successes. number of successes is = S/iXi — np

is a random

^

since each

= p.

Likewise, since any two
a set of

k,

(lb)

°f selecting

p(X

are obviously independent,

1

k)

=

.

and the rest 0 is pV~* but there are

whence the probability of exactly

successes, without

is

specifying when they should occur,

Then the

= npq

that any k of the Xf =

Now the probability ways

Sffij'

Xj

and

X>

k

mean

Then

= b(k;n,p)

pV-*

=

(177)

known as the binomial distribution, and has the mean np and the variance an a function of the argument and the parameters n and p. This example of a distribution function whose argument ranges over a finite number of discrete values, all integers from The parameter n can be any posi to n inclusive. tive integer, and to can be any number on the interval from Of considerable importance in the theory of radioactive decay the Poisson distribution p(k;m) = m*exp (-m)/JfeI (178) is

is

is

k

if

A;

is

Here the argument and can have any nonnegative integral value, while the parameter to can have any positive value. This represents the probability that counts will occur in a given interval of time the expected number of counts exactly (the mean) in that interval Both the mean and the variance of this distribution to. have the value to. An example of a continuous distribution given by the normal (or Gaussian)

and the

=

V(i)

normal distribution function *<*)

(2,r)-»exp (-x»/2)

-

f'm

function

density

(179)

#>(») dy

(180)

h)

a

is

h,

is

h.

is

is

is

A

of

a

a

if

of

variable

(x

h

is

a

it

is

h

i

the

+

a

zero mean and unit variance. If variable normally distributed, then will assume a value on to probability that particular interval from x — — — more precisely equal to *(x + approximately 2hip(x) and h). The approximation valid only for small but the expression by means of * rigorously correct for all common application of the normal distribution the following: Suppose that a series of repeated measurements made for the purpose of ascertaining the magnitude given quantity (length, duration, weight, or what have you). Assuming the measurements to be independent of one another and free from bias, they can be expected to scatter in some fashion about the true magnitude. Let to be the mean these measurements, and let y, be the deviation of these measurements from the mean. Hence the t/,- constitute an empirical distribution with a mean of 0. Let can be supposed to have been drawn at random from normal population. Now each y, can be regarded as a value assumed by a random variable F,-, and the are independent. If an are any fixed scalars, one can verify that for the random This has

y'i


is

1.

p

0

0

k

This

npq.

is

it

is

This

3-62

MATHEMATICS

[Sec. 3

one has, generalizing Eqs. (173) and (176),

In particular, if there mean.

Since

are

N variables Yi and one takes each the same for all, one has that

of

<*,•=

1/N, then

=
z

is the

(181) is

is,

the variance of the mean of .V independent measurements N'1 times the It also true that even though the yi may not themselves be normally distributed, in the limit for large jV the means are normally distributed. One can now ask by how much the mean m of the N measurements could be expected to deviate from the true value of the magnitude, and the question can be answered in the following sense: Suppose the true magnitude were given by some number m'. Then an error of m' — m has been made. If one made many sets of experiments, each set consisting of taking -V measurements and finding the mean, these means define random variable M whose mean m' and whose variance a*/N. Hence he random variable =

of Eq. (180). therefore 2*(x), where

is

m|

t

y/N/a

x = — |m' —

as much as

\/iV/
NOTES AND REFERENCES selected bibliography relating to the material of this section No given below. attempt has been made to select from the countless elementary texts on algebra, trigonometry, and analytic geometry. written as a text, Although Ref. contains to be found in American texts on the subject. considerably more material than Reference may be considered to be the definitive treatise on manipulative algebra, as are Refs. and, in some sense, 16 in their respective areas. Much of the older theory of determinants rephrased in more recent books as theory of matrices. Reference 15 indispensable for those interested in the manipula tive theory of determinants. While Ref. brief and elementary, nevertheless includes a considerable amount of material. Reference 20 the classical treatise on computational methods. Among the more recent publications Refs. and 13 give detailed computational layouts while Ref. 10 stresses the theoretical background. Among the many books on probability and statistics, most stress particular fields of No attempt has been made to select from these. application. Of the more general developments Ref. noteworthy for its success in combining rigor with lucidity.

it

is

it

is

&

2.

Aitken, A. C: "Determinants and Matrices," Oliver Boyd, Ltd., Edinburgh and London, 1946. Burnside, W. 8., and A. W. Panton: "The Theory of Equations with an Introduction Co., Ltd., London, to the Theory of Binary Algebraic Forms," Longmans, Green

*

1.

is

6

8

is

1

is

is

9,

2,

3

is

7

is

A

1886.

Chrystal, G.: "Algebra," A. & C. Black, Ltd., London, 1893. 4. Courant, Richard, and David Hilbert: "Methods of Mathematical Physics," vol. Interscience Publishers, Inc., New York, 1953. 5. Cramer, H.: "Mathematical Methods of Statistics," Princeton University Press, Princeton, N.J., 1945. 6. Feller, William: "An Introduction to Probability Theory and Its Applications," John Wiley
1,

3.

&


—

- mO

The probability of being in error by

has the distribution \m'

(M

is

is

is

a

X

m|

that

mean of each.

Sec. 3-1]


3-63

9. Hobson. E. W.: "A Treatise on Plane Trigonometry," Cambridge University Press, London. 1897. 10. Householder, A. S.: "Principles of Numerical Analysis," McGraw-Hill Book Com pany, Inc., New York, 1953. 11. Kendall, M. G.: "The Advanced Theory of Statistics," Charles Griffin & Co., Ltd., London, 1947-1948. 12. MacDuffee, Cyrus Colton: "Vectors and Matrices," Mathematical Association of America, 1943. Calculus," Princeton University Press, 13. Milne, William Edmund: "Numerical Princeton, N.J., 1949. " 14. Morse, Philip M., and Herman Feshbach : Methods of Theoretical Physics," McGrawHill Book Company, Inc., New York, 1953. 15. Muir, Thomas: "The Theory of Determinants," Macmillan & Co., Ltd., London, vol. 1, 1906; vol. 2, 1911; vol. 3, 1920; vol. 4, 1923. Green & Co., Inc., New 16. Salmon, George: "A Treatise on Conic Sections," Longmans,

York.

1929.

Schwerdtfeger, Hans: "Introduction to Linear Algebra and the Theory of Matrices," P. Noordhoff N.V., Groningen, Netherlands, 1950. Matrices, and Invariants," Blackie 18. Turnbull, H. W.: "The Theory of Determinants, Son, Ltd., Glasgow, 1929. 19. Turnbull,' H. W., and A. C. Aitken: "An Introduction to the Theory of Canonical Matrices,' Blackie & Son, Ltd., Glasgow, 1932. 20. Whittaker, E. T., and G. Robinson: "The Calculus of Observations," Blackie & Son, 17.

*


Ltd.,

Glasgow,

1940.

3-2

ANALYSIS BY

Ward Conrad Sangren 1

DIFFERENTIAL 1.1

AND INTEGRAL CALCULUS Differentiation


The derivative of a function is the limit, if it exists, of the ratio of the increment of the function to the corresponding increment of the independent variable when the increment of the independent variable approaches zero. The increment of a variable There are many notations for is the difference between two values of the variable. the derivative of a function y = /(x) with respect to x. Among these notations are

Pdx The definition of

a derivative

f

(*,)

y'

Dy

f

D,y

f'(x)

ax

can therefore be expressed,

-

lim

/(*'>-/(*■) Xt — X\

ari-.xi

.

assuming the limit exists, by

&

lim Ai-»o Ax

ii

Note that the derivative is defined only at a point and that the notation /'(x) implies that the derivative is defined at each point xi of the set of values of x under The process of finding a derivative is called differentiation. consideration. The derivative has both geometric and physical interpretations. The common geometric interpretation is that the derivative /'(x) represents the slope of the curve Physically the derivative is often inter y = /(x) at the point (x,y) on the curve. The velocity at any instant of time is the derivative of the preted as the velocity. distance with respect to the time. Since the derivative of a function y = /(x) with respect to x is also a function of x, it may also be differentiable. The function df/dx or /'(x) is called the first derivative of /(x) and its derivative, denoted by (d/dx) (df/dx) m d'f/dx1 or /"(x), is called the second derivative of /(x) with respect to x. In similar fashion further differentiation A few of the notations for the nth derivative of y leads to higher derivatives. f(x) are

"

pL ax"

„

*L

,<*,(*)

ax"

The concept of a partial derivative of a function of several variables corresponds to the fundamental concept of the derivative of a function of one variable. Let v = f(x,y), and take y to be the fixed value yr, then since v is a function of x only, its derivative with respect to x may be defined like that for an ordinary derivative. This derivative is called the partial derivative of v with respect to x and is denoted by such symbols as dv

£

df Tx

"■

8-64

.

U

p

analysis

3-65 is,

The defining equation for this partial derivative

/(*,

lim

„_//.,„ _ \

lim Air-.o

~

Af)

defined by

/fallgl)

Ay

v

c

= »(x)

(u + + w)' (uv)' = w'w + uv'

- nu*-1

V'

w'

(cuY

/fall

- «' +

v

1.

with respect to

Rules

General Differentiation

1.

u = u(x) 2. (u»)'

-/fa.,y.)

and Powers:

Sums, Products, Quotients, = cu'

Ar

In similar fashion the partial derivative of

Table

+ **,

y

m

+

yi)

v

Mxi

therefore,

is

Sec. 3-2]

+

= constant

«>'

u»

I

dx

dy

"

ax

dA/<>y

/dw\s Vdx/ \dx/

. .

=

= a,

*

*

0

/(x,y,z,

dv d*u

du dx'

/„

*

dy

d*»

du1

0

-3h

=

-

dx1

.)

h(x,y)

_

d!v

'

0

Implicit Functions:

(ir

_

dg

du dx

—

_

/.

_

u = ^(x)

and

f(u)

/»

du dx

df

dx

-

0

g

-J-

= — a, dx
13 '

dx

T" dx

•/«<*)

, .

16-

of

/"*/(') a

J

15.

4~

Differentiation

0

0(0

a,1

dx1

= *(n,n)

de dx

=

~ a"^'

—- =

= *>(x,y)

du dx

o«)

dy

whore u = u(x,y) and du dy

t>

—

dx

-d'x/dy* (dx/dy)'

a Parameter: x =

12 '

=

dx1

= i»(z,y)

dv dy

an Integral:

««

=/(x) d'

"

0'fa)fffa>f»)

- «'fa)fffa.«)

+

Functions

^

n'

dx/dy

*

^

=

dx

y

^

of

10

^

x = x(y)

0

Inverse Functions:

/


A)

dv du

dy

v—

Functions: v

of

Function

In u

^

5. («")' = u»(»ln u)' =

uv"

+

4. (uv)" = u"u + 2uV

+

*©'-H)' ©'-^

Mr ?fa.O # Ox

3-66

MATHEMATICS

[Sec.

3

A geometric interpretation of these partial derivatives is much like that for an The partial derivatives f* and /„, where v = f(x,y), can be ordinary derivative. interpreted as the slopes of the curves of intersection of the surface v = f(x,y) with the planes y = constant and x = constant, respectively. Since the first partial derivatives fx and/„ of v = f(x,y) are also functions of x and y, it may be possible to obtain the partial derivatives of /, and /, and thus obtain second derivatives. Among the notations used for the second partial derivatives are the following: a dx a^

ay dx

W

fdv\

= ~

\dy/

( —\ \dy/

aH'

ax'

"=

a'f to'

dy' =

f s/" =

Hr

dy'

a*" dx dy

mf "

= g =

j"'

m

a%v

dy dx

„ A (—} dy \dx/

The cross derivatives fly and /„, need not be identical, but if they are both continuous, be shown that they are identical. Higher-order partial derivatives are defined by repeated differentiations with respect to any of the independent variables. For functions of three or more variables, partial differentiation is carried out by holding constant all the independent variables except one and then taking the ordinary derivative of the resulting function of one variable. The differential of the dependent variable y = /(x), denoted by dy, is defined by

it can

- f'(x) Ax

It

is convenient to define the differential dx of the independent variable x as equal to its increment Ax. The total differential of a function v = f(x,y) is denoted by dv, and is defined by , dv

=

av A — Ax dx

. -\

av dy

Ay

Again it is convenient to define the differential of the independent variables x and y by

dx = Ax

dy = Ax

and

The definitions for differentials extend straightforwardly variables. 1.2

to functions of three or more

Integration

The concept of integration, or the integral concept, involves, as does differentiation, In most engineering problems the only type of the application of a limit process. In some applications in physics and integral encountered is the Riemann integral. mathematics it is desirable to consider other integrals such as Stieltjes integrals and The term integral, by itself, will refer here to a Riemann integral. Lebesgue integrals. Consider first a real-valued, bounded function /(x) defined 1.21 Definite Integral. Next divide the interval a < x < b into an integer on the interval a < x < b. number n of subintervals where the points of division x< are such that

Now form the sum

/(x,*)(x,

- xo)

< Xl ■ • • <

Xi-l

+/(x,*)(x,

- x,)

< Xi ■ ■ ■ < xn-i

+

• ■•

where Xi* is any value of x in the ith subinterval; that more conveniently expressed by

y

n

1=1

/(x.*)(x,-

- *_,)

<

+/(x„*)(x„ is,

O = Xo < Xi

is


dy

x,_i <

x,

= 6

- x„_,) x,-*

< Xi.

This

sum

analysis

Sec. 3-2]

3-G7

Finally let n increase without limit, and let the length of each subinterval Xi — x,_i tend to zero; then the value of the sum may approach some limiting value. When this limit exists, it is called the definite integral of f(x) from o to b and is denoted by 6

This definition

/(*)

dx

by

is expressed

- Xi-i)

n

fbf(x)

y

dx = lim

•

f(xi*)(xi

-1

Although the limit is indicated with only n tending to infinity, the important aspect of the limit process is that every subinterval Xi — tend to zero and therefore n tends to infinity. The most common geometric interpretation of a definite integral is that of an area.

If

y

where

—

/(i)

/(i) dx is the area under the curve y J is bounded by x = a and x =

> 0, then the

—

f(x)

v

Some of the important properties of definite integrals are given in Table understood in this table that the functions are bounded and integrable. and Improper

J

then the integral

fit)

Integrals.

dt, where x

Let f(x) be bounded and integrable;

any intermediate value in the interval (a,6),

is

Indefinite

is

1.22

It

is

di

2.

6

is

/>

if

is

a

is,

is

often written without the called the indefinite integral. The indefinite integral limits of integration, that This notation arises from converse theorem //(<) dt. a con to the fundamental theorem of integral calculus, which states that fix)

tinuous function in the interval, then the indefinite integral

An integral

called

fx,-n

~z

in the interval

/(<) dt

(a,6),

fb

exist.

then the

has

/(<) dt

a

/(<) dt exists and

necessary

is

J/

is

it

If f(x)

lim

that both limits

an infinite

said to exist

discontinuity,

if

J

lim

z-> »

is

x0

and

f(t) dt exist,

b

at a point

dt

In order that

said to exist

f

/(<)

/(<) dt

I

J/

/

J*

a

if

if

is

The improper integral

J/

.

in an interval,

it

(x) has fix) as its derivative for all values of

f(x).

defined as that limit. lim

J

F(x)

the interval of integration becomes infinite called improper either the integrand become infinite at one or more points of the interval of integra

tion.

x— ■

is

c

dx =

is

J/(x)

c

or

is

if

or

/(<) dt has the

+

- F(x)

x

dx

F

a function

a primitive for

Since F(x) where any constant, also has possible to write without confusion c,

fix).

as its derivative,

J/(z)

If

=
is

d

=

it

derivative

/(i)

both limits

«—

is

/(<) dt and lim

it

/

J"

/

assigned the value of the sum /(<) dt exist and «-*0 /«•+« Improper or infinite integrals are, therefore, limiting cases of of the two limits. proper or finite integrals.

lira

o


s

a

is,

and above the x axis which b. A customary physical interpretation of the definite integral relates distance and = f(t), then the velocity. If the velocity is given as a function of time, that covered from time to time given by distance

3-68

MATHEMATICS Table 2. of Limit*:

Interchange 1.

=

Jbf(x)dx

-

[SEC. 3

Formal Properties of Integrals

fl/Wdz

Addition of Intervals of Integration: 2.

dx =

Jbaf(z)

I ndependence

-

dx

fbf(x)

dx +

/*/«

dt =

fbaf(l)

dx

of Integration:

from Parameter

/*/(*)

3.

[CafW

of*

etc.

Linearity of Integration:

/

4.

fba

5.

IMx) + f,{x)\ by Parts:

Integration 6.

f

c/(x) dx = c

dx

Jbaf(x)g'(x)

-

-

dx

P/i(x)

dx +

-

[/(x)ff(x)]*

[/(x)

where

c is a constant

f(x) dx

/*/«(x)

dx

y*/'(x)8(x)dx

- f(a)g(a)

- f(b)g(b)

=

JbafWdx where »(()

Absolute-value

|

8.

JdAaU)]g'(t)dt

= x, »(d)

—

6,

7.

-

and g(c)

a

Inequality:

JbaXz) dx|< jha\f{x)\dx

x

ff, **

/a

if

/o

'

* /"ft

x <

dF/dx

1, y/

6

For g(z) =

°

/(x) dx

~

,7(6) -

A*)

and monotonic non decreasing,

dx

where/

is

for all x in (a,6), then

fXf(t)

dt

-

F(x) - F(a)

integrable,

and where a <

there exists a function

<

6

dx

9

bounded

Ax)

is

g(a) •

Fundamen/ai TAeorem o/ Integral Calculus; bounded and integrable in the interval (a,b) and /(x)

- /(x)

< b.

M(b — a), where m and Af are lower and upper bounds

/J

-

dx

if

where g(x)

dx is

jbaf{x)g{x)

f(z)

and o <

Value:

/"'

Second Theorem of Mean

a

in a < x <

6

<

for fix) in a

JI

f(8)(b — a) and m(6 — a) <

(x)

<

continuous and

6

o(x) dx > 0

- /(0)

+

fix)

dx

g

/"*/(*)*(*)

/*

Theorem of Mean Value:

where

12.

tora
fix)
„, J fb /,(l) d*

■

11.

-

, . "|i ^ dx

is

Firtt

/„(z>*

]

[

10.

o(x) dx

Inequality: fb

, ,

Schwarz's

f(x) dx <

/a

9.

jha

f*

Dominating-function Inequality:

If


Change of Variable:

Fix) such that

Sec. 3-2]

Differentiation

1.3

Table 3.

Ax)

No.


Differentiation

dz

CU

CU

3 4

u + v + to uv

u' + »' + w' ttn' + inj' — Uv'

c

dx

V1

9

w

6 7

W

ra>-iu< 4. u.„' ln u

8

fM

d/(u) -LL-i u',

9

e*

e'u'

10

In u

l-W

11 12 13 14 15 16

sin u cos u tan u cot u sec u CSC u

cos u • u' — sin u ■«' sec> u • u' — esc* u ■u' sec u tan u • u' — esc u cot u • »'

1. 2. o 3.

j J J[ /

7

du

u

Table

/(x)dx

-

"

ex

c dx

- e jZ

f

-

n ¥■

n + 1

-(1

21

sec"1 u

u-i(uj

22 23 24 26 26 27 28

C8C~>W sinh u oosh u tanh u ooth u secli u esch u

-u-'(u'

29

sinh-1

30 31 32

cosh-1 u tanh-1 u coth-1 u

33

sech-1 u

34 36 36 37 38

csch-1 u ln sin u ln cos u ln sinh u ln cosh u

b"

8.

9.

y

dx

a-'t"

dx

(o ln 6)"1k"

-1

ln x dx ■ x In x — x

Forms Containing a + 6x: 10.

J/

^

—

dx

- - In

2—

dx

-

a + bx

11.

J/

a + bx

12.

J/

(o + 6x)>

b

dx

i

b1

-

(a + 6x)

[a + bx

- a ln

u

A Short Table of Integrals

dx = In

7.

tan-1 u cot"1 u

(a + 6x)]

-! I ln (o 4T bx) H o + 6« L

—1 6xJ

/'(x)

u«)-Wu'

(1

cos"' u

J- i Jim\-jwmdz.luM J f -j 5.

-

sin"« u

du . v — dx dx

-

-£/(x) dx

17

u«) df

/

dx

4.

« = 2.7182818285,

18 19 20

fX/(t)dt

dv . u — dx — uv — I dx

i«

Ax)

du

2

u

Formulas

No.

0

c

5

and Integration Tables

u, v, and w are functions of x. toga = In, M = logic e = 0.4342944819)

(e and n are constants,

1

3-69

ANALYSIS

-

u»)-*V

(1 + u«)""u' -(1 + u»)-V

_

i)-Mu-

-

1)-W«'

cosh u ■u' sinh u • u' sech* u • u' - csch1 u • u' — sech u tanh u • ti' — csch u coth u ■u' (1 + «»)-Mu'

-

l)"û' - u')-'u' -(u> l)-'u' -u-'(l - u»)-*V

(«« (1

-u->(l + u«)-Wu'

cot u • u'

tanh u ■u' ooth u • u'

3-70

MATHEMATICS Table 4. *'

13.

J/

a 4- bx

14.

//

— dx (a + bx)'

/_L / _-L_

15.

i \l

dx =

b'

17. 18.

./

/ // 7

- i In

_J_

+ bx dx =

20.

JfxVa-+Yxdx

23.

*■

Va

'/ Va

+ 6x

-

-

„.

27.

/

'/

+ 6x + a

2

Va

+ bx

1

+ bx

i Va

1 . -— In

/Va

+ bx

-

VaA

VVa

+ bx +

Va/

1

Va

+ ox

- ~ tanh-"

dx

Va

+ bx

Va

\

x(a + 6x)=Wdx

=

1

.. /a + »* o

+ bx

b ~a

+ 6x

iI

i r
+ bx)'

Vo-T^

36'

. dx

Vo

tj)

2(2° ~

\/(a

'f x Va^= + .

i.w

1

J/■ xVa

Jf

12abx + 156'x>)

Va

.(j-

29.

-

2(8a'

V^^T^

156»

dx = 2

VST

■» i"

~ 3fa)

2(2°

+ bx

26.

-Lln»_±^

+ 6x)»

36

y^= / Va ./

6l

. a 4- 6x h — In a1 x

~ V(a =

j x'Va + bxdx f

^— 1 a + bz-1

2a In (a + bx)

4- bx:

y/

22.

J

2a(a + bx) + a' In (a + 6*) 1

a + 2bx , 26 . a + bx In H a» x a'x(a 4- bx)

dx =

Va

(Continued)

.6

1 ax

i»(a + bx)1-1

Va

a +

a (a 4- bx)

. dx =

19.

21.


'

x'(a + 6x)

-

dx

-

-

\ a + bx

dx =■

Ha + bx)'

Containing

Forma

-b'L

-

bx)

IB.

A Short Table of Integrals. (a + bx)'

2

[Sec. 3

4 ± n

,is

xVa _ a(a

+ bx +

taWI

J

2±n

Forms Containing c* ± x1 and Xs — c*; 30.

1

7 C + xt

/-L_ / ./-L_ /

31. 32

Forma

dx =

- tan-1 -

r'

. 1 . c 4- x dx = — In x« x 2c c

x»

c»

-

dx

X' x' '< c»

c

-

1 x <• — In 2c X 4- c

iooth-'-

x- > c'

Containing a 4- 6x and c 4- ex:

33 .

//

(a 4- 6x)(c + ex)

34.

Jf

(a + bx)(c 4- ex)

35.

Jf

86.

Vc' + tanh'-

/f

I

»

-

dx = dx

_dx

! (a + 6x)>(c + ex) x

—— (a 4- 6x)'(c + ex)

=

. dx —

,„ —L_ (1+^) Va 4- ex/ ae — c6 ?

- In

f ae-c6L6

(a + 6x)

- - In e

J

(c + ex) 1

_L_('_L_+— ^—Inl^ + bx' + ae — c6 Va

6x

ae — c6

—a

6(ae

- c6)(a

4- bx)

c.c4a

In

(ae

c6)«

c 4- es

a + 6x

analysis

Sec. 3-2]

A Short Table of Integrals.

Table 4.

Va

Forma Containing

j

1

Vu

-

39. A — L_ dx

VU

J Vx'

44.

/f

Vx'

I

Vx'

'f

x

Vx'

— a'

1

+ a'

48.

If

V'J>

i

J

V(x'

'

Vx'

J 52.

54.

"'

~

-

=

dx

-

Vx«

dx

*

b"

± a')]

gec-'-

V*+*\

+

/

«

a'

Vx'

-

a In

+

(°

a cm " ? x

^)

_±z o' Vx» ± a>

- - Vx'

Vx«

± a' + — In (x +

± o>)

±

x'Vz'

Va'

Va'

=

x

-i—

56.

/

iz

/ V(o> -

■»

A*

=

x«)'dx

— x')>

+ In (x +

VxTTT')

— x':

(x

Vo»

- - iln ■

^)

(lWEII*) / V 1

VaTTT'

-i

dx =

— *• + a' sin"' or — cos-1 -

■dx = sin

/35T

± a»

- VfE!!'

— x' dx = 1

V(a»

Vx'

:dx = T ± a'

dx '( x Va> - x»

58.

tan"' • tan-' \/-r^

ax

or

—

VxT + ^

± a')>

55.

57.

-

eiM"'

(

dx =

JfVz'±a'dx z'

j

/2

V-ke

Vu

± a')

ax -

dx = 2 In • V

'

Form* Containing 53.

=

± o> ± o« In (x +

Vx«

In (x +

dx =

/ ^"'dx /

-

± o«

x

/•

- i [x Vx«

dx

Vx»

47.

so.

bVi)

i

V-ke

± a':

± a« dx

1

./


43.

4».

du +

tan'

tVP

Formj Con/atntnp

if,

(V6

In

2

V-ke

V*ee

cb ae + -\- c6 - ^ - at f— /" 2(>e ^ Vur fce 'I -^dx Vuc

' iVui

4o.

Vie V*e —

- a In

V(a»

a« VaJ~ fa'

-

- x'

(°

x')>

± +

40.

Vu~,

Vke

—

-

i « Vu In e Vu +

2"* VaT^T'

22.'si„-.f

J]

J

1 -—

dx =

8oe y

+

o

(Continued)

+ ex, u = a + bx, o = c + ex, k ** at — be: k' /■ dx

2

/■

'/

k + 2bv î— ■Vu» 46e

4 L |~x

.o

38.

Vui> dx =

Vc

+ bx and

2

3,/

3-71

a

3-72

MATHEMATICS Table 4.

59. 60.

62. 63.


- z' dz - - - V(a' -T^ji + - (z Vo« - x» + l' — - idz - - - Vo« - x» + - rin-i 5

/ 7

61.

Va

-'

-

L

f^",

2

- V°^

dx =

J/" x> va«~

Vo»

-

Va«

-

— dx =

y/"

*

- 1')" dz

V(o'

Va

X»

sin-'

~

+ 6x + ex1: A" = a -f- 6x 4- ex*

/ VX V.Y

I

/ Va Generated for wjivans (University of Florida) on 2015-09-23 02:47 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

\

Vc

/•_L_dx--L.inh-(/

V

(Continued)

x> Va*

Forms Containing

fiS.

[Sec. 3

f

Vc

2"

V^c

!

-

—

VvV

/ VXdx

(2rx + 6)

VA

if c > 0

^ 6>/

- 4ac/1

2(2cx + 6)

dx

- 6J

2vV/ + »

\V4oc

ax = —-=L sin ! I

4oc

1

if c < 0

J

2kl y/x dx

VX .

y/■_^_rfx -x/a7

/ 2.

73

74.

75

76

-

. dx

dx

2c y

e

Va-

a*

^-A/-

=

2(fcx + 2a) (,

Vx

Va /xVvdx=^Y_^/'V*dx 3c 2c J a

7

'

x

v/a

^J_ JJ zy/X If

■/

1

xV.Y

.

. dx =

_

V-o 2

Va

Trigonometric

*

V

„„-,

(

2

2J VX — Vr 1

/ sin x dx = — cos x

70.

/

80.

/ tan xdx

81.

/

J

cos x rfx = sin x = — In cob x = In sec x

cot x dx — In sin x = — In cbc x

\

4ac/

xV.Y

ex T 6 . _, sin ■■ \Zbs + ac

Forms:

78.

-

if a = 0

**

, - ax = — ex* 2t ± 25x

Va/

»* + 2°

\x V6*

Va

*

•/

7

dx

Va" '

if a > 0 if „ < 0

a' sin"' 5 )

analysis

Sec. 3-2] Table 4.

J

sec x dx — In (sec x + tan x) = In tan

S3.

J

cbc x dx = — In (esc x + cot x) = In tan

^

f

86.

/ sec x tan z dj

esc1 x dx = — cot

y

88.

y aina x dx

89.

J

, .

91-

„„ 93.

J/

^ 2

= sec x

=

cos1 x dx ™ tan5

rdi

Mix

~ sin x cos x) = >gx — >i sin 2x

H(*

4" sin * cos x) — \£x 4-

^

, . . sin (m — «)x sin mx sin nz ax =

2(m ~ n) cos fm — n)x

f

95. 96.

j y

ain 2x

» tan x — x sin (m 4- n)x 2(m 4- n)

-

-

arcsin x dx — x arcsin x 4- a/ 1 — xs arccos x dx = x arccos x —

f arctan x dx

— x arctan

97.

y ainh

98.

y

cosh x dx — sinh x

99.

y

tanh x dx = In cosh x

100.

y

coth x dx = In Binh z

101.

y

sech x dx = arctan

102-

y

each x du — In tanh >£x

103.

y sech1 x dx

104.

y

each* x dx =» — coth x

105.

y

sech x tank x dx = — sech x

106-

y

csch x coth x dx = — each x

Exponential

x dx = cosh x

107.

y

e* dx = e'

IOS.

JI

a'

109.

y

x-e* dz

dx

-

(ainh x)

— tanh x

and Logarithmic.

Forms:

-^1In a

-

xv*

— n

y

— x1

x — >2 In (1 4- x5)

Hyperbolic Forms:

x*_ie* dz

mf*n

cos (m 4- n)x

2(m n) 2(m + «) . sin (m 4. sin fm (m — n)x + — — = —— — — mx cos nx ox cos 4~ ; 2(m n) 2(m + n)

X«rer*e Trigonometric Forms : 94.

^

i

, I am mx cos nx ax —

/f

+

esc x cot x dx — — esc x

87.

Jf

^-

(Continued)

sec5 x dx = tan x

So.

90.



82.

84.

3-73

mwn

3-74 110.

MATHEMATICS

J

111.

/[

112.

//

-

In x dz

zdz

x" In

—

115.

... 116. 117. .. 118.

-

— dz -


3

(Continued)

z

i"*' T-l"^ Ln

! ] (n + 1)»J

+ 1

ln (ln x)

J

. •*•(« sin nx — n cos nx)nx dx = a' + »«

JI

. e"(a cos nx + n sin fix) cos nx ax ■» a' + n«

,,,[..r« Definite

-

x In x

x ln x

,,, „ sin■ 113. /[ e" 114.

Table 4.

[Sec.

—

Integral*:

^0

*•-■« - dx =

f' x»—

J/

, 1

/fn0

dx

—

f\ (lni)'

1

m

-

r(»)

. , m > 1

1

-. if

a' + z'

-

'dx

2

a > 0; 0, if a = 0;

- -. if a < 0

l * sin mx dx = it ., „ „ if m > 0; 0, if m = 0; / 0n

J

x

2

2

2

.. . if m < 0

1.9.


120.

12L 122.

123.

/o-tan^d_x

/-«=L^.: /" « ■»•■dx - -2a Vr Jq

''dx

z'e

— I'

a > 0

4

Numerical Differentiation

1.4

and Integration

Given a set of numerical values of a function, the processes of numerical differenti ation and integration consist, respectively, of calculating the derivative (or deriva tives) by means of these values and of computing the values of a definite integral from the set of values of the integrand. In both numerical differentiation and integration the problem is solved by representing the function by an interpolation formula and then differentiating or integrating as desired. Interpolation formulas are discussed in Art. 3.5 of Sec. 3-1. It was noted in Art. 3.5 of Sec. 3-1 that a polynomial which agrees with f(x) at Xo, X\, x_i, Xj, x_j, is the central-difference formula

...

P(x)

= /(xo) + «m «/(*.)

jjHj^

+

^

iV(x„) +

tt(u' ~ P)

-

u.(M.

4!

- /(x.) where x

— xo

+ uA.

/'(xo) »

+

+ u Since u

PW

= 1

A"' +

=> 0

at

x0,

I*

A

-

iyM. 5!

V.

n a>/(*.)

+

3»)

(*lzL±^°

*

. . .

P) AV* + A'yM(W,37

the derivative at

_

+

x0 is

+

• • •

given by

A'»-.+A'y-,

+

. .

.)

Higher derivatives can be obtained in like fashion by further differentiating P(x). Near the beginning of a set of tabular values Newton's forward-difference formula

analysis

Sec. 3-2]

3-75

convenient and near the end of the set Newton's backward-difference formula convenient than the central-difference formula. There are a large number of quadrature formulas for the approximate integration of a function specified by a set of numerical values. As previously noted, any of the interpolation formulas can lead to quadrature formulas. The trapezoidal rule and Simpson's rule are the most commonly used. If h is the length of each subinterval, the trapezoidal rule is is more

is more

/. and

ydx

■

-

=

(i/o

+ 2y, +

2i/2

+

• • •

+ 2i/„_, + y„)

Simpson's rule is z»+nh V

•

where

dx

--

n

o

(tfo

+ 4y, +

2y2

+ 4y, + 2yt +

• ■■

+ 2y„., + 4y„_i + V.)

n in this last ease must be an even number; i.e., the number of subintervals is

even.

Gauss's formula (see references) is the most accurate of the formulas ordinarily can be used advantageously with high-speed machines.

used and


2

FUNCTION THEORY 2.1

Real Variables

The subject matter of real variables may include the following topics: the system of real numbers, sequences, infinite series, ordinal and cardinal numbers, set theory, functions/ and limits, continuity and discontinuity, differentiation and integration, and measure* theory. Infinite series and differentiation and integration are of such scope that tliey arc discussed in other articles. Ordinal and cardinal numbers and measure theopy are not felt to be essential here.

The

System of Real

Numbers

and Sequences.

The

foundation

of the

Although ,nepry of functions of a real variable depends upon the real-number system. !n!f refined concept of the real number is the starting point for any discussion of the ftyhdamental parts of higher analysis, only the more important concepts and results indicated. The concept of the natural numbers or positive integers — 1,2,3, . . . —may be taken as a starting point. The class of rational numbers is obtained from the positive integers by allowing the inverse operations of addition and multiplication, namely, subtraction and division. The totality of positive integers, negative integers, zero, and fractions constitute the class of rational numbers. ^*ill be

It is generally appreciated that certain numbers such as \/2 and r are not rational Irrational numbers and cannot therefore be represented by the ratio of two integers. numbers arc generally derived from the rational numbers by either Cantor's theory or Dedekind's theory. Cantor's theory of irrational numbers depends upon the concept of a sequence of rational numbers. If by some suitable process a first, a second, a third, . . . rational number can be formed successively, and if to every positive integer n one and only one rational number a* corresponds, then the numbers Hi, at,

. . . , On,

. . .

in this order, corresponding to the natural order of the positive integer, are said to form of rational numbers. The individual numbers that form the sequence are called the elements of the sequence. The sequence a sequence

Hi, os will be denoted symbolically

by |o„).

, a»,

. . .

MATHEMATICS

3-76

[Sec.

3

A sequence of rational numbers |a«) is called convergent or regular if for an arbi trary « > 0 there exists a number N such that for every n > N |a„

—

an+m\

where

< «

m = 1, 2, 3, . . .

The essential feature of Cantor's theory of irrational numbers is the assumption that corresponding to every convergent sequence of rational numbers there exists a uniquely Any real number can, therefore, be regarded determined object called a real number. Two real num as being represented by a convergent sequence of rational numbers. bers A and B defined by the convergent sequences of rational numbers \a„ \ and |6„] are said to be the same number or are equal if there exists an integer N such that for all values of n > N


\a„+„

—

m = 1, 2, 3, . . .

b„+„\ < €

where < is an arbitrarily small positive number. The real-number system consists of rational numbers, since any rational number a can be represented by a convergent sequence \cu\, where o» = a for all n, and the real numbers that are not rational, i.e., the irrational numbers. In the Dedekind theory of the real-number system the real numbers correspond to partitions of the rational numbers. A partition is formulated in the following manner. Divide all the rational numbers into two classes R and S. In class R every number is less than any number in S, and in class S every number is greater than any number in R. For an irrational number there is no largest number in R and no smallest number in S. For a rational number there is either a largest number in B or a smallest number in S. It is possible to show that Cantor's method of convergent sequences and Dedekind's method of partitions are equivalent in that starting with the rational numbers the same system of real numbers can be developed. If there exist a sequence \a„ | and a real number A such that the sequence {a„ — A | forms a null sequence, then the sequence \an\ is said to converge to the limit A and is denoted by lim

n —•to

a„ ■» A

\

\

\

t > 0 there exists an N such that for all n > (f, A\ < t. Every sequence that does not converge in the above sense is calli^ The Cauchy or general principle of convergence states that the necessary an } divergent. sufficient condition for the convergence of the sequence |a„| is that for every < > 0, there exists an N such that for n > N, \an+m — o,| < t, where m = 1, 2, 3, . . . . The system of real numbers can be considered sufficient for the needs of the theory \ of functions of real variables, since the real numbers form a closed system with respect to arithmetic operations, such as addition, subtraction, multiplication, division, extraction of roots of positive numbers, and powers, and the limiting process. A number system is called closed with respect to an operation or process if this process results in a number contained in the system. It is possible to set up a one-to-one correspondence between the points of a straight line and the real-number system. Because of this possibility, the properties and definitions of the real-number system have a geometrical interpretation. 2.12 Set Theory. The starting point for most mathematical developments is certain objects such as numbers or letters. A set (or class or aggregate or collection) is defined by any property that any particular one of these objects does or does not have. The objects that have the property are called elements of the set. This is symbolized by

This definition says that for every \an

—

s*S where s is an element and S is the set. An empty set does not contain any elements; i.e., there are no objects having the property of the set. Two sets Si and S2 are called equal, and in symbols Si = Si if every element of S, is an element of St and conversely if every element of Sj is an clement of Si. If all

ANALYSIS

Sec. 3-2]

3-77

of a set Si are simultaneously elements of a set Si, then Si is called a Sj, and this relationship is denoted by Si C Si. The notation S2 3 Si If Si C S2 and Si C Si, indicates the same relationship, and Si is said to include Si. then S, = St. If S, C S, and S, C S,, then S, C S,. If S, C S, but S, is not equal to St, symbolized Si ^ St, then Si is called a proper subset of Si. The intersection (or logical product or meet) of two sets Si and Sj is denoted by The Si C\ St and is the set consisting of all elements common to the sets Si and S2. union (or logical sum) of two sets Si and Ss is denoted by Si W Si and is the set con The definitions sisting of all elements that belong to at least one of the sets Si and St. the elements subset of

intersection and union hold for an arbitrary number of sets. If the intersection of Si and Si is the empty set, then the two sets are called disjoint or mutually exclusive. If Si is a subset of a set S, then the complementary set of Si with respect to S is the set of elements of S obtained by omitting the elements of S that are elements of Si. Generally the term complement of a set is used with respect to a fundamental, and therefore understood, set, such as the set of real numbers. Sets may first of all be classified into finite and infinite according to whether they contain a finite or infinite number of elements. An infinite set is called enumerable and only a one-to-one correspondence can be set (or denumerable or countable) Here the term countable up between the elements of the set and the positive integers. will be used to indicate either finite or an enumerable set. noncountable or a set that neither finite nor enumerable. The following results nonenumerable set dealing with countable and nonenumerable sets are well known: also countable. Any subset of a countable set The sum of a countable set of countable sets also countable. enumerable. The set of rational numbers The set of irrational numbers and the set of real numbers are nonenumerable. The set of all algebraic numbers An algebraic number enumerable. the root the polynomial equation of

is

is

is

is

is

is

5. 4. 3. 2. 1.

=

0

of

tux*

i-0

b.

b.

t, a

a

is

a

A

it

is

it

is

is

is

a

|lf ilf

S,

jS

if

is

a is

A

S if

is

is

S

is

S

is

S

S,

is

S

§

is

f

I

A

S,

V.

S,

S

S,

is

<

i

(o

f

a

S,

a

is

S

called limit point (or limiting point or accumulation point) of a set point a there exists point of the set different from a, in every neighborhood of the point — For a linear set a neighborhood of a point a means the open interval + «), there can be the definition of a limit 0. It shown from that every neighborhood > Kjint contains infinitely many elements of the set. If every point of an interval a imit point of a set then the set said to be everywhere dense. The set of all limit called the derived set (or derivative) of the set and denoted by >oints of a set and the limit points denoted by consists of all the points of The closure of = not a limit point that is, called an W S'. Any point of a set which totaled point. there exists a neigh point a called an interior point (or inner point) of set called an exterior point of a set point borhood of a containing only points of S. neither there exists a neighborhood of a containing no points of S. If a point called a boundary point of the set. then •n interior nor an exterior point of a set called closed. set contains all its boundary points and therefore its limit points, an interior point, the set called an open set. perfect set every point of a set Finally, a limit point of the set. closed set where every point of the set

A

[ .

A

if

is

x

.

A

is

is

6.

0

where a, ^ and all the en's are integers. The set of transcendental numbers nonenumerable. The real numbers that are not algebraic are called transcendental. In the discussion of the real-number system the set of points on a line was noted to This set of points called a linear point set or, correspond to the set of real numbers. all its points lie in a finite interval. linear set bounded briefly, a linear set. tAn open interval, symbolized by (a,6), consists of all points x such that a < x < < ■closed interval, symbolized by [a,b], consists of all points x such that a <

is a


is

a

A

if,

if,

two sets

3-78

MATHEMATICS

[Sec. 3

It may be noted that the continuum is a set that is perfect and everywhere dense. complement of an open set is a closed set and conversely. For a linear set S a point a is called an upper bound if s < a for every point s of the set S. The point a is a lower bound if « > o for every point s of the set. The point a is called the least upper bound for the set S if it is an upper bound and if for any « > 0 there exists a point of 8 greater than a — «. The greatest lower bound a for a set is similarly a lower bound such that there is a point of the set less than a + t, where


* > 0.

Two of the important theorems of set theory follow. Heine-Borel Theorem. Let the closed set of points S be covered by a set of intervals ; then there exists a finite number of intervals that likewise cover S. A set S is said to be covered by a set of intervals / if every point of the set S is interior to at least one of the intervals of the set /. Weierstrass-Bolzano Theorem. If S is an infinite bounded set there exists at least one limit point. 2.13 Functions and Their Limits. If in the course of a discussion a symbol may be assigned various numerical values, the symbol is called a variable. A constant A real variable has values in the assumes but one numerical value during a discussion. set of real numbers. Given two variables x and y, y is called a function of x if to every value of x in the domain of x there is determined a definite value or values of y. This functional relationship is denoted symbolically by y =» f(x). x is called the The vital aspect of the independent variable and y is called the dependent variable. definition of a function is that for every value the independent variable takes on, the corresponding value or values of the dependent variable are uniquely determined. The set of values that the independent variable assumes is prescribed and is called the domain. The set of values taken on by the dependent variable is called the range. A function is called single-valued if the dependent variable takes on but one value for If for any value of the independent variable each value of the independent variable. the dependent variable takes on more than one value, the function is called multivalued (or multiple-valued). A polynomial function has the form n

Oox"

+ OiX"-1 +

■■•

+ a,-ix

+

o»

where the a,- are constants and n is a positive integer. of two polynomial functions. An algebraic function

s

Y ail"-'

A rational function is defined

ratio of the

is the

by means

equation

n

7

/.(i)a._,

= 0

where the/;(x) are rational funct ions of x. Transcendental functions are functions that are not algebraic. The theory of functions of a real variable deals with correspondences between two sets of real numbers, designated the independent and dependent variables. The terminology of set theory therefore applies to the set y. The function /(x) is said to be If the least upper bound bounded, have a least upper bound, etc., if the set y does. of a function is a point taken on by the function, then it is called the maximum (or maximum value) of the function. The minimum is associated similarly with the

lower bound. The quantity f(x) is said to have a limit b as x tends to x<>if for any c > 0 there exists a i > 0 such that |/(x) — 6| < c for all x for which 0 < |x — Xo| < *. Sym bolically this relation is written greatest

lim /(x) =

b

Sec. 3-2]

ANALYSIS

3-79

From the definition of the limit of a function, it follows that if the limit exists, the value approached by fix) as x approaches xo does not depend upon the value of f(x) at io and also is independent of the particular set of values that x takes on in approach If the independent variable x is allowed to take on only values larger than ing xo. io or less than xo, then the respective limits are called right-hand and left-hand limits. These are symbolized, respectively, by

and

f(x<>+)

or

lim f(x) =

f(xt>~)

or

lim f(x)

x>x*

lim fix)

i
The Cauchy or general principle of convergence states that a necessary and suffi cient condition for the existence of a limit to fix) as x tends to Xo is that for « > 0 there exists a 8 > 0 such that \f(x") — f(x')\ < t for all values of x', x" for which 0 < |x" a\ < 4. a\ < \x' A function may depend upon the values taken on by two or more independent variables. Again the vital aspect of the functional relationship is that whenever each of the independent variables assumes a value, a corresponding value or set of values of the dependent variable is uniquely determined. Given a function of two or more variables, there exist two types of limits: iterated limits and simultaneous limits. Let fix,y) be the function of two independent variables x and y, and let (xo,yo) be the limit point; then and lim [" lim f(x,y)~l lim I" lim fix,y)~\ x—fXt |_V—*I/o V—'W Vx— >lo

-

-

J

J

An iterated limit indicates that first an are called iterated (or repeated) limits. ordinary limit is taken for one variable holding the other variable (or variables) fixed and then a limit is taken for the other variable (or other variable with the remaining The simultaneous limit fixed). lim f(x,y) x—*xt

has the value A if for c > 0 there exists a positive number 4 such that \f(x,y) — A\ < t for all x and y such that 0 < \x — io| < 4 and 0 < \y — yo\ < 4. If the simultaneous limit exists, then the two iterated limits exist and are equal. The converse does not hold, since the simultaneous limit can be nonexistent and yet the two iterated limits may exist and even be equal. A function fix) is said to be 2.14 Continuous and Discontinuous Functions. continuous at a point Xo if lim f(x) = /(xo) x—►Xo

t

0

4

-

if

<

a

is

if it

x—*a*

b,

is

S.

>

0,

t

for

—

is

|x

is,

such that |/(x) — f(x0)\ < for all x such that there exists a > In words this definition states that the limit shall exist at xo, that the Xt\ < continuous in the function defined at xo, and that these two values are equal. f(x) and at the end interval [a,b] continuous at every point x, where < x points lim /(x) and lim f(x) = f(b) /(a)

that

x—*b~

is

said to have an ordinary discontinuity (or jump discontinuity the rightor discontinuity of the first kind) at the point x0 hand and left-hand limits at point exist but are not equal; i.e., The quantity f(x)

a

if

or simple discontinuity

f(x)

*

lim

X~*Xa~

fix)

the right-hand and left-hand limits exist and are equal but the function has different value, i.e., lim fix) = lim fix) fix0)

*

a

lim %—*Xt*

If


lim f(x) =

3-80

MATHEMATICS

[Sec.

3

then the function has a removable discontinuity at the point Xi>. When the right-hand limit or the left-hand limit or both these limits fail to exist at point i0, then the function has a discontinuity of the second kind at the point x«. A function is continuous on the right at a ■point if the right-hand limit has the same value as the function at the point. Continuity on the left at a point and continuity on the right and left in an interval are denned in a corresponding fashion, /fx) is called uniformly continuous in the interval [a,b] if for « > 0 there exists a t > 0 satis independent of the xo in the interval [a,b] such that |/(x) — /(io)| < e for all fying \x Zol < S. Let f(x) be continuous in the interval [a,b], then the following results hold: 1. f(x) is uniformly continuous in the interval. 2. /(a) and/(6) have opposite signs; then there exists at least one value of x in the interval for which f(x) vanishes. 3. /(a) & f(b); then as x takes on all values between a and 6, f(x) takes on at least once all values between f(a) and fib). 4. If f(x) is single-valued in [a,b], then there exists at least one point of [a,6] at which Likewise there exists a value of x where the minimum f{x) takes on a maximum value. is attained. 5. The function is bounded in that interval. 6. The function is uniquely determined at every point of the interval by prescribing the function at a set of points everywhere dense in the interval [a,b].

i

-

Complex Variables

2. 4.

|zi |i|

6. 5.

|zi

7.

|zi

3.

+ Z2 = Zl z2. zTT2 = ziZj; (zi/z2) = 2,/22. Zi

-

—

zl

|z|.

=

y/x*

y*

=

and the con

•

±

1.

z

i

+

The following relations involve the absolute value — iy: = jugate

|z|

2.21 Complex Plane and Sphere. Complex numbers have been discussed in Art. 2 of Sec. 3-1. The set of complex numbers can be put into one-to-one corre This correspondence associates the complex spondence with the points of a plane. number z ■=x + iy with the point in the plane whose rectangular or Cartesian coordinates are (x,y). Because of this association this plane is called the complex or z plane. This geometric association for complex numbers not only gives a geometrical interpretation for operations involving complex numbers but also allows the use of geometric terminology such as points and distances when discussing complex numbers. When the improper point z = °o is added, the complex plane is closed. The number or point z can also be thought of as a vector that originates at the origin of the coordinate system and ends at the coordinates (x,y). The points of the closed complex plane can be mapped by stereograph ic projection one-to-one onto the points of a sphere called the Riemann sphere (or sphere of com In this mapping, the south pole is placed at the origin and correspond plex numbers). ing points for the sphere and plane lie on a ray that originates at the north pole.

|z|».

is

z

it

z

is

is

z

if it

z

z

z

is

of

a

if

if it

is

a

a

z2|

is

z2|

is

is

z2|

the distance between the points Zi and z2. farther from the origin than zi. |z2| > |zi| says that the point z2 < |zi| + |z2| corresponds to the geometric statement that no side of a + triangle greater than the sum of the other two sides. — > ||zi| — |z2|| states that no side of triangle less than the difference of the other two sides. 2.22 Functions of The concepts and definitions for real Complex Variable. variables generally have significance for complex variables, called a complex in the course of discussion variable assumes various complex values. Given two complex variables and w, w called a complex function the complex variable to every value of in the domain of there determined a value or values of w. Again w called single-mlued takes on only one value for each value of and w multiplevalued takes on two or more values for any value of z. Polynomial, rational, algebraic, and transcendental functions are defined for complex variables in the same is


2.2

if

Sec.

ANALYSIS

3-2]

3-81

The distinguishing feature is the use of complex con for real variables. the complex variable z in place of the real variable x. neighborhood of a point zi is the circular region

fashion as

-

|z

<

■

A

Zi|

stants and

> 0. The definitions of limit point, interior point, open set, and closed set An in the article on set theory are applicable to a set of complex numbers. any two points connected; that open region, or just region, denotes an open set that the set may be joined by continuous curve all of whose points belong to the set. called simply connected a region plus all its limit points. region closed region region every closed curve within the region encloses only points of the region. that called multiply connected. not simply connected an integral taken along closed curve that the boundary of a region, the called positively closed curve commonly called a contour integral. integral oriented traversed, the interior of the curve lies to the left of the curve as the curve counterclockwise. If only the initial and terminal points of a i.e., the direction curve coincide, then the closed curve called simple. for any > approaches zo The quantity /(z) said to have the limit u>o as — < there exists a for which < such that |/(z) — w<>\ < for all > = u(x,y) -f- iv(x,y), where = x Let h — /(z) = u + and u and are real functions of x and y; then the limit can be expressed by <

where

is

—»*«

for limits with

a

The results

u(x,y) +

zo|

|z

v

+ iy

Uo

*. 0

<

if

z

lim v(x,y) = x—»aro y—»yo

+ ivt

= u>o

complex variable follow almost directly, therefore, from complex variable

is

single-valued function /(z) of

a

A

corresponding results for real functions of two real variables.

called continuous at a point

tt if

lim /(z) =/(z0)

C—»*0

requirements of the definition are, again, that first the function be defined approached, and finally the limit value must the limit must exist as z0 equal the value of the function at zo. The results of real variables concerned with For example, a function continuity lead to analogous results for complex variables. uniformly continuous in that continuous in bounded closed region, then .Hz) a finite positive region; bounded in that region, that is, |/(z)| < M where M an inner point of the number; |/(z)| has a finite upper limit in the region; and z0 region such that /(z0) ^ neighborhood of z0 for which /(z) 76 0. then there exists The derivative of /(z) at the point zo defined by The three

is

z—»*o

z

is

is

a

0,

if

is

is

it

a

it

if

is

at zs, second

is

— Zo

t—»« Az

It,

defined by

f}

a

dz,

which

symbolized by

jcf(z)

l°J(z)dz

jcf(z)dz

=

lirn

£

n

=

/(*)(*.

is

the limits a and

is

between

0,

- *_,)

C C,

of

p

C

z

a

a

is

if it

The where /'(zo) denotes the complex number that the limit, exists, assumes. differentiation rules for real variables can be extended essentially without change to region derivative at every point of function that has variables. complex called differentiable in the region. let a and be region Let to = /(z) be a continuous single-valued function of in two points of R, and let be a curve of finite length connecting the two points and be a sequence of points on Furthermore, let zo = a, zi lying in R. z„ = wd let in be any point on the curve between z<_i and Zj. The integral /(z) along

A


f

x—*xt If—»yi

i

lim /(z) = lim

0

z

z

iy

t

0

5

is

is

is

is

is

if is

A

a

If

is

is

is

if

A

is

A

is

A

a

of

is

is,

as given

3-82

MATHEMATICS

[SEC. 3

J

[v(x,y) dx

+

i

+

u(x,y) dy]

may be represented by the real parametric equations x = g(t)

-

and

= h(t)

y

The curve

- v(x,y) dy]

\u(x,y) dx

/

/(») dz = C

/

i

\zi

where the limit indicates that n tends to infinity and that the absolute value of every — subdivision 2<_i| tends to zero. Since /(z) = u(x,y) + iv(x,y) and dz = dx + dy, the complex integral can formally be written as follows in terms of real integrals:

-

+

vh'(t)] dt

Wit) +

I

dz; that

/(z)

be noted:

dz; that is, the sum of two integrals

equal to the integral taken over the entire curve. if

dz — —

I

/(z)

/

2.

curves

f(z) is,

taken along two successive

dz m

is

f(z)

I

dz

+

/

1.

The following elementary properties for complex integrals may f(z)

p(0

uh'(t)] dt

the direction of integration

is

[ug'(t)

fl

0

-

dz

i

ff(z)

f^

1,

t

0

= j(l) + tVi(l). where < < Let the functions a s(0) + iVi(O), and and h(t) be single- valued and have continuous first-order derivatives; then

reversed,

M

h*

[ft(z)

for any

on

dz.

and

L

where |/(z)| <

dz

+

fcMz)

is

ML,

fc,

fc

+

<

-

C

dz

dz

z

/(z)

ktft(t)]

the length of the

I

4.

[fci/ito

/ / c

3.

curve C. is

;/

v

1.

+

is

if

is

A

a

is,

is

A

2.23 Analytic Functions. single-valued function f(z) which differentiable, has a first derivative at every point, in region that called analytic (or regular or holomorphic) in the region. function /(z) called analytic at a point z<> its derivative exists at every point of some neighborhood of zo. The concept of analytic functions, or analyticity, particularly unifying and important for mathematical physics. Two necessary and sufficient conditions for the function /(z) ■ u(x,y) iv(x,y) to be analytic in the region D follow: Cauchy-Riemann Equations. The four first-order partial derivatives of u and with respect to x and exist and are continuous in the region D, and they satisfy the Cauchy-Riemann differential equations

2.

dx

Cauchy-Goursat Theorem

_

dv

du

dy

9y

and Morera's

dv

_

dx

Theorem.

The integral

/

du

/(z)

dz of the

closed

Let /(z)

/CO dz

be continuous

in

simply

=0

curve lying within the region, /(z)

is

/. c for every

-

if

Morera's Theorem (Sufficiency).

nected region; then

dz

a

/(z)

0

c/,

is

C

continuous function /(z), when taken along the entire boundary curve of any subregion of the region D, zero. a. Cauchy-Goursat Theorem (Necessity). Let /(z) be single-valued and ana lytic within and on a closed curve C; then

b.


the value of the integral remains the same except for sign.

analytic in that region.

con

Sec. 3-2]

ANALYSIS

3-83

Although it is possible to formulate other necessary and sufficient analyticity, it is customary to consider the consequences of analyticity in turn, imply analyticity or not. Some important consequences follow. 1. Canchy's Integral Formula. If f(z) is analytic in a region D, integral formula

JC z

2m

—

conditions for whether they, of analyticity then Cauchy's

z0

valid for every simple, closed, positively oriented curve C and for every point z0 interior to the curve C. 2. Higher Derivatives. If a single-valued function f(z) is analytic in a region, then not only does the function by definition have a first derivative in the region but it also The formulas has all higher derivatives. is

V /(*„) 1

- £l ( _-/(*\ IC 2rt JC (z (z

zo zo)

,

dz

»+1

n

-

1, 2,

are valid with the same conditions used for Cauchy's integral formula. 3. Laplace's Equation. If /(z) = u + iv is analytic in a region, then the functions


u(x,y) and

v (x,y)

satisfy in that region the partial differential equation

dy*

dx1

This equation

is called Laplace's differential equation and is of great importance in mathematical physics. A function that has continuous second partial derivatives and satisfies Laplace's equation is called a harmonic function. If f(z) = u + iv is analytic, then u and v are called conjugate harmonic functions. ■i. Taylor's Series. If /(z) is analytic in a region D with a boundary C, then at each interior point zo, /(z) can be represented uniquely by a power series of the form

y

a„(z

-

z0)n

i-O ,fM(zo) o„ = —■

where

nl

This series, called a Taylor's series, converges and represents /(z) in the largest circle with center z0 that encloses only points of D. If z0 = 0, a Taylor's series is called a Maclaurin's series. 5. Laurent's Series. Let/(z) be analytic in the annular region D bounded by two concentric circles with center z0; then /(z) can be represented by the Laurent's series

V where

a„ =

^ jc

a„(z

(f

—

zo)"

- zo)-""1^)

<2f

and C is a simple closed curve lying in D and enclosing the inner circle. If two functions are analytic in a region, and if they coincide, 6. Identity Theorem. in any neighborhood of any point Zo of the region or any curve terminating at z0 or even for an infinite number of distinct points with the limit point Zo, then the two

functions are equal throughout the region. The maximum modulus of a function Modulus. 7. Principle of the Maximum analytic in a closed region always lies on the boundary of the region.

3-84

MATHEMATICS

[Sec. 3

8. Liouville's Theorem. If a function /(z) is analytic and its modulus |/(z)| is bounded for all values of z in the complex plane, then /(z) is a constant. 2.24 If a function can be Singularities and the Classification of Functions. made analytic at a point z0 by merely assigning the function a new value at the point An isolated singular Zo, then the function is said to have a removable singularity at z0. point of a function is a singular point that can be enclosed by a circle containing no other singular point of the function. An isolated singular point Zo of a function f(z) is called a pole of order n if a positive integer n exists such that (z

- *,)V(«)

is analytic at z = z0 and is different from zero when called a simple pole. An isolated singular point essential singularity of f(z) if lim (z r0)"/(z)

z = z0. Zo

of

a

-

In case n = 1, the pole is function /(z) is called an

t—*Zti

tends to infinity for all finite values of n. z = z0 is called a branch point of the function /(z) if /(zo + pe^) is not periodic in tp with period 2ir, where p is chosen so that Zo + p*** is in the region of analyticity of for all

/

=

/(*)

The expression

>

bn(z

—

)

z0)~"

a,(z

—

zo)"

is called the

+ y

b,(z

from the Laurent's expansion by

z0)~

principal part of f(z).

n = l

is defined as the residue of the function

—

f(z) at the point

Zo.

The coefficient b\

The formula for

b, is

given

where C is a simple closed curve enclosing zo. The fundamental theorem of the calculus of residues follows: Residue Theorem. Let f(z) be analytic except for a finite number n of isolated singular points within and on the closed curve C; then n

fcf(z)dz

=2W

£

Rt

where R,, . . . , Rn are the residues at the n singular points. When f(z) has a simple pole at z0 and has the form

3-85

ANALYSIS

Sec. 3-2]

0, h(z0) = 0, and ft'(zo) ^ 0, the residue of

where g(zo)

/(z) at

So

is

gfo) h'(z„) 2.26 Conformal Mapping. Let w = f(z) denote a mapping or transformation from the z plane to the to plane. If the transformation preserves the magnitude of If in a mapping both the angles but not necessarily the sense, it is called isogonal. magnitude and sense are preserved between every pair of curves through a point zo, then the mapping is called conformal. If a function is analytic at a point z0, then either f'(z) = 0 or the mapping w = /(z) is conformed at z0. A critical point z0 of a mapping is a point at which /'(zo) = 0. One of the most important results of a conformal mapping is that a harmonic function, that is, a function K(x,y) which satisfies Laplace's equation (d'K/dx1) + (d*K/dy*) = 0, remains harmonic under the change of variables that arises from the conformal map /(z); that to = u + iv a ping (d*K/du*) + UVK/dv') = 0. Furthermore, or of the type dK/dn = where dK/dn boundary condition of the type K(x,y) = the normal derivative, transforms into a boundary condition of the same type. Therefore by using analytic functions function possible to find in many cases that harmonic in given region and satisfies boundary conditions of the above type. The transformations w = (az + f))/(az + S), where aS — 07 ^ called linear fractional transformations, are conformal. In particular they map circles, which include straight lines, since the lines are circles with infinite radius, into circles. Other important properties of linear fractional transformations may be found in the references. The transformation a

is

- X,)-*'(Z - 2,)"*'

• • •

- Zn-O-'-'dz

+

Ci/(Z

(Z

W =

C2

="/(«)

2.27

Analytic

Continuation.

Riemann

+

+

1)

,

.

1,

2,

(»

,

2,

1,

z

is

is

where Ci and C2 are arbitrary constants and the integral an indefinite integral, called a Schwarz-Christoffel transformation. This mapping takes the x or real axis in the plane into a polygon of n sides in the w plane. The points Wi = /(x,), where = . . . n and x„ «■ «, are the vertices of the polygon. The exterior angles at the vertices u>< — are given by fcix. . . n — The exterior angle at tr» is given by ■• • fc„T — 2» — (*i + kt fc„-i)ir

i

Surfaces.

The identity

theorem listed

in a region D2.

Furthermore, let the regions Dt and D2 have

a

a

under analytic functions leads to the important concept of analytic continuation. Let /i(z) be an analytic function in region Du and let fi(z) be an analytic function

subregion in common

in which the functions f\(z) and /2(z) coincide completely (or even on a curve in the subregion). The functions /,(z) and /2(z) then define the same analytic function F(z). f,(z) and ft(z) are called analytic continuations of each other. Furthermore, fi
multiple-valued

function

is a

may

A

it

is

A

is

is

If

a

is

function

z

is a

is

it

that is not single-valued. In considering multiple-value functions convenient to introduce the geometric Kiemann surface concept of a Ilicmann surface. generalization of the plane consisting of a surface of more than one sheet arranged vertically. On each point A


0,

a

is

it

is

C

0,

is,

-

MATHEMATICS

3-86

[Sec.

3

of this surface the multiple-valued function has one and only one value, and the func tion is therefore single-valued on the surface. Special Functions

2.3

Since over a thousand special functions have been investigated, it is possible to consider here only a few of the more important functions of mathematical and reactor physics. Gamma Functions. Beta. Definitions. 1. The Factorial, 2.31 Polygamma, gamma function r(z) is an analytic meromorphic function of z with simple poles at The z = — n, where n = 0, 1, 2, . . . and with corresponding residues ( — l)n/n\. following conditions then determine T(z) uniquely: a. T(z + 1) = zT(z). b. If T(z) is real and positive, then (z) is real and positive. c.

r(i)

=

-

l.

d. [(dt/dzt)r(z)]T(z) 2.

Weierstrass

(dT/dz)1 > 0 when

z is real

and positive.

Definition:

n =1


where

y,

the

Euler-Mascheroni

constant, is defined by

J y =

3.

Jim

- In

(^

= 0.577215665

■■•

Euler's Formula:

■"-into +a'0 +r] n1

exists except for 4.

z =

— n, n = 0, 1, 2,

. .

. .

Euler's Integral:

f" e-H'~l

T(z) = holds for the real part of

F unctional

z

greater than zero (Re

z > 0).

Equations:

r(z + l)

-

r(z)r(i r(z)r (z For n

dt

+

^

r

(*

- 2 this becomes

+

1)

r(i)

r(Ji)

T'(l)

= n! = = o! =

r

(z

n(n l

r(2)

- 2)

=

=

+

zriz)

-A— jrZ

sin

~ir~0

=

C-ir)<"'1,/J"^"'r('«)

= (2jr)^^2^-,T(2z).

F(z)r(z + Vi)

Special Values:

r(n +

• • •

1)

z)

=

i

= = — 7, the Euler constant

•

■2 • 1

(n = 0,

1, 2,

.

.

.)

analysis

Sec. 3-2]

3-87

Derivatives:

r(z) **

dz

y- {fn-i(z)], n = 2, 3,

dz

=

*.(*) +

...

y — i— £l (j + *)"

(-D"n!

-

1)

is called a poly gamma function,

-

[(-l)*n!]z-«

Sterling's Formula or Asymptotic Formula for Large \z\: In

- -

r(z)

+ ft(z)

|arg z| < «■

and where

!*!>L

\Rn(z)\ <

- l)|z|»'v-'

2N(2N


2*-

+ «*(«)

Zrf n=1

where

The numbers B2„ are the Bernoulli Bi«+i = 0, and

In particular, n! = n"e~" Vîrn

(1

0 <

This

TAe Beta Function.

Re * »t > 0 and Re n Alternative definitions are

- J0f

B(m,n) B(m,n) 2.32 geometric

.

- re4*)

(cos|y'V and are defined by Bo =

Bi

=

Hi

2(2n)!

+ r(x)], where

r(i)

<

—

—

H

12x

288z'

=

- 0""'

<"-Hl

> 0.

» +[-)m+ndt

=

B(m,n)

/"ir/2

Jo

sin'-"-'

*> cos2"'"

... r(m)r(«) - = H(n,m), = —

—

T(m + n)

Hypergeometric Functions. Ordinary, General, Confluent. The hypergeometric differential equation Function.

z(l

- ^ + - z(a + «)

dz*

[c

6

+

at the origin the general solution u>(z)

1,

is defined by

B(m,n) where

(z

'

numbers

-

B„

has

- * + H In

V (-1)--' ~^—r. 2n(2n — 1)

ft(z) =

where

In *

(z

JV-l

= AF{a,b;c;z) + Bzl~
•The notation "Re" means "the reftl part of."

+

1

-

1)]

~

dz

c, b

+

- abw 1

Ordinary Hyper-

= 0

- c; - c; z) 2

MATHEMATICS

3-HS

where A and B are arbitrary constants and a,

„

E^+A> r(o)

1;(a + „

_

- • •

2)

-

(a),

+ 1)«

(a

The circle of convergence for this

function or series.

0

series

1.

]z|

in railed the hypergeometric = is the unit circle

_

(a + n

n!

(c).

Zrf

e

[0(0

can also be written in the form

- - z(e + a)(0 + 1)

+

The hypergeometric equation

=

b))w

0

-

(a)„

The

parameters.

c are

f Mfi-r

***** where

and

b,

3

[Sec.

1;

±

z)

z,

1;

±

0

1

\z\

<

tz)~' dt

The

6)

a

6)

T(c

Functions.

Hypergeometric

-

- - a)T(c -

r(c)r(e

f(o,6;c;l) Generalized

>

0

f'

<

-

< - - tz)~' dt

b

b)

JO

He

0

>

-

c)

+

b

(a

and lie

c

'

-

Re 1

z

When

IM_ -

—— T(6)r(c

0,

-

F(a,b;c;z)

generalized

hypergeometric

differential

1

g

+

if

#

and

at

TO

=

,F,(a;b;z)

' ■ • • •

~

'

-

and «

n^n =0

ffi= (bi)n

(b,)„ v

64;z)

-

p

0

It has singularities at 1). = The solution regular at

to

0

+

(0

i-i

a,)]

p

\\

z

I)

-

0 is

+

-

Y

ap;6,

+bj

z

q

A(a.

1.

if

q

of tho order max (p, — and 00 + p

0, is

y-i

(0

[0

l\

«

equation

1,

n!

<

p

1. z

+

q

p

if

0

z

1,

q

,

p

if

1

<

if

is

is

and called the generalized hypergeometric series. It assumed that the 6s are not negative integers. In general this series converges for all finite q, converges = for and diverges for all > 5* + Contiguous relations, integral representations, and relations among various argu ments for the generalized function may be found in the references. Con fluent Hypergeometric Functions. The confluent hypergeometric function, or Hummer fit net ion |z|


F

±

1,

1

0,

1,

0

s z(d/dz). The singularities (regular) of the equation occur at where the operator and «o . Because of these singularities solutions of the hypergeometric equations — are also often written in terms of the arguments l/z, 1/(1 — z), z/(\ — z), and Rummer's 24 solutions and various relationships among them involve (1 — z)/z. these arguments and may be found in the references. are 0; y; z), F(a, The six functions F(a y; z), and F(a, 0; y called contiguous functions to the functions (a,$\y;z). Gauss's 15 recursion formulas relate the contiguous functions by expressing one function in terms of two others. These formulas may also be found in the references. In the references are found various integral representations, both real and contour, The best known integral representation follows: for the hypergeometric function.

analysis

Sec. 3-2]

Rummer's differential equation

satisfies the

z Hummer's

Kummer's

first formula is

—

dho

.

+

dz1

,,

aw = n 0

dz

- a;

6;

+

H;

-z)

second formula is

eVi(a

=

l4z*)

independent recursion formulas or contiguous relations are

(a

- F(a - 1; b; z)

-

=

(a + z)F(a;ft;z) b

af>

-

+ l)F(a;b;z)

+ Z-F(a; —

+

1 ; 6; z)

+

z

af'(a +

1; 6; z)

+

(1

representation when Re

b

o

+

1; z)

^(«: 7? o

-

b)F(a;

b b

+

-

1; *) 1; «)

Re a > 0 is given by

6 >

Most of the functions of mathematical physics can be expressed in terms of general Examples of functions that can be expressed as special hypergeometric series. cases of a jFi or a if\ appear in some of the following articles. The cylindrical, or Bessel, functions 2.33 The Cylindrical or Bessel Functions. ized

are

solutions of the Bessel differential equation

,dlw —

z»

dz'

+

z

— dz

This equation has a regular singularity The

. , + («'

dw

-

= n 0

*s)u>

at z = 0 and an irregular singularity at z = oo.

functions

t-0 N,(z)

=

«V»(z)

=

Y,(z)

=

-

sin

vir

(./„(z)

y,(z) + ;r,(z)

cos

it

//,<»(«)

- /_,(z)] -

=

J,(z) is called a Bessel function of the Bessel differential equation. V, (or Nr) is called a The subscript v is the order of the function. Seumann function or a Bessel function of the second kind. HSl) and H „(J) are called It may be noted Jirst and second Hankel functions or Bessel functions of the third kind. that J,(z) and F»(z) are real if v is real and z is positive. If v is not an integer, it is customary to choose J,(z) and J-,(z) as the two linearly When v = n, where n is a positive integer, independent solutions of Bessel's equation. it is necessary to use Jn(z) and J'„(z) = lim V„(z) for the two independent solutions, solutions first kind.

are all

once

J..(z)

(-l)"./„(z)

that the Bessel function of the first kind of integer order, in the following expansion:

The Bessel coefficients, occur

=

exp

[Hz(t

- )]

=

£

of the

is,


z)

= e',F,(6

iFi(a;b;z)

F(a;b;z)

An integral

, dw

—

(b

,f,(o;2a;2z) The three

3-89

J„(z)<-

3-90

MATHEMATICS

[SEC. 3

The Bessel differential equation has the form, when z is replaced by is, z*

—

-

dz1

+

-

z

(z1

dz

The functions

IM

=

to

r*-J**»)

I

=

±

it — 0

-2

K,(z)

and

v*)w = 0

+

which are solutions of this equation, are called, respectively, modified Bessel functions of the first and second kind. Again if v is not an integer, l,(z) and /_,(z) arc taken as the two independent solutions, and if v is an integer, I,(z) and K ,(z) are taken as the two solutions. /»(z) and ZC,(z) are real when v is real and z is positive. Z, is used els Some of the more useful relations involving Bessel functions follow. an abbreviation for CiJ,(z) + cjK,(z), where Ci and ct denote arbitrary constants. Functional Equations: = —

+ Z,+\


J pj-p+l + J p-lJ

z

sin

2

-p

z, ■2

2

rz 1,1

—

Derivatives:

£'»

— — 2

—

\Zv-\

—

—

Zmi]

i

pjfp

i = Zq —

- J'.,J, JY

.

pi —p

1' pKp —

Integral Representations.

Jn(z)

Representation

—

- Z\

z

=

y

2

p

irz

i

_

.

—pi p —

—

2

sin

mr

TZ

K' pi p

-1 /fr cos

as a Hypergeometric

- Zv

z

oz'Z,_i(az)

=

J?'o ™ — iTi

J

sin nr

+ ^f-i =

z

[z'£,(az)]'

J' J.,

2

,~\

(z

" sin

Function:

1

t —

n^>) dV

—

Z,+\

analysis

Sec. 3-2]

3-91

General Differential Equation:

if

ft

m ?* 0 and b

zho" + azw' +

(bzm

+ c)z

= 0

0, w = z

r

where

= —

i-TLf z,

m

2

\/(l

—

V6 *»")/ (i \m — 4c

a)*

Special Cases: (K«*)>*

The

2.34 Legendre Functions. differential equation


£-*£

where

»

and

m

Legendre

+

functions

arc solutions

of Legendre's

[* + »-
are arbitrary parameters.

The functions

and

«^W

2,+1

r(„ + ^)

2,+(.+i

\2

T2

'2

2M

2*

2*«/

are linearly independent solutions of Legendre's equation. Ps(z) and Qs(z) are known respectively as Legendre functions of the first and second kind. Since Legendre's equation is not changed when z is replaced by — z, n by — p, and v by — {» + 1), it

follows that

P,**(±z)

P-,-^(±z)

Q,±*(±z)

Q_,_,=^(±«)

are also solutions of the equation. The most common Legendre functions occur when u = 0. The superscript 0 is dropped in this case, and the Legendre functions are written as P,(z) and Q,(z). In case it ^ 0, the functions are often called associated Legendre functions. The following simple relationships exist: P,*(Z) and

- 1)"» ^^ (*' - l)^^£l

= (*'

Q,*(z) =

ft?* dz"

fc = 1, 2,

. . .

When v is a nonnegative integer n, the functions P„(2) become polynomials In hypergeometric function notation the Legendre polynomials.

2»»(n!) and

called

[Sec. 3

MATHEMATICS

3~92 A convenient formula for

the Legendre polynomial

is

known as Rodrigues' formula. These polynomials form an orthogonal system on the interval [ — 1,1] and have all their roots real and simple and between — 1 and I. The Legendre polynomials occur in the following expansion:

-

(1

=

2zi +

£

P»(z)<"

71= 0

Some useful relations i fol follow:

-

Functional Equation: = z(2n + 1)P„

(n + 1)P„+, Derivatives:

= P'.+ 1 nP„ = z/J'„ (n + 1)P„ = P'„+.

Laplace's Integral:

- 1)»

*

- /

=

P.(z)

- P'.-i - zP'„

-

1)P.

(2n +

[(z8

cos «.

+

-

Pi(z) P„(l)

-

-

P»(-D

= 1

-

= x = cos *> 1) = W(3 cos 29 P«(x) = H(3i* >g(5 cos 3^ + 3 cos >) 3x) 30*» + 3) Hiî* ^4(35 cos 4? + 20 cos 2V + 9)

-Px(x) ^(5x>

1

P|(x)

-

(-D"

=

P»„+.(0)

= 0

P,„(0)

J'

Po(x) =

P„(z)s dz =

=

(-1)-

+ 1)

JHgL

4 Sunt Formula:

1-0 Orthogonality Relations:

Lp) m

m = n

0

az =

m = n =

re

z1 —

-

= ir

za)u>"

- zw'

+nht>

-

0

(1

The Tschebyscheff polynomials of the first and second linearly independent solutions of

-

arc cos

0

/

vi -

= sin

if

r„(z)7-„(z) —

0

/•l

y-i

i;

(n,

-n;

,

=

tr,,(z)

z)

r.«

Polynomials: (n

Tn(z) = cos (n arc cos

z)

Tschebyscheff

if

Orthogonal Polynomials.

j

n ^ to

if

P

2.36

P»(z)P«(z)

0

jl_x


Special Cases and Values:

z]«

kind,

Tn(z) and

U„(z)t

analysis

Hermite Polynomials:

-

Hn(z)

(-1)V'"^(«-'V1) dz"

a.i'N if

m ^ n m

dz =

e-''"Hn(z)H„(z)

= (27r)^n! nff„ =

+

- zff'„

ff"„

!)!_„/ if

+

l.z'\

0

_ (-l)"(2n

/:.

/

M _ (-D"(2n)!

H

3-93

-n

0

Sec. 3-2]

(a+,1)' n

+

+ nw

= 0.

- z)w'

r(l

+ a)

m =

n

if if

|AL-

+

1

zm>"

(a

satisfies

(SL± n!

Jacobi Polynomials:

- (-l)'(l

P„«.*>W

-»)-"(! +^»[(1.t)<^,(1 +f)M 2"n! dz"

n(n + a +

+ 1)»

=

0

+ 2)]i/'

0

+

+

- t*)y" + - a - z(c

0

satisfies

[0

(1

P .«•■*» (z)

Gegenbauer Polynomials:

is

is,

=

i#0

0

t(x) dx =

I

i(x)

such that

/:

but

0,

x

is

is

J

it

Dirac Delta Function. In many problems of physios and engineering 2.36 func expedient to introduce a quantity S(x) called the Dirac delta function (or just not a proper function in the sense of having a definite value for each tion). 6(x) Formally a Dirac. delta function i(x) value of the independent variable. defined = to be icro except at that is

i(x)

=

lim *a(x) A-.0

a

is

5

is

0

is

This formal definition does not give a clear picture. It may, however, be inferred function but zero outside very large near x = that the very small interval It not important in applications to know the precise variation of about x = 0. i(x) with x. For example, the function may be defined by S


» £„
m ^ n

0

/. 0

dz =

e-'zaL„('"Ln(a)(z)

■F.C-n; a +

!

=

1;

(e-'*"+«) ^" n az" !

=

^

L.""W

z)

Laguerre Polynomials:

3-94

[SEC. 3

MATHEMATICS

where

x<-\

■0

-|<*

be used with caution:

Laplace Transform:

*

dx =

[

S(x)e~"

a)

a)

= •r(a)

a)]

— 27ri

e"

A-<-

dx

S{x)eu'dx =

1

J'

Fourier Transform:

/_".*"*"*

»(*>

Other Functions.

Incomplete

7(a,x) =

Functions:

Gamma

f'e-'t'-^dt

T(o,x) =

»

x < = S'(x)

8(x) 2.37

- ,F,(a;

=

dt =

r(a)

1

0

=

<

-

0

S(x)

0

Let

Step Function.

1

Derivative

+ a;

-x)

- 7(a,x)

Error Functions:

=

= =

x*Fi(H;J*; -x«) H»M

- Erf

£,(x)

£<(-x)

=

^"

Exponential and Logarithmic Integrals:

e-T'(i<

=

r(0,x)

x

Erfc

fje-'dl = ^70$,**) «"'' * = Hr(^,x«)

j*

x = x

Erf

>

0

+

8(x

a

+

b)

-

[8(x

2a

dl

= 8(x)

o)

=

-

«(-x)

-

dx = 8(a

8(x) =

of


/(a)a(x

xi(x) =0

- a')

8(x»

=

_

/

- x)«(x -

i(a

-

b)

J'

/(x)«(x

/_".

"

dx

1

/frw*)

a)

/_".

The following formal formulas may

analysis

Sec. 3-2]

3-95

Sine and Cosine Integrals: si(x) =

-

Ci(x)

llE,(ix) 2i

=

(Z*^±dt J" t

[Xc-^di=\

- ft(-w)] Eii-ix)]

lEt(ix) +

=

W) -£

Second kind:

£(\,fc)

=

»(X,K,fc)

=

/

'*■

=

...

■■■

+

+

ai

of numbers, where

s0 =■ao, »i

= ao

+

Oi,

a„

ai +

a„

n-0 called

a

is

called the sum or value of the series.

arc called the partial

convergent, the limit of the sequence |«„|

If

series with positive terms.

all the terms a„ are such that a, > A

the infinite series

is

If

sn

series whose terms are alternately

is

The numbers

0,

J

ao

+

the sequence |s„j has some convergence property, the infinite series

said to have that convergence property.

>

f

Infinite Series

be an infinite sequence

>

is

\i*F{-HM ;l;fc«)

=

+

3.1

a.-

a,

__ - *')(! - fc*x')]H

«rf(H,«;i^*)

=

- K(k) £(*)

an = o0

if

+ •**)[(!

SERIES AND EXPANSIONS OF FUNCTIONS

3

Let «o, *i, st, and generally

sums.

dx dx

(1

F(l,*)

then

rff

Elliptic Integrals: J?(l,*)

then posi-

a,

called absolutely convergent

if

m

is

An infinite series

called an alternating series. >

tivc and negative

is

0

n=

the series of absolute values

n-o

is

|o»|

convergent.

If

an infinite series

is

0

n=

m >


Complete

_

f

Third kind:

cos

__*_,,

f

First kind:

t-Haintdt

1

Elliptic Integrals:

(2r)-M

r«

^

S(i)

/"*

C(x) = (2»)-M

[X

Fresnel Integrals:

convergent but not absolutely

con-

3-96

MATHEMATICS

[SEC. 3

The following results

vergent, it is called conditionally convergent.

deal primarily DC

The summation symbol 2 by itself will imply

convergence properties.

V

with

.

n =0

A necessary and sufficient condition that the series 3.11 Fundamental Theorem. 2a„ is convergent is that for any c > 0 there exists a number N = N(t) such that for every n > N and every integer m > 1, |s„+m

-

s„|

= |o„+i + a„+i + • • •

+

an+Jt|

< <

3.12 Let 2c„ and 2d„ be two series with positive terms, and Comparison Test. furthermore, let Ze„ converge and 2ri„ diverge. If a„ < c„ for all n greater than some N, then 2a„ converges. If a„ > dn for all n greater than some N, then 2a„ diverges. Root Test. If the series with positive terms 2a„ is such that for all n > N,

y/a*

< a < 1

However, if for all n >

then the series is convergent.

Ar,

> 1

?S!
the series 2an is convergent.

< 1

However, if for all n >

A^,

a*

the series 2a„ is divergent. 3.14 An alternating series Sa„ is convergent if |a„| < |o»_i| Alternating Series. and if lim a„ = 0. The error made in taking the sum of the first n terms of the n—* oo

alternating series 2o» as an approximation to the sum of the series is less than the absolute value of the (n + l)st term. 3.15 Rearrangement Theorem. The sum of an absolutely convergent series remains the same whatever change is made in the order of the terms of the series. In a conditionally convergent series it is possible to rearrange the order of the terms so that the new series converges to any desired value. 3.16 If two series 2c, and 2r4 are absolutely convergent Cauchy's Products.

i

«

with respective sums C and D, then the Cauchy's product

V

^

r*rf,_t = 2c,-2:dt is

(1/n)

is

The harmonic series

1,

DO

^

>

divergent.

=

1

Harmonic Series.

divergent

<

1.

r), and

r" is convergent if

n~0

71

3.18

—

V

The geometric series

if

with sum 1/(1

Series.

|r|

Geometric

is

3.17

|r|

also absolutely convergent and has the sum CD.

divergent when

series

n=l <

(l/«l)

is

The hyperharmonic 1.

Series.

k

ivnd

is

1

k

when

Hyperharmonic

^

on

3.19

>


then the series diverges. Ratio Test. If for all n > N, n„ > 0, and 3.13

convergent

Sec. 3-2]

ANALYSIS Infinite Products and Continued Fractions

3.2

3.21

Infinite Products.

An infinite product

djj,

=

W d,

i-l is a symbol

3-97

■■■ an

■■■

representing the sequence of partial products p„ = didtdt ■ • ■ dn

J|

An infinite product

i-l

d, is said to have a convergence property if for every

j greater

J

mm The

of

d,

is

is

it

1,

i-i

rf/.

It

common to write

in place of

i-i

.

if,

if

if,

+

Continued Fractions

3.22

a,) =

= e2'»<1+«/'

elD l»(>+"/)l

F„ =

/>„,. = oo

+

t(1

:

if,

is

it

if

is

+

is

+

called absolutely convergent the product x(l + |a,|) con product »(1 O/) It can be shown that the product ir(l and verges. absolutely convergent ay) only if, Say converges absolutely. More generally also can be shown that 2a,-2 converges absolutely, then Za„ and t(1 + a,) converge and diverge together. The value of an infinite product t(1 4- a„) unaltered by any rearrangement of its and only factors the product converges absolutely. It may be observed that formally

b.//),„

Dm = a, + />,/£>,» ■D.-i.» = a„_i + 6„/a„

= a0 +

bi/oi

+

F

a

is

a

it

0,

bi

if

...

is

If

is

,

. .

. ,

a„, bi, where a0, Oi, . . . The evaluation can be made by b„ are given. called a continued fraction. beginning with D,_i,„ and working back. The result F„ there are infinitely many a> and bi, and an infinite sequence F0, Fi, every ^ defined that may or may not have Ft, limit F. In case limit, this has limit will be called the value of the infinite continued fraction, which then said to converge and which can be represented

o»/oi +

• • •

,

,

is a

1

...

+

The slant line and bar may be understood here as a special sign of aggregation signify ing division by everything that follows. finite continued fraction Fn function of 2n a„, variables a0, . . . By the ordinary rules of algebra one can express bn. ft*

A

where

A* and

/?„ are polynomials in these 2n +

variables.

It

is

Fn = A„/B„ \


ir

PJ

[

|

<>i) instead

is

(1 +

y-i

ae

|^

if,

if,

than some ./ no factor d, vanishes and the sequence of partial products beginning at has the convergence property. It follows that a convergent infinite product has the and only one of its factors value zero zero. in Since the sequence of factors a convergent infinite product tends to customary to write the product as

easily verified that

3-98

MATHEMATICS

and in general,

it can

be shown inductively

[SEC. 3

that

+ biAi-z aA., + b,B._2 - diAi-i

At = Bt As functions of the variables a and represents

Ai

Ai and Bi

b, the

If

are called continuants.

one

K ( b" ■ ■ ■ 'b' ) \o0|a,, . . . ,
=

thus exhibiting the variables, then

Bi

K K(\aha2,

=

Two important properties of continuants

K

and

,e»-icb„\

by the identities

. . .

b„ . . . ,6. \ cJf /\ao,ai, . . . ,a„J

F

0

is

C

+

F

,

Fn

=

(-l)"-*6,65

• •

b./(B.B._,)

are some useful continued fractions:

tan

- x/1 - x'fi - x'/b -

-

•

The following

- Fn-t

•

is

a

it

b,

B,

i

is

...

in

it

1,

bi

b,-

for any ci, . . . , c„. In particular it is always possible to choose the rs so that every oi — at ** ■ • ■ — 1 or so that every bi = bi = • ■ • =1, provided only that no = in the other. a< — 0 in the one case and no = If every then said to be periodic or recurring in case for some positive > n. The value of any periodic integers n and v true that Oi+„ = a* whenever — Bty/C, where A, and continued fraction can be expressed in the form (A a„, and conversely. are polynomials the variables an, If the ai or the or both are functions of a variable x, then the continued fraction For questions of convergence the function F(x) wherever converges. defines often useful: following identity

■•

+

(1)

•

•_•

+

+

+ +

x

x*/7_ tanh x - xfl x'/S + x*/5 +_x*/7 tan-' x - x/l l»i>/3 3V/7 + 2*x'/B tanh"1 i xfl - l«i»/3 - 2siJ/5_- 3!i'/7 exp i - l/i - xfl + x/2 - z/3 x/4 - i/5 m

(2)

• • •

(3)

•

(4)

+

•_•

+

■■•

(5)

Power Series

3.3

The series ■

■

+

a„xn

• • • =

V

a^x1

+

+

•

a\X

+

+

«» ai,

ajcn

|x|

r

is

r

is

a

i

|x|

a

x

is

is

is

where x a variable and the numbers a„, called coefficients, do not depend upon x, for what values of called a power series in x. For power series the primary question properties. does the resulting series of constants have various convergence The following theorem applies to power series that converge for some value of x and diverge for other values. There exists positive number r such S.31 Fundamental Power Series Theorem. that ~a„i" converges absolutely for every but diverges for ever}' >■ r. < If real variable, the interval (—r,r) called the interval of convergence and is If the real variable x replaced by a complex called the radius of convergence. is


_

/

. . . ,c„a„

,o,/)

.

_ R / bit ... ,6: \ \ai,
bh . . . ,b( Va0,ai, . . . ,a
(c^c* \ a0,f iai,C2ai,

. .

are expressed

\

/

K

■ ■ ■ ,b<

analysis

—

a„(x

xo)"

called

a

series

is

The

V

is

of

=

r

3-99

variable z, the number r is again called the radius of convergence and the circle called the circle convergence.

\i\

Sec. 3-2]

—

power series in x

Letting x' = x

x0.

— xo,

n=0

y

0

n=0

±

(a„

6„)x"

(6)

=

o

n=

V

±

£

=

6„x»

=

V

be in the interval of convergence; then d_

a, and

0

Let x,

=

o„x"

0

>

£

:

a

of

|x

— = |x'| < r. this series converges by the above theorem when The quantity x0| it, is called the center the series. Some of the properties of power series in complex variable are discussed in the article on complex variables. Some operations for power series follow In the common interval of convergence for two power series Sa„x" and 2b„x",

a„xn^

dx

naÊ"-1

(8)

presents some of the common power series. Table

Sfaelaurin Series /or /(x):

Power Series

6.

5

Table

r.

0,x-

£ B=

0

1.

—

. /t">(0) — where a, — n!

Tii/Iot Srriet for f(z): 2.

V

-

a.(x

n)"

» =0 —

an

Ezpnnmtial

■

■

at x"

+

n->»V

_

(xlna)» —

IV n/

• • •

1

1!

Um

+

(xlna)'

-

(

a +

2!

.

S|+...

r In

+

2.718281828

-[:-Ti+iCTi),+iC-^),+-]

•>•

•

»-V

+

2)

+

~

2)

1

.+

r:

+,

In x

+

9.

■

"

^+ 3!

+

In x

the series terminate!)

^;

8.

+

>

-

«*.

integer,

|!

,

~

the series terminates at yn ~ n(" ~ x» + ■

nx +

positive

"'"

and Logarithmic:

-.#._i 7.

n

in a

When 5. • —

is

x).

n-^=Jl

integer,

a positive

-

+

(1

±

4.

„*-■„ +

±

- «-

+

nl

3. <* + »)■ When n

/(xo)

±

. where

1 +,


(9)

x
• ■

< x«

Power Series. (Continued) x> < 1 ytx' + ■ ■ ■ * -

Table 6.

- x - Hz* + Mi> -

10. In (1 + x) 11.

x) = In a + 2

ln(o +

-£l 7!

5!

3!

r» , j< — + 4! 2!

14. cos X = 1 15. tan x — x 4

H

IS

315

H


22.

x

-

x +

- ^ - -sL -

■; 45

x

x1

945

25. noah 20. tanh

• • •

-

— sin^1 x, cot-1 x —

2

,.,

x. < x.

3 • 5x' 3x> 4- — — + 2 • 4 • 0 ■7 2 •4 •5

+

2-3

-

2

■ ■•

xJ < 1

zt>\ ■• ■

x' < 1 3 •5 2 • 4 • 6 ■7*'

— tan"1 x. cscM x

x> < 1

-

2

— sec"1 x

and Inverse Hyperbolic:

+

+ £j +

|!

x.l+|!

f!+...

|| + |!4....

+

X-X-- + — - — + X*

°XS

17X7

3

15

315

27. ainh". x

-x-

28. ainh-' x

-

31. In sin x . 32. In cos

-

, 33. In tan x

L*—

2x —

2-4- 0-7

2 • 4 • 4x<

— -—

2 • 2x»

—

4

Xs

X*

X*

0

180

2,835

-

T1 4

+ ■ ■■

2 • 4 • ti • «x«

2 • 4 ■4x<

—

x' < 1

— • • •

xa < ir>

,

t' , x» < — 4

, x' , 7x« , 02x« , • • • In x + — + + 2,835 90 3

. _ t' x« < — 4

x«

x'

17x'

2

12

45

2,520

.

/. ^1

, x' 21

3x<

8x>

3x'

50x'

4!

51

61

7!

x» , 4x< — + —

—

31x« ,

+ • ■

\

-J

x« < 1

+ •••

— • ' ■ - —2 ■4 • 0 • 6x'

• • •

x>

14-x4

= f

? 2 ■2x>

x> <

— + — + — +■■ • 7 3 5

In x —

x-

34. <■»'"'=

Ior

-x+

t

30. tanh-'

• • •

+ • ■ 2 4 5 2 •3

loK 2x H

29. cosh-1 X — ±

35.

a > 0,

x' < —

4.725

2

23. cos-1 x —

24. sinh

J1

,.<-

!_+••• i - -2 - Ix + —3x> 5x» - Mx' + tan"1 x - x - Hx' + 'Si' 3 _I__L sec 1 x 2 • 4 • ox' x 0x>

Hyperbolic

■•

x«<,.

■

20. ten-' 21.

V+

x/

4

•••

+

- 1 - :3 -

• • ■

+

2,835

_+s-Ti+

+

+

sin''

+

+

«x.l - -+ _+

18. cot x

—r/V + o („ V2a

(— + 8\2a

+ 1

»' . +

—

4-

:<

l8.«e»-j+-

19.

— [— L2o + j

and Inverse Trigonometric: j--'

TrtffonoBw/rif

17.

[SEC. 3

MATHEMATICS

3-100

, • ■■ h

x > 1 x > 1

-a

< x <

«

analysis

Sec. 3-2]

3-101

Orthogonal Functions

3.4

. .

,o.)

the length

of the vector

given by

cosine of the angle between the two vectors

i-l

a,-1.

said to be normed or normalized.

and

b is

unit vector and

is

a

is

the vector

a

=

a

i-\

aj*

1,

}

n

>

square

a is

The

= (bi,bi, . . . ,b„).

of

and

b

N

.

a

space may be represented by rectangular = (at, at,

Two vectors a and b in coordinates; that is,

Sturm-Liouville Series and Other Expansions.

3.41

n-dimensional

If The

given by

n

~~.«-

>->

(10)

[(2")(2")r y-i y=i

is

denoted by (a,b),

called the inner product (or scalar product)

l

j

afij,

=

>

The quantity

of the two vectors.

a,-*

= (a,a)

is

n

called the norm of a.

The vectors

k.

j

k

j

if

is is

A

if

1

is,

%

5,

is 0 is,

b

a

and are orthogonal (or perpendicular) (a,b) =0. set of vectors |o,) called orthonormal, that Kroorthogonal and normalized, (a,,at) = 6,-t, where 8,* = necker's which or or '^ An arbitrary vector according to whether in n-dimensional space can be expressed as a linear combination of a set of « orthonormal vectors lay); that =

i-l

Cj&j

.

(g.ay)a,-

Y

n

n

y

g

=

j-1

(11)

such that

>

a?

= (ai.as, . . . ,ay, . .

.)

a

is

Most of the notations used for n-dimensional space can be extended not only to countably infinite spaces but also to certain spaces of functions. An example of a the special Hilbert space consisting of all points countably infinite space

= (a,a) < a

is

is

b;

in

it

In dealing with a function f(x) customary to speak of f(x) as a point or vector Let the real functions f(x) and g(x) be defined on the finite the function space. then the inner product of the two functions given by interval a < x 

<>

[/(*)]« dx =

(J,f)

(13)

3-102 A

MATHEMATICS

set of functions

|//(x)

)

is said to be orthonormal

/:'bfi(z)Mx)dx

= (/,-,/*)

[SEC. 3 on the interval a < x <

-«,»

b

if (14)

The concept of orthogonality for functions of a single variable can be generalized in two particular important ways. If f(x) and g(x) are complex functions of a real variable x on the interval a < x < b, then and g arc said to be orthogonal in the Hermilian

sense

/

if

b

i:

f(x)g~(x~) dx

= (f,g)

=0

(15)

where g(x) is the conjugate of g(x). Often there is associated with a set of real func tions /,(i) a function w(x) > 0, on the interval a < x < b, such that

/.

w(x)fi(x)Mx)

dx = «;,'

(16)

In this case the set is called orthonormal on [a,b] with respect to the weight function w(x). The orthogonality concept can also be extended to infinite intervals and to functions

a < x 

c/fj(x)

= 0 for all x

i-1

/

j,

- 0.

0,

is,

of the interval are the constants ci = o2 = • • ■ = c„ = 0. A set of orthogonal An orthogonal set functions is likewise a set of linearly independent functions. I //Of) j is called complete in a given function space if the only element of the space that (/,/,) = for all is orthogonal to every /,(x) is the zero function; that implies .),

/Or) =

Y

a

b,

. .

be a countable orthonormal set of real functions on the Let \
y-l

cm(x)

(/,**)

-

y

Formally multiply both sides of this equation by
y=i

dx

=

.

. .

l*f(x)v,(x)

2,

= (/,*>/) =

j

Ci

1,

The numbers (17)

called

a

y-i

generalized

Ci
where

<;,

\

are called the Fourier constants or coefficients of f(x) corresponding to the orthonormal The series system \(pj(x)}.

is


of more than one variable. The functions fi(x),fi(x),

=

(J,Vi)

Fourier series or expansion corresponding to /Or).

(18)

analysis

Sec. 3-2]

Given a finite orthonormal set j*>,(x) 10 = in the mean, or in the sense of least squares, a < x

,(x)

^

/-I

3-103

1,2,...,

71),

the best approximation f(x) on the interval

to the function

a,v, (x) is given by letting

a> be

the Fourier coefficients

dx; that is, n

is

minimized when The relation

a,- = c,. n

^

n

(/>/)*

=

T

c,» <

fablf(x))*dx

If

true for any n and is known as Bessel's inequality. normal set, then Parseval's theorem is

oo


J holds. 3.42

(/,/)

=

|*>;(x)

) is a

(19)

complete ortho-

00

(/,«)*

Sturm-Liouville Series.

=

£

c,« =

JflfteWdx

(20)

Systems of the following types:

W(x)]' + (\p + g)y = 0 aiy(a) + a,y'(a) + a,y(b) + aty'(b) 0 PiV(a) + P,y'(a) + ff,y(b) + Pty'(b) = 0

-

(21)

the prime (') denotes differentiation with respect to x, are known as SturmThe coefficients r(x) > 0, g(x), and p(x) are taken to be continuous The great functions of x in the interval a < x < b, and X is an arbitrary parameter. theory arises largely from the fact that many interest in S-L (Sturm-Liouville) boundary-value problems in physics and engineering lead to questions that can be answered by using S-L theory. Consider the general second-order linear differential equation where

Liouville systems.

Mx)v" +MxW

+ [/.(*) + Xgo(x)\y = 0

X is a parameter and where /2 > 0, fx, /o, and The general homogeneous boundary conditions where

go are

(22)

continuous functions of x.

a,y(a) + a2y'(a) + aty(b) + a,y'(b) = 0 0 0.0(a) + 0iV'(a) + P,y(b) + pty'(b)

-

are by

taken to be linearly independent. Upon dividing the general differential equation /2(x) and then multiplying the result by r = exp [{(Ji/fi) dx], the S-L equation

W(x)\

+ [Xp + g\y

-

0

(23)

It follows that the S-L system is where p = (ffo//j)r and g — (}o/Ji)r, is obtained. quite general. An S-L system is called sclfThe differential equation of an S-L is self-adjoint. adjoint when (a,0, 0,ajV(b) = (a,0, ari/9.)r(a)

-

-

The most common type of self-adjoint S-L system has two additional conditions. First p(x) is assumed not to vanish for o < x < 6. In this case both r(x) and p(x)

MATHEMATICS

3-104 can be assumed greater thai: zero. tions are of Sturmian type; that

[Sec.

3

+

|02| >

=

(24)

0

0,y(b) + 0,y'(b)

and and

0 0

a,y(a) + <*iy'(a) = |a,| + |a,| >

where

0

is,

The second condition is that the boundary condi

6.

+

The following three theorems summarize the primary results of S-L theory: The system of the Sturm-Liouville differential equation with Oscillation Theorem. Sturmian boundary conditions and positive p(x) and r(x) has an infinite number of real characteristic numbers that may be arranged in a monotone increasing sequence • • • that tends to 00 . Corresponding to each simple eigenvalue Xo < Xi < Xj < constant. X„, there exists an eigenfunction
2^

c.„

-

6

where

c„
p(x)f(x)
dx = (/,*>»)

(25)

0

71=

J

JC

uniformly and absolutely and has the sum /(x). The eigenfunctions fulfill the orthonormality conditions

j*

=

p{x)Pk(x)

dx = tik

(26)

The normality can easily be attained by letting

the weight function.

Vi(x) = Vi(vi,Vi)~H

=

W

where p(x)

is

(«,«)

Vi

[^'pWfi'W

-

ax

f*

and

dx

\

//

(e)»

/

-

Y(z) = (pr)Wt/

I

.

where

:

is

Theorem. Given that /(x) Equi-convergence integrable over the interval (a,b) then the S-L expansion behaves as regards convergence in the same way as an ordinary Fourier series. The transformation

[**

- g(z)]Y(z)

(27)

-

-

J'\

=

0

dz'

+

takes the S-L equation into the Liouville normal form

J\q/p). fc» = and q(z) (pr)-M(«i,/
it

is

r

if

is

If

is

is

multiregion and has coefficients or solutions. This arises because the problem interface conditions. The results of ordinary S-L theory can often be extended to neries

Hao

n=l

(«» cos

nx

+

JC

T

these cases. The trigonometric

+


converges

b„

sin nx)

(28)

analysis

J

&»■"-/ r J —* fit) =

x)

odd function, that /(— known as the Fourier sine series.

f*

1

and

cos nt dt ~r fit)

sin nt dt

and the corresponding —fix), a„ = an even function, that 0

T

/(x) is an is

When series

-1 /f'

is,

=

a.

in the interval (— x,t) if its coefficients

to/(x)

If fix)

is

railed a Fourier series corresponding given by

is

are

3-105

is

Sec. 3-2]

n—

b

*'*^

cos

—

a

dt

J" m

t^

b

— a

is

a

dt+r2-l

f

J" m

(29)

l

o

— a

b

_L_

More generally

a

is

he corresponding series called Fourier cmine series. < x < series corresponding to fix) in the interval

f»

and

the Fourier

t

=

0

6.

/(-I) -/(*)

is

form for a Fourier series

The exponential

A

number of theorems giving conditions under which a Fourier series corresponding function converges in some manner to that function are given in the references. The following an example of a Fourier theorem. Fourier Theorem. If fix) sectionally continuous on the interval (— *■,»•) and periodic with period 2r, then the Fourier series corresponding to fix) in the interval (-t,t) converges to the value j-£[/(x+) /(x-)] at every point where the right-hand and left-hand derivatives exist.

+

is

is

a

to

The set [1/\/2t, cos nx/y/r, sin forms a complete orthonormal set for the space of all functions that are sectionally continuous, are assigned the value

nx/\/r|

at the points of discontinuity and the value \ilfi— r+) /(*•")] at possess right-hand and left-hand derivatives at every point of the interval (— »•,*). Fourier series not only are special cases of S-L series but indeed arc the simplest \y = cases of S-L series, being associated with the differential equation y" tegendre Series. The Legendre expansion or series for function /(x) in the interval (-1,1) given by

+ fix')]

+

H

Ui*+)

of the intervals, and

is

a

0.

+

the end

+ nin +

=

gix) =

=

+

a

(32)

.)] forms an orthonormal ^)->»„

is

series

and

I)

+

pix)

=

(n

J^P„it)Pmit)dt

.

— **,

The legendre

n.

l)y(i)

(31)

.

= n{n 2,

=

0,

The set {in + }-i)l''P»ix),

in

X

S-L equation where r(x) =1

1,

seen to be an

- x')

^]

(1

d

[

Ax

dt

0

the Legendre polynomial of degree series. Legendre's equation

S-L

1,

2 is

PJx)

and where special case of an

~^ f^fiQP.it)

where a„ =

a.P.ix)

0,

no

is


i- _«

set; i.e.,

3-106

MATHEMATICS The

Fourier-Bessel Expansion.

Bessel function

[Sec. 3

J*(\x)

satisfies

the Bessel equation

£('g)+(--f)"-° 0

.)

2,

set.

1,

The

(j

each n this is seen to be an S-L equation where r = x, p = x, and g = —nl/x. = be the roots of the equation J„(\a) = and let . .

of functions

J.(X,x)\

I

For

X,-

Let

function f(x) in the interval (0,o)

fc/.(X,0/(0

a, =

and

^°

*

W0/(0

DIFFERENTIAL EQUATIONS

4

^°

=

4.1

Introduction

is

a

is

is

a

is

is

a

is

is a

is

is

if

is

is

A

a

is

is

is

a

is

is

A

is

an equation that involves differential or a derivative. differential equation The equation may contain algebraic and transcendental functions of a differential or a derivative. It assumed that a differential equation not an identity. It difficult to make simple yet complete classification of differential equations because not only the number of classes and subclasses needed immense but also any Nevertheless particular differential equation likely to appear in many classes. The certain broad classifications are commonly adhered to and have proved useful. primary division of differential equations results from the number of independent variables that are present. An ordinary differential equation an equation containing one independent variable, one or more dependent variables, and at least one derivative of By contrast, a dependent variable with respect to the independent variable. partial differential equation contains two or more independent variables, one or more dependent variables, and at least one partial derivative of a dependent variable with respect to an independent variable. differential equation also categorized by giving its order and degree. The order of an equation the order of the highest derivative found in the equation. (The word differential no confusion results.) The degree omitted in front of equation of an equation the power to which the highest derivative raised. To this defini tion should be added the requirement that in stating the degree of an equation implied that the equation polynomial in all the derivatives. If the dependent variables and their derivative occur only to the first power or called linear. For linear equation, the degree, and not as products, an equation coefficients of the dependent variables and their derivatives are therefore functions An equation that not linear only of the independent variables. called nonlinear. meant a set of functions of the independent By a solution of differential equation

it is


c,

given by

as

ao

where

dt = i,-,-

is

WW)

^-

Bessel expansion or series for

a

A

1°

is,

forms an orthonormal set in the interval (0,a) with respect to the weight function x; that

3-107

ANALYSIS

Sec. 3-2]

one function identified with each dependent variable, that when sub in the equation results in an identity in the independent variables. It is implied that the solution functions have at least as high derivatives as are needed in variables,

stituted

differential equation. solution of an ordinary differential equation of nth order and one A particular solution dependent variable contains n, and only n, arbitrary constants. of an ordinary differential equation of nth order and one dependent variable is obtained when the n arbitrary constants are given special values. Not all solutions of an A ordinary differential equation need to be particular cases of the general solution. solution that is not such a particular case is generally called a singular solution. The general solution of a partial differential equation of nth order, m independent variables, and one dependent variable is a solution containing n arbitrary functions of m — 1 variables. The question of the existence and the uniqueness of the solu tions is not considered here, and the references should be consulted. the

The general

4.2 1.21 of the

Ordinary Differential

Equations

Differential Equations of the First Order. first order has the form F(y',y,x) = 0

The general differential equation

prime denotes differentiation with respect to x. first-order equation. The equation

where the


for the

This

is the implicit

form

- f(y,x)

p m y'

There exist a number of special explicit form for the first-order equation. implicit and explicit equations whose solutions can be obtained by elementary methods. The following list presents the more common types and their solutions. is the

types of

f

Variables Separable:

P-f(x)g(y)

=

Ja

g(y)

[Zf(x)dx Ja

(35)

Homogeneous Equation:

log?-/'/e-*LJ a/a a f(v)

y-xv

p=fO>) \X/

(36)

v —

Now the equation P(x,y) dx + Q(x,y) dy = 0, where P(tx,ly) = t"P(x,y) and = t"Q(x,y), has this homogeneous form. 2. An equation p — f[(ax + by + c)/(ax + &y + t)] may be brought into the above homogeneous form, when afi — ab ^ 0, by the linear transformation x = u + d, y « » + e, where u and v are new variables and d and e are constants. If a/3 — ab = 0, the equation can be put in the variables separable form by either substitution 1.

Q(tz,ty)

ax + by +

v — or

substitution

v =

ax + fiy + y.

Linear Equation: V +/(*)
= (?(*)

=

V

Equation: p +

becomes

c

the linear equation p

+

(1

(« f(x)y —

=

-

^Vffdx)

g(x)r

n)f(x)y

h =

j'fdx

(37)

(38)

= (1 — n)g(x) under the substitution

3-108 Dependent 1.

[Sec. 3

MATHEMATICS Missing:

or Independent Variable

p = /(//) and p = g(x) are equations of the variables separable type.

-f(p);x

2. y i. x =

f(p);

y

-c

-

t

=

■=

f[f(p)/p]dp. Spf'(p) dp.

Clairaut Equation: y

- px + f(p)

either y = ex + f(c)

or

x + f'(p) = 0

(39)

a singular solution.

Exact Equation:

P(x,y)

dx + Q(x,v) rfu = 0 is an exact equation

J Pdx J (q - f Mdx\dy —

+

dy

if

—

«■

ay

— dx

(40)

= c

is the solution.

Integrating Factor:

A function H(x,y) is called

an integrating


HP(x,y)

dx

factor or Euler multiplier

+ HQix,y)

if

dy = 0

is an exact equation. There exist certain other well-known first-order equations whose solution cannot be The following types may be listed, and for further obtained by elementary methods. information concerning them the references should be consulted. Generalized Riccati Equation:

dx

Special Riccati Equation:

+ ^ dx Abel's Equation of

the

aj/» = bx"

First Kind:

*-

I «*■

i-0 Abel's Equations of

the Second

Kind: 3

\di(x)y + g„(x)]y

=»

} Mx)]/*

i-0 Binomial Equation:

4.22 Linear Differential the nth order has the form

©"-' Equations.

The general linear differential equation of

n

i-0 It is generally assumed that the coefficients /,(x) and g(x) either are continuous and one-valued or are meromorphic functions of x throughout some region. Furthermore,

ANALYSIS

Sec. 3-2]

3-109

If g(x)

it is usually assumed that/o(x) has at most isolated zeros in this region. Eq. (41) is called homogeneous. The expression

•

= 0,

-0

The following properties are com is called the linear differential operator of order n. mon to all nth-order linear differential equations: y =■ yi is a solution of the homogeneous equation L(y) = 0, then y = cy\ is 1. also a solution of the equation, where c is any arbitrary constant. 2. If Vi, J/., . . . , yp are p solutions of L(y) = 0, then

If

y = cij/i + ctyi

+

■■ ■

+ c„yp

is also a solution of the equation, where Ci, Ct, . . . , c, are arbitrary constants. 3. If i/o is any solution of L(y) = g and yi is any solution of L{y) = 0, then

+

11 =■ Vo

solution of L(y) = g. 4. The complete primitive of L(y)

Vi

is a

■■■

= cij/i + cti/t +

+ c„yn

...,!/„

is,

Y

• • •

y'i

„,<»-»

y* y'n

.

y'i

■ ■ ■

Vt

is

is

Y,

-\-

are n linearly where Ci, Ci, . . . , c» are n arbitrary constants and j/i, functions that are solutions of L(y) = 0. The general solution of independent where j/o a particular integral, that any L(y) = g{x) has the form y = yt the com solution of L(y) = g(x) containing no arbitrary constant, and where plementary function or complete primitive of L(y) = 0. Certain other concepts are useful in the discussion of linear differential equations. The determinant

a

0 is

A

if

,

.

,

is

known as the Wronskian of the functions y%, y% . . . yn. The vanishing of the Wronskian of j/i, . . yn at any point of the region being considered means that the the Wronskian does not vanish, the functions are linearly dependent. Conversely set of n linearly independent solutions of functions are linearly independent. called fundamental set or fundamental system. the nth-order equation L(y) = The adjoint equation to the equation

i-0 the equation

L

^ L*, called the formal adjoint operator to the operator L. If and the equation L(y) = are called self-adjoint. The relation 0

L

is

vL(y)

-

yh*{v)

=

[P(y,v)\

/

The operator L* then the operator

=

0

i'

is


Y(x)

= 0 has the form

dx

(42)

MATHEMATICS

3-110

[SEC. 3

where P is a linear function of y and v and their first n easily, and is known as the Lagrange identity. The equation

— 1

derivatives, can be verified

71

A(y)

where the a„_,- are constants, This special linear equation applications but historically solved. The substitution y auxiliary equation

=

= 0

£ a._.g ■=0

(43)

is a homogeneous linear equation with constant coefficients. not only is one of the most important and common in is the first equation of a general type to be completely = eml in this equation leads to the characteristic or n

Y

= 0

a^m*

t =0

If of

this auxiliary equation has n distinct roots, ru

A(y)

. . . , r„,

r2,

= 0 is

y = rie'>*

then the general solution

• • • 4- c„er-*

+ r.e'"x +


where the cs are arbitrary constants. If m of the roots of the auxiliary equation are equal to some value s, then the portion of the general solution corresponding to s is The equation

(ci + cix + csx* +

I

«.-<*'

i =0

• • •

+

cmx"~l)e'x

•-/<*) ^ dxx

(44)

where the a»_,- are constants, is known as Euler's differential equation. The substitu tion x — e* transforms Euler's equation into a linear equation with constant coeffi cients. This follows from the relation

"8-1(1-0

■(!-+>)'

Linear Second-order Differential Equations. 4.23 The second-order linear equa tion is a particularly important and common equation in physics and engineering. The general equation of this type has the form

f(x)y" + g(x)y' + h(x)y The differential expression

L(y)

has the adjoint expression

L*(v)

m

=

fy" +

fv" + (2/'

= r(x)

gy' + hy

- g)v'

+

(/"

-

/' = g, then L{w) = L*(w) and L(w) is self-adjoint. adjoint, the expression if

-e^Liy)

f

(45)

where F(x) =

g' + h)v

Although

L(y) is not self-

fX?dx J° f

is self-adjoint. Since any second-order linear homogeneous eqviation can therefore be made self-adjoint by the multiplication of a suitable factor, there is no loss of generality in considering only the self-adjoint case. The normal form for the homogeneous equation L(y) = 0 is u" + lu =0, where Hig/f)'. The left-hand side of the normal equation is Viig/f)1 /(i) = (h/f)

-

-

analysis

Sec. 3-2] obtained by letting y = u exp I

-

—

3-111

,dx\

/

in the expression L(y).

7(x) is called

invariant function of the equation. The method of variation of constants (or parameters) and the solution of equations For by the use of series are both applicable to nth-order linear differential equations. the

+

y"

hy =

+

gy'

r

is

+

0,

is,

illustrative purposes these techniques are considered for the second-order case. the general solution of the If the complete primitive cij/i(x) + ciyt(x), that known, then the solution of the equation gy' + hy = equation y"

Assume solution of the by the method of variation of constants. = Ci(x)j/i(x) ci(x)yi(x), where c\ and c2 nonhomogeneous equation of the form If c'ij/i c'tyt set equal to zero, then are functions of x that are to be determined. = cij/'i + cty'%. Furthermore, y" = C\y"\ + Ciy"2 + c'ty'i + c'tv't, and therefore substitution in the nonhomogeneous equation gives c\y'\ + c'ty'i — r. This last can be solved for c\ and c't. Direct equation and the equation c'ij/i + c'ty* = integration then gives 0

y'

is

+

y

+

a

can be obtained

mB +

rifnÛt

f

cj(a:)

and

The

are constants.

differential

B^(x)

equation by the use of power series consists

00

>

of assuming a solution of the form

,J§'dt + A^{x)

+

!/,(*)

anx"+'>,

substituting

it

-

f

f

¥£di

a

y

= v*(z)

The method of solving

formally into the differ-

y

0,

p

is

ential equation, collecting terms in like powers, and setting the resulting coefficients of each power equal to zero. The objective of this process to determine the and the = An elementary illustration follows. Consider y" + a„'s and thus a solution. =

Direct substitution

a„x"*i>, aa ^ 0.

>

and assume

y

oo

gives

n=0

n-0

n=0 00

=

a„xn+i>

a„_2]x»+"-' =

0

l)a„

=

0,

p

+

(p

0,

arbitrary

or

=

1.

=

0

satisfied

also satisfied for

p

a»-2 =

l)p"i

0

+

«■

if

=

—

is

The indicial equation p(p — In this case + l)pat == The remaining equation becomes

l)a„ 0

p

+

-

0 is 1)

2).

= 0.

(p

(n let

>

(n

The coefficients of the powers are zero when p(p p)(n +

-

p

+ p)(n +

1)

=

[(n

2

Y

+

+ Dpa.x"-'

n

(p

Daox"-'

+

terms results in

+

n=2

n=0

-

The collecting of like

a,,_2X'■+',~,.

\

The shifting of indices gives

>

00

„(p

=

a„z"+"

0

l)a„x"+<>-8

Y

x

-

+

p

+ p)("

+

("

Y

00

p


r is

is

where W = i/ii/'i — yiy'\ the Wronskian function and given by solution of y" + gy' + hy =

B

=A_f rJÛt

A

ci(x)

First

at, for example,

MATHEMATICS

3-112

_

-a-i nn

-

+C-4 - l)(n - 2)(n - «,..._

=

n(n

1

[Sec.

(_iw,?! n!

3)

for n even and a„ = 0 for n odd.

3

Consequently 30

y = a0

) Li

(

—

xim 1)" — — -;= (2m) !

m-0 This solution

x

a0 cos

is easily obtained or verified by using the properties of the cos x. to obtain a series solution, which is a0 sin x.

p = 1, it is easy

Partial Differential Equations

4.3 4.31

First-order Equations.

v = v(xi,x%,

Let

X\,

is,

. . . , x„ be n

independent variables, let

the dependent variables, and let

. . . ,x„) be

Pi

If

=

dv —

P2 =

OX\

dv —

• • •

OX J

Pn =

dv — dXn

The general partial differential equation of the first order has the form


F(php2,

. . .

,p„,v,xh

. . . ,x»)

= 0

(46)

When n = 2, it is customary to put x\ = x, xt = y, p = dv/dr, and the equation has the form F(p,q,v,x,y) = 0

q =

dv/dy, so that (47)

First-order equations arc called "linear" if the equation is of the first degree in the partial derivatives. Furthermore, they are called nonlinear if at least one partial derivative is present to some degree other than one. The linear equation for two independent variables has the form + Q(x,y,v)q = R{x,y,v)

P(x,y,v)p

and is called Lagrange's linear equatwn. equations

^ P

=

The

^ Q

=

(48)

simultaneous

ordinary

^

differential V (49)

R

called the subsidiary equations for Lagrange's equation. If f(x,y,v) = Ci and = c2, where ri and ct are arbitrary constants, are two independent solutions of the subsidiary equations, then any arbitrary functional relation >p(f,g) = 0 satisfies Lagrange's equation. (p(f,g) = 0 is called the general solution or integral of the For example, the equation ap + bq = 1 has the subsidiary equation equation. dx/a = dy/b = dv/1. The subsidiary equation has the two integrals are

g(x,y,v)

x — av m

f

—

ci

and

and the general integral is ip(x — av, y — bv)

y — bv m g

«

c>

=0

Subsidiary equations and the general integral are similarly obtained for the linear equation with n independent variables. Consider the nonlinear equation F(p,q,v,x,y) = 0. A complete integral of this equa tion is any solution that contains two arbitrary constants or parameters a and 0. The complete integral may be denoted by f(i,y,v,a,ff) = 0 and may be interpreted geo A particular integral is obtained metrically as a two-parameter family of surfaces.

analysis

Sec. 3-2]

3-113

complete integral by assigning a and 0 particular or definite values. eliminated from the three relations

from a are

/ then the integral

eliminant

*-0

= 0

^

da

(3

= 0

dP

if it satisfies the original equation, A singular integral, if one exists, may

>f>(x,y,v) = 0,

of the equation.

If a and

is called a singular also be found by

eliminating p and q from the relations

dp

dq

ipW so that f[x,y,v,a,(p(a)] = 0 represents a one-parameter family of The totality of solutions of the equation F = 0, which are derived from the

Next let 0 = surfaces.

equations

flx,y,v,a,,pM] =0 upon

eliminating a for all possible choices of
/

each choice

called a characteristic

curve. This Charpit's method may be used for solving nonlinear equations of first order. method is based upon determining a second equation of the type G(p,q,v,x,y) = 0 that, The p and q thus along with F = 0, can be solved for p and q in terms of x, y, and v. obtained can be inserted in p dx + q dy = dv and

an integxable

w(p,q,i>,x,y)

expression for dv may result. It can be shown that any solution p or q or both, of the system

= a containing dp

+

pf.

"_

f,

dq

+

_ =

-pF,

- qF,

=

-F„

_

,5Q.

-F,

is,

G

= u> — « = 0. where Fz signifies dF /dx, can be used for G; that This method to problems involving more than two independent variables. 4.3J Second-order Equations. The general second-order equation has the form .

,wI,x„,fx„

. . . ,vx.,v,xi,

. . . ,x„)

=

(51)

0

,vtiLi,

P(vMlXuvM„

. .

can be generalized

Ai,vXlIi

(52)

D

=

is

0

cv

+

Bivx,

(53)

t'-l

and D are functions of xi, x%, . . . x„. customary to deal with the to classify second-order linear equations

coordinate transformations

,

is

Aij, B„

it

In order

A{fl.„, +

C,

where the

v,

,

V

ij'-l

+

n

n

y

An

B

called xn, and are functions of the x t>*„, vtl Although most of the following concepts apply to quasilinear equations, easier to consider the linear equation

where the quasilinear. it

=

B

in

where »,,,,. signifies d*v/(dxi dxj). Almost all second-order equations of interest are the class of quasilinear equations. The equation

is


-0

|£ da

MATHEMATICS

3-114

[SEC. 3

"'I n

i-1

It is possible to find where the matrix o = (a«) is nonsingular with real elements. such transformations so that the above linear equation takes the canonical form n

£

A4,X.j

• • • = 0

+

w'-l


where point

/li,*

=

+1 if

i

=

j

< m < n and

Aij*

(54)

- 0 if t ^ j or if i = j

> m at a given

. . . ,x„").

With the use of this canonical form it is possible to classify a linear equation at a given point (xi°, . . . ,z„°) as follows: If at the point 1. All the An* ^ 0 and have the same sign, then the equation is called elliptic at the point. 0 and if all but one An* have the same sign, the equation is called 2. All the An* hyperbolic at the point. 3. All the An* ^ 0 and if there exist more than one positive An* and more than one negative An*, the equation is called ultrahyperbolic at the point. 4. Some An* = 0, the equation is called parabolic in the broad sense at the point. 5. Only one An* = 0, the other An* all have the same sign, and the coefficient of dv/dzi corresponding to the zero An* is not zero, then the equation is called parabolic at the point. A linear equation is called elliptic, hyperbolic, etc., in a region if it has that character at every point in the region. Next consider the equation Ar + 2Bs +

Ct + D(x,y,v,p,q)

= 0

(55)

where r = dh>/dx2, s = dh)/(ds dt), and I = dhi/df1 and where the coefficients A, B, and This equation C are functions of x and y that are twice continuously diffcrentiable. is called elliptic, parabolic, or hyperbolic according to whether the determinant A =

AB _ BC

=

AC

—

B* is greater than zero, equal to zero, or less than zero.

The

associated first-order ordinary differential equation

2B^ dx

+ C

has for solutions two one-parameter families of curves in the xy plane,

fi(x,y)

= a

fi(x,y)

and

= 0

In the hyperbolic case when A < 0, these two curves, called characteristics, are real. In the elliptic case, A > 0, the characteristics are complex. Finally for the parabolic case, A = 0, the curves coincide. The characteristics can be used to obtain the following normal forms:

Hyperbolic: or

V[(

Elliptic: vet

Parabolic:

-

!>„

= F(a,p,v,va,vp) vafi = F(r,Ti,v,v(,vv) + ij

a=f

+ tf„ = F(r,v,v,v{,vv)

V„ ij

arbitrary function of x and

= F(f,i),r,Pf,Pf) y.

a =

f

+

J7)

a = 0 =

{

f> =

t

0 =

f

—

V

- in

ANALYSIS

.... -1—^———FJ

M;

ko

< n; kQ

//,t(xi,

.

,X„)

...

—0,1,

should be noted that the number of equations

kt

,

=

(k

at*

.

—

.

the

which satisfies

•

. .

of finding a solution of the system

+

j

—

. . . ,vM,

. ,

XK,vh

[t,Xl

2,

= G<

(i,

—

The Cauchy problem consists

1,

Cauchy Problem.

3-115

+

Sec. 3-2]

• •

j

. •

+

fc„ < n.,)

n;_i)

= F(t,v)

differential equations

For partial equation

where

is

a

simple example dhi dt*

_™

= v0

given by the vibrating-string

dhi dx* is

Ox1

—

dhi

dy'

=

or

!'„

Vyy =

0

+

0

dhi —

+

is

.

a

if

if

is

where v(U,x) = H\(x) the initial displacement and where v,(tu,x) = H2(x) the initial velocity. The Cauchy-Kowalewski theorem states that all the Gi arc analytic in a neighbor hood for all their arguments, and the Hn are analytic in the corresponding neighbor hood of the point (xi°, . . ,x„°), then the Cauchy problem has unique analytic solution in the neighborhood of (<°,xi°, . . . ,x„°). 4.33 Elliptic Equations. The simplest representative of the class of elliptic equa tions Laplace's equation in two dimensions and rectangular coordinates, namely,

a

This equation often describes steady states, such as the steady temperatures in body or the equilibrium form of a membrane stretched over a curve. function called harmonic or a potential function in has continuous region D derivatives of the first two orders and satisfies vIZ + vn = at every point of D. Consider an elliptic partial differential equation L(v) = in finite region D with a boundary r. The boundary-value problems for elliptic equations are classified as follows IHrichlet's Problem, or the First Boundary-value Problem. Find that function satisfies L(v) = in the region D and equal to prescribed continuous function on the boundary r. The term on means that the limit equal to a function approached by as the boundary approached by points interior to D Xeumann'8 Problem, or the Second Boundary-value Problem. Find function that satisfies L{v) = in the region D and whose outward normal derivative dv/dn at every point on the boundary equal to a prescribed function/. Mixed Problem, or the Third Boundary-value Problem. Find a function that in the region D and, a prescribed function on the boundary satisfies L(v) = s or a = then the mixed problem ar -f b(dv/dn) = on the boundary r. If reduces to Dirichlet's problem or Neumann's problem. few of the important results associated with elliptic equations follow: bounded inimum- Maximum Properly. If harmonic function in The then the values of in D cannot exceed continuous on the boundary region D and its maximum on or be less than its minimum on r.

A

if it

a

v

is

/

r

/

a

r,

is

if

r

a

0

0,

v

is

r,

is a

/

6

if

v

/

0

v

is

r

0

v

a

/.

is

v

r

v

0

is

a

:

0

0

a

is

i>

homogeneous

A


v(t0)

is

at

a

^

a

is

of

I,

is

if

is

It

equal to the number of unknowns; variable time in physical problems, the order singled out; n* the highest derivatives of tu that are present, then d*>u/dtH< must also be present; and the functions Hm are all given in the same region in the (ii.xi, . . . ,x„) space. The simplest example of that of finding solution of Cauchy problem the independent

3-116

MATHEMATICS

Poisson's Integral.

_, V2v and

v =

problem for the circle of radius R,

The solution of Dirichlet's

where =

ft

1 itv ■

Or1

r Or

[SEC. 3

1ft

,

0

r* d6*

f(0) when r = /J, is given by "(r'9)

"

•2irfi f'-'K io

1

%rR

/(8)

fit

-fl *

r2

r'-2flrcos[(s/r)

+

ft>

_

(56)

Mean Value Properly. If u is a harmonic function in the region D, then v has the mean value property in D. The function v is said to have the mean value property in D if at. every point P in D v(P) equals the average of v either over the circumference or over the area of every circle contained in I) with center at P. Harnack's Theorem. If Vt(x,y) (k = 1, 2, . . .) are a sequence of functions har monic in a finite region D, and if this sequence converges uniformly in D, then the limit function v(x,y) is harmonic in D. Requirement for Neumann's Problem. In order that there exist a solution of the second boundary-value problem it is necessary that the integral of /over the boundary r vanish; i.e.,

The equation of a vibrating string

ft

ft

_~

Of

dx1

The Cauchy problem for this equa

is the simplest example of hyperbolic equations. tion with the initial conditions —

and

f(x)

has the solution, called d'Alembert's =

M\J(x +

formula,

t)

-

+f(x

*

(57)

formal solution of the wave equation

Vu =

^

n

A

v(t,x)

= g(x)

v,(0,x)

t)]

v(0,x)

+

Hyperbolic Equations.

Yi

4.34

VI(ri

with the initial conditions . . . ,x„)

=

f(xi

u,(0,ii,

and

xn)

. . . ,x„)

= g(xh . . .

,x»)

"

F»r--'

p
/o

- '")<"-,/*W*)r) *

('S

ai, (r>

- p!)«

.

.

xn

+

0„r) dV'„ ■ ,

...,*,

+

0T3a)i

/■

2

=

(Sir,

y

Hx, +

■■

I

-y

■

Qh(.x,r)

f

where

£

dr

f

('2

7

&nh)i

+

*

fo

given by

alS

is

v(0,xh

"


/r(2)*-/r'«*-°

«„)

,

(58)

analysis

Sec. 3-2]

3-117

K„ = — — — -

where

r(n/2)

and p* = nti* sional space.

+ • ■ • + «„s is the mean value of the function h in the n + 1 dimen In three dimensions (n = 3) this formula is called Kirchhoff's formula,

and in two dimensions

the formula is often called the l'oisson formula. Methods for solving hyperbolic equations are discussed in connection with bound ary-value problems. Parabolic Equations. The simplest parabolic equation \» the one-dimensional 4.36 heat-conduction or diffusion equation

_ ~

dv

dh)

at

ax1

This equation with the initial condition and the boundary

v(0,x) = f(x)

conditions

- gi(t)

v(t,Q) is the first boundary-value


4.4

and

v(t,L)

- g,(t)

problem for the heat equation.

Differential

Equations of Mathematical

dy1

dx1

Physics

dz*

in Cartesian coordinates. 4.41

1.

Laplace Equation

:

W

= 0

Gravitational potential

2. Steady not on time

temperature,

in a region free of mass i.e., temperature in a body that depends on position but

3. Velocity potential of the irrotational flow of an incompressible 4. Magnetic potential 5. Electrostatic potential in a region free of charges 4.42

1.

Poisson Equation

:

W

p(x,y,z)

Electrostatic potential in a region containing charges temperature with internal sources

2. Steady 4.43

k1

Helmholtz Equation

:

W

+ fcV

= 0

may be a constant (or parameter) or a function of position. 4.44

or Diffusion

Heat-conduction

Equation

:

1 3*>

K 4.46

Wave Equation

W 4.46

Telegraph

i)t

:

Equation

= c1 at*

:

ai1

at

fluid

MATHEMATICS

3-118 Maxwell's

4.47

Equations

V •

V

•

:

H

B = 0

curl

D = irrp

curl E

-—

=

c

dt

+

[SEC. 3

i 4tJ

D=«E

c

1 r)H -— c

B =

dt

a*H

where E is the electric field vector, H is the magnetic field vector, c is the velocity of is current density, p is the density of the charge, p is permeability, and « is the light, dielectric constant.

J

Equation for the Transverse

4.48

Motion of a Plate or Bar

0.AAv,

Equation of Continuity

4.49

p is

4.410

+ V

■ (pv)

= 0

the density and v is the velocity vector. Navier-Stokes

Equation p


= o

+

:

£ dt where

:

where p is the pressure,

^ at

=

F

:

-

Vp + „V2v + £ V(V .i

m is the coefficient

4.6

•

v)

of viscosity, and F is the external force.

Boundary-value

Problems

Although the term boundary-value problem can be used in connection with ordinary differential equations, difference equations, integral equations, and variational prob lems, the term generally denotes a partial differential equation and a set of auxiliary conditions. Some authors separate such problems into initial-condition or boundaryvalue problems according to whether the conditions are specified at some initial time or all the conditions depend upon the space coordinates but not upon an initial state. The following two comments should be noted when dealing with boundary -value First, the number of boundary conditions is usually equal to the sum problems. of the orders of the highest derivative with respect to each independent variable. For example, in Problem 1 of Art. 4.51, the number of boundary conditions is four while in Problem 2 the number of boundary conditions is three. The second and most important comment is that, whenever possible, the problem should be broken into simpler problems that can be dealt with separately. In par ticular, if nonhomogeneities occur in more than one of the equations, it is often possible to consider a sot of simpler problems each with one nonhomogeneity. For example, consider the problem with the boundary conditions V(0,y) = Mv) V(x,0) -Mx)

V„ + V„

V(a,y) V(x,b)

= .4(1,,/)

= Mv) =/,(*)

0 < y < b

This problem can be broken into five simpler problems. Let V = Ui + Ui + Ui + f/« + W, where U\, Ut, Ui, and Ut satisfy the equation Furthermore, E/*» + U„ = 0 and where W,x + Wyy = A. W(fl,y) = W(a,y) = W(x,0) --= W(x,b) = 0 Ut(0,y) Ui(a,y) UAx,0) = Ui(x,b) = 0 U,(a,y) = My) U,(0,y) = UAx,0) = Ut(x,b) = 0 UA0,y) = Vm(o,v) U,(x,0) = Mx) U,(.x,b) = 0 U,(x,b) = Mx) ut(0,y) = Uda,y) = f'«(x,0) = 0

-Mv)

-

-

analysis

Sec. 3-2]

3-119

problems are the method of variables and the method of transforms, particularly the Laplace The techniques of conformal mapping, integral equations, and calculus transform. of variations can also be used for solving boundary-value problems. Two examples of both the method of separation of variables and the use of the Laplace transform The two most common methods of solving boundary-value

of separation

follow.

Separation

4.51

of

Variables.

Problem

Solve the two-dimensional

1.

Laplace

equation

a*v

di«

dy*

Formal Solution.

0

V(0,y)

= 0

V(a,y)

V(jc,0)

=

Vix.b)

fix)

Let V(x,y)

=0 =0

Xix)Yiy),

=

0 < y < b 0 < < a

i

and substitute into the Laplace equa

This gives

or upon

dividing by

X Y, d'Y/dy'

d'X/dx' X

left-hand side of this equation is a function only of x and the right-hand side function only of y, both sides must be independent of x and y and therefore equal to some constant X. The term separation of variables is seen to be derived from having separated all the functions in an equation that depend only upon a certain variable from the other functions that depend upon the other variables. The constant X is called the separation constant or parameter. The use of the separation constant results in the two equations Since the is a

*5

=

dx*

If

X =

— <**, the

as

X

XX

^-

and

-Xy

dy*

general solutions of these two equations can be written, when a ^ 0,

= A sin ax +

B

Y

and

cos ax

=

E

sinh ay +

F

cosh ay

Ey + F. X Ax + B and Y At this point it is necessary to apply the boundary conditions. First consider XiO)Yiy) = 0. This implies that B = 0 for all a. Next consider V(0,y)

and when a = 0, as

=

=

-

Via,y)

=

-0

X(o)K(y)

In order that A 0 and therefore X j*- 0, it follows that sin aa = 0 or a = «7r/a, where n is an integer. The conditions Vix,b) = Xix)Yib) = 0 imply that

The

F

cosh aft = 0

= — D cosh ab and

F

quantity

+

=

C„ sin

F

= D sinh ab.

D sinh a(b -

— a

x sinh

— a

Hence

)/)

E

afc

—

y)

This will be satisfied if

sinh

(b

E

b.

y

is

for any integer n, seen to satisfy the Laplace equation and the boundary conditions — along the sides x = x = a, and

0,


_

rectangular boundary conditions

with the

tion.

aw

MATHEMATICS

3-120

[SEC. 3

In order to satisfy the remaining nonhomogeneous conditions usually necessary to consider the expression

/ i-i

—

Cn sin

—

x sinh

a

a

V'(x,0)

=

fix), it is

— y)

(6

It was noted that each term and indeed the sum of the C'„ are constants. any finite number of terms of this series satisfy all the conditions, except perhaps = C, can be chosen so that when V(x,0) = f(x). The question now is sin — x = a

/(i)

known that

if

a

seen to be a Fourier scries, and

0,

y

b

—

sinh

Cn

is

This

is

i-i

it

/

if,

where

Fourier coefficients,

fa

2 a

=

/(<) sin

JO

—

C„ sinh (nx/a)b

=

c are the

dt

.

.

.

Solve the heat equation for one-dimensional flow

-

—

™

dt

k

Problem

2.

1

n=»

sinh (nir/a)(b — sinh n-rb/a

x

y)

nir —

—

3xs

»(ir,0

Formal Solution.

Let v(x,t)

0

=0

>

< x < w

X(x)T(t), and substitute into the

=

heat

equation.

gives

_ dT

.

d*X

_

7"

or

Xlu"k~d7T

=

rr

A'"

X and

is

is

Therefore, since in this last equation the function on the left a function only of the function on the right a function only of x, the equations

t

This

0

- /(i)

I

K0,<) = p(x,0)

0

with the boundary conditions

and

7"

the separation constant.

-

XkT

is

= XA"

It

is

are obtained, where

X

X"

easily seen that the

quantities

Ce-"1*' sin (nx) n

an integer, satisfy these equations and the homogeneous boundary Again consider a series of such terms

2^

where tions.

is


is

is

then indeed the series converges to/(z), where /(z) nearly an arbitrary function. The solution of the boundary-value problem therefore given by

n-l

C.e-*'*1 sin (nx)

condi

analysis

Sec. 3-2] When

3-121

= 0, this reduces to the Fourier series

t

/

c„

sin (nx)

n= \

and

to/(x),

provided the

are chosen as the Fourier coefficients,

e„

c„ =

=

-

2

Laplace Transform.

4.52

/(*) sin

I

J"

(ns) ds

is given by

The solution of the problem

v(X)l)

-

V

/

f(s) sin

d*U

_

oi

e^"1*'

Solve the heat equation

Problem 3.

dU

sin (nx)

(ns) da

ax1

l/(x,0) Formal Solution.

t/(f,<)

I'm

[7(0,0 = *X0 = 0

=0

/ > 0

0 < x < 1

-L|f/(x,<)l

Let u(x,«)

-

/

e

"U(x,l)

The application of

dt.

the Laplace transform to the above problem results in the equations

-k—

8U(X,d)

/(«), lim u(x,s)

problem is u(x,s) =

I

-x f- j

^ [-*(*)"]- Ml ^-*

the use of the convolution

a constant, then l/(x,<)

Problem 4.

f* r-*F{t

-

Solve the vibrating-string Ytt

with the boundary

conditions K(x,0) K(0,«J

= =

/(x)

erfc

- r) exp (-

|£)

(x*)-H

dr

(«)"*]

equation

-

a'F„

F«(x,0)

=0

Y(i,t) =0

>

<

1

|

<

A,

■*(■-£)}

x

=

j

gives

[|

U(x,t) =

If F(0

exp

0

Since

/(.<)

0

this transformed

The solution of

where /(s) is the transform of F(l).

=■ 0,

<

=

4

u(0,«)

0


with the boundary conditions

3-122

MATHEMATICS

The transformed problem

becomes

- s/(x)

s*u(x,s)

y(0,s) = 0

[SEC. 3

= alyxx

.v(l,s) = 0

where ;/(x,s) = L[Y(x,t)]. The solution of this transformed problem can be obtained either by the use of variation of parameters or by transforming with respect to x. In either case the solution becomes ,

*,

o sinh

where = sinh

—

+ sinh — /

— dz

/(z) sinh

/ J"

a

s/a

a

a

Jx f(z)

sinh

a

— dz

is,

In order to invert the transform and obtain Y(x,t), it is necessary to use an inversion the The terms in the inversion series correspond to zeros of sinh s/a, that The sum of the two residues of euy(x,s) at = ±s„, where «» = inrra poles of y(x,s). .), is

.

.

2,

=

1,

(n

s

series.

given by

+ exp (— in-rat))

J.£c„ sin (rnrx) [exp (in-ral)

Y(x,t)

f(z) sin (nrz) dz

I

r„

The formal solution therefore

becomes

=

c„ sin

(nrx)

cos (nrat)

this

is

=

0,

When

t

l

n=

4.6

Fourier

seen to reduce to the

series

Numerical Solution of Differential

for/(i). Equations

J/-.+1

»

yn

y'n Ax

!/'o

y(xo

Ar)

« ».

+

y.

m

yt

Ax

In like fashion n =

. . .

This, the oldest, simplest, and crudest technique, was devised by Euler. Euler method lets yi* ~ yo + y'o Ax and

= y(.xi)

VJ*±M1 a,

A

2/o

2,

«

shorthand notation for/(x0,!/o).

+

is

»i where y'o

An approximation for

= i/o.

1,

y(xo)

easily seen to be given by

+

with the initial condition

f(x,y)

=

+

V'

is

it

A

it

is

is

it

Since relatively few differential equations can be solved (or integrated) in finite terms, generally necessary to consider numerical methods in order to obtain Even when the solution of a differential equation can be approximate solutions. obtained in infinite terms, the solution usually difficult to evaluate and hence often of limited practical value. The advent of high-speed digital computers has made relatively easy to obtain the-numerical solution of difficult equations, such as multigroup reactor equations. large number of useful techniques for solving differential Some simple techniques follow. equations may be found in the references. 4.61 Differential Ordinary Equation. Although this Modified Euler Melhod. method as well as the following Runge-Kutta method can be used for a system of easier to consider a single first -order equation in order to indicate the equations, method of solution. Consider the differential equation

is


=2

p

where

modified

3-123

ANALYSIS

Sec. 3-2]

(yi*)' = /(ii,j/t*). Although this modified Euler method is slow and of accuracy, it is very simple and easily used. Method. Again consider y' = f(x,y) subject to t/(xo) = J/o. This Runge-Kutta method consists of computing in sequence fci, k2, k,, kt and then the desired 3/1, where where

limited

Ve

+

+

y0

Ax, 1/0 + k,) Ax 2k, + k,) H(.ki + Mi

+

/do

«

y,

+

=

+

kt l/i

(x0

f

k, =

^)

ki = f(xo,y0) Ax

is

it

.),

2,

1,

(n

In

= order to obtain y„+i . . necessary only to replace the x0 and yo right-hand side of the above relations by the previously obtained x„ and y„. This technique one of the most commonly used and of sufficient accuracy for most is

is

on the

applications.

Partial Differential Equations. Elliptic Equations. For illustrative pur = consider the Laplace equation vxx with Dirichlet conditions on a „„ square boundary; that v(0,y) = f,(y), v(a,y) = My), v(x,0) = f,(x), v(x,a) = f,(x). Let the square of side o be divided into a network or lattice of squares of side where an integer multiple of This can be accomplished by drawing two families of lines parallel to the sides of the original square and spaced at integer multiples of The intersections of these families of lines are called lattice points. The numerical technique consists of solving a set of associated difference equations for a function that takes on values only at the lattice points and using as an approximation to the funrtion
-

-

-

h)

v(x,

y

- 2v(x,y)

+

+

h)

v(x,

y

-

y)

2v(x,y) + v(x

0

+

h,y)-

h,

v(x

+

equation

_

N

-

1

...

,

=

2,

j

i,

"a = H(Vi+i.i + Bi-ui + vt.j+i + Vi.j-i)

1,

,

2,

1,

0,

i,

j

is

+

h, h, y)

y

+

h)

y

y)

h,

a

+

y)

h,

+

y)

h,

+

— or v(x,y) v(x, This associa + v(x, v(x H\v(x h)]. — w(x,y)]/h tion results from approximating a first derivative dw/dx by [w(x + — 2w(x,y) — and second derivative dhv/dx* by [w(x + The y)]/h*. w(x last equation for v(x,y) indicates that the value at an interior lattice point the arith metic mean of the values at the four nearest points. With the notation v(ih,jh) = = where . . . N, and nh = a, the difference equation becomes

is

0

j

»

Whenever the subscripts or X, the corresponding va or equal set equal to the appropriate boundary value; for example, voj = fi(jh). The problem now consists of solving a system of linear equations. There are two general methods, relaxation and iteration, that can be used to obtain

Rtj

-

Vi+l,j

Vi-ui

+

quantities

+

approximate solution to this linear system.

an

The relaxation method deals with the

»<./+!

+ "<./-!

is

is

1)

,

2,

j

1,

(t,

In

= this method a set of . . N — chosen and the ft,-/ are com puted. The Kij are called the residuals, and the object of the relaxation process to try to alter (or relax) the v,,0 so that the residuals all reduce, as nearly as possible, to

it

formula

„(/ - H[vi'\+Ui

+

is

is is a

not systematic and may proceed in any fashion hand method and not a machine method. Iterative methods of solving linear equations are discussed in Art. 4.1 of Sec. 3-1. The Gauss-Seidel method Consider the a commonly used iterative technique. Since the relaxation process order to reduce the residuals,

zero. in


h.

a is

h.

h,

is,

v

0

+

4.62

poses

»<•+»,_,.,

+

»c-)<>J+1 +

»<"■•-»

3-124

MATHEMATICS

3

[Sec.

i\,

where the superscript n denotes the nth iteration. is chosen for the Again a set interior points. New values t><,(,) are obtained from the formula by starting at the lower left-hand corner and proceeding to the right and then upward. are The process is continued until the t>,-/"+1> are similarly obtained from the t\/">. sufficiently close to the » 0, 0 < x < L be covered with a net U(0,t) = ffi(<), U(L,t) = gt(t). of equal rectangles with sides Ax — h and At = k, where L is an integer multiple of h. U(x,t)]/k and t/« is replaced by [U(x + h, i) If U, is replaced by [U(x, I + k) 2U(x,t) + U(x — h, t)]/h* the heat equation is expressed by

-

-

U(x,

t

+ k) = rU(x + h,

+

t)

- 2r)U(x,t)

(1

+ rU(x

-

h, t)

k

h

is

if

r

h

0

L

-

Ui+Ui

2

+

f',-,„

A'

-

-

0,

- 2r.,

Jfc1

-

1,

j

,

,

+ t/,.,-1 = ^

°

- 2Lfo

2,

0,

t

I

Ui.i+i

1,

is

L

I

0,

-

k,

0

r

/

is

indeed, any U(0,t) and U(L,l), since they are specified by the boundary conditions. Having obtained U(x,k), the values U(x,2k) can be obtained, and so on, step by step the time interval to be covered. to U(x,nk), where — nk One precaution must be followed in using this explicit method. The value of must < < otherwise the numerical solution obtained may have little con satisfy nection with the actual solution. This fact, rioted by Courant, Friedrichs, and Lewy, fixed, the time mesh length the space mesh length cannot states, roughly, that be too large. Consider the wave equation Ua = a'Uit with the boundary Hyperbolic Equations. gM. Again /,(*), U(0,l) = g&), U(L,l) conditions U(x,0) = /,(*), Ut(x,0) cover the strip < x < with rectangles having sides Ax = and At = > where an integer multiple of H. Let the quantity U(x,t) be denoted by Ua when = jk, where — If the second x = ih and . . . N and = derivatives are replaced by second differences, the wave equation becomes

U(x,k)

-

f/0,0)

r

= k'a*/h*. The initial or Ui.j+i = rt/,+1„ t7„y_,, where 2(r W/„ + rf7._,., conditions t7(x,0) «»/i(ar) and t7i(x,0) = /-(i) or, with forward differences,

-

kf,(x)

t/

+

1

is

+

U

r

is

it

is

z

is

is

U k,

+

and therefore U(x,k) = A/2(x) f,(x) supply the information needed in order to start the step-by-step process. — f/.,,_i. Note that when ka = the equation reduces to C/<,/+i = f7<+i.y + i— t.y — = are at), It easily verified that where and any two g(x g(z) + at) f(x /(z) functions of with second derivatives, a solution of both the wave equation and this Therefore any solution of either the wave equation or the last difference equation. = f(x — at) + g(x difference equation a solution of the other, since a at) general solution. More generally necessary when dealing with hyperbolic equations to make sure that the numerical process makes sense; for example, high-frequency component of a = fc!a2//is must be restricted by wave motion will always be distorted. Here < in order to ensure convergence of the numerical method.

r


is,

With this formula it is possible to calculate the values of U on the where r = ka/h*. Now the values of V on the line t + k line if the values of U are known on the I line. t = 0 are given by U(x,0) — f(x), and therefore it is possible to calculate the values It of course, not necessary to calculate the values U(0,k) and l!(L,k) or, U(x,k).

ANALYSIS

Sec. 3-2]

3-125

OTHER TOPICS

5

Vector Analysis

6.1

of vectors appears in Art. 1 of Sec. 3-1. The primary objective here is of the more common and useful formulas and results of vector analysis. Scalar and Vector Products. The scalar (or inner or dot) product of two 6.11 vectors a and b is denoted by a • b or (a,b). The magnitude or length of a vector a is denoted by |a|. The vector (or cross) product of two vectors is denoted by a X b. It is customary to express a vector in 3-space in terms of three unit vectors i, j, k along the positive x, y, and z axes. e2, and In Sec. 3-1 the unit vectors are denoted by The two vectors a and b, e3. ta denotes a unit vector in the positive a direction. therefore, can be written as a = aii + a2j + ask and b = bii + bjj + 6jk; the follow relations may be noted: ing elementary A discussion

to present some

= a - a = a,s + a2» + b2> |a|> a • b = b • a = Oi6t + u.62 + a3bj = |a||b| cos (a,b) * • b = 0 means a = 0 or b = 0 or a is perpendicular to b = k- k = l a X b = — b X a = i(aj>3 — a3o2) + j(a3(»i — biat) + k(aib2 — 6iaj)

i-i

i-j=j-k=k-i=0

j- j=

iXi=jXj=kXk=0

iXj=k Generated for wjivans (University of Florida) on 2015-09-23 02:47 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Triple Scalar Product.

(63)

Volume of parallelepiped with edges a, b, and c:

= a • (b

(abc)

kXi-j

jXk=i

X

X b)

c) = (a

-

•c

X

b • (c

a) =

di

Oj

Oi

bi

bt Cj

C|

Ci

Triple a (a (a

X

X

Vector Products:

X

•

b) b) X

(b (c (c

X

X X

-

-

•

- b) -

c) b(a c) c(a • • d) = (a c)(b d) • d) = [a (b X d)]c

The Differential Operator

(a

•

[a

d)(b

■

(b

Vv = grad v = l The gradient is a measure

v

•

X

c)

c)]d = [a

dy

F

dx

h

J

X d))b

-[b-(cXd)]a

dy

|-k

—

dy

- »MI dz

J

„ ^ V • /„ (Vti) =

F

dz

+ j

\

(67)

(68)

dz

of the rate of change of the scalar field

- rot F - i \ VH

(c

(65) (66)

dz

v-.F=divF=^+^+^ dx dy

X F = curl

•

Rectangular Coordinates:

V.

(64)

6a

•

dx

few).

(59) (60) (61) (62)

dz

1 1 del squared =

=

9*V

dx'

F,i+ dx

/

. d2V 1

dy'

v

at the point

Fvj + FJc

+k

\dx

(69)

- ^)/ dy

, d'V 1

dz'

(70) (71)

Differentiation Formulas V(uv) (rF l V X (fF) V

•

= =

V- (F X Gi

r(F-

G) V X (F X G) V • (V X F) V X (V X F)

= =

-

uVv + vVu •F + i>V • F (Vv) (Vv) X F + W X F F • {Y X G) G • (V X F) F ■VG + G • VF + F X (V X G) + G X (V X F) G • YF F • VG + F(V ■G) G(V • F) 0 V X (Vv) = 0 V»F V(V • F)

-

-

-

-

(72) (73) (74) (75) (70) (77) (78) (79)

MATHEMATICS

3-126

[SEC. 3

Cylindrical Coordinates: x = r cos 8

z = z

y = r sin 8

ds* = dr! + rs d9« + dz2 ,

dv

,

D = er —

grad

+

dr

divF

(80)

- — +

e»

,„.,

dV

,

e, — dz

r d$

(81)

-IA(Pfr)

curl F = er V«ti

\ dv

(

-

\r

— de

(r

= 1 — r dr

(82)

\

/I

dz

—

\

-—

4-

dr)

+

e»

\

dr

dz

I

Ir

dr

Mo)

r

90

J

—

+

r* d8*

I—

(83) '841 ( *

3z»

Spherical Coordinates:

rlin-9

"

^

—I— 1

r sin 9 39

*

(sin Of,)

+

^ If ~

llr

Lsin

/

a

39

.

sin

VI

sin2

9

,

d
1

9

Bin*

r2

d*v

1

r*

I

dr/

H,

.

dv\ rs —

(

r! 3r

VI

1

a

„. = V*o

(87)

aA — 39/

is

n a unit outward normal vector to Integral Theorems, of surface, and dV an element of volume. +

= x\ +

yj

r

surface .4, dA

zk

•n

dA =

F

X

|c

JJa(V

• dr

(90)

is

taken over the area bounded by the closed curve C.

The double integral Divergence or Gauss'

Theorem:

The triple integral

is

ffJyV-rdV- Jfj-ndA

(91)

taken over the volume inside the closed surface A.

dV

=

dV

=

»(Vto)

^'VW

•n

~

dA

-

jjy

Aid)

J

•

''V,W ~ U>V!"^

(i

ffv

(V»

lis Jja

Theorem:

IIjv

Green's

an

Theorem:

F)

Stokes'

(89)

is

element

dr

dip

is

+

r sin 9 d«>

J

(rVr) + — ?—

(80)

r 08

r

C'

-—

ee

\

-

1

+

J

F

1

r* dr

Lai

-

(85)

r2 d92

8

8 3ys

8

I

+

r cos

a

fV

—— - e, -dr + er — r sin

div F CUrl

r2 sin2 9

9

p

/


grad

- dr2 +

r

ds*

2 =

y = r sin
x — r cos
WV"^

'

W*w dV

n dA

The triple integrals are taken over the volume inside the closed surface A.

(92) ^93)

ANALYSIS

Sec. 3-2]

3-127

Integral Transforms — Operational Mathematics

5.2

The integral transform

f(s) of

a function

/(«) =

J*

is defined by the integral equation

F(t)

K(s,t)F(t)

dt =

T.[F\

(94)

The function K(s,t), called the kernel of the transform, is taken to be a known function. It is apparent from the definition that

cf(s) =

jb K{s,t)cF(t)

dt =

T.[cF\

and

/.(*)

+Ms)

=

fb K(s,l)[Fi(t) +Ft(t)]dt

=

T.\Fl +F2\

=

T.\F,\

+

T.\Ft\

two relations indicate that the transformation between the functions F(t) and linear. The defining transformation above then, a linear integral transform, although customary to leave out the word linear. The use of integral transforms particularly useful technique for solving bound ary-value problems. In essence, this technique can be described as follows: First, apply an integral transform to differential equation and its boundary conditions. This, in effect, reduces the number of independent variables by one and introduces parameter. The resulting, or transform, problem then to be solved for the trans form of the dependent, or wanted, variable. It then necessary to invert this solu tion, that is, undo the result of the transform, in order to obtain the solution of the given problem. An example of this process appears in the article concerned with problems. boundary-value Each type of integral transform particularly appropriate to the solution of certain kinds of linear boundary-value problems. The Laplace transform has proved to be especially useful for solving transient problems of the type arising in the conduction heat in solids and in vibration theory. In contrast to he Laplace transform, where the transformed variable usually the time, the Fourier transforms are generally taken with respect to The space variable over an infinite or semi-infinite interval. Hankel transform applicable when the problem has symmetry about an axis and a radial variable from to w similar to the Fourier present. The Mellin transform transform in its application. should be noted that the transform of the product of two arbitrary or unknown functions cannot be obtained usefully in terms of the transforms of the individual functions. successful only when the Consequently, the integral transform method coefficients involved in a boundary-value problem are either independent of or ele mentary functions of the transform variable. 6.21 defined The Laplace transform f(s) of a function F(l) Laplace Transform. is,

These

is

is

is

0

is

is

a

is

t

of

is

is

It

f"

by

f(s) =

dt

~L\F\

(95)

is

C

is

if

of F(t) exists sectionally continuous in every finite F(t) of exponential order, that is, there exist constants and

if it

0

I

The Laplace transform interval in and > such that

e-"F(D

M

The inverse Laplace transform

is

l— >

lim

00

e~"\F(t)\ <

M

denoted by L-1| /(»)).

L\F(t)\ -/(«)

and

This indicates that

F(t) = L~l\f(»)]

is

The inverse transform not always easily obtained. The use of tables of trans The general forms the most convenient method of obtaining the inverse transform. is


is

a

a

is a

it

is

is

fit)

3-128

MATHEMATICS Table 6.

! 2

F'(f)

3

)

4

fofoFMixdx<

5

ft qF(x)

flQF,(t

/(.)

r)Fi(f)

<"A'(()

- b) where F(« - 0 if ( < 0

/(«

F«


-

■■•

- F«-»(0+)

"

"/(»>(«)

ax)

- a)

«»•/(.)

/(«)

4

-

<;(«)

q'(a.)

,-

-

(«

- a:)

a0(s

■• ■

^C*+1>0,

1 ■3 • 5

C-'e-'

15

cob a(

10

sin a(

17

sinh at

18

cosh a/

19

1 cos at

20

t sin af

21

f" sin at

22

• •

i, 2, • ■ ■)

<» + o» a «'

-

a'

-

«* + a' s= o> (s' + o')i 2as (•■ + a')' 2"a n!«" («» + a')-+> sin 2

(wt)-54

/o(2 Vfa)

^»(o()

_

t

24

/o(al)

(„

s1 + a' a

23

26

1) is — a)*

coa 2

25

-

(2n

(r(V*>o)

14

27

(-1)

y

12 13

«"-»F'(0+)

/■((•)

convolution

10

11

- «"-'F(0+) -

>

oV

7 H
L|F(D)

-X»)

dz

-

-

- F(0+)

»/(s) »"/(«)

= 6

Laplace Transforms

= I."1 !/(«)!

F(0

No.

[SEC. 3

t'J,(<*)

• ~>V»/.

Vtf

(Re ► >

.-'»«-*/<

!.-*/. -1)

(Re * >

28

(jrW)

29

erfc

30

M-»-p(-g

-

(.' + o>)-W 0-'|r,i + aj)W _ ,]»(,,

H) r-lM» + e-*V« ,t - e

•

-

H)(2o)

V

> o)

(* > 0) >0)

+

ol)-M

+ <.=)-'-Ji

(t

- u.v)

analysis

Sec. 3-2]

3-129

inversion procedures, however, involve inversion theorems and most often inversion The references contain tables of transforms as well as more integrals or series. information concerning the following transforms. The inversion integral for a function f(s) is given by

LrM/WI -;r-.

fy

lim

+

'0e"f(z)dz

(96)

is,

where the integration is along a line parallel to the imaginary axis in the complex z The following theorem, essentially the same as the one found in Churchill's plane. "Modern Operational Mathematics in Engineering," relates the inversion integral and inverse transform: If /(s) is an analytic function of s and is of order 0(s-t) in some Inversion Theorem. half plane R« * > c, where k and c are real constants and k > 1, then the inversion integral L,~H/(s) \ along any line Re z = y, where y > c, converges to a function F(l) that is independent of y and whose Laplace transform is/(«); that

>

and

is

for all

0,

of order 0(«1")

-/(«) /

is

>

0,

for each

ITOI

and

such that

0.

is

continuous Fit) F(0) =

I

-LrM/«)

F(f)

is

It

often possible (see references) to express the inversion integral in terms of an series or an equivalent infinite integral. Fourier Transforms. The Fourier transform f(y) of a function F(x) 6.22 defined by

The function /(;/) also called •ailed the formally states that

\"

-"

for F(x).

- « F(<)e'»

dy

(97)

The Fourier integral

dt

(98)

J

2jr

J

=

-i

>t

F(x)

e'»F(t) dt

spectral function

is

theorem

"
fr

(2t)-«

f"

/(!/) =

This leads to Fourier's inversion formula =

F(x)

continuous

=

fix)

dt

and

(100)

\fix)\

dx con

has right- and left-hand derivatives

"

-

* dy

/

+

\ 2

is

If fix)

F(x-)\

-

in every finite interval,

verges, then at every point where [F(x+)

F(t) cos 2iry((

if /

scctionally

J"x

dy

TT J™

— °°

F(t) cos 2*y(.t

- x)

dt

(101)

an odd function, the Fourier sine transform /(?/) =

'

J('

If fix)

J~

-

(99)

side of the Fourier integral theorem has the form z)

the right-hand

Jf

Fix)

(2»)-» f~Kf(y)e-'ydV

(2J

For real

is

Fit) sin

j/
(102)

Fix)

=

Jo"

with the inversion formula

(2)


is

infinite

fiy)

sin xy dy

(103)

MATHEMATICS

3-130 Table No.

-

F(x)

™

(2i)"

^

[Sec.

3

Fourier Transforms

7.

^ /(»)«-•'»

rf»

/(V) = (2r)

Wy*

b

ax /■•(x)e'>«

1

7 8

-•'«/(»)

ax

a) /(?/ «-"»»o/(j/) »(y)

e'«F(x) F(x + xo) 1

(2ir) -M

j"

_ F«)0(z (2x)-«|x|-W

-I)^

/(»)»(») l»|-w

©»W<.

l»l>a

0

9

10

sin ax1

11

cos ax3

kl

|x|

16

c-"»

0 < Re s < 1

-

(;)M|»I'-T(1

Re « > 0

»)sin

(2»)-*V»V<«

1

-

(a» 0

20

(*> + <.')

p»k)

is

H

kl < kl >o

V-

kl k|

o

f(x)

H

x«)

19

(5)

x» + o«

If

is

obtained.

i(2x) -4(><> +» + «)"'

[(a3 + »«)** + alW(a> + y>)'M

17

21

i(2r)

ll/l -

IE

18

+

1_

J_

\a\

M

p < x < 9 x < p, x > g where p < q 0 g-e»+iw« x > 0 0 x < 0

(-)

■'•<«»)

K.(a

|y|)

 1

an even function, the Fourier cosine transform f(y) =

(-)

13

/0"f«)

C104)

cos«,«
with the inversion formula

f(y)

of

a

of order

function. F"(

j)

xJ,{xy)F(x)

dx

(106)

yJ,(xy)f(y)

dy

(107)

given by =

J0"

The inversion formula

=

f(>j)

y

The Hankel transform

("

obtained. 5.23 Hankel Transforms. defined by

is

is

(105)

is


12

«->•-.(£

j.

4 5 H

f/0/>

:)

s

«F(x)

+

2

Sec. 5.24

analysis

3-2]

3-131

The Mellin transform fiy) of

Mellin Transform.

fiy)

[J

=

a function

Fix)

is defined by

dx

x>~lF(x)

(108)

The inversion formula is given by

-

-i-.

Fix)

fC+" f(y)x-'dy

(109)

The finite sine transform f,(n) and finite Finite Fourier Transforms. are defined, of a function respectively, by F(x) d n) transform

cosine

F(x)

fc(0)

sin nx

-

/,(n)

0

-

/.(")

n-l

.

(110)

. . .

(Ill)

.

2,

2,

0,

1,

-

.

are given, respectively, by

these transformations

--

=

n

< x <

cos nx

0

and

cos nx dx

F(x)

F(x)

(112)

< x < *•

(113)

1

n *»

Linear Integral Equations

6.3

The following four types of linear integral equations

are commonly

noted:

:

Volterra equation of the first kind

-fix)

J*

K(x,s)y(s)d»

(114)

:

Volterra equation of the second kind

-

[*

y(x)

K(x,s)y(s)

ds =

/(*)

(115)

f*

Fredholm equation of the first kind:

K(x,s)y(s)

ds =

f(x)

(116)

Fredholm equation of the second kind:

Kix,s)yis)

ds =

fix)

(1

-

J*

yix)

17)

is

b

is

is

The problem for these linear integral equations to determine the unknown function y(x) that satisfies the equation in the desired interval. It assumed that the func K(x,s) tions Kix,s), f(x) and the limits o and are known. called the kernel of the These equations are called integral equations, since the unknown function equation. appears in the integrand. Similarly the equations are called linear, since the unknown function y(x) occurs- linearly. related to nonlinear integral equations, most of the existing Although some theory theory deals with linear equations. is


The inversion formulas for

n =

)

=

sinnxdx

Y

/.(n)

('f(x)

+

and

=

f"

f.(n)

1,

f

6.26

'

MATHEMATICS

3-132 y(x) +

[SEC. 3

K(x,s) sin [;/(«)] ds = /(x)

Jb

is an example of a nonlinear equation, since the unknown function appears nonlinearly Some theory has also been developed for systems of linear through a sine function. integral equations and linear integral equations in more than one independent variable. This article, however, deals only with linear integral equations in one independent

variable. An integral equation is called singular if either one or both of the limits of integration become infinite or if the kernel becomes infinite at one or more points of its domain. e~"y(s) ds

[x — -

and

f J"

1

Vi

— s

-

y(s) ds =

f(x) /(x)

are examples of singular equations. There are t wo common methods of relating linear differential equations Consider the linear differential equation integral equations.

and Vollerra

/<

;/(x) one lets z(x) =

- JOL

y{x)

(n

—

1)!

*

z(s)

-

J* (i

-

»)*(«) ds + a„_,x + a„_s

+ a»-' ,

(n — im 1)!

+

• ■■

+ a°

in the differential equation leads then to the integral equation

Direct substitution

j*

[a,

*(*) +

i/("-2,(-r)

t—Hrr (x

and finally

z(«) ds + a„_,

= *>(*)

Ai(x)y<"-»

- s)A,

■■■

+

A„]

Jj

+

!/<"-"(z) =

then

£ i=1

+

*(«)
/(x)

—

r

r

The other method consists of repeated integration of the differential equation. not difficult to verify using the formula

/•*

■■ ■

It

is

/;*4j£/w*

that the differential equation will then become ds =

(118)

/(*)

t,(s)(x

—

s)', the A\(s) being functions of Ai(s).

=

0

i

where K(x,s) =

V

b-

K(x,s)y(s)

+

l

;/(x)

j*

-

X

K(x,s)y{s) ds =/(x)

exhibits the customary location of a parameter.

s

b

is

X

it

s

is

y(x)

is

a

a

Most developments of linear integral equations deal with Fredholm equations, since Volterra equation can be considered as special case of Fredholm equation with a < b. < x < and zero for a < x < defined for a < kernel K(x,s) which convenient to introduce such a appears naturally or Often either a parameter The equation parameter in the discussion of a integral equation.

a


If

+

(119)

ANALYSIS

Sec. 3-2]

3-133

There are three general methods leading to the solution of Fredholm equations of the kind with a parameter. The first method is called the method of successive, substitution. In this method the unknown function is obtained as a power series in x with the coefficients being functions of the independent variable as. This series is usually called a Neumann series and converges for certain values of X. Method two is due to Fredholm and will be called the Fredholm method. The unknown function in this technique is obtained first as the ratio of two power series in X. The power series in the numerator has coefficients that are functions of x, while the denominator power series has coefficients that are independent of x. This method considers the integral equation as the limit of a set of n linear algebraic equations in n unknowns when n tends to infinity. The Hilbert-Schmidt theory provides the third method for obtaining a solution. In this theory the unknown function is obtained in the form of a series of or characteristic functions. These characteristic functions, or eigenfundamental funrlions, are solutions of the homogeneotis integral equation and are associated with second

particular values of the parameters called characteristic numbers or eigenvalues. Before indicating some of the results obtained by these three methods, it is worth while to consider the special case where the kernel has the form

-t

,)

Ai(x)Rds)

-4,(jr) and B<(y) are continuous and linearly independent.

where

equation

of the first kind it is necessary

that f(x)

=

V

equation have a solution. in the form

For

a Fredholm

C,-.4i(j) in order that the i

I

The general solution of this equation can then be written n

y(x) =

o.fl.Cr)

£

+ h(x)

to all the B<(as); that

h(s)Bi(s) ds =

0.

h(x) is any function orthogonal

J

where

is,

•— 1

For Fredholm equations of the second kind with the above type of kernel a solution . form

1

aAdx)

a

n.

2

—

1,

Bi(s)Ai(s)

dsj

A

is

A

is the Kroneoker delta and

jb

-

= det

j

«,-,•

i-

necessary and sufficient condition that there exist unique solution that a certain determinant not equal to zero. This determinant

A where

+

i,

of r, defined by

-/(x)

A

assumed.

and set

is

is

y(x)

£

n

,

of the

is

written as

where

L(x,s)

L{x,s)f{s) ds

AAx)

■

/(*) +

J*

=

0

y


n

■■

A.(x)

This solution

can

be

3-134

MATHEMATICS

[Sec.

3

If A = 0, solutions may also exist. This situation parallels the linear algebraic case and corresponding results hold. 6.31 Successive Substitution. The method of successive substitution describes in its name the process used to obtain a solution. Successive substitution for yix) gives X

£ K(x,s)y(s)

- fix)

d»

+

f(x) +

X

f Kf + \* f

K.f

• • • =

+

=

J*

X

K(x,s)

Kf+

[/(*)

+

X

J*

Kb.sJyM

dst

J

■■

•

y(x) = f{x) +

x»

X"JA

lim *

n— •

$K

=

lim

n—+M

(\fK)"y

-

0

If, then,

indicated symbolically. y

where the integration

is

i+i

f

(x

y

a-)"/

The series involved in this solution converges Tn the case of the Volterra equation

if

=

fix)

(120)

A')"/

absolutes and uniformly.

Kix,s)yis)

J'

-

X

yix)

b,

(a

f

+

(x

'fix)

d»

given by

V

has a unique continuous solution that

yix)

is

if

is

Kix,s)yis)

J*

-

is

yix)

X

|X|

satisfies

+

gives the essential result for this method. If the kernel K(x,«) real and continuous in its domain < x < continuous, and the parameter A(x,s)| < M in this domain, fix) < l/[A/(b — a)], then the equation 1.

Theorem < b), a <

- fix)

w

if |

Theorem

1

vix)

a

ds =

fix)

the theorem holds without the restriction on the size of |X|. 5.32 Fredholm Method. The Fredholm method evolved

from considering a of linear equation that replaces the integral equation. Divide the range of and consider the unknown function and the integration into n equal parts of length kernel at The system of n corresponding values of the independent variable. equations in the n unknowns yiii) h,

system

/(»)

=

2,

=

1,

hKixi,Si)yisj)

»

-

X

yiii)

2

n


then formally

analysis

Sec. 3-2]

3-135

to the integral equation

corresponds

- j* K(x,s)y(s)

ds =

X

V(x)

The solution of these linear equations, providing A ^

=

iKxv)

f(x) 0, is

given by

A

1 det (i,, — X/iA,,), A;, = K(xi,Sj), and A,, is the first minor of the element where A in the ith row and jth column. The determinant A can be expanded in the form

A, +

(-**)• n! A,.

as n tends to infinity

lim

A

a

leads formally to

£)(X)

j

Ad,,.?,)

• • ■ A'(x,-,«i)

xA

/xi

where

A(xi,«,)

A(x,,s,)

In a corresponding fashion

Li =

The solution

k\

/

• • '

given above for y(xi) can

lx>

/K '

•

(*** Vx,

*'■)

V

*

h

% n—>m

X

* ,

- D(*,*;X) -

0

lim

Xk/

dxt

rewritten in the form

=

i

■

a'

t/(x)

T,

y{x)

~x

continuous in its domain,

K(x,s)y(s)

ds =

f(x)

if

the kernel A(x,s) then the equation

is

If

/(x)

in Theorems

is 2

D(x,s,X)

ja

D(\)

2.

Theorem

ds

primary results of the Fredholm theory can be expressed

and 3.

0,

The two

L(x,s,\)f(s)

is

where

and

=/(x) +

it

b,

i

j,

Hy starting with this solution, by taking the where the prime indicates that ft limit as n tends to infinity, and by noting that x,- can be any point of a < x < possible to obtain the formal result

if


The limit

continuous,

3-136

MATHEMATICS

[SEC. 3

has the unique continuous solution y{z) = where,

as obtained formally

/(x) +

j* L(x,8;\)f(s) ds

(121)

before,

j.

L(x's'x)

.

=

D(x,s,X)

~WT

K(x,s)

0.

= K(s,x)

A

Before some of the results of Hilbert-Schmidt theory are indicated, the types of com mon kernel may be noted. Hermitian kernel satisfies the condition

K(x,s)

K(s,x)

A

where the bar indicates the complex conjugate.

skerv-Hermitian kernel satisfies

A

A'(x,s) = —K(s,x) polar kernel has the form p(s)K(x,s) is

A

Any equation that has a polar kernel may be kernel A"(x,s) called symmetric kernel. dsdt >

Jbf(t)K(l,s)f(s)

0

jb

if

positive definite

a

is a

where A'(x,s) symmetric kernel. transformed into an equation with

z(x)*dx

=

1

z{x)z(x)dx =

jb

jb

if

is

is

is

is

is

If the inequality reversed, the kernel called for any continuous bounded fix). If the equality also permitted, the kernel called semidefiniie. negative definite. The following concepts appear in the results of Hilbert-Schmidt theory. A function normalized z(x)

said to be orthonormal r6

if

|z,(x)

)

set of functions

is

0

,

if

Two functions As usual the bar indicates the complex conjugate. said to be orthogonal 'b z,(x)z,(x) dx = A


A

is

is

it

is

0,

is,

0,

is,

The power series D(\) converges absolutely for all X, and the power series D(x,s,\) converges uniformly with respect to x and s in its domain and converges absolutely for all X. Theorem 3. There exists at most a denumerable number of values of X, called char In general no solution will exist acteristic values or eigenvalues, for which D(\) = 0. for the nonhomogeneous integral equation, that for X0 such that £)(X0) = f(x) On the other hand the homogeneous integral equation, that has no nonf(x) = trivial solution except for those Xo where Z)(X0) = 0. These theorems and other existing results or theorems are analogous to theorems for systems of linear algebraic equation. Hilbert-Schmidt Theory. The Hilbert-Schmidt theory developed with 5.33 the same conditions on the kernel K(x,s) and the function f(x) that are used for the In addition symmetric Fredholm theory. assumed the kernel symmetric. kernel satisfies the condition

z,(x)z*(x) dx = Sit

21(2)

and Zj(x) are

ANALYSIS

Sec. 3-2]

3-137

S,t is the Kroneckcr delta. A set of functions Zi(x) defined on the interval < b is called complete if the only function of the type being considered that is A value X0 such that the homogeneous orthogonal to every z,(x) is the zero function. whore

a < x

equation

y(x) = \0

Jb

K(.x,s)y(s) ds

nontrivial solution is called a characteristic value or eigenvalue. A correspond function is called a characteristic function or eigenfunction. Some of the more important results of Hilbert-Schmidt can be summarized as Given a real, symmetric, continuous, nonzero kernel, the following properties follows. has a ing

hold: 1. 2. 3. 4.

There exists at least one eigenvalue X0. All the eigenvalues are real, and the eigenfunction can be assumed real. There exists a complete orthonormal set of eigenfunctions {yi(x)\. A continuous function f(x) can be expressed in the form

where

if infinite

convergent on a < x <

is uniformly

b.

Calculus of Variation

5.4

The methods of calculus of variation are generally separated into direct and indirect In the direct method a suitable approximate variational problem is formu lated in which a finite set of n constants or parameters is to be determined. It then may be possible to let n tend to infinity in the solution of the approximate problem and thus lead to the solution of the original problem. In the indirect method a related differential equation, often called the Euler equation, is usually obtained. Some solution of the differential equation can then be shown to solve the variational problem. Only some elementary results of the indirect method follow. methods.

The simplest general problem of the calculus of variation a function ;/ = y(x) that gives a minimum (or maximum) value

-

I

(x,y,

—

\

integral

f

to the

Euler Equations.

of obtaining

Jxi

6.41 consists

dx =

f(x,y,y') dx

taken to bo values X\, x2, y(xi), and y(x?) are given. The given function Furthermore, the continuously differentiable with respect to its arguments. assumed to be twice differentiable. sought function y(x) Let y{x) denote the minimizing function for the integral and let tj(z) be function de6ned for xi < x < such that that possesses a continuous second derivative and otherwise arbitrary. Consider then the set of comparison liii) = = 77(22) but functions Y(x) defined by

/

is

where the

Y(x) = y(x) + enix) = y(x)

+

0

is

is

ij

a

/,

is

twice

ty(x)

Since ij(x) vanishes at the end points X\ and xj, Y(x\) = y(x\) tis a parameter. = First, The set of functions Y(x) have two important properties. y(x«). Y(ii) and i;(x) proper choice of possible to represent any suitably differentiable Second, no matter what jj(x) chosen, y(x) function satisfying the end conditions. = member of the set for The quantity Sy = ti){x) called the variation thefunction y(x). Consider next the integral where

/(«) =

[X'f(x,Y,Y')dx

of

is

0.

t

s

is

is

it

t

by

and

a


The scries

3-138

MATHEMATICS

[SEC. 3

Since letting < = 0 is seen to be equivalent to replacing Y and dY/dX and y and dy/dx, the integral /(«) is a minimum with respect to e when < = 0. It follows that a necessary condition for a minimum is given by

4

d

/(«)

I |€ = 0

=

/'(0) = 0

Now

and therefore

™»-/:(if'+£'')*-« The integration by parts of this last integral gives

/.:

v(x)H(x)

dx

-0

(122)

0

is

valid for all functions ir(x) that vanish at the end with H (x) a continuous function, point and are twice continuously different iable, then H(x) = identically in Xi < < Xi. Proof of this lemma may be found in the references. The use of this lemma with the above equation I'(0) = leads to the Eider (or Euler-Lagrange) differential equation 0

x

£-£(&)-*-£*■-• v7»'»

+

/»'«

-

(,23>

0

l/"/»V +

ft

or in detail,

The term

a

or a stationary value (or inflection point) for I(t). extremum to the value of /(«) applied to all three cases. a necessary condition for the existence of an equation is

a maximum,

0

a

is

0,

a

is

This seen to be second-order differential equation for the unknown function y(x). The two constants of integration can be evaluated from the end-point conditions. The condition /'(0) = which leads to the Euler equation, not sufficient condi tion for minimum of /(«). This condition /'(0) = indeed may imply a minimum, The Euler

- -I/,,.]

+//

iydx

ej = 4// does not vanish at the end points, even variation) of the integral The problem discussed above may be generalized in first the integral

8;/]"

called the variation (or first

l,Xl,

. . ,Xn,Xu

.

a

. .

/>

.

/.

is

[/,

SI = «/'(0) =

J"

is

is

differential extremum. Every solution of the Euler equation called an extremal (or extremizing function). The sufficient conditions for the minimum (or maximum) of 1(e) are quite involved and may be found in the references. Often the physical situation indicates the type of extremum obtained. The expression

if


which is valid for any of the chosen r/(x). The basic or fundamental lemma of the calculus of variation states the following: If the equation

number of ways.

,i„)

dt

Con.sider

analysis

Sec. 3-2]

3-139

where the superior dot denotes differentiation with respect to the independent variable Proceeding in a fashion similar to the above problem, the Euler equations becomo (.

M(£)-'~-5'-°

•

Ox,

For the integral

I the

Euler equation

h

=

J\x,y,y',v",

\

■■■

,y™]dx

becomes

- t-/,< d

dx

-

d*

+ T-J>" ax'

• • •

+

d" (-V"t-J,W dx"

= 0

Finally consider the double integral

I

f(x,y,v,vTv„) dx dy

1 1

D is the region of integration and v takes on prescribed values on the boundary The Euler differential equation becomes

dx

dy

or in detail

f,,*,Vxz + 2/„,,vx„ +

/,,,,»„

+ f„vt>, + /,,,»» +

6.42 Lagrange's and Hamilton's Equations. is conservative, a variational principle, called

/„,

+ /,,„

- /,

= 0

In classical dynamics, when a system Hamilton's principle, can be used to

is,

For a system with n degrees of freedom it is determine the equations of motion. possible to choose n independent quantities gi, . . . , qn that specify the configuration of the system. The corresponding velocities are given by
«2

«Wi9i

of the qs. For conservative systems the external scalar potential function or energy V. The kinetic gradient of — = defined by the difference between the function energy. Hamilton's principle states that V,

T

L

dt =

(124)

0

s:L

L

be functions

is a

where the a,,s may force given by the potential or Lagrange kinetic and potential is

2

-

1,

-

i

-

0

(gO

^

The Euler equations that result from this principle are

«

(125)

it

is

q

is

is a

is

of

These equations are customarily called Lagrange's equations motion. For a conservative system, the total energy E, which equal to the sum of the kinetic and potential energy, constant throughout the motion of the system. When the total energy and the momenta p, expressed in terms of the coordinates called the Hamillonian function H. The momentum p< for the ?'th coordinate given by is


where of D.

=

3-140

MATHEMATICS

[SEC. 3

n

The variational

= S

-

(T

V)

dl =

f''(2T

5

f' Ldl

*

f''

principle now gives

- H)

. .

. ,

_

2,

.

=

1,

aH

p,- =

and

dpi

i

aH —

,

qi =

,

Without giving the details, the resulting Euler equations, called Hamilton's equations, become

canonical

n

dqt

=

'

K

/

"

f(x,y,y') dx

/

I

is

Auxiliary Conditions. In many problems a function sought that renders one integral an extremum and that causes one or more other integrals to take on pre scribed values. Variational problems that involve such auxiliary integral conditions are often called isoperimetric problems. Consider, for example, minimizing the integral

g(x,y,y') dx

+

=

/

/*

dy

dx

defined by

^

an undetermined

(-ailed

is

r

function

or Lagrange multiplier,

the Euler

\dy'J

The three constants, the two integration constants, and end conditions and by giving K its prescribed value.

X

X

is

where the constant equation becomes

a

If

having a given prescribed value.

can be evaluated from the

BIBLIOGRAPHY Some references

Art.

for various sections

follow:

9,

8,

7,

6,

4,

3,

2.

6, 1, 2, 2,

1.

Refs. 11, 16 10, 15, 17. 22, 25, 27, 36, 37 Refs. Art. 3. Refs. 4. 21, 25, 34 12, 13. 19, 28, 30, 32, 38 Art. 4. Refs. 12, 14, 15, 18, 23, 24, 31, 33, 35, 39 Art. 5. Refs. General: 12, 20, 26, 29, 40

Art

Batcman, H.: "Partial Differential Equations of Mathematical Physics," Cambridge Pniversity Press, New York, 1932. 2. Carslaw, II. S.: "Theory of Fourier Series and Integrals," The Macmillan Company, New York, 1930. 3. Carslaw, II. S., and J. C. Jaeger: "Conduction of Heat in Solids," Oxford University Press, New York, 1947. " Fourier Series and Boundary Value Problems," 4. Churchill, R. V.: McGraw-Hill Book Company, Inc., New York, 1941. 5. Churchill, R. V'.: "Introduction to Complex Variables and Applications," McGrawHill Book Company, Inc., New York, 1948. 6. Churchill, R. V.: "Modern Operational Mathematics in Engineering," McGraw-Hill Book Company, Inc., New York, 1944. 7. Coddin^ton. K. A., and N. Levinson: "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York, 1955. 1.


subject to the integral

ANALYSIS

Sec. 3-2]

Behandlung von Differentialgleichungen," Springer-Verlag

8. Collatz,

L.

9. Collatz,

L.: " Eigenwertaufgaben mit

"Numerische

:

OHG, Berlin, 1951.

T.: "Theory York, 1935.

Anwendungcn," Mathematik und Keihe A, Bd. 19, Akademische Verlags-

tcchnischen

in Physik und Technik, ihre Anwendungen gesellschaft, Leipzig, xvii + 466 pp., 1949.

10. Copson, E. Press, New

3-141

of Functions of a Complex Variable,"

Oxford University

R.: "Differential and Integral Calculus," vols. 1 and 2, Interscience Pub Inc., New York, 1946 reprint. 12. Courant, R., and D. Hilbert: "Methoden der Mathematischen Physik," vols. 1 and 2, Publishers, Inc., Springer-Verlag OHG, Berlin, 1937. (Vol. 2 reprint Interscience

11. Courant, lishers,

1943.) 13. Courant.

R., K. Friedrichs, and H. Lewy: tlber die particllen Differenzengleichungen

Physik. Math. Ann., 100: 32 (1928). der mathematischen 14. Doetsch, G.: "Theorie und Anwendung der Laplace-Transformation,"

Springer-Verlag

OHG, Berlin, 1937, reprinted Dover Publications, New York, 1943. 15. Erd61yi, A. (Bateraan Project): "Higher Transcendental Functions," 3 vols., "Tables of Integral Transforms," 2 vols., McGraw-Hill Book Company. Inc., New York, 1953. 16. Goursat, E. : "Cours d'analyse mathematique," vols. 2 and 3, Gauthicr-Villars et Cie, Paris, 1915. 1". Hobson, E.

W.: "The Theory of Functions of a Real Variable," Cambridge University London, 1907. 18. Householder, A. S. : "Principles of Numerical Analysis," McGraw-Hill Book Com pany, Inc., New York, 1953. 19. Ince. E. L.: "Ordinary Differential Equations," Longmans, Green & Co., Inc., New York, 1927, reprint Dover Publications, New York, 1945. 20. Jeffreys, H. J., and B. S. Jeffreys: "Methods of Mathematical Physics," Cambridge University Press, New York, 1946. 21. Koopp, K.: "Theory and Application of Infinite Series," Blackie & Son, Ltd., Glasgow, first issued 1928, reprinted 1944, 1946. 22. Knopp, K.: "Theory of Functions," vol. 1, 1945, vol. 2, 1947, Dover Publications, New York. 23. Lass, H.: "Vector and Tensor Analysis," New York, McGraw-Hill Book Company, Inc., 1950. 24. Lovitt, W.

V.: "Linear Integral Equations," McGraw-Hill Book Company, Inc., New
York, 1924, reprint Dover Publications, New York, 1950. 25. MacRobert, Co., Ltd., London, 1927, T. M.: "Spherical Harmonics," Methuen reprint Dover Publications, New York, 1948. 26. Madelung, E.: "Mathematischen Hilfsmittel des Physikers," Springer-Verlag OHG, Berlin, 1936, reprint Dover Publications, New York, 1943. 27. Magnus, W., and F. Oberhettinger: "Special Functions of Mathematical Physics," Springer-Verlag OHG, Berlin, 1943, reprint, Chelsea, New York. 1949. 28. Milne, William Edmund: "Numerical Solution of Differential Equations," John Wiley Sons, Inc., New York, 1953. -*9. Morse, Philip M., and Herman Feshbach: "Methods of Theoretical Physics," Parts and McGraw-Hill Book Company, Inc., New York, 1953. 30. Moulton, F. R.: "Differential Equations," pp. 179 -231, The Macmillan Company, New York. 1930. Sons, Inc., New York, 1933. 31. Phillips, H. B.: "Vector Analysis," John Wiley 32. Scarborough, J. B.: "Numerical Mathematical Analysis," 2d ed., Johns Hopkins &

2,

1

&

Press, Baltimore, 33. Sneddon. N.:

I.

34.

1950.

"Fourier Transforms,"

York, 1951. Szego, G.: "Orthogonal 1939.

35. Titchmarsh,

E.

C:

versity Press, New

C:

Polynomials,"

"Introduction

York,

McGraw-Hill

Book Company, Inc., New

American Mathematical Society,

New York,

to the Theory of Fourier Integrals," Oxford Uni

1937.

9,

"Theory of Functions," Oxford University Press, New York, 1939. "Treatise on the Theory of Bessel Functions," Cambridge University Press, New York, 1944. 3S. Webster, A. G.: "Partial Differential Equations of Mathematical Physics," StechertHafner, Inc., New York, 1933. v:<. Weinstock, R.: "Calculus of Variations," McGraw-Hill Book Company, Inc., New York, 1952. to 40. Whittaker, E. T., and G. N. Watson: "A Course in Modern Analysis," chaps. Cambridge University Press, New York. 1940. 38. Titchmarsh, E. •i". Watson, G. N.:

1


Press,

3-3

PRINCIPLES OF HIGH-SPEED COMPUTING MACHINERY BY Ward Conrad Sangren and Alston S. Householder

It is apparent that many of the practical problems of science and engineering are The last decade has seen the viewpoint toward computa essentially computational. tions change radically because of the advent of high-speed machines. This should be expected in any field where an improvement of high order, say 10* or more, has The improvment here is in the length of time required to carry suddenly taken place. out certain mathematical operations.


1

DIGITAL COMPUTING MACHINERY

The abacus and the desk calculator are well-known examples of digital computers. By an arrangement of beads on The abacus is a computer in a limited sense only. wires it can represent numbers in the decimal base, and additions and subtractions are performed by manipulating the beads in accordance with certain simple rules. Rules for multiplication and division are somewhat more complicated. Desk calcu Thus, for lators perform one or more of the arithmetic operations automatically. addition, one enters the commands by pressing keys, and then one presses the appro priate operation button at which the machine adds and exhibits the result on a dial. It is explained elsewhere that transcendental functions, square roots, etc., can be represented approximately by means of interpolation polynominals, truncated power Given such a representation, one can series, continued fractions, and the like. evaluate the function approximately by carrying out a finite number of arithmetic Techniques have also been developed in some cases for obtaining these operations. approximate representations; e.g., when the function is known to satisfy a given differential equation, these techniques themselves require the performance of arith metic operations, finite in number (numerical solution of differential and other equa One uses a digital computer to "solve" a mathematical problem in the sense tions). The result, is that the arithmetic operations can be carried out on the computer. generally an approximation only to the theoretically correct result, but in most physical problems an approximation, if sufficiently close, is quite as satisfactory as the theoretically correct result. Operands must be entered manually into the desk calculator, and results copied manually on paper. Punchcd-card machines, however, can read the operands from cards by sensing the location of holes punched in the cards. They can also punch the results out on cards. These results can, in turn, be printed by a printer that reads the cards itself, or if the results are intermediate and not final, they can be returned to the calculating machine as operands for subsequent steps in the calculation. In some punched-card machines the cards contain not only the operands but also the coded designation of the operation to be performed. Thus the machine reads the card to determine both operands and operations. Operating speeds are of the order of magnitude of seconds or possibly tenths of a second. The electronic digital computer is almost completely automatic. Division, the slowest operation, requires on some machines only a fraction of a millisecond and on Typically, such a machine con some under construction only a few microseconds. sists of a memory or More, an arithmetic unit, a control, and various input and output units. In terms of function, the elementary constituents, besides the connecting wires, are of two classes (realized electronically, however, in many different ways, 3-142

high-speed computing machinery

Sec. 3-3]

3-143

within the same machine). Elements of the first class are bistable, or often called toggles or flipflops. Those of the second class are gates and The on-off elements may be thought of as emitting or not emitt ing a signal. inverters. A gate that has t wo or more An inverter converts a signal to no signal or vice versa. An "and" gate transmits only when all inputs carry inputs is one of two types. Clock signals super signals; an "or" gate transmits when any input carries a signal. imposed upon the entire system maintain the synchrony of operation of the system. A set may Typically the store consists of one or a few thousand sets of flipflops. include 16 (as in the Whirlwind at MIT) but more often contains about 40 flipflops. In the Princeton-type machines, such as the Oracle at Oak Ridge, it is exactly 40. An ordered set of 40 bistable elements is capable of representing 240 distinct "messages" Any one of these messages is (since each of the 40 elements can be either off or on). known as a word, and a word can, in turn, stand for a number or for an instruction or combination of instructions for the machine. Thus, when a computation is started, the dates and the instructions, all properly coded and properly organized, are read into the store. Each instruction must specify an operation and the location(s) in the store of the necessary operand (s). Some machines go sequentially through the store until the sequence is interrupted by a special jump (or transfer) command. Most com or "three-address," although "two-address" and monly, machines are "one-address" In a three-address machine, the typical command "four-address" machines exist. is made up of the code symbols for the operation, the locat ions of two operands, and the location to which the result is to be sent in the store. In the one-address machine only one location is associated with an operation, and addition, for example, must be broken into three steps: bringing the augend from the store to the appropriate register in the arithmetic unit, adding the addend to and returning the result from the The last step arithmetic unit to the specified location in the store. unnecessary when the sum not needed for itself but only as an operand in the operation next to be performed. most effective) Many computations (and these are the ones for which the machine consist mainly of cycles of operations, each performed many times, though, of course, on different Thus, in applying an iterative scheme (such as Newton's operands. method) for solving an equation, one obtains consecutive terms of a sequence Xo, Xi, by a formula X% Xi+i =
is

it,

is

is

...

is A is

is

a

is

a

a

is

a

a

a

A

a

A

is

a

if

is

if

it

a

is

if

it

is

As soon as Xi has been evaluated from one evaluates A',+t from Xi by the same formula, and the process continues until some convergence criterion satisfied. particular location in the memory will be selected for storing each A", until replaced by Xi+\. The machine will be given the sequence of instructions needed for Next, there will be instructions for evaluating the

on-off,


3-144

MATHEMATICS

[SEC. 3

incorporated in any program that requires them. The larger the library of such precoded routines, the less the labor of coding a new problem, in general, and the more feasible it is to use the machine for any given problem. For all machines, addition, subtraction, and multiplication are built-in operations. For some, it is necessary to program division, although this is built in for most. But. also certain logical operations are necessary, the jump operations being the most Machines differ considerably in their logical operations. They also important. Most computers built for scientific purposes differ in t heir representation of numbers. use the base 2, but others, like the Univac (built by Remington Rand), are decimal. Binary representation permits a very simple arithmetic unit but requires initial con version of the data into binary form and a final conversion of the results into decimal form. Both of these conversions can be coded for the machine. A decimal machine will use four flipflops to represent each decimal digit. Of the 16 possible states of the set of four binary elements, only 10 are utilized, the other 6 This, together with the more complicated arithmetic, necessitates being meaningless. more circuitry. A further difference among machines is that some (like the Univac and the Prince ton-type machines) are fixed-point and some (like the NORC, Navy, Dahlgren, In a floating-point machine, one set of flipflops is used Virginia) are floating-point. This represents an integral to designate the location of the decimal or binary point. exponent of the base, whether 2 or 10. In fixed-point machines, it is presumed that all numbers handled by the machine lie in a limited range generally but not always (e.g., the SEAC, National Bureau of This requires that all quantities Standards, Washington, D.C.) between — 1 and +1. be scaled so as to fall in range, and the programer must take care not to call for an operation that would yield a result outside this range. Some machines have an "overflow" device of some sort that detects the occurrence of such an illegitimate It is always possible to program floating-point operations (at considerable operation. expense in terms of speed of operation). For some problems the number of digits representable by a Bingle word does not Thus the Oracle's 40-digit word provides 1 binary digit provide sufficient precision. This is equivalent to about 11 decimal digits. for the sign and 39 for the magnitude. For rare problems the rounding errors build up to such an extent that higher precision For such it is possible to program "double-precision" operations by is required. As with floating means of which two words can be used to represent each number. point operations, however, the speed is greatly reduced. As already mentioned, the advantage of the binary representation lies in the fact that the arithmetic is simpler and that all states of the flipflops, instead of only 'P-ê To illustrate, consider the following binary multiplication: of them, are meaningful. = 2"1 + 2-3 + 2~« = = 2-' + 2"2 + 2~«

lHa

= 2~' + 2~' + 2~« +

2"' + 2-"

-

0.1011

0J101

»Ji6

1011 1011 1011 0.10001111

As a further illustration simple fractions:

2-'

= 14;,-2'5r,

of the system itself, consider the binary representation of sonic

- 0.10000 -

. .

.

3-1 = 2 2(1 2-')-' = 2-'(l . . . 0.01010101 4-' = 0.010000 . . . 5-1 = 2-2(1 + 25)1 = 2-H1 0.001 1001 10011 . . . 6-' = 2-'0 • 3"1

-

- 2-»)"1 - 2-'(l

= 0.001010101

7-1 = 2-»(l

- 0.001001001

. .

+ 2 •' + 2-" +

• •

•)

- 2-« + 2"« - 2"«

+

• • •)

+ 2-» + 2-« +

+

• •

.

. . .

2~«

•)

HIGH-SPEED COMPUTING

Sec. 3-3]

MACHINERY

3-145

further difference among binary machines lies in the treatment of negative In some, the number is represented as a magnitude and a subtraction. sign. In others (e.g., those of the Princeton type) a complement representative is used, as in the common practice of representing a negative logarithm by a positive Thus, in these machines all numbers are of mantissa and negative characteristic. A still

numbers and

(his form:

-co +

a =

a.0.2-1 + «20.2-! -f-

• • •

+ a.,0.2-"

with each a either 0 or +1. Thus a = -1, one takes ao — 1, on = a» 0 U0O

...

- 1 4-

is

+1 cannot be represented, although to represent ■* • • ■ =0. What appears in the machine as is 2"1 +2-» = %, but. what appears in the machine as 1.1100 . . . 2~l + 2~5 = —Y\. The representation of — % would be obtained by

writing

-i/x

_1 +

m

(1

_

- -l

»4)

+

_l

=

yx

+ 0.0100

....

. . .

1

4,

write

-o

taking

-

= 1.0011 +0.0001 a =

write

- 0.1100

-a

-%i

= 1.0100

= 1.1100

= 0.0011 = 0.0100

+

Again,

—a

%

1

by

o To obtain

0.0001

is

reduced to complementation and addition, since to perform Thus, subtraction subtraction a — one can, instead, obtain first —6 by the above rule and thus obtain + — 6). For paper work the binary system has the disadvantage of requiring many char ters. In computing laboratories that use binary machine customary to group

7

8

1000 1001 1011

F

A

in

0000 0001 0010 0011 0100 0101 0110 0111

.1 9

convenient.

triads (octal system) or in tetrads (base 10). When group to six additional symbols are required, and for this, the letters Thus, one has the following representation: 6 5 4 3 2 1 0

ire

digits either

into tetrads,

F E DC B

ing

the binary

is

a

it

(

a

b

the

1010 1100 1101

1110

mi

is

is

is

if

a

is

t

if

a computing job required less han two The recommendation was once made that major machine. This hand, one should not consider trying to use being good advice now. Even a two-day hand job might better be put, the program can be made up mostly of existing subroutines and hence machine Nevertheless, close at hand. •semblcd quickly and provided, also, the machine 'y-mze of the job to be measured in certainly the first consideration, and the size a»W that increase with the remoteness of the most available machine. The units ■rein-years by ^ttt far from ' s -

a

a

if

a

if

however, with increased adequacy of the subroutine library for that machine with increased repetition in the computation or the number of runs of the same short nonrepetitive problem, Naturally, wMem (as in parameter studies). the repetitions are long repetitive problem, even •"'teafced many times, becomes '•dtly spaced. It hard to define precise criterion, especially since this changes is


is, 0

A simple rule for obtaining the appears in the machine as 1.0100 by and representation of —a is the following: Replace every at by 1 — a, (that and take Thus, suppose a were equal to 0), and thus add 2"» to the result. and this

3-146

MATHEMATICS

[Sec.

3

constantly because of the rapid increase in the availability of machines and in the number of tested programs. The on-off elements, as physically realized in the store, are always different from those in the rest of the machine. Some machines (e.g., the Princeton type) use a rectangular or square array of spots on the face of a cathode-ray tube. All possible charge distributions within any spot are separated into two classes. Any distribution of the one class is taken to represent "on," and any other "off." Magnetized spots on a rotating drum are used in others; acoustic signals in a mercury tank or nickel bar in others. The drum is the favored device in the less expensive and slower machines. The mercury tank is used in the Univacs, and nickel in the Elliott machines in England. These are intermediate in speed. Newer machines are utilizing magnetized cores of iron oxides somewhat faster than the cathode-ray tube and transistors, which to date are slower. Most of the high-speed digital computers are truly general-purpose computers. In theory these machines are capable of finding approximate solutions to any mathe matical problem requiring a quantitative answer. In addition, they arc capable of solving most business, military, and governmental problems that involve either logical decisions or data processing. The mathematics carried out by digital computers is largely that associated with the field of numerical analysis. Since the customary engineering or science major is not trained in numerical analysis, it is usually necessary to have a group of people especially trained who work only with the digital computer.


2

ANALOGUE

COMPUTING MACHINERY

The common distinction between digital and analogue computing machines is that digital machines deal with discrete quantities represented, for example, by counters or toggles while analogue machines deal with continuous quantities, represented, for The slide rule is an example of an elementary example, by voltages or shaft rotat ions. analogue machine. The cost of an analogue machine is closely related to the accuracy desired from the However, all analogue machine. Accuracy of 1 per cent is relatively cheap and easy. machines, primarily because they depend upon continuous measurements, have been limited in accuracy to less than 0.01 per cent. Analogue computers can be characterized according to whether the primary com Because of the ease of ponents are mechanical, electromechanical, or electronic. operation and maintenance and the relative cheapness, electronic, or d-c, analogue computers are by far the most numerous. The following remarks therefore concern d-c analogue computers. A d-c analogue computer uses d-c voltages, which vary with time, to represent vari The independent variable of a system of ordinary differential equations is ables. generally associated with the independent variable, physical time, of the computer. The solution of a system of differential equations by a d-c computer consists, therefore, of preparing the computer so that the circuit equations or equations of motions are similar to or analogous to the given system. Analogue computers that are able to solve ordinary differential equations are commonly called differential analyzers. Before indicating in more detail how an electronic differential analyzer is used to solve ordinary differential equations, it is desirable to consider the representation of mathe matical quantities in and the computing elements needed for such a computer. As already indicated, a set of mathematical variables x, y, . . . , z is represented in a computer by corresponding d-c voltages or machine variables X, Y, . . . , Z. The machine variables are generally taken as simply proportional, the proportionality factors being scale factors to the mathematical variables. A problem requires that certain relations among the machine variables be set up in the computer. For example, if y is a function of x, then the voltage Y must be set up as a corresponding function of the voltage X. Physical time r serves as an input to certain computing devices, such as circuits containing capacitors, in order to produce voltages that The independent variable I of the mathematical problem is depend upon time.

3-147

HIGH-SPEED COMPUTING MACHINERY

Sec. 3-3]

taken proportional to the physical time t; that is, t = at, where a is a scale

generally factor.

electronic analogue computers, complicated relations among variables and therefore mathematical variables are obtained by combining a limited number of simple relations or operations. It can be shown that a wide class of problem can be solved by using computing elements that carry out the following In most commercial

(he machine

operations:

Multiply a machine variable by

a positive or negative constant. two or more machine variables. 3. Integrate or differentiate a machine variable. 4. Obtain the product of two or more machine variables. 5. Obtain a desired function of a machine variable. A potentiometer or voltage divider is used to multiply a voltage by a positive For convenience, potentiometer dials are usually adjusted by constant c;0 hand and then locked in position. Direct-current amplifiers with feedback, called operational amplifiers, are used for In an operational four purposes: amplifying, inverting, summing, and integrating. Cus amplifier it is possible to multiply a d-c voltage by constants larger than 1. tomarily the factor of amplification is chosen to be 1, 4, or 10, whichever is desired. Since the sign of the output voltage is opposite that of the input voltage, the amplifier acts as an inverter. The use of potentiometers and operational amplifiers thus allows a machine variable to be multiplied by any real constant. It is possible to apply several different input voltages to an operational amplifier and with the appropriate circuit arrangement obtain the sum of the voltages as the output. It is also possible to connect the same input voltage to two or more input terminals. The second operation of adding machine variables is thus accounted for by using operational amplifiers. Arrangements can usually be made so that integrators are used in place of differen tiators on a differential analyzer. The use of a capacitor with a d-c amplifier gives an electronic circuit that integrates quite accurately. The integration of d-c voltages is generally carried out with respect to the physical t ime t. The product of two machine variables and the generation of nonlinear functions of a machine variable are more difficult to obtain than the above linear operations. The most common types of multiplication and function generation devices are servomechanisms, that devices consisting of an amplifier or relay and an electric motor that positions potentiometer shaft in the manner desired. Although such devices are convenient and accurate for slow variations in voltages, they do not handle variables having large frequency range. Purely electronic devices for multiplication and function generation rely on the nonlinear characteristics of circuit elements, and the device economical, has to date lacked precision. consequently, The computing elements of an electronic analogue computer consists in general of operational amplifiers, potentiometers, servomechanisms, and the associated circuitry. addition, a computer contains control circuits, recording devices, and power 1.

2. Add

is,

a

is

it

a

if

In

supplies. .

Z,

,

Y,

is

a

is

is

Z

,

is , z, I

is

The first step in setting up a problem on an analogue computer to establish the transformation relating the mathematical variables x, y, . . and the machine variables X, . . . t. Ordinarily the transformation given by X = aiX, = any, . . . = aiZ, t = at, where the as are scale factors. Two conflicting First, the machine variables f-onsiderations determine the choice of scale factors. are generally restricted to the range + 100 volts in order to guarantee linearity of the linear circuitry. the accuracy; Second, the larger the machine variables, the better that is, stray voltages cause a smaller percentage error. The choice of a scale factor for time also complicated by the fact that short computing times usually minimize the errors associated with the operational amplifiers but may increase the errors asso ciated with the computer servos. The machine equations are derived from the mathematical equations by using the transformation relating the mathematical and machine variables. In order to arrange the computer elements to represent the machine equations, block diagram usually

Y


ScSl.

MATHEMATICS

3-148

[Sec. 3

A block diagram consists of various symbols that represent constructed on paper. computing elements and the connections between the elements. Such block diagrams serve both as visual aids for the computer setups and as records of the setups. The following are representative of symbolisms that, are customarily employed: Xo-

POTENTIOMETER AMPLIFIER (INVERTER)

-©-

-oY

Y=CX,0£C=l

-OY

Y=(X0+4X2)

-oY

Y=-JXdr

SUMMING

INTEGRATING

Fio.

1.

X,o-

AMPLIFIER Xc

Representative analogue-computer symbolisms.

For illustrative purposes consider the differential equation d*Y

=

dr1


where 0 £

C, C, £

1,

F(0)

= 0, and

_ "

dY dr

dY/dr

The following block diagram presents

CZY

= 0.

a solution for this problem:

RECORDER

Fio.

2.

Solution for

a

differential equation.

An analogue computer can be used as a computer, a control instrument, or a simu lator. Analogue computers are usually considered special-purpose machines because, although they can solve other problems, they are used primarily to obtain the solution of systems of ordinary differential equations. As a control instrument the output of an analogue computer can be used to turn switches off or on in accordance with computed results. Consequently, it is possible to stop or start motors at prescribed intervals and, in fact, carry out most control operations with analogue computers. When an analogue computer is operated on a 1-to-l time scale, it may be used as a simulator. Two noteworthy fields for such simulation are aircraft flight and the kinetic behavior of nuclear reactors. When used as a simulator, an analogue com puter can be used as a training device without the expenses and dangers inherent in the simulated situation. It is relatively easy to train an engineer or science major to use an electronic analogue computer.

BIBLIOGRAPHY Hartree, Douglas R.: "Calculating Instruments and Machines," University of Illinois Press, Urbana, III., 1949. 2. Wilkes, M. V., D. J. Wheeler, and S. Gill: "Preparation of Programs for an Electronic Digital Computer," Addison-Wcsley Publishing Company, Heading, Mass., 1951. 1.

SECTION

4

NUCLEAR PHYSICS BY C. HOYT, Ph.D., Manager, Nuclear and General Physics Division, Lockheed Aircraft Corporation, formerly Alternate Division Leader, Los Alamos Scientific Laboratory.

FRANK


CONTENTS 1 Elements of Atomic Theory 2 Elements of Nuclear Theory 3 Basic Particles and Their Interaction

with Matter

PAGE 4—2

4-33 4-54

4-1

4 Nuclear Reactions 5 Neutron Physics References

PAGE 4 0.">

4-90 4-102

4

PHYSICS

NUCLEAR BY

F. C. Hoyt 1

ELEMENTS OF ATOMIC THEORY

questioned. Although apparently insurmountable computational difficulties prevent an exact calculation of physical properties from first principles in all but the simplest cases, it has nevertheless been possible to find suitable approximations that lead to a reasonably quantitative understanding of a very wide variety of phenomena. In the field of nuclear physics there is as yet no convincing reason to doubt the validity of the principles of quantum mechanics, although the range of their successful quanti tative application is more limited than in atomic theory and they can in many instances serve only as a guide to the interpretation of experimental information. Electromagnetic Radiation and Relativity is,

1.1

1,

k

E

1.

a

7

E

is

is

The earliest indications of the inadequacy of classical (that Newtonian) mechanics at the atomic level came to light in the study of the properties of radiation. It therefore appropriate to begin with a brief account of some of the important features of the theory of radiation. 1.11 The Classical Wave Theory of Radiation.f An electromagnetic field and the magnetic field described by two field vectors, the electric field strength strength H. Through the partial differential equations of Maxwell, these field vectors are determined when any charges and currents that are present are specified as functions of position and time. In atomic theory the most important electromagnetic fields are the oscillatory ones resulting from the periodic motion of charges. Such wave fields constitute the classical description of the light (or X or rays) emitted by an atom or a nucleus. At great distance from its source, the field will have approximately the form of Fio. The field vectors and H and the an infinite plane polarized wave, as shown wave number vector in a plane polarized electromagnetic wave. in Fig. or at least may be analyzed into such. In an electromagnetic wave of this simplest type (solution of the Maxwell equations in free space) the electric and magnetic vectors oscillate harmonically in phase in a plane perpendicular to the direcSec Rcf. 25 for a relatively elementary account references are identified at the end of lie section.) t

t


The structure of the atom, as distinguished from that of its central constituent the nucleus, has been established since the first decade of the present century. The physi cal principles governing the behavior of atoms and radiation (quantum mechanics) were gradually developed between 1900 and 1930. The validity of these principles and the adequacy of the model in the field of atomic physics cannot be seriously

4-2

and Rcf. 45 (or an advanced

treatment,

(The,

ELEMENTS OF ATOMIC

1]

tion to the vectors may

(point) source and have the same amplitude, t be written as the real parts! of

E

=

H

A

=

exp

2«

Q

4-3

THEORY

-

The magnitudes of these

rf)

Art.

c

is

X

k

a

H

=

exp

i(k

•

r

E

=

A

k

a

E

r

a

k

is

is

is

is

is

the frequency. c

i

z

distance perpendicular to the plane of the vectors, the wavelength, and — X», The wave progressive, and the speed of propagation It customary also to write the expression for the where the velocity of light. = 2*-/X and the angular frequency u = wave in terms of the wave number In terms of position vector from an arbitrary origin and using wave number vector = 2ir/X and direction normal to the plane of and H, the having the magnitude wave may be alternatively written where

- oil)

(1)

of the wave will coincide with that of the charge oscillations consti its source; the polarization of the wave, that is, the orientation of or H in their plane, will depend on the detailed directional properties of the source. The intensity of plane electromagnetic wave (the energy passing per second and

E

The frequency

a

tuting

centimeter through surface perpendicular to its direction of propagation) given by (c/i*)E* = (c/4r)£T*. Next in simplicity to the infinite plane wave the spherical wave in which the and H, at a sufficient distance from a point source, diminish inversely amplitudes of as the distance from the source and have a variation with angle over the surface of any sphere which depends on the details of the charge motion. Classical electrodynamics provides the mathematical apparatus for the calculation all details of the radiation field from given charge and current distribution. The situation of greatest importance in atomic and nuclear theory that of the radiation from such distribution oscillating harmoni If the frequency cally with the time. v, = c./v, and the wavelength emitted if

X

is

is

a

is

a

of

distribution of charge density and confined to region of dimen sions the field at distances very large rather com compared with depends in plex way on the space derivatives of the a

=

pve*-«

J

A(P)

potential

i

electromagnetic

a

R

«

X, is

R

velocity

v

p

the

dV

(2)

is a

Fio.

2.

Illustrating

calculation potential.

of

an

the geometry for electromagnetic

is

best displayed through the use this turned out to be part icularly significant to expand the expoThe procedure is

y

in

is

k

in is

k

a

s

in

2, r is

As illustrated in Fig. the distance from the source to a point P the field, vector giving the position of volume ele ment dV in the source distribution, and = 2r/X the wave vector of magnitude the direction from the source to the point P. In calculating the intensity of the emitted radiation the symmetry of the source distribution of first importance, and of the so-called multipoU expansion,^ which has the treatment of radiation from a nucleus.

5.

8,

is

-

-

i

is

t {

is

These are the so-called the uauss. The unit for B the electrostatic unit (eau), and that for // Gaussian units." cos a + sin a, so that the real part of the above expression It will be recalled that eia eos 2*[(z/M «f). chap. 13. See Ref. 19. chap. or Ref. (


E

is

U

a

per square

NUCLEAR PHYSICS

[Sec.

4

nential in the integral of Eq. (2) in the form exp (ik • s) — 1 + iks cos 9 — %(ks cos 0)2 +

where 6 is the angle between the form

s and k.

■■•

The integral then becomes a sum of terms of

J>(fcr)<

cos'

6dV

and each term is roughly of the order of magnitude (kit)1 = (2irR/\)', since the inte gration is confined to a region of radius R. The field corresponding to a term with In general index I is called a multipole of order 21 (I = 1, dipole; 1 = 2, quadrupole). R/\ is an exceedingly small quantity, so that only the first nonvanishing term is of importance, the total intensity of the radiation falling off roughly as (R/X)u. The case which occurs most frequently in radiation from an atom is that in which the first nonvanishing term is that with I = 1, i.e., dipole radiation. The field then turns out to depend only on the quantity

D

= /ps cos B

dV

(3)

which is called the dipole moment. The total intensity of the radiation Q (energy emitted per second) is then given by


Q =

tLD>

(4)

In radiation from nuclei the dipole moment is ordinarily zero, and the properties of the radiation will depend on quadrupole or higher moments, which are integrals over the distribution of a more complicated kind than in Eq. (3) above. The radiation from a moving point charge may be treated as the limit of that from a continuous distribution or by direct WAVELENGTH X IN CM methods and is given byt

I

—

£5

\

5 I

k r>

V T"

2000* K

2 e'

3c»

(5)

where Q is the total energy emitted per second and w is the acceleration of the Thus a uniformly moving point charge. point charge does not emit energy — an acceleration is required. 1.12 Thermal Radiation. This term V T- 1500' K is applied to the electromagnetic radiation emitted by matter at a given tempera ture. The simplest case is that in which the radiation is in equilibrium (established T- 1000* K by a balancing of emission and absorp tion) with matter. Such an equilibrium i 2 3 will be set up inside an enclosure with FREQUENCY J/xlO-" IN SEC-1 completely absorbing (that is, "black") Fig. 3. The energy distribution in the walls maintained at a constant tempera black-body spectrum for three different ture, and the radiation is for this reason temperatures. commonly called black-body radiation. From general thermodynamic reasoning it can be shown that all properties of blackbody radiation, including the spectral distribution of its energy, depend only on the temperature of the matter with which it is in equilibrium. To account theoretically for the experimentally observed distribution of energy in the black-body spectrum (see Fig. 3) is thus a fundamental problem for theoretical physics. The quantum theory had its origin in the complete failure of attempts by Wien, Planck, and Rayleigh (1895-1901) to account theoretically for the observed blackt For

an plementary

\

derivation see Ref. 50, Appendix I.

ART.

ELEMENTS OF ATOMIC

1]

4-5

THEORY

on the basis of general principles of statistical mechanics and electro these general principles lead rather directly to a unique distribu tion lawf which is clearly incorrecj. The procedure in deriving such a distribution law is as follows: It is legitimate to assume that the radiation is emitted by an assembly of linear harmonic oscillators, each having a definite frequency v, which are in tempera ture equilibrium with each other and with the radiation in the enclosure. The average energy E, of the oscillators of frequency v can be shown to determine the energy density u,t per unit frequency range of radiation of that frequency by the relation body spectrum

dynamics.

In fact,

«,

=

— c3

E,

(6)

standard procedure of statistical mechanics (average over all energies, by the Boltzmann factor c~E/iT) gives Ev = kT in accord with the principle of equipartition of energy. The temperature dependence of the energy density in the radiation field is then given by inserting E, in Eq. (6), resulting in the distribution law Use of the

weighted

u,(T)

=

~kT c'

(7)

known as the Rayleigh-Jeans law and is wholly at variance with experiment low frequencies. This complete breakdown of classical theory led Planck to propose an ad hoc assumption; namely, the oscillator energy is "quantized" to the discrete values E = hv, E = 2 hv, . . . with h a new universal constant . § A statis tical average over these energy values gives the so-called Planck distribution which

is

_

8jt;'2

hv

,Q.

u, dv

=

aT*

a

Jo"

for the black -body spectrum. With proper choice of the constant h the observed spectrum is accurately reproduced. It remained for Bohr to extend Planck's assump tion of discrete energy values to t he general concept of stationary states (see Art. 1 .22). An integration of Eq. (8) over all frequency gives = 7.652

X

10"'5 crg/(ems)(deg4)

for the total energy per cubic centimeter in the black-body mann law). 1.13

Dualism.

Wave-particle

spectrum (Stefan-Boltz-

The hypothesis of Planck

and its generalization

is

E

by Bohr imply that the emission of monochromatic light of frequency v occurs indis = hv. crete amounts of energy There now exists convincing experimental evidence that, as first suggested by Einstein in 1905, this energy localized in space in the

is

it

is

;

it

a

is

is

if it

is

is

'

is

v

by

||

a

electromagnetic field, that is, that radiation has corpuscular structure. The earliest evidence for the existence of light corpuscles, or photons, came from the photoelectric effect. The energies of the photoclectrons ejected from an atom light of frequency found that the maximum may be directly measured, and hv and nergy This imme independent of the intensity of the incident light. diately understandable assumed that the light consists of a stream of particles of energy hv and that the ejection of an electron due to an individual collision lx-tween photon and an atom quite inconsistent with the picture of the light as

h

I

/

/,

is

is

t

is

t

For a more detailed account of the classical and quantum theories of the black-body radiation see Itef. 50. chap. 4. The energy referred to that of the electromagnetic field, which (Ei + //s>/8t ergs^cm'. It in erics per square centimeter per second, of the rudintiun from a small opening related to the intensity = lacu. in the wall by the equation The The dimensions of are those of the dynamical quantity called action, that is. energy X time. approximate value is 6.62 X 10~17erg sec. See Ref . 50. chap. 3. f


except at

4-6

NUCLEAR PHYSICS

a wave motion with a continuous flow lation of energy up to an amount hv required when the light is extremely electron is what is observed. Closely related to the photoelectric

[Sec.

4

of energy. Any hypothesis requiring accumu is ruled out by the long time that would be weak. An, immediate ejection of the photo-


effect is the more recently (1923) discovered In the scattering of X rays by atoms there is observed a small Cornplon effect.] increase in the wavelength of the scattered light which is dependent on the angle 6 between the directions of the incident and scattered beams, the shift in wavelength at a given angle being given by AX = (h/mc)(l — cos 8), where m is the mass of the Again the process is directly understandable if electron and c the velocity of light. it is pictured as due to a collision between a photon and a free electron. The assump tion of conservation of energy and momentum, J with the photon having a momentum hv/c as well as an energy hv, leads directly to the above expression for the wavelength shift in its dependence on angle. The decrease in frequency of the scattered radiation is attributable to the loss of energy of the photon in the encounter. Arguments similar to those for the photoelectric effect show that the existence of the Compton effect is irreconcilable with any reasonable picture of radiation as a continuous wave

phenomenon. The picture of a monochromatic beam of light as a stream of particles (photons) moving with velocity c and having energy hv and momentum hv/c is thus one which is supported by direct experimental evidence. On the other hand the picture of light as a form of wave motion is equally well established experimentally, primarily through the phenomena of interference, which requires for its direct interpretation a picture of In fact, vacillation between a picture light as some form of continuous wave motion. of light as waves and a picture of it as particles can be traced through nearly three hundred years of the development of physical thought. It is now believed, however, that a decision is impossible and that a complete description requires the compli mentary use of both pictures and, indeed, that such a paradoxial dualism is a necessary element in any theory of matter as well as of radiation (see Art. 1.31). 1.14 The Special Theory of Relativity. § Electrodynamics and the Lorentz Trans The origins of the theory of relativity are to be found in the difficulties formation. and paradoxes encountered in the problem of determining the frame of reference in which the Maxwell equations are valid, for these equations, which govern all optical and electromagnetic phenomena, do not have the same form in two coordinate systems that are moving relative to each other with a uniform velocity. Consequently, a coordinate system in which they have their usual form (and with respect to which light will therefore be propagated with velocity c = 3 X 1010 cm/sec) will be singled out as a unique frame of reference in the sense that the velocity of any other frame of reference moving relative to it could be determined by optical experiments in the moving system. The simplest of such experiments would be the measurement of the velocity of light in the second system, which should differ from the velocity c in the first system. It was originally thought that such an absolute frame of reference existed in the form of a physical medium called the ether and that the motion of the earth through the ether could he determined by optical and electrical measurements. A long series of experiments, i| however, failed completely to detect any of the consequences to be On the other hand, expected from the orbital motion of the earth around the sun. the conclusion that the earth itself is an absolute frame of reference (or in terms of the ether hypothesis that the earth is at rest in the ether) is in direct conflict with astro nomical observations on the apparent motions of the fixed stars (stellar aberration). t

See Ref. 50, p. 380. for a more complete account. theory a radiation field of energy density u lias a momentum of u/c per cubic X In electromagnetic centimeter, which suggests hv/c for a photon of energy hv. This assignment of momentum was shown by Einstein to be a consequence of the theory of relativity. \ Elementary accounts are to be found in Ref. 14 and Ref. 50, chap. 2. For a more advanced treat ment see Ref. 3. In essence it consisted II The most notable of these was the Michelson-Morely experiment (1880). to, the direction of of a measurement of the difference in the velocity of light along, and perpendicular motion of tlio earth. See Ref. 50. p. 50, for a description of the experiment.

Art.

ELEMENTS OF ATOMIC

1]

THEORY

4-7

in 1905. The failure to detect the motion by purely terrestrial experiments he proposed to interpret as indicating that or ether, in which this motion has a sig there exists no absolute frame of reference, nificance and to require that the laws of electrodynamics, like those of mechanics, have the same form in all reference systems, differing only in their relative velocity. Failure to detect the motion of a reference system (like the earth) by measurements within it can then be attributed to the fact that the equations governing the phenomena are unaffected by the motion of the reference system — a situation already well known and understood in ordinary mechanics. Since the Maxwell equations are not independent of the motion of the reference system, it would seem that the above requirement could be satisfied only by an alteration in these equations. However, Einstein was able to show that the form of these equations may be retained and that the "invariance" can be achieved by a modification of the previously accepted way in which the quantities entering into them are related to the reference systems involved. In order to understand something of the nature of the radical solution proposed by Einstein and embodied in the special theory of relativity, it is necessary to look a little more closely at the formal aspects of transformations between moving reference sys tems. A set of equations like the Maxwell equations is said to be invariant if it is unaltered in form when the quantities entering into it are transformed from one refer ence system (usually a set of rectangular coordinate axes) to another moving with uni form velocity with respect to the first. Among the quantities to be transformed are the coordinates themselves, say x', y', z' in the "moving" system and x, y, z in the original system. It seems kincmatically obvious that the coordinate transformation should be simply x' — x — Vt, y' = y, z' — z when the relative motion is along the common x axis with velocity V . The Newtonian equations of motion, for example, are invariant in form under this transformation, which is known as the Galilean The dilemma was resolved by Einstein

—

y(x

- Vt)

-

V

x'

V'

transformation. The required equivalence of moving reference systems as regards the laws of electro dynamics implies that the Maxwell equations should be relativist ically invariant. This can be achieved by a formal modification of the Galilean transformation, provided the field vectors E and H are at the same time suitably transformed. The modified coordinate transformation has the form (9)

where

known as the Lorenlz transformation. It reduces to the Galilean transformation V/c — * 0 and so is equivalent to it unless the systems involved have relative veloci ties comparable to that of light. It differs from the Galilean transformation, however, in that time, as well as the space coordinates, has different values in the two reference systems. While the mathematical properties of the Lorentz transformation were already known, it was Einstein who first gave it physical reality by showing that its apparently paradoxical consequences are philosophically acceptable. In particular, the assignment of different times in the two reference systems for a single event is physically understandable as soon as time is recognized as a measured quantity and not an abstraction. Among the kinematical consequences of the Lorentz transformation is the shortening of distances in the direction of motion known as the Lorenlz contraction: A body at rest in a given reference system and having an extension I in the x direction will have and is

as

I y/ 1 — {V/c)1 as measured in a reference system moving with velocity V direction. Relatirislic Mechanics. With the establishment of the relalivistic invariance of the Maxwell equations (that invariance for the Lorentz transformation) all problems concerning the effect of motion on optical and elect rodynamic phenomena are satis However, new questions are raised in mechanics, since the New factorily solved. tonian equations of motion are invariant only for the Galilean transformation and an extension in the x

is,


of the earth

4-8

NUCLEAR PHYSICS

[Sec.

4

not for the Lorentz transformation. This situation led Einstein to propose a modifi cation of the Newtonian equations which gives them the required property of rela tivistic invariance. Such relativistic equations of motion must, of course, be equiva lent to the ordinary ones except at very high velocities, and this requirement, together with that of invariance, is practically sufficient to single out the following reformu lation of mechanics, the validity of which has been amply demonstrated in experi mentation with high-speed electrons and other elementary particles. In place of the Newtonian equation of motion (rf/d/)(m0v) = F for a particle of mass too and velocity v acted on by a force F, the relativistic formulation requires

ii_^!!l_=F -p

(10)

dl\

«

where fl = v/c. For velocities small compared with that of light (0 1) the deviation from the Newtonian equation is of order p; for speeds comparable to that of light the result may be interpreted as an increase in mass of the particle according to the equation

— —mo

.

.

(11)

since with this definition of a variable mass Eq. (10) has precisely the Newtonian form (d/dt)(mv) = F. This variable mass increases indefinitely as v—► c, and the velocity of light is accordingly a limit that cannot be exceeded. As in ordinary mechanics the equation of motion leads to an expression for the rate F • v at which work is done on the particle and hence to a definition of energy. In place of (d/dt){\imtjv2) = F • v Jhe new mechanics gives d

*

a/1

- p -Fv

(12)

The appropriate modification of the definition of the energy

«

%p

1)

(f(

+

+

(13) use of the expansion . . .

»

w„c2

+

p gives

E

with neglect of terms higher than

=

l

(1

For velocities small compared with that of light

- p)-H

of the particle is thus

Vi -p +

E _

E

(13a)

J^mon1

p is

is, a is

+

is

is

the Newtonian expression for the kinetic energy with the significant exception which that corresponding to the ''rest mass" m0 there a "rest energy" mac*. Another important consequence of the relativistic formulation of the equations of motion the existence of an "invariant" combination of momentum and energy. that quantity which has the same numerical value in all Lorentz frames of refer From Eq. (10) the appropriate definition of momentum ence. clearly

is

is

-f-

-f-

' 1

is

it

In the mathemntica! formulation of relativistic mechanics (and electrodynamics) convenient to regard z, y. z, and ict as the components of a four-dimensional vector (4-vector) specifying an event " in space and time. The Lorentz transformation can then be made to appear as a rotation of the fourdimensional coordinate system that leaves unchanged the quantity r- + y* z* — cH* which can regarded as the square of the length of a 4-vector. Quantities like the above which have the same value regardless of the motion of the reference system are known as rclali>nstic invariant*. The invariance of x1 + I/1 z3 — c1*1 equivalent to the statement that the velocity of litiht will appear the same to all observers regardless of their motion relative to its source. In the mathematical formalism there appear other 4-vectorn, one of which has the momentum components px, pv, p.-, and iE/c as its components. — The invariant p1 the square of the length of this 4-vector. E*/c*

t


m =

Art.

ELEMENTS OF ATOMIC

1]

4-9

THEORY

P =

(14)

In a system in shown that the quantity pl — E*/c* is an invariant. particle is at rest (» = 0) it has the value — wio'c1, and hence this is also its in any system in which the particle has an arbitrary velocity v. Thus,

and it can be which the value

p1 is valid

E*

—

=

TOo'c'

generally and gives E* = moV +

cV

(15)

relativistic relation between energy and momentum for a free particle. The kinetic energy T is customarily defined as the excess of E over the rest energy; i.e., T = E — moc*, and from Eq. (13a) T « ^nw* for 0
E* =

+ vioc')* =

m0V<

(l \

+

—Y mac*/

« m.V

fl \

+

—\

wiocV

substituting this into Eq. (15) gives, to terms in (T/mac1)1,

-

2£-0

T

(16)

1,

is

c

E

- cp

is

it in

is a

is

it

is

the nonrelativistic relation between kinetic energy and momentum. When <3C and nonrelativvalid, readily shown that approximation (T
this

of

(17)

Art.

*!

The definition of the energy the form

E

Eq. (13) may be written

in relativity

theory as given in

in

Mas* and Energy.

e

=

c

=

E

it

is

E

= hv (see The assumption of Einstein that the photon energy quantized to has momentum 113) thus implies, according to the theory of relativity, that

= rocJ

(18)

—

a

a

is

is

it is

it

E

m

for the case

is

While this relation the total mass mo/(l actually derived only 0')*4. that the sum of rest energy phis kinetic energy, was proposed by to the status of a universal relation between energy and mass, Einstein to elevate the total internal (kinetic plus potential) energy of any valid also when the energy This assumption has been validated system of particles such as an atom or nucleus. experimentally by measuring the change in mass of an atomic system (by means of the mass spectrograph) when its energy known amount in a nuclear changed by The Einstein relation, sometimes interpreted as an equivalence of mass reaction. widespread practical application in nuclear physics, in the and energy, has now fnse that changes in energy may be measured by changes in mass and conversely 'cf. Art. 2.15). where

is


and

(T

4-10

NUCLEAR PHYSICS 1.2

[Sec.

4

The Bohr Atomf

By 1905, the importance of a quantum of action in the theory of radiation had become apparent through the work of Planck on the black-body spectrum and that of It was not until nearly ten years later, however, Einstein on the photoelectric effect. that the significance of this same quantum of action in atomic physics was brought to light by Bohr, who, in 1913, formulated a quantitatively correct theory of the line The genesis and development of the Bohr theory will be spectrum of hydrogen. briefly outlined in this section, which is largely historical in content. 1.21 The Rutherford Model of the Atom. The discovery by Thompson in 1897 that particles of negative electricity can be isolated from atoms in a discharge tube led shortly thereafter to the first definite picture of the constitution of atoms and its relation to the sequence of chemical elements. The charge e of these particles, called electrons, was estimated at a value close to that now accepted, and their mass was shown to be roughly 2,000 times as small as that of the lightest element, hydrogen. {


J. J.

It was evident that the emission, absorption, and scattering of light were to be attributed primarily to electrons as the lightest and hence most mobile constituents of atoms. In particular, the scattering of X rays by an atom provided a measure of the number of electrons it contained and gave a clear indication that their number increases roughly proportionally to the atomic weight. It was originally assumed by Thompson that the positive electricity which atoms must contain in addition to electrons was distributed uniformly throughout a sphere of atomic dimensions, i.e., with a radius of the order of 10~' cm. In such a model an electron would have a simple harmonic motion with a single definite frequency which would be identical with a frequency that the atom could emit or absorb. The com plexity of the line spectra of atoms made it highly improbable that so simple a model could be adequate, and it was, in fact, shown very clearly by Rutherford in 191 1 that the positive electricity in atoms, and hence also all but a minute fraction of the atomic mass, must be concentrated in a central nucleus of radius about a ten-thousandth that of the atoms. The evidence for such a structure of the atom came from the scattering of a particles (helium atoms) of very high energy by thin foils. Such scattering must be attributed to individual encounters between atoms, and deflections through large angles were found to be much more frequent that could be accounted for in any other way than by the assumption that an inverse square law of repulsion operates down to distances of the order of 10-IJ cm and hence that very large forces are involved in close encounter. (In the Thompson atom with its continuous charge distribution* the force would diminish toward the center.) Furthermore, the angular distribution of the deflected a particles was in good agreement with that to be expected from an inverse square (coulomb) law of force and a quantitative measure could be obtained of the magnitude of the positive charge of a nucleus. § Since the atom as a whole is neutral, this positive charge is a measure of the number of electrons, and for a variety of ele ments it could be shown that the number Z of electrons is about equal to the atomic number, that is, to the ordinal number giving the position of an element in an arrange ment according to increasing chemical atomic weight. On the basis of such evidence and similar indications from the scattering of X rays the hypothesis that the number of electrons in an atom is exactly equal to its atomic number was proposed by Ruther ford in 1911 and immediately accepted. In the Rutherford atomic model, then, the lightest element, hydrogen, has one elec tron; helium, the next in order of atomic weight has two; and in uranium, the heaviest atom then known, there are Z = 92 electrons. These electrons are acted on by an t For brief elementary accounts of the Bohr theory see Ref. 3fi, chap. 7, and Ref. 50, chap. 5. more detail on spectroscopic applications see Ref. 30. t The currently accepted values are Charge of the electron: Mass of the electron: Mass of the hydrogen atom:

| For

some detail

on this "Rutherford

-

e -4.802 X lO-" nu 9.10G6 X 10 11 mH — 1.6725 X 10""

scattering" see Ref. 50, p. 144.

P811

For

Art.

1]

ELEMENTS OF ATOMIC

THEORY

4-11

electrostatic force of attraction to a nucleus of charge Ze of radius less and are to be thought of as in motion in orbits about the central nucleus. 1.22 The Bohr Theory of the Hydrogen Atom. While the Rutherford model clearly gave the correct picture of the structure of atoms, fundamental difficulties were at once encountered in any attempt to account for the properties of atoms on the basis of the dynamics of the "planetary" electrons moving under electrostatic forces. Even such an elementary fact as that atoms have a well-defined size appears anomalous, since "orbits" of any dimensions are dynamically possible. Furthermore, no orbital motions of the electrons would be stable, since an accelerated charge emits energy in the form of electromagnetic radiation (cf. Art. 1.11), and the motions would be damped until there was complete collapse of electrons into the nucleus. An elementary problem in atomic theory, in which difficulties of the above kind appear sharply, is that of accounting for the observed frequencies of the spectral lines emitted by the hydrogen atom. If the damping effect of radiation, which is quite small, is neglected, the single electron in such an atom would move in an ellipse about the nucleus. The frequencies emitted would be integral multiples of the fundamental frequency of this periodic motion. Not only is the observed spectrum quite different in structure from such a set of equally spaced lines, but the very existence of definite frequencies is unexplained, since nothing in the model determines what this funda mental frequency should be. It was this dilemma which led Bohr, in 1913, to propose a basic revision of the laws of atomic dynamics and the radiation process at the atomic level while at the same time retaining as much as possible of the laws of "classical"


inverse square than 10" l* cm

mechanics. The theory of the hydrogen spectrum proposed by Bohr was based on two postu lates. The first of these was the assumption that atoms can exist only in stationary " conditions from among the states of *lates which are selected by "extramechanical motion permitted by classical mechanics. Some such hypothesis is clearly called for by the very existence of definite properties of atoms. The second postulate was that the emission of radiation can occur only by the instantaneous transition of an atom from one stationary state to anot her and that the frequency v of the radiation emitted is given bv the relation

U

-

E'

- E"

(19)

where E' and E" are the energies of the two stationary states involved and h is the quantum of action first introduced by Planck in his theory of the black-body spectrum. This Bohr frequency condition is clearly a kind of reversal of Einstein's equation for the photoelectric effect, with the frequency of the light determined by the energyavailable instead of the energy of the photoelectron being determined by the frequency of the light. The form of Eq. (19) was also suggested by an empirically known rule in spectroscopy, the combination principle, according to which the frequencies of spec tral lines can be represented as the differences of a relatively small number of term values — these term values are now to be identified as the energies in the stationary states divided by h. These drastic assumptions would probably not have been so readily accepted as they were had it not been that Bohr was able to show that a quantitative theory of the hydrogen spectrum could be based on them in the following way: It was known that the frequencies of the lines of atomic hydrogen are given by the simple formula

,

-

lie

(-L Vrii2

- —)

n-V

(20)

where cf is the velocity of light, R is an empirical constant known as the Rydberg If Eq. (20) is to be regarded as a conse constant, and ni and n2 are positive integers. quence of the frequency condition of Eq. (19), the energies of the stationary* states of the hydrogen atom must bet

t The occurrence of c in this formula is due to the fact that in spectroscopic practice R is measured in reciprocal wavelengths, so that Rc is a frequency. X The energies of the states are negative because of the convention of taking the potential energy to be zero when the electron is infinitely far from the nucleus, which corresponds to using V — — e*/r for the potential energy. The total energy of an electron bound in an atom must then be less than zero or it would have enough energy to escape to infinity.

4-12

NUCLEAR PHYSICS E,

=

n =

na

1, 2, 3,

[Sec. 4 . .

.

(21)

" These are precisely the energies that are obtained if the "extramechanieal condition fixing a stationary state is taken to be that all orbits are circular and have the angular With this rule of "quantization" of the angular momentum, which momenta nh/2*. was a purely ad hoc assumption having only a vague analogy to Planck's quantization of the energy of an oscillator, it can be readily shown by elementary mechanics that the energies of the stationary states are

E

(22)

kT~n*

which leads to

R

=

?4^-4 h'c

(23)

as an expression for the Rydbcrg

constant in terms of atomic constants m„ e, and h. The agreement with the empirical value of It was limited only by the accuracy with which the atomic constants were known. In addition Bohr was able to show that the radii of the circular orbits were

_

n'h*

(24)

The binding enerjry of a particle from the rest of the atom.

is the amount of work required

to remove

it

is

of

it is

a

a

is,

giving for the smallest orbit with n = 1 the radius 0.53 X 10~8 cm, which was of the correct order of magnitude for atomic dimensions. A radically new picture of the hydrogen atom and the process of emission and absorption of radiation was thus proposed by Bohr. Only certain mechanical motions of the electron, defined by a rule of quantization and labeled by a quantum number, were assumed to have a real existence, every change in state of the atom consisting in a transition (assumed instantaneous) between two such stationary states. Among these states, called also energy levels, there is one of minimum energy (or maximum binding energy t) called the ground stale or normal state. This is the state with quan tum number n = 1, and from it the atom may be raised to an excited state of higher quantum number if energy is communicated to it, as by collision with a particle or by The process of light emission involves a transition from such an absorption of light. excited state to one of lower energy. The frequency condition of Eq. (19) was assumed to hold for absorption as well as for emission; that is, the atom can absorb in a transi tion from an energy level to one of higher energy only radiation which it can emit in the reverse transition. It was soon found that additional details of the hydrogen spectrum could be accounted for on the basis of the Bohr theory. For example, allowing for the very small effect of the motion of the nucleus as well as of the electron gives rise to a slight displacement of the frequencies of the hydrogen lines which is in exact accord with the observations. Again, the fine structure of the hydrogen lines, that the fact that with high resolution they appear as groups of closely spaced lines, was shown by Sommerfeld to be consequence of allowing for the effect on the electron motion of the variation of mass with velocity required by the theory of relativity. 1.23 Further Development of the Bohr Theory. The initial conspicuous success of the Bohr theory of the hydrogen atom led to decade of development which must be accounted one of the most fruitful epochs in the history of physics. The final state of development of the Bohr theory need be only very briefly sketched here, as the interest in mainly historical. The rule originally proposed by Bohr that angular momenta Rules Quantization. be quantized to nh/2-r insufficient to fix the energies of stationary states except in t


a

to an infinite distance

Art.

4-13

THEORY

ELEMENTS OF ATOMIC

1]

Hence, a basic objective of the theory was the formulation of cases. rules of quantization. This problem was never satisfactorily solved except for mechanical systems possessing certain periodic properties, such as those of an electron in a central field differing from an inverse-square (coulomb) field or an elec tronic orbit perturbed by an external electric or magnetic field. In such cases, the with an inde energies were fixed by a series of quantum numbers, each associated very special general

frequency of motion. of quantization of the ground state for two electrons in a helium atom was eventually solved, but the result proved to be in disagreement with the measured ionization potential. A complete theory should lead to a method for Intensities and Selection Rules. calculating at least the relative intensities of spectral lines as well as their fre quencies. This the Bohr theory was never ahle to do satisfactorily, but certain selec tion rules predicting zero intensity for certain lines could be made highly plausible by These took the form of restrictions analogy to the classical theory of light emission. on the occurrence of transitions between certain pairs of stationary states — there were said to be ''allowed" transitions and "forbidden" transitions. Such considerations also led in some cases to correct predictions of the polarization and relative intensities of certain groups of lines. Atoms with Many Electrons and the Periodic System. Very considerable progress was made within the framework of the Bohr theory in the qualitative understanding of the electronic structure of atoms and its relation to spectra and the chemical and physical properties of the elements. In the absence of any method applicable even in principle to a many-body problem it was assumed that each electron in an atom could be treated as moving in an approximately spherically symmetrical (central) field resulting from an average of the forces on it due to all the other electrons and the nucleus. The stationary states of the electron could then be classified by three quan tum numbers. One of these, the principal quantum number, determines the energy to a first approximation, another determines the angular momentum, and a third the The corresponding component of the angular momentum vector in a given direction. electron orbits, when the deviation from a coulomb field is small, are ellipses modified by a precession of the major axis in the plane of the motion. The stationary states with the same principal quantum number n] have in general nearly the same energy, given approximately by the formula pendent

n2

(25)

n1

is

1.

it

in

is,

it

L

is,

is,

which is the appropriate modification of Eq. (22) when the nuclear charge is Z*e instead of e as in hydrogen. % The deviation of the "effective atomic number" Z* from the true Z gives a measure, which can be roughly estimated, of the screening It was in this way possible to divide the electronic effect of all the other electrons. orbits into "shells"; those of lowest energy for n = 1 wore said to form the K shell, those with n — 2 the L shell, and so on. The number of possible different orbits in a given shell, that the number of possible sets of quantum numbers, could be shown to he «*, that shell, nine in the M shell, and one orbit in the K shell, four in the so on. On such a picture could be expected that all electrons in an atom would be found in the K shell with n = orbits of the lowest possible energy, that This quite at variance with the character of the spectra which atoms are observed to emit and with the fact that the size of atoms increases rather than decreases with increasing Z. In order to obtain any proper coordination between electronic structure and was found necessary to introduce a new assumption, the Pauli physical properties

Z

is

t

n

determined the angular momen In the original Bohr rule for quantization of the hydrogen atom That the energy levels in hydrogen could be fixed by the tum. The notation waa later changed. angular momentum was due to the restriction, later abandoned, that the orbits were circles rather than ellipses. In general, the angular momentum does not. determine the energy even in a coulomb field. Equation (25) accurate with Z* = for atoms which have been stripped by ionization of all but one electron. It thus gives the energy levels of singly ionized helium or of doubly ionized lithium, etc. X


The problem

4-14

NUCLEAR PHYSICS

[Sec. 4

a

it

a

it

a

a

z»m exp

(2*-i»„mf)

(26)

each corresponding to a transition from a state n to a state m, with x„m an amplitude giving the intensity of the corresponding spectral line of frequency »„„. The theory would thus from the beginning deal only with such observable quantities, and he was able to lay down the rules of a true "quantum mechanics" involving only such arrays.

a

it

Very shortly thereafter was shown by Born and Jordan that matrix algebra was the proper mathematical language in which to formulate a theory of the kind proposed by Heisenberg. In parallel with the development of a matrix mechanics on the above lines an appar ently radically different approach was being pursued by Schrodinger, who showed that all results of the Bohr theory for the hydrogen atom could be obtained by treating the electron, not as a point, but as a vibrating continuum. This wave mechanics and the matrix mechanics were, however, very soon shown by Schrodinger to be mathemati cally equivalent. There followed a period of intensive development of the new theo ries at the hands of Bohr, Born, Heisenberg, Dirac, Schrodinger, and others which, in few years, led to the establishment of quantum mechanics in essentially its present form. t


a

it

is

A

a

is

t

is,

a

is,

exclusion principle, restricting the number of electrons which can have a given set of " a given shell. It might quantum numbers and hence the number which can "occupy have been expected that the number in a given shell would be n', since this is the Instead, it was quite clear that the number must be 2n*, number of different orbits. two giving 2 electrons in the K shell, 8 in the L shell, 18 in the M shell, etc., that This double occupancy of the orbits was soon found to be electrons in each orbit. associated with the existence of an internal rotation of the electron about its axis, the so-called electron spin which had been postulated by Goudsmit and Uhlenbeck to account for certain features of the fine structure of spectral lines and the magnetic properties of atoms. It was found necessary to assign to the electron spin quantized one-half of the quantum of angular momentum angular momentum of J-«j(A/2ir), that for orbital motion. The electron has then a fourth degree of freedom in addition to the three specifying its position in space and a corresponding quantum number which specifics whether the spin axis (direction of its angular momentum) parallel or antiThere are then four quantum numbers required to given direction. parallel to specify completely a stationary state, three for the orbit and one (having only two The exclusion prin possible values) for the spin orientation with respect to the orbit. ciple may then be stated as the rule that no two electrons in an atom can be in the same stationary state. table showing the classification and occupancy of the electron orbits according to the exclusion principle to be found in Sec. 10-1. 1.24 The Breakdown of the Bohr Theory and the Origins of Quantum Mechanics. By 1925 had become evident that though highly successful in leading to a qualitative correlation between spectral lines and atomic structure, the Bohr picture was never theless inadequate as a basis for true theory of atomic dynamics. Its most obvious defects were the lack of any satisfactory method of calculating intensities of spectral lines, the incorrect result which gave for the energy levels of the two-electron system of the helium atom, and its inability to deal with the details of aperiodic processes such as the collision of an electron with an atom. In addition there was growing conviction that the idea of applying "classical" mechanics to the electrons and then introducing quantum conditions only at a later stage was in principle unsatisfactory and that true atomic dynamics would have to be based on a still more radical depar ture from classical concepts. The first decisive step toward new theory was taken by Heisenberg in 1925. He kind of philosophical defect of the theory, namely that emphasized particularly attempted to calculate the observable frequencies and intensities of spectral lines in terms of the unobservable orbital motions of the electrons. Starting from the idea that a fundamental theory should contain only observable quantities, he proposed to represent an atom symbolically by an array of quantities of the form

For further dctuil on electron spin and magnetism

see Art. 1.33ff.

ART.

1]

ELEMENTS 1.3

OF ATOMIC THEORY

4-15

Quantum Mechanics!

The earliest formulations of quantum mechanics (matrix mechanics) were expressed abstract mathematical language. Following the experimental discovery of wave properties for the electron, however, it has become possible to state the fundamentals It is this approach that will be used here, with of the theory in more physical terms. no attempt to follow the historical order of development. 1.31 Foundations of the Theory. That radiation has the properties of both waves and particles has long been recognized (cf. Art. 1.13). It is now believed that such a wave-particle dualism is fundamental for matter as well, and the basic equations of quantum mechanics (also known as wave mechanics) are suggested by the necessity of ascribing certain wave properties to all elementary particles. Their ultimate justi fication, however, is to be found in the agreement with experiment of deductions from in

them.


Wave Properties of Particles. That electrons have wavelike properties is most clearly shown by the experimental fact that a beam of electrons is diffracted by the regular arrangement of atoms in a crystal in the same manner as is a beam of X rays. This was first demonstrated by Davisson and Gcrmer in 1927, verifying earlier theoretical

speculations by de Broglic (1927), who proposed to apply to matter the Einstein equa tion for radiation, p = hv/c (cf. Art. 1.13). While Einstein used this equation to determine the momentum p associated with a wave of frequency v and wavelength X — cjv, it was "inverted" by de Broglie to associate a wavelength X with a particle having a momentum p = tnv. As a formula for this so-called de Broglie wavelength the Einstein equation may be written X

--

(27)

P

For an electron accelerated by a potential difference tum mv is (nonrelativistically) given by the relation

V

expressed in volts, the momen = eK/300, where e is the

},£mv*

charge of the electron in electrostatic units (the factor 300 converts from Gaussian units to volts for V). Its de Broglic wavelength is then X = h/y/meV /150, J and the above formula is verified experimentally in the sense that electrons so accelerated and allowed to fall on a crystal will produce a diffraction pattern geometrically identical with that produced by X rays having a wavelength equal to the de Broglie wavelength of the electrons. A diffraction experiment such as the above cannot bo interpreted as demonstrating that the picture of the electron as a point charge is to be replaced by a picture of it as a wave, for while it is true that the formation of an interference pattern can be under stood only by assuming that in some sense the electron extends over the whole crystal, it is equally true that in many other respects, for instance in its ability to impart all its energy at once to an atom, the electron behaves like a particle at least as small as an atom. The viewpoint adopted in quantum mechanics is that a complete mental picture is impossible; at best, certain aspects of the behavior of the electron can be visualized by appeal to one picture or the other. The mathematical formalism does, however, provide a means of calculating what can be measured. There is every reason (including direct experimental evidence for protons and neu trons) to believe that all material particles behave like waves (called de Broglie waves) in the same sense as the electron, with wavelength given by X = h/p. Any theory which takes account of the wave properties of The Wave Function. particles must associate with them a continuous function of position and time, called a wave function, which can form interference patterns as a light wave does and has the qualitative feature that it is large where many particles are found and small where there are few. In the case of photons the electromagnetic field (the vectors E and H) t Elementary accounts of quantum mechanics are to be found in Ref. 50, chaps. 6 and 7, and in Ref. For more complete coverage see Refs. 0. 44, and 54. X For example, for V ■ 100 volts, A » 1.22 X 10_l cm, which is of the order of magnitude of the distance between atoms in a crystal. 55.

NUCLEAR PHYSICS

4-16

[Sec.

4

plays the role of such a wave function; for matter, quantum mechanics introduces a single scalar function }/,(x,y,z,t), the Schrodingerf wave function, associated with any material particle. The wave function to be associated with a particle of constant momentum p and energy E moving in the direction of a z axis should, in order to account for the observed interference patterns, be an infinite progressive plane wave, analogous to a light wave, with wavelength X = h/p. The frequency of this de Broglie wave does not enter into the experiment but is reasonably assumed to be given by the Planck relation r = E/h. The form of this wave function assumed in quantum mechanics is = A exp i(kz

where

k =

— X

=

\n

-

ut)

u = %rr =

(28)

f

(29)

n

The type of interference observed is accounted for if such a wave is to be diffracted according to Huyghens principle by the atoms of a crystal and the intensity in the interference pattern is taken proportional to |^|2 = W*< where The use of the complex exponential, rather than ^* is the conjugate imaginary of its real part, in Eq. (28) greatly simplifies the mathematical formalism of quantum mechanics. For example, is independent of the time while the real part of ^ would not be. Fluctuations in time with frequency a in the intensity would be an undesirable feature of theory intended to represent a stationary state. The de Broglie wave of Eq. (28) is, of course, a quite special wave function, referring to a uniformly moving free particle. Quantum mechanics assumes that the most For general "state" of a particle can also be "represented" by a wave function. example, the state of an electron in an atom is represented by a wave function which must effectively vanish outside the atom. The relation between wave function and particle is taken to be as follows: It is assumed that certain attributes of the particle, such as its position, its angular momentum, or its energy (called dynamical variables), are quantities that can be measured. From the wave function representing some defi nite state of the particle the results of such measurements can be predicted, but only statistically. That is, it is only the relative frequency with which, in a very large number of experiments, various possible results of the measurement will be obtained that can be deduced from the wave function. An alternative way of saying this is that the wave function determines the probability of any value of a dynamical variable. The way in which the wave function determines position has already been qualita tively described. A more precise statement is the following: ^| (x,y,z,l)\* dV gives the probability that a measurement of the position of the particle at time I will locate it in the volume element dV at the point x,y,z. In a de Broglie wave representing a par ticle of fixed momentum |^|* is constant, corresponding to the fact that the position of the particle in the beam may have any value. The way in which the wave function specifies the statistics of other quantities is stated briefly in Art. 1.32. The Uncertainly Principle. From the fact that quantum mechanics regards quanti ties like position and momentum (or velocity) as measurable to\any desired accuracy, and

ft = h/2-r.


assumed

it might appear that it describes something having actually all ithe properties usually assigned to a particle. That this is not the case is seen most pearly from the fact that both position and momentum cannot be thought of as simultaneously being How this comes about can be seen from a vcipr simple example. sharply defined. Suppose that the position of a particle in the x direction is determined to be in the range Ax by the procedure of passing a beam of particles through « slit of width Ax The de Broglie wave of the particle will be diffractild by the slit, and (see Fig. 4). consequently there will be an uncertainty in the momentum of the jjarticle after it has passed through the slit due to the fact that it can be said only tlij**- its direction of motion lies somewhere in the diffraction cone. If the slit is sufficietl^ly narrowed, the diffraction cone opens out and eventually the direction of motion can become eomt First introduced by Schrodinger (192ft). who showed that the enerjry levels of/Btoms puted from a partial differential equation involving this function. If F + iG. with F and 0 real functions. F F' + GT
J

f -

- -

-

could he

com

Art.

ELEMENTS

1]

OF ATOMIC THEORY

4-17

Ax Ap,

■="Ax

ap,

»

h

is

of

is,

it

X

is

X

is,

It it

pletely undetermined. Such considerations show how, in quantum mechanics, the applicability of the particle picture is limited by the simultaneous applicability of the wave picture. For instance, it is not possible to think of an electron in an atom as moving in an orbit, since this would imply a definite direction of motion at a point. The above example of diffraction by a slit illustrates how, in quantum mechan the partial determination ics, of one variable disturbs a previous knowledge of another variable. in fact, a There semiquantitative relation between the "uncertainties" in the two variables and the momentum p, in the x direction which may be deduced as follows: The Fio. 4. Diffraction of a de Broglie wave by a particle has a de Broglie wavelength slit, leading to an uncertainty Ap, in the = h/p„ where p, the momentum in component of the momentum p, transverse the direction of the beam. Like a light to the slit. will be diffracted into a cone wave, then, Ap, = ap. (for a small angle of The uncertainty in p., angle a ■" \/Ax. deflection), and the product of the two uncertainties

a fundamental relation in quantum mechanics which may be either by deduction from the formal principles of the theory or by critical examination of the inherent limitations of physical measuring instruments, The result from either approach expressed in the uncertainly relations first formulated Heisenberg in 1927: It coordi possible to specify simultaneously the x, y, and nates of particle and the corresponding momentum components p,, p», p, only within minimum ranges or "uncertainties" Ax, Ay, Az and Ap,, Ap„ Ap, that mutually restrict each other by the relations a

t

is

a

z

is

by

Ax Ap,

Ay Ap,

Az Ap,

(30)

is

in

is

a

a

limiting case, exact specification of momentum component means that the cor responding position coordinate entirely undetermined, as in the example of a de Broglie wave cited above. The uncertainty relations in the above form hold also for other pairs of variables, like angle and angular momentum, that bear a certain formal relation to each other the equations of dynamics and are said to be canonically conjugate. The com monest example that of energy and time, for which the uncertainty relation reads As

AE

At

(31)

is

a

is

Much of the essence of quantum mechanics expressed in these uncertainty rela tions, and they provide handy tool for the rough estimation of quantum mechanical effects as will be seen in examples to follow. The Schrodinger Wave Equation. The form assumed in Eq. (28) for the wave func tion of a free particle reasonably well justified by appeal to the experimental facts

electron diffraction. For the more general situation of a particle acted on by forces, no such direct guidance from experiment. However, the analogy to optics suggests that the wave function in such a case can be calculated by assuming to be solution of a partial differential equation, just as light waves for photons are solutions the Maxwell equations. The general form of the wave equation of quantum mechanics (the Schrodinger equation) can be made plausible, though, of course, not uniquely established, by gen eralizing a wave equation satisfied by the de Broglie wave of a free particle. This wave function may be written directly in terms of the momentum and energy of the particle instead of its wave number and frequency as in Eq. (28). At the same time, of

a

it

there

is

of

t


This equation expresses

generalized,

For a complete discussion

see Ref. 29.

4-18

NUCLEAR

PHYSICS

[SEC. 4

it may be generalized to represent a particle moving in an arbitrary direction in space with a momentum p. It then becomes an infinite plane wave 4, =

A

- Et)

ex-p -' (p • r

n

(32)

Differentiating this expression twice with analogous to the light wave of Eq. (1). respect to the space coordinates and once with respect to the time gives

©'W

-(*)'*-*

PV

=

(33)

=

±p*

+

2

m

V

E

V

is

it

ifr

where Vs = di/dx1 + dt/dy1 + dl/dz* and p is the magnitude of the momentum vector The relation E = (l/2m)p* between the energy and momentum of a free particle p. then gives immediately a relation between the derivatives of that constitutes a wave At the same time equation for a free particle. strongly suggested that the gen can be obtained by eralization to a particle in a force field with potential energy using (34)

In fact, multiplying Eq.

(34) through

by

2m

V

t

dt

i/

is

u(r)e*

Once this assumption

- — Vhi

(36) is a

function of position i<(r) which into Eq. (35) gives

only.f

Vu = Eu

(37)

2

m

"

+

an "amplitude made, substitution

=* Ku>and

is

l'lanck relation

=

simple harmonic dependence on the time with a frequency given by the

E

have

a

that

is,

+

It

is

which the now accepted form of the Schrodinger wave equation for a single particle in the nonrelativistic approximation. The wave equation (35) finds immediate application as a tool for the determination of the properties of the energy levels (stationary states) of atoms, replacing the "quan plausible for such stationary states tum conditions" of the Bohr orbital theory. that their wave functions are of the form

u(i) of the wave function. determined by the potential energy The atomic system described by the equation — —e'/r the equation that for the electron in a hydrogen V(r). For example, with V

is

is

ns the differential equation governing the spatial distribution

atom.

E

,

»

a

is a

r

E

a

is

a

For any value assigned to E in Eq. (37) there will be corresponding solution u(E,r). However, only those solutions for which the wave function w(r) essentially confined to closed region can represent states of a particle bound in an atom and under certain conditions (cf. Art. 1.33) this requirement can be satisfied only by those solutions corresponding to a discrete set of values of E. In fact, for intermediate values of the solution u(E,r) will in general increase indefinitely as — 00 and such solutions are clearly unacceptable for the representation of a particle bound in an atom. For the hydrogen atom there discrete set of (negative) values of deter mined by such requirement on the corresponding solutions, and these values corSuch states are also "stationary" in the sense that the probability distribution of position ^* is independent of the time, since ^* — wu*. It should be noted that the complex form of the wave function essential for this interpretation. is

t


+

a

in place of the expression for free particle. and substituting from Eq. (33) give

Art.

1]

ELEMENTS OF ATOMIC

THEORY

4-19

In other cases there exactly with those given by the Bohr orbital theory. physically significant solutions for a continuous range of values of E. These correspond to aperiodic motions such as that of an electron deflected by, but not bound to, a nucleus, a situation that could not be adequately treated by the Bohr theory. In mathematical language Eq. (37), together with any requirements on its solu tions (called boundary conditions), is called an eigenvalue problem. The values of E for which solutions satisfying boundary conditions exist are known as the eigenvalues of E, and the corresponding solutions as eigenfunctions. For the extension to more than one particle of the above method of "quantization" see Art. 1.34. The emission and absorption of light can he handled in quantum mechanics on the basis of the general principles of the theory extended to the radiation field and its interaction with matter. f Additional assumptions, such as the frequency condition of Eq. (19) in the Bohr theory, are unnecessary, and it is possible to develop a complete which provides a definite prescription for the calculation of quantum electrodynamics the intensities as well as the frequencies of spectral lines. The most useful single result of quantum electrodynamics is that for any pair of stationary states there is an electric charge and current distribution from which the properties of the radiation associated with transitions between these two states can he calculated by the methods of classical electrodynamics. For a pair of stationary states with wave functions J/i and d>2, which will have the form given in Eq. (36), the associated charge density is the real part of respond


can be

(38)

the charge of the electron. The frequency of oscillation of this charge dis tribution is the energy difference between the two states divided by h in accord with the Bohr frequency condition. In the simplest cases the intensity of emission and absorption will depend only on the dipole moment of the charge distribution as given by Eq. (3). It is frequently necessary, however, to make use of the multipolc expan sion sketched in Art. 1.11. While radiation processes may be calculated in quantum mechanics from the picture of an oscillating charge distribution, the idea of emission and absorption as connected with transitions between two stationary states remains essentially as described for the Bohr theory in Art. 1.22. It is customary to describe light emission where e is

absorption in terms of transition probabilities. In order to proceed beyond the cal Further Development of the Theory. culation of energy levels and transition probabilities for a single particle it is necessary to introduce new assumptions into the theory. These are few in number and lead to an abstract mathematical scheme for the calculation of the results of observations on atomic systems. No attempt will be made here to describe this mathematical formalism in any precise way, but some of its principal features will be briefly noted. It has been seen that Eq. (37) determines the energy values which it is possible for a system to have and at the same time the corresponding wave functions. In the complete formalism of quantum mechanics there is a similar equation for any dynami cal variable, that is, for any function of the coordinates and momenta, which deter mines its possible values and the corresponding wave functions. The procedure for setting up the general equation (for a single particle) is as follows: To any dynamical variable \f(x,y,z; p,,py,Pt) is assigned an operator M formed from it by replacing the coordinates and momenta by operators according to the rule J and

1.32

y-* v (39)

tSe* Ref. 53, chap. 14. I For the coordinates the operation implied is simply multiplication by the corresponding lad for any function Jix.y.z) simply multiplication by this function.

coordinate

4-20

NUCLEAR PHYSICS

The equation which gives the generalization of Eq. as

(37) for the energy is then

Mf

Nty =

[SEC. 4 written (40)

where M is an operator and M a number which corresponds to E in Eq. (37). With appropriate boundary conditions Eq. (40), which generally, although not always, will be a differential equation, defines an eigenvalue problem like that for the energy and the eigenvalues of M are the values which it is possible for M to have. It can be verified that the above rule gives Eq. (37) when the dynamical variable is taken to be the energy expressed in terms of the coordinates and momenta, an expression which is known as the Hamiltonian H of the system. t For a single particle in a force field with potential energy V it is

B

"

and Eq. (37) can be written

JT" 2 in

CP*»

(with

^

+

Pv%

H

is the operator formed from

(41)

replacing u)

Ity where

+ P.!) + V(x,y,z)

H

=

Ef

(42)

by the rule (39).

Two examples will serve to illustrate how the above procedure is applied: For the z component of the angular momentum, Lt = xpv — ypz, the operator


formed by the rule (38) is

•■-*(•£-'=) It

can be shown that the eigenvalues of the equation L,^- = Lî are Lt = mh, where ±1, ±2, . . . , corresponding to the quantization of angular momentum in the Bohr theory. The operator corresponding to the x component of the linear momentum is The eigenvalues of the equation (n/i)(d/dx).

m = 0,

Vd

=

t

--

I dx

- Pd

consist of all values of pz between — » and + », since for any value p, there is a solution (eigenfunction) ^ = Aei/hpxi. As illustrated in these two examples, the "spectrum" of eigenvalues may be either discrete or continuous. There are cases, including the hydrogen atom, in which the spectrum consists of a discrete and a continuous part. An eigenfunction of Eq. (40) for a particular eigenvalue, say M = M', is taken in the formalism of quantum mechanics to represent a state of the system such that a The theory also measurement of M on the system will always lead to the value M'. provides a rule for calculation of the result of a measurement of M for a state which is not an eigenfunction of M. In such a case various values of M may be obtained in individual measurements, and quantum mechanics is capable of calculating only the statistics of such observations, that is, a probability distribution for M. The procedure for such calculations will not be described here| other than to state that it follows essentially from the rule that the average value or expectation value M of Af for any arbitrary state is given by the integral (the integration is over all space)

f

M when

ip is so

=

JV'M^dK

(44)

"normalized" that

fo**dV<=l See Ref. 24 for the precise definition of the Hamiltonian those concepts of classical mechanics that are of significance 5.42 of Sec. 3-2. X See Rcf. 54, chap. 3.

t

(45) and a relatively elementary for quantum mechanism.

treatment of See also Art

Art.

1]

ELEMENTS OF ATOMIC

4-21

THEORY

1.33 A basic problem in quantum Stationary States of a Single Particle. mechanics is the determination of the energies and wave functions for the stationary states of an atomic system. The problem is denned mathemat ically by the eigenvalue = E<(i, but only for certain cases of a single particle in a force field (or equation for two particles with a central force between them) is it possible to obtain analytical However, these one-body problems that can be treated simply form the solutions. starting point for any method of attacking the many-body problem. Many situations in atomic and nuclear physics Single Particle in a Central Field. can be approximated by the problem of a particle acted on by a force which depends In such a central field the potential only on the distance r of the particle from a point. energy is a function V(r) of r only. The quantum mechanical treatment of such problems is based on solutions of the wave equation written in polar coordinates r, 6, where 6 is the polar angle and the azimuthal angle, the direction of the polar axis in space being necessarily arbitrary. The wave functions of the stationary states can all be obtained from those solutions = R(r)Y(e,4>). of Eq. (37) which have the form u(r,$,4>) When Vs is transformed from rectangular to spherical polar coordinates, it turns out that it contains a term which is just the operator L2 corresponding by the rule, Eq. (39), to the total angular As a consequence, Eq. (37) then momentum of the particle about the force center. into two equations, one for R(r) and one for Y(B,), that for Y being the separates eigenvalue equation L«K = TJY (46) for the total angular momentum. The eigenvalues L% of this equation give the possihle values of the angular momentum, and it can be shown that these eigenvalues are

L'

= 1(1 + l)yis

l

- 0,

1, 2,

.

. .

(47)

In a central field the angular momentum is thus quantized f to the values y/l(l + 1) h. For historical reasons the states with the "quantum numbers" I = 0, 1, 2, 3, . . . are known, respectively, as s, p, d, f, . . . states. The corresponding eigenfunctions Yi(8,) are spherical harmonics which, for any value of I, can be written as sums of the eigenfunctions of a component of the angular momentum, say the component L, in the direction of the polar axis. The eigenvalues of the operator L« which cor respond to a given value of / are mh, where to = 0 ± 1, . . . , ±1. Each of these may be thought of as giving an orientation in space of the vector representing the total angular momentum. The "space quantization" obtained in this way differs from that of the Bohr theory in that the state of lowest angular momentum, / = 0, is one in which this quantity has the value zero rather than ft. The energy levels in a central field are given by the radial part of the wave equation, that is, by the (ordinary) differential equation for It(r) which will be different for each value of / and has the form

=

V(r) +

+

V*

-(*

The energy E is thus determined by an eigenvalue equation which is identical in form with that for motion in one dimension except that R is replaced by rR and the actual potential energy V by an effective potential

V*'

(49)

It is customary to aay that the angular momentum •tit*m«nt that L' has the value f(Z + !)»«.

L

has the value Ih, replacing

a

r

5

The term added to the actual potential which depends on the angular momentum. V predominates for small as illustrated in Fig. and corresponds to repul se force, reflecting the fact that to decrease the radius of a particle, keeping its

energy

t


,

the more accurate

NUCLEAR PHYSICS

4-22

[Sec. 4

angular momentum constant, requires an increase in its velocity and hence that work be done on it. The energy levels of a particle in a central field can be computed as the eigenvalues There will be a set for each value of /, as illustrated in Fig. 5, and of E in Eq. (48). within each set the levels can be ordered according to a principal quantum number n = 1, 2, 3, . . . which is such that n — I — 1 is the number of nodes (exclusive of the origin) of the corresponding wave function. The pattern of the energy levels in a central field which does not differ too much from a coulomb field with V ~ 1/r is shown qualitatively in Fig. 5. Characteristic features are that the energy depends in first approximation only on the principal quantum number n and that for a given / negative energy levels can be formed only for n > I. A basic problem which can be treated approximately by the method sketched above The central field acting on it is then taken is that of a single electron in an atom. to be the average repulsion of all the other electrons modified by the field of the central

V*


UJ

X-

1-0

i r

0

n-2

n

n-3

-1

V* Fia. 5. Energy levels for a particle in an approximately coulomb field. The horizontal There is an infinite number of lines give the negative energy values for n » 1, 2, 3 them converging to E *= 0, of which only a few are shown. For E > 0 there is a con tinuum not indicated on the diagram. nucleus. This gives the fundamental classification of the electron wave functions in an atom by the quantum numbers n and I which corresponds to that of the Bohr It is still incomplete, however, in that the spin of the electron has so far theory. been neglected (see below). In general, analytic solutions cannot be found for the radial part of the wave func tions in a central field. However, for the simplest case of a single electron in the field of a bare nucleus of charge Ze, Eq. (48) does turn out to be soluble in terms of In this special known functions, the so-called confluent hypergeometric functions. case of a pure coulomb field the eigenvalues of the energy are independent of I and depend only on n, being given by the expression

E'-Z^K

»-l,2,...

which is identical with that of the Bohr orbital theory. In a small external electric or magnetic External Fields and the Bohr Magneton. field, the energy of an atom will depend on its orientation with respect to the direct ion of the field. In the case of a single atomic electron in a magnetic field the additiona

ART.

ELEMENTS OF ATOMIC THEORY

1]

4-23

the product of the magnitude of the magnetic field H, the magnitude of the momcntf V of the electron and the cosine of the angle between them, or vector notation. According to electromagnetic theory the magnetic moment of a particle of mass tn0 and charge e moving in a central field is determined by its angular momentum L, the relation being energy is

magnetic H ■v in

V

The "interaction

L

=

(50)

energy" of such a particle in a magnetic field is thus

AE

=

=

~U

(51)

As a consequence of this it is to be expected that the different orientations of the angular momentum L given by the values of L, will correspond to states of different This is born out by the quantum mechanical calculation, energy in a magnetic field. and the added energies are those given by Eq. (51) for the values L, = mh, with « 0, ±1, , ±1, namely,

-

...


AE

=

lie — mil 2moC

(52)

The quantum number m is accordingly known as the magnetic quantum number. The spectroscopic corollary of this is a splitting of spectral lines into a number of closely spaced components, a phenomenon known as the Zeeman effect. According to Eq. (52) an atomic electron behaves in a magnetic field as though it had the magnetic moment corresponding to the classical relation, Eq. (50), for an angular The magnetic moment, momentum component of L, = h in the direction of the field.

mb

=

^ 2c

-* m

(53)

with e/m the charge to mass ratio for an electron, is thus a natural unit of magnetic moment in quantum theory. It is known as the Bohr magneton. In an external electric field also each state of an atomic electron of a given angular momentum is split into a number of states differing in energy. The phenomenon is known as the Stark effect. It can happen that for a given energy level (i.e., for a given eigenvalue Degeneracy. of the energy) the wave function is not completely determined but can be written as 3 sum of independent wave functions. Such an energy level is said to be degenerate and to have a degree of degeneracy equal to the number of independent wave func tions required to represent any wave function "belonging" to it. In the central field example discussed above, an energy level Eni is degenerate and the wave functions belonging to it were chosen as the eigenfunctions of the z com ponent of angular momentum. They are 21 + 1 in number, corresponding to m — 0, ±1, . . . , +/, and the level is 21 + 1-fold degenerate. This particular choice of wave functions for a given n and / is to a certain extent arbitrary, but it can be shown that the degree of degeneracy is an intrinsic property, depending on the symmetry of the energy of the system in its dependence on momenta and coordinates. In an external field the energy of a single particle in a central field will depend on three quantum numbers, as n, /, and m in a magnetic field. Eacli will have a wave that is completely determined (except for multiplication by a constant). function It k said that the degeneracy is removed by the external field. t That an atom has a magnetic moment y means that it behaves in a magnetic field like an infiniThe direction of y is that of the axis 'ramally short magnet with pole strength times length equal to The magnetic moments of atoms and nuclei can be directly measured from their defleeftht magnet. magnetic fields. The simplest experiment of this type is the Stern-Gerlach aoas in inhomogeneous For a description of the experiment see Kef. 50. p. 298. *3$cnmet]t, 6rst performed in 1921.

4-24

NUCLEAR PHYSICS

[Sec. 4

where S is the angular momentum of the electron spin. At the same time, in order to account for the details of the anomalous Zeeman effect it is necessary to assume that S has the single quantized value x/ih, rather than the set of values h, 2h, . . . (cf. Art. 1.23) found for orbital angular momentum and that its component S, in any direction is quantized to one of the two values S, = ± %h. It must suffice here to state briefly the results of incorporating spin into the theory of an electron in a central field without stating the modifications of the wave equat ion which lead to these results.! The total angular momentum is now the sum of the angular momentum L and a spin angular momentum S. This total angular momen = I — J^, = L + S has the two quantized values§ jh, with j > = I + \% and tum which correspond to orientations of S (quantized to l£ft) parallel and antiparallel to L. The magnetic interaction between L and S is small, and the states n,l,j = l -+= I — differ but little in energy, giving rise to a doublet structure for and n, I, certain spectral lines. intimately connected with the question of the The existence of electron spin In fact, as was shown by Dirac in relativistic formulation of quantum mechanics. 1928, electron spin automatically introduced by suitably modifying the Schrodinger wave equation, Eq. (42), to make In setting up such an relativistically invariant. is, of course, necessary to make use of the relativistic relation between equation energy and momentum, namely, as follows from Eq. (16),

j

=

=

+

VniJc

+ c*p* +

V

H

E

it

it

is

is

j

}>%

J

(55)

is

is

is

is

a

it

is

However, a special interpretation of the square root, after the operator replacement of the momentum, required to achieve relativistic invariancc, and in the resulting necessary to regard the wave function as equation, known as the Dirac equation, vector with four components rather than with two as in the Pauli theory of the connected with the ambiguitv spin. The existence of the two additional components the fact that there are found to exist in the sign of the kinetic energy in Eq. (55), as solutions of the Dirac equation for which the kinetic energy negative. It was the

is

is

X

t

As noted at the end of Art. 1.23 the necessity for introducing such an electron spin was appu-en* even within the framework of the Bohr theory. due to Pauli (1927) and involves regardingr the The theory of electron spin in its original form There an analogy to the resolution of a light wave function as a vector having two component*. wave into components polarized at right angles to each other. For the electron wave function the two components correspond to parallel and antiparallel orientations of the spin. Sec footnote on p. 4-21.

J


The above-described results of quantum mechanical Electron Spin and Relativity. theory as applied to an atomic electron are not in accord with the facts. For example, an electron in an s state (/ = 0) is predicted to have zero angular momentum and Actually an atom like the sodium atom in which the hence zero magnetic moment. total angular momentum can be attributed to such an s electron is observed (in a Stern-Gerlach experiment) to have an angular momentum different from zero. Again, the frequencies of the spectral lines in a Zeeman pattern are found not to agree with the predictions of the theory in its above form (anomalous Zeeman effect). Such disagreements are not to be attributed to an inaccuracy of the theory but They are, in fact, removed by to a deficiency of the model to which it is applied. assigning to the electron an "intrinsic" angular momentum in addition to that attributable to its motion as a point charge (called its orbital angular momentum). This intrinsic angular momentum may be thought of as arising from rotation of the electron about an axis (though such a viewpoint is not necessary) and is called the electron spm.t With it must be associated a magnetic moment. Agreement with experiment can be obtained, however, only if the ratio between magnetic moment /i and angular momentum is taken to be twice that for orbital motion as given by Eq. (50), that is,

Art.

4-25

ELEMENTS OP ATOMIC THEORY

1]

of these negative energy states which led Dirac to the prediction of the later verified experimentally, of a positively charged electron or positron. Not only is the spin J^ft for the electron correctly accounted for in the Dirac theory but also the double magnetism, that is, the relation, Eq. (54), bet ween magnetic moment and spin angular momentum. The relativistic fine structure of the hydrogen atom is also given correctly by the In a pure coulomb field the degeneracy of the nonrelativistic energy Dirac theory. — I + levels is removed; the energies are a function of J£ only, and the separations are of the order of magnitude of the universal dimensionless constant occurrence

existence,

j

■-S-

(56)

157

as the fine-structure constant and has proved to play a fundamental many other fields of application of quantum theory. The rules of quantum mechanics as stated above for 1.34 Systems of Particles. a single particle have been formulated in such a way that they can be immediately generalized to a system of N particles. The wave function must then be written as a function ^(ri, r2, . . . , Tn) of the positions n, r2, . . . , r*, of all the particles, that The analogy to a true "wave" in physical the "configuration" of the system. lost, but the interpretation of |^|* as the probability of finding simultaneously space all the particles at the indicated positions retained. The Schrddinger Equation for the Many-body Problem. For system of particles masses mi and momenta p,- the Hamiltonian will have the form is known

which

N

i

H

=

+ V(Tu

u

tn)

(57)

i

I

p
neglected and the potential energy depends only on the positions of the wave equation of the form of Eq. (42) can at once be formed by an It an equation in 3Ar space vari obvious generalization of the rule given in Eq. (39). not susceptible for N > ables (three space coordinates for each particle) and as such to solution by exact analytical procedures. For some account of the available methods approximation see Ref. 50, page 266, and Ref . 54, Chap. 6. It Angular Momentum. impractical to obtain, even by numerical methods, anything but approximate solutions for the wave equation for more than two par ticles. There are, however, certain exact statements about the stationary-state solutions which depend only on the symmetry of the Hamiltonian from which the equation formed and which can be rigorously derived, most elegantly by the use of the methods of group theory, the statement that the Primary among these iota] angular momentum has discrete quantized values when the forces depend only on the distances between the particles of the system. It can, in fact, be shown that any spin which the particles may have neglected, the total orbital angular momen tum has one of the quantized to the values Lh, where the quantum number in values Just as in the case of a single particle the component of +L, corresponding ±1, where m = any direction can take on the values m to 2L + These results are exact and depend only possible space orientations. on the fact that for such forces the total energy of the system unchanged when rigid body. rotated as assumed that each particle has a spin Mh, When spin taken into account and that the above result modified to the following: The total angular momentum and the sum of all the particle 'he sum of the total orbital angular momentum S.v, spins. Si, Si, quantized to the values Jh, where has the integral values even and the half-integral values the total number of particles %, odd. The number of possible space the total number of particles spin particles.

L

L

0,

h,

it

is

is

is

./

is, 0,

L

is

,

...

if

...

if

I,

2,

is

J,

is

is

it

a

is

1

,

...

2

1,

L

0, is

if

is

t

is

is

is

of

2

is

is

A

is

when

t


of

a

is

is

of

is,

role in

For a brief account of Kroup-theory methods in quantum mechanics

see Ref. 40. chap.

15.

4-26

NUCLEAR PHYSICS

[Sec. 4

orientations is in either case 2J + 1, the "magnetic" quantum number m specifying in any direction having integral values when is an integer and the component A proof of these results would be out of is a half integer. half-integral values when However, they appear reasonable when it is noted that place here.

J,

J

J

S = S, +

S2

+

■• ■

+ Sa,

the sum of all the particle spins of which are lined up in the same or opposite directions, is even for N even and odd for N odd, so that the same is true of the maxi = L + S. mum magnitude of The wave function for a state of given angular momentum Jh can be shown to have definite symmetry properties depending on the value of J. For example, when = 0, the wave function has spherical symmetry, i.e., depends only on the distances of the particles from the origin. As a consequence of these symmetry properties, it is possible to demonstrate the existence of certain selection rules governing, for example, transitions from one stationary state to another with emission of radiation. One important rule of this sort is that such transitions can occur (at least to an appreciable changes by ± 1 or 0. extent) only when In all cases of interest the wave equation of a system of particles will be Parity. unaltered when the rectangular coordinates of all its particles are changed in sign It can be shown to follow from this that the wave functions (reflection in the origin). of stationary states can always be chosen so that they are either unchanged or multi plied by —1 by such a reflection. Those which are unchanged are said to have even parity; those which are changed in sign only are said to have odd parity (or simply called, respectively, even or odd states). The significance of this classification of energy levels, which has no analogue in classical theory, lies in the fact that it implies a very general selection rule; namely, transitions can occur only between states of different parity, f Identical Particles and the Exclusion Principle. The invariance of the wave equa tion under rotations and reflections has been seen to lead to the classification of states There is a third kind of invariance according to their angular momentum and parity. of the wave equation for a system of particles some or all of which are identical with one another, namely, its invariance when the coordinates of any two such particles are It is possible to show that as a consequence of this invariance of the interchanged. wave equation there will always exist, for any stationary state, an eigenfunction which is symmetric (remains unchanged when the coordinates of two identical particles are interchanged) or antisymmetric (is multiplied by —1 when the coordinates of two identical particles arc interchanged). \ For particles with spin this statement is true, however, only when the orientation of the spin is counted as a coordinate in addition to the three specifying the position of the particle in space. The electrons in an atom must, of course, be regarded as identical particles, and it is an empirical rule§ of fundamental importance that for electrons only the anti symmetric eigenf unctions occur in nature. This rule must be regarded as an inde pendent hypothesis, not deducible from the principles of quantum mechanics. The restriction to antisymmetric states for electrons has a simple consequence when the electrons individually are described by single-particle wave functions so that each can be said to be in its own state. Under these conditions it is impossible to form an antisymmetric eigenfunction for the atom as a whole when there are two

J

J


J

t It has. however, been demonstrated experimentally that the conservation of parity leading? to this selection rule is violated in the case of 0 decay. This does not imply a breakdown of the principles of quantum mechanics but rather indicates a lack of invariance of the wave equation for reflection in the origin which can be attributed to the existence of a right-left asymmetry for elementary particles. The possibility that parity is not conserved was pointed out by T. D. Lee and C. N. Yang [Phyn. Reg. , 104 : 25-1 Rep., suggested experiment by [Phyt. who the 8. Wu decay on 0 carried out C. 105 : 1413 ( 1956)1. (1956)]. t It is not implied that all eigenf unctions are cither symmetric or antisymmetric but only that eigenalways of this can simple functions kind be constructed. ( The existence of such a rule was 6rst discovered in connection with the two electron systems giving rise to the helium spectrum. In the case of two particles there can occur only symmetric or antisym metric eigenfunctions. The number of energy levels found in helium would be twice as great cks it is if both kinds occurred.

Art.

4-27

THEORY

ELEMENTS OF ATOMIC

1]

in the same state, since the occurrence of two electrons in the same state that the wave function is unaltered when these two electrons are interchanged. Thus the restriction to antisymmetric functions rules out the occurrence of two elec trons in the same state. This is clearly the quantum mechanical analogue of the Pauli exclusion principle of the Bohr theory (cf. Art. 1.23). The symmetry rule has been stated above only for electrons. The form which it takes for other part icles is discussed in Arts. 2 and 3. The Independent-particle Model and the Periodic System. In quantum mechanics the problem of the electronic structure of atoms is approached, as in the Bohr theory, by way of the independent-particle model. That common average central field potential assumed for each electron, to which can then be assigned set of four quantum numbers: n, m describing, as in Art. 1.33, the central field eigenfunctions and a fourth quantum number m, = + giving the orientation of the spin. quantum mechanics, in contrast to the Bohr theory, the independent-particle approximation may be justified and controlled by appeal to the accurate many-body wave equation, and there are practical and reasonably accurate methods (notably that of "self-consistent" fields) for calculating the dependence of the energy on n electrons

is,

and

/.

In

1

I,

a

is

a

implies

0 is

/

1,

-

I.

,

.

0,

/

1,

n

+

1

is

/

0

n

2, /

/

a

As in the Bohr theory, the exclusion principle leads to shell structure of atoms. The maximum number of electrons which can "occupy" an electron state with = ™ quantum number only 1), independent of n. For possible; 2(2/ = = = = for and and in general . . n The symbol (nl)'

L,

K

L

shell: (Is)1 shell: (2«)J(2p)6 M shell: (^'(Sp)'^)" N shell: (4s)s(4p)'(4d)'°(4/)u

is

is

a

It

The electronic structure of the lightest atoms (Z < 19) consists of completed, or "closed," shells and the outer electrons in the partially completed outer shell. incompleted shell, called the valence electrons, that determine the chemical properties and optical spectra of the atom. For example, the electron configuration of the sodium atom (Z = 11) Na: (ls)'(2*)I(2p)8(3s) is

The 3« electron the valence electron. In the heavier atoms (Z > 19) there are, in general, incomplete inner shells as well as the outer shell of valence electrons. Problems. 1.35 Perturbation Theory, Matrices, and Time-dependent For only few special problems does the Schrddinger equation have exact analytical solutions, and in general The most widely necessary to resort to approximate methods. is

it

a

the perturbation method, which can be applied when the Hamilof these and the solution of the problem with Hamilput in the form H = Ho 4tonian Ho known. The term regarded as a perturbation to the unperturbed problem defined by Ho. When each energy level of the unperturbed problem has only a single eigenfunction when the energy levels of Ho are nondegenerate), the effect of the perturba (that tion If the unperturbed problem on the energy levels can be very simply stated. H«u — Eu has the eigenvalues E„ and the eigenfunctions ua (where a can stand for an array of quantum numbers), the energy levels of the perturbed problem are, to a first approximation (neglect of terms in F'), given by

F

is

is,

is

F

is

tonian

is

applicable

F

Ea +

/«.*Fu.

dr

(58)

where dr a generalized volume element and the eigenfunctions u« are "normalized" so that /u.'u, dr = The integral which gives the change in the energy levels

is,

1.

is


I

r

is

customarily used for an electron state occupied by electrons. With the spec the structure of the com troscopic notation of Art. 1.23 for the angular momentum M, N shells can be indicated as follows: pleted K,

4-28

NUCLEAR PHYSICS

[Sec.

4

by Eq. (44), just the average value of the perturbation energy F for the unperturbed state u.. A simple expression can also be found for the perturbation produced in the eigenfunctions. t The perturbation calculation can be carried out to higher order in F. The results depend on the array of quantities of the form

/ua*Futdr

(59)

is,

which define an infinite matrix of which the integral in Eq. (58) is a diagonal element. In the case of degenerate systems the perturbation in the energy is again given by Eq. (58), but with a particular choice of the unperturbed eigenfunctions, which are to a certain extent arbitrary when there is degeneracy. While the matrices that can be associated with any dynamical quantity by Eq. (59) appear here only as an aid in the formalism of perturbation theory> they are actually of fundamental importance in quantum mechanics. In fact, it is possible to express all the laws of quantum mechanics as relations between matrices rather in terms of wave functions and operators. Such a matrix mechanics was actually the original formulation discovered by Heisenberg in 1925. Perturbation theory is also extensively used in time-dependent problems, for example, in calculating how transitions between stationary states are induced in atoms by bombarding them with electrons or photons The process can be described in terms of a transition probability w„-.h, which is the number of atoms undergoing transitions per unit time divided by the total number of atoms bombarded. (Quan tum mechanics can describe the process only in this statistical sense.) Only one If F is the inter simple but quite general result of the theory will be quoted here. action energy between the atom and the bombarding particle (or any external field or perturbation), the transition probability to first order, given by

to.-*

=

vn

lî'P

(60) p

is

is

the matrix element defined by Eq. (59) and where Fai> the number of states per unit energy range for the bombarding particle assumed to have a continuum of such states. 1.36

a

is

a

is

a

is

is

A

Collision Processes. While quantum mechanics was developed mainly in connection with problems of the stationary states of atoms, its methods are equally applicable to the collision processes which arc of such great current interest in nuclear physics. Scattering in a Force Field. simple but fundamental problem which may be treated completely by quantum mechanics that of the deflection or scattering of a beam of particles by a spherically symmetric force field. The scattering of a particles by the field of nucleus an example. The analysis of more complicated problems, such as those of nuclear reactions, commonly in terms of this elementary problem of scattering in central field or of some modification of it. The general procedure in quantum mechanics for the above type of scattering to seek a solution of the wave equation (37) which has problem given energy E and which satisfies the boundary condition that at a great distance from the force center the wave function has the form

- e""™»

4-/(9)

pikr —

(61)

r

v

is

6

r

where and are polar coordinates (cf . Fig. 6) with origin at the scattering center and axis along the direction of the incident beam. The first term the space part of the

See Ref. 54. chap.

7,

is

fc

= \Z2mE /h, and de Broglie wave of the incident particle with the wave number the second that of an outgoing spherical wave representing the scattered particles. The factor f(6), called the scattering amplitude, gives the angular distribution of the t


Fab =

for this and other details of the perturbation method.

ELEMENTS OF ATOMIC

1]

scattered particles. This distribution is usually expressed cross section da, that is, the scattering number of particles, for an incident beam of unit intensity, that are scattered into a solid angle da about the direction 0. It is readily shown that do = |/(9)|» dw

For the calculation

4-29

THEORY

in terms of the differential SCATTERED

WAVE Ikr <{B)-f-

f

(02)

Tf

of the scattering

INCIDENT . Ikz

is

is

of

amplitude f(0) the so-called method par tial waves has found the widest application. The essence of this method the separate treatment of the scattering of those parti cles of the incident beam that have a given (quantized) angular momentum about the force center. Formally this accomplished

.

Ler

Art.

WAV

Fig. 6. Schematic representation of the incident and scattered waves correspond ing to a solution of the wave equation the form of Eq. (61).

of

by expanding the wave function in the

form

1

ff,(r)P,(cos

(63)

9)

£ 0

BP

-

u(r,9)

=

are Legendre polynomials that are eigenfunctions of the square of the Pi(cos Each term in the sum then angular momentum for the eigenvalues L1 = 1(1 l)h*.

+

6)

where

f

a

(21

l)eta sin MMcos

(64)

9)

~

+

-

y

in

f(0)

1-0

Si

4i

r)

where the are phase shifts defined by the asymptotic form (for large of the solutions of the wave equation for the ffj with the potential energy V(r) of the scatter ing field. When V(r) vanishes at infinity, the asymptotic form of the Ri can differ only in phase from the solution (1/r) sin [kr — (\/2)ln] in free space. The are the shifts in phase between the solutions in free space and those in the force field, as illustrated in Fig. 7.

Substituting from Eq. (64) into Eq. (62) gives for the differential scattering cross section

+

l)e,i8 sin iiPi(co8

(65)

0)

(21

sin

«,

+

1)

(21

(66)

1

a = 4*X«

Y

is

is

is,

\

is

= X/2x = l/k where the de Broglie wavelength of the incident particles. The total scattering cross section a, that the cross section for the total number of par ticles scattered out of the incident beam, given by integrating da over all directions. The result

See Sec.

4.

in

pp. 7tt-77, for some remarks

on the interpretation of this decomposition.

a

is

a

a

it

The chief merit of the method of partial waves lies in the fact that often few, and sometimes only sufficient approximation to take into account only single, term the expansions for da and a. For example, in the scattering of slow neutrons t


"

"

definite angular momentum. Each partial wave corresponds to particles having Eq. (63) must satisfy a Schrodinger equation of the form of Eq. (48), and the complete wave function must have the form given in Eq. (61) for sufficiently large r. These conditions lead to a determination of f(0) in the form

4-30

PHYSICS

NUCLEAR

[Sec. 4

discussed in Sec. 5, it is sufficient to consider only the / — 0 component, that is, particles of zero angular momentum. Two-body Collisions. An actual scattering process will in general involve a col lision between two particles, and when these are of comparable mass, the motion of both must be taken into account. However, in quantum mechanics as in classical mechanics this two-body problem can be reduced to a one-body problem of scattering by a fixed force center. The two-body-collision problem is greatly simplified by transforming its description from the coordinate system in which the collisions are actually observed, called the laboratory system or L system, to a coordinate system moving with the velocity of the center of mass C of the two particles, called the C system. The C system for any number of particles has the following two basic properties: (1) The total momentum 2


O

>

o

a: LlI

z

UJ

Fig.

7.

Illustrating the zero order (s-wave) phase shift for an attractive potential.

of the particles is always zero in the C system, f and (2) the total kinetic energy of the particles is that of the sum of their masses moving with the velocity of C plus the kinetic energy of the motion of the particles relative to C. For two particles with masses mi and m2 at fi and r-, the center of mass is at re

where M = m, + m». limv1, where v = vi mass defined as

miti + m&t

M

It —

is readily shown that the kinetic energy in the C system is Vj is the velocity of Wi relative to »ij and m is the reduced

mitni

(67)

mi + m?

The total kinetic energy T of the system

T

= X^mv1

+ HMvc1

m,vt + mjv2 vc =

where

M

t This

tivists

can hence be written

as

(68) (69)

statement can be used to define the center of mass in a way that can be taken over to rela and hence applied to photons as well as material particles.

mechanics

Art.

ELEMENTS OF ATOMIC

1]

THEORY

4-31

the velocity of C. When the force between the two particles depends only on the between them, vc remains constant and the relative motion of the two particles is given by the equation of motion for a single particle with the reduced mass m. The two-body problem in classical mechanics is thus reduced to an equiva lent one-body problem, and the same is true in quantum mechanics, since the form of the wave equation is based on the classical expression for the energy. The Schrodinger equation has the form of Eq. (37) if m is taken as the reduced mass and V(r) as the potential energy as a function of the distance r between the particles. is


distance

(b)

Fio. 8.

A

velocities velocities

two-body collision in the C system is shown schematically in (a). The final The vector additions giving the »i and vt are the same as the initial velocities. in the L system are shown in (b).

In the usual two-body-collision problem one particle, say mi, can be thought of as projectile fired with velocity vo at m:, which is initially at rest in the L system. The velocity of the center of mass C is then a

(70)

The total energy, which remains constant in an clastic collision such as is being assumed initial kinetic energy Ei. = ••£mit>oJ. The energy of the relative motion, is that of the equivalent one-body problem, is which however, here, is the

The relation between

these two energies

is then

(71)

The collision as it appears in the C system is illustrated in Fig. 8a. The particle m, velocity vc = (mt/M)v0, and m2 a velocity vc — wo = (ms/A/)yo, both toward C.

has a

4-32

NUCLEAR PHYSICS

[Sec. 4

When the particles are separated again after the collision, conservation of energy requires that their velocities be unchanged, and since the momentum remains zero, they must move apart in opposite directions as shown. In the C system the deflection is thus specified by the V(r) single angle 8 between the direction of motion before and after collision. In the L system, however, the two particles move in different directions obtained, v-v0 as illustrated in Fig. 8I>, by adding vectorially the E velocity vc to the velocities in the C system. The _ cross section for scattering through the angle 9 in the C system is obtained by solving the wave equation — a—. for the equivalent one-body problem with the energy E of Eq. (71) and the boundary condition of Eq. (61). Transformation to the L system in the manner indi cated above then gives the observed cross sections, t Penelralion of Potential Barriers. In classical me z2 e2/R chanics a particle with a given total energy E — T + V is excluded from regions of space in which T = E — V is negative, since T = y^mv1 is always positive for any real value of the velocity v. For motion in one dimension this implies that a particle can encounter an impenetrable potential barrier as illustrated in Fig. 9a. A particle impinging from the left on a discon tinuity in the potential V(z) where there is a jump from V = 0 to V = V0 would be reflected at the discontinuity if E < Vu and transmitted (with reduced velocity) if E > V„. (b) In quantum mechanics the situation is radically Fig. 9. Two types of potential different. The wave function does not vanish when barriers: (a) a rectangular bar T «■ E — V < 0, and a penetration into classically rier, (b) a coulomb barrier for forbidden regions is possible. = The behavior of the wave function when T < 0 however, qualitatively being exponential in the first case and oscillatory in different from that when > the second. As a consequence, just as in the reflection of light at an interface where the index of refraction changes, there will be partial transmission and partial reflection at a sudden change in the potential energy, with the wave function falling off expo nentially in the classically forbidden region. Of particular interest the case illustrated in Fig. 9o of barrier of finite width a. When Va, the wave function will fall off exponentially as cxp — ■\/2m(V0 — E) = = a. Thus, unless \/2m(Vt, — E) a» in the region from to the wave function to the right of the barrier where need not be negligible. > Particles incident on the barrier can then be transmitted through it. The complete quantum mechanical solution§ shows that there will be transmitted and reflected wave with transmission coefficient (ratio of transmitted intensity to incident intensity) which thick enough barrier diminishes exponentially with y/lmiV* thus com E) a. pletely impenetrable as in classical theory, but a thin one will allow "leakage" of particles through it.

_ _

0.

1,

is

a

0

A

a

T

z

z

0

[

z\

a

<

E

is

{

T

0,

is,

I

a

-

t

5,

t

For the relations between the cross sections in the two sy items and a more detailed account of the The relati viatic generalizations are given whole problem see Ref. 54. chap. or Ref. 16, Appendix B. in Ref. 43. Thus in the one-dimensional example illustrated in Fig. 9a the wave equation

solution* exp V 2m(V*o — E) Sec Ref. 54, chap. 5.

2m

2mE/k, in the classically z\

kz, where

k

has solutions

| 0 ±

rfJu

5


'

allowed

to the right of the barrier.

region to the left of the

Art.

elements of nuclear theory

2]

4-33

potential barrier problem which has numerous important applications in nuclear that of the coulomb barrier illustrated in Fig. 96. It enters decisively in the treatment of the collision between two charged particles (such as an a particle and a nucleus) when there is coulomb repulsion at large distances but an attractive force when the particles are close together. The potential energy as a function of the dis tance r between the particles may then be idealized as shown. If the attractive force sets in at r = R, the top of the barrier is at an energy B = ZiZê'/R when the particles have charges Zi« and Zie, both of the same sign. When E < B, where E is the energy in the C system, there is then a barrier of width b — R, where 6 is the so-called classical There will, in this case, turning point, that is, the value of r at which E = ZiZê'/b. be leakage through the barrier, whereas in classical theory the particles could not approach each other to within a distance less than b and the repulsive force would have no influence on t he scattering. The transmission coefficient To of a coulomb barrier for the I = 0 or s-wave com ponent! of an incident beam of particles of velocity v can be shown to be given approxi mately by A

physics is

\

hv

E

n*

/

«

B. The transmission coefficients for the coulomb barrier is frequently the Gamow factor. The second term in Eq. (72) is small compared with the first when E
when

q

= 2*Z\Zze* hv

_

2lcZ\Zl

.yg.

137/3

v/c and 1/137 is the value of the fine-structure constant a = e*/hc. readily shown that the transmission coefficient of the barrier for particles coming from the inner region r < ft is the same as that for particles coming from the outside. Thus particles with E < B that, in classical theory, would be trapped inside can in quantum theory' leak out through the barrier. where 0 =

It

is

ELEMENTS OF NUCLEAR

2

THEORY \

While only charge and mass of the nucleus played an}' role in the early development theory, it was evident after the discovery of radioactivity in 1896 that nuclei could not be elementary' particles. The problem of the constitution of nuclei, however, was not satisfactorily solved before the discovery by Chadwick in 1932 of the existence of an uncharged particle the neutron, having approximately the mass of the hydrogen nucleus, or -proton. The hypothesis that the elementary constituents of nuclei are neutrons and protons, proposed by Heisenberg immediately after the discovery of the neutron, was the starting point of the first serious attempts to develop a quantummechanical theory of the nucleus. of atomic

2.1

Basic Properties

of Nuclei

is

Z

a

a

is,

A 2.11 The Classification of Nuclear Species and the Constitution of Nuclei. given chemical element mixture of atoms, called isotopes, with few exceptions, which have the same atomic number and hence the same chemical properties but differ in nuclear mass. an average of these masses. The chemical atomic weight is a

0

I

it

0

I

t

For components with > the centrifugal potential energy 1(1 + l)nV2mr' muat be added to the However, in most applications good approximation to take into shown in Fig. 96. » component. account only the An elementary treatment of nuclear physics, with emphasis on the historical development, will be viewpoint, are found in Ref. 50. General references on nuclear physics, largely from the experimental Kefs. 16 and 36. detail The most exhaustive treatise on nuclear theory with full mathematical Ref. 8. nontechnical account by Heisenberg (Ref. 28). Monographs on There an interesting special topics will be referred to in appropriate places. potential

is

is

t


called

NUCLEAR PHYSICS

4-34

[Sec.

4

a stable element into its isotopes was first carried out by J. J. Thompson in 1912, using the deflection of ions by electric and magnetic fields in an apparatus known as the mass spectrograph. t With the use of modern techniques^ it is possible to measure nuclear masses with a high degree of precision. The masses are always, to within a fraction of a per cent, equal to an integral multiple A of the mass of the proton. The possible nuclear species are thus specified by two integers, the atomic number Z and the mass number A. Isotopes have the same Z but differ in A. Atoms or nuclei The term nuclide is used to having the same A but differing in Z are called isobars. designate an individual atom or nucleus with given Z and A. The commonly used symbol for a nuclide is the chemical abbreviation for the element of which it is an isotope with A as a right superscript and Z as a left subscript, as in uMg" and »jU"8. The atomic number Z is frequently omitted, as it is already determined by the chemical symbol. The Constituents of Nuclei. The hypothesis! is now accepted without question that of neutral particles and positively nuclei are made up of neutrons and protons, that hydrogen atom. The term charged particles all having very nearly the mass of neutron or proton. nucleon used for one of these constituents, which may be either — Z, and the A, the number of neutrons The number of nucleons in a nuclide Z. number of protons 2.12 All naturally occurring elements with above 80 and a Radioactivity. few of lower have isotopes which are unstable and disintegrate with emission of an Such natural radio a particle (helium nucleus) ,or particle (negative electron). rays (photons) and results in activity normally accompanied by the emission of the transformation of one nuclide into another according to the scheme a

a

A

is

y

-

in a decay in decay

4)

A A)

0

2,

I,

-

(Z,A) -► {Z (Z,A) -* (Z + is

X,

is a

The number of atoms dN that decay statistical process. Radioactive decay in time dl to the number N which are present. proportional, with decay constant Thus dN = — \N dl, which integrates directly to (74)

A

ments see Ref. 2.

methods

and a compilation of nuclear

2,

a 0

For a brief description see Ref. 30, chap. 9. For a comprehensive review of experimental

1,

A

1

t X

is

is

A

a

0)

A

is

is

Z

Z

is is

is

I

when N0 atoms are present at = 0. The reciprocal of the decay constant t — 1/X the constant ordinarily listed, called the mean lifetime and (In 2)t = 0.693r, which known as the half-life. The naturally radioactive nuclides with > 80 have half-lives from 10~7 sec to The few (some eight) feebly radioactive elements with lower have very 10'° years. long lifetimes. In addition to those which occur naturally, a large number of radioactive nuclides They are known as artificially radioactive nuclides (it is, can now be made artificially. The most com of course, the preparation and not the activity that "artificial"). in fission products. mon source of such radioactivity which results from an a decay implies that a chain The decrease of four units in = 4n, where n is an nuclide of mass decays starting from of successive a (or also of the form in, the integer n integer, can lead only to nuclides for which in an a decay and remaining unchanged in decay. The same decreasing by = 4n + true of decay chains starting from a nuclide with 4n + or 4n + 3. mass measure

is

p

p

If

is

S

The alternative hypothesis that nuclei are composed of protonB and electrons, which was the most natural one before the discovery of the neutron, can be ruled out by a number of theoretical arguments. an electron exists inside a nucleus of dimensions of the order of 10 1! cm, the following: One of these must be greater than the uncertainty in its momentum implied by the unoertainty its momentum That is, The energy corresponding relations of Eqs. (30). > k/Ar with Sr <* 10~" cm. relativistically to Ap about 50 Mev, whereas the electrons emitted from a nucleus all have energies less than Mev. There are even more convincing arguments against the existence of electrons in nuclei. For

4


is

f)

Z

Z

is

is

a

is

a

is,

The separation of

an enumeration

see Ref. 16, chap. 8.

Art.

ELEMENTS

2]

OF NUCLEAR

4-35

THEORY

The decay chains of the heavy radioactive nuclides can thus be classified as belonging to one of these four categories. The in series is known as the thorium series, the in + 1 as the neptunium series, the in + 2 as uranium series, and the An + 3 as the actinium series. The genetic relations within these series are shown in Figs. 10 to 13. 2.13 The Stable Nuclides and Relative Isotopic Abundance. the Although distinction between stable and unstable nuclides is not sharp, it is a convenient one Tl

Pb

Bi

Po

At

Em

Ff

Ro

Ac

Th

Po

U

Np

Pu

81

82

83

84

85

86

87

88

89

90

91

92

93

94


P" !240 24° !6500yrl

33.7% ThC" 3lmin

6&3%

ThO 00 82

■208

83

84

85

86

87

88

89

90

91

92

93

94

Atomic number

Fio. 10. The decay chain of the in or thorium scries. The solid boxes denote naturally occurring nuclides. Diagonal arrows indicate a and horizontal arrows /J decay. (From R. D. Ecant, Ref. 16, p. 518.) to make. It is appropriate to classify all naturally occurring nuclides with Z < 83 stable, as although a few are weakly radioactive (r > 10' years). (Bi) For hydrogen, Z = 1, there are two naturally occurring isotopes with A = 1 and A =« 2. The nucleus of the first is the proton; that of the second, composed of one Atoms with the deuteron as nucleus proton and one neutron, is called the deuteron. are known as deuterium and occur only to the extent of 0.0149 atomic per cent in There is a third isotope of hydrogen with A «• 3 made up of two natural hydrogen.

4-36

PHYSICS

NUCLEAR

[Sec.

4

Tl

Pb

Bi

81

82

83

Po 84

At

Em

85

86

Fr 87

Fta

Ac

88

89

Th 90

Pa 91

U

Z

is,

neutrons and a proton which is 0 active with a half-life of 12.4 years and does not occur Its nucleus is called the triton; its atoms tritium. naturally. Among the lighter elements (Z < 30), stable nuclei occur only when the number of neutrons is roughly equal to the number of protons, that is, when .4 — Z « Z or however, always in excess of the number of .4 « 1Z. The number of neutrons protons for > 20, and this tendency to an excess of neutrons increases steadily with Z. Np

Pu

Am

92

93

94

95

I

Î4yrn470yrj

Ua7' 'Np237i

241

-237

r

2.2xl0V

Po*»T[u«»l

-233

]

[274daysJ|

-229

I

ProîTac*25"!

-225 » E

days]

I

[MdaysjTlO

!~A>7

-221

5minJ

i

!

IFr"'i_

-217

!0.02secJ

rBi2l3"MPo2l31. sec! [47 minjj4 u

-213

98%

LZjninJlShrjl 81

82

Bi209 CO

83

84

-209 85

86

87

88

Flo.

11.

The decay chain of the 4'i +

1

Atomic

89

90

91

92

93

or neptunium series,

(From

in

p. 520.)

The stable nuclides occurring

94

95

number

R

2%

nature

aire

shown diagramatically

D. Evans. Ref. 15,

in Fig.

14.

It

a significant fact that over half of these stable nuclides have an even number — N = of neutrons and an even number of protons and that practically none The remainder are about equally distributed between has both and JV odd.

Z

Z Z

A

is

odd Z-even Ar and odd .V-evcn Z. This can be taken as a rough indication that even Z-even N nuclides, in which nucleons of the same kind can occur in pairs, are particu larly stable and that odd Z-odd jV nuclides, in which there can be an unpaired proton and an unpaired neutron, are in general unstable. Intermediate in stability are the It clear that the pairing nuclides in which nucleons of only one kind can be paired. is


1

Th2», 7340 yr

Art.

ELEMENTS

2]

OF NUCLEAR

4-37

THEORY

of like nucleons is of significance, which suggests the operation of a Pauli exclusion

principle for nucleons as well as for electrons. There is convincing evidence for an abnormally

large frequency of occurrence of nuclides in which either the number of protons or the number of neutrons is 20, 28, It has recently become clear that these numbers form part of a larger series 50, or 82. Tl

Pb

Bi

Po

At

Em

Fr

Ro

Ac

Th

Po

U

Np

Pu

Am

Cm

8:

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

r Bk

Cf

97

98

Cfz

36hr

Puz

. jCm242 1162days'

5x05yrj

UI

246

-242

iPu5*8'

-238

|90yr i'

>^45xl0Vf/'

fuxTiruii


UXi UX2 24.1days 1.2mm

/

234

i/25xl05yr Io

-230

8xK34yr

/

Ra

-226

I620yr

/

RaB RoC 27min 20mn

fc

2

E

Rn 3.8 days

RaA 3 miri

CNJ + c

-222

i

-218

RoC

-214

64psec

Q04% 9996% RoC I 3nin

RoD 22yr

RaE -RoF(Po) 5doys I38dcys

-210

RoG 00 81

82

-206 83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

Atomic number

Fig.

12.

p. 521.)

The decay chain of the

in

+

2 or

uranium series.

(From H. D. Evans, Ref.

16,

called "magic numbers" and that the abnormal properties of nuclides for which Z or A' is one of these numbers reflects the occurrence of a "shell structure" of the nucleons in a nucleus analogous to that of the electrons in an atom (Art. 2.31ff.). The relative amounts in which the isotopes of a given element occur in natural samples vary only exceptionally with the source of the sample. These relative isolopic

4-38

[Sec. 4

NUCLEAR PHYSICS

abundances arc of practical importance for isotope separation processes and of great interest from the standpoint of geology and cosmology. The bombardment of a "target nucleus" with accel 2.14 Nuclear Reactions. erated particles that have a low charge will very frequently result in a transformation Tl

Pb

Bi

Po

At

Em

Fr

Ro

Ac

Th

Pa

U

Np

Pu

Am

Cm

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96 ;Cm*«;

jPu*

243

239

24x10 yr AcU

235

7.1x10 yr

UY

Pa

-231

25hr

Ac 22yr| 1.2%

-

RdAc !86doys

227

98 8%

AcK

AcX

2lmri|

ll.2doys

223

0004%, At2"

/

09min

97% B,215 8min

AcB 36min

AcC

/

An

■219

39sec

Ac A

-215

000l8sec

AcC

2 2min 05sec

99.7% 03% AcC |4.8min

AcD 00

82

81

-207 83

84

85

86

87

88

89

90

91

92

93

94

95

96

Atomic number

Kin.

13.

The decay chain of the

in

+ 3 or actinium series.

(From R. D. Evang,

p. 522.)

Rrf. 16.

is a

of the target nucleus into a new one with the ejection of a particle different from the incident one. The charge of the incident particle must be low, with the accelerating voltages ordinarily available, in order that it may override or appreciably penetrate the coulomb barrier (cf. page 4-32) opposing its penetration to the target nucleus. neutron, and This repulsive force of course, absent when the incident particle consequently reactions are more easily produced with neutrons than with charged is,


3.4x10 yr

Art.

4-39

ELEMENTS OF NUCLEAR THEORY

2]

A very great variety of

such nuclear reactions have now been observed Similar processes, such as disintegration in radioactivity and the fission of a nucleus, either spontaneous or produced by bombardment, are also classified as nuclear reactions. Such nuclear reactions have some features in common with chemical reactions, and it is customary to describe them by equations resembling those for chemical reactions. particles.

and studied.

240i

1

1

1

1

1

1

50 NUMBER. Z

70

1

1

Fl

200

160 < of

I


z

120

in <

3

.•I

80

.!":

40

0^

10

30 ATOMIC

Flo. 14. The stable ath ed., p. 460.)

nuclides

occurring in nature.

(From F. K. Richtmyer

90 et al.,

Ref. 50,

The system used is sufficiently well explained by an example:

,N'« + .He'

- .O"

+

,H'

The reaction described is the collision of N1* with an a particle (He4), resulting in the formation of 0" and the ejection of a proton (iHl). (This is actually the first arti Another example is afforded by the equation ficially produced nuclear reaction.)

,Bc' + iHe4

-> „C'S + on1

which corresponds to the bombardment of beryllium with a particles to produce C1J and neutrons (on1)(It was by this reaction that neutrons were first produced.) In all such reactions, there must be conservation of the total number of nucleons, requiring that the sum of the superscripts be the same on the left and right sides of the equation. Conservation of charge requires a similar balancing of the subscripts.

An abbreviated

notation

X(a,b)Y

4-40

NUCLEAR PHYSICS

[Sec. 4

is also commonly used. The target nucleus is X, the residual nucleus is Y, the incident particle is a, and the ejected particle is b. The two reactions above would, in this notation, be written and N"(«,p)C17 BeHcr.nJC"

is

is

a

X

is

ic

Physical at. wt = 1.000279

chemical at. wt

In both

1

amu

-

(931.162

+ 0.024)

Mev

is

in

is

included. scales, the mass of the electrons in the neutral atom electron volts and in atomic mass units The relation bet ween energy measured

(75)

A

B =

A

Z

is

it is

a

a

a

is

thus of the order of 105 Mev. This The total energy of, say, uranium atom includes, however, the rest energy of all constituent nucleons. Of more practical nucleus and its significance are the relatively small differences in energy between measure of the total kinetic and Thus, disintegration products, real or imaginary. given by its so-called binding energy B, potential energy of the nucleons in a nucleus — Z which protons and the work required to disintegrate completely into neutrons. This binding energy may be calculated from the equations

- Z)M„ - M

= c*A

ZMH + (A

(76a) (766)

Z

is

is

M the mass of the nuclide, Mh the mass of the hydrogen atom, and M» the The quantity A mass of the neutron, all on the physical atomic mass scale (amu). known as the mass defect. The masses of the atomic electrons are included in M, Mh, and M„ but cancel out§ in the expression for A, since the mass of electrons enters with positive sign in the term ZJlfn and with negative sign in the term ilf.

where

nuclide with binding energy

B(.\,Z),

(77)

A

a

/-*

The average binding energy per nucleoli of that is, the quantity

is

is a

is

2.

jt

3,

or Ref. 36, chap. 9. For a brief description of experimental methods see Ref. 16, chap. Table 37 of Sec. It actually only the rest masses which cancel, since M and Mn include the mass equivalent of the binding energies of the electrons. The error introduced in the binding energy calculated from E

is

is

%

is

is,

is

it is,

The subject of nuclear reactions is more fully treated in Art. 2. 15ff. and in Art. 4. 2.16 Energy and Mass. The total energy of a nucleus can be determined by a measurement of its exact mass M, which is related to its energy E by the Einstein equation E = Mc* (cf. Art. 1.14). Precision measurements of M by means of the mass spectrograph! are available for a large number of the known nuclides and are In all, supplemented by direct determinations of the energy from nuclear reactions. the masses (and hence the energies, the two terms being used interchangeably in nuclear physics) are known for over two-thirds of the stable nuclides and for many of the unstnble ones. The exact mass M of a nucleus differs by only a few hundredths of a per cent from an integral number A of proton (or neutron) masses. It however, just this small deviation which of significance for nuclear theory, since measures the difference in energy between the nucleus and its separated components, that its binding energy, more precisely defined below. The energies of nuclides are recorded in the tables in one-sixteenth of the mass of the neutral terms of their masses. The mass unit atom of the nuclide Ou. This unit known as the atomic mass unit (abbreviated to These physical atomic weights are almost, but not quite, equal to the chemical amu). atomic weights, the difference being due to the fact that the oxygen used in the chemi cal determinations mixture of isotopes (99.759 per cent, however, O") whereas, measurements, the single isotope O'* in mass-spectrograph used for reference. The relation between the two

Art.

2]


4-41

as the binding fraction, t A closely related concept is the separation energy Sa(A,Z) of a particle a (such as a nucleon or a particle) from a nuclide (A,Z), which is defined as the energy required to remove the particle to an infinite distance from the nucleus. For example, the separation energy «S„ of a neutron is is known

S(A,Z)

=

B(A,Z)

- B(A -

l,Z)


As illustrated in Fig. 15, the binding fraction is close to 8 Mev per nucleon for all In other words, the total energy of a nucleus is nuclides except those with Z < 20. nearly proportional to the number of particles which it contains.

On a nucleon weight

macroscopic scale, the energy corresponding to the binding energy of even one Thus, for 10 Mev per atom of an element of atomic per atom is enormous. 50, the energy per gram is Energy /g =

-

fifty * 50 1.9

in*3

X

10

X 10" ergs/g

X

1.6

= 2.4

X 10-' X

10"

erg

kwhr/lb

In this calculation 6.0 X 10" is Avogadro's constant, the number of atoms per gram of an element of atomic weight 1, and 1.6 X 10~8 is the number of ergs in 1 Mev. A calculation of the binding energy of a nucleus is usually carried out by first obtain ing the mass defect from the measured masses and then converting to energy by Eq. For example, the simplest composite nucleus is the deuleron, made up of one ("5). proton and one neutron. The calculation of A runs as follows: t A parameter frequently encountered in the older literature is the packing fraction/? It does not have ao direct a physical significance aa the binding fraction /.

= (m —

A)/ A.

4-42

NUCLEAR

PHYSICS

[SEC. 4

Mn = 1.008 982 amu

Mu

+Mn M(H') A = Mi,

H

= 1.008 142

- 2.017

124 = 2.014 735

-

+ M. M{\V) = 0.002 389 B = c'A = 0.002389 X 931.2 Mev = 2.225

Mev

The binding energy of 2.22 Mev is the work required to disintegrate a deuteron into a neutron and a proton, a process which can actually be carried out by bombarding deuterium with y rays. The minimum energy required for the y photons is 2.22 Mev. Conversely, when a neutron is captured in hydrogen, 2.22 Mev of energy is "set free.'1

It appears as a photon of this energy. The experimental verification of these con clusions constitutes one of a number of similar proofs of the validity of the Einstein relation between mass and energy. Corresponding to the heat of reaction in chemistry, there is a net energy change in a nuclear reaction which can be computed from the masses of the nuclei involved. It is generally called the Q value of the reaction, and for a reaction X(a,b)Y, it is defined

by

Q =

The energy

c*(Mx + M.

(78)

Q is often included in the equation for the reaction, as in a +


- My - Mb)

X

-

Y +

b

+

Q

If Q is positive, energy can be released in the reaction, for example in the form of If kinetic energy of the particle h. Such reactions are called exothermic or exoergic. Q is negative, energy must be supplied, say through kinetic energy of the particle a, to produce the reaction which is then said to be eruiothermic or eruloergic. The calculation of the Q value of the reaction Be' + a — ► C11 + Q is given below as an illustration. = 9.015 043 amu A/(Bc») = 4.003 873 A/(He«) A/(C")

M.

= 13.018 916 = 12.003 804 = 1.008 982

A/(C12) + M„

= 13.012 780

Af(Be8) + M(He4)

Q = A/(Be") Q

- 0.006

+ M(He<)

130

X

931.1

-

M (C")

= 5.708

- M.

= 13.018 916

-

13.012 786 = 0.006 130

Mev

The reaction

is thus exothermic, and the Q value of 5.708 Mev can appear as kinetic energy of the neutron. The way in which the energy balance in nuclear reactions is modified when the kinetic energies of the reactants is taken into account is discussed in Art. 4.22, along with further details of the theory of nuclear reactions. 2.16 There are a number of other quantities, in Further Properties of Nuclei. addition to the nuclear masses, that have been measured and tabulated for a large number of nuclides. These are briefly described here, their theoretical interpretation being reserved for Arts. 2.2 and 2.3. Size. That heavy nuclei have dimensions of the order of magnitude of 10~11 cm Refine was first shown by experiments on the scattering of particles (cf. Art. 1.21). ments in nuclear theory and experiment permit a more accurate estimate of size and shape and show that nuclei are very nearly spherical with a radius R given by the approximate formula It = 1.5 X 10-'»AM cm (79)

That R is proportional to the cube root of the number of particles implies that the volume of the nucleus is proportional to the number of nucleons A which it contains.

Art.


2]

4-43

for

particular nuclide

1,

0,

is

/

/

2

The value of

a

/

is

J-^

The density of nuclear matter is thus approximately constant and its value is readily calculated to be about 10" g/em*. There are a number of methods of measuring the nuclear radius which do not agree to better than about 20 per cent. Such discrepancies are to be expected, however, since the radius can be only loosely denned as a distance at which specifically nuclear forces become negligible. The earliest measurements of nuclear radii come from the study of the half-lives of a emitters. The principle involved is described in Art . 4.11. There Scattering of fast neutrons also measures the nuclear radius (cf. page 4-9(5). are a number of other methods which are reviewed, for example, in Ref. 16, Chap. 2. A considerable amount of indirect evidence as well as Angular Momentum (Spin). direct measurement has established beyond doubt that both neutrons and protons have a spin of as do electrons. The general principles of quantum mechanics then imply (cf. Art. 1.34) that a nucleus made up of such nucleons will have a total angular which momentum has one of the valuesf quantized to Ih, where

called its nuclear spin, although

A

is

is

pt

-—

=

(80)

m fC

2

MO

is

the magnetic moment that classical theory would give for a proton moving in orbit with one unit of angular momentum, just as the atomic Bohr magneton [cf. Eq. (53)] the corresponding quantity for an electron. The nuclear magneton thus smaller by obtained from the atomic magneton by replacing m, by tnp and factor 1/1,840 than the atomic magneton. The observed magnetic moments arc not integral multiples of the nuclear magneton, nucleus compounded in complex fashion out indicating that the magnetism of the spins and orbital motions of its constituents. For just this reason, the study of magnetic moments proving extremely useful in the elucidation of nuclear struct ure. Qiiadrupole Moment. While the magnetic moment of a nucleus determines (to first approximation) the part of its interaction with an external magnetic field that comes from the magnetism of internal currents, its interaction with an external electric field similarly determined by the qiiadrupole moment of its charge distribution. might have been expected that the interaction of a nucleus with an electric field would depend in first, approximation (after subtracting the effect of the resultant elec However, trostatic force Ze%) only on the electric dipole moment. theorem in quantum mechanics that the dipole moment in any stationary state vanishes exactly. Consequently, the next higher approximation must be used, in which the interaction field E(z) in the For energy depends on the qiiadrupole moment. direction this of the form interaction energy depends only on the quadrupole moment which

a

a

-

((3 cos>

0

-

is a

- DprVi

(81)

e

Q

Q

z

a

it

E

It

is

is

of

a

is

is

is

is

an

2.

5,

+

More accurately, 1* has the value /(/ .Sec footnote on p. 4-21. DA1. For a detailed account of methods of measuring both spin and magnetic moments see Ref, 17. principles of the methods are briefly described in Ref. 1(1, chap. par.

t t


It

A

is

is

It

a

/

a

it

may contain contributions from orbital motion of nucleons as well as from their spin. The nuclear spin can be measured in number of ways, most of which depend on the fact that a nucleus has magnetic moment associated with its angular momentum and that this magnetic moment gives rise to an interaction with external fields. have integral spin while those found that without exception nuclei with even with odd have half integral spin. This in accord with the predictions of quantum in addition an mechanics for a system of particles all of which have spin empirical rule that nuclei with an even number of protons and of neutrons have zero angular momentum. Magnetic Moment. Very extensive and accurate measurements of the magnetic of nuclei are now available. The natural unit in which to express the moments^ the nuclear magneton magnetic moment of a nucleus

The

4-44

NUCLEAR PHYSICS

[Sec. 4

where e is the electronic charge, r and 6 are polar coordinates with origin at the center of the nucleus, and the average is over the charge density distribution p in the nucleus. The interaction energy is — (\i)(dE /dz)Q. The quadrupole moment Q is easily seen to vanish when the charge density is spherically symmetric, and a finite value of Q hence measures the deviation of the charge distribution from such symmetry. Quadrupole moments have been measured for many nuclides by methods (see Ref. 47 for a description) similar to those which give the magnetic moments.


2.2

General Features of Nuclear Theory

Nuclear theory is based on the still unchallenged assumption that the broad general principles of quantum mechanics are applicable to the nucleus and that the elementary constituents of the nucleus are protons and neutrons. There is ample evidence that this viewpoint is justified in the sense that nuclei display the general characteristics of quantum mechanical systems such as the existence of stationary states, quantization of total angular momentum, and the emission of radiation accompanying transitions from excited states. In spite of such indications of the correctness of the general method of approach, nuclear theory has, however, by no means reached the satis factory state of development attained in atomic theory. This can be attributed to two main causes: 1. A law of force between nucleons is not a priori given as is the electrostatic force between the constituents of atoms. 2. Mathematical methods are not as yet sufficiently highly developed to solve the quantum mechanical problems of a system of more than two nucleons. In atomic theory this difficulty exists to a very much smaller extent because of the first approxi mation afforded by considering all electrons as having stationary states in a common force field dominated by the nucleus (independent particle approximation). Difficulties like these have so far prevented the working out of a systematic and In fact, the basic law of interaction between consistent theory of nuclear phenomena. nucleons is still not established, and empirical prescriptions for nuclear forces which at one time have seemed adequate have often had to be abandoned when new experi mental information has become available. 2.21 The observed spins of the Spin, the Exclusion Principle, and Statistics. nuclides are resultants, formed according to the laws of quantum mechanics, of the intrinsic spins and their orbital angular momenta, each nucleon having a spin of Thus a nuclide with A nucleons will have an integral total spin if A is even and a half integral spin if A is odd, since the individual nucleon spins can be oriented only parallel or antiparallel to any reference direction and the orbital angular momenta have always an integral resultant. This rule is verified for all nuclear spins that have been meas ured, as already noted in Art. 2.16. There is a tendency for nucleon spins to cancel each other out by forming parallelantiparallel pairs, so that resultant spins (in the ground state) have not been observed to be greater than In fact, with very few exceptions, nuclides with even A have zero spin. The way in which the exclusion principle governs the extranuclear electronic struc ture of atoms was outlined on page 4-26. Nuclear structure also is dominated by this principle, somewhat complicated, however, by the fact that there are two different kinds of identical particles in nuclei, the protons and the neutrons. It is very well estab lished in a variety of ways that for a system of protons only the antisymmetric wave functions occur in nature, just as for electrons. That the same is true of a system of neutrons cannot be so directly established but is in accord with what appears to be a perioral rule that particles of half-integral spin have antisymmetric wave functions and is supported by much indirect evidence. Consequently it must be assumed that the wave function describing a nucleus is antisymmetric with respect to the exchange (of both position coordinate and spin direction) of either two protons or two neutrons. As in the case of electrons, the exclusion principle has a rather more intuitive formu lation in terms of the independent-particle picture, which is often used for nucleons,

Art.

2]

ELEMENTS

OF NUCLEAR THEORY

4-45

with less direct justification than for atomic electrons. In this formulation principle states that no two protons can be in the same state of orbital motion unless their spins are antiparallel and that the same statement can be made about neutrons. The compounding of spins and the operation of the exclusion prin ciple for the nuclides made up of two, three, and four nucleons can be illustrated as in Fig. 16if it is assumed that there are energy levels for the orbital motion that can be occupied by the nucleons as in an independent-particle model. The lowest of these levels would be L expected to have zero orbital angular momen tum (/ ■» 0). The nuclear Bpin / is then the H3 He5 He" sum of the nucleon spins of ±%. In the I I - 1/2 I -1/2 I " 0 deuteron the two unlike particles can have Fig. 16. Schematic representation of equal or opposite spins, giving / = 1 or 0. the spins of the nuclides with two, The observed spin is 1. In the triton the two three, and four nucleons. The open neutrons in the lowest state must have oppo circles represent protons; the black ones neutrons. site spins, giving a resultant spin of J£ from the proton in accord with the observed spin of H- For He4, the a particle, the resultant spin must be zero as observed, since the two proton spins and the two neutron spins must be opposite when they are in the though

the exclusion

I-

same state.

behavior of nuclei affords further evidence of their structure and conthat they are composed of neutrons and protons obeying the gen mechanics. That elementary particles (or aggregates of particles as rigid bodies) show two distinctly different types of behavior of large numbers of them are described by the methods of statis tical mechanics is a quantum mechanical phenomenon which has no analogue in ''lassical theory. It must suffice here to say that the two types are characterized by the number of particles that can be assigned to a given stationary state of an indi vidual particle. When this number is 1, as for electrons governed by the exclusion The statistical

hypothesis eral laws of quantum that can be treated »hen the properties


fans the

When an unlimited principle, the statistics is said to be Fermi-Dirac (FD). of particles can be assigned to a given state, as is the case for photons, the is said tively,

number statistics These two types of behavior correspond, respec to be Bose-Einstein (BE). to the description of systems of particles by antisymmetric or symmetric wave

functions.

well known that protons, like electrons, obey FD statistics, and there is con Composite (though less direct) evidence that the same is true of neutrons. nuclei are observed t to obey one or the other type of statistics, with the general rule that nuclei with half-integral spin obey FD statistics and those with integral spin obey BE statistics. This is completely in accord with the predictions of quant um mechanics if the nucleus is assumed to consist of neutrons and protons both obeying FD statistics. It would be in disagreement, however, with a hypothesis that nuclei are composed of protons and electrons. 2.22 Nuclear Forces. The current picture of the nature of nuclear forces has been developed partly from consideration of the over-all properties of the stable nuclei and partly from a detailed study of processes (mainly scattering) which involve only two nucleons. It is possible here to give only a brief sketch of the resulting theory J which is far from having reached a stable form. Saturation. There are two striking facts which in themselves give qualitative information about the general characteristics which nuclear forces must have: The average density of matter is about the same for all nuclides as noted in Art. 2.16 saturation of nuclear density), and the binding energy per nucleon, except for the lightest elements, is also very nearly constant (saturation of binding energy). The saturation of density is an indication that nucleons are packed into a nucleus in someIt

is

vincing

t It is actually the symmetry of the wave function, as determined from the band spectra of molecules, that is "observed" in most cases. For a complete account of this and other methods see Ref. 47. ♦For a simplified treatment of nuclear forces see Ref. 50, chap. 10.

NUCLEAR PHYSICS

[Sec. 4

if

a

and

4.

3,

rhapn.

2.

See Ref.

8.

is

is

it

a

is

a

is

it

if

is

it

is

a

it,

what the same way that atoms are in a liquid or solid with about a constant average distance between particles. This is in contrast to the open-shell structure of the elec trons in an atom. The saturation of the binding energy is also a property suggesting strongly that "nuclear matter" is in some way analogous to ordinary matter, for it implies that the total energy of a nucleus is proportional to the number of nucleons it contains just as the internal energy of matter is proportional to the number of atoms. There is further implication that the nuclear forces are "short range," that is, a nucleon by and large interacts only with its nearest neighbors, for if each nucleon interacted with every other nucleon in a nucleus, the binding energy of the nucleus would be more nearly proportional to /Is than to .4. Electrostatic forces varying as the inverse square of the distance would, for example, lead to an energy accurately proportional to the square of the number of charged particles (as is true for the contribution to the Such forces are called "long-range binding energy of the repulsion between protons). forces" in contrast to the specifically nuclear attractive forces which must fall off much more rapidly than the inverse square of the distance between nucleons. There is an additional property of nuclear forces strongly Charge Independence. suggested from a consideration of the binding energies and stability of nuclides; namely, the nuclear force is the same between any pair of nucleons. This hypothesis of charge independence means that the interaction is the same for a neutron-proton pair as it is for a neutron-neutron pair or a proton-proton pair, except that for a protonproton pair the electrostatic force of repulsion must be added to the nuclear force. Evidence for the equality of n-n and p-p forces is to be found in the fact (cf. Art. 2.13) that the stable lighter nuclei contain nearly equal numbers of neutrons and protons. The argument is that if p-p forces were much larger than n-n forces, nuclei with an excess of protons would be energetically favored and more likely to occur than those with equal numbers of neutrons and protons. Similarly, an excess of neutrons would That an excess of neu be expected if the p-p force is much less than the n-n force. trons is actually observed, increasing steadily with Z for Z > 30, is accounted for by the electrostatic repulsion between two protons which effectively reduces the attractive This argument may be made at least partly quantitative. Evidence for p-p force. the equality of the n-p with the n-n and p-p forces can be obtained from a study of the binding energies of some of the lighter elements but is more convincingly demon strated by a direct comparison of the n-p and p-p scattering. t In summary, then, it may be said that the saturation of nuclear density and binding energy requires that nuclear forces be of short range compared with electrostatic forces and that there is considerable evidence in favor of the simple hypothesis that the basic law of force is the same for any two nucleons. The Law of Force. The correctness of the above simple qualitative picture of the character of nuclear forces can be proved or disproved only by the success or failure of a quantitative theory built on and the first step toward such a theory should be the establishment of a law of force between nucleons. Unfortunately, the construction of such quantitative theory has proved extraordinarily difficult, so that at the present time cannot be definitely said whether such a simple picture adequate or radically new ideas must be introduced in dealing with nuclear matter. It abun dantly clear, however, that there exists fundamental law of interaction between of nucleons, very complicated nature compared with the law of gravitation or of electro-magnetic interaction. In the attempt to work out law of force by trial and error a basic difficulty encountered at the beginning: The simple two-body problems, which are the only ones that can be handled mathematically, prove to be insensitive (with the present accuracy of the experiments) to all but gross features of the law of force. Thus the deuteron should occupy the same position in nuclear theory as the hydrogen atom does in atomic theory, but this does only to a limited extent. If a law of force between assumed, its binding energy can be computed and compared neutron and proton with the observed value of 2.22 Mev. However, many laws of force can give the same result for the binding energy and also correct values for other properties of the deu teron. An assumption dictated by simplicity therefore commonly made; namely, t


4-46

or Ref. 48.

Art.


2]

4-47

between neutron and proton has the form of a "square well potential" as in Fig. 17. The law of force is then characterized by two parameters, the The binding energy and other properties of range b and the depth of V0 of the well. the deuteron are insensitive to further details of the shape of the potential energy the force

shown

curve.

With the above assumption as to the form of the potential the only statement that on the basis of the binding energy of the deuteron alone is that radius and well depth are related by the equation can be made

b*Va

1.02

X 10-"

Mev-cm«

(82)

information can, however, be obtained from the interaction between free and protons as manifested in scattering experiments, and this shows that the value of b is approximately 2 X 10~l> cm, which cor responds to Vo ~ 25 Mev. This attractively simple picture of nucleon interaction has, however, proved quite definitely to be inadequate for a complete description of the properties of even twoIt is necessary to complicate the law of body systems. force in at least two respects. First, there is clear-cut evidence of a spin dependence of the nuclear forces. That 0 the force between two nucleons different when their spins are parallel (triplet state) from what when they are antiparallel (singlet state). This effect plainly evident in the scattering of slow neutrons by protons. The triplet and singlet potentials do not differ greatly, but the difference sufficient to rule out the existence of stable singlet ground state for the deuteron, so that its binding energy depends only on the triplet potential. the existence of Second, quadrupole moment of the Flo. 17. The square well deuteron requires a further complication in the law of potential for the force force, since indicates between a neutron and a nonspherical distribution of proton. electric charge in the ground state. This can be quanti tatively accounted for only the n-p force assumed to depend on the orientation of neutron and proton spins wit respect to the line joining the two particles. The deviation of such forces Such forces are called tensor forces. however, from central force in which the potential depends only on distance quite small. spin prescription can be found for a law of force between two nucleons which dependent and slightly noncentral and which will reproduce all essential features of nuclear two-body problems, including the fact that there exists no bound state of two protons, the coulomb repulsion between them overbalancing the nuclear force of attraction. unable However, this law of force suffers from the fatal defect that to account for the saturation of density and binding energy to be found in more com plex nuclei, for although an accurate calculation of the binding energy of heavier nuclei not possible, can be shown quite convincingly that a collection of nucleons with forces between each pair of range will collapse to a sphere of approximate radius 4/2 and that its binding energy will be proportional to A1. In actual fact the binding energy proportional to A, corresponding to the constancy of the binding energy per particle, and the heavier nuclei have radii considerably larger than half of the value determined from the two-body problems. In order to explain the phenomenon of saturation, additional complications in the nucleon law of interaction are thus required. possible approach to the problem of saturation which still remains the preferred one was proposed first by Heisenberg It based on a somewhat vague shortly after the discovery of the neutron in 1932. analogy to the theory of (homopolar) chemical binding which also shows the phe that the forces between nomenon of saturation. The new assumption required This type of force has no analogue in ordinary classical nucleons are exchange forces. Additional

is

is

is

a

is,

i

is

if

a

it

a

a

it

b

it

is

is

is

is

A

is

A

of 6


is

it

is

is,

neutrons

NUCLEAR PHYSICS

[Sec. 4

is

B

a

is

is,

is,

mechanics, although it does appear in a quantum-mechanical computation of the The binding forces in a composite system such as the hydrogen molecule ion (Hs+). characteristic of exchange forces is that they can be attractive or repulsive depending on the symmetry of the wave function describing the motion of two particles relative to each other, f It is just this possibility of the occurrence of repulsive forces that can prevent a collapse of heavier nuclei. With this more general type of force it is possible to give a qualitative explanation of saturation without its being possible, how ever, to decide uniquely between various forms which the prescription for the exchange forces can take. While the introduction of exchange forces can be made without appreciably influ encing the law of force which accounts for the binding energy of the deuteron and the scattering of neutrons and protons at low energy ( < 10 Mev), this is no longer true for scattering experiments with particles of much higher energy, such as the 350-Mev pro tons from the Berkeley cyclotron. In such experiments exchange effects should be apparent even in two-particle systems, but this has not been borne out by the observed facts. It can thus be questioned whether exchange forces are actually the true expla nation of saturation. On the other hand, the meson theory of nuclear forces (cf. Art. 2.33), which constitutes at present the only approach to nuclear forces other than the empirical one, does lead to forces of exchange type, although their exact calculation is still impossible because of technical difficulties. 2.23 Although The Liquid-drop Model and the Semiempirical Mass Formula. appreciable progress has been made toward an understanding of nuclear forces through a phenomenological approach like that of Art. 2.22 (as well as through a direct theo retical approach in the meson theory), the problem of computing the properties of In complex nuclei from the nuclear forces is only beginning to be seriously attacked. the meantime relatively crude models for the behavior of nuclei have proved successful at least in coordinating nuclear data. One of the simplest and most useful of these The central idea in this approach is that the forces between is the liquid-drop model. nucleons resemble those between the molecules of a liquid in that they are forces of attraction extending only to a relatively small number of near neighbors. The prin cipal justification for such a viewpoint lies in the fact that the binding energy of nuclei is in first approximation proportional to the number of particles A rather than to ,4' Furthermore, the approximate constancy of the den (saturation of binding energy). sity of nuclear matter is a parallel to the incompressibility of liquids, and a part of the deviation of the nuclear binding energy from proportionality to A can be interpreted That is, the nucleons on the as analogous to the surface energy of a drop of liquid. surface of a nucleus are attracted to roughly half as many other nucleons as are those in the interior. This surface effect results in a contribution to the binding energy of a nucleus which is proportional to the square of the radius (that to A5i) whereas There the main contribution to A). real evi proportional to the volume (that dence for the occurrence of such surface term in the observed binding energies oi nuclei. The liquid-drop model suggested by such simple qualitative ideas has been elabo rated in an attempt to reproduce the general features of nuclear energy levels as analo Certain aspects of nuclear fission also find a simple gous to surface waves on a drop. interpretation in terms of the liquid-drop model. The general ideas embodied in the The Semiempirical Rinding Energy Formula. liquid-drop analogy as outlined above suggest that the main contribution to the bind of a nuclide containing a large number of nucleons can be written in the ing energy form a\A — ajA*5. The first term represents the volume energy, and the second its reduction due to the lower binding of particles on the surface. An empirical determi However, nation of ai and ai could be made by fitting the observed binding energies. If

Exchange forces are formally introduced into quantum mechanics in the following faslucra. u(ri.ri:
t


4-48

Art.

4-49


2]

depend on Z us well as A, and with the aid of further general ideas elaborate formula to represent the binding energies of any combination of neutrons and protons can be constructed. This takes the form the binding energies

a more

- atA* - a,

~

- a, Z(Z~

2Z)'

1}

aui

A"

A

=

-(A

B(A,Z)

+ S(A,Z)

(83)

third term corresponds to what It generally called the "symmetry effect." account of the empirical fact that stable nuclei tend to have equal numbers of — 2Z neutrons and protons, for the number of unpaired neutrons and the whole term represents a diminution in binding energy that increases with deviation from The exact form used, including the inverse proportionality neutron-proton equality. to The fourth term represents the effect of the has some theoretical justification. coulomb repulsion between the protons. It is, in fact, simply the classical electrostatic proportional to A** [cf. Eq. (79)]. energy $iZ{Z — \)et/R with the nuclear radius Finally, the term d(A,Z) in Eq. (83) takes account of the so-called "pairing energy." As noted in Art. 2.13 the even Z-even A' nuclides are more stable than those in which either or N or both are odd so that there can be at least one unpaired nueleon. The lowering of the binding energy for unpaired nucleons can be attributed to the operation the exclusion principle in forcing such nucleons into states of higher energy than those occupied by pairs. This effect introduced empirically into the binding energy formula by assigning to the following values: is

The

S

is

of

Z

R

A,

A

is

takes

135

0 0

A

S S

=

Mev

(A even),

=

1

-

.

OK

—-

Mev

A

N

=

i

Odd Z-odd

— ——-

o

Even Z-odd .V (A odd), Odd Z~even N (A odd),

In

this way nuclei with no unpaired nucleons (even Z-even Ar) are raised and those two unpaired nucleons (odd Z-odd AO are lowered in binding energy compared with those with only one unpaired nueleon (A odd). determination of the constants! in Eq. (83) based on a comparison with the observed binding energies and a certain admixture of theory leads to the expression

-

14.0A

-

13.1AW

-

18.1

(A ~ 2Z)A

B(A,Z)

-

0.584

Z{Z ~

X)

A

with

A"

+ «(A,Z)

(85)

+

M

4- 0.0141A*

0.021

^-2Z)' A

- 0.99395A - 0.00084Z

-

+

A'*

1)

i

B

M(A,Z)

is

in Mev and has the values given in Eq. (84). The energies of the nuclides are also commonly expressed in terms of their masses rather than binding energies, the two being related by ICqs. (76). When converted to give the masses M (A,Z), Eq. (85) becomes where

_

f(AiZ)

m

d

is

= — 0.

For a complete

discussion

see Ref. 16, chap.

it

A

B

i

is

= 0.1 43 /A amu for even Z-even in atomic mass units and N, zero. 145/A amu for odd Z-odd A' and otherwise The above formulas reproduce only the general features of the "energy surface" or M as a function of and Z. In particular, representing does not include in any way the peculiar stability of nuclides with "magic numbers" (cf. Art. 2.3 Iff.) of neutrons or protons. 2.24 The proton and neutron magnetic moments Magnetic and Electric Moments. have been measured with high precision. The values found, expressed in nuclear

where

t


Even Z-even N (A even),

11, Art. 3.

4-50

NUCLEAR PHYSICS

magnetons (cf. Art. 2.10), are

Mp =

M. =

[Sec.

4

2.7934

-1.9135

is

a

it

is

is

the deuteron. The large volume of information on the magnetic and electric Complex Nuclei. moments and spins of both stable and radioactive nuclides which now available Their theo exhibits many significant regularities which cannot be reviewed here.t retical interpretation still largely qualitative but has proved extremely valuable in For example, suggesting suitable models of nuclear structure. seems possible to single attribute the moments of odd-yl nuclides to the orbital motion and spin of nucleon, the others being paired in a "core" of zero spin and magnetic moment. The observed spins and magnetic moments are then found to be intermediate between the values that can be computed for the parallel and antiparallel alignment of spin and orbital angular momentum of the odd nucleon. The deviation from the extreme values thus computed (the so-called Schmidt lines; see Ref. 16, page 157) can be attributed to a modification of the magnetic moment of the nucleon which qualita tively understandable on the basis of meson theory. 2.3

Further Developments in Nuclear Theory

8,

2,

is

it

is

is

2.31 The Shell Model. As pointed out in Art. 2.13 there evidence for particular stability of nuclides with certain numbers of neutrons or protons. This fact suggests that closed "shells" occur in nuclei as well as in atoms, and while a direct approach through the theory of nuclear forces as outlined in Art. 2.22 does not suggest in any way an analogy between nuclear shells and atomic shells, the evidence for the signifi cance of the magic numbers so overwhelming that has justified development of a theory of nucleon orbits based on ad hoc assumptions. The Magic Numbers. The complete sequence of magic numbers 14, 20, 28, 50, 82, 126

is

and the "magic properties" are found for any nuclide in which the number of neutrons or the number of protons one of this sequence. The empirical evidence! for the 4

5.

».

a

t

4,

For a condensed accotint see Ref. 1G, chap. Arts. and where the theoretical discussed. For more detailed account of the empirical evidence see Ref. 42, chap.

J


is

a

(I

a d

s

a

0) is,

(/

where the minus sign for the neutron moment means that it is in the opposite direction These moments are often referred to as "anomalous," since, if the proton to the spin. and neutron were elementary particles obeying a Dirac relativistic wave equation as the electron does, the moments would be 1 for the proton and zero for the neutron. Nuclear theory is at present unable to give a complete and quantitative explanation of this anomalous behavior, but it is qualitatively understandable on the basis of the meson theory of nuclear forces (cf. Art. 2.33ff.). The Deuteron. The measured magnetic moment of the deuteron is 0.8576 nuclear magneton. On the basis of the picture of the deuteron as made up of a neutron and proton with their spins of \$ aligned to give a resultant of 1 and in a state of zero orbital angular momentum, the moments would be expected to add to give a resultant of 0.8799. The observed value is about 2 per cent less than this. Although small, this deviation from additivity is greater than the experimental error and is attributable This to a tensor component (cf. Art. 2.22) in the force between neutron and proton. state then, not exactly force central force, and in addition to the predominant = 2). = The state of the orbital motion, there will be a small admixture of orbital motion, then, contributes slightly to the magnetic moment, which accounts for the deviation from additivity of the magnetic moments due to nucleon spin. The deuteron 1ms also small quadrupole moment (Q = 0.273 X 10~" cm1) which This can again be attributed to a tensor was first detected and measured in 1939. component in the neutron-proton force and quantitatively accounted for by the same prescription for the force as that which gives the observed magnetic moment of

interpretation

is

Art. special


2]

significance of these

categories: 1.

Binding Energy.

numbers falls principally

In addition to the

general

4-51

under the following

three

evidence from relative abundance

(see Art. 2. 13) a detailed study of the binding energy as a function of Z and Ar shows that among the heavier elements the binding energy is generally higher by 1 to 2 Mev for a magic nuclide (Z or A' a magic number) compared with its immediate neighbors. " " Among the lighter elements the abnormally high binding energy of the doubly magic nuclei {Z and N both magic) is particularly striking. The most conspicuous examples 2), Ol»(Z N = 14). are He'(Z N N 8), and Si"(Z 2. Alpha and Beta Decay. When the end product of a natural radioactive decay is a nuclide containing 82 or 128 neutrons, the emitted aortf particles have abnormally corresponding to the high binding energy of the daughter nucleus. high energies, the emitted particles have low energy when the parent nuclide is Correspondingly,

- -

-

-

-

magic.

Capture. The singular properties of the magic numbers are very clearly in the fact that nuclides with Ar = 50, 82, and 126 have cross sections for cap ture of fast neutrons (energy about 1 Mev) that are lower by as much as a factor of 50 than those of neighboring nuclides. There are similar anomalies for magic proton numbers and for slow neutrons. The evidence from the dependence of fast neutron cross section on the neutron number is the more significant, however, for reasons 3. Neutron

shown

on page 4-99. While the gross features of the dependence of nuclear binding energy Shell Theory. on composition (as well as many of the general aspects of fission (Art. 4.3) and nuclear reactions (Art. 4.23) suggest a picture of the nucleus which likens it to a drop of liquid, there is equally strong evidence from the properties associated with the magic numbers for a picture in which individual nucleon orbits play an important part, for it has proved possiblet to account not only for the particular stability of magic nuclei but also for many other properties by means of an independent-particle model which it seems impossible to reconcile in any simple way with the liquid-drop picture. The starting point of this so-called "shell theory" is the assumption that each nucleon moves independently of the others and is acted on by strong forces only near the surface. This situation is represented by a "potential well," that by a potential constant out to the nuclear radius and then energy, the same for each nucleon, which increases rapidly by some 30 to 40 Mev. There will then be quantum states for each nucleon, and each (nondegenerate) energy level can be occupied by two neutrons and two protons of opposite spin direction in accordance with the exclusion principle. As for electrons in the approximately coulomb field of an atom, closed shells will occur when certain numbers of nucleons have been added; that is, the levels are arranged order of increasing energy (decreasing binding energy), they will occur in groups approximately the same energy with relatively large energy differences between different groups. These groups of energy levels constitute the "shells" (like the K, M shells of atoms), and nuclei in which shell just filled will have particular stability, since an additional nucleon (of the kind occurring in the closed shell) will have lower binding energy. The filling up of the shells should thus occur at numbers neutrons and protons equal to the magic numbers. The ordering of the energy levels in potential of the kind described above shows well-defined shell structure with the numbers of particles in closed shells quite dif ferent from those for electrons in coulomb field. However, the occupancies of the closed shells are not those corresponding to the magic numbers except for the first three shells. The proper ordering and occupancy can be attained only by the addi tional ad hoc assumption of strong spin-orbit coupling, by which meant that the interaction between the spin of a nucleon and its orbital motion provides an energy which comparable to the energy differences between orbits of different angular is,

is

if

by

a

a

a

is

of in

L,

a

of

is

is

a

a

»

is

momentum. In this ordering of the levels the energy determined by three quantum numbers, (number of radial nodes), the usual orbital angular momentum quantum

The basic ideas of shell theory were developed independently and in almost identical form by Haxel. Suess and Jensen. M. G. Mayer. Feenberg, and Nordhctm, in 1949. For detailed treatments uf tbe theory see Rcfs. 17 and 42. t


discussed

4-52

NUCLEAR PHYSICS

[Sec.

j

4

number /, and the total angular momentum quantum number which can have only the two values I + \$ and I — \i. Each such level is 2j + 1-fold degenerate and can thus be occupied by no more than 2j + 1 neutrons or by 2j + 1 protons. Agreement of the occupancy of the shells with the magic numbers requires the additional assump = I + tion that a state with more (spin parallel to orbital angular momentum) — — strongly bound than the state with and the same and For the existence of strong coupling with this property there has recently been found independent experimental evidence in the scattering of nucleons by light nuclei (cf . Ref. 42, page 54). The quantum numbers of the orbits making up the closed shells are shown in Table together with the occupancies of the shells. a

Energy Levels in

Rectangular Potential Well with Strong Coupling

that of decreasing binding energy, although within a given shell this ordering

(

2

X H H H %

2 2 a 3 1

"«

2

H

2

H

0

M

Hi

>«A

%

l»,4

H

Vjl

H

50

5

H

20

2pH

5

4

14

28

3

4

\dH

3

1

K

Number of particles up to and including shell

VH

2dH 82

3p?4

"ti

1

(!

VI

1

, 1

1

1

3

2sH

H H H

2

IV

14*

m

,2

i

Ilia

iB not

established.

H

1

C i

III

clearly

Spectroscopic symbol (>■/,)

1

Ha

W

0

I

II

V

126

1.

j.

I

a

The assignment of individual-particle states to the nucleons in nucleus which forms the foundation of the shell theory not only takes account of the preferred position of the magic numbers as corresponding to closed shells but has proved unexpectedly suc cessful in providing an interpretation of many other properties of stable and radio active nuclides. Thus, with the assumption that the angular momenta of an even number of like nucleons will nearly always combine to give zero resultant angular momentum, the spins and magnetic moments of odd-.4 nuclides can be attributed to a single odd nuclcon in a state of given and The observed spins of the stable nuclides are, with a few exceptions, correctly accounted for in this way, confirming the assignment of The observed values for the odd nucleon as given in Table magnetic moments are also in nearly all cases in qualitative agreement with the values to be expected from the and values of the single odd nucleon. /

j


Shell designation

*

always

8

The ordering

is

Table

1.

1

/.

v

\^

I

j

is

J-2

j

Art.

2]

ELEMENTS

OF NUCLEAR THEORY

4-53

There have also been striking amount

successes of the shell model in interpreting a large of experimental information on 0 decay and nuclear isomerism (cf. Arts. 4.12

and 4.13).

The interpretation of quadrupole moments as due to a single odd proton, when this by the shell model, has not met with the same success as has the cor of the magnetic moments. The observed quadrupole responding interpretation moments are much too large to be explained in this way, indicating that there is a significant contribution from the "core" of nucleons in closed shells. is indicated

2.32 The Unified or Collective Model. In spite of its spectacular success in accounting for many characteristic nuclear properties the independent-particle model of shell theory cannot be accepted as a complete picture, for at present the only approach to an understanding of the general features of nuclear reactions (cf. Art. 4.23ff.) and of nuclear fission (cf. Art. 4.32ff.) is through the serniclassical liquidwhich appears in principle to be irreconcilable with the existence of drop model, single nucleon orbits in the nucleus. Although the paradox which this situation presents has by no means been resolved, there have recently been attempts to con


struct a

"unified" modelf combining features of the two extreme pictures.

The unified theory may be thought of as a modification of shell theory in which the force on an individual nucleon is no longer represented by a static potential well but as an interaction with a "core" which contains all or nearly all the other nucleons and is capable of oscillations and deformations known as collective motions. In this model there is a coupling between individual particle motions and collective motions which is similar to that existing in molecules between electronic motion and vibration and rotation of the system as a whole. In such a theory closed shells are no longer spherical but can be deformed by interaction with nucleon orbits. necessarily They can thus contribute to magnetic and electric moments, and in this fact an explanation can be found for the large quadrupole moments which were anomalous in the original shell theory. The unified model has also been conspicuously successful in correctly predicting the occurrence of low-lying excited levels of nuclei that can be attributed to rotations of the core. 2.33 The Meson Theory of Nuclear Fprces. Much information about the general characteristics (short range, charge independence, spin dependence, etc.) of nuclear forces may be extracted from the available experimental material on nviclei in the manner briefly described in Art. 2.22. In addition to such an empirical approach there exists a still imperfect but rapidly developing theory which it is hoped will ultimately be able to deduce the laws of nuclear forces from general physical prin The starting point of this "meson theory" of nuclear forces is a relatively ciples. The well-understood electromag simple idea first formulated by Yukawa in 1935. netic forces between charged particles (which are long-range forces, varying as the inverse square of the distance between particles) may be thought of as resulting from the emission of a photon by one particle and its absorption by another, although this is not the simplest viewpoint in the limit of static forces. The essential idea of the Yukawa theory is that a kind of interaction, different from that of electro'lynamics, can be obtained if the photon in the above picture is replaced by a particle In such a theory having, in contrast to the photon, a rest mass different from zero. the electromagnetic field associated wit h photons is to he replaced by a field associated with particles of finite mass (and possibly charge) which were later called mesons. A plausible form for the new law of force was obtained by Yukawa in the following way. The behavior of photons in free space is governed by an equation of the form

W-iÔ for a component

v>of the

electromagnetic potential (cf. Art. 1.11).

(87)

Such an equation

A brief review of t The moat extensive work in this field is that of Aage Bohr and his collaborators. the theory with references to the original literature is to be found in chap. 7 of Ref. 4. See also Ref. 31 " " " " for a parallel treatment. has been used by Bohr, collective model unified model The terminology ''S Wheeler.

4-54

NUCLEAR PHYSICS

[Sec.

4

is analogous to the Schrodinger equation for material particles with v playing the role of a wave function for photons. It may, in fact, be derived from the relation E* = csp* between energy E and momentum p for photons by the standard sub

An stitution (cf. Art. 1.32) E— > (—h/i)d/dt, p— » (ft/j')V of quantum mechanics. equation analogous to Eq. (87) but applicable to particles of finite mass can be obtained by the above procedure applied to the energy-momentum relation E1 = Mlc* + c*p' The equation thus obtained [cf. Eq. (15)] appropriate for a particle of finite mass M. is readily seen to be (88

a

is,

which may be interpreted as the wave equation or field equation for mesons of mass M. Comparison of Eqs. (87) and (88) now gives at once the suggested change in the law of force, for the solution of Eq. (87) which corresponds to a point charge e at the in the electrostatic limit, ipo — e/r, with origin interpreted as the electrostatic The meson equation (88) has potential. precisely analogous solution 4>a

= ge-'''>

(8!)

is

a

it

of

is

is

is

it

a

is

is

is

it

a

of

a

is

it

3

||

a

§

interaction remains to be established. The meson theory also affords at least qualitative explanation of the "anomalous" magnetic moments of the neutron and the proton, attributing the deviations from the "normal" values (cf. Art. 2.21) to the fact that nucleons are capable of emitting mesons whose motion contributes to the magnetic moment.

BASIC PARTICLES AND THEIR INTERACTION WITH MATTER is

used here to designate certain simple particles which The term "basic particles" have an independent existence outside atoms or are used as projectiles in the bombard ment of nuclei. Their general properties are assemblied in this article, together

2, 2,

7. 7,

t

The imrticles first discovered in cosmic rays (*i mesons) were later found to be different from those Cf. Art. 3. Iff. responsible for nuclear forces (» mesons). For a complete treatment see Ref. 7. vol. Art. 47, and Ref. 48. Art. 2F. See Ref. Art. 46. Sec Ref. vol. II 5 X


g

is

is

g

" an arbitrary constant. where It natural then to identify with the " potential con of a nucleon just as
Art.

BASIC PARTICLES

3]

an account of such of their interactions questions of attenuation.

with

3.1

4-55

with matter as are of importance in

General Properties of the Particles

The masses given here for the particles are in grams. The more common unit in physics is He of the mass of the Ou atom (the atomic mass unit, written The masses on this scale are recorded in Table 37 of Sec. 2. amu). Spins are given in units of h. Magnetic moments are given in terms of the Bohr atomic magneton, eh/2m,c = 9.273 X 10-S1 erg/gauss or the nuclear magneton eh/2mpC = 5.050 X 10~" erg/gauss, where m, is the mass of the electron and m, that of the proton. 3.11 The Photon. Mass: zero rest mass. Charge: zero. Spin: 1. Statistics: Bosp-Einstein. nuclear

are the particles associated with the electromagnetic field (radiation) in that material particles are associated with de Broglie waves or Schrodinger functions. Energy E and wavelength X are related by the equation

Photons the way wave

_

1.239

X

10"'

E is in electron volts and X in centimeters. The corpuscular properties of radiation become increasingly prominent as the wavelength decreases and the photon In the optical region (X roughly 10~4 to 10-6 cm) the photon ener energy increases. gies are a few electron volts; in the X-ray region (X roughly 10-6 to 10~10 cm) they Photons of energies above about range from a kilovolt or less to more than an Mcv. 1 Mev (X < 10~10 cm) are generally classified as y rays, at least when they originate in nuclear rather than electronic processes, t Photons are produced when atoms are raised to excited states by bombardment


where

particles or when charged particles are decelerated or accelerated. When with energies in the kilovolt range impinge on a target in an X-ray tube, both of these processes occur. The first leads to emission of lines of definite fre X rays); the second, due to quencies characteristic of the target atoms (characteristic deflection by atomic nuclei, leads to radiation with a continuous spectrum known as The bremsstrahlung has an upper limit in frequency given by hremsslrahlung. »o»i = E/h, where E is the energy of the electron. 3.12 -4.803 X 10-'° esu. The Electron. Mass: 9.108 X lO"18 g. Charge: Spin: Yi. Statistics: Fermi-Dirac. Magnetic moment: one negative Bohr atomic with

electrons

magneton. The electrons emitted by naturally or artificially radioactive atoms are called fi rays. It is also possible to produce elec They have energies up to several Mev. trons of these energies in the laboratory by means of high-voltage machines such as the betatron and the van de Graaff electrostatic generator. Charge: equal in 3.13 The Positron. Mass: equal to that of the electron. magnitude to that of the electron but opposite in sign. Spin: J^. Statistics: FermiDirac. Magnetic moment: one positive Bohr atomic magneton. Positrons were first observed in cosmic rays by C. D. Anderson in 1933, following a

It was later found that they could by Dirac. in matter by 7 rays with energies greater than 1 Mev and that they are emitted in the radioactive decay of a number of artificially produced nuclides. Positrons are unstable in the presence of electrons, an electron and a positron being This annihilation capable of annihilating each other with the emission of radiation. radiation, when the electron and positron are at rest and the electron is not bound in an atom, consists of two photons, each of energy m«c* = 0.511 Mev, emitted in opposite directions as required by conservation of linear momentum. Emission of theoretical prediction of their existence

be produced

t In general the distinction between X rays and y rays is on the basis of origin rather than energy, photons of relatively high energy from atomic transitions being clossi6cd as X rays and those from nuclear processes as y rays.

NUCLEAR PHYSICS

4-56

[Sec.

4

a single photon of energy 2mIc1 = 1.022 Mev is possible in the presence of a nucleus which can take up the recoil momentum. An electron and a positron attract each other and can form an unstable compound, known as positronium,t which exists for times of the order of 10~10 sec before annihila tion takes place. The occurrence and properties of positrons are currently Theory of the Positron. explained in terms of the Dirac relativistie wave equation for a free electron (see Art. 1.31). This theory requires the existence of states of negative kinetic energy less than — m^c8, the complete energy spectrum having the appearance shown in

It is assumed Fig. 18. There are no possible states between — mj? and +mecs. that normally all negative energy states are doubly occupied by electrons in accord ance with the exclusion principle and that only deviations from this normal state of affairs lead to observable consequences. A positron is interpreted as the hole left by the removal of a single electron from a negative energy state. It can be shown that such holes actually have the properties of positively charged electrons. Accord ing to this "hole theory," a minimum of 2m„cJ = 1.02 Mev is required to raise an electron from a negative to a positive energy level across the gap between — m«c2 and -\-mtC1. The result is the creation of an electron and a positron. This process is known as pair production. Similarly, the annihilation radiation is interpreted as a transition of an electron from a positive energy state into an unoccupied negative energy state with an energy release of at least 2»i,cs. POSITRON

Fig.

18.

1

Illustrating schematically the energy spectrum by a y ray of energy hv.

and pair production

—

ENERGY

(heavy line) of the Dirac electron

/3

is

J>2

In spite of its seemingly artificial character, the hole theory provides a satisfactory method for the calculation of details of processes involving positrons. The theory of the positron implies that it has a spin of and obeys Fermi-Dirac statistics. Charge: zero. 3.14 The Neutrino. Mass: rest mass zero. Spin: J.^. Statistics: Fermi-Dirac. Magnetic moment: none observed. The existence of a particle with the properties of the neutrino postulated in order to preserve the principles of conservations of energy, charge, spin, and statistics in decay (see Art. 4.12). It must be assumed to be emitted simultaneously with the particle and to be undetectable by ordinary means because of an exceedingly small The conservations laws are satisfied or nonexistent interaction with matter. the neutrino It not required assigned charge, spin, and statistics as given above. have zero rest mass, but measurements of the energy balance in decay showthat that any rest mass which less than 0.05 per cent of the electron mass. may have Any interaction of the neutrino with matter known, from both theory and experi ment, to be exceedingly small. has must be less than 10~7 magnetic moment, Bohr magneton, as shown experimentally by the absence of detectable inelastic col lisions between neutrinos and electrons. There should theoretically be small interaction of neutrinos with matter due to the process which the inverse of the 0-decay process; that is, a proton and a neutrino should interact to give a neutron and a positron. The cross section for this process can be estimated to be of the order of 10~M barn. Recent experiments by Cowan, Reines, and Harrison, using the strong neutrino source provided by the decay of fission fragments in reactor, seem to indicate that this disintegration of the proton actually occurs. 3.16 The Proton. Mass: 1.672 X 10~" g, which 1,836 times the mass of the electron. Charge: a positive charge equal in magnitude to that of the electron. Spin: Yi. Statistics: Fermi-Dirac. Magnetic moment: 2.793 nuclear magnetons. it

is

0

a

J

is

a

a

it

If

is

it

is

/S

it

is

is

if

0


9

—t"

ELECTRON

-I

tSco Ret. 13. tSoe Rrt. 12.

Art.

4-57

BASIC PARTICLES

3]

moment of 1 nuclear magneton might have been expected for the this would result from assuming it to be governed by a relativistic wave That the measured magnetic equation like the Dirac equation for the electron. moment differs so greatly from this value can probably be taken as an indication that in some sense the proton is a composite particle (cf. Art. 2.33). If negative energy states exist for the proton as they do for the electron, the absence Just of a proton in such a state should appear as a negatively charged proton. " antiprotons " have been created in the proton beam of the Berkeley recently t such Bevatron. As is to be expected theoretically, energies above 4 Bev (1 Bev = 10' ev) There is also experimental evidence for the occur are required for their production. rence of the annihilation process analogous to that for positrons and electrons. Spin: H- Sta 5.16 The Neutron. Mass: 1.67247 X 10-" g. Charge: zero. Magnetic moment: —1.9135 nuclear magnetons. tistics: Fermi-Dirac. The mass of the neutron is about 0.08 per cent greater than the sum of the proton and electron masses. The spontaneous decay of a neutron into these two particles is thus energetically possible and has been observed to occur with a half-life of 12.8 ± 2.5 min (cf. Art. 4.12 and Ref. 51). A neutrino must be assumed to be emitted simul taneously with the electron as in any other /S decay. Whether or not antineutrons exist, in the sense of the Dirac hole theory, is not at present known with certainty. The generic term nucleon is used to designate a particle that is eit her a neutron or a It is at present no more than a vague speculation that these two particles proton. will in some future comprehensive theory turn out to be different quantum states of the same particle. 3.17 The Deuteron. Mass: 3.344 X 10~" g. Charge: that of the proton. Magnetic moment : 0.8576 nuclear magneton. Spin: 1. Statistics: Bose-Einstein. The deuteron is a stable nuclide (an isotope of hydrogen of mass number 2) made up of a neutron and a proton whose parallel spins of l£ add to give a resultant spin of I. Its magnetic moment differs by about 2 per cent from the algebraic sum of the neutron and proton magnetic moments (cf Art. 2.24). It. is disintegrated The binding energy of the deuteron is 2.226 ± 0.004 Mev. into a neutron and a proton by 7 rays with energy greater than this binding energy. Conversely, a proton can capture a neutron with formation of a deuteron and emission A magnetic

of a photon.

and is prepared by isotope separation. 3.18 The Triton. Mass: 5.0078 X 10-" g. Charge: that of the proton. Spin: Magnetic moment: 2.97863 nuclear magnetons. Statistics: Bose-Einstein. The spins of the two neutrons The triton made up of two neutrons and a proton. cancel each other, as required by the exclusion principle for two equivalent particles The magnetic moment the same state, leaving from the proton. resultant of as to be expected, very nearly equal to that of a free proton. halfdecay into He* with The triton an unstable nucleus, disintegrating by life of 12.4 years. The binding energy of the triton 8.49 Mev. It can be known as tritium. The hydrogen molecule with two tritons as nuclei n — H3 + a. made from reactor neutrons by the reactions Li* 10"" g. Charge: twice that of the 3.19 Mass: 6.64424 The Alpha Particle. proton. Magnetic moment: zero. Statistics: Bose-Einstein. Spin: zero. It made up The a particle the nucleus of the helium atom of mass number 4. two neutrons and two protons whose spins cancel each other in pairs and has binding energy of 28.2 Mev. J.110 Mesons and Hyperons.J Extensive research in the last 10 to 15 years has revealed the existence of large number of unstable particles with from 200 to about 2,500 times the mass of the electron. All of these have plus or minus the charge of The hydrogen molecule with two deuterons as nuclei is known as deuterium

►

7

a

tSee Ref. 11. For more extended but atill elementary account of the properties Detailed theoretical treatments are to be found in Refs. and 41.

of mesons see Ref. 50. chap.

1 1 .

a

a

of

is

is

X

+

is

is

a

p

is

a

is, in

is

lj.

readily

t


proton, since


4-58

NUCLEAR PHYSICS

[Sec. 4

the electron or are neutral. Those with masses intermediate between that of the electron and a nucleon are known as mesons, and those with masses greater than that of /a nucleon are known as hyperons. The existence of a "heavy electron" having a mass of some 200 Mev was first postulated by Yukawa in 1935, as the result of speculations on the origin of nuclear forces (cf. Art. 2.33). Particles of about this mass were actually observed in cosmic radiation in 1936, and 10 years later it was shown that similar mesons could be artificially produced in the Berkeley cyclotron. As a result of experimentation using Wilson cloud chambers and photographic emulsions as detectors and cosmic rays and accelerators in the billion-electron-volt range as sources, it has now been found that a whole family of mesons and hyperons exists. These particles are in general unstable, decaying into each other and into electrons, neutrinos, and photons. An enumeration of all particles known at the present time, the number of which is still increasing, would be out of place here. It will have to suffice to give a brief account of some of the main results which have been well established in a rapidly changing field. The lightest mesons are of two distinct types: the ir mesons (positive, negative, and neutral) which interact strongly with nucleons and the ii mesons (positive and nega tive) which interact only very weakly with nucleons. The primary decay process for the ir mesons is into j» mesons for the charged particles and into photons for the neutral ones. The n mesons are themselves unstable, decaying with the emission of an electron. The process is similar to £f decay in a nucleus (cf. Art. 4.12ff.), the electrons having a continuous energy spectrum and the simultaneous emission of two The lifetime of x neutrinos being assumed to balance energy and momentum. mesons is 2.6 X 10~8 sec for the charged particles and 10~H sec for the neutrals; that of n mesons is 2.1 X 10-' sec. The ir mesons have masses of 273 for the charged and 264 for the neutral particles; the it mesons have a mass of 207, all expressed as multiples of the electron mass. The mesons first discovered in cosmic rays were n mesons. Those first produced artificially by bombardment of nucleons were x mesons. Beams of jt mesons are readily produced by the high-voltage proton accelerators Their operating at Brookhaven, Berkeley, Chicago, and Birmingham (England). interaction with nucleons, as mainly evidenced in their scattering by nuclei, is being intensively studied for the information which can thus be obtained concerning nuclear forces. It is believed that these forces are related to the emission and absorption of ir mesons, in very much the same way as electromagnetic forces are related to the emission and absorption of photons (cf. Art. 2.33). Since 1939 evidence of the existence of particles with mass intermediate between At least a that of the ir meson and the proton has accumulated in great profusion. dozen such particles have been identified in cosmic rays, and they are collectively known as K particles. There is no reason to assume that the list is complete. Rela tively few hyperons (also known as H particles) have so far been observed. Both K particles and hyperons have now been produced artificially with the 3-Bev protons available from the Brookhaven Cosmotron. 3.2

The Passage of Photons through Mattert

Photons interact with matter through the forces which the electric and magnetic sectors of a light wave exert on charged particles, either on electrons or on the pro tons in nuclei. The interaction with electrons is predominant in the attenuation of photon beams, since the heavier protons are relatively immobile and do not take up energy so easily as the electrons. The attenuation of light in the visible and ultraviolet is due mainly to absorption The absorption coefficient, as a function of photon of photons in atomic transitions. energy in this region will have many resonances, the positions of which depend on For high-energy photons (X rays and y rays), the the atom or molecule involved. t For a detailed treatment of the attenuation of both Ref. 6, or Rcf. 10, chaps. 21 through 25.

photons

and charged

particles

in matter-

stfe

Art.

4-59

BASIC PARTICLES

3]

which vary with the atomic number Z in a relatively way, the most important being:

attenuation is due to processes smooth

L The photoelectric effect Compton scattering 3. Pair production 2.

Each

of these results in the complete elimination of a photon from a collimated beam, is consequently attenuated exponentially in passing through matter; the intensity 7(j) at penetration distance x given by is

a

I(x)

N

(90)

the number of atoms per cubic centimeter, a the atomic cross section in = No, the linear absorption coefficient in cm-'. centimeters, and

60 PHOTON

IN

ENERGY

80

100

—

is

1

It

40

1

20

, I

p,

I

KMOi

120

KEV

L

Fig. 19. The photoelectric edge has a structure corresponding effect in lead. to The three closely spaced energy levels of the relativistic fine structure energy level with principal quantum number n — 2. the

convenient to write the exponent giving the attenuation in the form (j>/p)(px), where the density of the material, so that p/p, called the mass absorption coefficient, gives the absorption in terms of the amount of material px penetrated per In many cases p/p will be less dependent than on the nature square centimeter. p

p

is

frequently

the

K

shell

which

is

in

is

tion edges.

usually absorption by electrons

L,

a

• ■ ,

is

is

is

A

of

the absorbing material. brief description of the above three processes follows. S.S1 The Photoelectric Effect. The process involved here the absorption of a photon followed by the ejection of an electron from a bound state (or, in some cases, its removal to an unoccupied bound state). It the predominant effect when the quantum energy hv of the photon not too far removed from one of the binding M, . . . shell. The absorption of an electron in K, f-nergies, Ek, El, Em, ■ '•ross section as a function of frequency has the general appearance shown in Fig. 19 for lead, having a sharp maximum whenever hv = Ek, El, Em, . . . and falling off rapidly (approximately as r~*) beyond each of these maxima, which are called absorp

It


7(0) e-f*

is

where square

-

= 7(0) e~s"

I

that

is,

and the beam

the most

important

4-60

NUCLEAR PHYSICS

[SEC. 4

This absorption edge has approxi photoelectric process for the attenuation of y rays. mately the energy EK = (Z 1)' 13.5 ev (91) is,

-

for an element of atomic number Z, that the square of the effective nuclear charge times the energy of the first Bohr orbit in hydrogen. Quantum mechanics provides a fairly accurate calculation of the atomic absorption sk far from such K absorption edge, giving zb /wy* „,-*4V2— (_

(92)

=

(—

= 0.6651

J

„

where

^

)

,

,-n

X 10-" cm'

(93)

\

a

is

the classical cross section for scattering of light by free electron, known as the Thompson cross section. Large corrections must, however, be applied to this formula near the absorption edge and when relativistic effects are of importance. 3.22 As the photon energy hv increases beyond the energy Compton Scattering. Ek of the K absorption edge of a given element, photoelectric absorption becomes of decreasing importance, since
INCIDENT

PHOTON

h»

h».

RECOIL ELECTRON

Fio.

20.

Directions and angles in the Compton effect. is

order of m,c' = 0.511 Mev, since Ek even for the heaviest elements no larger than about 0.11 Mev. The well-known classical calculation of the scattering of light by a free electron gives for the scattering cross section per electron the Thompson cross section of Eq. (93) in the nonrelativistic case (hv The quantum mechani m^»). cal treatment of this process leads to the same cross section in the nonrelativistic case but requires that the process be one in which energy and momentum of the photon and electron are conserved. That is, the characteristic corpuscular properties of the photon are involved and the scattering must be treated as two-body-col lision process (cf. Art. 1.13). As a consequence of the conservation laws, the energy and frequency of the photon are diminished (Compton effect) by the encounter and kinetic energy imparted to the electron. The frequency of a scattered photon originally of frequency va which has been scattered through an angle given by (cf. Fig. 20)

«

A\

in wavelength

A\

=

— COS

WljC*

0)

«(1

(94a)

by

Ml

- cos

For a simplified derivation aee Ref. 50, p. 350.

6)

1

and the increase

+

0

is

v

is

a

«

t


y

»

Xc =

nigC

(946)

Art. The the

The kinetic energy T of

wavelength Xc is known as the Compton wavelength. recoil electron is

T and

4-61

BASIC PARTICLES

3]

,

= h(»

v)

= hvi, — (1

the angle of recoil, is related to cot

4>

6

—

2< coss

+

- t' cos2

«)»

(95)

by (1 -+- «) tan

(96)

The cross section for Compton scattering per electron is the Thompson cross of Eq. (93) and is independent of the frequency as long as relativistic effects are negligible, which means restriction to relatively low-energy y rays. The relativistic quantum mechanical calculation has, however, been carried out with use of the Dirac wave equation for the electron. There is no change in Eqs. (93) to (96), which are already relativistic, but the cross section is modified to section

„

-£<[-HlfB]«'*+»

+5

(97)

«

known as the Klein-Nishina formula. 1 it may be written in the For < form 0o[l — 2* + (26 /5«)2], so that it gives a reduction of ac below the Thompson cross section of nearly 20 per cent even for photon energies as low as 50 kev. For « ~ \,ac = 0.43<£o and ac continues to decrease steadily as the photon energy increases. which is


♦*-«£»}

The Compton cross section per atom, rather than per electron, is Zac. The number per cubic centimeter may be written as N — No(p/A), where A'o is AvoThe mass absorption gadro's number. is then n/p — No(Z/A)
y/

->/

ap =

Z*rW%

In 183Z-W

- %7)

hv

»

'l37m«c»Z-W

(98)

where r0 = e'/mj;* = 2.818 X 10-13 cm is the classical radius of the electron. 3.24 The Total Cross Section. Although there are a number of effects, t such as Compton scattering by the nucleus, which y rays produce in matter, their influence t For an enumeration of them see Ref. 6, sec. 3E.

4-62

NUCLEAR PHYSICS

[Sec. 4

is negligible and the total cross section can be taken as the sum of those for the three processes discussed above; that is,

on attenuation

c =

oph

+ Zac +

op

(99)

is

is

where apt, the cross section for photoelectric absorption, gives the total cross section per atom for removal of a photon from the initial beam. The linear absorption coeffi cient [cf. Eq. (90)] then ii = Ntr.

L

iooo

\\

\\

isA

z UJ

u

'

liu. UJ

z

OGE

N.

Pb

o

■

K

•' 1

10.

Al

v>

s

^

to

».oi

PHOTON

effect

ENERGY

IN

100

MEV.

The total mass absorption coefficient for a heavy element (Pb) and a light element Note that r/p about the same for both in the region around Mev where Compton predominant (cf. Fig. 21).

22.

1

is

Fro. (Al).

10

I

I

01

is

is

Z

The relative importance of the three partial cross sections will depend on the as indicated in Fig. 21. The variation with energy of the photon energy and on mass absorption coefficient n/p shown in Fig. 22 for some typical materials. 3.3

The Passage of Charged Particles through Matter

is

is

it

is

is

a

a

A

is

(a

is

a

is

The loss of energy by fast-charged particles in passing through matter due pri marily to two processes: (1) inelastic collisions with atoms resulting in ejection of electrons (ionization) and (2) deflection of the incident particle by nucleus with the acceleration of the particle giving rise to the emission of radiation (radiative collisions). For heavy charged particles particles, protons, deuterons), the second effect entirely negligible compared with ionization, since those particles arc too heavy to have many large deflections. For light particles (electrons, mesons) ionization predominant at lower incident energies, but radiative collisions become increasingly important as the energy increases. 3.31 neutral Heavy Charged Particles. fast, heavy particle passing close to atom loses only very small fraction of its energy in ejecting an electron from the slowed down almost continuously until atom. It practically undeflected and its velocity small compared with that of the atomic electrons, at which point can pick up an electron from an atom. The result that the incident particle is


o

\

\

£10

K '

sl

z o

o

EDGE

Art.

basic particles

3]

4-63

is,

The relation between energy stopped at a quite well-defined distance called its range. and range has been extensively investigated both theoretically and experimentally. Measurement of the range is a commonly used method of determining the energies of particles. The theory of the ionization loss by energetic heavy particles is relatively well under The basic formula comes from a quantum mechanical calculation by Bethe stood. the (1930) generalizing a classical formula due to Bohr (1913), which expresses the energy loss per centimeter of path in the form stopping power, that

dx

1

mtv'

is

a

is

is

is

is

/3

is

6,

Z,

a

Z

5

is,

is

is a

It

0

2irel nicV1

XZ

"^

- In

- p) -

0»]

_

(1

dx

[in

dE

is

it

a

is

«

is

(101)

The = v/c and the other symbols have the same meaning as in Eq. (100). where about at minimum broad has the of the particle function of energy energy loss as a

0

a

0


is

/

is

is

is

/

v

is

its velocity, N the number of atoms where Ze the charge of the incident particle, the average ionization potential per cubic centimeter of the medium traversed, and The formula valid only for nonrelativistic energies and when the of the atom. The range determined by integration of the energy loss not too small. velocity proportional to the mass of the incident particle, whereas the per centimeter and stopping power (—dE/dx) depends only on its charge and velocity and the ionization may be approximately The average ionization potential potential of the medium. constant to be deter usually treated as calculated by quantum mechanics, but mined by experiment. The stopping-power formula of Eq. (100) gives only the average energy loss per Individual particles will deviate statistically from the average, resulting centimeter. in a distribution of the ranges at which the particles are completely stopped. This readily observed in cloud-chamber photo spread in the individual ranges, which known as straggling. graphs, remarkable experimental fact that the energy loss per ion pair (ion + elec very nearly independent of the energy and charge of the incident charged tron) It has the approximate value of 30 to particle and of the nature of the medium. 35 cv per ion pair for a particles, protons, and electrons, a fact for which there no simple theoretical explanation. of course, valid only as long as the The stopping-power formula of Eq. (100) Corrections due to capture incident particle remains stripped of all its electrons. and loss of electrons from the atoms of the medium are important for a particles Mev and for protons below 1.25 Mev. The effect increases with the atomic below nucleus with a larger charge captures elec number of the incident particle, since the capture and loss For fission fragments with relatively high trons more easily. The theory of this process has been extensively of electrons are predominant effects. Sec. 14). too involved for brief summary (cf. Ref. developed but rays The loss of energy of 3.32 The Passage of Beta Rays through Matter. in matter due almost exclusively to two processes: ionization of atoms as dis cussed above for heavy particles and emission of radiation accompanying deflection of the particle. 0.5 Mev) Eq. (100) qualita Ionization Loss. For nonrelativistic energies (E tively correct for electrons as well as protons, although there are minor corrections (reduced mass of the electrons, identity of incident and recoil particles) which amount to some 10 per cent only, since they result in small factors in the argument of the For higher energies, however, relativistic effects are of importance and logarithm. few Mev, whereas for protons they are unim in fact become predominant above The relativistic form of Eq. (100) region. portant except in the billion-electron-volt sufficiently somewhat complicated and contains some minor uncertainties, but well approximated by the expression

NUCLEAR PHYSICS

4-G4

[Sec.

4

t

3

S

5

For higher energies it rises slowly as illustrated in Fig. 23. 1 Mev in all substances, with the term In (1 — fl3) predominating as 0 — > 1. Deflection by atoms without loss of energy (elastic scattering) is much more important for electrons than for heavy charged particles, and the path of a 0 par „~6 ticle through a thick absorber will be 5 The thick zigzag rather than straight. 0 ness of absorber required to stop a 0 ray is thus not simply related to the path length to zero energy obtained by integration of Eq. (101)4 o The deflection of elec Radiation Loss. trons by atomic nuclei (and to a slight extent by atomic electrons) gives rise to the emission of radiation (bremsstrahlung). The rate of energy loss due to this process Â IR o is proportional to the acceleration [cf. Eq. " *-A and hence to Z1, whereas ionization (5)]. CTTT loss is proportional to Z. The radiation IONIZATION LOSS energy loss per centimeter is proportional or RADIAT 0NL0SS_-J to the energy of the incident particle, while ionization loss increases only logarithmi cally with energy. Consequently, radia KINETIC ENERGY OF ELECTR0N.MEV tion loss will become greater than ionization Flo. 23. Energy loss by ionization for loss above a certain energy which will be rays in air, aluminum, and lead. The lower for high Z than for low Z. The /p) (dEldx), the energy loss ordinate at which the two are equal is called energy per unit surface density. The radiation the critical energy and is about 7 Mev in The loss also shown for comparison. lead, 47 Mev in aluminum, and 340 Mev theory. are calculated from curves in hydrogen. (From R. D. Evans, Ref. 16, Fig. 2.1 on p. 609.) The radiation energy loss per centimeter ran be calculated with reasonable accuracy Sec. 2A, for the cross-section for by the methods of quantum mechanics (cf. Ref. rough estimate of the radiation loss given by the formula mulas). i

5

4

0

(1

is

A

f>.

is

is

0

Radiation loss /em is

in which

E

Ionization

_

loss/cm

EZ

(102)

800

in Mev.

Radiation. was first observed by Cercnkov in 1034 that very high particles in passing through transparent dielectric give rise, under certain speed conditions, to visible light which has properties quite different from those of ordinary 0

a

It

Cerenkov

is

is

a

8

is

a

is

is

9

fi

1,

v

is

v

is

a

The features which distinguish this Cerenkov radiation from fluorescent radiation. any previously known type of radiation are the following: (1) Radiation of given emitted only when the speed of the particle greater than the phase frequency The condition can be written velocity of light in the medium for that frequency. the index of refraction of the medium for the given frequency, or as > c/n, where n = v/c. where emitted given frequency, fin > (2) The radiation, again of with the direction of motion of the particle. only in a direction making an angle = l/j3n. determined by the relation cos This direction of emission (3) There characteristic polarization of the radiation which related to the direction of motion of the particle.

Sec Ref. 16, chap. 21. or llcf.

li,

sec. 2E, for a discussion

of the theory

it

is

in

a

is

1

The theory of the Cerenkov radiation (cf. Ref. 16, page 589, and Ref. 46, page 309 The emission of radiation from now well understood. uniformly moving charged greater than that of light the medium through which particle when its speed t


3

12

3

2

\^

and experimental

material.

Aet.

NUCLEAH REACTIONS

4]

4-65

of the classical electrodynamics of material media, and the detailed properties of the radiation are in agreement with the theory. The theoretical of the radiation is such that in the visible region short wave spectral distribution The 0 particles from radioactive "slugs'" immersed in water lengths are preferred. passes is a consequence

have been

observed to produce a bluish glow attributable


3.4

to Cerenkov radiation.

The Passage of Neutrons through Matter

While the interaction of charged particles with matter is due almost entirely to electrostatic forces and the emission of radiation, the interaction of neutrons with matter is entirely dependent on the force between a neutron and a nucleus. This specifically nuclear force has a short range compared with electrostatic forces and comes into play only when a neutron passes very close to a nucleus. Consequently, neutrons travel long distances, compared with charged particles, before they are absorbed in matter. The passage of neutrons through matter resembles that of photons in that each collision of a neutron with a nucleus results in the removal of a neutron from the incident beam. There is, consequently an exponential attenuation of the beam. The collisions of neutrons with nuclei can be divided into three classes: 1. Elastic scattering, in which a neutron is deflected without loss of energy except for the kinetic energy which it imparts to the recoiling nucleus. 2. Inelastic scattering, in which there is an interaction between the nucleus and the neutron resulting in the emission of a neutron with an energy different from that of the incident one. 3. Capture, in which a neutron is absorbed into a nucleus which then undergoes some reaction (other than the reemission of a neutron), usually the emission of a photon.

The details of these three processes are given in Arts.

4

and

5.

Energy loss from emission of Cerenkov radiation is negligible compared with ionization loss in gases, but for very-high-speed particles in dense media it may become of importance. In such cases the Cerenkov energy loss is not a phenomenon inde pendent of the ionization loss; it can rather be regarded as a consequence of inclusion of the effect of the medium in the ordinary theory. Cerenkov radiation is emitted by fast protons and mesons as well as by electrons, and counters based on its detection by photomultiplier tubes are in common use. 4

NUCLEAR

REACTIONS

A nuclear reaction may occur as a spontaneous disintegration of a nucleus [as in natural radioactivity (cf. Art. 2.12) and in fission] or as an induced reaction resulting from the collision of two nuclei (cf. Art. 2.14). The energy change in a single nuclear reaction will be of the order of a few Mev (binding energy of a nucleon) as contrasted to a few ev in chemical reactions (binding energy of an electron). Energy release from nuclear reactions on a macroscopic scale became possible only after the con struction of the first chain-reacting pile in 1942. 4.1

Radioactive Decay

Some elementary aspects of radioactivity have already been described in Art. 2.12. be found here, together with a brief sketch of the theory. 4.11 Most of the nuclides with A greater than about 200 decay Alpha Decay. with the emission of a particles having sharply defined energies in the range from 4 to 10 Mev. In many cases the energy spectrum of the a particles has a fine structure; that is, they are emitted with one of a number of closely spaced but well-defined energies. The energies of a particles are frequently measured and expressed in terms of their

Further details will

4-66 ranges (cf. energy is

NUCLKAR PHYSICS Art.

3.31).

[Sec.

A rough empirical formula for

R

the relation between

4

range and

= 0.32£«

R is the range in centimeters in normal air and E is the energy in Mev. is a characteristic feature of a activity that the lifetimes for decay decrease exceedingly rapidly with increasing energy of the emitted particle. Thus for 4-Mev a particles the lifetime is about 1010 years while for 9- Mev particles it is about 10~5 sec. The empirically established relation between the decay constant [reciprocal of the lifetime (cf. Art. 2.12)) and the energy is known as the Geiger-Nuttall rule and can be written in the form log X = o + 94 log E (103) where

It

\

where E is in Mev and a = —77.7 for the 4n series of natural a emitters (cf. Art. 2. 12), a = —75.8 for the 4n + 2 series, and o = —79.6 for the 4n + 3 series, f The conservation of energy in the decay of a nuclide The Energetics of a Decay. {A,Z) with emission of an a particle having a kinetic energy Ea requires that

c*M(A,Z)

=

cHi (A

- 4,Z -

2) + A/ (He4)

+ Ea


where M(A,Z) is the atomic massj of the parent nuclide and M(A — 4,Z — 2) is the atomic mass of the product nuclide, which may be in an excited state so that the mass equivalent of the excitation energy is included. Rearranging gives for the energy of the a particle §

Ea = c»[M(A,Z)

- M(A - A,Z - - Af(He4)] 2)

(104)

The fine structure of the energy spectrum of the a particles which occurs in a number of cases reflects the fact that M(A — 4,Z — 2) may have a series of values cor The spacing of such nuclear states can thus responding to different excited states. be determined from measurements of a-particle energies, and the energies of y rays accompanying « emission correspond to transitions between these states. The criterion that a decay be energetically possible is Ea > 0. With the use of Eq. (104) and the semiempirical mass formula of Art. 2.23 it can be shown that this That a decay is actually not condition is satisfied for A greater than about 150. observed below A = 200 finds an explanation according to the theory as outlined below. The general features of a emission are paradoxial in The Theory of a Decay. classical theory but have found a satisfactory interpretation in quantum mechanics. The potential energy of an a particle in and near a nucleus would be as shown schemati cally in Fig. 9; that is, the coulomb repulsion between an a particle and a nucleus that are completely separated becomes an attractive nuclear force when the distance r between the two is less than a distance R which roughly defines a nuclear radius. For an a particle bound in a nucleus there would thus be a "potential barrier" of height B = 2Ze*/R, where Z is the atomic number of the product nucleus, and in classical theory the particle could not escape unless its energy were greater than . Actually, scattering experiments with a particles show that in many cases the height of this coulomb barrier is several times greater than the energy of an a particle that the nucleus can emit. In quantum mechanics, however, there can be leakage of particles through such a classically impenetrable barrier in the manner described in Art. 1.36. The rate at which particles leak through the barrier determines the rate of a decay and decreases exponentially with li — E, the difference between the height of the barrier and the particle energy, and with /) — R, the thickness of the barrier " Since both of these (cf. Fig. 96 for the definition of the "classical turning point b). quantities increase as Ea decreases, it is qualitatively understandable that, the rate

It

t For a more extended aecount of the experimental material on a decay 10, chap. 111. The musses of the atomic electrons are included in .W, X Cf. Art. 2.15. This is not necessarily the case in trons remains unehanired in a decay. | There is a smnll correction for the recoil momentum imparte 1 to the

see Uef. 30, chap.

13. or

but the total number of 0 decay (cf. Art. 4.12). product nuclide.

Ref . elec

Art.

nuclear reactions

4]

4-67

of a decay decreases very rapidly with decreasing energy of the a particle. When the energy available for the a particle is small, as for nuclides between A = 150 and A = 200, the a decay rate becomes so low as to be unobservable.

A quantitative relation between the decay constant X and the energy or velocity a particle can be derived from a quantum mechanical calculation of the trans The decay constant may be expressed mission coefficient of the coulomb barrier. in the form X = Xo7\ where T is the transmission coefficient as defined in Art. 1.36 The trans and X0 is the (hypothetical) decay constant in the absence of a barrier. mission coefficient (for zero angular momentum of the q particle) is given approxi mately by the Gamow factor of Eq. (73), that is, To = e-0, where G = 4-rZe*/hv, which gives a relation of the form of the

Iog x =

_

Cl

i%

nearly independent of Ea. Such an equation, although it does not form as the Geiger-Nuttall rule of Eq. (103), represents the experi A more refined calculation or even the use of Eq. (72) as a mental data equally well. better approximation to G shows that X is strongly dependent on the nuclear radius li. It in fact, possible to determine the nuclear radii of the heavier elements with some accuracy from the relation between the observed decay constants and a-particle

d and

with

cs

is,

have the same

is

is

Closely related of frequent occurrence for artificially produced nuclides. and /3+ emission the mode of radioactive decay known as electron capture in which nucleus absorbs one of its own orbital electrons, usually from its K shell. Such a process directly detectable only by means of the X-ray emission following the loss of the orbital electron. In 0~ emission the nuclear transformation involved (A,Z) —* (A,Z — (A,Z) — (A,Z + 1); in both f3+ emission and electron capture conversion of a neutron into a In all three cases the net effect in the nucleus 1). proton, or the inverse process. The electrons and positrons emitted in decay have a continuous distribution in This continuous spectrum well-defined maximum value. energy from zero up to of in sharp contrast to the monoenergetic character of a rays and has had rays important consequences for the development of nuclear theory as noted below. The lifetimes in f}~ and $+ decay are related to the maximum energies of the ejected particles in qualitatively the same fashion as in a emission; that is, the lifetime not so strong, being However, the dependence decreases as the energy increases. The curves for the approximately as the fifth power of the energy in many cases. function of energy are known as "Sargent curves" (cf. disintegration constant as Ref. 36, page 305). The energy Eg- available in a 0~ disintegration The Energetics Processes. E(A,Z + the total internal energy of mjc\ where E(A,Z) Ea- = E(A,Z) the parent nucleus (without its electrons), E(A,Z + that of the product nucleus, the rest energy of the created which may contain nuclear excitation energy, and m
it

is

a

is

0

is

a

0

is

*

is

is

a

to 3~

is

is 1) is

1)

0

of

-

1

+

+ (Z

=

c*\M(A,Z)

- M(A,Z

+

)wi.Jc»

See Ref. lfi, chap.

Ea* = c'[M(A,Z) for a detailed

- M(A,Z -

account of the method.

(105)

1)1

disintegration the corresponding expression for the available energy

to be

1)

0+

2.

For

+

gives for the available energy

Ea*een

[M(A,Z

- 2m.]

is

Making this substitution

=

1)

E(A,Z +

1)

s,

-

t


t

energies. 4.12 Beta Decay and Electron Capture. While the naturally radioactive elements emit only negative electrons (/3~), the emission of positrons (/3+) as well as negative

readily (106)

4-68

NUCLEAR PHYSICS

[Sec. 4

In both cases there are small changes in the binding energies of the outer electrons, due to the change in Z, which are completely negligible. In electron capture the available energy is E(A,Z) — E(A,Z — 1) — (m^c* — E.), where E, is the binding energy of the captured electron which cannot be neglected in the case of heavy elements and capture from the K shell. Expressed in terms of atomic masses the energy EK available in electron capture is =

c*[M(A,Z)

- M(A,Z -

- E.

1)]

(107)

v »

+

v

p

+

> >

$

p p

:

+

:

:

/3

(v

is

is

J-£

It should be noted that when /3+ emission is energetically possible (Ep* > 0) elec tron capture will always be a competing process with greater available energy. The Neutrino and the Theory of 0 Decay. That /3 rays are emitted with a continuous energy spectrum would seem at first to imply the existence of a continuum of energy This viewpoint is, however, untenable, since the monoenergetic levels in the nucleus. character of « and 7 rays clearly indicates discrete energy levels in both the parent and product nuclei involved in 3 decay. This paradox was finally resolved in 1934 when Fermi, following a suggestion by l'auli, proposed a theory of /3-ray spectra based on the idea that in each decay process there is available a definite amount of energy corresponding to a transition between two stationary states of the nucleus but that only a fraction of this, varying from one process to another, appears as In order to retain the principle of conservation of energy energy of the 0 particle. the balance of the energy was assumed to be carried off by a second emitted particle called the neutrino, which had to be neutral in order to preserve the balance of charge. Since no such second particle is observed, it was postulated that the neutrino had an exceedingly small interaction with matter and was thus undetectable by ordinary There is, indeed, strong experimental support for such a hypothesis in the means. fact that calorimetric measurements show that only a fraction of the energy available in 0 decay appears as the total energy of the f) particles. It is only a 0 particle emitted with the maximum energy at the upper limit of the spectrum that receives the full energy available, none being imparted to the neutrino in this case. It was soon apparent, from theoretical considerations, that the neutrino has a very small rest mass, and it has since been established from considerations of the energy balance in /S decay that to within a very small experimental error this res', mass is zero as it is for the photon. Conservation of angular momentum and sta tistics in 0 decay is satisfied by assigning to the neutrino a spin of and FermiDirac statistics. inconsistent with the general Since the existence of electrons as such in the nucleus principles of quantum mechanics (cf. Art. 2.11) the basic process in the Fermi theory taken to be the conversion in the nucleus of a neutron into a proton, or the reverse Thus process, with emission or absorption of a B~ or a p+ particle and a neutrino. stands for a processes are the nucleon conversions involved in the three types of neutrino) n — 0~ + fi~ decay — n + P+ + f)+ decay ~ —* n Electron capture:

,V(«) d* = g*M*(t* account ments will he found in Hef.

S

l)M(<0

of the theory see Hef. 16, chap. chap. 12, and in Ref. 38.

8,

For a simplified

-

-

(108)

t)*tdt

17, or Ref.

5,

0

is

/S

A

f

is

/3

is

based on an analogy to light emission by an The Fermi theory of these processes atom but must of necessity contain factors which can be determined only by com particle and parison with experiment, since the interaction of the nucleon with the a priori unknown. the neutrino strikingly successful feature of the theory has been its prediction of the forms of The theory, when the coulomb emission (3-ray spectra). the energy distribution in neglected, leads to an expression for the number of force on the emitted particle particles .V(<) (it emitted per unit time in the energy range dt from a single 0~ or 0" decay process which has the form

t


EK

chap.

16.

Det&'Ied

treat

Art.

NUCLEAR REACTIONS

4]

4-69

t is the total energy (including rest energy) of the particle in units of nttC1 is, < = E/miC*), «o is the energy available from the nuclear transformation measured also in units of m*c2, M is a factor which contains the matrix element for in which (that

nucleon transformation and depends on the nuclear wave functions, and g is a universal constant (the Fermi constant) having the dimensions of energy X volume. The matrix element in M cannot be accurately evaluated, and in fact, even the form of the interaction entering into it is still uncertain, although the possi bilities are restricted in number and all lead to selection rules analogous to those governing light emission. Only the order of mag nitude (10~48 erg/cm3) of the constant g is experi mentally determined. The shape of /3-ray spectra is primarily given by the factor («! 1)«(h «)' in Eq. (108), which has a quite general theoretical basis. The factor A/1 is of a different order of magnitude for allowed transitions and forbidden transitions, the former being those in which angular momentum and parity of the nucleus remain unchanged and the latter those in which at least one of these KINETIC ENERGY In allowed transitions M1 is independent changes. Flo. 24. Illustrating the general of the energy of the 0 particle, while for forbidden shape predicted by theory for " transitions it has a typical energy dependence. In the (3 and /3+ spectrum in j3 Eq. (108) the effect of the coulomb field of the decay for an allowed transition. nucleus has been neglected, and it applies to either 9" or 0+ emission. This coulomb effect can be approx mately calculated, and when it is included, the 0~ and 0+ spectra have a different appearance as illustrated in Fig. the

new

-


-

24.

The observed shapes of /3-ray spectra, at least for allowed transitions, are in good with the Fermi theory. Conformity to the predicted shape is usually tested by plotting the quantity agreement

KM

™

=

function of «. Such a diagram is known as a Kurie plot. It follows from Eq. that K (e) ~ M(e<> — t), so that for allowed transitions and with neglect of the coulomb effect the Kurie plot is a straight line intercepting the t axis at c = e0. Many observed spectra give linear Kurie plots to a high degree of accuracy, and deviations from this behavior indicate the occurrence of forbidden transitions or coulomb effects. The total lifetime in 0 decay is obtained by integrating Eq. (108) over all energies which gives as a

(108)

l

where

/

=

=

f"

j

N(t)

-

l)H(eo

(««

?'A/y

- ()Ude

(109a) (1096)

and t is the mean lifetime. The product fx, or more usually ft, where t = (In 2)r is the half-life, is called the comparative half-life or simply the ft value. Since ft = 1 /
f

5*ee Ref. 57. Ref. 51.

I Se-


4-70

NUCLEAR PHYSICS

[Sec.

4

The emission of y rays directly 4.13 Gamma Emission and Internal Conversion. from the nucleus very frequently accompanies a or 0 decay or electron capture, since the nucleus is often left in one of a number of excited states from which it can decay As illustrated in with the emission of radiation. the decay scheme for La140 in Fig. 25 the -y-ray line Ce La spectrum may be of considerable complexity. Several hundred decay schemest have now been worked out, and the characteristics of the excited ■3 0 Mev. nuclear states involved, such as spin and parity, have frequently been determined. As an alternative to decaying with emission of a ■2 51 y ray, a nuclide left in an excited state may decay -2 42 with ejection of an orbital electron. The process, which differs from 0 decay in that the nuclear 2.09 charge remains unchanged, is known as internal The kinetic energy of the ejected elec conversion. 1.60 tron is equal to the energy available from a nuclear transition diminished by the binding energy of the electron. The energy balance is thus as though a y ray were emitted and then produced a photo electric effect in the atom, although the process must be treated theoretically as a direct exchange of energy between the nucleus and an orbital electron. Isomerism. 4.14 Nuclear Ordinarily -y-ray Fig. 25. The decay scheme of emission by radioactive nuclides occurs with life La140. times that are too short to be measured. This is in accord with the theoretical estimate of lifetimes of the order of 10-la sec for dipole radiative transitions when the energy change is large. However, a great many cases are now known in which the 7-ray lifetimes are much This implies the existence of relatively longer, sometimes even of the order of years. stable isomers, since the parent nuclide in the y emission has the same nuclear con stitution (A,Z) as the product nuclide, differing from it only in having an excitation energy.

The existence of nuclear isomers is understandable theoretically, since the lifetimes for y emission can become quite long when the energy changes involved are small and when the transition corresponds to multipole emission (cf. Art. 1. 11) of high order. Such multipole emission is to be expected when there are large changes in spin between initial and final states, the multipole order being determined by the change in angular momentum. The study of the properties of the y rays accompanying transitions between isomeric states has led to a considerable amount of information on the spins and parities of excited states of nuclei.} 4.2

Induced Nuclear Reactions

The simpler aspects of induced nuclear reactions have already been taken up in Further details, some account of the theory, and a brief survey of the Art. 2.14. different types of reactions are contained in this section. § While the first induced nuclear reactions to l>e 4.21 Introductory Material. observed were individual events in cloud chambers produced by particles from naturally radioactive sources, our present knowledge of nuclear reactions is liasod almost exclusively on the use of artificially produced|| beams of light particles impingSee also Ref. 45. See the compilation in Ref. 32. X A comprehensive review is that of Ref. 23, chap. 16. I Largely descriptive treatments of nuclear reactions will be found in Ref. 50, chap. chap. Hi. For a more extended survey of theory and experiment see Ref. 43. II For a description of the various kinds of accelerators see Ref. 30, chap. 21.

t

10, and Ref.

36.

Art.

NUCLEAR REACTIONS

4]

4-71

target of the nuclei to be studied. The reaction products may be observed individual particles detected in counters or photographic emulsions, as currents ionization chambers, by means of the activity produced in the target and in other

ing on a as in

ways.

The particles most commonly used as projectiles arc protons (p), deuterons (
incident beam of given intensity and is defined as follows:

^

_

processes/unit time (intensity of incident beam)

X

(target nuclei /unit vol)

(110)

The intensity / of the incident beam is specified by the number of particles passing The unit time through unit area perpendicular to the direction of the beam. frequency of occurrence of a process with cross section a is then, according to Eq. (1 10), given by Events/unit time = aIN (111)


per

where A' is the number of target nuclei per unit volume. A cross section is readily seen from Eq. (110) to have the dimensions of an area.f The common unit in nuclear physics is 10 s4 cm2, known as the bam. The process described by a cross section a may be the occurrence of any disintegra tion of the target nucleus or simply a deflection of the incident particle without change in the target nucleus. It is then usually called a total cross section. Frequently more than one nuclear reaction can occur with a given projectile. To each such reac tion there is then assigned a partial cross section. When the angular distribution of the direction in which a disintegration product (or a deflected particle) comes off from a reaction is observed, it is described by a differential cross section da which specifies the number of emissions (or deflections) per unit time into a solid angle dw

direction (cf. Art. 1.36). The cross section for a given process will in general be a function of the energy of the particles of the incident beam, and a differential cross section will also be a function of in a given

angle.

The total yield of a nuclear reaction is defined as the number of disintegrations per incident particle that occur in a target thick enough to stop all such incident particles. The various types of nuclear reactions (with the exception of those resulting directly in more than two disintegration products which occur at very high energies) can all be represented

by the symbolism o +

X

Y +

b

or more briefly X(a,b)Y, already introduced in Art. 2.14. It must be understood, however, that for a given incident particle a and a p;iven target nucleus A' more than one process may occur, each with a given partial cross section. It is often convenient to count as distinct processes those that differ only in the quantum states in which the residual nucleus Y is left and in which the particle b emerges in a particular state. Such a specification of one of the possible sets of states in which the reaction products can occur is said to define a channel of the reaction. Subscripts on Y and b may be used to indicate different channels (cf. Ref. 8, Chap. 8, Sec. IB). The common types of induced nuclear reactions are the following: Nuclear Disintegration. The typical reaction of this type is one in which a light particle as projectile comes into intimate contact with a nucleus which then dis integrates into another light particle and a residual nucleus. Examples are Fl0(;>,a)O16 and N"(n,p)CM. Reactions of this kind were first observed with charged particles

t It is possible to think of the area
area of the nucleus such that every

NUCLEAR PHYSICS

4^72

[Sec.

4

In such cases a relatively high energy of the incident particle is required in order to bring projectile and target nucleus close together, since the coulomb repulsion must be overcome or the barrier penetrated. Such reactions ran now, however, be much more readily produced by neutrons which are not prevented by electrostatic repulsion from coming close to a nucleus even when their kinetic energy is low. Elastic Scattering. The deflection of a particle a by a nucleus X, when there occurs only the dynamically necessary exchange of kinetic energy between them and no change in the internal energy of either, is called elastic scattering. Strictly speaking, no "reaction" is involved, but since such scattering can occur as a channel in any nuclear reaction, it is convenient to treat it in the same category as true reactions. Such a scattering may be written a + A' — > .Y + a. Inelastic scattering differs from elastic scattering only in that internal energy is transferred to the target nucleus X with a corresponding loss in energy of the incident Such a process may be written a + X— * X* + a, where the asterisk particle. denotes that X is in an excited state. Inelastic scattering is common with high energy neutrons and is treated in more detail in Art. 5.24. Radiative Capture. A particle, usually a neutron, may unite with a target nucleus to form a new nuclide, the binding energy of the incident particle appearing as excita tion energy of the new nucleus. If emission of a y ray follows, the process is called radiative capture. A typical reaction of this kind is the capture of a neutron by hydrogen, H1(n,T)HJ. Incident y rays may also produce nuclear disintegration. Photodisintegration. Such reactions are always endothcrmic, since no binding energy of the photon is A typical reaction of this type is Be°(7,n)Be8. available. Fission. The products of a nuclear reaction involving a heavy target nucleus may be two nuclei of comparable mass instead of a heavy and a light particle. The It is here discussed separately in Art. 4.3. process is then called fission. 4.22 The Q value Consequences of the Conservation of Energy and Momentum. of a nuclear reaction, as already pointed out in Art. 2.15, measures the difference in the internal energies of the particles between initial and final states. However, the kinetic energies (of the centers of mnsB) of the particles must be taken into account in an actual experimental situation. Since the total energy, that the sum of internal and kinetic energy, must be conserved, the gain in kinetic energy must equal the loss in internal energy which measured by This leads to the equation is,

Q.

is

Q

=

EY +

Eb

-

Ea

- Ex

(112

is

is

is

is

0,

is

E

where the kinetic energy of the particle indicated. In the usual experimental since the target nucleus at rest. situations Ex = Of the kinetic energy Ea of the incident particle only a fraction available as Accord energy of relative motion in the two-body collision involved in the reaction. the energy in the laboratory reference system ing to Eq. (71), remembering that E„ (L system), this energy of relative motion Ea'

-

MJM

Ea

(113)

It only this energy, which will be referred to as the channel energy, that enters into the two-body-collision problem, the remainder being energy of motion of the center of mass of the system as a whole. It is, of course, only when the projectile and target nucleus have comparable masses that a distinction has to be made between the channel system, since energy and the measured energy in the M«
Q

+

Q

is

it

0)

Q

is

if

L

is


as projectiles.

EJ,

Art.

4-73

NUCLEAR REACTIONS

4]

reaction products to separate. The threshold energy Etk, that is, the minimum that must be supplied to produce the reaction, is thus

for the energy

Eth

„M.

+ MxQ Mx

(m)

L system. reaction linear momentum as well as energy is conserved. The method of treatment of the two-body scattering problem as outlined in Art. 1.36 is thus applicable except that the reaction energy Q must be added to the energy after collision, t Of particular practical importance is the fact that the kinetic energy of the particle 6 when it is observed to come off at an angle 6 in the L system is com pletely determined by the Q value of the reaction and the energy Ea of the incident particle. This is the basis of one of the most convenient methods of obtaining monoThe rela energetic beams of neutrons and charged particles (Art. 2.11 of Sec. 5-2). tion between the particle energies and the Q of the reaction can be written in the form in the

as measured

In any nuclear

(l \

+

Mr'

-E.(x-

Mr'

V

-

2

(M«g°M^ Mr

cos S

(115)

of

the

is

if,

This equation may be solved (best graphically) to give the energy Et in terms of Q, >, and Ea. 4.23 The Theory of Nuclear Reactions. J Even with a complete knowledge of nuclear forces a direct wave-mechanical calculation of the course of a nuclear reaction would be prohibited by the mathematical complexity of the problem. Consequently, it is necessary to base any discussion of nuclear reactions on highly simplified models which are useful for the general interpretation of experimental results rather than for detailed calculations. General Qualitative Ideas. The theoretical discussion of nuclear reactions is greatly as was first proposed by N. Bohr in 1936, the process conceived of as Amplified taking place in two stages: (1) the formation of a compound nucleus by the combination

incident particle and the target nucleus and (2) disintegration of the compound By nn extension of the symbolism already

b

-»

Y

->

+

X

C

a

+

nucleus into the final reaction products. 'i**d the reaction may then be written

a

it

is

a

is

is

is

C it is

very large class of nuclear the compound nucleus. Furthermore, for reactions appropriate to assume that these two stages, the formation of the com pound nucleus and its disintegration, can be treated as independent processes in the sense that the mode of disintegration (in general there will be more than one mode of 'leray of C) does not depend on the way in which the compound nucleus was formed. The plausibility of the Bohr assumptions and a rough estimate of their range of applicability can be established by the use of a semiclassical picture closely related to the liquid-drop model. thought of as a collec In this picture the target nucleus tion of nucleons separated by distances of the order of magnitude of the range of nuclear forces, the whole having known of the well-defined surface. From what readily shown that a particle impinging scattering cross sections of free particles where

is

is,

a

is

For a detailed treatment see Kef. 16, chap. 12. The most complete theoretical treatment of the theory of nuclear reactions will be found in Ref. chips. and 9. 8

8.

is

it

A

if

not too

if

its energy a target nucleus will collide with nucleon near the surface not too small. That the mean free path of the the nucleus high and particle inside the nucleus will be small compared with the nuclear radius, although at sufficiently high energies this condition may no longer be fulfilled, since scattering 'mss sections diminish with increasing energy. The range of applicability of this picture can be roughly set at > 10, and particle energy less than 50 to 100 Mev. then to be expected that the target nucleus will be excited Within this range on such

t t


Q = Eb


4-74

NUCLEAR PHYSICS

[Sec. 4

by a process in which the incoming particle makes many collisions within the nucleus, losing a fraction of its energy in each and thus sharing its energy with many nucleons. t The compound nucleus will thus be formed in an excited state which is above the ground level by an energy equal to the separation energy Sa (cf. Art. 2.15) of the incoming particlej plus any kinetic energy which it may have. For the further course of the reaction the character of the motion in such excited states (with an excitation energy of at least 5 to 8 Mev except when the incident particle is a photon with no separation energy) is a decisive factor. If the target nucleus contains a large number of particles and the excitation energy is high, the motion can be expected to be of a complicated kind, involving continuous interchange of energy between nucleons by means of collisions. The situation can be likened to that of the heat motion of the molecules of a liquid, and disintegration of the compound nucleus will take place through statistical fluctuations which concentrate on one particle a, sufficient fraction of the excitation energy to eject it from the nucleus — a process analogous to evaporation from a liquid. Reactions for which such a picture is appropriate are called statistical reactions. The energy level spectrum of the compound nucleus, when the above statistical picture is applicable, may be expected to consist of a large number of closely spaced levels corresponding to the many possible complicated types of motion of nearly the same energy which differ only in the division of energy between different nucleons. There are a number of crude models of a complex nucleus which permit an approxi mate calculation of such excited energy levels. All of them show the same general features, namely, a close spacing of the energy levels which decreases with increasing excitation energy, the spectrum finally merging into a continuum at sufficiently high With such close spacing of the levels the disintegration of the compound energy. nucleus can be thought of as depending on the average properties of the levels over a small energy range rather than on those of one sharply defined energy level. In fact the experimental uncertainty in definition of the energy of the incoming particle justifies such a statistical treatment. The qualitative picture of an excited nucleus as analogous to a fluid with nucleons as its molecular constituents can be made more precise by applying the very general methods of statistical mechanics. The usual thermodynamic quantities, including entropy and temperature, are introduced in this way, and it is possible to derive an expression

p(E)

-

Ceig'T

(116)

for the number p(E) of levels in the range dE near E in which E is the excitation energy. C is a constant characteristic of the nucleus involved, and T a quantity of the dimen sions of energy that is analogous to the usual kT of the kinetic theory of gases. The constant C and the "nuclear temperature" T are to be regarded as quantities to be determined empirically.! The above statistical picture is, however, not always a suitable one. If the avail able excitation energy is low and the number of nucleons of the target nucleus that are involved is small, it may happen that the energy levels of the compound nucleus are relatively widely spaced, resembling those of an excited atom. Under these circumstances the course of a nuclear reaction is governed by the properties of indi vidual energy levels of the compound nucleus. The situation is analogous to the absorption and reemission of light by an atom which shows the typical phenomenon of dispersion in which the absorption and scattering have sharp peaks when the energy of the photon coincides with that of the difference between two stationary states of the atom. Correspondingly, nuclear reactions in such situations display strong

t In the light of the recent successes of the shell theory (cf. Art. 2.31) there is reason to question the validity of arguments like those above which imply a close-packed model of the nucleus. There ia now evidence that a nuclcon entering a nucleus need not interact with nucleons as much accumulated The implications for the theory of nuclear reactions of such a modified strongly as previously assumed. " clouded viewpoint have not yet been quantitatively explored. See. however, Ref. 20 for a theory (the " theory) of neutron reactions based on ideas suggested by the applicability of a.n inde crystal ball particle model. } The term binding energy rather than separation energy is frequently used. 5 For a derivation see Ref. 4.1, sec. 3B, or Ref. 8, chap. 8, sec. 6.

pendent

NUCLEAR REACTIONS

4] is,

Art.

4-75

7

is

is

of

a

a

resonances; that the cross section as function of energy has a more or less sharply defined peak which can be attributed to coincidence of the sum of the kinetic energy (in the center-of-mass system) and the separation energy of the incident particle with the energy of some individual level of the compound nucleus. The quantumtreatment of such problems closely resembles, at least in its mathe mechanical matical form, that of radio waves received by tuned electrical circuit. The behavior reaction and scattering cross sections near resonance given by the Breit^Wigner formula, which discussed in Art. 5.22 below. The energy levels of nuclei are accessible to direct experimental observation through region as well as from the observed peaks in reaction cross sections in the resonance the frequencies of the rays emitted from excited compound nuclei. An enormous amount of experimental information of this kind for the lighter nuclei has now been

X

it

is,

(117)

See, for example,

Refs.

1

is

and 15.

+

r

is

If

is

is

is

is

is

7

a

is

is a

is

r

is

is

a

r

is

h,

r

may be interpreted as the uncertainty in the energy of a quasiThis level width stationary state which, according to the uncertainty relation [cf. Kq. (31)] AE At Also, in any detailed calculation of the a consequence of its finite lifetime t = At. will actually appear as the half-width decay of a compound state, the energy value of a typical distribution in energy of the emitted particle. The decay of a compound nucleus once formed can frequently take place in a To number of ways — by emission of material particle or of a photon or by fission. each such process can be assigned a lifetime t< and correspondingly a partial width = ZI\ and that readily seen that the total width of the level Ti = A/tj. It r,/r. the relative frequency of occurrence of any one process or mode of decay The idea of competition among different modes of decay of the compound nucleus illustrated, for typical feature of the Bohr theory of the compound nucleus. It example, in the scattering and absorption of neutrons by a nucleus. For sufficient energy of the incident neutron the compound nucleus may be formed in an excited state which can decay in three different ways: (1) return to the ground state with ray, (2) reemission of a neutron with the residual nucleus left in its emission of a ground state, (3) reemission of a neutron with the residual nucleus left in an excited that of radiative capture; the second, in which the neutron state. The first process necessarily emitted with its original energy diminished only by the kinetic energy elastic scattering; the third, in which the energy imparted to the residual nucleus, of the neutron diminished also by the excitation energy of the residual nucleus, the partial widths for these processes are, respectively, inelastic scattering. = Ty r„, + rni and their relative rT, r„, and r.j, the total width of the level frequencies of occurrence are proportional to their respective r's. increased, other processes may also become As the energy of the incident, neutron possible, for example the emission of charged particles which will, in general, be In special cases the impossible at lower energies because of the coulomb barrier. fission process may also compete with those already mentioned. t


is

A

is,

a

is

is

accumulated and available in summary form.f The energy levels of a compound nucleus arc not, Decay of the Compound Nucleus. strictly speaking, stationary states, since material particle emission or photon emission generally possible. They can, however, be thought of as quasistationary, having lifetimes that are long compared with any estimate of period of vibration of an individual nucleon in a nucleus. In this respect they actually differ only quantita tively from the excited stationary states of the electrons in an atom which can decay with photon emission. The disintegration or decay of a compound nucleus according to general prin given ciples of quantum mechanics, a statistical process like radioactive decay. the decay constant (cf. state will have a finite mean lifetime r = 1/X, where however, the usual practice to express the rate of Art. 2.12). In nuclear theory decay in terms of the level width

4-76

NUCLEAR PHYSICS

[Sec.

4

It is evident .that with increasing excitation energy the total widths of the levels of the compound nucleus will be expected to increase, since the number of possible Since the total frequency of occurrence of dis decay processes becomes greater. integration of the compound nucleus is limited by the cross section for formation of the compound nucleus, the absolute frequency of occurrence of any one process will be diminished as the number of competing processes increases. Analysis of the Scattering and Reaction Cross Sections. In considering the inter action of a beam of particles with nuclei it is impossible to neglect the scattering. The total cross section at, that is, the cross section for removal of a particle from the incident beam (deviation from the original direction is counted as a removal), may be written
*. +

(118)

is

I

(I

it

X

I

R

(i

is

t

A

a

v is

d is

is

rfmv, where distance from The angular momentum of a particle in the beam the perpendicular moving. the center of the target nucleus to the line along which the particle of mass m and velocity

t


a

is

is,

is

is,

The first term a, is the cross section for elastic scattering, that for all events in which the internal state of the target nucleus remains unchanged; the second term or the reaction cross section, that the cross section for all events in which the internal state of the residual nucleus differs from that of the target nucleus. It should be noted that inelastic scattering included in o> rather than in into contributions from incident particles of different angular momentum with respect to the target nucleus plays a fundamental role in the theoretical treatment of nuclear reactions, which closely resembles that of elastic scattering discussed in Art. 1.36. The plane de Broglie wave which represents in quantum mechanics the beam of incident particles may, as in elastic scattering theory, be treated as a superposition of waves each of a definite angular momentum lil with respect to the target nucleus. Thought of classically, the beam contains But as particles of all angular momenta, illustrated in Fig. 26 those which would pass within a distance IX of the nuclear center have exactly the quantized angular momenta Ih, since IXmv = Ih. Hence the closest one can come to a classical analogy to think of the particles in the region between IX and + 1)X as all having the The great advan angular momentum Ih. tage gained from such a decomposition of the beam comes from the fact that for a only those particles with nucleus of radius < R/X would angular momenta for which come close enough to the nucleus to interact with it, and the quantum mechanical Fio. 20. Illustrating the decomposition treatment also has this feature. Since of an incident beam into components of different angular momenta. increases as the energy of the incident par ticles decreases, often possible, as is always the case for slow neutrons, to consider only those neutrons of zero angular = 0). As the energy of the incident particles increases, more and momentum more values of must be taken into account. Scattered particles of a given angular momentum have a typical angular distribution, those with 1=0 showing spherical symmetry. In manner analogous to the decomposition of the elastic scattering cross section

AliT. in Eq. sums

(65)

4-77

NUCLEAR REACTIONS

4]

both the scattering and reaction cross sections may then be written a, =

)

a, — } ov.j

o..i

as

(119)

I

I

from particles of angular momentum I. each term giving the contril)iition It is frequently sufficient to consider only the first terms with !=()(« xvavc). It is possible to treat theoretically the process of elastic scattering, including removal of particles by compound nucleus formation and their reemission, by a modificationf of the method of partial waves of definite angular momentum described In this procedure the effect of the nucleus on the outgoing spherical in Art. 1.36. wave is completely specified by the change in phase and amplitude of the incoming A single complex number expresses this change for a wave at the inner surface. given /, and although it cannot be computed from the conditions inside the nucleus on which it depends, the theory nevertheless imposes certain restrictions on the scattering and reaction cross sections and their relation to each other. For example, the maximum partial scattering cross section occurs when the corresponding reaction cross section is zero and has the value =

(o„.l)m.<

D>rX!

(120)

reaction cross section occurs when a,,i (
— ('21

+ lVX2 and

, = (2/ + l)rXa

This last condition, that for as can be seen from Fig.

has the (121)

R

a a

A

is

is

is,

in fact, understandable classically, a>.i < (2/ + 1)ttX2, 26, (2/ + l)jrX' just the area of the ring through which pass the incident particles of angular momentum In, and the inequality expresses the fact, that the nucleus cannot absorb more particles with this angular momentum than exist in the beam. that of nucleus of radius that absorbs all particles special case of interest In the case of such "black" nucleus and particles of very short which strike it. wavelength a classical picture might be expected to be applicable, leading simply to Actually, the detailed calculation the geometrical area rli* for the total cross section. gives o,

» 2tR*

(122) /.

The added cross section and a, « aT for the partial cross sections summed over all classical wave, picture of diffraction by the can, however, be understood from The phenomenon known as shadow scattering and has been observed nucleus. experimentally. According to the general assumptions of the Bohr theory the reaction cross section a, can be written, for specific reaction X (a,b) Y, in the form a

\

is

a

tW

i

i

+

b

__-__ ■

•>(a,6) =
L

+

r,

+

(123)

■■■

8,

See Rcf. 43. eec. ;")fc,for the details. p. 324. See Ref. 43, p. 57. or Ret.

A

3

is

a

6,

is

is

is

the cross section for formation of the compound nucleus by the incident where ac(a) the partial width for decay of the compound nucleus with emission of particle a, Tt the total width of the level excited, written as the and the denominator particle sum of the partial widths for all possible modes of decay. The Continuum Region. Approximate expressions can be derived for the formation and decay factors in Eq. (123) when the incoming particle has sufficient energy to state in which its excited levels are relatively closely form the compound nucleus in known as "continuum theory" and can be spaced. The theory for this situation Mev and elements with In used for incident energies greater than about > 50.

t t


and the maximum value

+

1(2/

4-78

NUCLEAR PHYSICS

[Sec.

4

general ac will vary smoothly in this region, in contrast to its behavior in the resonance region at lower particle energies. The behavior of ac as a function of energy in the continuum region is largely gov It may be written erned by extranuclear forces and reflection at the nuclear surface.

in the form aCj =

ac(a) = Zacda)

(21

+ l>rX«r,(a)

(124)

where (21 + l}r\* is the maximum possible reaction cross section for particles of angular momentum I [cf. Eq. (121)] and Ti a transmission coefficient, the fraction of the incident particles that penetrate to within the nuclear surface which is assumed to have a sharp radius R. In Tt are included the effects of the coulomb force for charged particles, the centrifugal potential h'KI 4- l)/2.Vr* for both charged and neutral particles, and the effect of the sharp change in potential at the nuclear surface. t There are available extensive calculations of Ti for charged partirles and for neutrons (cf. Ref. 8, Chap. 8, Sec. 5). For charged particles the main effect is that classically the coulomb barrier is impenetrable when the energy of the incoming particle is less than the barrier energy

B(R)

=

(125)

Z'Zê*

R is the nuclear radius. Quantum mechanically, however, there is always a probability, diminishing rapidly with energy, for penetration even at energies below the barrier energy. The penetrability is given approximately by the Gamow factor of Eqs. (72) and (73). For incident neutrons, compound nucleus formation is mainly governed by the ability of the neutron to escape reflection by the potential drop at the nuclear surface and to penetrate the centrifugal barrier. For I => 0, there is only the surface reflection effect, and assuming a constant potential energy just inside the nucleus which cor responds to a wave number K for the neutron inside, the transmission coefficient can be shown to be lb A* (126. r.(n)


where

-

--^

where k is the wave number of the neutron outside. When I is not zero, there is an additional factor vi < 1 expressing the penetrability of the centrifugal barrier, and

Ti(n) The values of

vi

for

v0

=1

/

=0,

1,

Vi

-

and

(1271

2 are

x5 1

= r,7'o(n)

+ x*

Vt =

x* 9 + :ix' + x*

i

=

kR

(128)

The wave number K of the neutron inside the nucleus is not a sharply defined concept, since a description by a single-particle wave function is a very crude approximation. However, from general considerations it can be shown that it is reasonable to assume as an order of magnitude K = 10u cm-1 for neutrons with initially zero kinetic energy. In the continuum region an approximate formula for the compound nucleus forma tion by neutrons of angular momentum / is then eci

=

(21

+ l)VXHi

--

(129)

The decay of the compound nucleus when it is once formed [second factor in Eq. depends on the lifetimes (as expressed by the level widths) of the various

(12li)|

t

See Art. 1.30 on the transmission

of potential barriers.

Art.

NUCLEAR REACTIONS

4]

4-79

that can occur. The qualitative ideas of the Bohr theory, of not permit a detailed calculation of these lifetimes, but they do lead to a roughly justifiable relation between the lifetimes and the average level spacing, that the average energy interval D between neighboring levels of the compound nucleus. fact, a number of crude models of the complex nucleon motion ol the kind assumed Bohr lead to the expression processes

emission

by In

is,

course, do

-

= T(b)

£

r„ =

Tb

(130)

2jt

is

is

*

("0)

(131)

The Resonance Region. The continuum theory as outlined above can be applied the energy levels of the compound nucleus in the energy range involved are so number of them. closely spaced that processes depend only on averages over This, general, implies incident particles with energies of a few Mev and target nuclei that are not too light. At lower energies the properties of individual levels become important and resonance peaks occur in the reaction cross sections as functions of the energy of the incident particle. These are most readily observed for neutroninduced reactions, since at lower energies reactions induced by charged particles have small cross sections because of the low penetrability of the coulomb barrier. An expression for the reaction cross section in the neighborhood of resonance in its dependence on the energy of the incoming particle and on the widths for particle emission can be derived by the use of quite general methods of quantum mechanics, as was first shown by Breit and Wigner in 1936. The result, known as the BreitWigner one-level formula, is

f

a

in

a

ivhen

= gi(2l

+

„(«,*)

1W (g_^(r,/2),

(132)

E

is

is

a simplified

derivation see Rcf.

chap.

8,

for

8.

is

b

a

is is a

is

is

it

it

b

is

is

a

is

where E, the energy of level of the compound nucleus, the energy with which the compound nucleus formed by the particle o, IV, IV are respectively the widths for emission of particles a and from the level with energy and \„ the de Broglie The factor gi wavelength (in the center-of-mass system) for the incident particle. enters only when necessary to take into account the spins of the particles involved and is essentially a statistical weighting factor which will not be further considered The interpretation of the Breit- Wigner formula here in this general connection. and the way in which used in the study of resonances taken up in Art. 5.22. Equation (132) cannot be applied to give the complete elastic scattering, that is, to the rase in which the particle identical with the particle and has the same The reason that there further contribution to such clastic scattering energy.

t


it

is

is

is

is

is

a

6

is

a

a

6.

is

The factor T(b) width for emission of the same transmission particle The way in which such coefficient as in Kq. (126). relation established some what as follows: Recmission of the particle supposed to depend on the relatively long time required for the complex nucleon motion to concentrate energy on single This time can be estimated to be 2-jrh/D. There are thus D/2irh impinge particle. ments per second of the particle on the nuclear surface from the inside, and T(b)D/2rh Equation (130) then results the number that penetrates to the outside per second. immediately. As inde to be expected, the transmission coefficient of the barriers (which pendent of whether the particle comes from the inside or the outside of the nucleus) enters in both the formation and decay factors of the reaction cross section. Its occurrence in the width for emission accounts for the fact that neutron emission from the compound nucleus usually favored over charged particle emission. to emission from the compound nucleus of The simplest application of Eq. (130) reduces to neutrons of zero angidar momentum, in which case for the

sec. 7D.

NUCLEAK PHYSICS

4r-80

[Sec.

4

from the extranuclcar forces which is cohcrentf with the scattering resulting from compound nucleus decay. The statistical theory may be derived from the resonance theory by the process of averaging over many levels, the effect of each of which is expressed by the BreitWigner formula of the resonance theory. 4.24 It will be possible here to describe only a few Charged Particle Reactions. typical features of some charged particle reactions. Neutron-induced reactions are treated separately in Art. 5.2. Proton- and Alpha-induced Reactions. For charged particles incident on light nuclei (A < 25) the important factors are the penetrability of the coulomb barrier and the existence of relatively few and widely spaced levels of the compound nucleus. The reactions depend on the properties of these individual levels and afford informa tion on their energies, spins, and parities. A much studied example is afforded by the reactions resulting from the bombard ment of Li7 by protons. The compound nucleus formed is Be8* which can disintegrate in a number of ways corresponding to the following reactions:

is,

The numerous resonances observed in the cross sections of these reactions as a function of the bombarding energy of the proton correspond to excited levels of the Be8 nucleus. The energies of these levels may be computed by adding to the separa tion energy of the incident particle its channel energy Ed as defined by Eq. (113). For example, there is a resonance in the neutron yield of the (p,n) reaction at 2.25 Mev in the laboratory system as illustrated in Fig. 27. The corresponding channel energy.' by Eq. (113),

M(H')

The separation energy of the proton S„ = c»[A/(Li7)

E.

+ Af(H')

=

——

2.25 = 1.97

Mev

1

+

7

M(Li')

+

=

is

Ed

- M(Be8)]

= 17.24

Mev

is

a

a

it

is

is

is

3

a

7

a

+

and the observed resonance at 2.25 Mev then indicates the existence of an excited level of Be8 at an energy of 17.24 1.97 = 19.21 Mev. The cross sections as a function of the energy for the reactions listed above are shown schematically in Fig. 27 together with the energy levels of Be8 to which the radiative capture, and the first resonances correspond. The (p,y) reaction resonance gives rise to a ray of 17.57 Mev by return of the Be8 nucleus to its ground The (p,a) reaction shows Mev with an exponen state. single broad resonance at tial rise of the cross section that typical of the effect of the Gainow factor in charged particle reactions. Other reactions induced by a particles and protons show features similar to those discussed above, with a large number of resonances for target nuclei of intermediate At high incident energies (p,n) and (a,n) reactions are predominant and are mass. endothermic with thresholds of several Mev. There preference for neutron emission when energetically possible, which can be attributed to the effect of the coulomb barrier in inhibiting the ejection of charged particles. At sufficiently high energies (a,'2n) and (p,'2n) reactions are observed; that is, the emission of one neutron may leave the residual nucleus in state with sufficient excitation to permit the ejection of second neutron.}

8,

8,

is

8,

t

That is, has a definite phase relation to the nuclear scattering so that amplitudes rather tha.il cross sections must he added. For the complete formula see Kef. chap. 8. For very general deri%-stt.ions of dispersion formulas, including the case in which interference between neighboring levels chap. 10. tance. see Ref. Charged particle reactions are discussed in detail in Ref. chap. 9. and in Ref. 43.

t


(p,n): Be8-> Be7 + n (p,7): Be8-> Be* + y (p,a): Be8 —» a + a (p,p'): Be"-» Li'* + p'

of ir

Art.

NUCLEAR REACTIONS

4]

4-81

Reactions. With deuterons as projectiles much larger cross (d,p) reactions are observed than for other charged particles For this reason deuterons from accelerators are frequently energy.

Deuteron-induced sections

for (d,n) and

at the same used

for the production of radioisotopes.

The relatively large cross sections for deuteron-induced reactions, which are of the order of those produced by neutrons, suggest that the inhibiting effect of the coulomb

199k

18.14 1763


Li7 +p

17.242

>

S

Fio. 27. The energy levels of the compound nucleus Be11 that are indicated by the bombard ment of LiT with protons. All energies are in million electron volts. The cross sections of the Li7 reactions are shown schematically as a function of the channel energy of the protons. There are a large number of additional levels of Be* revealed by other reactions leading to it as a compound nucleus. The complete energy-level diagram will be found in Ajzenberg and Lauritsen, Energy Levels of Light Nuclei V, Rev. Mnd. Rhys., 27 : 91 (1955). A simple qualitative explanation of this barrier has in some way been circumvented. is to be found in the fact that the deuteron is a loosely bound structure and can be readily broken up when it enters the field of a nucleus, leaving the proton free to Reactions of this kind arc known as stripping penetrate alone into the nucleus. reactions, and at least at low energies, the breaking up of the projectile is known as the Oppenhcimer-Phillips process. The theory of deuteron reactions is complicated by the fact that compound nucleus formation may take place in three different ways: by absorption of the entire deuteron,


4-82

NUCLEAR PHYSICS

[Sec.

4

At high deuteron energies where the coulomb of a proton alone, or of a neutron alone. barrier is unimportant, the relative occurrence of these methods of formation can be calculated from simple semiclassical considerations; at lower energies the calculation presents difficulties which have not yet been satisfactorily overcome, although the observations can be qualitatively understood (cf. Ref. -13, Sec. 8). At bombarding energies of 50 Mev or more Reactions at Very High Energies. nuclear reactions involve the emission of a number of nucleons or a particles, and as the energy increases, large numbers of different lighter nuclear species are among the reaction products. The process of separation into a number of fragments which is involved in such reactions is called spallation. For example, the bombardment of copper with 340-Mev protons yields, as reaction products, some 35 nuclides from sodium to zinc. At energies above 50 to 100 Mev, as already noted, the Bohr theory of compound nucleus formation is no longer expected to be valid in the sense that the incident The particle is captured and shares its energy with a large number of nucleons. situation is rather that the mean free path of such a particle inside the nucleus is of It then may rather be expected to the order of magnitude of the nuclear radius. communicate energy to a single nucleon or even to traverse the nucleus without a Many qualitative features of spallation reactions can be accounted for collision. by a theory based on this modified picture (cf. Rcf. 43, Sec. 11). 4.25 For nuclear reactions initiated by y rays there is Photodisintegration. always a threshold energy, since if a material particle is to be ejected from the com pound nucleus, its separation energy must be supplied by the energy hp of the photon, The commonest reactions there being no binding energy of the photon available. observed are (y,n), (y,p), and (y,a) reactions. f Thresholds for the (y,n) process have been observed for a large number of nuclides and are principally valuable for the fact that they give directly the separation energies of the neutrons. The photodisintegration which has been most extensively studied and the only one which can be calculated with any accuracy is that of the deuteron. Its threshold at 2.226 Mev measures the binding energy of the deuteron. Study of the cross section as a function of energy and of the angular distribution of the reaction products has given valuable information on the neutron-proton interaction. 4.3

Nuclear Fission

In 1939, Hahn and Strassman discovered the quite unexpected fact that among the reaction products resulting from the bombardment of uranium with slow neutrons there are elements with Z in the neighborhood of 50, indicating a fission of the uranium nucleus into two approximately equal parts. In the same year, Meitner and Friseh pointed out that the energy release in such a process would give to the fragments a kinetic energy of about 100 Mev and were able to detect such fragments directly by observing the ionization which they produced. The early development of the theory of fission is due to Bohr and Wheeler (1939) (Ref. 10). The practical importance of fission lies in the fact that in the process neutrons are emitted and that under certain conditions they may be utilized to produce further fissions, thus leading to a self-sustaining, or "chain," reaction. It is thus possible to "burn" uranium, to give a source of power with an energy release per atom some 108 times greater than from a chemical fuel. 4.31 The Basic Facts about Fission. J Since its discovery in uranium it has been found that fission can be induced in most elements with Z above about 30 by means of sufficiently energetic bombardment such as with neutrons above 100 Mev. How ever, it is only for Z > 90 that fission is produced by neutrons of low or moderate energy (< 10 Mev) and with appreciable cross sections. Fission may also be induced by protons, a particles, deuterons, or y rays and is a spontaneously occurring process in a few instances. t For the theory, which is not so well understood as that of charged particle reactions, see Ref. 8s chap. 12, sec. 7E. t See Ref. 3*1, chap. 19, for additional reading, including an account of the history of the discovery of fission.

Art.

4-83

NUCLEAR REACTIONS

4]

and Thresholds for Neutron-induced Fission. The cross sections for by neutrons show large variations with energy and with the character of the nucleus. The slow neutron cross sections, for example, vary from 10~2 to

Cross Sections fission target

10sbarns.

It

typical of the fission process that some nuclides are fissionable with thermal while others require neutrons of at least 1 Mev. It is a general rule that nuclides with an odd number of neutrons are thermally fissionable or have low thresh olds while those with an even number of neutrons have much higher thresholds. This is well illustrated by the naturally occurring isotopes of uranium. The principal isotope Us" has a threshold near 1 Mev, while U236, which makes up less than a per cent of natural uranium, has a cross section of 580 barns for thermal neutrons. This [act, of great practical importance, was actually predicted theoretically before it was observed experimentally. The general form of variation with energy of the fission cross sections of some prac tically important nuclides is shown in Fig. 28. is

n

i

.i

Fig. 28. Fission

jo* lb too io* ENERGY IN EV

cross sections

io1

»'

'»*

.i

i

io too to5" k>4 ENERGY IN EV

of U"s, U238, and Pu"* as a function of neutron

energy.

is

a

y

is

is,

The fission disintegration will bo in competition with other modes of decay after compound nucleus formation, and the competing process of most importance, at /east at not too high excitation energies, is y emission. Such radiative capture occurs with a cross section of 107 barns in U235; that in about 20 per cent of the encounters neutron emitted. captured to form U238 and a ray Nuclides Formed in Fission. The division of nucleus in fission does not, as would have been expected, lead to two fragments that are as nearly as possible identical. On the contrary there are, for the common fissionable nuclides, some 30 different and a

hr

022 hr

Xe,3>

r—

>

I'"

»

0-

;

lOmin

Te1"

>

r

1

,,Sh"»

is

A

a

is

if

/3

a is

is

highly unsymmetrical ways in which the fission takes place. In U235, for example, the range of observed mass numbers from 72 to 158, with the symmetrical division occurring in only 0.01 per cent of the fissions. The primary fission products are highly unstable nuclides as to be expected from tfie fact that they contain an excess of neutrons, coming from parent nucleus con active taining some 50 more neutrons than protons. They are accordingly highly and may occasionally emit neutrons sufficiently high. Each their excitation /3-decay chain and about 60 such primary product of fission forms the beginning of chains are known. typical decay chain

r»


neutrons

5.3 days

ssCs'" (stable)

4-84

NUCLEAK PHYSICS

[Sec. 4

The end result of


a chain is a stable or nearly stable nuclide of the same mass as the The relative number of chains with a given mass thus primary fission product. reflects the frequency with which the corresponding fission occurs. Typical fission yield curves are shown in Fig. 29. The Neutrons Accompanying Fission. The neutrons emitted in the fission process are of two kinds: the prompt neutrons observed within the limits of experimental error to be emitted simultaneously with the act of fission and the delayed neutrons which make their appearance within a matter of seconds after fission. The prompt neutrons are believed to have their origin in the ejection of a neutron from an excited fragment at the instant of fission. The delayed neutrons, on the other hand, are emitted in the course of the successive 0 decays that constitute the chains. The process involved is a 0 emission leading to an excited state of the product nucleus

60

80

100

120

MASS NUMBER.A.OF

140

160

180

FISSION PRODUCT

Fig. 29. The distribution in mass number of the fission products from the thermal fission of Um. The yield curves for U2" and Pu"' are qualitatively similar. The curve for an a-induced fission is shown for comparison. (From R. D. Evans, Ref. 10, p. 393.)

3 4 3 ENERGY IN MEV.

Fig. 30. The distribution in energy of the prompt neutrons from the thermal fission of UJ". The measurements of Watt. Phyt. Ret., 87: 1037 (1952), are well represented from 0.4 to 17 Mev by the expression c~g sinh (2£)W for the relative number of neutrons.

from which a neutron is ejected. The rate of emission of the delayed neutrons is thus governed by /3-decay rates and can be of the order of seconds or minutes. the case of U23i only 0.72 per cent of the neutrons are delayed. The average number of neutrons emitted in fission, usually designated by v, is a number of the greatest practical importance, since the possibility of sustaining a chain reaction depends on its value. It is about 2.5 for the thermal fission of U*** and about 3 for the thermal fission of Pu8'9. Gamma Rays from Fission. The emission of prompt neutrons is accompanied by the emission of prompt y rays, presumably coming directly from excited primary The energy of these prompt y rays is about 5 Mev. fragments. In addition, delayed y rays accompanying the fission decay chains are observed. About 5 per cent of the total energy release in fission is in the form of y rays. The emission of a neutrino must be assumed to accompany each 0 Neutrinos. Some 10 per cent of the available energy in fission ran emission in the decay chain. be estimated to appear in this form. The neutrinos so emitted and their energy are not detectable by ordinary means. Attempts have recently been made, however, with at least partial success, to detect the theoretically predicted neutrino flux in the

In

Art.

NUCLEAR REACTIONS

4]

4-85

neighborhood of a reactor through observation of the positrons and neutrons that a neutrino is expected to produce (with a cross section of about 10~20 barn) in hydrogen, t The Energy Release in Fission. The total energy release in the fission of a heavy nucleus is readily estimated from the difference in the binding energy per particle in the middle and at the end of the periodic table, to be approximately 200 Mev.

result of a calorimetric measurement for U236, which, however, does not detect of y rays, neutrons, or neutrinos, is 177 ± 5 Mev. The observed (or estimated) energy release in the fission of Um is broken down as follows: The

the energy

168 5 10 5

II

T99

j

§

it

a

y

is

t

y

is

Z

is

it is

^

of

g

It

a

is

See Ref. 49 and also Art. 3.14. for a review of this rather special field of high-energy pliysics Reference 10 is the basic paper on the theory of fission.

is

is

it

§

7

I t t


a

is

is

in

is

2

it

of course, not available in any usable form. The kinetic energy of the prompt neutrons from thermal fission shows a broad dis tribution from a fraction of an Mev up to some 15 Mev and has the general shape shown in Fig. 30. The distribution has Mev, beyond which peak at about falls off exponentially. a controlling factor The form of this energy distribution the practical utilization of fission energy. The average energy of the fission neutrons not sufficiently high to induce further fissions in a nuclide like U"s with high fission threshold, nor low enough to take advantage of the high fission cross sections at low energies of the thermally fissionable nuclides. The distribution in kinetic energy of the fission fragments has been measured in number of cases. has the general form shown in Fig. 31 for U236 and reflects the asymmetry of the fission process. ir Other Types Fission. While fission due to neuIrons of energies up to a few Mev has been the most studied because of its practical impor intensively tance, possible to produce fission in a variety of other wavs. In fact, the addition of excitation — — — energy to any target nucleus in the upper half of the "^q 50"6b^TO 80 90 160 110 energy in mev periodic system may be expected and observed to give fission, although the cross sections are very low Fio. .31. Energy distribution of the fission fragments from U»". for < 83 (Bi). The distribution in mass and in energy of the fission products, the distribution in energy of the prompt neutrons, and similar properties vary only slightly from those of neutron-induced fission when the bombarding particle a proton, deuteron, a particle, or ray, at least at moderate bombarding energies. At very high energies '>100 Mev) more radical changes appear, Fission induced by rays, known as photofission, of special theoretical importance, since the threshold always observed in this process gives a direct measure of the pxcitation energy required to produce fission by a neutron. Among the heavier a emitters, including the uranium isotopes, spontaneous fission of frequent occurrence. It can be thought of as a type of radioactive decay gov The life erned, like a decay, by penetration of coulomb barrier (cf. Art. 4.321T). times are extremely long, for example, 10" years in U238, which corresponds to about X 10~' fission per gram per second. 4.32 The Theory of Fission. According to the general picture of nuclear reac tions given in Art. 4.23 the fission reaction can be regarded as a two-stage process, formation of a compound nucleus followed by its decay. Thus in the fission of U235 the division of the compound nucleus U2'6 that involved. by neutrons a

The neutrino energy

Ma

energy of fission products energy of fission neutrons of 7 rays of 0 rays of neutrinos

is,

Kinetic Kinetic Energy Energy Energy Total

see Ref. 52.

NUCLEAR PHYSICS

4^86

[Sec.

4

The cross section for neutron-induced fission [cf. Eq. (123)] may be written


»r(nj)

=

oc(n)

l'

(133)

when the only processes competing with fission (width T/) are y emission (width r7) and neutron emission The cross section
Ef

=

E(A)

- 2fi(^)

m

-Af(A)

+

2|/(|)

=

A

[/(|)

-/(.!)]

(134)

Consequently, the fission reaction is exothermic as long as A/2 is greater than the A at which f(A) has a maximum. If the maximum is taken at A = 60, the above estimate would imply that all elements with A > 120 would lead to energy release on division into two equal parts. A more careful estimate, based on the liquid-drop model, which takes account of the fact that the fission fragments do not have the same Z to A ratio as the stable isotopes, places the limit at A =81 (bromine). It will be recalled that the fundamental reason for the decrease with Z of the binding energy per particle which makes fission of the heavy elements possible is that the negative contribution to B of the electrostatic repulsion between protons increases rapidly with Z (cf. Art. 2.23). The order of magnitude of the energy release in fission can be obtained directly The maximum value of f(A) is about 8.5 Mev, while for an element from Eq. (134). A more with A = 230 it has gone down to about 7.5 Mev, giving Ef ~ 230 Mev. accurate estimate of the energy available, taking account of the asymmetry the process and using the measured masses of the products, gives 200 Mev. Even when fission is energetically possible, its spontaneous occurrence is a rare event and, in general, excitation of the nucleus is required to initiate fission. The existence of such a threshold is understandable on the basis of the liquid-drop model. In its original spherical form the nucleus is stable for at least small deformations. To distort it to an elongated form, however, as a first step toward division requiros the

of

Art.

NUCLKAR REACTIONS

4]

4-87

addition of energy (such as binding energy and kinetic energy of a neutron), since the surface-to-volume ratio is increased in such a process. In terms of the usual picture of short-range nuclear forces there has been an increase in the number of "unsaturated" nucleon bonds at the surface, which implies a rupture of bonds in the interior (surface-tension effect). Sufficient excitation energy can produce com plicated vibrations and distortions of the drop as a whole, including its partial separa tion into two halves to a point where the electrostatic repulsion takes over and the fragments fly apart. This situation may be described in terms of a potential barrier for fission. As illustrated in Fig. 32 the poten tial energy is simply that of the coulomb repulsion (plus the constant internal energy of the fragments) as long as the two parts 5 are well separated. When they merge, the energy can, of course, no longer be plotted as a function of distance alone, but the curve may be completed in a schematic fashion to the fission energy Ef, leading to a poten tial barrier closely resembling that for a separation of fission fragments

J

emJ*slon-

....

,

representation

Fici. 32. Schematic

,

of the

in hsswn may be po((,Iltia, barrier for fission estimated as the coulomb energy (Ze/'2) '/r,, where r„ is the separation of the fragments at the top of the barrier. With rc taken as of the order of magnitude of the radius of a heavy atom, the energy is again about 200 Mev. The potential barrier picture provides a good qualitative explanation of many of the general features of fission. For example, spontaneous fission is expected to occur as a consequence of quantum mechanical "leakage" through the barrier, in close The probability of spontaneous fission however, quite analogy to a emission. low because the large masses of the fragments imply short do Broglie wavelength and hence nearly classical behavior. The threshold for photofission measures directly the excitation energy required to involved for photons. The measured produce fission, since no binding energy threshold energies are about Mev for the common fissionable nuclides, that of the order of neutron binding energies. Fission by thermal neutrons, which provide only binding energy, thus a possibility. The fact that the neutron binding ener gies and the excitation energies are so close together makes understandable the rule that fissionable nuclides with an odd number of neutrons have lower fission thresh olds than those with an even number of neutrons, for neutron more strongly attracted to nucleus containing an unpaired neutron, as noted in Art. 2.23. From the semiempirical binding energy formula [cf. Eq. (91)] can be seen that the differ ence nucleus with an even number of neutrons and one binding energy between = 238, with an odd number of neutrons at least 135/^1, which, for 0.57 Mev. An energy of this magnitude can be sufficient to make up the difference between the neutron binding energy and the excitation energy. This exemplified in the case of U*** and U"6. The binding energy of neutron in U"9 (the compound nucleus involved in Ua3B fission) less than the excitation energy of 5.08 Mev observed neutron in the even Z-cven The increased binding energy of photofission. nucleus Usse, however, must be greater than the observed excitation energy of 5.31 0.27 Mev. (The accuracy of the theoretical estimates of the neutron binding energies not quite sufficient to make the comparison quantitative.) Extensive calculations! have been Further Development the Liquid-drop Model. nucleus as undertaken for the purpose of evaluating the energy of function of The model used that of classical liquid parameters specifying its distortion. drop with surface energy and coulomb energy; the parameters are the first two nonfunction vanishing coefficients in the expansion in zonal harmonics of the radius as release

is

A

0. 1 5

±

±

a

in

is

a

is

is

A

a

in

it

a

a

is

is

5

is,

is

a

is,

energy

a

a

a

is a

of

is

t


The

Set Fief. 10.

4-88

NUCLEAR PHYSICS

[Sec. 4

it

J

is

is

is,

From a study of the energy surface thus obtained the excitation energy of angle. This energy will correspond to a "saddle required for fusion can be computed. point" over which the system must pass. Calculations of this sort give, however, only qualitatively correct results and do not account for the asymmetric character of the process. Calculations on high-speed computcrsf have extended the accuracy of the determination of fission excitation energies. One simple consequence of the liquid-drop model of fission is that the excitation energy in the two-parameter representation should be a function only of Z'/A, that of the ratio of the coulomb energy Z'/A^ to the surface energy AH of the original spherical drop. The asymmetry of fission remains without any clear explanation, although there are indications that in some way related to shell structure in that the two broad peaks of the fission product yield curves lie in the vicinity of the magic neutron num bers 50 and 82. The problem of interpretation of fission on the basis of the collective model at present receiving much attention. 4.4

Fusion

is

is

§

if

is

is

it

a

a

1

is

a (k is

E

T

'\

it is

is

I t t

See Rcf. 21. See Rcf. 31. the Atomic Energy Commission has released a considerable Since this handbook was undertaken, snmMuch of this amount of information on thermonuclear reactions that was previously classified. marised in an article by R. F. Post [Rev. Mod. Fhyt.. 18: 338 (1950)]. more convenient to measure Hnthcr than expressing temperature in degrees in such a connection ■=kT This energy two-thirds of the temperature the Boltzmann constant). by the energy A average kinetic energy of a particle for Maxwell distribution of velocities at the temperature T. temperature of ev corresponds to 11,000°K.

ft


/

A

is,

is

A

is

A

The possibility of using the heaviest elements as a source of power has been seen greater than about 50 the (Art. 4.32) to depend on the fact that for nuclei with binding energy per particle/ increases as this which decreases (cf. Fig. 15). It makes fission an exothermic process. just For the light elements the situation = 50. increases up to about reversed; that Hence the union of two nuclei to form a single one increases the total binding energy. Such a fusion thus leads to a release of energy. The essential reason for the increase in binding energy with fusion from the standpoint of the liquid-drop model that in the larger nucleus a smaller fraction of the nucleons are at the surface where they are less tightly bound. 4.41 Thermonuclear Reactions. Release of energy on a macroscopic scale through the fusion of light nuclei can take place only when both reactants are present at densities higher than those which can be obtained in accelerators and with kinetic The energies high enough to overcome the effects of the coulomb potential barriers. conditions for such "thermonuclear reactions" will exist in matter at necessary sufficiently high temperature, the kinetic energy being supplied by thermal agitation. That fusion of light elements the principal source of stellar energy now generally accepted. The temperatures required to produce thermonuclear reactions with even the lightest elements would be of the order of millions of electron volts|| the average thermal kinetic energy were required to be higher than the top of the potential barrier. However, appreciable reaction rates can be expected at temperatures of a few kilovolts when taken into account that many nuclei will have energies far above the average of the Maxwell distribution and that barrier penetration takes place below the classical threshold. The calculation of the rate of a thermonuclear reaction involves the averaging over energy of the product of two factors: the cross section as function of the energy which contains the barrier penetration factor and the number of particles per unit energy range in the Maxwell distribution at the given temperat ure. Because of thoir lower couloznb barriers the very lightest elements are the most favorable for the production of thermonuclear energy. The fusion of two protons would lead to the formation of the unstable nucleus He2, followed by emission of a neutrino, and does not have an observable cross section. Clearly the positron and

Art.

nuclear reactions

4]

4-89

likely reaction is that of two deuterons which takes place probability in two different ways: most

with nearly equal

ll2 + Ha -» He' + n + 3.2 Mev H2 -> H3 + H1 + 4.0 Mev

IIs +

rate at which these reactions would liberate energy in deuterium at its liquid estimated as a function of temperature. The number of reactions per cubic centimeter and per second is given by \£NvNv in deuterium with N particles At a per cubic centimeter of relative velocity v and for a reaction cross section a. given temperature the product av must be averaged over the Maxwell distribution of velocities. The rate of energy generation per cubic centimeter is then The

density can be

Energy /(cm3) (sec) =

^JV'oiQ

(135)

the above average and Q is the energy release per reaction. The cross less than the top of the coulomb barrier is a very rapidly increasing function of the temperature, since it is governed primarily by the Gamow factor which depends exponentially on the velocity [cf. Eq. (73)]. The actual values of the energy production rates in liquid deuterium can be estimatedt with sufficient accuracy for illustrative purposes. The results at a few selected temperatures are shown below: where 55 is

section a at energies

°K I0*(10cv)


Temperature,

3

X

10«(86ev)

10" (260 ev) 10' (860 cv)

Energy

production,

ktp/ff

10" I I0» 10'"

While at temperatures readily attainable in the laboratory the energy production entirely negligible rate, at temperatures of the order of 100 ev it attains explosive proportions. Thus, although deuterium is a potential source of energy, the problems of containment (the pressure at 100 ev and the density of liquid deuterium is about 10* atm) and control are of a different order of magnitude than any yet suc cessfully solved. is at an

Quite early in the development of modern by the stars could not gravitational contrac for the main source of known from geological evidence to have been producing energy at about its present rate of 2 ergs per gram and per second for a period longer than 10' years. Chemical burning could have produced energy at this rate for only about a millionth, and gravitational attraction for only about a hundredth, of this known lifetime of the sun. Natural radioactivity and fission can be eliminated as energy sources if for no other reason than the absence of heavy elements in stars. Fusion of the lightest elements (cf. Art. 4.41), of which the sun and stars are mainly composed, was speculatively considered as a possible source of stellar energy even The before nuclear reactions were understood to the extent that they are today. growth during the last 20 years of a detailed theory of nuclear reactions and the accumulation of measurements of nuclear cross sections have been paralleled by the development of a theory of stellar structure and evolution based on fusion as the primary energy source. It is generally assumed that, in the course of evolution, con traction under gravitational forces has produced central temperatures and densities which can sustain thermonuclear reactions. The commonest stars (the so-called "main sequence," which includes the sun) are composed almost entirely of hydrogen and helium, heavier elements amounting to only 1 or 2 per cent by mass. Central temperatures are deduced to be of the order 4.42

Energy

Production

in Stars.

astrophysical theory it became clear that the energy radiated be attributed to chemical burning or to heating accompanying tion. The evidence excluding these two obvious possibilities stellar energy is most convincing in the case of the sun, which is

t Reference 22, chap.

10.

4-90

NUCLEAR PHYSICS

[Sec.

4

of a kilovolt (10'°K), and mean densities range from 0.3 to 7 ft/cm', with central Under these conditions the burning of densities higher by a factor of 10 or more. deuterium, if it were present even in small amounts, would proceed at too rapid a rate to be consistent with the evidence for a relatively low rate of energy production over a long period of time. The currently accepted source of stellar energy is rather Such a the synthesis of helium out of four protons, which is a much slower process. synthesis cannot take place directly but can occur by a sequence of nuclear reactions in a variety of ways. It is currently believed that the two most important of these are the proton-proton chain and the carbon-nitrogen cycle. The reactions involved in the prolon-proton chain are

H' + H' H2 + H1 He3 + He'

-» HJ + 0+ + -» HcJ + T

v

He4 + 2H1

The first two reactions must occur twice to where fl+ is a positron and v a neutrino. produce the two He' nuclei for the third, so that the net result is


4H' —

He4 + 2/3+ + 2y + 2v + 26.7 Mev

All the thermonuclear reactions involved are dependent on temperature through the coulomb barrier penetration factor in the same way as the d-d reaction discussed in Art. 4.41. The first p-p reaction has a mean reaction time, under typical stellar conditions, which is much slower than that of the succeeding two. It accordingly Although the cross section for this governs the rate of hydrogen consumption. reaction is too small to be measured in the laboratory, it can be calculated with some The hydrogen lifetime of about 109 years which results from such a calcula accuracy. tion is of the order of magnitude indicated by astrophysical theory and measurements. The carbon-nitrogen cycle consists of the following sequence of reactions: C» +

N»

C" N14

+ + +

N"

+ y + v H1 -> N'» + y H'-> O" + y N" + /3+ + v H' -* C11 + He4 C'» +

0ls->

N"

H> ->

0+

The net result is again the synthesis of He4 from four protons, with no consumption of carbon or nitrogen. The first reaction is the slowest and governs the rate of hydrogen consumption, which is less by a factor of about 10_s than in the protonproton chain. While thermonuclear reactions involving other light elements may be of importance in some cases, the bulk of the energy production in the sun and similar stars is now attributed to the two processes described above. The establishment in astrophysics of a definite energy source, calculable in its dependence on density and temperature, has supplied a previously lacking element necessary for a systematic development of a theory of stellar constitution and evolution. f 6

NEUTRON PHYSICS

Since the neutron is so intimately involved in almost all aspects of nuclear physics has, of necessity, been included elsewhere in this article much that could be called "neutron physics." The additional material in this articlef is mainly con Neutron diffusion is cerned with further details of neutron-induced reactions. treated in Sees. 6-1 and 6-2. there

t For an account of such theories and a more detailed treatment of nuclear reactions in stars see Ref . 52 and Ref. 22, chap. 10. The inter X For additional reading on the material of this article see ReiB. 18; 30, chap. 18; and 35. It is covered, action of neutrons with matter in the bulk has not been included here for lack of space. for example, in Kefs. 18 and 33.

Art.

NEUTRON PHYSICS

5]

4-91

Sources of Neutrons

6.1

The principal sources of neutrons for experimental purposes are enumerated below: Radioactive (a,n) and (y,n) Sources. The first neutron sources to be developed made use of the or and y radiation from naturally radioactive elements to Such sources are still exten produce neutrons by bombardment of light elements. sively used because they are relatively inexpensive and of small size. Alpha-neutron Sources. The commonly used reaction is Be,(a,»)Cl,> which is the reaction in which the neutron was first discovered. Radium and polonium are most Neutron sources of a centimeter or so frequently used as a source of the a particles. in diameter can be prepared by mixing these radioactive substances with beryllium The energy spectrum of the neutrons is broad, extending from 1 up to 10 powder. Mev or more. The Po-Be source has the advantage over the Ra-Be source in that it is very nearly free from y rays. Photoneutron Sources. Natural y emitters may be used in place of a emitters to However, much stronger y produce neutrons by bombarding the lightest elements. 5.11

All can be produced artificially in reactors and are now used exclusively. make use of the Be*(-y,n)Be* or Hs(-y,n)H' reaction, the y rays being provided by a variety of substances activated in a reactor. Photoneutron sources produce approximately monoenergetic neutrons (20 to 60 per cent spread in energy), the energies ranging from 100 to 600 kev depending on the y source and target used. Neutrons of higher energy can be produced by using the 7 rays from high-energy sources

such sources

Neutrons from Particle Accelerators.

6.12

Accelerated protons, deuterons, and

tritons produce neutrons copiously with the lightest elements as targets. The great advantage of such neutron sources is that for a definite bombarding energy the neu trons observed at a definite angle are very nearly monoenergetic (cf. Art. 4.22). With the use of Van de Graaff or cyclotron accelerators it is possible in this way to produce neutrons from a few kev up to about 20 Mev. The principal reactions used arc the following: U>(p,n)l\e\ Li7(p,n)Be', II2(rf,n)Hcs, II3(
distribution in energy. Thermal neutrons relatively free from higher energy components are obtained by allowing neutrons from the interior of the reactor to pass through a block of moderat ing material (water, paraffin, or graphite). Such an arrangement is called a thermal

is, a

column. A pure fission spectrum may be obtained from a reactor by the use of a converter, that foil of fissionable material bombarded by neutrons from the thermal column.

5.2

Interaction of Neutrons with Nuclei

it

it

6.21 Introduction. The unique properties of the neutron which make of such decisive importance in nuclear physics and engineering come primarily from the fact that electrically neutral, so that its electrostatic interaction with electrons and is is

is

it

it

It

almost completely negligible. for this reason that even low-energy neutrons can enter a uranium nucleus to produce fission without being hindered by electrostatic the absence of charge which makes repulsion. Again, possible for free neutrons (in contrast to free protons, which would pick up electrons and make nuclei

is


accelerators.

hydrogen molecules) to exist for relatively long times in equilibrium with matter.

NUCLEAR PHYSICS

4^92

[Sec. 4

Wave Properties. The de Broglie wavelength X of a neutron of momentum p is In the nonrelativistic region (energy less than 50 Mev) the wave given by X = h/p. length may be written, for a neutron of energy E and velocity v, X =

4-

After inserting the numerical values of written X = 2.86

where

E

i—

=

(136)

h and the mass

M

of the neutron, this may be

'/VE

cm

(137)

X

10"

is in electron volts.


This neutron wavelength varies from about

10~8 cm for thermal neutrons to about 10_" cm for 100-Mev neutrons. The wave properties of the neutron, as of any particle, become more important than its particle properties when it interacts with an object having dimensions (or inhomogeneities) extending over distances that are of the order of or smaller than the wavelength of the neutron. The wide range of variation of its wavelength for attainable energies makes the neutron a most flexible tool in nuclear research. Thus slow neutrons may be used for the investigation of crystal structure, since X is of the order of interatomic distances (10~8 cm). At higher energies X becomes of the order of nuclear radii and at the highest attainable energies even of the order of magnitude of the distance between nucleons in a nucleus. Classification According to Energy. A number of classifications and terminologies have been used to describe neutrons of different energies. One that is now gaining some general acceptance is the following: Thermal neutrons: This term is applied to neutrons in thermal equilibrium with matter at a temperature T which have a Maxwell distribution in velocity and energy For T = 20°C the velocity has a maximum at (modified slightly by absorption). 2,200 m/sec and the corresponding neutron energy is 0.0252 ev.

E = 1,000 ev to X = 0.904 X 10" 10 cm 1 to 500 kev Intermediate neutrons: E X = 0.904 X 10"" to 4.04 X 10"11 cm E = 0.5 to 10 Mev Fast neutrons: X => 4.04 X 10"11 to 0.904 X 10~" cm Very fast neutrons: E = 10 to 50 Mev X 0.904 X 10-" to 4.04 X lO'12 om

Slow neutrons:

E

0 to

X = =

Ultrafast neutrons:

E

*

> 50

Mev

X < 4.04

X 10-"

cm

In discussing the characteristic interactions of neutrons in these groups with nuclei it is convenient to make use of the following arbitrary classification of nuclei according to mass number: Light nuclei: A = 1 to A = 25 Intermediate nuclei: A = 26 to A =80 A > 80 Heavy nuclei: The Nuclear Interactions. The basic reactions of neutrons with nuclei are the following: (n,n) Elastic scattering : (n,y) Radiative capture: (»,«') Inelastic scattering: Fission : (n,/) Charged particle production: (n,a), (n,p) Neutron production: (n,2n), (n,3n), . . .

Art.

NEUTRON PHYSICS

5]

4-93

Brief descriptions of the first three processes have already been given (cf. Art. 4.2)), Charged particle and of fission have been discussed in Art. 4.3. neutron production are processes occurring at higher neutron energies and are treated in this section. and some aspects

It was the discovery in 193G of slow neutron Theory of Neutron Resonances. that the fact that the cross section for capture of such neutrons sharply peaked at certain energies characteristic of the target nucleus, that led to the It true that development of the current form of the theory of nuclear reactions. such resonances are to he expected on the basis of even the simplest model, such as a potential well representing the action of the target nucleus on the incident neutron. With model of this kind, however, impossible to account for the observed notably for the fact that at a resonance for capture properties of neutron resonances, the scattering only rarely abnormally high. The necessary flexibility for the inter of neutron resonances is, however, provided by the compound nucleus pretation In this theory (cf. Sec. page 75) capture and scatter theory of nuclear reactions. ing (reemission of neutron) are competitive and independent processes, the ratio between them depending on the relative widths for neutron and 7-ray emission once The resonance the compound nucleus has been formed. then due to an abnormally high probability for compound nucleus formation when the binding energy of the neutron plus its available kinetic energy exactly equal to the excitation energy of level of the compound nucleus. for a reaction cross section when only one The general Breit-Wigner expression level In the form given involved has already been written down in Eq. (132). there applies directly to the capture process (n,y) but requires an extension, as given below, when applied to elastic scattering (n,n) in order to take account of the interference between resonance scattering and at least part of the potential scattering. For nearly all observed resonances the neutron energy low enough so that only the = 0) component of the incident beam s-wave involved (cf. Art. 4.23), and the formulasj for the capture cross section a{n,y) and the scattering cross section o-(n,n) will be written down here explicitly only for this case: is

is

is

it

((

"*»V-B™W X I

±

(l

+ 4xK»(l

_L_j

-

g„)

(139)

for/>0

(140)

meanings:

energy provided

particular level of the compound nucleus by incident neutron (separation energy plus

a

energy above ground state of

excitation

„

+R

(138)

R is is

r

is

kinetic energy) T. and Vy are widths for neutron and 7-ray emission, respectively = T, 4assumed that only neutron emission (It rT = total width. getically possible) Xis de Broglie wavelength of the neutron

is

E

En

is ia

The symbols have the following

p0

iT/2

i

fOr/=0

1

go =

HV„

Eo

+

-

ener

radius of target nucleus

J

I

H

/

0

•

J

is

a

statistical factor which gives the probability that the neutron and the target nucleus have a relative orientation such that the compound nucleus formed with the spin of the level involved in the resonance. The only possible values of for = — }£, corresponding respectively to the neutrons are ar>d upper and + lower signs in Eq. (140) go

The form of Eq. (139) for the scattering cross section o(n,n) requires special cornThe first term in the absolute square gives the amplitude for resonant reSec Ref. 43, p. 65. for details. Derivations will be found in Ref. 43, sec. 6B, and Ref.

8,

went. t t


is

a

is

is

a

4,

is

t

a

it

is

is

is,

5.22

resonance,

chap. 8.


4-94

NUCLEAR PHYSICS

[Sec.

4

3X10-1

-600

-400

0

-ZOO

200

400

600

r Fig.

(b) 33b.

resulting

The scattering cross section farther from resonance showing the asymmetry from interference between resonance scattering and potential scattering.

emission of a neutron from the compound nucleus (resonance scattering) and the second, U/\, the amplitude for reflection of a neutron at the nuclear surface of radius Ji without compound nucleus formation (potential scattering). These two amplitudes are coherent and can interfere with each other; hence they must be added before squaring to give their contribution to the cross section. There is in addition the last term Airli1^ — g0) which gives an incoherent part of the potential scattering coming from relative orientations of neutron and target nucleus that do not lead to formation of a compound nucleus with the correct value of the spin. The potential scattering is here assumed to be that of an impenetrable sphere with It
4-95

NEUTRON PHYSICS

Akt. 5]

The Breit-Wigner formulas given above are not in their most general form, being to neutrons with / = 0 and to the case in which only a single level con The extension to I > 0 involves, among tributes to the absorption and scattering.

restricted

multiplication of both a(n,y) and 0) have so far sufficed for the analysis and interpretation of the great majority of the observed neutron resonances. The general form of the cross sections as given by Eqs. (138) and (139) in the immediate neighborhood of a resonance is shown in Figs. 33a and 336. Methods for determining the parameters E„, F„, Ty, and q0 from the observed cross sections an a function of E are described in Art. 3.21 of Sec. 5-2. The ratio of capture to scattering cross sections for a given resonance is of the order of T,/r7, as can be seen from Eqs. (138) and (139). Accordingly there can occur resonances in which one or the other predominates: scattering resonances when P, » Ty and capture resonances when r„. The experimental study of capture and scattering cross sections near resonances has given striking evidence of the validity of the Breit-Wigner equations. This " neutron liranch of spectroscopy" is providing a rapidly increasing volume of informa tion on neutron and radiation widths and on the energies, spins, and other properties of the excited levels of nuclei. Aside from their importance for an understanding of the theory of nuclear reactions, such studies are of considerable significance in reactor largely determine the behavior in physics, since the properties of the resonances pile materials of the neutrons released in fission. 5.23 Elastic Scattering and Radiative Capture. These two processes are so Their cross intimately connected that it is profitable to consider them together. sections are governed by the positions and properties of the energy levels of the com pound nuclei involved (resonances) and by the widths r„, r7 (together with the widths for any other processes that may occur at higher neutron energies). The general tendency is for the level spacing to decrease with increasing excitation energy and increasing atomic number. The order of magnitude of the neutron widths for / = 0 neutrons and for inter K, a condition that is mediate and heavy nuclei is given by Kq. (131) which, for k ■atisfied except for ultrafast neutrons, may be written other things,

of the

»

(141) where

k is the

channel wave numbcr + of the neutron outside the nucleus, K (of the number inside the nucleus, and D the average level spacing. for the neutron widths Eq. (141) may be written

order 10~'*cm~') its wave As a crude approximation

r, «

1.4

X

10-* £W

D

(142)

E

The level spacing D (in electron volts) is the (channel) energy of the neutron. levels of the same total angular momentum and parity. The observed neutron widths are only roughly iu accord with this relation, and there is an obvious A first step in this direction lias recently been necessity for revision of the theory. taken in the so-called "clouded crystal ball" theory} which attempts to modify the usual form of the compound nucleus theory in a direction suggested by the strong evidence for the partial applicability of the independent-particle model (cf. Art. 2.31). Its essential feature is the introduction of an absorption of neutrons which implies that compound nucleus formation takes place in only a fraction of the processes. where

is that between

'2MB/h,

where

E

is

=

V

is.

t That the wave number roast. See Ref. 20.

k

t There is an extensive scries of papers in the Physical Review by Winner and collaborators, lent of which is summarized in Ref. 8. chap. 10.

I


I\ »

the channel

energy of the neutron

the conand

M its

4-96

NUCLEAR PHYSICS

[Sec. 4

While the neutron widths can be at least estimated on the basis of present theory and can be expected with some confidence to be proportional to E'<1, the radiation width rT must be treated as an essentially empirical constant in the Breit-Wigner formulas, although the theory can give a not unreasonable estimate of its order of The observed radiation widths vary from 1 to 30 ev for light and inter magnitude. mediate nuclei and are usually only a fraction of an electron volt for the heavy ele ments. They do not vary in any systematic way with excitation energy. The Cross Sections Near Zero Energy. The 1 /v Law. As the energy of the incident neutron approaches zero, the capture cross section shows a typical behavior which can be deduced from the Breit-Wigner formula. It may be assumed that the capture is governed by a nearest resonance at an energy En and that the resonance denominator (E — En)2 + (T/2)! may be considered as practically independent of the neutron energy E as E approaches zero. The only significant variation in a(n,y) is then that due to X2 and r„, making a(n,y) ~ X2r„, the radiation width being independent of energy. From Eq. (142) and the fact that X2 ~ l/E it follows that

l\


. .
=

const

—

iV

=

const

.

...

(143)

where v is the velocity of the incident neutron. This l/v law " for capture of neutrons at energies below the first resonances is well verified experimentally. Although derived here from the Breit-Wigner formulas it has actually a much more general justification, depending only on the fact that for very slow neutrons the wavelength X is practically infinite compared with the nuclear radius. The rate of absorption of neutrons can then depend only on their density at the nucleus, which is measured by the square of their wave function I/-!2. Since the incident current is then v\i//\2, the cross section is a — (absorptions /sec) + (incident current) ~ |^!2/t|^!2 = l/v. It is clear from this general argument that the l/v law is not confined to radiative capture but applies to any nuclear reaction (except scattering) that is initiated by very slow neutrons. In contrast to the capture cross section the scattering cross section at vanishing neutron energy is expected to approach a constant value of the order of 4rR* as illus trated in Fig. 336. Potential Scattering. The Scattering Amplitude. Neutron scattering at energies far from a resonance and when, as is very frequently the case, radiative capture is negligible, can be approximately treated as scattering in a potential field which is not known in detail but which has the genera] character of a rectangular well some 20 to 30 Mev in depth and having a width of the order of the nuclear radius. Such a model for pure potential scattering has proved adequate for the description of the elastic scattering of slow neutrons on lighter nuclei where it may be considered as a refinement on the treatment by resonance theory which predicts simply a scattering cross section of the order of -ItcR1. For the cases of most interest the neutron wavelength is some orders of magnitude larger than the nuclear radius, so that only the I = 0 interaction need be considered. Under these circumstances the method of partial waves (cf. Art. 1.36) gives for the scattering cross section a, = 4xX2 sin2 So, where So is the f = 0 phase shift [cf. Eq. (66)1. When X is very large, the phase shift is simply So = — 2ira/X = — ha, where a, called the scattering length,] is the intercept on the r axis of the external wave function as illustrated in Fig. 34. In terms of the scattering length the scattering cross section for slow neutrons far from resonance is simply a, ' 4ira2. It is independent of energy, and the scattering is symmetric in the center of mass system. It can be shown that for heavy nuclei a is approximately equal to the nuclear radius For lighter elements a is to be considered as a parameter to be ]{, so that
discussion

of the scattering

amplitude and ttie part it plays in "neutron

Art.

NEUTRON PHYSICS

5]

4-97

calculated when a law of force is assumed between nucleons. The experiments scattering have accordingly played an important part in the working out of nuclear theory. Unfortunately, however, this scattering, like the binding energy of the deuteron (cf. Art. 2.22), is not sensitive to the detailed shape of the potential for the n-p force, t Radiative capture of neutrons by pro tons has a quite small cross section (see below) and may be neglected in the discus sion of the scattering. The observed cross section for scattering by hydrogen when the target is a hydrogenous substance such as 1 ' 1 ■^ //A1 water varies with energy as shown in Fig. < / i 35. To account theoretically for the ap proximately constant cross section of 20 bams between 1 ev and 10 kev, it is neces R sary to consider the strong spin depend -i ence (cf. Art. 2.22) of nuclear forces (in I fact it was in this connection that the Fio. 34. Illustrating the relation between existence of such a spin dependence was the scattering length a and the zero-order phase shift 4o •» —ka. first discovered by Wigner). In an actual scattering experiment three-fourths of the collisions occur with neutron and proton having parallel spins (triplet state) and onefourth with the particles having antiparallel spins (singlet state). An approximate value of the scattering amplitude for the triplet-state collisions can be deduced from the theory of the ground state of the deuteron, which is a triplet state (spin 1). The rately


on n-p

100

80)

ENERGY

Fio.

35.

The scattering

IN

ev.

cross section of hydrogen

in water.

cross section thus estimated is only about 3 barns compared with the observed 20 barns. The difference must be attributed to singlet-state scattering which is much stronger than that in the triplet state. The decrease in the n-p scattering from 10 kev to 10 Mev is well represented by Above 10 Mev, angular momenta extrapolation of the theory for lower energies. greater than zero begin to have an effect, exchange forces should come into play, and their influence can be clearly seen in the angular distribution of the scattering between 10 and 260 Mev. Extensive theoretical studies of the n-p scattering, together with the p-p scattering at high energies, have, however, so far failed to give a quanti

tatively satisfactory recipe for the fundamental law of force between nucleons. The abrupt rise of the scattering cross section to about 80 barns which occurs below 1 ev cannot be accounted for on the basis of the nuclear forces. It is an effect t For a detailed

account

of the theory

see Ref. 8, chap. 4.

NUCLEAR PHYSICS

4-98

[Sec.

is

is

2

is

1

is

E

is 1

A

E

a

it

is

is

a

is

is

A

a

is

is,

is

a

is

is

is a

it

is

y

is,

of the chemical binding of protons in the target, and for free protons the cross section would remain constant below 1 ev at the low-energy value of 20 barns. The scattering experiments are necessarily carried out with the protons bound in some hydrogenous This is of no consequence as long as the neutron substance such as water or paraffin. energy is much greater than the proton binding energy of a fraction of an electron volt, since the protons arc ejected by recoil with practically the same energy as a free proton would have. However, at lower neutron energies the molecule as a whole The result of the will recoil with a very low velocity because of its large mass. inability of the proton to "run away" from the incident neutron is a closer collision It can be shown in a with an increased interaction and hence larger scattering. number of ways that the result is multiplication of the cross section by the square of the ratio of the reduced mass of the neutron-molecule system to that of the n-p system. For a molecule of mass M this factor is 4/[l + (1/Af))*. Radiative Capture of Neutrons by Protons. This capture is the inverse of the photodisintegration of the deuteron and is one of the few nuclear processes that can be The computed value of 0.32 barn for 2,200-m/sec neutrons is in calculated in detail. The cross good agreement with the experimental value of 0.313 ± 0.013 barn. section has a 1 /v dependence, and capture is hence extremely small except for thermal rays of an The capture of course, accompanied by the emission of neutrons. energy at least equal to the deuteron binding energy of 2.2 Mev. Light nuclei have relatively few and widely spaced levels which give rise almost The concept of a well-defined nuclear radius exclusively to scattering resonances. cannot be applied to such nuclei, and there are no general rules for the variation of the cross sections with mass number and energy. Neutron absorption can result in The processes other than 7-ray or neutron emission even at relatively low energies. an (n,a) rather than an (71,7) reaction. absorption in boron, for example, The Resonance Region. For all nuclei possible to distinguish a neutron energy With medium nuclei range in which there occur numerous well-resolved resonances. this resonance region occurs for slow and intermediate neutrons and with heavy At higher energies the resonances are unresolved nuclei for slow neutrons only. (continuum region). Analysis of the resonances in this region rapidly expanding field of investigat iont which providing fundamental information on level spacing and neutron and radia tion widths. The general pattern one of radiation widths relatively independent of energy and mass number, being usually of the order of a fraction of an electron volt (high thermal absorption cross sections, when they occur, are an effect of the 1/c fraction of an electron volt Neutron widths, on the other hand, vary from law). to some hundreds of kev. found to be roughly The relation between the neutron width and level spacing of the order of 10~4 when the neutron in accord with Eq. (142); that r„/E^-D However, its variation by factor of or more energy expressed in electron volts. as a function of of significance as a test of nuclear models, resembling qualitatively the predictions of the "clouded crystal ball" theory. For low-energy neutrons on heavy nuclei, absorption predominates over scattering. to be attributed to decrease in r» with Analysis of the resonances shows that this Such a decrease in I"„ to be expect«i, Ty remaining nearly constant at about 0.1 ev. since proportional to the level spacing which diminishes with increasing A. The Continuum Region. In contrast to the resonance region there exists what may be called a continuum region in which resonances are no longer observed and in which the general ideas of compound nucleus formation and decay are still at least partially this region sets in when so high that the levels are not For given valid. instrumentally resolvable or have widths comparable to their spacing. This occurs, Mev for intermediate nuclei and at kev for heavy nuclei. very approximately, at the 25 to 100 Mev at which the compound nucleus theory The upper limit on breaks down because the neutron mean free path in the nucleus no longer small compared with the nuclear radius (cf. Art. 4.23). In the continuum region all cross sections are relatively slowly varying functions t


4

See, for example.

Rcfs. 20 ami 39.

Art.

4-99

NEUTRON PHYSICS

5]

With increasing energy, processes competitive with elastic scattering capture begin to appear. The most important of these is inelasticscattering (cf. Art. 5.24JT.), since neutron emission is preferred over charged particle Measurements in this region emission as the mode of decay of the compound nucleus. are almost exclusively of the total cross section at = a, + o>, where a, is the cross section for elastic scattering (n,n) and aT, the reaction cross section as defined in Art. 4.23, includes the radiative capture processes (n,y) and the inelastic scattering (n,n'), with the latter predominating at higher energies. To a rough first approximation the total cross section in the continuum region approaches 2-kR* with increasing energy, which is the limiting value theoretically /f. The predicted for compound nucleus formation for a "black" nucleus when X elastic scattering in this region is almost entirely potential scattering of the kind called "shadow scattering" At still higher energies (see below) the (cf. Art. 4.23). total cross sections eventually begin to fall off from the approximate value of 2x7^. A more refined theory of the total cross sections in the continuum region gives as the next approximation 2jr(fi + X)s rather than 2xfts. At present, the most successful theory of the cross sections in this region is the "clouded crystal ball" theory of Feshbach, Porter, and Weisskopf (cf. Art. 4.23). It is able to account for the general features of the variation of
and

and E. radiative


«

a{n,y)

»*(«

+X)'^

(144)

There are observed exceptionally small values of a(n,y) for nuclides with the magic This is to be attributed to abnormally large neutron or proton number 50, 82, 126. level spacing D for these magic nuclei, and these large values of D have also been directly observed in some cases. The Very High and Ultrahigh Energy Region. The general nature of nuclear reac tions in this region where the theory of compound nucleus formation breaks down has already been noted (cf. Art. 4.23). Above about 25 Mev neutron cross sections fall off with energy below the theoretical limit for a "black" nucleus, indicating the "transparency" to be expected when the neutron mean free path inside the nucleus is no longer small compared with the nuclear radius. The scattering is shadow scat tering, and the reactions are of the spallation type, involving emission of several neutrons and charged particles at sufficiently high energies. X 6.24 Inelastic Scattering. When the energy E0 of the incident neutron exceeds the excitation energy of the first level above the ground state of the target nucleus, inelastic scattering becomes energetically possible, since the product nucleus may then be left in an excited state (possibly higher than the first). Target and product nuclei differ only in that the product nucleus is in an excited state. The energy balance equation is E,> + S„ = Ei + S„ + E, where S„ is the separation energy of the neutron in the compound nucleus and E the kinetic energy of the scattered neutron which, from the above equation, can have one of the values E = En — Ei. Since, for sufficiently high Eo, the product nucleus may be left in any one of a number of excited states, the scattered neutrons will, in principle, have a discrete In energy spectrum, the spacing of the lines being that of the energy levels Ei. practice, however, the spectrum can usually be thought of as continuous, because of either poor experimental resolution or close spacing of the energy levels. The excited product nucleus will emit one or more y rays in its return to the ground state, and

t t

See Ref. 34. See Ref. 43, chap.

11.

NUCLEAR PHYSICS

4-100

[Sec.

4

observation of these y rays provides a valuable source of information on excited nuclear levels and on the inelastic scattering cross section.! A complete description of the inelastic scattering process would be given by the differential cross section dot, = a(E
Jo

(the cross section for scattering to an energy less than E') for isolated values of E<, and E' and of the total cross section for scattering to any energy less than Ed. The threshold for inelastic scattering (energy of the first excited level) is at about 1 Mev for heavy and intermediate nuclei and somewhat higher for light nuclei where the level spacing is large. The general behavior of the total inelastic cross section as a function of energy is in qualitative accord with theory. Just above the threshold energy E,k, the total inelastic scattering cross section is proportional to (Eo — En,)^, where Eo is the incident neutron energy, provided only 1=0 neutrons contribute. This follows directly from the proportionality of r„ to wave number of the emitted neutron [cf. Eq. (131)]. At higher energies the cross section approaches the approxi mate value r(R + X)*, that the total cross section for compound nucleus forma to be expected when inelastic scattering becomes the predominant mode of tion, as decay. At sufficiently high energies, however, other processes, such as (n,2re) or diminished inelastic scattering cross section. (n,p) may compete, resulting in The statistical theory of nuclear reactions provides an approximate expression for the energy distribution of inelastically scattered neutrons which applicable when the product nucleus may be left in any one of a large number of closely spaced levels, as to be expected for fast neutrons on heavy nuclei. The result of such a is

is

is

theory

most conveniently expressed in terms of a nuclear temperature, in which

it

the concept of case gives

do (Eo,E)

=°cyt

e~E/T

dE

(145)

where ac the cross section for compound nucleus formation at the energy Bo and the temperature measured in the same energy units as E. An ap proximate value of the temperature may be ob tained from the level density [cf. Eq. (116)], or may be regarded as an empirical constant. In the few cases where measurements of the energy spectrum are available, Eq. (145) in quali tative agreement with the data. As illustrated in = distribution has maximum at and falls off

2.0

2.5

30

Fio. 36. The type of energy dis tribution expected from theory for inelastically scattered neu temperature

is

trons for a nuclear of 0.5 Mev.

it

1.5

ENERGY IN MEV

T

is

I

.5

T

B

is

6

6

+

B

a

a

is

a

E

Fig. 36 the theoretical energy exponentially at high energy. 6.26 The Angular Distribution in Scattering. The angular distribut ion expressed in terms of the differential scattering cross section da for scattering into the solid angle da. Since the scattering must be axially symmetric with respect to the direction of the incident beam (except for particles of definite spin orientation), the solid angle may be taken as the area on unit sphere included between d6, where and is the polar angle with respect to the incident beam. In this case do = 2r sin d0. The angular distribution, as stated in Art. 1.36, usually expressed in the form do = \f(6)\'da,

where f{9), the scattering X t


a

is

is,

J

amplitude,

See, for example, Ref. 37. For an necount of the theory

see Ref. 27.

has the zonal harmonic expansion in

terms of

Art.

NEUTRON PHYSICS

5]

4-101

shifts as given in Eq. (64). When only the / = 0 component (s wave) of the incident beam is important, as for slow neutrons, is independent of 8 and the scattering is said to be isotropic. When the I = 1 component (p wave) also con tributes, (6) has the form a + 6 cos 8. In the scattering of neutrons by light elements it is necessary to take account of the fact that the above description of the angular distribution applies only in the renter-of-mass reference system. For example, the scattering of s-wave neutrons, which is isotropic in the center-of-mass system, is no longer as observed in the labora The consequences of this and of the transfer of energy to the recoiling tory system. nucleus are of importance in neutron diffusion and are discussed in Art. 3.1 of Sec. 6-2. For fast and very fast neutrons when X
J

predictable way on the parity and spin of individual levels. When many levels involved, a predominance of isotropic scattering is to be expected, since s-wavc neutrons are most readily emitted. 5.26 Neutron Induced Charged Particle Reactions. The principal reactions of this type are the (n,p) and (n,a) reactions. The kinetic energy available to the charged particle 6 is E>, = Q + En, where E„ is the channel energy of the incident neutron. If the reaction is exothermic (Q > 0), it may occur even with thermal neutrons; if it is endothermic (Q < 0), there will be a threshold at E„ = \Q\. | Energy of the neutron in the laboratory system = [(A + l)/A]Q, where A is the mass number of the target nucleus. | The cross section for the reaction as a function of energy is governed mainly by the Gamow factor as long as Eh is not (6) in Eq. (130) for greater than the coulomb barrier height. For a given value of Ei, proton emission will always be more probable than a emission because of the lower height of the barrier for the proton. There are a number of (n,p) and (n,a) reactions for slow neutrons on light nuclei, notably the reactions Li6(n,o)H» with Q = 4.785 Mev, the B10(n,a)Li' with Q = 2.791 Mev, and N"(n,p)C14 with Q = 0.626 Mev. The first two have large thermal cross sections. Such reactions obey the 1/ti law for a considerable range above thermal, since the first resonance will be at a relatively high energy for light nuclei with large level spacing. The Bl0(n,a)Li7 reaction, for example, follows the l/v law up to in a


are

rj

10 kev.

For heavier nuclei, the greater height of the coulomb barrier prevents appreciable emission of charged particles with low energies. Hence, even though the reaction may be exothermic, the cross section for (n,p) and (n,a) processes will be extremely small for slow and intermediate neutrons, becoming appreciable only in the millionelectron-volt range. In general, the cross sections increase rapidly with energy in a way that can be readily computed from the Gamow factor, approaching a con stant, which is small because of competition with other processes, after the barrier height has been exceeded. 6.27 The (n,2n) Reaction. This reaction is a two-stage process: After an inelastic scattering the residual nucleus may be left in a state of excitation sufficiently high to permit the emission of a neutron. The observed result is two scattered neutrons (of different energies) for each incident neutron and a residual nucleus of mass number -4 — 1 for a target nucleus of mass number A, with Z unchanged. The residual nucleus is frequently radioactive, usually a positron emitter. In order to emit a neutron, the nucleus remaining after the inelastic scattering which is identical with the target nucleus) must have an excitation energy at least mual to the separation energy S„ of a neutron from the target nucleus. Since this excitation energy can come only from the kinetic energy of the incident neutron (the binding energy having been lost in the scattering), an (n,2n) reaction has a threshold equal to S„ (and a Q value —Sn). The separation energy of a neutron is at the same t

S«

R?f. 43, p. 57.

4-102

NUCLEAR PHYSICS

[Sec.

4

time the threshold energy for the (-y,n) reaction on that nucleus, so that the (y,n) and (n,2n) thresholds are identical. The thresholds of the (n,2n) reactions are thus well known from nuclear mass data or from reaction energetics. They are highest for lighter elements (10 to 20 Mev) and around 5 to 7 Mev for the heaviest elements. When the statistical theory and the concept of a nuclear temperature are applicable, as will in general be the case for heavier nuclei, the energy distribution of the inelastically scattered neutrons is governed mainly by the factor D = l/p(E) in Eq. (130), with the density of energy levels p(E) given by Eq. (116) in terms of the nuclear This leads, as might be expected, to a Maxwellian distribution in temperature. energy at a temperature T.t If there is no appreciable competition with charged particle emission, as is the case for heavier elements and not too far above the (n,2n) threshold, the (n,2n) cross section as a function of energy and nuclear temperature The result is may be obtained by integration over all emitted neutrons. „(n,2n) =

,c

[

1

-

(l

+

exp

(

-

*^*»)

]

(140)

Just 2?o is the threshold energy and E the energy of the incident neutron. above threshold this gives o(n,2n) ~ (E — E0)'. The general form of variation of
5.

dam, 1955. Bethe, H. A.: "Elementary

3.

Nuclear Theory," John Wiley

4.

Ajzenberg, F., and T. Lauritsen: Ret. Mod. Phya., 27: 77 (1955). Bainbridge, K. T. : Charged Particle Dynamics and Optics, Relative Isotopic Abun dances of the Elements, Atomic Masses, in E. Segr6 (ed.), "Experimental Nuclear Physics," vol. 1, John Wiley & Sons, Inc., New York, 1953. Bergmann, P. G.: "Introduction to the Theory of Relativity," Prentice-Hall, Inc., Englcwood Cliffs, N.J., 1942. "Beta- and Gamma-ray Spectroscopy," North-Holland Publishing Company, Amster

1. 2.

Sons, Inc., New

York.

1947.

15. 16.

J. Ashkin: Passage of Radiation through Matter, in E. Segre, (ed.). "Experimental Nuclear Physics," vol. John Wiley
17.

Fcenbcrg,

Bethe, H. A., and

11. 12. 13. 14.

V

9. 10.

J.

8.

J.

7.

2,

1,

6.

1955.

N.J.,

E.: "Shell Theory of the Nucleus," Princeton University Press. Princeton,

1955.

Feld, B.: The Neutron, in E. Segrrfi (ed.), "Experimental Nuclear Physics," vol. 2. John Wiley & Sons, Inc., New York, 1953. Orear, A. H. Roscnfeld. and by 19. Fermi, E.: "Nuclear Physios," notes compiled R. A. Schluter, University of Chicago Press, Chicago, 1950. 20. Feshbach, H., C. E. Porter, and V. F. Weisskopf: Phys. Rev., 96: 448 (1954). 21. Frankel, S., and N. Metropolis: Phys. Rev., 72: 914 (1947). 22. Gamow, G., and C. L. Critchfield: "Theory of Atomic Nucleus and Nuclear Energy Sources," Oxford University Press, London 1949. " 23. Goldh:iber, M., and S. W. Sunyar: Beta and Gamma Ray Spectroscopy," Intersoience Publishers, Inc., New York, 1955. The derivation

is

J.

18.

t


REFERENCES

Riven in Ref. 18, p. 354.

REFERENCES

4-103

H. : "Classical Mechanics," Addison- Wesley Publishing Company, Cam bridge, Mass., 1950. Harnwell, G. P.: "Principles of Electricity and Electromagnetism," McGraw-Hill Book Company, Inc., New York, 1949. Harvey, J. A., et al.: Phys. Rev., 99: 10 (1955). Hauser, W., and H. Feshbach: Phys. Rev., 87: 366 (1952). Heisenberg, W.: "Nuclear Physics," Philosophical Library, Inc., New York, 1953. Heisenberg, W.: "The Physical Principles of Quantum Theory," University of Chicago Press, Chicago, 1929. Herzberg, G.: "Atomic Spectra and Atomic Structure," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1951. Hill, D. L., and J. A. Wheeler: Phys. Rev., 89: 1102 (1953). Hollander, J. M., I. Pcrlman, and G. T. Seaberg: Rev. Mod. Phys., 25: 469 (1953). Hughes, D. J.: "Neutron Optics." Interseience Publishers, Inc., New York, 1954. Hughes, D. J.: Phys. Rev., 75: 1781 (1949). Addison-Wesley Publishing Company, Hughes, D. J.: "Pile Neutron Research," Cambridge, Mass., 1953. Kaplan, I.: "Nuclear Physics," Addison- Wesley Publishing Company, Cambridge, Mass., 1955. Kiehn. R. M., and C. Goodman: Phys. Rev., 95: 989 (1954). Konopinski, E. J., and L. M. Langer: Ann. Rev. Nuclear Sci., 2: 201 (1953). Levin. J. S., and D. J. Hughes: Phys. Rev., 101: 1328 (1956). Margenau, H., and G. M. Murphy: "The Mathematics of Physics and Chemistry," D. Van Nostrand Company, Inc., Princeton, N.J., 1943. Marshak, R. E.: "Meson Physics," McGraw-Hill Book Company, Inc., New York,

24. Goldstein, 25. 26. 27. 28. 29. 30. .11. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.


1952. 42. Mayer, 43. 44. 45. 46. 47. 48.

49. 50. 51. 52.

M. G.. and J. H. D. Jensen: "Elementary Theory of Nuclear Shell Structure," John Wiley & Sons, Inc., New York, 1955. Morrison, P.: A Survey of Nuclear Reactions, in E. Segre (ed.), "Experimental Physics," vol. 2, John Wiley & Sons, Inc., New York, 1953. Mott, N. F., and I. N. Sneddon: "Wave Mechanics and Its Applications," Oxford University Press, New York, 1948. "Nuclear Data," Nail. Bur. Standards Circ. 499, 1950 and supplements. Panofsky, W. K., and M. Phillips: "Classical Electricity and Magnetism," AddisonWesley Publishing Company, Cambridge, Mass., 1955. Ramsey, N. F.: Nuclear Moments and Statistics, in E. Segr6 (ed.), "Experimental Nuclear Physics," vol. 1, John Wiley & Sons, Inc., New York, 1953. Ramsey, N. F., Nuclear Two-body Problems and Elements of Nuclear Structure, in E. Segre (ed.), "Experimental Nuclear Physics," vol. 1, John Wiley & Sons, Inc., New York, 1953. Reines, F.. and E. L. Cowan Jr.: Phys. Rev., 92: 1088 (1953). Richtmyer, F. K., E. H. Kennard, and T. Lauritsen: "Introduction to Modern Physics," 5th ed., McGraw-Hill Book Company, Inc., New York, 1955. Robson, J. M.: Phys. Rev., 83: 239 (1951). Sachs, It. G.: "Nuclear Theory," Addison-Wesley Publishing Company, Cambridge, Mass., 1953.

53. Salpeter, E. E.: Energy Production in Stars, Ann. Rev. Nuclear Sci., 54. Schiff, L. I.: "Quantum Mechanics," 2d ed., McGraw-Hill Book

New York,

"Quantum Theory of Matter," McGraw-Hill Book Company, Inc., New York, 1951. Spence, R. W., and G. P. Ford: Ann. Rev. Nuclear Sci., 2: 399 (1953).

55. Slater, 56.

1955.

J. C:

2: 41 (1953). Company, Inc.,

57. Wu, C. S.: Rev.

Mod. Phys., 22: 386 (1950).


SECTION

5

EXPERIMENTAL TECHNIQUES BY

BURTON

J. MOYER, Ph.D.,

and Physicist,


DONALD

University

Professor of Physics, University of California Berkeley, of California Berkeley Radiation Laboratory.

J. HUGHES, Ph.D.,

Brookhavcn

Senior Physicist,

National

Laboratory.

JOHN A. HARVEY, B.Sc, Ph.D., Senior Physicist, Oak Ridge National Laboratory, formerly Brookhaven National Laboratory. HERBERT KOUTS, Ph.D., Experimental Reactor Physics Group Leader, Brookhaven National Laboratory. G. K. GREEN, Ph.D., Senior Physicist, Brookhaven National Laboratory.

R HUFFMAN, Ph.D., Assistant Manager, Technical, Atomic Energy Division, Phillips Petroleum Company. F. L. McMILLAN, M.S., Assistant Superintendent, Operations Engineering Branch, Atomic Energy Division, Phillips Petroleum Company.

JOHN

CONTENTS INSTRUMENTS FOR MEASURING RADIATION

5-1

BY BURTON J. MOYER 1 Principles of Radiation 2 Applications to Specific References

Detection Radiations

2 Slowing-down and Diffusion Measure ments 3 Experiments with Reproducing Assem blies 4 Exponential Experiments 5 Neutron Economy References

PAGE

5-2

5-28 5-49

EXPERIMENTAL NEUTRON PHYSICS DONALD J. HUGHES AND JOHN A. HARVEY 6-2

1 2 3 4

Neutron Interactions Fast Neutrons Intermediate-energy Neutrons Thermal Neutrons References

5-3

5-4

1 Experimental

Methods

5-105 5-108 5-117 5-119 5-121

ACCELERATORS BY G. K. GREEN

1 Introduction 2 Straight-line Accelerators 3 Magnetic Accelerators References

5-53 5-57 5-72 5-83 5-99

6-5

MEASUREMENT OF REACTOR CONSTANTS BY HERBERT

PAOI

ENGINEERING TESTING IN REACTORS

BY JOHN R. HUFFMAN AND

KOUTS

1 Static Tests 2 Dynamic Tests

5-102

5-1

5-123 5-125 5-129 5-137

F. L.

McMILLAN 5-140 5-141

5-1

INSTRUMENTS FOR MEASURING RADIATION BY Burton


1

PRINCIPLES

J. Moyer

OF RADIATION DETECTION

The detection of both particle and photon radiation depends usually upon one of the following observable effects of their passage through matter: (1) ionization, cither by the incident radiation or by secondary particles produced through inter action of the primary radiation with the medium, (2) fluorescence of selected mate rials from the decay of short-lived metastable states excited in certain molecules by their interaction with the incident radiation or its secondary progeny, (3) production of developable grains in photographic emulsions. These three phenomena provide the most important principles for the detection of radiation, though other effects are occasionally employed in special cases, such as the metallic photoelectric emission for ultraviolet photons and the Cerenkov radiation for relativistic charged particles. In the succeeding paragraphs the useful nature of the three principal phenomena will be elaborated, and thereafter applications to Since the basic physics pertinent here is specific radiation types will be described. presented in Sec. 4, such information will be quoted without development or derivation. 1.1

Production

of Ionization

by Radiation

1.11 The moving electric fields associated with swift By Charged Particles. charged particles can impart to the electrons of a medium sufficiently strong impulses to cause some of them to escape from their normal atomic binding and thus produce a The collection of these succession of ion pairs along the path of the moving particle. ions from a gaseous medium under the influence of an electric field which controls their migration can give rise to an electrical current pulse whose effect upon electrode potential is detectable. The specific rate of energy loss for a moving charged particle heavier than an electron is given, in ergs per centimeter, by

where m = electronic rest mass e = electronic charge ze = charge carried by moving particle 0 = velocity of moving particle r = velocity of light NZ = number of electrons per unit volume of medium / ■>=mean "excitation potential" for atoms of the medium, approximate value of / for a medium of atomic number Z is

/

= (13.5Z)(1.60

The logarithms are natural logarithms.

6-2

X 10-")

erg

ergs.

The

instruments for measuring radiation

Sec. 5-1]

6-3

2v

1

1

>!J

more apt to be known

0

mc' = (7.

T

E

is

it

7

(1

— = w/c and = where Since electron energy than velocity, more useful to write

—

1

4

& ^± + is

in

+

('";)' /

(*z>

mi;'

I

ax

\

«-

L

_

[in

For electrons the corresponding expression is

+ mc* mc*

_ l)H

1.

1.

0

is

it

1.

3

2

1,

is

is

a

A

Table

Mean Energy Loss w in Electron Volts per Ion Pair Po alphas*.*

340-Mev protons*

36 42 36 32.5 36.8 26.4

35.3 29.9 33.6

6 7 3

Gas

Oxygen

Air

35.5

en.

CO,

Superscript numbers

Electrons

31.5 28.6 25.5

26 . 9«

33.3

32.0'

29.2 27.5 34.5

CIl, •

refer to References

at end of subsection.

The value of w for argon remains essentially constant as the energy of the ionizing Consequently, the number of ions produced in argon closely varied. particle This convenient char related to the total energy lost by the particle in the gas. page 236) where the value not quite true for a particles in air (see Ref. acteristic of uj increases by about 16 per cent as the energy of the a particle approaches zero. is

1,

is

is


is

is

1.

is

is

a

is

=

T

T

-

£

electron kinetic energy + 1H
EXPERIMENTAL

5 4

TECHNIQUES

[Sec.

5

For slow electrons the effect is even larger. Nevertheless, since these effects arc present only at low energies, the values given in Table 1 serve as very useful approxi mations for relating ionization to energy loss in practice. It will be sufficient to present as illus trative examples plots of range and > dE/dx in standard air for a particles (Fig. 2), protons (Fig. 3), and specii'ic ionization of electrons (Fig. 4) up to a few Mev in 6. to i energy. The important characteristics O 41 ■R ANGE for the present concern are the concen tration of energy loss near the range end, the dependence of energy loss upon elec tric charge, and the approximate numbers of ion pairs produced per unit path at various energies. Extensive data are available in Refs. 6, 7, and 8, and in Ref. 7 4 5 8 3 6

\

1, pages

ENERGY (MEV)

Range, and energy loss per centi meter for a particles in air (15°C, 760 mm


180 to 222.

In Art. 3.2 of Sec. By Photons. 4 the processes of photon interaction with matter in the region of X rays and y rays Hg). were shown to be the photoelectric effect, Compton scattering by electrons, and (above 1.02 Mev) the production of electronpositron pairs. The energy dependences and kinematics of these processes are there described. In the photoelectric effect the photon is absorbed by an atom, resulting in the emission of an orbital electron which, after surrendering its atomic binding energy, then dissipates its residual energy in exciting and ionizing the atoms or molecules of Fio.

2.

2

;!

ENERGY (MEV) SCALE 11 < 4 7 5 3 ) 6

/ /

/ / 1 ,«/ 7

10

1.12

ENERGY fMEV) SCALE H 7 5 6 8 9

4

/

/

f

/

'

/ w\ r

f\fi>

0

10

20

ENERGY (MEV) SCALE

Fio. Hg).

3a.

I

3.0

10

Proton range in air (15°C, 760 mm

10

2.0

ENERGY (MEV) SCALE

I

3.0

Fio. 3b. Proton energy loss per centimeter in air (15°C. 700 mm Hg).

the medium. This effect is predominant as photon energies increase up through the region of the if-shell ionization energy of the medium and thereafter decreases in probability in favor of the Compton effect. In Compton collisions photons, with energies distinctly greater than electron binding energies, scatter from individual electrons with conservation of energy and momentum

Sec. 5-1]


5-5

Thus the scattered photon is degraded in energy by that collision partners. The scattered photon may then proceed to expe amount imparted to the electron. or photoelectric absorption or to escape from the collisions Compton further rience probability is proportional to the electron density of Compton-collision medium. in the energy region beginning well above the it-absorppredominates It the medium. threshold, but it falls off uniformly tion edge and continuing past the pair-production energy ap increasing with in efficiency ZbO proximately as the reciprocal of the pho

by the

tr UJ 1UJ

ton energy.

For photon energies above the selfS 200 of an electron pair, 2mc* = 1.02 iz Mev, the photon may convert into a pair u 150 in the electric field of an atomic nucleus a: in the probability, tor, with much smaller a The pres electric field of an electron). (A 100 cc provide to ence of the nucleus is required g conservation, though momentum-energy 2 50 its because of its relatively large mass O recoil energy is negligible for practical 1 ~~ 1 The probability of pair considerations. i.o \s "as _ increasing with increases KINETIC ENERGY (MEV) ELECTRON production photon energy, and it rises in a Za Fio. 4. Specific ionization of electrons in air In dependence upon atomic number Z. (15°C, 700 mm Hg). the case of a Vb medium, the pair-produc thereafter without tion process predominates after 5 Mev and rises rapidly with energy Mev are attained. of hundreds in the energies high extremely until off leveling by the The production of ionization leading to detection methods is accomplished three the preceding of or all some by medium in a generated secondary electrons Basic analyses of these processes are widespread in textbooks and may be processes. found in thorough detail in Ref. 9.


energy

1.2

Instrumental

Types Based upon Ionization

Frequently the desired information about 1.21 Ionization Current Detection. a relative measure of the rate of produc is simply radiation of or fields sources steady It is then sufficient to provide an insulated collecting tion of ionization in a gas,. electrode within an enclosed volume of the gas and to apply a potential difference to

enforce migration of the ions. Figure 5 illustrates such an ionization chamber. The ionization current collected is meas ured by a suitable electrometer, types of which will be discussed below. Such a simple chamber device serves to follow relative changes in radiation intensity from a given source or to compare intensities of radiation fields of a given type of radiation, provided the intensity and the chamber size are so related as to produce readily measurable currents. COLLECTING =- VOLTAGE When an absolute measure of ioniza SUPPLY TO tion rate per unit volume of a given gas ELECTROMETER is required, the above simple device is not adequate because an undetermined frac Fio. 5. Schematic diagram of simple ioni tion of the ionizat ion is due to secondary chamber. zation electrons from the chamber walls and electrode. This fraction will vary as the character of the radiation field is changed. It is necessary then to eliminate effectively the walls of the chamber and to provide a precisely known volume of gas out of which the ions are extracted.

_

6-6

EXPERIMENTAL

TECHNIQUES

[Sec.

5

For a beam of radiation, such as a defined pencil of X rays from a target or a beam of protons from an accelerator, of known cross-sectional area, this may be accomplished by the scheme of Fig. 6, in which the collecting volume is defined by the parallelplane geometry and guard-ring feature, together with the known beam cross section. The entrance and exit foils are sufficiently far from the collecting volume so that negligible numbers of ionizing secondary particles from them enter. But if the For an extensive radiation field the wall cannot be thus eliminated. wall material possesses an average atomic number close to that of the enclosed chamber gas there are two extremes in which simple interpretations of observed ionization are possible. (1) If the chamber dimensions are large compared to the range in its gas of the prevalent ionizing secondary particles, then the effect of the wall is negligible, and the ionization charge collected is appropriate to the enclosed gas volume. (2) If the chamber dimensions are sufficiently small, and the wall thickness greater than the range in wall material of the prevalent ionizing secondary particles, one may regard the chamber space as a small cavity in a medium consisting of the wall material. In this case it is possible to directly relate the cavity ionization to the In either of these extremes, with air energy absorption per gram of the wall material. as the chamber gas, and with wall material of low atomic number, the chamber may be calibrated in "roentgens." (One roentgen of X or y radiation develops ionization


I

Fig.

6. Schematic

diagram

»-

TO ELECTROMETER

for absolute ionization chamber.

This implies a to the extent of 1 esu of charge of either sign in 0.001293 g of air. See Art. 1.21 specific energy absorption of 83 ergs/g by air from the radiation field. of Sec. 7.) Estimates of ionization current to be expected are essential to ionization chamber design. For charged particle radiation whose identity and energy are known, an evaluation of energy lost within the collection volume and division of this by the value of w will give the ion contribution and hence the current or charge to be col The chamber is then to be designed with particle path length and collecting lected. volume such that a current magnitude is obtained appropriate to the galvanometer or For X or 7 radiation an intensity of 1 r/sec will electrometer to be employed. develop 0.33 X 10~9 amp/cm3 of collecting volume of standard air. The photon flux density corresponding to 1 r/sec depends markedly upon photon energy but may in air (Sec. 7.3, Table 3). be calculated from X- and 7-ray absorption coefficients But the wall effect is typically predominant in photon detection unless chambers are specially designed to minimize it. Because of the cosmic radiation, a certain background current is always present At sea which depends upon the amount of material over and around the chamber. level an unshielded chamber will develop approximately three ion pairs per cubic centimeter per second per atmosphere pressure of air, so the current from this source in a 2-liter chamber may be approximately 10-" amp for air at 1 atm. But local radioactivity in the earth and nearby materials and a-particle emission from the chamber walls may cause the background in such a chamber to be 2 to 10 times this figure. Statistical fluctuations in the measurement of this background impose a fundamental limitation on accuracy in evaluation of weak radiation. When Recombination of ions operates to reduce the observed ionization current. high pressures of gas are employed, and when ionizing events of high specific ionization


Sec. 5-1]

6-7

prevalent, this can lead to appreciable errors in measuring the ionization produced. experimenter should observe ion current vs. collecting voltage and operate with his collecting voltage sufficiently high to produce the "saturation" value of ion cur rent. The attainment of such a saturation ion current is greatly facilitated if the chamber gas is free of molecules which tend to attach the free electrons to form The mobility of the latter is much less than that of free electrons, negative ions. and their tendency to neutralize by recombination with positive ions is greater than that of electrons. Thus the elimination of negative-ion-forming gases such as ()2, HjO, and Cl2 from the chamber gas may be desirable when saturation is difficult to achieve because of high pressures and high specific ionization events. Reference 10 gives theory and calculations on recombination effects. Electrometers for measuring ionization currents or for measuring charge collected in an interval of time may be of the electrostatic type, electronic-tube-circuit type, dynamic-condenser type, or, for relatively large currents, galvanometers or microarc

The

ammeters.

MICROSCOPE

VIEWS

"T" CROSS-BAR


QUADRANT PLATE (FAR PLANE) QUADRANT ~-» PLATE (NEAR PLANE)

QUARTZ SUPPORT BOW

PLATINIZED QUARTZ FIBER, BEARING QUARTZ "T" TORSION INDICATOR VANE, ALSO PLATINIZED FROM COLLECTING ELECTRODE

^

Fig. 7. Illustration of principle of Lindemann electrometer. Distribution of collected charge upon the vane causes latter to deflect against fiber torsion under influence of field due to

quadrant plate system.

type is the Lindemann electrometer illustrated in Fig. 7. quartz-fiber instrument of this kind, when viewed by a microscope with con venient magnification and an eyepiece reticle of 100 divisions, can achieve stable sensitivities of one division deflection per millivolt potential change of the insulated fiber system. If the insulated system consisting of the chamber collecting electrode, the quartz fiber, and the conductor joining them possesses a total capacitance of 100 ji/if, then at this sensitivity one division deflection corresponds to 10-13 coulomb. A deflection rate can be ascertained by timing a deflection interval of several divisions, and thus charge collection rates as low as 10~15 amp are reasonably measurable. If the observation time is not too much limited, 10~la amp may be achieved with the best instruments. By use of a high-resistance leak, across which the electrometer indicates the millivolts of potential difference, a steady current reading as low as 10~" amp is accessible. Reference 11 provides detailed information on various types of electroscopes and electrometers and their characteristics. In Ref. 12 are examples of the application of quartz-fiber electrometers, including special balanced circuit techniques and means of rendering them absolute in their measure of charge collected. In Fig. 8 is shown a schematic circuit for a particularly useful method of obtaining The electrometer absolute measurement of charge collected in an ionization chamber. is used simply as a very sensitive balance indicator, the collected charge being drawn into the calibrated capacitance C by application of the measured "balance voltage" A much-used electrostatic

A good


5-8

EXPERIMENTAL

TECHNIQUES

[Sec. 5

Thus the electrometer is used only as a null-reading device, and its character Vc. istics of linearity or of nonlinearity are of no consequence. The charge collected is measured by the product CVe. Eleclron-lube electrometer circuits are ELECTROMETER familiar in the following ranges and types: in (1) For portable kmization-chamber struments, involving ion currents down to 10-11 amp, the circuits of Figs. 9 and are useful. The balanced-bridge 10 features of Fig. 10 contribute to stability. (2) For smaller currents, down to 10-" amp, the inverse-feedback circuit shown It is schematically in Fig. 11 is typical. here shown as used in a charge-integrat ing manner, the collected charge being stored 1"ig. 8. Null-reading electrometer circuit for in the input capacitance C by the feeding absolute measurement of collected charge. back of the "balance voltage" V in such The capacitor C is conveniently of concen a manner as to annul the input signal. tric-cylinder type, with guard rings, so that For steady current readings rather than is known from dimensions. C charge integration, the input capacitance is simply replaced by a high resistance, and the recorder then presents a steady read ing rather than a time-dependent sweep. Principles of such circuits are fully dis cussed in Itef. 16a. The input sections of these circuits, consisting of the ion collector electrode, input resistors or capacitor, input switching devices, and the control grid of the electrometer circuit, must all be mounted with excellent insulation (see Insulator materials below).

M : 0-20pa.

Flo.

9. Schematic

of simple electrometer

The input shorting switch of Fig.

circuit for portable instrument.

11 should involve negligible capacitance and con A fine needle point tact potential when charge-collection technique is being used. with weak, spring-loaded pressure gives good performance. This involves low Electrometer tubes are chosen primarily for low grid current. In the past tetrodes such as the surface leakage and low internal grid current. FP-54 and D-96475 have been designed to operate with grid currents as low as 10_1* amp under proper conditions, and with good technique in achieving a gridcurrent plateau, currents as low as 10~lr' amp may be obtained in charge-collection However, the dynamic-condenser electrometers (see Kef. 1M and chap. lOof lief. 11). described in the next paragraph are to be preferred for the very weak currents. Some electrometer tubes in frequent present-day use are the following: 5803 (triode), 5800 (tetrode), and CK-5889 and 954 (pentodes). a-c amplification and synchronous involving Dynamic-condenser electrometers, The "vibrating detection, are becoming widely used in automatic recording form.

instruments for measuring

Sec. 5-1]

radiation

5-9

reed" instrument14 is commercially available and provides an excellent instrument for ,e routine work at collection currents down to 10 The operating principle amp.

type of instrument involves the generation of a modulated potential by means variation of the capacitance of the insulated collecting system through its proximity

uf this of

22.5*


Fig.

10. Balanced-bridge electrometer

circuit for portable instrument.

T, = 5803

V, = 22.5 v.

■R4

= I00K

Tt» IU4

V2= 45v. V3= 22.5v. V4= 67.5v.

= 30K R5 = I MEG. R6 R7 = I50K

R, = I MEG.

Re

= 50 K = I MEG,

Ts= 3V4 C, = INPUT CAP FOR CHARGE COLLECTION Cj= 05pf. R = INPUT HIGH RESISTANCE FOR CURRENT MEASUREMENT

Fio.

11.

R2= 100 K

R3=I00K

Schematic of inverse-feedback

electromoter

R9

Rl0= 20 MEG. R,, = I0K

circuit or d-c amplifier.

to a vibrating reed or diaphragm as illustrated in Fig. 12. After amplification the signal is detected by a system sensitive only to the vibrating reed frequency. Usually the instrument i9 employed in the null-reading technique, an adjustable balance voltage being fed back to a charge storage capacitance C, which may be the dynamic

condenser

Then, as in the case of itself, so as to maintain zero signal generation. the balance voltage and the storage capacitance measure the charge

Fig. 8 above, collected.

5-10

EXPERIMENTAL

TECHNIQUES

[Sec.

5


Insulator materials for ionization chambers and electrometer insulation where small currents and low background effects are necessary should have volume resistivi They ties of the order 1017 to 10IS ohm-cm and comparable surface resistivities. should be relatively free of piezoelectric effects and of dielectric "soakage." Standard materials for many years have been fused quartz and amber. To these may be added polystyrene, lucite, polyethylene, teflon, fluorethene, and synthetic Desirable combinations of properties of machinability, thermosapphire and ruby. plasticity, and resistivity can usually be found. Teflon and related materials may slowly outgas, yielding undesirable electronegative contaminants in some situa tions, but their surface resistivity is typically very high and stable against humidity conditions. When there is evidence of humidity-dependent surface leakage over materials such as glass or quartz, great improvement may be achieved by thoroughly drying the surface and immediately covering it with natural ceresin wax, or with a silicone material such as Dry-Film whose vapor will deposit a moisture-repellent layer upon the surface. Leakage currents are minimized by the guard-ring construction shown in Figs. 5 and 6, which prevents the existence of large potential differences across the collector This may be particularly important for the ionization pulse electrode insulator.

cs=F

Fio.

12.

Illustration of vibrating-reed

INPUT TUBE OF AC. AMPLIFIER

electrometer

principle.

chambers discussed below, since fluctuations in small leakage currents can give spurious electrical pulses. Reference 15 provides a summary of insulator "noise" effects and of techniques and materials which minimize them. 1.22 Ionization Pulse Counters. The electrical pulses arising from collection of the ions produced in traversal of the chamber by individual particles may be amplified and detected. Mobilities of positive ions and of atomic and molecular negative ions are typically 1 to 2 cm /sec per unit field gradient in volte per centimeter. Elec tron mobilities are much greater, by a factor of about 103. Consequently, if the electrons remain free from molecular attachment, the total charge-collection process will involve a rapid electron-collection phase, extending characteristically over a period extending over few microseconds, followed by the positive-ion-collection milliseconds of time. In a gas in which the electrons largely become attached, only the slower pulse is available. Electrical amplification adapted to the slower pulse will need to possess an input time constant RC not less than 10~s sec, and it must provide good amplifying proper These two requirements carry the undesirable ties for frequencies in the audio region. features that the allowable input pulse rate is limited by the RC recovery time and the amplifier sensitivity in the kilocycle region permits microphonics disturbances. Pulse counting in an air-filled chamber or with other negative-ion-forming gases requires the slow-pulse technique and calls for shockproof mounting of the chamber and amplifier input system to decrease microphonics. In the fast-pulse, free-eleetron-collection case the input time constant usually need not exceed 10-s sec and correspondingly higher pulse rates may be accommodated. The amplifier need not be sensitive for frequencies in the microphonics region. The

Sec. 5-1]


5-11

short input time constant makes the system essentially insensitive to the motion of

positive ions. Familiar counting gases allowing free-electron collection are argon, methane, C()«, The electron mobility is particularly favorable in a mixture of 95 per N*:, and BF3. cent argon plus 5 per cent C(>2 (precise percentages not critical) in which the COj with its relatively low molecular excitation energy levels keeps the average energy of the migrating electrons sufficiently low, by inelastic collisions, to permit them to drift rapidly through the argon by virtue of a Ramsauer effect, giving large mean free paths in argon at low energies. The lowest excitation level in the argon atom is 11.5 ev, so that in pure argon inelastic collisions cannot keep the average electron energy low enough to benefit from the Ramsauer effect. Pulse character, apart from modifications due to associated electronics, is deter mined in the simple electron-pulse chamber such as illustrated in Fig. 13 by the total


the

Fig. 13. Simple electron-pulse chamber, with illustration of dependence of electron-pulse shape and amplitude upon ion puir distribution. TO

PULSE AMPLIFIER

Fio.

14.

Grid chamber with typical voltage division.

number of electrons collected, their initial average distance from the anode, and the orientation of the original ionization track. The total electrical energy in the pulse is related to the work done in collecting the electrons and thus depends upon the number of electrons and the potential difference through which they move in the The rate of rise of the pulse will be sensitive to track orientation, collecting process. as may be seen by comparing the time distribution of the work done in collecting the electrons from tracks a and b in Fig. 13. These details of pulse rise will be seen only if the input time constant is shorter than the total collection time of the electrons and if the amplifier has sufficient bandwidth to reproduce the pulse shapes faithfully. Usually it is desired that pulse size be proportionately related to the number of

formed in the original track so that discrimination may be achieved among two or more types of ionizing events in the chamber. The dependence of pulse size upon the position of the track is removed by the chamber construction shown in Fig. 14. The grid electrically screens the anode from effects of the motion of elecion pairs

5-12

EXPERIMENTAL

TECHNIQUES

[Sec.

5


trons below the grid, and the pulse is developed by the work done in moving them from grid to anode. Thus the contributions of all electrons to the pulse energy are equal, regardless of their points of origin. Rise time effects dependent upon track orientation may still exist, but if the input time constant is made somewhat longer than the possible range in collection times, the voltage height of a pulse will be simply proportional to its energy and hence to the original ion population. Clearly this method is possible only when particle trajectories can be limited to the region below the grid, such as in the case of counting the or-particle emission from a localized sample or in counting the passage of a defined beam of particles through the lower part of the chamber. Pulse amplification usually involves a preamplifier located in close proximity to the chamber, frequently mounted directly on the chamber structure, followed by a pulse amplifier which may be located at a distance dictated by circumstances. The function of the preamplifier is to accept the signal from the high-impedance source in which it is generated and feed it into the relatively low impedance line extending to the pulse amplifier and to boost its amplitude to a value large compared with any "noise" or pickup which develops in the line leading to the pulse amplifier. A typical preamplifier circuit is illustrated schematically in Fig. 15.

Fio.

15.

Preamplifier circuit containing feedback.

(From Ref. 16.)

Input pulse sizes range from tens of microvolts to several millivolts in amplitude, and the final output pulse of the amplifier system is required in volts or tons of volts so as to drive such devices as oscilloscopes, pulse-height discriminators and analyzers, Over-all voltage gains as high as 106 may scaling circuits, or coincidence circuits. thus be required, of which a factor of 10 to 100 is provided by the preamplifiers. Successful analysis of the smaller input pulses demands great care in amplifier design and use to eliminate spurious pulses caused by microphonics, insulator leakage noise, power-line transients, inductive and radiative pickup, and the piling up of back ground pulses which are singly below signal level. Microphonics problems are alleviated by adjusting bandwidth so that audio fre quencies are not strongly amplified; insulator noise is discussed in a preceding para graph, the elimination of humidity-dependent surface effects being important; powerline transient elimination requires adequate line filtering and r-f chokes in power supply leads; pickup suppression involves adequate shielding and elimination of multiple ground connections; the piling up of background pulses can be diminished by limiting pulses to short duration through clipping or time-constant adjustment. When small pulses are to be studied in the presence of large background pulses, the overload characteristics of the amplifier are critical. Poor characteristics usually involve an overshoot-return pulse which trails after the original overload pulse and which may spuriously count above the setting of the discriminator level. Proper relations among time constants in the various amplifier stages can minimize this difficulty.

Sec. 5-1]


6-13

When analytic counting is required in which selection of pulse amplitude is employed of discriminating between various types of ionizing particles, or if a dis tribution of energies of a given typo of particle whose ranges are expended within the chamber is to be inferred from the distribution of output pulse sizes, the pulse ampli fication throughout must be linear; i.e., the amplification factor must be independent of pulse height. The construction of high-gain, stable, linear amplifiers has involved much effort over several years and is presented thoroughly in Refs. 16. The ability to approximate the characteristics of the rapid rise of the pulses in fast counters demands good high-frequency characteristics with substantially constant In order to reduce noise and micro amplification extending out to beyond 1 Mc. limit is chosen to be consistent with the input phonics problems the low-frequency RC product. 1.23 When an anode of small dimensions, usually a wire, Proportional Counters. is mounted within an enclosed volume of gas, the high field intensities near the wire can, if the applied voltage is sufficient, impart enough energy to electrons during their free paths between collisions to make them capable of ionizing upon collision with the Thus the electrons produced within the chamber by the passage of an ws particles. ionizing particle may, after migrating into the region of strong field, multiply them " selves by an average factor A, known as the "gas-amplification factor. If the applied voltage is raised yet higher, a condition will finally be obtained where an electrical breakdown into gaseous conduction occurs which can be triggered by an ionizing event. This is the operating region of the Geiger counter. But in the present section the regime of gas amplification, or of "proportional counting," is of as a means

the term "proportional" implying that the final number of electrons reaching wire after the multiplying avalanche is proportional to the original electron release by the radiation event traversing the chamber regardless of the size of the original release. This proportionality is not strictly true for events of very high specific ionization or for operation involving very large gas multiplication, probably due to space charge effects of the slow-moving positive ions. Also, if significant electron attachment occurs, this simple description is not true to fact. Representative values of electron mean free path at proportional counting pressures would be near 10-3 cm. If an electron is to gain ionizing energy, say 10 volts, in this At threshold for gas distance, the field required is of the order of 10' volts/cm. amplification this field intensity must exist at a distance of roughly one mean free path from the wire surface. The applied voltage threshold should then be, assuming concentric cylindrical geometry, 'he anode

V

E

=

E

(r, + X) In

r-.

= required field intensity

~104 volts/cm wire radius and wall radius, respectively X = mean free path ~10~3 cm = 0.0125 cm ^ 810 volts. In practice, counter of If cm, mil) and r2 = these dimensions, per cent filled to about 20 cm pressure of 95 per cent argon and COj, will give value of near 1,000 when operated at an applied voltage of 2,000 Tt =

a

A

5

if

a

(5

V

rii

5

where

<•]

volts.

is

is

a

is

(2

is

A

V

with increasing should be greater for smaller Evidently the rate of increase of consistent with fact, but wire diameters wires and for larger mean free paths. This less than 0.005 cm mil) are not recommended. within few or several mean free Since the multiplying region, even for large A, paths of the anode wire, the fraction of counter volume represented negligible. The character of the pulse thus nearly independent of the position of the original production, except that the delay time bet ween particle passage and beginning of pulse rise will vary with migration distance of the original electrons, and the rising rate of the pulse will depend upon the distribution of the original ionization loci. But this last effect revealed only when amplifier time constants of 10"* sec or less ion

is


concern,

are

employed.

5-14

EXPERIMENTAL

TECHNIQUES

[Sec. 5

pulse shape for a hypothetical counter with very long input time Representative The migration phase contributes a slight pulse, as constant RC is shown in Fig. 16. The rapid collection of the electron avalanche from regions in an ionization chamber. close to the wire surface causes a very fast upward break, and the motion of the positive ions contributes a continuing rise which is rapid at first while they arc in a strong field close to the wire but is later very slow as they move through the weaker

ELECTRON MIGRATION PERIOD -ELECTRON MULTIPLICATION AND COLLECTION - POSITIVE ION MOTION

0

I

2

TIME IN MICROSECONDS Generated for wjivans (University of Florida) on 2015-09-23 02:49 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Fig.

10.

Characteristic pulse from proportional counter.

field. Common practice, with gases for which electron attachment is negligible, is to make RC 1()~8 to 10"' sec so as to use the initial rapid rise but not be subject to long recovery times. In practice it is convenient to have the counter wall at ground potential as indicated in Fig. 17. Pulse amplification involves the same features as described in connection with pulse ionization chambers, except 1 allows the use of less amplifier that A gain for counting similar events. References 17 present theory and char acteristics of the proportional counter. Constructional Features. Unless a high degree of constancy of .1 is required, pro portional counters permit wide variation in shape and size, since the character of the multiplying region near the wire is not Proportional Flo. 17. counter with wall at strongly affected by the wall as long as ground potential. (Note: The coupling the wire diameter is very much smaller condenser C must be of high quality with than other dimensions. Knd effects will low leakage.) involve regions of nonuniform .4 and of uncertain limitation of the counting volume unless "guard tubes" or field-shaping features are employed as suggested in Fig. 18. Support of the wire and connection to the outside is conveniently accomplished with kovar-glass seals, carefully cleaned and coated with ceresin or Dry Film. Counter gas considerations are essentially the same as for pulse ionization chambers. The argon-CG2 mixture has the same virtues here as there. Hydrogen, methane, and BF» are employed in neutron counting. If electron-attaching gases must be involved, the circuit must be adapted for "slow" counting, as, for example, in the air propor tional counters for a-omitting contamination surveys. Since pressures of 1 atm or more may be used, it is not necessary always to vacuumFlowing the counter gas through the counter shell with slight seal the counter walls. excess pressure is a feasible technique. Applications of proportional counting include 0 and y counting as well as that of the

»

Sec. 5-1]


6-15

particles to which the pulse ionization chamber is practically limited in its absence of gas multiplication. The excellent features of reliability of proportional counters and the opportunity of rejecting some background events upon a pulseheight selection basis are resulting in an increasing supplanting of Geiger counters by proportional counters for radioactive sample counting. Counters for this purpose may be constructed after the manner of Fig. 196. The gas-flow technique allows design of counters allowing quick insertion of a sample directly into the counter so that very low energy particle emission may be detected. The much shorter duration of the proportional counter pulses, a few microseconds in length, permits use of counting rates an order of magnitude greater than the Geiger counter allows (o) FIELD SHAPE AT WIRE END IN ORDINARY with its longer recovery time of the order CONSTRUCTION heavy

of 10~* sec. 1.24

Geiger-Miiller Counters.

When

voltage applied to a proportional counter is further elevated, a failure of the

proportionality will eventually transpire large ionization events do not Kb) FIELD TUBE" PRINCIPLE FOR develop so much ga.i multiplication as the PROVIDING UNIFORM GAS smaller events. This is understood in AMPLIFICATION IN KNOWN COUNTING VOLUME (SEE terms of space-charge reduction of the SECOND ENTRY IN REF 47) field intensity near the wire by the posi tive ion sheath which is more dense for large pulses and begins to limit multipli Fio. 18. End effects in proportional coun cation. in this region of However, ters. limited proportionality evidence is found for the onset of more extensive discharges than the simple Townsend avalanches, M if a mechanism such as photo-ionization were causing a spreading of ionization l)eyond the locale of the original avalanche. This is manifested by a more rapid increase of pulse size with applied voltage than was characteristic of the proportional


in which

region.

Continued elevation of the applied voltage brings about eventually a condition an ionizing event can trigger an electrical breakdown into a state of gaseous conduction, and all pulses are now of uniform size regardless of the nature of the The conducting state is developed by the rapid spreading of originating event. ionization and gas multiplication along the wire through production of photoelectrons by photons released from deexcitation of gas molecules and recombination of ions. The "break" into the conducting state develops typically within 10~' sec from the beginning of the original gas multiplication. If the applied voltage is not excessive, this discharge is quenched by the fact that the sheath of slow-moving positive ions produced near the wire suppresses the local field intensity below the value necessary for further development of an electron ava lanche. Until this sheath has migrated radially to such a distance as to allow a multiplying field to exist at the wire surface, the counter is insensitive to further events. This is the "dead time." After restoration of threshold field intensity for multiplication the counter sensitivity gradually increases toward its initial value as the positive ions are swept out to the cathode during the "recovery time." This complete sequence occupies typically 10_< sec or more. If the counting gas is a pure, monatomic gas such as argon, for which the lowest excitation level lies at 11.6 ev, the ultraviolet photons from deexcitation may produce photoelectron emission from the cathode wall. Also, the recombination of positive ions with electrons at the cathode surface forms excited neutral atoms which con tribute further deexcitation photons to the possibility of photoelectron production. This photoemission at the wall in such a counter tends to maintain the discharge by producing multiple afterpulses, and in such a pure gas counter it is necessary to aug ment the internal quenching mechanism by some external means of suppressing the where


5-16

EXPERIMENTAL

TECHNIQUES

[Sec. 5

applied voltage below discharge threshold until the photoemissive processes have sufficiently subsided. In early work a series resistor of 10* to 109 ohms between the anode and its potential supply served to drop the anode potential by 10 to 100 volts when a discharge occurred. The recovery time for the anode voltage would then be of order 10-2 sec even for low capacitance of the wire and associated signal transmission conductors. This recovery time may be much shorter if an electronic circuit is devised to impose a square-wave voltage depression for a limited time of about 10-4 sec upon the anode accompanying each pulse. The development of the "self-quenching" counter, however, has eliminated the need of such external quenching aids. Addition of a small percentage of a complex molecular Self-quenching Counters. vapor such as alcohol, methane, ethylene, amylacetate, ether, etc., prevents a sus tained discharge and eliminates most afterpulses by virtue of the ability of such molecules to absorb various amounts of energy upon collision and by photoabsorption and their tendency to dissociate from an excited condition rather than emitting A representative counter gas is 9 cm Hg pressure of argon plus 1 cm of photons. Clearly such a counter gas may have temperature-dependent proper C^HsOH vapor. ties if vapor-condensation conditions are approached. The discharge spreading occurs by the strong absorption of the photons arising from an initial electron avalanche. Photons from excited argon will possess energies Such photons between its lowest level, 11.6 ev, and its ionization potential, 15.7 ev. will be able to ionize the alcohol whose ionization potential is 11.3 ev. The mean free path for the photons with respect to alcohol absorption is so small that the suc cessive processes of ionization and avalanche are localized near the surface of the wire and spread over its entire length with a propagation velocity near 107 cm/sec. Quenching is initially accomplished by the positive ion sheath, as previously described. The generation of afterpulses is prevented by the following phenomena: Argon ions migrating outward from the wire experience collisions with alcohol mole cules. In the wire-to-wall migration time about 10' such collisions can occur. Since the argon ionization energy exceeds that of alcohol by 4.4 ev, it is possible for an electron to transfer from an alcohol mol KOVAR-GLASS SEALecule to the argon ion in collision, and 6LASS SLEEVE the probability of such charge exchange is "V sufficiently great to allow unimportant WIRE. TYPICALLY survival of argon ions at the cathode. 004" TUNGSTEN-' The 4.4-ev photons are absorbed by sur FILLNG TUBES* rounding alcohol molecules and produce CYLINDRICAL WALL, TYPICALLY BRASS, COPPER OR SILVERED GLASS and the alcohol ions decomposition, (a) CYLINDRICAL COUNTER result ing from the charge exchange, when neutralized at the wall, decompose rather than yielding up their excitation in pho KOVAR-GLASS SEAL tons. Thus the photoelectric processes yielding afterpulses are effectively sup GLASS SLEEVE pressed, although at the high-voltage end of the Gcigcr region they are not com TUNGSTEN ANODE WIRE A thorough descrip pletely eliminated. METAL SHELL tion of these processes is given in Chap. GLASS BEAD 4 of Kef. 17a. LOW MP GLASS, OR Counter Construction. The two most VACUUM WAX SEAL familiar forms of the Geiger counter are MICA WINDOW illustrated in Fig. 19. (b) END-WINDOW COUNTER Kovar seals are regularly employed for wire support and Fio. 19. Two familiar types of Geigerinsulation; aluminum, stainless steel, cop rounter I lilies. per, brass, and silvered glass are commonly The wire is of a few mils diameter, and, in the case of the free-end used wall materials. construction where some stiffness is desirable, it is usually tungsten. The free end carries a glass or metal bead to prevent excessive field intensities at the wire tip. Counters for cosmic-ray studies are usually of type a and may be of any convenient

X

Sec. 5-1]



wall thickness; counting of

radiation

6-17

/3 particles requires thin walls, and in the counting of radioactive samples, the end-window type 6 counter is most familiar. The efficient counting of y radiation requires wall material of high atomic number so as to optimize photoelectric and pair-production conversion. The sealing of the thin mica window to the metal or glass shell in a type b counter has been traditionally done with one of the low-vapor-pressure vacuum sealing waxes. A recent technique uses a low-melting-point glass which will fuse both to the lip of a metal or glass shell and to the mica. Metal for this purpose should be stainless steel or oxygen-free copper. The glass may be obtained as a powder or in rings of thin It fuses at about 600°C. Corning supplies such a sheet glass of various diameters. Mica sheet with uni glass powder of 325 mesh size under the Code Number 7570. form surface density as low as a few milligrams per square centimeter is successfully used. Preparation of the counter tube for its gas filling involves thorough removal of extraneous substances such as dust, lint, solder flux, and grease. Glass-shell counters can be cleaned with acid cleaning solutions and rinsed thoroughly with distilled water. Metal-shell counters may allow only thorough mechanical cleaning in course of assembly, followed by ammonia wash and grease-free organic solvent cleaning. The cathode surface should be chemically stabilized against any reaction with the filling mixture which is to be employed. Care in preparation of the anode wire is of particular importance, since dust par ticles, die scratches, and any irregularities can produce electric field distortions which Examination of the wire under magnification is affect operational characteristics. recommended, and heating the wire to incandescence during the evacuation of the tube is highly desirable, though this is easy to accomplish only with the double-ended tube design. Evacuation should be accompanied by heating to whatever temperature the con struction materials allow, and pumping must be prolonged until outgassing of surfaces Mercury vapor from manometers need not be trapped out. is complete. Storage of the component gases in previously outgassed glass bulbs attached to the vacuum system through stopcock controls and with manometers attached to the vacuum system permits filling and subsequent testing of counting characteristics. Reevacuation and refilling in the event of finding poor characteristics are thus facili Discussion of preparation and filling procedures is to be found in Refs. 11, tated. 17, and 18. Since the Geiger pulses in size are independent of Operational Characteristics. origin, the amplifying system need not be linear, and since they are of the order of magnitude of a few volts at the amplifier input, the gain required is small. It is necessary to provide output pulses carrying sufficient energy and with proper time characteristics and impedance matching to operate the scaler or recorder desired. The counting rate in response to a conz slant radiation field should demonstrate a plateau with respect to applied voltage, which may extend for typically 200 volts £ PLATEAU SLOPE OF above threshold as suggested in Fig. 20. ORDER OF 1% PER 100 VOLTS The pulse size, of course, increases as the ^ voltage is raised through the plateau region; §° and at the high-voltage end afterpulsing seta in, followed for yet. higher voltage by ^qo Qq0 800 MOO 1200 Good tube performcontinuous discharge. VOLTAGE j-IQ. 20. Plateau characteristic of typical ance provides a plateau whose slope is Geiger counter, not more than a few per cent per 100 volts, so that regulation of the applied voltage can Slight amounts of air or other electroneasily provide negligible error from this cause. attaching gases can destroy the desirable flatness of the plateau. After a large number of counts the quenching gas becomes sufficiently decomposed This to give impaired performance and the tube approaches the end of its useful life.

J

6-18

EXPERIMENTAL

TECHNIQUES

[SEC. 5

10" and 1010. If an overvoltagc is applied so that continuous discharge occurs, decomposition of the vapor rapidly proceeds and In this connection it should be mentioned the counter lifetime is markedly decreased. that Geiger counters of essentially unlimited lifetime are produced using an argonchlorine mixture." The halogen gas molecules are able to deexcite the metastable The argon atoms and to perform the quenching function of the organic molecules. percentage admixture of chlorine is about 0. 1 per cent, and too much chlorine increases the probability of negative-ion formation to the point of slowing the pulse develop Because of their unlimited life they have been considerably used, but the ment. experience of some with the halogen-quenched counters has indicated lower efficiencies and poorer plateaus than are typical of organically quenched counters, probably due to negative-ion formation. The efficiency of a Geiger counter for detecting the passage of an ionizing particle may be considered as follows: The average specific energy loss for the particle moving through the counter gas is given by the expression for — dE/dx previously stated. The average number of ion pairs produced in a path length x through the gas is then

limiting number of counts lies between

V =

-dE x -j ax w


If p is the average where w, as before, represents mean energy loss per ion pair. number of ions, the probability of producing some number n other than p is obtained from the Poisson formula P{n) = The probability

I n\

p»e-p

of producing zero ions, when the average number is p, is then

P(0)

= e-''

and this represents the probability of failing to count the passage of the particle. The efficiency for its detection is then 1 — e~". For a high-energy electron the value of —dE/dx in argon at 10 cm pressure is approximately 340 cv/cm, and with to = 27 ev per ion pair the specific ionization is Thus for a path length even 13 ion pairs per centimeter. seen to be approximately as short as 3 mm the detection efficiency would be 1 — r1"1"1'1 = 1 — e-3-9 = 0.98. Consequently, the detection of high-energy particles moving through the tube is with essentially 100 per cent efficiency. The detection of y rays occurs with an efficiency mostly determined by the pro This duction of photoelectrons, Compton electrons, and pairs in the wall material. efficiency can be estimated from knowledge of the y photon energies, the wall element and thickness, electron ranges in the wall material, and the fundamental data or For a counter with a 1-mm-thiok formulas for the photon conversion processes. copper wall presented to a broad flux of 1-Mev photons a fraction of about 1 per cent Much higher efficiencies in y detection are of the incident photons will be counted. available by scintillation counting described in a following section. The possibility of rendering visible the trajectories of 1.25 Cloud Chambers. charged particles has made accessible the detailed analysis of many nuclear phenomena not easily subject to interpretation or even to observation by ionization counters alone, and combinations of counters with a cloud chamber have provided means of selecting desired types of nuclear- and cosmic-ray events. Visibility of track is accomplished by the scattering of light from minute water droplets formed by condensation from a supersaturated vapor upon the ions dis tributed along the path of the ionizing particle through the chamber gas. A thorough discussion of the conditions under which condensation growth of droplets to visible size will selectively occur upon ions is given in Kefs. 20 nnd is also reviewed more briefly in Part I, Sec. 2G of lief. 1. The phenomenon is due to the fact that for a charged droplet of very small radius, less than about 10~; cm, the

Sec. 5-1]


5-19

equilibrium vapor pressure at its surface is less than that for an uncharged droplet same radius as a result of forces arising from electrical polarization of the In both cases this equilibrium pressure is greater than that for an infinite liquid. Consequently, for a certain range of supersaturation, plane surface of the liquid. charged droplets can grow indefinitely whereas uncharged droplets will evaporate. If foreign particles such as lint or dust exist in the chamber atmosphere, they will provide surfaces of large effective radius upon which condensation can readily occur for degrees of supersaturation even below that required for growth of a charged If the supersaturation is too high, the random coalescence of vapor mole droplet. cules in the course of their kinetic behavior into statistical agglomerates will provide This is the threshold for fog formation in the nuclei which can grow into visibility. chamber. The chamber must thus be free of foreign particles and should be operated at supersaturations somewhat below the condition for fog formation. Two methods for producing the supersaturation are available: (1) The chamber expanded, causing the internal region to pass may be suddenly and adiabatically from a standby condition near saturation to one of supersaturation by the sudden


of the

This clearly involves intermittent, pulsed operation. cooling. (2) A vertical tem perature gradient may be maintained between a relatively warm upper plane con taining an excess of the liquid state of the vapor substance and a cold lower plane A continual mass transfer of the upon which condensation is constantly occurring. vapor substance progresses by virtue of evaporation at the top and condensation near the bottom, and in a limited intermediate region a condition of supersaturation can exist in a steady state. These two types of cloud chamber are termed expan CLEARING FIELD GRID WIRES XENON ON INNER FACE OF GLASS LID DISCHARGE and diffusion sion chamber chamber, V TUBE respectively. Either type of chamber is frequently operated in a strong magnetic field to pro vide momentum determination of the energetic particles through measurement of radii of curvature of their trajectories. Expansion chambers will involve the features shown schematically in Fig. 21, though the mechanism has been designed in

various

manners.

The

vapor

sub

stance is usually ethyl alcohol or water, and the atmospheric gas is frequently air Flo. 21. Schematic diagram of expansion cloud chamber. or argon. Values of the expansion rat io, i.e., the ratio of expanded volume to initial volume for various gas mixtures, are quoted in the section of Itef. 1 referred to above and in Ref. 21. For two representative mixtures the expansion ratios recommended and the factors of supersaturation produced are the following (where initial pressure and temperature are 1.5 atm and 20°C, respectively):

Mixture Argon plus 40 % IljO and 60^ CtHtOH Air plus HiO

Exp. ratio

Supersaturation

1.09 1.31

5.7

1.9

Expansion ratio values generally decrease as operating pressures are raised. Recycling time will depend upon the interval required to restore thermal equilibrium Typically it lies between '2 a"d 2 min, depending after the adiabatic expansion. upon the rate at which heat can flow back into the chamber and its atmosphere to reduce the supersaturation below droplet formation levels.


6-20

EXPEKIMENTAL

TECHNIQUES

[Sec.

5

Sensitive time, i.e., the interval during which droplet growth upon ions occurs, may be as short as a few milliseconds for low-pressure chambers and very rapid expansions and as long as a few seconds for high-pressure chambers and slow expan sions. A typical time is Ho sec. An ion clearing field must be established in the chamber after each expansion to remove all charged particles so they will not produce a general distribution of back ground droplets in the subsequent expansion. This field intensity should ensure sufficiently rapid ion migration to sweep clean all parts of the sensitive volume. A minimum field intensity of 10 volts/cm is recommended for atmospheric pressure chambers. Illumination for photography of tracks may be provided by photographic lamp bulbs, overvoltaged to produce a pulse of light during a fraction of the sensitive time, with condensing lenses to gather light and deliver it through the chamber wall win dows. A better light source is the xenon discharge tube available commercially. The energy from a capacitor charged to 2,000 volts is discharged through it at the desired instant by initiation of conduction with a high-voltage transient in a "tickler" electrode or coil outside the tube. Cylindrical condensing lenses and cylindrical reflectors are employed. Track photography requires adequate depth of focus to include the sensitive volume and hence involves small aperture, intense illumination, and sensitive film. Stereo scopic photography is employed when the spatial orientation of tracks is to be GRID WIRES FOR CLEARING LIQUID SUPPLY FIELD TROUGH-7 measured and is accomplished by use of two cameras or by obtaining a second direction of view with a single camera by use of a plane mirror. A survey of operational factors is pre sented in Chap. 15 of Ref. 22. Diffusion chambers are schematically described in Fig. 22. Since these operate -LIGHT SOURCE AND REFRIGERANT continuously there are present at any REFLECTOR FLUID instant both old tracks which are diffuse Flo. 22. Schematic diagram of diffusion due to ion diffusion and new tracks which cloud chamber. of a clearing are sharp. Application field sweeps the ions out rapidly during standby conditions between pictures, and brief removal of the field at the time of illumination and exposure prevents a severe general background of old tracks. The rate of vapor supply to the sensitive volume is limited, and it is important to avoid intensities which condense so many droplets that the super-saturation is lost. Otherwise "dead" regions or intervals will occur. A general discussion of this type of chamber is to be found in Refs. 23, and a descrip tion of some special types for high-pressure operation with light gases is given in Ref. 24. Clearly the diffusion chamber can operate only in the horizontal plane, since stability of the sensitive layer, typically 1 to 2 in. deep, depends upon the steady "raining" of droplets from its lower side facing the cold plate. It is used frequently for experiments with particle accelerators where the desired beams of radiation arc usually horizontal. 1.26 Bubble Chambers. A very recent development in visible track methods i< the production of pulsed conditions of superheat in certain liquid media under which the passage of a charged particle will generate a sequence of tiny bubbles of vapor along the ion trail which rapidly grow to visibility and permit photographic recording. Bubble chambers employing organic liquids such as ether and pentane" are in use in accelerator physics experiments, and successful operation of liquid-hydrogen chambers is reported. " The former operate at temperatures and pressure ranges convenient to the average laboratory, while the latter involves cryogenic facilities not generally available. The superheated condition is produced by a sudden pressure reduction, allowing


Sec. 5-1]


radiation

5-21

bubble tracks to appear if a particle passage occurs. After bubble formation has relieved the superheat, restoration of original pressure and return to thermal equilib rium by heat flow within the liquid prepare the system for a further cycle. Recycling can proceed within small fractions of seconds, so that rapid accumulation of data is possible if the photography techniques and nature of the radiation under study allow. As with the cloud-chamber expansion and its supersaturation, so also here the range of superheat must be properly adjusted. For values too low the bubbles do not grow to visibility, and for values too high the statistical fluctuations of molecular spacings in the liquid state giving momentary microscopic voids will grow into bubbles giving boiling throughout the volume. Foreign particles, sharp points, and dis continuities of surface may also serve as bubble-forming loci. Advantages of the bubble chamber are (1) large stopping power compared with a gaseous medium, (2) rapid recycling, (3) relatively simple medium compared with photographic emulsions, and (4) the fact - ETHER that time relationships are preserved so VALVES OPERATED N TANDEM that counter control is possible and simul taneity of events may be ascertained. As yet the literature on these techniques is scarce, though several laboratories have OPEN TO begun their use. ATM. Typical operating parameters for an ^FROM TEFLON ether bubble chamber are given in Refs. DIAPHRAGM PRTANKRE (NITROGEN) 25 as follows: Temperature is held con stant + 0.5°C somewhere in the range TEMP. CONTROLLED OIL BATH 138 to 143°C by surrounding the ether Flo. 23. Schematic diagram of ether bubble chamber with an oil bath. The begin chamber. ning pressure is 300 psi, and the decom pression is to atmospheric pressure. Figure 23 shows Tracks were illuminated by a 5-/isec flash from a xenon discharge schematically the features of the system. Tracks of minimum ionization are visible, and under reliable conditions of super heat and of timing of the flash relative to particle passage the bubble count is uniquely related to specific energy loss. 1.3

Photographic

Photographic

Emulsions in Particle Detection

films and plates are employed in radiation

detection and measure

ment in two distinct categories: (1) X-ray film may be calibrated in terms of density vs. exposure and employed to detect and measure the intensities of fields of X and 7 rays for the purposes of radiological evaluation, or such film may be used to detect the positions of analyzed beams of particles in a mass spectrograph and X-ray or

8-ray spectrograph. (2) Special "nuclear plates" are available in which the indi vidual tracks of ionizing particle radiation may be microscopically observed and analyzed. The use of films for radiation field evaluations is widespread in radiological survey work and will be discussed under Health Physics in Sec. 7. The second category, use of nuclear plates, is prominent in nuclear physics because it provides the following advantages: (1) A lasting record of individual particle trajectories is made which can be examined subsequently with relatively simple equipment. (2) The sensitive time is arbitrary; it may be exposed within a fraction of a second to particles in an accelerator experiment or over many weeks in cosmicray work. (3) Its high stopping power frequently permits containment of the ranges of energetic particles within small volumes available to study. (4) Its sensitivity may be adjusted within limits to favor desired particles and to suppress background. (5) Within limits the grain densities of the tracks may be related to specific ionization. The chief disarlmnlages inherent in emulsions are (1) time relationships of events are unknown, (2) use of magnetic fields for momentum determinations on particle

EXPERIMENTAL

6-22

TECHNIQUES

[Sec.

5

tracks has not been achieved, (3) the atomic heterogeneity of emulsion is frequently undesirable, (•)) data are not quickly available. Nuclear plate emulsions differ from optical photographic emulsions by their higher content of AgBr and the much smaller size of developed grains. By weight the nuclear plates are over 80 per cent of AgBr, and the sizes of developed grains are a few tenths of microns as compared with several microns in optical emulsions. Nuclear track plates are made with emulsion thicknesses ranging from 25 to 2,000 microns, though those commonly used are 100 or 200 microns. The larger thicknesses facilitate containment of ranges of energetic particles but require more critical development techniques. Emulsions can be loaded with such elements as Li, B, Bi, U, and Th for the study of nuclear reactions by soaking in solutions of certain compounds of the desired element. The use of boron-loaded plates for slow neutron detection through the B,0(n,ar)Li' reaction is an example. Density of developable grains along a particle track is proportional to dE/dx until values of the latter become high enough to saturate the available grain production in the emulsion involved. Likewise the smallest value of dE/di which can provide an identifiable track will depend upon the type of emulsion as indicated approxi mately in Table 2, in order of increasing sensitivity. Table


Eastman film

NTC NTA NTB NTB-3

2

Ilford film

Sensitivity limits

D-l E-l

Sensitivity poor for or particles For fission fragments only. Fission fragments and alow ar's a tracks up to 200 Mev; protons up to 20 Mev Proton tracks up to approximately 50 Mev; mu mesons of 4 Mev barely visible; a particles to 400 Mev Protons up to over 100 Mev; electrons up to nearly 100 kev Sensitive to minimum ionization, giving about 20 grains per 100 microns

F-3

(C-2 \

lc-3

G-5

along track

of high-energy

electron

For the most sensitive emulsions the grain density along the track of a minimumionizing particle is typically four times the density from random grains, depending upon background conditions. In the G-5 emulsion the grain density reaches a saturation value of about 120 grains per 50 microns when dE/dx is 20 kev/micron, and departure from linear relationship between grain density and energy loss begins at about 3 kev/micron where the grain density is about 60 grains/50 microns.500 Developing techniques arc critical, since nonuniform development can falsify relative grain densities within a single film or between batches. For the very thick emulsions uniform development is accomplished by careful control of temperatures and pH according to procedures such as described in Ref. 27. Developing recipes for nuclear plates are also quoted in Chap. 14 of Ref. 22; in Chap. 1, Sec. 2H, of Ref. 1; and in Refs. 28, which are general discussions of emulsion techniques. Temperature changes and drying must be uniformly and slowly accomplished to prevent distortions of the emulsion which could modify apparent ranges and directions of tracks. Eradication of the latent images of background or unwanted tracks can be nearly completely accomplished by an oxidation of the emulsion in a high-humidity atmos It should be noted also that latent phere after a process described in Ref. 28a. images gradually fade under normal conditions, and this is a factor of consideration in long exposures. Grain counting and measurements upon the tracks of angle, length, and scattering are done under oil immersion with over-all magnifications in the range of 1,000 to Scanning in search of tracks or events of interest or simple counting of heavy 1,500. In the study of tracks is typically done with dry objective and lower magnification. thick emulsions a long-distance objective is necessary and yet one of small focal depth These two require is required for accurate track location and measuring dip angles.

Sec. 5-1]


5-23

an objective lens of large aperture. Much ingenuity has entered into to provide freedom from backlash, accuracy in range and angle measure ments, reproducibility in the relocation and registering of positions of events, and the ability to measure validly the small lateral displacements of track inherent in mul tiple scattering analysis. Range vs. energy data for emulsions applying to protons and a particles are given in Refs. 28. More recent values are found in Refs. 29. Ranges of other charged particles (not including electrons) may be inferred from these by the relationship merits demand stage designs

M, E, and R refer, respectively, to charge, mass, kinetic energy, and range. Identification of particles under satisfactory conditions may be accomplished through For the high-energy particles of cosmicmeasurements of grain density and range. ray studies, where ranges are not usually contained, a technique of measuring the where Z,

MAGNETIC


SHIELD -

PHOSPHOR

• PHOTOMULTIPLIER

PULSE

HEIGHT DISCRIMINATOR

TUBE

-VOLTAGE OIVIDER PRE-AMPLIFIER

AMPLIFIER

AND

Flo. 24. Block diagram of elements of

a

• SCALER OR RECORDER

scintillation counting system.

multiple scattering of the track together with grain density may allow the distinguish ing of masses of various particles*0 if their charges are the same. This method is especially useful in distinguishing meson from nucleon tracks. References 28 contain full details of techniques and possibilities in use of nuclear plates, and Ref. 31 illustrates many types of nuclear events as registered by emulsions. 1.4

Scintillation

Counting

In the beginning of experimental nuclear physics the scintillation

counting of a

particles by the dark-adapted eye viewing a zinc sulfide screen was a fruitful technique, but it could not be developed into a general method of charged particle detection until the production of photomultiplier tubes was achieved.

The elements of a scintillation

counting

system include

the phosphor

within

which the energy loss from swift charged particles excites short-lived metastable states whose decay results in emission of visible and ultraviolet light, the photomultiplier tube which receives a fraction of the photons and develops an electrical current pulse, the pulse amplifier, pulse-height discriminator, and the recording device (see Fig. 24). 1.41 Phosphors for scintillation counting include a variety of fluorescent materials, inorganic, organic, organic phosphors.

liquid solutions of organic phosphors and plastic solids containing

Selection of the phosphor is made upon the basis of such con siderations as (1) photon yield per unit energy loss in the desirable spectrum region, (2) decay times of the light emission, (3) efficiency for detecting y radiation, (4) linearity in the relation of light output to energy loss, and (5) miscellaneous factors such as size, shape, machinability, stability, etc. In general, the inorganic phosphors are more efficient y detectors than organic phosphors because of their higher average atomic number and density upon which 7-conversion processes strongly depend. Also their photon yield is typically better for a given loss of particle energy. But their lifetimes for decay of the photon emission are too long for some purposes, and they are difficult to produce in the form of large single crystals.

EXPERIMENTAL

6-24

TECHNIQUES

[Sec.

5

The organic phosphors, including the liquid and plastic solutions, have short decay times of the order of 10-8 sec and thus permit fast counting and sharp time resolution They contain considerable hydrogen, which is desirable for of particle passage. They are relatively easy to grow in large, clear crystals, fast neutron detection. and the liquid and plastic phosphors may be of arbitrary size and shape. Some properties of phosphors in wide use are listed in Table 3. The data of this table are a selection from various sources which do not completely agree on the data in the last two columns. References 32 and 33 are summaries of phosphor information and scintillation counter information to date. Table

Name of phosphor

Composition and density of medium,

g/cm'

3

Pulse Maxima in

Refrac tive

emission spectrum, A

index

strength for 0 particles

Decay time, see

relative to anthracene

Inorganic: Zinc sulfide


Sodium

iodide

(Tl-activated) Calcium tungstate Cadmium tungstate Organic :

ZnS(Ag), 4. 10

p-Terphenyl in toluene (with a-n.p.o.)* p-Terphenyl in polysty rene (with t.p.b.)*

io->

1 0

1.7745 1.918

4.100 4,300 5.300

2 5 X 10-' 3 X 10' IO-«

C..H.,, CuHn,

1.595 1.622 1. 497

4,120. iAW, 4.720 4,080. 4,200 3.900, 4.050, 4.300

3 2 X I0» 0 68 X I0~«

CiiH,., 0.94

1.525

4.500

0.8 X 10-'

0 35

CtH., 0.87

1.498

4.150

0.3 X 10"

0 42

CM,,

1.595

4.450

0 4 X 10"

0 39

Nal(Tl). 3.67 CaWOi, 6.06 CdWO., 7.90

1.25 1.16 (trans) p-Terphenyl in xylene. . . . CsHio, 0.86 p-Terphenyl in phenylcyclo-hexane (with Stilbene

4.500

1.09

0.7 X I0n

2. 1

0.36 0. 21 1.0

0.6 ~0 2

* See recipes given below.

Liquid and plastic phosphors listed in Table tively: 1.

5

2. 3

3

have the following recipes,

g/liter p-terphenyl in xylene. g/liter p-terphenyl plus 0.01 g/liter d.p.h. in phenylcyclohexane (d.p.h.

respec

=

1,6-

diphenyl-l,3,5-hexatriene). 3. 5 g/liter p-terphenyl plus 0.02 g/liter a-n.p.o. in toluene [a-n.p.o. = 2-(lnaphthyl)-5-phcnyloxazole]. 4. 36 g/liter p-terphenyl plus 0.2 g/liter t.p.b. in styrene monomer, which is then polymerized to form the plastic phosphor (t.p.b. = l,l,4,4-tetraphenyl-l,3, butadiene !. The mechanism of light emission from these solutions involves an efficient transfer of the excitation energy which solvent molecules receive from passage of the charged particles to the terphenyl molecule whose excitation will frequently bring into play The additional small the metastable states whose decay yields the useful light. additives are molecules which strongly absorb the terphenyl light and reemit in a more favorable spectrum region with regard to the spectral sensitivity of photomultiplier tubes. Recent evidence indicates that a yet more efficient band shifter is in concentrations near 0.1 g/liter of solution. l,4-bis-2(5-phenyloxazolyl)-benzene The mechanism of light emission by such materials as Nal(Tl) is a complicated solid-state phenomenon in which the impurity centers (Tl) are able to absorb some of the energy of excitation of the Nal crystal lattice and to reemit it in the visible region. The reemitted light is negligibly absorbed in passing through the Nal, The Tl is present only to the extent of a few tenths of 1 per cent (by atoms).

Sec. 5-1]


5-25

of the phosphors, in terms of useful light yield per unit energy loss, As dE/dx becomes large, a saturation upon dE/dx for the particle radiation. in light yield sets in which is generally more important in the organic than in the Consequently the relation between output pulse height and inorganic substances. In Ref. 33 there energy loss is typically better for Nal than for the organic phosphors. is a summary of light yields from various phosphors from the stopping of electrons, pro For anthracene the depend tons, deuterons, and a particles of given initial energies. ence of luminous efficiency dL/dx upon dE/dx is shown in Fig. 25 from the work of Ref. 34, and the incremental light output per unit energy loss from various particles moving through anthracene can be obtained therefrom. Absolute efficiencies are not well known, but for a particle of minimum ionization (electrons over about 100 kev) moving in anthracene the energy loss per useful photon is in the neighborhood of 70 ev /photon. Other phosphors may be roughly compared by the last column of Table 3. L4S Photomultiplier tubes are commercially available with photosensitive areas ranging from 0.3 to over 100 in.', though those which have received most use have a Efficiencies

depend

50^

Iao"

'

30


%

NUMBERS MAF !KED ON CURVE ARE PI ?OTON

A*

ENERGIES

id

_., 1000

gm

dE/dx

cm* Mev

Fig. 25. Relative light per unit energy losa in anthracene

as

function of specific energy loss.

circular sensitive area somewhat under 2 in.1 forming an end window to the cylindrical tube envelope. Light transmitted through the thin glass of the end window liberates photoelectrons from the semitransparent emitting surface which coats the inner window face, and these are focused by electrical potentials upon the first of the chain of multiplying surfaces. Another type in common usage receives light through the side wall of the glass envelope on a rectangular photocathode mounted within. The electron multiplication per stage of the multiplier is typically 4 to 5, so that in a sequence of 9 or 10 stages a current amplification (or charge amplification in terms of an integrated pulse) will be of order 10*. Over-all sensitivities, in terms of anode output, of 20,000 to 80,000 m« per microwatt of 4,000 A incident light upon the photocathode are specified, or in terms of output amperes per incident lumen, values ranging from 20 to 80 amp/lumen are stated by manufacturers for various tubes. Using nominal values of over-all sensitivity and amplification, the photocathode inversion efficiency is found to lie between 5 and 10 per cent; i.e., 10 to 20 photons will be incident per useful electron released at the cathode. (However, opaque For selected photocathodes, as in the 1P21 tube, may be over 10 per cent efficient.) tubes the voltages may be elevated above manufacturers' specifications so as to provide amplifications up to an order of magnitude greater than nominal. Spectral sensitivities for the most frequently used tubes are of three types: (1) the S-4 response, extending from 3,000 to 7,000 A with its maximum at 4,000 A, (2) the

EXPERIMENTAL

6-26

TECHNIQUES

[Sec.

5

S-9 response, extending from 3,000 to 6,500 A, with its maximum at 4,800 A, and (3) the 8-11 response, extending from 3,000 to 6,500 A, with its maximum at 4,400 A. The matching of spectral emission of phosphors with tube response is obviously desirable, and the use of band-shifting additives as described above is toward this end.

Manufacturers array the multiplying electrodes, or "dynodes," in at least three different systems. These will involve transit times for the electron pulse which differ, and the greater or smaller distances from photocathode to first dynode in various tube designs will affect transit times. The experimenter must be aware of Dynode these diversities when accurate pulse timing or coincidence work is pursued. structure and multiplicity will also affect the time spreading of the electron bunch, which in turn determines the pulse-rise time. Some pertinent data on a few widely used photomultipliers are displayed in Table 4. Table

4

Nominal gain

Nominal sensi tivity. amp/ lumen

2X 10" 0 5 X 10"

80 100 20-30

Photocathode Tube

No. stanes

Manu facturer


Type

Shape

Position

IP21

RCA

Opaque, rect.

WX?io"

Side

9

5819

RCA

Semitransp.

1W diam

End

10

6342

RCA

Semitransp.

I1 Ms" diam

End

6655 6810 6292 6364 5311

RCA RCA DuMont DuMont

Semitransp. Semitransp. Semitransp. Semitransp. Semitransp.

I1 Ho" diam I'Mo" diam 1W' diam 4W diam

End End End End

EMI

Spectral response

1

10 I (Ag-.Mg) 10 14 10 10 14

/

S-4 S-9

18-4

S-ll

0.6 X I0>

35

S-ll S-ll S-ll S-ll

0 .5 X 10" 6 6 X 10' 2X I0« 2X 10" 10'

25 4000 120 120 500

Approximati* pulse-rise time, sec

io-» 10'

5X

5X10-1

5X 10^ 5-10 X 10 « ~io-> ~io-»

7X

10 •

(British)

Magnetic sensitivity of multiplier tubes is great because of the critical paths of lowenergy electrons from photocathode to first dynode and in the successive multiplying The end-window tubes, especially stages. those of large cathode diameter, are strongly responsive to orientation even in the earth's field if they are without shielding. 5 0.D For fixed locations and orientations the tul>e space can be made field-free by correcting _=___^____ f* j coils. But usual practice is to reduce weak fields to negligible values within the tube vol LI ~l 1 ume by a jacket of annealed mu-metal whose permeability is very high at low flux densities. L SOFT IRON When work near particle accelerators and Flo. 26. Example of magnetic .shield other magnetic devices is involved, it is neces system for 5,819 photomultiplicr tubes sary to mount enough concentric shells of soft in fields up to approximately 200 iron about the tube to reduce the flux within to gauss. values which the mu-metal can shunt around the tube. The shells must extend sufficiently far beyond each end to eliminate end effects within the tube volume. Figure 26 illustrates a shield system which has enabled 5,819 tubes to work in fields up to 100 gauss near a cyclotron. 1.43 The statistical accuracy with which the electrical output Light Collection. pulse from the photomultiplier represents the light production in the phosphor will depend primarily upon the number of photons collected. For example, if a fas* electron traverses a 5-cm thickness of stilbene, it will dissipate about 10 Mev, which will give rise to approximately 10' photons. If it is required that the photomultiplirr deliver a pulse which represents this energy loss with a probable accuracy of 3 ]


Sec. 5-1]

5-27

(standard deviation), it will be necessary for the electron release at the photocathode to be at least 1,000. If the conversion efficiency of the photocathode is 10 per cent, the incident photons muBt number 104, or one-tenth of the total photon production. Similarly, if y spectroscopy by total absorption of photons in Nal(Tl) is to strive cent

per cent line breadth at 1 Mev, the pulse must initiate at least 2,500 photoFor a y-line quan the collection of over 2.5 X 10* photons. tum of 1 Mev which is completely absorbed in the crystal approximately 3 X 104 photons will be produced, so it would be necessary to capture essentially all of them. Since, as mentioned earlier, Nal(Tl) does not appreciably reabsorb the visible light coming from the thallium impurity centers, it would be possible to collect much of the emitted light if a perfectly reflecting surface could surround the crystal and photocathode. The best reflectors for this purpose have proved to be diffuse reflectors for 2

electrons, which means

SHIELD

Olpf ANODE

iPHOTOCATHODE

\\

I80K

_

©OK ©OK

NOTE:

©OK ©OK IOOK

©OK

IOOK] IOOK ©OK

TO MATCH TRANSMISSON CABLE IMPEDANCE R IS CHOSEN

TRANSM. CABLE

IOOK

HhHhHl Olpf

.Oljjf

ARRANGEMENT, WITH NEGATIVE (o) GROUNDED CATHODE DIRECTLY INTO TRANSMISSION LINE TO AMPLIFIER Generated for wjivans (University of Florida) on 2015-09-23 02:49 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

I

DYNODES

PULSE CLIPPING

+HV.

.Olpf

SIGNAL PULSE FED

LINE

NOTE: R, IS CHOSEN TO MATCH PULSE CLIPPING LINE. R, IS NEARLY ZERO, BUT IS CHOSEN SO AS TO OPTIMIZE PULSE SHAPE. Rj IS CHOSEN TO MATCH OUTPUT SIGNAL LINE (b) GROUNDED ANODE UNIFORM OUTPUT

Fig.

ARRANGEMENT PULSES.

27. Schematic

diagrams

WITH PULSE CLIPPER

AND LIMITER

TO PROVOE

of typical circuits for photomultiplier tubes.

MgO and A1203 surfaces. Excellent technique of this type is described in Ref. 35 for y spectroscopy by pulse-height methods. The piping of light by lucite from phosphor to phototube is often desirable in order to avoid large discontinuities in refractive index, with consequent back reflections, in optically coupling the phosphor to the photosensitive surface. The lucite-glass and lucitc-phosphor interfaces are joined through use of mineral oil, Canada balsam, or a clear and viscous silicone grease. Light piping is a necessity in cases where the phosphor must be inserted into regions inaccessible to the tube and its accessories or when counting is to be done in a local magnetic field in which magnetic shielding of the phototube is impractical. Absorption in lucite is very small over the spectral region of importance, and internal reflection along a polished column preserves a useful fraction of the light for delivery to the tube. 1.44 Photomultiplier circuits must distribute the required potentials to dynodes, match the impedance of the output signal cable, and in some cases function in shaping the output pulse. Two typical arrangements are shown schematically in Fig. 27, in which 6 involves a pulse-clipping and limiting feature which is sometimes desirable for coincidence counting with short-time resolutions. such as

5-28

EXPERIMENTAL

TECHNIQUES

[SEC. 5

Voltage per stage is usually in the region of 100 to 150 volts, and the gain of the tube (nine-stage) rises by roughly a factor of 2 per 10 volts' increase per stage. Selected Presumably tubes may operate without prohibitive noise at 200 volts per stage. identical tubes will often differ greatly in noise level and in the dependence of noise In case very small signals are to be treated, a reduction in upon applied voltage. noise can be achieved by cooling the multiplier, but precautions against affecting liquid phosphors and against unwanted moisture condensation must be recognized. counter work, and especially with inorganic Amplifiers for much scintillation phosphors, involve the same principles as applied in fast pulse chamber and propor But when it is desirable to take advantage of the fast rise of tional counter cases. scintillator pulses and the rapid decay of the organic phosphors in order to achieve precise timing in the 10~8 to 10~'-sec region, it is necessary to use amplifiers of band width extending to very high frequencies. For this purpose the distributed amplifiers with bandwidths extending to the 100- to 200-Mc/sec region are much used. The experimenter who employs scintillation counting in radioactivity measurements and in 0 and 7 spectroscopy will be especially concerned with linearity, pulse-height discrimination and analysis, and stability of voltage and gain, and the techniques and references of the article on amplifiers will apply.

2

APPLICATIONS TO SPECIFIC RADIATIONS Alpha -particle Counting

is is

a

a

is

a

if

Selective counting of a particles in the presence of a background of /S and 7 2.11 radiation makes use of their relatively large values of dE/dx compared with those Thus in an ionization chamber or proportional counter characteristic of electrons. in which, for example, the maximum possible electron path length is 10 cm in argon at 1 atm the greatest energy loss to the gas from electrons will be about 0.1 Mev, whereas a particles up to about 9 Mev can deliver all their energy within the same Consequently, the amplification and output pulse-height discriminator path length. are so adjusted as to reject all electron pulses and retain all a events delivering more than 0.1 Mev to the chamber gas. The scintillation counting of as does not always provide so pronounced a separation from background as indicated for the gas pulse counters because of the fact that light yield is not proportional to energy loss for large dE/dx, as discussed under Scintillation Counting. However, thin surface layers of inorganic phosphors spread upon the tubes or viewed by the tubes are frequently employed in faces of photomultiplier a counting. Identification of a tracks in photographic emulsions is usually easily made in terms of range and grain density, since ranges of natural as are less than emulsion thick At high energies where ranges are much greater than emulsion thicknesses, nesses. distinction between a segment of a track and a segment of proton track having the sufficient portion of path length of course, not possible, but same dE/dx available to observe the rate of change of grain density, the identification may be accomplished. In cloud chambers the distinction among made upon the basis of droplet particles count (that is, dE/dx), range, and curva ture of track in known magnetic field. Any two of these three quantities can yield the identification. Fio. 28. Illustration of limiting angle for 2.12 Absolute o-pnrtiele detection from given depth x. counting of or-particle emission from radioactive solid sample most often done in a chamber of the type illustrated in Fig. 13 or 14. The sample must be sufficiently thin that self-absorption of the particles involves only a small correction, and the back scattering into the chamber of a particles originally emitted into the hemisphere away from the sensitive region must be determined. These is,


2.1

Sec. 5-1]

instruments

for

measuring

radiation

5-29

corrections must be applied to the otherwise simple assumption that half the emitted a's will enter the sensitive region and be counted. The self -absorption correction is made as follows: For particles originating at depth x with energy E there exists a certain limiting angle Bc (Fig. 28) within which the particles must be emitted if they are to emerge with residual energies equal to or greater than t, which is necessary to provide a pulse above the discriminator bias. So of the upper hemisphere the fraction

/(*)

-/:

sin * de

cos $e =

R(E)

-

ft(«)

will contribute counts, where R(E) is range of emitted particles in the source material and R(f) is the range in source material of particles with energy «. So /Or)

R(E)

- RW

Then the efficiency against self-absorption, for particles hemisphere, for the entire sample of thickness t will be


-;/,>x) dx

= 1

-

emitted

into

the tipper

it

R(E)

- «(«)

The observed counting rate should be divided by 8 to obtain the corrected emission of particles into the upper hemisphere (i.e., one-half the total emission). The back-scattering correction is dis cussed in Chap. 6 of Ref . 36. It can be experimentally evaluated by varying the thickness of the material upon which the source is deposited and extrapolating the measurements to a value corresponding to zero thickness. This correction is THN ORGANIC FILM by use of material of low minimized .BEARING SAMPLE DEPOSIT NEAR CENTER atomic number. Its value is typically of the order of 1 per cent. Elimination of the back scattering, to gether with achievement of 4x steradians of effective detection solid angle, is ac complished by a double-chamber propor tional counter such as that indicated in Fig. 29. The source is deposited upon a very thin organic film such as collodion. 2.13 Energy spectroscopy of a parti Fio. 29. Double proportional counter for cles may be accomplished by three meth 4w geometry in a counting. ods: (1) pulse-height analysis of the output pulses from a grid chamber with linear amplification, (2) magnetic bending of particle trajectories to give momentum analysis, (3) range measurement. Pulse-height energy analysis employs the grid chamber so as to eliminate the depend ence of pulse height upon location of track as discussed under ionization pulse counting previously. Obviously the chamber dimensions must be such as to contain completely the ranges of the particles emitted from the source. The energies of a groups may be At 5 Mev a line breadth compared by this means to an accuracy of 1 or 2 per cent. at half maximum of 0.1 Mev is readily obtained, and 0.04 Mev is quite possible. The position of the peak of the line can be relatively assigned to a precision of ±0.01 Mev This method is applicable to small samples and when lines are not too close together. low intensities of a emitters; because of its simplicity and convenience it is the most used technique unless greater resolution or precision is required than it provides.

EXPERIMENTAL

5-30

TECHNIQUES

[Sec.

5

Clearly the sample must be very thin so that self-absorption will not seriously modify the energies of the particles. Reference 37 described this technique of spectroscopy, and Ref. 38 is a discussion of grid-chamber design and theory. Magnetic analysis of particle momenta allows absolute energy determination through the relation r =

~ y/2ME

where r = radius of curvature of particle path, cm q = esu of charge carried by particle = 2e for a particle B = magnetic flux density, gauss c = velocity of light, cm /sec M = particle mass, g E = kinetic energy, ergs This type of analysis can thus yield absolute energy determinations of high precision, but it requires an extensive, uniform magnetic field and vacuum chamber, and because of the very small solid angle and source size necessary for high resolution it is limited to rather intense a emitters if lifetimes are short. Photographic recording of the - MAGNET


VACUUM TANK WALL

Flo.

30.

Illustration of

a 180° type of magnetic spectrometer

POLE

FACE BOUNDARY

for energetic charged particles.

analyzed "lines" allows long exposures for long-lived emitters and relieves the require ment of high intensities. Line widths of In Refs. 39 techniques of magnetic spectroscopy are presented. 0.01 Mev and less are achieved, and energy assignments with accuracy somewhat better than this can be made, with solid angle efficiencies of the order of 1/5,000. A schematic illustration of the 180° type of a-particle spectrometer is shown in Fig. 30, in which the recording of analyzed particle groups as lines upon photographic emulsion is indicated. Counting techniques employing a slit aperture may be employed when intensity of source allows. Range determinations have historically played an important role in energy determina tions, and this method is often the most feasible for a beam of particles from an acceler ator. In principle it may be applied to any source yielding sufficient intensity. Considerable precision is possible by employing a very shallow chamber, such as Adjustment of the pressure of the argon in illustrated in Fig. 31, with a grid wall. which the system is immersed allows the distance measurement to be convenient for accuracy, and the increased ionization near the end of the a trajectory permits a A plot of shallow chamber (about 1 mm) to register satisfactorily large pulses. counting rate vs. distance or vs. gas pressure for a given distance permits exploration of the range-end region with its characteristic straggling; then recourse to range vs. The method is generally used for energy data allows the energy to be assigned. comparisons of unknown with known a energies.

Sec. 5-1]


5-31

When more than one a-energy group is present, it is best to use a double-chambered The shallow ionization chamber, with the two chambers operating differentially. thin dividing wall, or grid, forms the collecting electrode, and charges of opposite Thus a particle which passes sign are received from the two chambers, respectively. through both shallow volumes will develop only a small net pulse, whereas a particle stopping near the dividing wall will contribute a large pulse, either positive or nega It is thus tive, depending upon the volume in which it expended the most energy. possible to identify the range endings of a mixed group of a's. ADJUSTABLE PRESSURE

-THIN

FILM BEARING

SOURCE

r

OF ARGON

TO

DEPOSIT

-5


Fig.

31.

Shallow accounting

AMPLIFIER TRACK

CALIBRATED

chamber

and source arranged

for range measurement.

Analysis of this method is thoroughly given in Ref. 40, and examples of chamber and data are to be found in Chap. 6 of Ref. 36 and in older sources such

construction as Ref. 41.

2.2

Beta-ray and Electron Counting and Spectroscopy

Because of its simplicity Geiger Counting, Applicable Counting Techniques. simplicity of its associated electronics, the Geiger counter is most widely used in the 8 counting of radioactive samples. Its limitation in counting rate due to dead time and recovery time has been previously discussed, and simple experimental methods exist for evaluating the loss of counts at a given count rate due to this cause. A simple, first-approximation technique to evaluate the effective dead time is the divided-sample method in which a ^-active sample foil is divided into two parts, A and B, which can be mounted singly or together in the counting position with identical When A counting efficiencies from the standpoint of solid angle and absorption. alone is counted, the observed rate is Ci = IcNa/(1 — C\T), where iV,i is the 0 emission of sample .4, A: is a constant determined by solid angle and absorption in sample and When B alone is counter wall, T is the average effective dead time per count. counted, the observed rate is C2 = kNa/(l — C,T), and with .4 and B together the From these three relations one observed rate is C» •= 1c(Na + iVa)/(l — CiT). 2.21

and the

obtains

j,

& cv

—

-

Ci

cy

—

Ci

- cy

and the correction for dead time to be applied to any observed counting rate C con sists of dividing this rate by the net fraction of time (1 — CT) within which the counter is sensitive. A more accurate method of ascertaining the loss of counts at any counting rate consists of activating a material of convenient decay period (say a few hours' halfThe sample life) to a level of activity which causes the counter to count very rapidly. must contain only one decay [>■i ■■ 13 rates are observed as the activity decays down into regions where the count loss is certainly negligible as evidenced by the proper time dependence of the observed rates. The latter are plotted vs. time on a semilogarithmic sheet, and the line representing the known decay of the activity

(•..,.

is

drawn through the data it fits at the lower counting rates.

From the departure

of the observed rates below the line at higher counting levels the correction is obtained

5-32

EXPERIMENTAL

TECHNIQUES

[Sec.

5

0

is

w

J

ww

ANGLE

CONE

energies. is

It good practice to limit the solid angle subtended by the counter at the source region in such a manner that particles pro Fig. 32. Example of solid angle limitation jected from the source into the counter will surely enter only the sensitive volume, as in absolute ^/-particle counting. illustrated in Fig. 32. An alternative to this, which also eliminates the scattering correction, the use of the 4x steradian counter such as illustrated in Fig. 29. Mounting of the sample on thin plastic film, such as polystyrene of to mg/cm1 thickness, can reduce back scattering from the mounting to the order of per cent, but in some cases the specific activity of the active material so low that samples of considerable thickness, and hence of considerable self-scattering and self-absorption, are demanded for useful counting rates. Much effort has been devoted to absolute 0-activity determinations with Geiger counting, and Refs. 42 are a summary of the techniques which apply. Similar considerations apply to the use of either proportional counters or scintilla tion counters in absolute counting. 2.23 The most frequent type of measurement in the Spectroscopy of Particles. analysis of emission the determination of the maximum energy, and usually is 0

is

3

1

2

is

SOURCE

0


0

a

it

a

if

is,

for any desired rate. The experimenter should make sure that counting-rate loss due to blocking of the mechanical register is not occurring by employing a sufficiently large scaling factor ahead of it to stay well below its maximum rate of operation. For sample counting, the end-window type of Geiger counter shown in Fig. 19 is widely used, together with a sample holding system which allows accurate reproduci bility of geometry. Proportional counters are being increasingly employed in 0 counting in view of their reliability and their short dead time. The gas-flow technique of supplying the count ing gas allows the use of very thin windows or of inserting the sample within the tube envelope. Internal counting of activities in the gaseous phase is frequently possible in proportional counters when the sample gas allows rapid electron collection or is present in such dilution as to allow it. Proportional counter application to lowenergy jS spectroscopy is described below. Scintillation counting of electrons can be applied up to very high rates, the time limitation existing in the electronics rather than in the detector itself. It however, useful only the light pulses from the electrons are large compared with the noise inherent in the photomultiplier tube, so in continuous spectrum of incident elec trons some at the lowest energies must be biased out with the noise. The possibility of surrounding the active sample with the phosphor by placing cavity or between plane blocks of phosphor has been exploited where weak into samples were involved or large solid angle desired. 2.22 Absolute counting of activities involves the evaluations of corrections which may be quite severe: (1) self-absorption in the sample, (2) absorption in counter window or wall, (3) effective solid angle subtended by the sensitive volume at the sample, (4) back scattering by the mount ing material and possibly by the source material itself. (1) and (2) are compli ESTIMATED LIMIT OF UNIFORMLY cated by the fact that spectra are continu SENSITIVE VOLUME ous down to zero energy with characteristicshapes, so that the calculation of the frac tion surviving (or lost) difficult, especially in view of the fact that various obliquities of path will involve different cutoff energies. The scattering correction may be large THICKNESS EXCEEDS and difficult of evaluation because of the MAXIMUM RANGE OF DEFINING ease with which electrons with their low 0-PARTICLES APERTURE mass are scattered by nuclear coulomb fields and because of the spread of electron -EFFECTIVE SOLID

Sec. 5-1]


5-33

done by the absorption method. However, for purposes of testing theories decay it is necessary to investigate the energy spectrum of the emission. This can be performed by various species of magnetic spectrometers and by pulse-height counting with proportional counters and scintillation counters. These methods also provide precise determinations of the maximum energy when analyzed by the method of the Fermi (or Kurie) plot familiar in 0 spectrum examination.4' Absorption Method for Energy Limit. In principle, the experimenter seeks that limiting thickness of a convenient absorbing material, such as aluminum, which when Determination interposed between source and detector will barely prevent detection. this

is


of 0

Flo. 33. Range of p particles vs. electron energy.

ELECTRON ENERGY IN MEV or electrons in milligrams per square centimeter

of aluminum

of this "end point" is facilitated by plotting counting rate vs. absorber thickness on a semilog plot and then attempting to identify the point at which the counting rate Figure 33 relates aluminum range to has become equal to background counting. electron energy. Of course, it is essential to add the aluminum equivalent of the air path and of the counter window or wall to the thickness of absorber inserted. In favorable cases the end point may be determined to the accuracy desired by

but there are factors which may make the visual assignment difficult: of the 0 spectrum is such that the intensity of particles goes to zero The shape (1) at the energy limit, (2) electrons scatter considerably in matter so that their paths are not rectilinear, and (3) y radiation which may accompany the 0-decay transition or which may be present in the background makes questionable the judgment of just where the detection of /S's ceases. A number of analytic techniques for treating the plot of counting rate vs. absorber inspection,


6-34

EXPERIMENTAL

TECHNIQUES

[Sec.

5

thickness have been devised to make the assignment of the end point more precise than simple inspection. One of these, the Feather technique, compares the shape of the absorption curve of the /3 emission under investigation with the shape of the absorption curve of some other f) emission whose end point is well known and whose spectrum is assumed to have the same proportions as that of the unknown (which may not be true). This and other techniques are presented in detail in Refs. 44. Accuracies possible in absorption determinations depend upon 0 energy, back ground or 7 intensities, and strength and lifetime of sample (which may determine the statistical accuracies of datum points). With good technique and favorable conditions, 2 per cent accuracy in the end-point assignment may be possible. For low-energy 0's and for the electron lines from 7 conversion, absorption tech niques generally give way to pulse-height methods with scintillators or proportional counters. The very low energies, such as the region of Hs jS's, are particularly well accommodated by proportional counting. Magnetic spectroscopy of & particles makes use of a choice of three types of spec trometers: (1) plane orbit type involving 180° bending or deflection through a smaller angle by a magnetic field of sector outline (in simplest form these magnetic fields are uniform, but shaping of the fields to provide desired types of focusing is possible and can give increased aperture for a given resolution), (2) the magnetic lens type in which electrons emitted from a small source within a limited solid angle can be focused through helicoid paths into an image of the source located on the lens axis at a distance from the source depending upon energy (momentum), (3) the longitudinal magnetic field type in which helical paths connect a point source to an axial image point for emission within a certain cone. Souree-to-image distance again is uniquely related to electron energy. In certain cases, to favor a desired combination of resolution and angular aperture, the collector openings are ring-shaped around the axis rather than circular holes on the axis. For any of these types the collector or detector may be a Geiger, proportional, or scintillation counter or photographic emulsion. A thorough description and analysis of resolutions, line profile forms, and solidangle efficiencies of these types of spectrometers and of several of their modifications are found in Ref. 45. Momentum resolutions among the various types range from 0.1 to 10 per cent, and solid angle efficiencies from 0.01 to 10 per cent. For a given type these two characteristics are, of course, inversely correlated. Virtues of resolut ion of the spectrometer proper are of no significance in determining spectrum shape unless the source preparation technique is adequate. Self-absorption must be minimized, and back scattering by source and its supporting material must be as small as possible. Both of these factors tend to accentuate lower energies in the observed spectrum; this is so in the case of scattering because of the stronger deflections of low-energy particles and the increased path length in absorber owing to scattering. Very thin films of plastic carrying a thin, localized deposit of the sample are desirable. Sample preparation is widely discussed and review of such techniques is given in Ref. 46. Magnetic analysis of electron momenta is accomplished also in the cloud chamber by imposing a uniform magnetic field normal to the median plane of the chamber. The radii of curvature of 0 tracks from a sample on very thin backing material may be measured with accuracy which is limited by the effects of multiple scattering and turbulence. This technique, however, provides large effective solid angle and may thus be desirable for very weak samples or for activities in gaseous form which could be introduced into the chamber atmosphere. Proportional counter spectroscopy of /3 particles is particularly well adapted to the low-energy, low-intensity /3 emitters, of which H3(£mol = 17.9 kev, T\i ^ 12 years) and C14(£m»r = 155 kev, Tyi ^ 6,000 years) are representative, which require a technique combining high efficiency; elimination or minimizing of absorption in sample, windows, and air path; and sufficiently good absolute resolution to provide desirable percentage resolution at low energies. When the entire ranges of the electrons can be contained within the region of uni form gas multiplication, the output pulse-height spectrum will represent the /3-energy

Sec. 5-1]


5-35

Containment spectrum, provided again that source preparation technique is good. of the electron paths is aided by use of large counter volume and high gas pressure, and by the use of pressures as high as 700 psi this technique has been applied to /3 When a longitudinal energies as high as 0.6 Mev from a small internal source. magnetic field is imposed such that particles from the source are bent in helical paths to their transverse momentum components, the containment of total appropriate electron energies from a small source up to a few Mev can be accomplished with a counter long enough to ensure small percentage loss due to particles of high energy moving in nearly axial directions.

For the very low energy 0 emitters for which self-absorption and mounting would critical, a gaseous phase of the material is desirable which can be injected in small concentration into the counting gas without deleterious effects. Through use of this technique a low-energy limit of analysis near 200 ev has been achieved. Energy resolution in the proportional counter method is basically determined by the statistical variations in number of primary ion pairs formed by an electron of given and electronic amplification need also energy, but variations in gas multiplication to be considered. Sorting and recording of pulse heights are accomplished by elec tronic pulse-height analysis or by oscilloscope photography. Reference 47a provides a summary of the proportional counter spectroscopy tech give specific details and applications. niques, and Refs. 476 to Beta spectroscopy with scintillation counters is occasionally a useful technique, though it lacks the uniqueness of the y spectroscopy which derives from the large T-absorption properties of phosphors such as Nal(Tl). References 48 and 32 describe the usage of the scintillation counter as a 0 spectrometer through pulse-height analysis of its output after linear amplification. The high-energy portions of Fermi plots are satisfactorily linear in the data published by this method, making it a convenient and rapid way of comparing cutoff energies for various 0 spectra. Techniques of preparing and mounting the phosphor are similar to those to be described for y be


/

spectroscopy.

The low-energy portions of the 0 spectrum as rendered by the scintillation counter may be accentuated because of the scattering out of electrons so that they do not surrender their full energy within the phosphor. However, a 4x geometry arrange ment in which the thin sample is sandwiched between scintillator blocks can remove this problem. This technique of spectroscopy retains the high efficiency of the proportionalcounter method and is more favorable for high-energy (S's, whereas the former is better adapted to the low energies. That this last should be true is apparent from considerations of statistical fluctuations in the fundamental events leading to detec tion; e.g., a 70-kev electron in anthracene will produce about 1,000 photons of which considerably less than 100 per cent will be collected at best and only about 5 per cent will produce photoelectrons, whereas the same electron in the argon of a proportional counter will produce about 2,600 ion pairs, all of which will be collected. 2.3

Detection and Spectroscopy of y and X Rays

2-31 Detection of 7 radiation is accomplished, as outlined previously under Principles of Radiation Detection, through the production of secondary electrons by the photoelectric effect, Compton effect, or pair production in the chamber or counter walls or the material of the detecting medium. Because of the strong dependence of these processes upon atomic number Z, it is necessary for optimum efficiency to employ materials of high Z in counter walls and counter gas or to select scintillator materials of high average Z. If the energies of X or 7 rays to be detected can match or overlap the K -absorption energy of an element in counter wall or medium, the photoelectric sensitivity will be enhanced. For the detection and relative measurement of 7-ray Gamma-sensitive Chambers. intensity levels, an ionization current chamber may be employed whose sensitivity can be made high by a multiple-plate construction, such as illustrated in Fig. 34, of high-Z material such as tantalum (since lead is not a desirable structural material

5-36

EXPERIMENTAL

TECHNIQUES

[Sec.

5

in thin sheets). A heavy gas at high pressure makes a further small contribution to 7 conversion, and its pressure and the plate spacing should be so related as to allow the electrons to expend as great a fraction of their range as possible within the Such a chamber will not follow extremely rapid variations in intensity because gas. of the long collection times of the ions, and improvement in this regard is possible by use of the free-electron gases such as the argon-C02 mixture familiar in pulse chambers and proportional counters. Chambers of this type are discussed in Chap. 5 of Ref. 36. Gamma Sensitivity of Geiger and Proportional Counters. Empirical results are quoted in Chap. 5 of Ref. 36 for the approximate detection efficiency of a counter whose wall thickness is at least equal to the range of the most energetic secondary electron produced in the wall material but yet not so large as to significantly attenuate the y flux. The formula there given for efficiency E is


E

= 0.135(w»ftp»

+

HeRc

+ 2/lpr/fpr)

where n's and R's indicate, respectively, the 7-absorption coefficients and mean ranges of secondary electrons for the three processes of photoelectric absorption, Compton This formula is in essential agreement with the data of effect, and pair production. Fig. 35, which come from page 103 of Ref. 36 and which illustrate 7-energy depend ence and Z dependence of counter sen sitivity. Except for the high-Z wall materials for energies in the region of nuclear 7 rays, the efficiencies are less than 1 per cent. 020,

-HY

TO D C.

AMPLIFIER

GAMMA RAY ENERGY (MEV)

Fio.

34. Scheme of multiple-plate ionization current chamber for 7-radiation detection.

Flo.

35. Energy dependence of 7-counting efficiencies for cylindrical counters of two wall materials. (From Ref. 36, p. 103.)

Gamma efficiencies of scintillation counters are characteristically large, especially for the inorganic materials, such as CaWOj and Nal(Tl), with their high densities and high average values of Z. The total attenuation coefficient for 7 radiation passing through Nal as a function of photon energy is given by Fig. 36. At minimum absorp tion energies near 5 Mev, 1 — e-w.u>u.M) ~ 28 per cent of the photons will experience interaction in passing through 1 in. of Nal, and at lower and higher energies the percentage is larger (at 1 Mev it is 43 per cent). For CaWO* the absorption is yet much stronger, and this material has found use for this reason in the clinical detection of I131 7 radiation. For anthracene as a representative of the organic phosphors the attenuation coeffi cient is also plotted in Fig. 36. In this case, at 1 Mev, 19 per cent of the photons will experience interaction in passing through a 1-in. thickness. 2.32 Spectroscopy techniques for 7 and X rays make use of the following physical principles: (1) In the photoelectric effect the photon is completely absorbed and ejects an electron with kinetic energy E =• k — B, where k is the energy of the photon and B is the energy of binding (or ionization energy) for the shell from which the electron is ejected. (2) In a Compton collision the photon is degraded in a scattering event


Sec. 5-1]

radiation

6-37

- k'

U

tan1

+

+

1

+

+ k/vw1)a)2

(1

= (1

tan*

2k(k/mc') + t/fflf1)' 2k/mc*

J

- - k' k

E 4/

\f>

with an electron and recoils at an angle 8 from its original direction with energy k' = k/\l + (k/mc')(l — cos 8)]. The companion electron recoils at angle with kinetic energy:

= rest energy of the electron. Thus the maximum energy imparted by a = of in a Compton collision energy (setting photon Em„ = k/(l + Hmc'/k). The scattered photon may proceed on to further events in the medium. (3) In pair 0)

AT Rl 3H

r)

(SCAL

X

Is

\J

0 I

o

-A NTHRACEN'E

E

9

z

•JoKSC :ale AT LEFT

)-

O.I

10

10

GAMMA

Fio.

36. Energy dependence

RAY ENERGY (MEV)

of -y-ray absorption in Nal and in anthracene.

E-

=

k

E+

+

a

k

is

= 2mr' = 1.02 Mev, the photon exchanged production, which can occur only above for an electron-positron pair which will then carry away total kinetic energy

- 2mc2

is

y

0

a

is

0

k

for high This process depends upon Z* and rises rapidly with increasing values of values of Z. Photon energies are inferred from identification of the conversion processes and measurement of the kinetic energies of electrons produced. All the various techniques of spectroscopy may be here applied to the measurement of electron energies. Gamma spectroscopy at tow energies by proportional counters an adaptation of lowvery thin con energy spectroscopy by this method accomplished by illuminating Frequently verter within the sensitive volume by the X or rays to be analyzed. the gas itself the converting material. Pulse-height analysis allows identification photoelectron groups associated with ejection from various shells plus a broad distribution of Compton electrons with a definite upper energy limit. of


^

is

mc1

k

where

EXPERIMENTAL

TECHNIQUES

[Sec.

5

Following a photoelectric conversion event, typically involving the K shell, the emitting atom can radiate its characteristic K X radiation, or alternatively, it may surrender this energy in the form of an Auger electron. In the latter case the total counter pulse, adding the contributions of both photo- and Auger electrons, will represent essentially the original photon energy, whereas if the K X radiation escapes without conversion, the pulse height represents the original photon energy minus Both pulses are often present in the pulse-height spectrum the K binding energy. as well as pulses from electron ejection from other shells. Photon spectroscopy by this method is employed from ultraviolet energies up to With good technique resolution may be but little worse than the statistical 0.5 Mev. uncertainties in ion formation by the ALUMINUM OXIDE OR ejected electrons. Gas multiplication is MAGNESIUM OXIDE typically in the region of 100 to 1,000, METAL AGAINST FROSTED CONTAINER SURFACE OF and it must be uniform, implying care CRYSTAL At with end effects in counter design. low energies, comparable to X-shell po tentials in the converter, the Compton GLASS References contribution is insignificant. WINDOW SEALED TO on this technique are listed under Refs. CONTAINER 47 and 49. Gamma spectroscopy by scintillation •MINERAL counting has been developed particularly OIL FOR END WINDOW OPTICAL The typical with the Nal (Tl) phosphor. OF MATCHING PHOTOMULTIPLIER met hod of conserving photons is indicated TUBE by the arrangement in Fig. 37 and is clearly discussed in Ref. 35. Crystals Flo. 37. Example of light-containment and companion photomultiplier tubes up features and optical coupling of Nal crystal to a few inches in diameter are commer to photomultiplier tube. cially available, and because of the great difficulty in growing large, clear Nal crystals and of "canning" them in a moisturefree and impervious container lined with high-reflecting material, the prospective experimenter is advised to purchase this item. The various energy depositions within the crystal by an incident photon of energy k may be classified as follows: (1) By pair production (if k > 1.02 Mev), with contain ment of both pair particles and conversion and containment of both photons from positron annihilation: AE = k. (2) Same, except escape of one or both annihilation photons: AE = k — 0.51 Mev or k — 1.02 Mev, respectively. (3) By a single Compton collision in which the scattered electron is contained but the scattered photon escapes: any value of AE up to AE(max) = Jfc/(1 + \imc*/k). Since the probability of a particular value of Compton energy transfer is highest at the maxi mum value, the "Compton cutoff" would be fairly well defined from a single event alone, but further collisions of the scattered and degraded photon within the phosphor blur the sharp cutoff. (4) By photoelectric conversion in a K shell, with contain ment of the electron: AE = k — Bk, where Bk is the K-shell binding energy. (5) Same, but with Auger conversion of the K X-ray energy and containment of the Auger electron: AE = k. Other combinations are possible, and the uniqueness of these events is modified by partial escape of electrons. But it is evident that the pulse-height spectrum of the linearly amplified multiplier output can display characteristic maxima and a Compton Relative emphasis of certain maxima may be effected by choice of crystal limit. size and limitation of regions of primary conversion by y collimation. The lack of definition inherent in electron escape is greatly reduced by closely surrounding the crystal with another phosphor (plastic or liquid) in anticoincidence with the Nal (TP signals so that pulses involving electron escape are rejected. Identification of the origins of the various maxima allows assignment of photon energy relative to a calibra ray in the same energy region. tion effected by use of a known The article of Borkowski and Clark" contains specifications for crystal preparation and mounting for optimum resolution. frosted crystal surface They recommend a

->■


5-38

Sec. 5-1]


6-39

for the viewing face in contact with the tube or light pipe) in intimate contact The crystal cover glass is optically blended as a diffuse reflecting surface. to the phototube surface by mineral oil or a clear silicone grease. The authors cited have obtained, with selected tubes and crystals, peak half widths ranging from 4.3 per cent at 2. 70 Mev to 21 per cent at 41.6 kev. Peak positions with such resolution can be assigned with a precision of 1 per cent for energies over 0.5 Mev. The relative importance of the various conversion processes varies with photon energy, but as far as spectroscopic identifications are concerned, the photoelectron peak typically predominates until the pair-production threshold energy is considerably References 35, 48, and 50 are pertinent. exceeded. Gamma Spectroscopy by Magnetic Analysis of Electrons. Replacement of the source in a 0-ray spectrometer by a thin foil of converting material irradiated by the photons to be analyzed provides a widely used means of precise y spectroscopy. Clearly the efficiency will be low, for besides the characteristically small, angular aperture of the spectrometer there is to be considered the conversion efficiency of the foil, which must be thin enough to involve suitably small absorption of energy from the conversion (except

with

AljOj


electrons. Pair spectrometers which bend the positron and electron in 180° plane orbits in Photon energy is opposite directions have been employed in the work of Refs. 51. equal to the total energy of the pair, and the momenta of the latter are determined by their radii of curvature in the known uniform magnetic field. Thus

k

=•

y/ c*p+*

+

m*c*

+

\/c'p-'

+ m*c*

= Ber±, where p+ and p_ are, respectively, the positron and electron momenta; rest energy of an electron, B the magnetic flux density, e the electronic With cgs units charge (esu), and r± the radii of curvature of positron and electron. and B in gauss, the photon energy k is in ergs. Energy assignments with +0.05 per cent precision have been reported with this type spectrometer when source intensity was unlimited, but more typical results in the region of 4 to 12 Mev gave energy assignment precisions of 2 to 5 per cent with efficiencies ranging from 10~8 to 2 X 10~7 detected pair per quantum emitted from source.'1" Resolution widths were much broader than the energy assignment precision. Pair spectrometers involving magnetic lens analysis of electrons are reported in Refs. 52. Placement of the converter and source slightly off the lens axis results in formation of separated positron and electron image positions on opposite sides of the axis, since the usual arrangement involves a 90° portion of helicoid rotation which will be in opposite sense for the pair members. Coincidence counting of the pair members with short time resolution (~5 X 10~* sec), using scintillation counters with organic and rp±

mc* is the

background effects. Energy assignments with + 1 per cent phosphors, suppresses precision and peak resolution widths of 6 to 10 per cent are reported with efficiencies of 1 to 4 X 10""8 pair count per quantum emitted for 7 energies between 4 and 10 Mev.

Magnetic spectrometers utilizing photoelectric conversion for 7 rays below pair pro threshold energies have been widely employed for many years in special cases for energy assignments with precision under 0.01 per cent and with the order of 0.1 per cent as quite common accuracy. All the techniques and refinements per Reference 53a discusses a number of taining to 0 spectroscopy apply here also. these, and Refs. 536 to d give recent work with this method and contain additional bibliography. emulsions has been applied to photon Gamma spectroscopy by deuterium-loaded Photographic emulsions •mcrgies greater than the binding energy of the deuteron. soaked in heavy water to build up the H* content will develop characteristic proton tracks from the reaction H,(7,n)HI in which proton energy (range) is uniquely related to the photon energy and to the angle of proton ejection with respect, to original photon direction. Thus if the photon direction is determined, the known differential of the deuteron as a function of photon energy '•ross sections for photodisintegration duction

5-40

EXPERIMENTAL

TECHNIQUES

[Sec.

5

allows calculation of the spectrum. This technique is presented in Ref. 54. Energy assignments to the nearest 1 per cent were made, though peak widths at half maximum were 0.4 Mev at energies of 6 and 7 Mev.


2.4

Neutron Detection and Counting

The subject of neutron measurements is discussed in Sec. 5-2; consequently this article will identify and comment upon instrumental methods and principles of detec All neutron detection necessarily depends upon a nuclear interaction of the tion. neutron, and the production of secondary particles subject to observation by their ionization, fluorescence production, or photographic effect. Consequently, neutron counting efficiencies are not large, especially when analytic counting is desired which will preserve information concerning original neutron energy. It is convenient to classify neutron energy regions into (1) thermal, i.e., in kinetic equilibrium with surroundings; (2) resonance, extending roughly from 0.1 to a few hundred ev, within which interval characteristic capture resonances occur; (3) fast, from resonance up to around 20 Mev; and (4) high energy. The distinction between (3) and (4) is not precise, but it generally relates to a change in character of nuclear collisions near 20 Mev in which the compound nucleus concepts seem clearly to become inadequate. These various regions involve characteristic instrumentation and detection principles. The thermal and resonance regions demand exothermic neutron capture reactions which will release amounts of energy into the counting In the fast and high-energ;/ regions, elastic system clearly above background effects. collisions with protons are most important, since in the n-p encounter the equal masses imply equal sharing of energy on the average and such sharing produces detectable Unique among the exothermic reactions is the production of fission proton energies. in certain heavy nuclei. 2.41 Thermal and near -thermal neutron detection is most widely accomplished by utilizing the reaction B10(n,a)Li7 which is exothermic by 2.34 Mev (to a 0.44-Mev state in Li7) and thus yields a recoiling a particle and Li7 nucleus which are readily detected by ionization or scintillation counters or by tracks in boron-loaded photo graphic emulsion. The neutron capture cross section for B10 varies accurately with reciprocal velocity between the energies of 0.01 and 1,000 ev. At the representative point of the velocity distribution for thermal neutrons at 20°C, namely, 2,200 m/sec, the capture cross section for the B10 nucleus is 3,990 barns (1 barn = 10~" cm') and the average value for a natural isotopic mixture (18.8 per cent B10 and 81.2 per cent B") at the same velocity is 750 barns. This 1/v dependence, together with the v factor inherent in flux density, means that a boron detector delivers a response proportional to the numerical density of neutrons with respect to the energy region below 1,000 ev. Above 1,000 ev the capture cross section departs from the l/v dependence, but it is by then very small. Ionization chambers and proportional counters for slow neutron detection either are rilled with purified BF3 gas or have a thin layer of boron adhering to their walls and are filled with the argon-COj mixture. Pulses from the capture reaction are large compared with electron pulses in a properly operating counter, and the latter are excluded by the discriminator bias. Purity of the BF3 is important if free electron collection is to be assured, and this is best accomplished by releasing BF3 by thermal decomposition of CaFs-BFj or of C5H6N;BF4, but satisfactory results can sometimes be obtained by fractionally dis tilling tank BF3 from a frozen state ( — 127°C). It is most important that the gashandling system and counter to be filled be free of moisture, and it is not advisable to store BF3 in contact with stopcock grease. A clean brass vacuum system, thor oughly dried and outgassed, is satisfactory for evacuating and filling the chambers. If the gas is to be liberated from either of the compounds above, they should decomposition be thoroughly outgassed at a temperature just below point before generating the BF3. Reference 55 describes the technique pertaining to BF> usage.

Sec. 5-1]


5-41

The boron coating of a metallic surface for a chamber or counter wall is performed decomposing B2C« (borane) on heated foils of tantalum. The thickness need be only equal to the a-particle range, or about 1 mg/cm'. Both this type and the BF, type of proportional counters are commercially available. Because of the well-known boron capture cross-sectional values, absolute determina tions of neutron density in the thermal and slow neutron regions can be made, though a measurement of flux density would require precise knowledge of the energy spec trum. Chapter 8 of Ref. 36 discusses absolute calibrations of pulse chambers and proportional counters, and in Ref. 56 the approxhnate calibration of the response of an ionization current chamber to a slow neutron flux is described. It is clearly necessary to avoid strong neutron absorbers in the construction of the counter; in particular, glass walls of pyrex contain boron and should not be used. Scintillation counting of slow neutrons by use of Lil(Sn) crystals is described in Ref. 57. The effective reaction is Li*(n,a)Hs, which is exothermic by 4.63 Mev. The capture cross section has a 1 /w-dependent behavior at low energies between 0 and 100 ev and is about 70 barns at the mean thermal energy. Liquid phosphors loaded with boron or with cadmium have been devised and reported in Ref. 58. The cadmium capture displays a strong l/v behavior with a resonance superimposed at 0.180 ev, and detection is accomplished through scintillations due to secondary electrons developed within the liquid by the capture y radiation from the reaction by

Cd»>(n,-y.)Cd»».

Foil activation methods for slow and thermal neutrons involve usually the use of foils, with and without cadmium envelopes (sec Art. 1.1 of Sec. 5-3 for discussion). All foil activation methods, if they are to give absolute data, will depend ultimately upon techniques of absolute 0 counting, but comparisons can be made with simple equipment and procedures. Fission counting of thermal neutrons utilizes the large thermal fission cross section of Um which is deposited in a thin layer upon the cathode plate of a pulse chamber. The U"*, which of course, present in 99.28 per cent abundance in the natural isotopic mixture, will not undergo thermally induced fission, but the stopping power introduces against emergence of fission fragments into the counting gas greatly reduces efficiency, so desirable that material isotopically enriched in LTJ" be available. Fast neutrons will produce some fissions in Uz" and in U238 for energies above the latter's threshold of about But the Um thermal cross section Mev. 580 barns compared with fast fission cross sections of 0.5 to barn in U238. The effective value of thermal fission cross section for uranium of natural isotopic con stitution 4.18 0.06 barns. Since fission fragments ionize most heavily early in their range and then become less ionizing at lower velocities as they pick up orbital electrons, desirable to plate the uranium upon the cathode so that most electrons will be collected from almost completely across the chamber and thus contribute toward large size and uniformity of pulses. Multiple-plate fission chambers and spiral fission chambers, together with a general discussion of this technique, are to be found in Chap. of Ref. 36, including methods of depositing the uranium. 2.42 Neutron detection in the resonance region will typically involve the selection material displaying a capture resonance at the desired energy, lifetime of the induced activity of convenient length, and a physical form (e.g., metallic foil) suitable to handling and counting. Table summarizes data on substances frequently used. The Cd listed to illustrate that with its large thermal absorption and the lowable effectively to shield the other materials energy position of its resonance, against sensitivity to thermal neutrons when they are surrounded by a Cd envelope of typically 0.020- to 0.030-in. thickness. Their response will then be essentially at their resonance energies. The Cd cross section decreases rapidly with increasing down to 20 barns at energy after its resonance passed and ev. In view of this conventional5' to evaluate the resonance absorption integral, defined as is,

is

9

it

is

±

is

1

1

it

is

it

a

1

is

is

is

it

is

5

of

a

it is


indium

EXPERIMENTAL

5-42

Table Lifetime Element

of induced

activity

Mn

Rh Ag"»

In

Au

hr sec min min days

[SEC.

5

Resonance Detectors for Neutrons

Resonance energy, ev

300 1.25 5. 1 1.44

Approx. resonance

width, ev

Peak cross section, barns

50 45.000 7.600 28.000 15.000

200 0. 1

0.2

4.9 o. ia

Resonance integral, barns

Cross sec tion at r = 2.200 m/sec. barns

14.3 589 98.3 2.580 1.300

12.6 12 44 145 95

activation

250

7.000

7

is

it.

which represents the effective cross section for activation by a neutron atmosphere with a "slowing-down" spectrum for energies above the "Cd cutoff," which is arbitrarily called 0.5 ev. Boron and thermal neutron counters can detect neutrons in this energy region with good efficiency if a moderating jacket of hydrogenous material (water or paraffin) surrounds the counter to degrade the energies of some of the neutrons by n-p collisions. 2.43 Detection Methods for Fast Neutrons. The mass equality of neutron and proton and the fairly large value of the n-p collision cross section result in this col lision process providing the predominant mechanism whereby energy is extracted from a flux of fast neutrons moving in a hydrogenous medium. A plot of the n-p collision cross section vs. neutron energy is given in Fig. 38. In the fast neutron region the cenler-of-mass angular distribution of the neutrons and protons recoiling from mutual collisions is spherically symmetric, resulting in an energy distribution of recoil protons in the laboratory reference system with equal probability of any energy between zero and En, the energy of the incident neutron. The probability per unit solid angle for the proton to recoil in the laboratory at angle 9 from the incident neutron direction is just (1/x) cos 6, and the energy of recoil is E($) = E„ cos' 6. Detection and analysis of the recoil protons in pulse chambers, proportional counters, organic scintillators, photographic emulsions, and cloud chambers constitute the outstanding principles of fast neutron measurements. Other principles leading to unique detection of fast neutrons are fission of elements with definite fission threshold energies and neutron reactions, such as (n,p) and (rt,2n), which manifest thresholds in the fast region. A survey of fast-neutron-measurement techniques is provided in Ref. 60. Proton recoil counting in a hydrogenous gas such as CH4 or in Hi itself, by either pulse chamber or proportional counter techniques, provides a method for both detec If the chamber is large com tion and analysis, within limits, of neutron radiation. pared with recoil proton ranges, the effects of walls in absorbing some of the path lengths will be small, and monoenergelic neutrons will produce a proton recoil energy The pulse-height spectrum from proportional spectrum as shown ideally in Fig. 39a. counting or from slow ion-pulse counting (not fast electron collection) in a chamber will then be similar to the recoil spectrum except for a low-energy cutoff due to the If, now, the neutrons electronic bias applied against detection of background events. are not monoenergelic, the recoil energy spectrum will be a superposition of the differen tial rectangular distributions arising from the populations in the various differential neutron energy intervals weighted by the n-p collision cross section at each energy and will appear typically like Fig. 396. Except for bias effects, the pulse-height clear that spectrum will also look like Fig. 396. From the foregoing description a representation of the energy spectrum of the incident neutron flux can be obtained by taking the derivative of such a spectrum and then dividing the value of the deriva tive at each energy by the n-p collision cross section. Wall effects and the con tributions of carbon recoil nuclei in the use of CII( will modify somewhat this simple of Ref. 30. picture. Detailed discussion presented in Chap. The mathematical analysis of the preceding paragraph allows calculation of counting is


Cd

2 .6 44 2. 4 54 2. 7

6.

TECHNIQUES

Sec. 5-1]


radiation

NEUTRON ENERGY (KEV)-CURVE


10

6

Fig.

38.

Neutron-proton

.8 1.0

I

10

2

4

6

N(E) =NUMBER OF RECOIL PROTONS PER UNIT INTERVAL PROTON ENERGY ENERGY OF ,/MONOENERGETIC ' NEUTRONS ENERGY OF RECOIL PROTON (o) UNIFORM SPECTRUM OF RECOIL PROTONS FROM COLLISIONS WITH MONOENERGETIC NEUTRONS N(E)

ENERGY OF RECOIL PROTON (b) SPECTRUM

OF RECOIL PROTONS ARISING FROM A 0F NEUTRON ENERGIES. ILLUSTRATING SUPERPOSITION OF DIFFERENTIAL UNIFORM SPECTRUM ELEMENTS

SIJSiyBSE!

39. Illustration

100

8 10

NEUTRON ENERGY (MEV)-CURVE TX collision cross section vs. neutron energy.

Fio.

5-43

of proton recoil spectra.

6

(1

Barn

=

8 100

10-" cm!.)

EXPERIMENTAL

5-44

TECHNIQUES

[Sec.

5

10

(MEV) Fio. 40. Efficiency in units of 10-4 counts per cubic centimeter per unit flux density of neutrons at energy En for a hydrogen gas counter. The values of Eb are the " bias energies." i.e., the energies of protons barely detectable at the electronic biases employed.

and proceeds as follows: The ordinate of the curve in Fig. 396 at any value of recoil energy E' from the argument above, is,

efficiencies

N(E')

=

dE

B»>»*

VN

per mev and per sec

a(E)Ê)^

V

chamber volume, cm' hydrogen nuclei /cm n-p collision cross section, cms, for neutron energy "(E) flux density per unit energy interval of incident neutrons, neutrons/ 4(E) (cmJ)(sec)(Mev) number of proton recoils per sec per Mev interval at E' N(E') upper energy limit of incident neutron flux Enmx The total chamber counting rate will then be where

VNH

dE'

Jr

=

JEi

X(E')dE'

a(E)4>(E)

dE

per

set-

itself and If only

is

is

This analysis

is

is

the smallest recoil energy detectable as determined by electronic bias. not seriously attenuated by the chamber assumed that the neutron flux uniform throughout the volume V. single energy of neutrons En present, the counting rate

Eh

a is

where

=

JEi

C

E

>

N„

C(En) = VNH
0-1)

per second

and the efficiency in terms of counts per unit volume per unit flux density at energy En =

C(En)

V(E„)

N„a(E,l)

-

f)

£(£„)

(•

is


E„ =NEUTRON ENERGY


Sec. 5-1]

5-45

This is plotted in Pig. 40 as a function of En for a pressure of 1 atm of H2 and bias of Eb = 0.1, 0.2, and 0.5 Mev. Wall effects, of course, are neglected. Counting of proton recoils from a thin sheet of hydrogenous material mounted in a counter filled with argon ( + C02) is frequently more desirable for analytic counting energies

in a

unidirectional

flux, particularly

if absolute flux determination is required, since

sheet can be so situated that the neutrons enter it normally and the recoiling protons are contained within the sensitive volume of the chamber or proportional counter. This localization of recoil protons permits the use of the grid ionization chamber with fast electron collection if the sheet is placed upon the cathode surface or in such a position that all the recoil proton paths are in the collection volume between grid and cathode the

(see Fig. 14) tt;„ a, T„ In rig. 41 .

w let

'N

l

EN^RgasÊST ..•

j

,— U

,

p_

J.

S ^SjJ 0F 2

— \£s

r 2.

"*—

R(E„)-Rb

1

i T OF INCIDENT NEUTRONS

DIRECTS,

ii

for discussion of maxiFlo. 41. Illustration „neutrons „r „ . rp ibe , , . ot energy . , , , ., a„ mum angle for detectable recoil protons . incident normally from below upon a from a given depth x_ sheet of hydrogenous material of thick ness I. For recoil protons originating at depth x there will be a maximum angle Bx for which they will expend sufficient energy Eb in the gas to count above the elec tronic bias, where

.j

R(EX)

- Rt

W'

cos B, = .

/—

\E.

R(EX) and Rb being the ranges in the sheet material of protons with energies EM

= En cos' 8X and Eb

respectively. Since all recoils lying within the 8X cone will count, the pulse rate contributed from the incremental thickness dx at x will be, per unit area of sheet, dC(x)

=

N„ dx(En)o(En)

J°'2

sin 9 cos B d6 =

Nn

dx4>(En)o(E„)

-

y)

JITrt(J.V(B.)

/Q'

=

(l

C(J?„,<)

-

(l

The total counting rate per unit area from integrating over the thickness will then be

t

[
dx

a

is

7

a

/.

I

(p

is

5

is

is

pt

is

a

6

6

is

is

is

where Ex It useful to assume obtained from ■\/EI/E„ = x/[R(Ex) — Rb]. functional expression for R(EX) to facilitate further calculation, and this frequently taken R(E) = aE^ + bE, where a and are constants characteristic of the material which may be evaluated from range vs. energy data such as that shown for CH2 in For many purposes may be omitted because of its smallness with respect Fig. 42. to a. The efficiency for detecting flux of neutrons of energy En incident normally upon the sheet, expressed as recoil counts per square centimeter per second per unit flux An example of this counting efficiency density, simply S(£„,<) = C(E„,t) /<£(£„). = density), = 0.010 g/cmJ of CH2 as a function of En shown in Fig. 43 for E, — shown with greater than the Also a curve Mev, and Eb = 0.2 Mev. ft(E,'), in which case the upper limit of integration R(En) — Rb instead of of of Ref. 36 presents calculations pulse-height spectra and efficiencies Chapter for various recoil counter arrangements, and Refs. 61 discuss the theory of the hydrogenous solid recoil radiator. the evaluation of fast neutron radiation field from the frequent concern A


■

6-46

EXPERIMENTAL

TECHNIQUES

[Sec.

5


standpoint of biological hazard. The applications of recoil counting techniques to this problem are discussed in Refs. 62. Recoil counting in scintillation counters provides sensitive detection of fast neutrons Analysis of efficiencies and due to the hydrogen content of the organic phosphors. of recoil energy spectra will be essentially identical with that for the hydrogen gas counter, since this is also a distributed volume detector, but attention should be given to the carbon recoil pulses which can be calculated in a similar manner, and the effects of finite crystal size here are analogous to wall effects in the chamber in that particles which leave the phosphor do not contribute their full quota of light. Because of the much higher densities and consequent higher hydrogen concentrations the organic phosphors are more efficient fast neutron detectors than hydrogenous gases by a factor of about 1,000 on a volume basis.

01

0 2

0.5

10

PROTON

Fig. 42. Range

20

50

10

20

ENERGY IN MEV

vs. energy

for protons in CHi.

The failure of light output to follow proportionally the increase in dE/dx as protons slow down in organic phosphors creates a serious distort ion in pulse-height representaI ion of recoil energies. For this reason and also in order to develop more total light, mixtures of plastics and inorganic phosphors such as ZnS(Ag) have been frequently The relatively large y efficiencies of scintillation counters present a employed. serious problem of separating the counting due to neutrons from that due to y's. Several accounts of the use of scintillation detection of neutrons are given in Refs. 63. Efficiencies of a few per cent are typical under conditions which exclude y background for the counting of neutrons of a few Mev energy. Accounts of survey instruments are given in Refs. 64 and 62. Examples of methods of neutron spec troscopy by scintillation counters are found in Refs. 65. Not always are the protons developed within the phosphor volume, but in manv cases the angular and energy distributions of proton recoils out of a hydrogenous "radiator" or from a reaction target are sampled by a small piece of phosphor such as Nal(Tl) at a specified angular position or by a scintillation counter coincidence "telescope" in which the passage of a recoil proton through two or more scintillation

Sec. 5-1]


5-47

in line gives simultaneous pulses identified as such by coincidence-detecting electronics. These methods are widely employed in scattering and reaction studies in the fast and high-energy neutron regions. Fast-neutron detection with photographic emulsion provides the means of identifying both the energy and the direction of the recoil proton, providing the track does not leave the emulsion. So if the incident neutron direction is known, its energy can be calculated. Also, since the n-p collision cross section is known as a function of energy, additional knowledge of the hydrogen content of the emulsion enables one to make


counters

~4

6

8

10

NEUTRON ENERGY (MEV) Fig. 43. Proton recoil detection efficiency vs. neutron energy for normal incidence upon sheet of CHs and recoil bias energy of 0.2 Mev. Curves for thickness of 0.010 gram per square centimeter Burface density, and for thickness greater than maximum proton recoil range are shown. absolute evaluations of intensity and spectrum for unidirectional flux. Energies of recoil protons can be determined from the range vs. energy data of Refs. 29, and for energies above 8 Mev by £(Mev) = 0.251 (R microns)0-58 in dry Ilford emulsions.

A very complete presentation

of neutron-measurement

techniques for nuclear

Techniques emulsions, together with an extensive bibliography, is found in Ref. 66. of this kind have been the most successful in such typical measurements as the neutron energy distribution from a Po-Be source. Cloud chambers applied to fast neutron measurements carry also the advantage of both range and direction data for the proton recoils in a hydrogenous atmosphere. When the neutron energies extend so high that proton ranges are not sufficiently well contained

within

the sensitive volume, energies

are determined

from momenta as

5-48

EXPERIMENTAL

TECHNIQUES

[Sec. 5

observed by track curvatures in a known uniform magnetic field applied to the This is a possibility which only the cloud chamber allows with facility. chamber. For a neutron Hux from a given direction, absolute measurements of intensity and energy spectrum are possible if the hydrogen density in the chamber is known at the instant in the expansion cycle when the observed tracks are produced. As in photo graphic emulsion, the protons from n-p collisions are usually distinguishable from those emitted from other nuclei (C, O, etc.) under neutron bombardment because of the short, dense nuclear recoil tracks present in the latter events. Fast neutron detection by fission counting involves the same techniques as that for fission detection of slow neutrons. The fissionable material on the cathode surface of the pulse chamber should be thorium or uranium. Since natural U carries with it ..Or

08, 06

/

0.4

ÛRANIUM


02

0.1

/

/

\\J

I

OS

/

06|

/

/

.04

< T> 0 »ll M

j 1

/

/

02

f I 0.8

I

1.5

2

NEUTRON

Flo.

44.

3

4

5

6

7 8 9 10

ENERGY (MEV)

Fission cross sections vs. fast neutron energy for uranium and thorium.

0.71 per cent Um whose fission response to thermal neutrons is very large, it is neces sary to surround the chamber with cadmium or boron to filter out the thermal neu trons. Thresholds and cross sections for Th and U fission are presented in Fig. 44. Varieties of fission chambers and their characteristics are found in Chap. 9 of Ref. 30. Reactions with known energy thresholds and leading to radioactive products offer means of ascertaining the presence of neutrons above the respective thresholds. Only in the cases where the absolute cross sections and energy dependences of the reactions are known can rough absolute data be secured, and this will involve absolute A summary of such reactions is presented in 0 counting of the product 0 activity. Ref. 67, and some selected examples are given in Table 6. The asterisks upon the threshold values for the (n,p) reactions imply that, owing to the suppression of yield by the coulomb barrier, the "effective" thresholds are considerably higher than these About 3 Mev is usually given as the effective values which are energetically allowed. threshold in these reactions. The (n,2n) yields are not suppressed by coulomb effects and thus rise much more abruptly to their maximum values.

Sec. 5-1]

instruments for measuring radiation Table

6

Neutron threshold energy, Mov

Reaction

S"(n.p)P" Al«(n,p)ME" Fe"(n,p)Mn" Ag'"(n,2n)A|t'" Cu«(»,2n)Cu"

1.0* 2. 1* 2. 1*

9.6 12-13 20. 2

C»(n.2n)C"

5-49

Half-life of product activity

14.3 days 10 .2 hub 2.59 hr 24.5 min 10 1 min 20. 5 min

Experimentally determined values of reaction cross sections at various neutron are periodically summarized by the Atomic Energy Commission and pub lished in AECU-2040 and its supplements,"8" or, more recently, in "Neutron Cross

energies

Sections."581


Energy insensitive neutron counting for neutrons below 5 to 10 Mev is approximately accomplished by the "long counter" described in Ref. 67 and illustrated in Fig. 45. It obviously has directional properties which are desirably employed in detecting a unidirectional flux or the neutron emission coming directly from a point source of

Flo. 45.

The "long

counter";

energy-insensitive,

directional neutron

detector.

(See

Refa. G9.)

Reference 69ft describes sensitivity vs. neutrons such as an accelerator target. neutron energy which is nearly constant over a span of several Mev but which is markedly dependent at very low energy on the specific design of the sensitive face of the counter. This device is extremely useful in making approximate neutron yield comparisons when spectra are not identical. Other forms of jacketed slow neutron counters with adjustable thicknesses of sur rounding paraffin are useful as fast neutron detectors of relatively good efficiency compared with proton recoil counters, but it is difficult to extract absolute flux data or even rough spectrum data from them.

REFERENCES 1.

Segre, E. (ed.): "Experimental New York, 1953.

J.

2. Jesse, W. P., and 3. Bakker, C. J., and 4. 5. 6.

Sadauskis:

Nuclear Physics," vol. I, John Wiley & Sons, Inc., Phya. Rev., 90: 1120 (1953).

E. Segrd: Phya. Rev., 81: 489 (1951). Nicodemus, D. B.: Thesis, Stanford University, 1946. See quotation in Ref. Gray, L. H.: Proc. Cambridge Phil. Soc., 40: 72 (1944). Livingston, M. S., and H. A. Bethe: Reva. Mod. Phya., 9: 245 (1937).

1, p. 238.

6-50 7. 8. 9. 10. 11. 12a. b. c. 13a. b. 14. 15. 16a. b. c. d. 17a.


b. 18. 19a. 6. 20a. b. 21. 22. 23a. b. c.

EXPERIMENTAL

TECHNIQUES

[Sec. 5

Aron, W. A., B. G. Hoffman, and F. C. Williams: Rept. AECU-663, Oak Hidge, National Laboratories, Oak Ridge, Tenn., 1949. Rutherford, E., J. Chadwick, and C. D. Ellis: "Radiations from Radioactive Sub stances," Cambridge University Press, London, 1930. Heitler, W. : "Quantum Theory of Radiation," Oxford University Press, London, 1947. Jaffe, G.: Physik. Z., 30: 849 (1929). Boag, J. W.: Brit. J. Radiol., October, 1950, p. 601. Strong, J. D., el al.\ "Procedures in Experimental Physics," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1945. Millikan, R. A., and H. V. Neher: Phys. Rev., SO: 15 (1936). Compton, A. H., E. O. Wollan, and R. D. Bennett: Rev. Set. Instr., 5: 415 (1934). Jesse, W. P., L. A. Hannum, H. Forstat, and A. L. Hart: Phys. Rev., 71: 478 (1947). Janney, C. D., and Moyer, B. J.: Rev. Set. Instr., 19: 667 (1948). Penick, D. B.: Rev. Set. Instr., 6: 115 (1935). Nielsen, C. E.: Rev. Set. Instr., 18: 18 (1947). Palevsky, H., R. K. Swank, and R. Grenchik: Rev. Sci. Instr., 18 : 298 (1947). Glass, F. M.: Methods for Reducing Insulator Noise and Leakage, Rept. AECU-27, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1948. Elmore, W. C, and M. L. Sands: "Electronics: Experimental Techniques," McGrawHill Book Company, Inc., New York, 1949. Elmore, W. C: Nucleonics, 2: 16 (1948). Jordan, W. H., and P. R. Bell: Rev. Set. Instr., 18: 703 (1947). Schultz, M. A.: Linear Amplifiers, Rept. AECD-275A, Oak Ridge National Labora tories, Oak Ridge, Tenn., 1949. Korff, S. A.: "Electron and Nuclear Counters," D. Van Nostrand Company, Inc., Princeton, N.J., 1955. Corson, D. R., and R. R. Wilson: Rev. Set". Instr., 19: 207 (1948). Hegener, V. II.: Rev. Set. Inslr., 18: 267 (1947). Liebson, S. H., and H. Friedman: Rev. Set. Instr., 19: 303 (1948). Liebson, S. H.: Rev. Set. Instr., 30: 483 (1949). Wilson, J. G.: "The Principles of Cloud Chamber Technique," Cambridge University Press, London, 1951. Das Gupta, N. N., and S. K. Ghosh: Revs. Mod. Phys., 18: 225 (1946). Beck, C: Rev. Sci. Instr., 12 : 602 (1941). Fretter, W. B.: "Introduction to Experimental Physics," Prentice-Hall, Inc., Engle wood Cliffs, N.J., 1954. Nielsen, C. E., T. S. Needels, and O. H. Weddle: Rev. Sci. Instr., 22: 673 (1951). Snowden, M.: "Progress in Nuclear Physics," Vol. 3 (O. R. Frisch, ed.), Academic Press, Inc., Now York, 1953. Cowan, E. W.: Rev. Sci. Instr., 21 : 991 (1950). R. P. Shutt: Rev. Sci. Instr.. 22 : 730 (1951).

Fowler, W. B., R. P. Shutt, A. M. Thorndike, and W. L. Whittemore: Rev. Sci. Instr.. 25: 996 (1954). 25a. Glaser, D. A.: Phys. Rev., 91: 762 (1953). 6. Glaser, D. A.: Phys. Rev., 97: 474 (1955). 26a. Hildebrand, R. H., and D. E. Nagle: Phys. Rev., 92: 517(L) (1953). b. Dittler, H. C, and T. F. Gerecke: Liquid Hydrogen Bubble Chamber, Unit. Calif. 24.

Rod. Lab. Rept. 2985, 1955. 27. Dilworth, C. C, G. P. S. Occhiallini, and R. M. Payne: Nature, 162: 102, 1948. 28a. Yagoda, H.: "Radioactive Measurements with Nuclear Emulsions," John Wiley
J.

Sec. 5-1] c. Menon,

instruments for measuring radiation M. G. K., K. Gottstein, J. H. Mulvey, C. O'Ceallaigh,

Mag., 43: 708, 932, 1050, 1089, 1232 (1951). 31. Powell, C. F.. and G. P. S. Occhiallini: "Nuclear 32.

Book Company, Inc., New York,

1953.

R. K.: "Annual Review of Nuclear Science" (J. G. Beckerlcy, ed.), vol. 4, Annual Reviews, Inc., Stanford, Calif., 1954. 34. Taylor, C. J., W. K. Jentschke, M. E. Remley, F. S. Eby, and P. G. Kruger: Phys. Rer., 84: 1034 (1951). 35. Borkowski, C. J., and R. L. Clark: Rev. Sci. Instr., 24: 1046 (1953). 36. Rossi, B. B., and H. H. Staub: "Ionization Chambers and Counters," McGraw-Hill Book Company, Inc., New York, 1949. Research, 26A: 243 (1948). 37. Cranshaw, T. E., and J. A. Harvey: Can. Research, 27A: 191 (1949). 38. Buneman, O., T. E. Cranshaw, and J. A. Harvey: Can. 39a. Chang, W. Y.: Phye. Rev., 69: 60 (1946). b. Reynolds, F. L.: Rev. Sci. Instr., 23: 749 (1951). 40. Holloway, M. G., and M. S. Livingston: Phys. Rev., 54: 18 (1938). 41. Lewis, W. B. : "Electrical Counting," The Macmillan Company, New York, 1943. — 42a. Gleason, G. J., J. D. Taylor, and D. L. Tabern: Nucleonics, 8: 12 (1951). b. Baker, R. G., and L. Katz: Nucleonics, 11: 14 (1953). c. Zumwalt. L. R.: AEC Rept. AECU-567, 1950. d. Houtermans, F. G., et al.: Z. Physik, 134: 1 (1952). L., and D. H. Vincent: Z. Physik, 134: 9 (1952). e. Meyer-Schutzmeister, 43a. Kurie, F. N. D., R. Richardson, and H. C. Paxton: Phys. Rev., 49: 368 (1936). b. Orear, J., A. H. Rosenfeld, and R. A. Schluter: "Nuclear Physics" (Fermi), Uni versity of Chicago Press, Chicago, 1950. c. Bethe, H. A., and R. F. Bachcr: Revs. Mod. Phys., 8: 195 (1936). 44a. Katz, L., and A. S. Penfold: Revs. Mod. Phys., 24: 28 (1952). b. Glendennin, L. E. : Nucleonics, 2: 12 (January, 1948). e. Von Bleuler, E., and W. Zunti: Helv. Phys. Acta, 19: 375 (1946). 45. Persico, E., and C. Geoffrion: Rev. Sci. Instr., 21: 945 (1950). 46. Cook, C. S.: Nucleonics, 11: 28 (December, 1953). 47a. Curran, S. C: Atomics, 1: 221 (1950). b. Cockroft, A. L., and S. C. Curran: Rer. Sci. Instr., 22: 37 (1951). c. Insch, G. M., and S. C. Curran: Phil. Mag., 42: 892 (1951). d. Fulbright, H. W., and J. C. D. Milton: Phys. Rev., 76: 1271 (1949). f. Curran, S. C, A. L. Cockroft, and J. Angus: Phil. Mag., 40: 36, 929 (1949). /. Curran, S. C: Physica, 18: 1161 (December, 1952). 48. Jordan, W. H., and P. R. Bell: Nucleonics, 6: 30 (October, 1949). 49a. Bannerman, H. C, G. M. Lewis, and S. C. Curran: Phil. Mag., 42: 1097 (1951) (erratum in 42: 1450). 6. Hanna, G. C, D. H. W. Kirkwood, and B. Pontecorvo: Phys. Ree., 75: 985 (1949). c. Bernstein, W., H. G. Brewer, Jr., and W. Rubinson: Nucleonics, 6: 39 (1950). 50. Hofstadter, R., and J. A. Mclntyre: Nucleonics, 7, September, 1950. 51a. Terrell, J., and G. C. Phillips: Phys. Rev., 84 : 891 (1951). b. Terrell, J.: Phys. Rev., 80: 1076 (1950). c. Walker, R. L., and B. D. McDaniel: Phys. Rev., 74: 315 (1948). Research, 31: 537 (1953). d. Kinsey, B. B., and G. A. Bartholomew: Can. 52a. Alburger, D. E.: Rer. Set". Instr., 23: 671 (1952). b. Same, S. J., and L. M. Baggett: Phys. Rev., 84: 891 (1951). c. Siegbahn. K., and S. Johansson: Rev. Sci. Instr., 21 : 442 (1950). 53a. Hedgran, A.: Arkiv Fysik, 5: 1 (1952). 6. Hornyak, W. F., T. Lauritsen, and V. K. Rasmussen: Phys. Rev., 76: 731 (1949). c. Bell, R. E., and L. G. Elliott: Phys. Rev., 79: 282 (1950). d. Brown, W. L.: Phys. Rev., 83: 271 (1951). 54. Gibson. W. M., T. Grotdai, J. J. Orlin. and B. Trumpy: Phil. Mag., 43: 457 (1952). 55. Fowler, I. L., and P. It. Tunnicliffe: Rer. Sci. Instr., 21: 734 (1950). 56. Moyer, B. J.: Nucleonics, 10: 14 (April, 1952). 57. Schenck, J.: Nucleonics, 10: 54 (August, 1952). 58. Muehlhause, C. O., and G. E. Thomas: Phys. Rev., 86: 926 (1952), and Nucleonics, 11: 44 (January, 1953). 59. Hughes. D. J.: "Pile Neutron Research," Addison- Wesley Publishing Company, Cambridge, Mass., 1953. 33. Swank,

J.


and O. Rochat: Phil.

Physics in Photographs," Clarendon

Oxford, 1947. Birks, J. B.: "Scintillation Counters," McGraw-Hill Press,

5-51

J.

J.

EXPERIMENTAL

5-52 60.

Barschall,

24:

TECHNIQUES

H. H., L. Rosen, R. F. Taschek, and

J.

H. Williams:

[Sec. Revs.

5

Mod. Phys..

1 (1952).

61a. Axner, Y.: Arkiv Fysik, 3: 491 (1951). 6. Barschall, H. H., and H. A. Bethe: Rev. Sci. Instr., 18: 147 (1947). Nuclear Energy, 1: 299 (1955). 62a. Skjoldebrand, R.: b. Moyer, B. J.: Nucleonics, 10: 14 (May, 1952). c. Hurst, G. S., and R. H. Ritchie: Radiology, 60 : 864 (1953). 63a. Segel, R. E., C. D. Swartz, and G. E. Owen: Rev. Sci. Instr., 25: 140 (1954). 6. Seagondollar, L. W., K. A. Esch, and L. M. Cartwright: Rev. Sci. Instr., 26: 689 (1954). c. Emmerich, W. S.: Rev. Sci. Instr., 25 : 69 (1954). d. Rae, E. R., and E. M. Bowey: Proc. Phys. Soc. (London), 66A: 1073 (1953). 64a. Thompson, B. W.: Nucleonics, 12: 43 (May, 1954). b. Handloser, J. S., and W. A. Higinbotham: Rev. Sci. Instr., 25: 98 (1954). 65a. Curran, S. C: "Luminescence and the Scintillation Counter," Butterworth's Scientific

J.

Publications, London, 1953. Draper, J. E.: Rev. Sci. Instr., 25 : 558 (1954). 66. Rosen, L.: Nucleonics, 11: 32 (July, 1953) and 38 (August, 1953). 67. Cohen, B. L.: Nucleonics, 8: 29, 45 (February, 1951). 1, November, 68a. "Neutron Cross Sections," AECU-2040, May 15, 1952, Supplements: 1952; 2, June, 1953; 3, April, 1954, Office of Technical Services, Department of b.

Commerce.

D, J., and J. A. Harvey: "Neutron Cross Sections," McGraw-Hill Book Company, Inc., New York, 1955 and Government Printing Office, Washington, 1955. 69a. Hanson, A. O., and J. L. McKibben: Phys. Rev., 72: 673 (1947). 6. Nobles, R. A., tt at.: Rev. Sci. Instr., 26: 334 (1954). Generated for wjivans (University of Florida) on 2015-09-23 02:49 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

b. Hughes,

6-2

EXPERIMENTAL NEUTRON PHYSICS BY Donald 1

J.

Hughes and

John A. Harvey

NEUTRON INTERACTIONS

is

1.1

Neutron Energy Classifications

1.13

is

is

of

a

The energy classifications in most common use are as follows: 1.11 Thermal Neutrons. These are neutrons in thermal equilibrium with matter Maxwellian distribution commonly near room temperature, which therefore have The most probable velocity of this distribution (for a temperature of velocities. It often suffi 2,200 m/sec corresponding to an energy of 0.0253 ev. 293°K) ciently accurate, for example in calculations of production of radioisotopes, to consider thermal neutrons as monoenergetic with an energy of 0.0253 ev.

Intermediate-energy

Neutrons.

The

intermediate-energy

region extends

is

a

is

is

is

from thermal to about 10 kev and the energy region in which neutron absorption exhibits sharp "resonances" for most elements; for this reason the neutrons in this region are usually called "resonance" neutrons. Intermediate-energy neutrons are most commonly obtained from a moderating material in which fast neutrons are slowed down by elastic collisions, with a resulting neutron flux inversely proportional to energy — the well-known "dE/E spectrum." 1.13 Fast Neutrons. The energy of fast neutrons from 10 kev up to about 15 or 20 Mev, the upper limit being fixed rather vaguely by the fact that energies involved in nuclear reactions usually are not higher than about 15 or 20 Mev per particle, that thi3 energy the upper limit of several useful sources of neutrons, and that relativistic effects begin to become evident above this energy. The nuclear interactions for fast neutrons usually vary rather slowly with energy as compared with the lower energy regions, but greater variety of reactions becomes possible. 1.14 Relativistic Neutrons. The region above 15 to 20 Mev has no well-estab lished title but distinguished from the lower energy neutrons by the fact that particles of this energy show relativistic effects. For example, for ionizing secondary protons produced by recoil, the ionization approaches the minimum value with As relativistic neutrons are not produced increasing energy, then increases slowly. Also reactors, the available intensities are much lower than for slower neutrons. in


it

is,

Neutrons have a high probability for interaction with matter over a wide range of neutron energy. The range of large interaction extends to very low energies because the lack of electric charge enables neutrons of low velocity to penetrate the electric field surrounding nuclei. Neutrons are available from various types of sources in sufficient intensities for study of interactions over an energy range from about 10~* to 10* ev, a 1013-fold range of energy. of course, a versatile neutron The reactor covers most of this range, namely, from 10-4 to 10' ev, as we source, for by itself shall see in succeeding sections. The types of interaction vary markedly with neutron energy as do the neutron sources themselves. It therefore convenient to classify neutrons according to their energy, the classifications corresponding to differences in the primary types of interactions with matter and to some extent to differences in types of neutron sources and methods of experimental study of neutron interactions.

6-53

5-54

EXPERIMENTAL

TECHNIQUES

[Sec.

5

the nuclear interactions are much more complicated and may involve the production of several particles, as in the spallation reactions, for example. 1.2

Cross Sections

The many possible interactions of neutrons with matter are expressed quantitatively The concept of a nuclear cross in terms of the cross sections for these interactions. section can be most easily visualized as the cross-sectional area, or "target area," If we consider nuclei as spheres of presented by a nucleus to an incident neutron. radius R cm and the neutrons as point projectiles, then the target area, or cross section a, of each nucleus will be given by a =

tR*

cm*

(1)

For actual nuclei the cross section may be larger or smaller than *■/?', depending on the neutron energy. For an incident beam containing n neutrons per cubic centimeter, moving with velocity v toward N nuclei, the rate of nuclear interactions of all types can be expressed as

Interactions/sec

This result

nwN

can be rewritten to give a simple definition of total cross section or-

_ Generated for wjivans (University of Florida) on 2015-09-23 02:49 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

=

total interactions/sec

The quantity nv, which is an important item in most neutron experiments, would be called flux density in strict analogy with electrodynamics but the shorter term flux Thus the total cross section is has come to be generally used in nuclear physics. just the number of interactions of all types per second for a single nucleus and unit incident flux. The statement is often made that a certain pile flux represents the number of neutrons crossing a plane 1 cm* in area in 1 sec. If we assume an isotropic flux, however, and integrate over all directions, we readily find that the number of neutrons crossing a plane 1 cm' in area in 1 sec is just nv/2. The flux nv is the number of neutrons crossing a square centimeter normal to the neutron velocity, and if we desire a geometrical representation, we should say that the flux is the number of This definition will neutrons passing through a sphere of 1-cm' cross-sectional area. give the correct flux for any angular distribution. 1.3

Partial Cross Sections

The simple mechanical picture used above involves only the collision cross section, There arc without reference to the course of events ensuing after the collision. many possible nuclear interactions, but the cross-section expression used applies In addition, the cross section for any particular to the sum of all these interactions. process considered, whether neutron scattering, capture, or fission, is in itself also proportional to the number of such processes occurring per cubic centimeter per unit incident flux. The scattering cross section
Sec. 5-2]

experimental neutron physics

5-55

It is transmits or absorbs energy of the lattice vibrations of crystalline materials. possible to measure the sum of all interaction processes other than elastic " scattering; this particular cross section ar — "*i is now usually called the "nonelastic In many cases it is the same as the inelastic scattering because no cross section. other nonelastic processes can take place, but in general the nonelastic cross section includes not only inelastic scattering but various reactions, such as (ra,p), (n,a), (n,2n). often

1.4

Measurement

of Total Cross Sections

Id general, total cross sections are the simplest to measure, for they do not require Whatever the neutron energy of the neutron knowledge of the absolute neutron flux. Neutrons of the desired source used, the principle of the measurement is the same. energy are detected by some type of counter placed to receive them in the appropriate direction. For energies above 1 Mev the detector is often one in which proton recoils are counted, but for lower energies, for which the recoils would be too weak, a The counter in which a particles from boron disintegrations are detected is used.1'* total cross section of a material is obtained from the "transmission" of a sample placed

in the beam delineated by the source and detector.

The counting rate of the


detector is recorded for the open beam, then again with the sample interposed between If these counting rates are denoted by /0 and /, respectively, source and detector. the transmission T is equal to J//o and the total cross section is obtained from the relation T = (3) with n the number of nuclei per cubic centimeter in the sample of thickness x. Although the principle of a transmission measurement is very simple, in practice in order to attain accurate, reproducible results. several precautions are necessary In determination of the counting rates / and /0, for example, the background rate, arising from any neutrons not. passing through the sample position, must be subtracted. The background is usually measured by blocking the beam near the sample position with a thick absorber, but the possible modification of the background by this absorber must always be considered. As the simple form of Eq. (3) is based on the assumption

that any nuclear interaction in the sample prevents the neutron from reaching the detector, corrections must be made for those that succeed in doing so because of multiple scattering or the finite angle subtended by the detector (departure from Fortunately, the corrections can usually be kept down to a few "good geometry"). per cent by careful choice of experimental conditions. As only the relative counting rates are needed in determining the transmission, This no knowledge of the absolute counter efficiency or of the neutron flux is needed. fact, together with the high intensity available when only the transmitted beam is measured (in contrast to the scattered neutrons), is the reason for the relative sim plicity and possibility measurements.

of high accuracy (as high as 0.1 per cent) in total cross-section

1.6

Measurement

of Partial Cross Sections

Determination of the partial cross sections (scattering and reaction) that make up the total cross section requires higher flux and is much more involved than measure In principle, the scattering cross section as a function ment of total cross sections. of angle (differential scattering) is measured simply by placing the counter in the However, the great reduction in desired position to detect the scattered neutrons.

intensity resulting from distribution of the scattered neutrons into 4r solid angle and possibility of confusion with background make accurate intensity measurement difficult. Absolute differential cross sections require knowledge of the actual solid angle subtended by the counter as well as measurement of the intensity of the beam Relative scattering cross sections, i.e., angular distributions, incident on the scatterer. are somewhat easier to perform and have been carried out at specific energies for many materials, f * Superscript numbers refer to References at end of subsection, t Examples and references may be found in Ref. 2.

6-56

EXPERIMENTAL

TECHNIQUES

[Sec. 5

Although the reaction cross sections, (n,p), (n,a), and (n,y), can be measured by detection of the reaction products, the work is difficult and requires knowledge of absolute efficiencies of detectors. The situation is simplified somewhat if relative cross sections alone are measured, the standard being a material such as boron, whose cross section has been determined by other means, for example, extrapolation from lower energy. If a radioactive element is produced, the cross section can be deter mined from its specific activity ; this method is not difficult if high intensity is avail able. It has been used with Van de Graaff neutrons, mainly for heavier elements of large cross section, and with high-intensity fission neutrons for many elements.' Neutron Activation

1.6

The fundamental equation for neutron activation follows from Eq. (2), written for activation alone instead of total cross section,


Activations/sec

= nvaaclN

dN'

= —— at

(4)

where N is the total number of atoms that are in the neutron flux nv and N' is the number of activated atoms. Although the flux in Eq. (4) is the number of neutrons that cross a square centimeter per second, the activation rate dN' /(it depends on the total number of atoms present regardless of the volume over which they are dis tributed. Equation (1) was considered only for a beam of neutrons, but it and Eq. (4) hold for isotropic fluxes, such as those within a pile, equally well. If a range of velocities is present, it is necessary to consider appropriate average values of flux, which are discussed in more detail in the section on thermal neutrons later. After a time t the number of activated atoms will, of course, be given by nvaactNt if the number of activated atoms that have decayed by time t is negligible; that is, If the activated atoms have a reason the activation will increase linearly with time. ably large radioactive decay constant X, the decay of the activated atoms will decrease the rate of formation as time progresses. The number of activated atoms present A" will increase until the rate of radioactive decay, given by N'\, is equal to the rate of formation: N'\ = nw.e,N (5)

The number of activated atoms will remain constant with further irradiation. The disintegration rate of the sample, or intensity of radioactivity 7, will then be equal to nv
~ at

- A"X

= nva^N

(6)

and the growth of the number (N') of activated atoms or of the activity (/ = A"X) is obtained by integration of Eq. (6):

/

= /0(1

-

e-X<)

(7)

The formation of a typical radioisotope, Na", by neutron where 70 = ni»«i.V. irradiation is given in Fig. 1. The "half-life" of Na" is 14.8 hr; hence its mean life is 14.8/0.693, or 21.3 hr, and its disintegration constant X equals 1/21.3 = 0.047 hr-1. It is seen in Fig. 1 that half the final activity is produced in one half-life, three-fourth in two half-lives, etc., and that the amount of activity formed approaches the satura tion amount very slowly after the first few half-lives. The decay of a radioisotope of initial activity 70 is usually quite simple as compared because there are few complicated parent-daughter with natural radioactivity, If the radioisotope gives rise to a stable daughter, as is usually the relationships. case, the activity will be observed to decay as 7 = 7<*rx'

(8)

EXPERIMENTAL

Sec. 5-2]

NEUTRON PHYSICS

5-57

Na" is shown in Fig. 1. Combination of Eqs. (7) and (8) gives a result for the calculation of the saturated activity of a radioisotope when the activity is produced in a neutron flux by irradiation for a time U and then measured at a time U after the end of the irradiation, giving an activity / at that time: The decay curve for

useful

Io.tal

The

e-X(,(l

_

(9)

e-M,)

activation cross section, in terms of the observed activity and flux, is given simply

by 0 net

(10)

nvN

(9) and (10) are those commonly used in measurement of activation

Equations

cross

sections.

The few cases in which radioisotopes give rise to daughters that are also radio can be handled by solution of the differential equations5 for the particular

active


10:

i-e"0047'

I T. 0 5

e- 0.0471

.20

10

30

40

t, HOURS

Fig. 1. Formation of radioactive

Na" by

neutron

irradiation and its radioactive

decay.

situation involved. When the half-lives of parent and daughter are markedly dif ferent, one may be negligible compared with the other, with the result that the situa tion is greatly simplified. For instance, a parent of extremely short half-life can be studied by measuring the long-lived daughter if the latter has a more convenient halflife. A long-lived parent followed by a short-lived daughter will exhibit the parent's half-life and an activity just double that of the parent. 2

FAST NEUTRONS

The cross-section curves in the region of fast neutrons, as given, for example, in smooth variation of total cross section with energy for most elements. Although for light elements it is true that the level spacing is sufficiently great so that individual resonance levels can be seen in the curves, for most of the elements the level spacing is small relative to the instrumental resolving power and the observed curves represent a smooth average of individual resonances. Thus, although the curves appear simple compared with those at lower energy, in fact many resonances are present and a variety of processes occur relative to low-energy cross sections. In addition to elastic scattering and capture, at high energies reactions such as (n,p), Further (n,a), and (n,2n) may take place as well as inelastic neutron scattering. complications ensue because the angular distribution of the emitted particles is not The various reactions that make up the total cross section isotropic as at low energy. Ref. 2, show a

6-58

EXPERIMENTAL

TECHNIQUES

[Sec. 5

are of importance to reactor design; hence it is not sufficient merely to measure the total cross section, but the various partial cross sections as well as angular distributions of reaction products must be determined as well. These latter cross sections are in general difficult to measure, and for many of them very little accurate information ia available at the present time.


2.1

Fast Neutrons from Accelerators

The availability of accelerators such as Van de Graaffs, which produce highintensity beams of charged particles, makes possible the production of monoenergetic sources of fast neutrons of accurately known energy. By bombarding various thin targets with variable energy protons and deuterons from accelerators and observing the monoenergetic neutrons at different angles, the entire neutron energy range from a few kev to 20 Mev can be covered. In the neutron energy range up to 1 Mev, beams of neutrons that are monoenergetic to within a kilovolt can be obtained. At higher energies some energy resolution is sacrificed to gain intensity, and at the highest energy of 20 Mev the resolution is generally about 100 kev. A CockcroftWalton generator with a tritium target is an intense source of 14-Mev neutrons. The neutron detector that is used often depends on the type of measurement to be made. Recent developments of scintillation counters have greatly improved effi ciencies of detectors. Total neutron cross sections have been measured to an accuracy of 0.5 per cent, since the absolute neutron flux need not be measured. For some of the partial cross sect ions such as fission, absolute flux measurements limit the accuracy to 4 per cent. Neutron sources above 23 Mev are not monoenergetic, and detectors for selecting neutrons of the various energies with about 10 per cent energy resolution are used. 2.11 Accelerators, Sources, and Detectors for Fast Neutrons. The accelerators* that are commonly used for cross-section measurements are the Van de Graaff generator, the Cockcroft- Walton or cascade generator, the cyclotron, and the syn The early Van de Graaff generators* were capable of accelerating chrocyclotron. charged particles to a few Mev, but some present-day machines* attain energies of from 6 to 8.5 Mev. A Van de Graaff can supply up to tens of microamperes current of accelerated protons or deuterons whose energy can be varied continuously from a few hundred kev to the maximum energy of the machine. This charged particle current is well focused, is a few millimeters in diameter, and has a spread in energy of only about 2 kev. By employing either a magnetic or an electrostatic analyzer, focused beams with less than 1-kev energy spread and of a few microamperes intensity can be obtained. The analyzer also serves to separate monatomic and diatomic hydrogen ions, which may be present with roughly equal intensities. Cockcroft-Walton accelerators range from 50 to 500 kev. They can produce cur rents of several hundred microamperes of charged particles with only a few kev spread in energy. Cyclotrons range from a few to about 20 Mev, and the external beams are of fixed energy. External beam currents of approximately 100 /iamp spread over a few square centimeters with a few per cent energy spread have been obtained. In contrast to Van de Graaff and Cockcroft-Walton generators, which are continuouscurrent machines, cyclotrons give a pulsed output of only a few milli-/isec duration at a repetition rate of 10 to 20 Mc. Synchrocyclotrons range from 20 to 400 Mev and produce currents somewhat less than 1 jiamp. For a given machine, bombardment by lower than maximum energy accelerated particles is accomplished by inserting the target at a radius smaller than maximum. Neutrons are produced from the charged particle beams of the various accelerators by a variety of (p,n) and (d,n) nuclear reactions. The most useful (p,n) reactions are those on lithium and tritium ; the most useful (rf,n) reactions are those on deuterium The energies of neutrons available from these four reactions as a func and tritium. tion of the energy of the bombarding particle are shown in Fig. 2. This figure is taken from a comprehensive review article on monoenergetic neutron sources by * Section on accelerators

in this handbook'.


Sec. 5-2]

6-59

Taschek, and Williams.* The neutron energy is shown only for neutrons The energy angles of 0 and 180° to the direction of the incident particle. of neutrons emitted at other angles can be readily obtained from nomographs.4 From Fig. 2 it is obvious that, with incident particles up to 7 Mev, neutrons of all energies up to 22 Mev, except the small range from 10 to 12 Mev, can be obtained from these sources. Both the Li7(p,n)Be7 and the T(p,n)He3 reactions have negative Q values and hence have thresholds for neutron production. With a Li target, the threshold energy is 1.882 Mev and the neutron yield rises very rapidly in about 30 kev more and is At the threshold energy the neutrons approximately constant for higher energies. are all forward and have an energy of 29 kev. From the threshold proton energy to Hanson,

emitted at

50 20 10 9

>

5


I >

2 1.0

o

0.5

£

0.2

g

0.10

cc UJ

or

h 0.05 uj

0.02 0.0 I 0 0.005

0.002 0

12

3

4

5

6

7

ENERGY OF ACCELERATED

PARTICLE-Mev Fio. 2. Energies of neutrons emitted at 0 and 180° for the most important monoenergetic (p,«) and (d.n) reactions.4 The upper curve of each pair is the 0° energy; the lower curve the 180° energy. l .;>2Mev two groups arc emitted at each angle in the forward direction. For example, with 1.900-Mev incident protons, 80- and 3-kev neutrons are present at 0°. Above 1.92 Mev only single-energy neutrons are present at a given angle. To obtain monoenergetic neutrons below 120 kev it is necessary to observe the neutrons at back At this angle the spread in neutron energy is only one-half the angles such as 120°. spread in proton energy, while at 0° they are nearly equal. At 120° for 30-kev solid angle)(Ma) are obtained with a Li neutrons, about 5 X 104 neutrons/(sec)(unit The yield rises approximately as the 0.6 target 2 kev thick f (for 2-Mev protons). power of the energy of the neutron up to 150 kev and then decreases. However, since the yield of neutrons is a maximum at 0°, neutrons with energies greater than At 0° the yield is approxi 150 kev are usually obtained in the forward direction. mately constant for neutrons greater than 100 kev and is about 2 X 106 neutrons/ of protons for a 2-kev-thick target. For neutron energies (sec) (unit solid angle) * See Ref. 4. Reference 1 also reviews sources and detectors of fast neutrons. t The target thickness means the loss in proton energy in passing through the

Li

target.


6-60

EXPERIMENTAL

TECHNIQUES

[Sec.

5

greater than 600 kev the Li'(p,n) reaction is not truly monoenergetic, since the Be' nucleus is left in the 435-kev excited state about 10 per cent of the time at these higher Often the detector may be biased so as not to count this lower energy energies. The Li targets are group, and the Li source may be used for neutrons up to 1 Mev. generally thin metallic films evaporated onto tantalum backing foils, but they are sometimes oxidized to LijO to stand higher temperatures. The targets are stable for about 100 hr. The T(p,n)He3 threshold is 1.019 Mev. At this energy the neutrons are all for ward and have an energy of about 60 kev. Two energy groups of neutrons are present in the forward hemisphere up to a neutron energy of about 300 kev at 0°. Monoenergetic neutrons with energies >300 kev can be obtained at 0°. Since there arc no known excited states of He1 up to many Mev, these neutrons are truly mono energetic. The yields from a tritium target are generally better than from a lithium target. Two types of tritium targets have been developed, tritium gas adsorbed in a thin film of zirconium metal welded to a 10-mil tungsten backing and a tritium gas target. In the adsorbed target about one atom of tritium is present per atom of zirconium. The gas targets are a few centimeters long and require thin windows (~0.0001 in. Al or 0.00005 in. Ni) to hold the tritium gas up to atmospheric pressure. For a given target thickness, the gas target, if it can be used, has the advantage over the adsorbed target because of the larger number of tritium atoms present and hence a higher yield of neutrons. The D(tf,n)He3 and T(
Sec. 5-2]

6-61

for neutrons with energy < 1 Mev and hydrogen recoil gas counters, liquid hydrogenous scintillation counters for higher energies. However, in order to measure the energy distribution of inelastically scattered neutrons a neutron spectrometer must be used such as photographic plates, cloud chamber, or the timeFor partial cross sections such as fission, a fission detector must of-flight technique. be used. Where the description of a particular detector is pertinent to a specific measurement, it will be described later in outlining the experiment. 2.12 Measurement of Total Neutron Cross Sections. Of all the neutron cross sections the total cross section can be determined most easily and most accurately. A comprehensive review article on the methods of measuring fast neutron cross sections has been written by H. H. Barschall.5 The total cross sections of practically all elements have been measured in the Mev energy However, only for region.1 the lightest nuclei has the energy resolution been sufficiently good to resolve all the resonances For intermediate-atomic-weight nuclei, although some up to a few Mev. structure is observed in the total cross section below 1 Mev, many resonances are not resolved. For heavy nuclei (A > 100) which have level spacings ~50 ev, the total cross section in this neutron energy range is the average over many resonances and no "fine structure" is observed. The measurement of a total cross section consists of a simple transmission experi ment in good geometry in which the sample is inserted between the neutron source and the detector. Good geometry means simply that very few of the neutrons scattered by the sample will reach the detector. The sample thicknesses are usually such that they have transmissions from 0.3 to 0.7. If the transmission varies strongly with energy, samples of different thicknesses have to be used. In most measurements no shielding is used around the source, sample, or detector. The walls and floor of the room and other neutron-scattering material are kept as far away as possible to reduce background neutrons. The diameter of the sample must be sufficiently Only with low-energy neutrons below large to shadow the detector completely. 200 kev has a directional counter* been used. This counter consists of a massive shield around the detector and a collimator from the sample to the detector. The detector consists of BF3 counters embedded in a paraffin moderator in order to increase the efficiency, and the massive shielding is necessary to reduce the background. WTienever available, samples of the element in the solid form are used. If the element is not available in elemental form, chemical compounds are used. However, "chemically pure" compounds frequently contain an unknown amount of water, which decreases the accuracy of the measurements. Measurements on gases may he made using the gas at high pressures or on liquefied gases at low temperatures. The samples are generally about 1 in. in diameter and require hundreds of grams of material. Some work has been done with separated isotopes, particularly in the low-energy region, where the experimental arrangements are such that measurements can be made on enriched isotopes which are available in 10-g quantities. In general two corrections have to be made to the measured transmission. The first is made because some neutrons scat tered by the sample will reach the detector. This correction is kept as small as possible by using good geometry, since there is some error in making the correction. For low energies where the angular distribution is isotropic, the correction is simply calculated from the solid angle subtended by the detector at the source and is generally of the order of 0.1 per cent. However, at higher energies such as with 14-Mcv neutrons, the angular distribution of the scattered neutrons is strongly forward and inscattering increases by an order of magnitude and may be several per cent. Measurements of the angular distributions have been made for several elements at a few energies, and t hese, together with optical diffraction The second correction which must be theory, are used to make the correction. applied to the observed transmission is for background neutrons being scattered into the detector from the floor, etc. This background is determined by inserting a 50-cmlong "shadow cone" such as copper, which has negligible transmission of neutrons from the source, in between the source and the detector. This background is generally less than a few per cent For the low-energy measurements which are made in the backward direction, the background would be very serious if a highly directional counters

and solid



5-62

EXPERIMENTAL

[Sec. 5

TECHNIQUES


counter* were not used which sees only neutrons from the source and its nearby Since there is always some uncertainty in the determination of both surroundings. the inscatteriug and the background, thick samples giving transmissions of less than 30 per cent are seldom used. Since total cross sections count a large fraction of the neutrons in a given solid angle from a source in contrast to partial sections such as the angular distribution, in which only a fraction of the neutrons are counted, the total cross sections have been made with much better accuracy and much better energy resolution than any other Since reactions in even lightest nuclei, except H, D, T, He3, and fast cross section. He', have some resonances which are only a few kev wide, it is necessary to have monoenergetic neutrons with a very narrow energy spread. The energy spread of a neutron beam depends on three factors: (1) the spread in energy of the incident charged particles upon the target, (2) the energy loss of the charged particles in the target due to its thickness, and (3) the angle subtended by the neutron detector at the source, since the neutron energy varies with angle. In principle, all three factors

cc < m

O.OI

2

4

6

80.1

2

4

6

81.0

2

4

6 8 10

4

2

6 8 100

E„-MEV

Fio.

3. The total neutron cross section of oxygen from

Itef.

2.

may be made as small as desirable with a resulting loss in intensity. Measurements have been made from a few kev up to 1 Mev with energy resolutions of <2 kev. From 1 to 8 Mev measurements are generally made with a resolution of from 30 to 100 kev, and better resolutions are used only in specific ranges for particular reso nances. From 12 to 20 Mev resolutions of about 100 kev are generally used. Above 20 Mev measurements are made with about 10 per cent energy resolution. Since synchrocyclotrons produce neutrons with a broad energy spread, it is necessary to use a special detector such as a proton recoil counter telescope or the time-of-flight technique to obtain a 10 per cent energy resolution. The absolute energies of the neutrons below 20 Mev are determined from proton The proton energies from the Van de Graaffs are calibrated from known energies. threshold reactions to about 1 kev. Neutron energies can be determined to 1 kev at 10 kev, a few kev at 1 Mev, 10 kev at 8 Mev, and about 20 kev at 20 Mev. Above 20 Mev the energy is known to a few per cent. The measured total cross section of oxygen from the Neutron Cross Section Com pilation1 is shown in Fig. 3. In the energy range up to 4 Mev there are several resonances per Mev and they vary in width from a few to several hundred kev. In order to see the true shapes of the narrow resonances very good energy resolution However, it is not necessary to use the best resolution over the whole must be used.

Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

5-63

The resolution used from 4 to 8 Mev was range, for example, below 1 Mev. For all light Above 8 Mev about 10 per cent energy resolution was used. nuclei except the isotopes of H, He, and Li several resonances have been found up to a For nuclei ranging from few Mev and the same general energy resolutions are used. atomic weight 30 to 60 the separation of levels decreases from a few hundred to tens of kev and very good resolution must be used. For heavy nuclei with atomic weights A > 100, except for a few magic-number nuclei, the level spacing is ~50 ev and it is For these elements it is pointless to use good not possible to resolve the resonances. resolution, and the measurements are made with resolutions of about 100 kev.' Since it is not possible to resolve the fine-structure resonances above 3 Mev for medium and heavy nuclei, an energy resolution of 10 per cent is sufficient to reveal the slow variation in the total cross section. Some valuable total cross-section measurements8 have been made from 3 to 12 Mev, using a source of neutrons with a broad energy spread from a fast reactor and a Usss converter plate (see Art. 2.22). The neutron energy is determined from the pulse height in an ionization chamber of energy

100 kev.

The pulse-height collimated recoil protons from a thin polyethylene radiator. distribution is recorded with a pulse-height analyzer where each channel represents a small neutron energy interval. The sample is inserted in the beam, and the trans Since the neutron mission can be determined for many neutron energies at one time. flux falls off rapidly with energy, a factor of 10 for every 3-Mev increase in energy, the upper limit for this method is about 13 Mev. Below 12 Mev it is possible to determine cross sections to 5 per cent statistical accuracy in several hours. Generated for wjivans (University of Florida) on 2015-09-23 02:49 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

2.13

Angular

Distributions.

The

angular

distribution

of elastically

scattered

neutrons is of considerable interest for reactor calculations and to nuclear theory. At low energies neutrons are scattered iso tropically. However, at higher energies the

angular distribution

of the scattered neutrons departs from isotropy.

For inter

For light nuclei where mediate and heavy nuclei there is a strong forward peaking. the scattering is due to individual the angular distribution depends on resonances, the angular momentum of the level of the compound nucleus. For these light nuclei it is necessary to use good energy resolutions ~10 kev; for heavy nuclei resolutions of 100 kev are sufficient. The scattering samples must be quite thin (~70 per cent so that multiple scattering in the sample can be corrected for with transmission) reasonable accuracy. Since the detector subtends less than a per cent solid angle at the source, the scattered intensity is low. The basic problem is to optimize the ratio of scattered neutrons from the sample to those coming from the source. Since neutron sources emit neutrons in all directions, except at threshold, the detector must be shielded from the source. In addition the detector should have a step func tion sensitivity near the primary neutron energy, so that lower energy background neutrons scattered by the shield or the walls of the room and even the air are detected with a much lower efficiency. If neutrons are inelastically scattered from the sample, this sharp threshold detector will be quite insensitive to these inelastically scattered Backgrounds are neutrons and will detect only the elastically scattered neutrons. determined by removing the sample. The detector must also be insensitive to Various detectors 7 rays so that it does not detect the y rays from inelastic scattering. with suitable characteristics have been used, such as a hydrogen or helium recoil gas counter or a hydrogen recoil scintillation They are biased at a high counter. level, about 90 per cent of elastic peak. The most obvious way to measure the angular distribution is simply to scatter neutrons from a small cylindrical sample and measure the counting rate as a function of angle.' The experimental arrangement is shown in Fig. 4 for a particular angle of scattering. The sample is some 50 cm from the source, and the detector ~5 cm in diameter and about 10 cm long. Angular resolutions of about 5° are used, and the distribution can be measured from 15 to 150°. A suitable paraffin or polyethylene slab to shield the detector must be used at each angle. The background is determined by removing the scattering sample. It varies with angle but averages ~40 per cent of the scattered neutrons. The effect of the shielding wedges on the primary neutron beam at the sample position can be shown to be negligible. The differential elastic scattering cross section can be obtained from the angular distribution, the distance

EXPERIMENTAL

6-64

TECHNIQUES

[Sec.

5

from scattcrer to detector, and the counting rate in the detector at the sample position. measures the product of the unknown counter sensitivity and the primary neutron flux at the scattcrer. A second arrangement for measuring angular distributions using the ring geometry 10 is shown in Fig. 5. The scatterer is in the form of a ring on whose axis lie the source and detector. A shadow cone shields the detector from the source. The scattering angle may be varied either by moving the detector or by moving the rings and varying the size of the ring. Ring diameters average around 6 in., and the average distance from source to detector is about 50 cm. However, the angular range is limited in the backward direction, since the detector must be moved closer to the source than the scatterer, which makes the shielding problem very difficult.

This

CYLINDRICAL CHARGED PARTICLE BEAM

NEUTRONS

SCATTERING

SAMPLE


TARGET

Flo.

4. Schematic using a

neutrons

diagram

for measuring

the angular

cylindrical scattering sample.

distribution of elastically scattered

RING SCATTERER

Flo.

5. Schematic diagram for measuring using a ring scatterer.

the angular

distribution of elastically

scattered

neutrons

Differential cross sections at small angles can be obtained from transmission measurements in poor geometry. When the sample is made larger in diameter so that more scattered neutrons reach the detector, the apparent transmission increases. From the change in transmission as a function of sample size the differential cross section of the elastically scattered neutrons can be determined. Since sources are not strictly isotropic, the method is limited to small angles up to about 15°. For light nuclei where inelastic scattering is not important and where the recoiling nuclei have sufficient energy, the angular distribution of the scattered neutrons can be determined from the angular distribution of the recoiling nuclei. The latter can be determined by two methods. The first method is to measure the angular dis tribution of the recoils from a thin foil or gas cell by means of a counter telescope or in a photographic plate. However, the recoil does not have a sufficient range unless the neutron energy is above a few Mev. For light gases a cloud chamber can be filled with the gas under investigation and the angular distribution of the recoil Angular distribution of neutrons scattered from protons have observed directly. also been observed from recoil protons in a photographic emulsion. Measurements have also been made on the angular distribution of protons, deuterons, tritons, and

Sec. 5-2]


5-05


a particles by loading the photographic emulsion with the element to be investigated. The second method is to measure the energy distribution of the recoiling nuclei." The angular distribution of the scattered neutrons is readily computed from the energy distribution of the recoil nuclei. High-energy recoils correspond to neutrons scattered in the backward direction, while low-energy recoils correspond to neutrons scattered through small angles. This second method is limited to gases, and measurements The energy have been made on hydrogen, deuterium, helium, nitrogen, and oxygen. of recoil nuclei is determined from the pulse height from a proportional counter filled with the gas being investigated. Care must be taken that the counter does not Small pulses saturate or the pulse height will not be proportional to the energy. from neutrons scattering through small angles arc lost in pulses produced by electrons or recoils from other heavy nuclei. Also any inelastic scattering will produce lowenergy pulses. Successful measurements have been made on several resonances in different elements, and the angular momentum of the neutron exciting the level has been determined. In addition to elastic scattering, scattered neutron measurements have been made Several wellrecently on the angular distribution of all neutrons from a target. shielded identical detectors are arranged around the target so that several angles The detectors have high efficiencies, about 15 per cent, can be measured at once. region. For light nuclei where the and a flat response in the million-electron-volt inelastic scattering is small, the measured distribution is simply the elastic angular distribution. For nuclei where the inelastic scattering is appreciable, the results can be combined with elastic scattering measurements to give the angular distribution of inelastically scattered neutrons. However, very little experimental data on the inelastic scattering are available at the present time. 2.14 Nonelastic and Inelastic Cross Sections. In addition to elastic scattering, neutrons can also lose energy upon scattering and excite low-lying energy levels in the target nucleus. This interaction is called inelastic scattering. The measurement of the inelastic scattering cross section as a function of incident neutron energy for the neutron to be inelastically scattered with a specific energy at a particular angle is a very difficult problem. The spectrum of the y radiation following the inelastic interaction is also of interest for shielding studies. Several experimental methods have been developed in which it is not possible to distinguish between neutrons which lose most of their energy by an inelastic scattering interaction and neutrons which are absorbed by other processes such as (n,p), (n,«), (n,y), (n,2n), (n,/), etc. The "nonelastic" cross section is defined as the total cross section minus the elastic scattering cross section {a„onti = or —
5-66

EXPERIMENTAL

TECHNIQUES

[Sec.

5


technique the roles of the source and the detector are interchanged, so that the sphere surrounds the detector. From Fig. 6 it is obvious that for each possible neutron path with the sphere around the source there is an equivalent one when the sphere is around the detector. The sphere must be small compared with the distance between source and detector (d > 3r) to minimize energy and intensity changes across the It is necessary that the detector have the same sphere from anisotropic sources. efficiency regardless of the direction of neutrons relative to the source-detector axis. Several types of neutron detectors have been used for these sphere experiments. There are threshold detectors such as fission counters using those nuclides which have thresholds in their fast fission cross sections, hydrogen- or helium-gas-filled propor tional counters, and a specially designed organic scintillation counter. Fission counters have quite sharp thresholds but have inadequate sensitivities for accurate measurements. The threshold for a particular fissionable nucleus is fixed, which is generally undesirable if one wishes to make measurements as a function of incident neutron energy. The fission thresholds of the various nuclides now known go only to a little over 1 Mev, which limits the range of measurement. Other threshold

Fia. 6. Geometry for nonelastic cross-section measurements with the sphere surrounding the source S and the inverted-sphere technique with the sphere surrounding the detector D. reactions, such as the CuH(n,2n)CuM reaction which has a threshold at 11.5 Mev, Gas-filled hydrogen or helium have been used, but their efficiencies are quite low. recoil proportional counters12 have been developed which have thresholds somewhat sharper than fission counters. The threshold can be readily changed by varying the A bias. The counters operate at high pressure and have reasonable efficiencies. third type of counter consists of several small spheres of plastic phosphor or stilbene in oil of the same refractive index to facilitate the transmission of light produced by the proton recoil in the scintillator to the photomultiplier. This type of detector seems to be the most favorable at the neutron energies above 3 Mev. In addition to the multiple scattering correction, corrections must be made for counter anisotropy, for the energy loss in elastic collisions with light nuclei and effects due to geometrical asymmetries from source to sphere. When these corrections arc made and if the nonelastic cross section is quite large, it can be determined to an accuracy of about 5 per cent. The total and nonelastic cross sections* for bismuth For low-energy neutrons the nonelastic scattering is very small. are given in Fig. 7. Below 0.9 Mev the inelastic scattering is zero, the capture cross section at 0.9 Mev For neutrons above 0.9 Mev is only 0. 1 barn and other cross sections are negligible. For energies above 6 Mev the nonelastic the inelastic scattering cross section rises. cross section is about one-half the total cross section which is shown by the dashed line in Fig. 7. At the higher energies the nonelastic cross section also includes (n,2n), (n,3n), etc., reactions.


Sec. 5-2] As mentioned

6-67

earlier, direct measurements of inelastieally scattered neutrons is

A few measurements have been made on the energy distribution of very difficult. scattered neutrons using the proton-recoil photographic plate technique. 13 Large amounts of shielding must be used to reduce background in the plates. Since the exposure times are long and the scanning of the plates very tedious, the method is limited. Recently, successful measurements have been made on the energy dis tribution of inelastieally

scattered neutrons by the time-of-flight

technique.

The

efficiency is quite high, and the spectrum is recorded on a multichannel time analyzer. This technique is being pursued at several laboratories and should yield some valuable results in the near future. Whenever inelastic scattering takes place the bombarded nucleus is left in an excited state and will return to its ground state with the emission of one or more Trays. Observation of the y rays emitted from inelastic scattering yields information on inelastic scattering cross sections. In a few cases the y ray is not emitted instan taneously but decays at a measurable time later, i.e., from 1 sec to several hours. \

1

I

1

1

8


6

TOTAL

-v»' 2 NON-ELASTIC

1.0

2

4 E-Mev

I

1

7

1 i

10

The dotted curve is Flo. 7. The total and nonelastic neutron cross sections of bismuth. just one-half the total cross section and agrees with the experimental nonelastic crosssection points above 6 Mev. The techniques used for these met ast able states will be described later. The tech nique of observing the prompt y rays14 has been feasible only because of the develop ment of sensitive, high-resolution scintillation counters. Two types of geometry have been used similar to those illustrated in Figs. 4 and 5. The neutron detector in these figures is replaced by Nal(Tl) 7-ray spectrometers. The yield of a particular 7 ray from inelastic scattering is measured at a given angle as a function of neutron Correction must be made to allow for the multiple scattering of the neu energy. trons in the sample and also for the self-absorption of the y ray in getting out of the Absolute cross sections are obtained by measuring the efficiency of the sample. detector and geometry using a calibrated source of 7 rays and from a determination of the neutron flux, which is described in the next section. Present absolute cross The cross section for the production of the sections are good to about 15 per cent. Since the first excited level in bismuth 895-kev 7 ray from bismuth is shown in Fig. 8. is at 895 kev, no inelastic scattering is possible until neutrons have energies greater than 895 kev. Since the second excited level in bismuth is at 1.56 Mev, the cross section for this 900-kev 7 ray up to 1.56 Mev is identical with the inelastic scattering cross section. Since the 1.56-Mev level does not decay to the 895-kev level, the total inelastic scattering cross section above 1.56 Mev and up to the next excited level is simply the sum of the two cross sections for the 7's. This agrees with the non-

scattering curve in Fig. 7. If, as is often the case, the second excited level all to the first level, the yield of the 7 ray from the first level is a measure of the inelastic scattering. In a few cases the excited state is metastable, for example, the elastic

decays

5-68

EXPERIMENTAL

[Sec. 5

TECHNIQUES

The sample is irradiated with mono7.5-soc mctastable state at 410 kev in Aum. The background is con energetic neutrons for several half-lives and then counted. siderably lower with the neutron source off than in the prompt y measurements. The cross section is generally lower than the total inelastic scattering, since only In some cases like Au"7™ about half transitions to the isomeric state are observed. the nonelastic cross section for neutrons from 1 to 5 Mev arises from transitions to the 410-kev metastable level. Absolute cross sections are obtained from a measure ment of the flux and the absolute activity produced. 2.16 Reaction Cross Sections. Reaction cross sections are measured in a variety of ways: for fission the fission fragments are detected; for (n,p), (n,a) reactions in light nuclei the protons or as are detected; many (n,p), (n,a), (n,2n), and (n,y) reactions can be measured from the activity produced ; and a few (n,2n) cross sections Although the variation of the can be measured by a modified sphere technique. cross sections with energy can be measured accurately, an absolute measurement of For most of the reaction cross sections it is necessary the cross section is very difficult. OS

04


c 5

o

foe b

04

"08

1.2

"1.6"

^0

2.4

28

E-Mev Fio. 8. The cross section for the production of the 895-kev and 1.56-Mev y rays from Since the 1.56-Mev y ray does not decay into inelastic neutron scattering from bismuth. the 895-kev level, the two cross sections should be added to get the inelastic scattering cross section.

There are several methods" for to make an absolute neutron flux measurement. making an absolute measurement of the flux, but only the two most useful and accurate The first is the associated particle method in which the will be mentioned here. For example, for T(d,n)He4 source, charged particles from the source are counted. the number of neutrons emitted in a small solid angle can be determined from the number of He4 particles emitted in the corresponding solid angle. Measurements of the absolute flux have been made by this method to 5 per cent. The second method is the recoil particle method in which proton recoils from a thin radiator are counted in a chamber. Corrections must be made for the small recoil pulses which The best measure are lost in background noise and for several other small effects. ment made by this method is accurate to 3 per cent. Relative fission cross-section measurements have been made in two ways: first, by measuring fission counts relative to a "long counter" and, second, by measuring A long counter is an arrangement of a BF> the ratio of two fission cross sections. counter located in a lnrge paraffin block such that it has almost a constant efficiency Two types of fission counters have for neutrons from a few hundred kev to 8 Mev. When the ratio of boon used: the spiral counter and the parallel-plate counter. two fission cross sections is measured, the fissionable foils are mounted back to back in a parallel ionization chamber. The fission counter is located about 20 cm from the

Sec. 5-2]


5-69


The counts in the detectors must be corrected for background as explained source. in the measurement of total cross sections. For measurements with the long counter the thickness of the fissionable foil need not be known, since only a relative crossHowever, with the back-to-back chamber, the section measurement can be made. absolute ratio of the two fission cross sections can be measured if we know the thick Foil thicknesses can nesses of the two foils, since they are in the same neutron flux. be obtained by counting a particles from the foil and by careful weighing to an After small effects like nonuniformities of the accuracy of better than 2 per cent. foils, inelastic scattering, and room background have been corrected, the ratio of two The measure fission cross sections can be determined to an accuracy of 3 per cent. ments relative to the long counter can be determined to an accuracy of 2 per cent. The absolute fission cross section of U"5 has been measured to an accuracy of 4 per This measurement is used as the cent at 1.25 Mev by means of the recoil method. standard for all fission measurements from 100 kev to 8 Mev. The energy variation of the fission cross section of U"6 has been measured by means of the long counter. This U"b cross section is then used as the standard for the ratio measurements. Rela tive measurements with the long counter must be normalized to a value from an An absolute fission measurement has been made at 14 Mev, absolute measurement. and the other fission cross sections at this energy are measured relative to this absolute value. Experiments have also been performed on the number of neutrons per fission as a function of neutron energy. A small multiplate fission counter is located in the center of a large liquid scintillator loaded with cadmium 30 in. in diameter and 30 in. long with a 2-in.-diameter hole along the axis. The number of fissions is given by the counts in the fission counter; the fission neutrons in delayed coincidence with the fission counts The fission neutrons are slowed down in the are detected with the large scintillator. liquid scintillator, and the y rays given off when the slow neutron is captured in cad Fission neutrons are detected with an 80 per mium are detected in the scintillator. The mean life of the neutrons in the liquid scintillator is about cent efficiency. 10 ^sec; hence about 95 per cent of the fission neutrons have been captured by cad Thermal neutrons are used to calibrate the mium 30 /jsec after the fission pulse. scintillator. Hence the number of neutrons as a function of incident neutron energy is measured in terms of the number of neutrons per thermal fission. Reactions producing light charged particles from light nuclei may be observed in However, the smaller energy of the reaction products the same manner as for fission. The targets to be investigated may be in makes their observation more difficult. the form of thin foils in an ionization chamber or if possible as a gaseous compound When thin foils on heavy backings are used, to be the gas in the ionization chamber. the foil is set perpendicular to the neutron beam and the detector is generally biased Since the foil must be thin, disintegrations produced to count only the light particle. in counter gas are likely to be troublesome. Since nitrogen and neon have appreciable Also disintegration cross sections, they cannot be used as the gas in the chamber. disintegrations produced in the counter walls or recoil protons from water or organic The difficulties materials absorbed in the walls can give prohibitive background. encountered with thin foils are much reduced if the element to be investigated is a All reactions of the same reaction energy will gas or forms a gaseous compound. For gases like nitrogen fast electron release the same amount of energy in the gas. collection is used. For electronegative gases like oxygen ion collection of both the positive and negative ions is necessary. For both the foils and gas samples the absolute neutron flux must be measured as outlined for fission in order to obtain absolute reaction cross sections. Frequently a radioactive nucleus is formed as a product of a nuclear reaction. Observation of the activity, if it has a half-life of from 1 sec to several days, may be In addition to (n,p) and used for the determination of the reaction cross section. (n,a) reactions, (n,y) and (n,2n) cross sections can also be measured by the activation Activation measurements give cross sections for a particular isotope rather method. than for the clement. In order to obtain absolute cross sections the absolute ft activity of the radioactive nucleus as well as the absolute neutron flux must be meas-

6-70

EXPERIMENTAL

TECHNIQUES

[Sec.

5

Absolute 0 counting" usually involves a comparison with Ra-DEF standard For nuclei involving K capture, /3 counting can still be used if a known fraction of the disintegration involves ^-particle emission. With considerable care absolute activity measurements can be made to an accuracy of 6 per cent. When we include errors arising from the neutron flux and the source thickness, the best activa tion measurements have an accuracy of 10 per cent. Since they do not involve absolute p counting, relative measurements as a function of energy are accurate to a few per cent. In a few cases it is possible to obtain absolute (n,y) cross sections from a measurement of the radioactivity produced with thermal neutrons monitored with a BFa counter and then monitoring the fast neutron flux with the same BFS counter. In some cases relative measurements from 12 to 20 Mev are made based on the angular distribution of the T(d,n)He4 neutrons which are known only to about 10 per cent; in other cases the 0° yield must be known as a function of energy. For nuclei where the (n,2n) reaction does not result in radioactivity, a modification 17 of the sphere method for measuring nonelastic cross sections may be used. In this method the sphere surrounds the 14-Mev source and the detector is an energy-insen sitive detector such as a long counter. This measurement gives the multiplication of the sphere. With a standard sphere measurement of the nonelastic scattering and assuming that the cross section of the (n,3n) reaction is zero, upper and lower limits can be placed on the (n,2n) cross section, the inelastic scattering cross section, and the sum of the cross section such as (n,p), (n,a), (n,y), etc., in which a neutron is lost. Thus if one of these three cross sections is known from other work, the other two may be solved. The spectrum of neutrons which arise from (n,2n) or (n,n') with 14-Mev neutrons has also been measured with photographic plates" using the sphere tech nique. Absolute cross-section measurements of the neutron spectrum are accurate to about 20 per cent. This differential cross section can be integrated over the out going neutron energy and may be greater than the nonelastic cross section because of (n,2re) reaction. ured.


sources.

2.2

Fission Neutron Sources

The experimental techniques just described utilizing accelerators have furnished most of the neutron data in the high-energy range used for reactor design. The fast neutrons in the pile itself, however, which are primarily unmoderated fission neutrons, have also been effective as sources. Although the fission neutrons show a spread in energy of several Mev, they have proved to be useful as tools for fast neutron research because of the high intensities available. Fast neutrons are found in highest, intensity at the center of the pile lattice, but at that point they are accompanied by many resonance neutrons. However, the fission neutrons can be obtained uncontam mated with slower neutrons, but at a great loss in intensity, by use of a uranium "converter" outside the pile. 2.21 Threshold Reactions in Pile. Because of the great range of neutron energies present in the pile flux, irradiation inside the pile lattice is not useful for measurement of those fast neutron cross sections that have an appreciable value in the resonance The presence of neutrons of intermediate energy creates no difficulty, energy range. however, for such threshold reactions as (n,p), (n,a), and (n,2n) reactions, which usually have thresholds well over 1 Mev. These reactions, when produced in the lattice, can be conveniently measured by means of the radioactivity of the product nucleus, a procedure that unfortunately cannot be applied to those reactions in which the final nucleus is stable. Irradiation inside the pile lattice is particularly well suited for production of certain radioisotopes by threshold reactions because their low cross sections require the high flux characteristic of in-pile irradiation. Because of the spread in energy of the neutrons, the interpretation of the observed rate of a threshold reaction produced by fission neutrons, in terms of the cross section, which is a function of the energy, is not simple. For a light element, usually exhibit ing strong resonance structure in its cross section, the interpretation is extremely In the case of a heavy element, however, the dominant feature deter complicated. mining the cross section is the barrier penetrability of the outgoing proton or a particle, which determines the emission probability of the outgoing particle. The fast neutron

Sec.


5-2]

5-71

are so closely spaced for all but the lightest nuclei that they overlap; as a section rises smoothly with energy, from a value of zero at the threshold This assumed shape of the plot of cross section vs. energy, together with energy. the known energy distribution of fission neutrons, can be used to relate the measured reaction rate to the cross section.* In order to obtain the highest possible intensity of unmoderated fission neutrons resonances

result the cross

production of a threshold reaction in the pile, it is important to place the sample possible to the uranium fuel rods in the lattice of a heterogeneous reactor. A material can be irradiated with a high flux of fission neutrons by placing it inside one of the central lumps of uranium or, better still, inside a piece of U1". A uranium U»") fuel rod, constructed in the shape of a cylindrical shell to hold samples, usually called receptacle slug. The flux of fission neutrons produced will be of the same order of magnitude as the thermal flux because practically all the thermal In principle, the fast neutrons will be converted to fission neutrons in the uranium. flux could be calculated from the known constants of uranium, but the calculation rather involved and self-protection. Actually, because of flux depression with another fission flux simple matter to obtain the fission flux by comparing produced in simpler geometrical arrangement, such as the "converter" described in Art. 2.22. As the fission flux in the converter arrangement can be calculated quite easily from the thermal flux because of the simple geometry involved, the pile fission Hux follows directly. The thermal neutrons that are present in the pile can be pre vented from reaching the sample by wrapping in sheet cadmium, of the usual 0.03-in. thickness. 2.22 Use of "Converter" for Production of Fission Neutrons. It sometimes necessary to obtain fission neutrons that are uncontaminated with resonance neutrons; for example, in the measurement of (n,y) cross sections at fission neutron energies the It possible to obtain contamination of resonance neutrons must be extremely low. fission neutrons free from intermediate-energy neutrons by irradiating uranium with well-thermalized neutrons outside the pile. The fast neutron flux produced in this the only one nay will be much less intense than that inside the pile, but the method that removes the resonance neutrons effectively. The thermal neutrons cause fission the uranium with the emission of 2.5 fast neutrons per fission, and the uranium thus acts as neutron "converter," creating a fission flux that can be several times as large as the thermal flux. The unmoderated fission neutrons produced in the converter can be resolved into different energies used, as already described. Without an energy sensitive detector however, they are valuable for measurement of capture cross energy resolution, for

it

is

is

is

if

a

in

sections.

cadmiumcarried out by activating section measurement foil of the material of interest with the fission neutrons from the plate, while an identical bare foil simultaneously irradiated with thermal (plus fission) neutrons. After the foils have been counted on a Geiger-Muller counter, the fast activation cross section obtained directly from the counting ratio of the cadmium-covered to the bare foil, a

is

The capture cross

=

— R

at

t

is

is

covered

Iblre

—

(U)

led

-SeeRef. 18. Most of the pile techniques Lilted in Ref. for example. 2,

•

is

A

is

is

R is

is

the known thermal cross section for the particular activity detected! and In this the calculated ratio of fast to slow flux at the plate (of the order of 2). method no corrections are necessary for self-absorption of 0's in the foil, complicated decay schemes, or efficiency of the counter. shown in Fig. 9. It typical converter, one at Brookhaven Laboratory, arranged in such a way that irradiations with fission neutrons can be accomplished without increasing the general neutron background in the pile room. The U235 plate located in a space formed by removal of one of the large shielding blocks on the top where
t


is

it

a

it

is a is

a

is

(or

as close as

we shall sketch are treated

in detail in this reference,

EXPERIMENTAL

&-72

TECHNIQUES

[Sec.

5

of the pile, and this cavity is made sufficiently large so that no difficulty is experienced with neutrons that are slowed down in the shield and return to the foil as resonance neutrons. The purpose of the design is to obtain a high flux of thermal neutrons REMOVABLE SHIELD (which do not need to be collimated) at the U"* plate with as few resonance neu — \ X N N \ ■• 3' trons as possible and to shield the plate / FOILIN Cd-y / U1'* so that irradiations can be accomplished / without pile shutdown. bjvberoil-'T It is possible to withdraw the plate, through an opening in the removable shield, to a shielded space outside the PILE LUM TANK irradiation hole and there remove foils SHIELD from the plate without radiation hazard while the pile is operating at full power, 28 Mw. At 28 Mw, the thermal flux at the plate is 5 X 10", the fission flux about 10s, and the total emission rate, or Q, of the \ x \ \N Because plate about 10" neutrons/sec. of the leakage of fast neutrons from the COOLING AIRcenter of t he pile through cooling channels, there is an appreciable flux of resonance neutrons at the point where the U"* plate Wgraphite reflectoris located. The activation of a foil caused Fig. 9. Experimental arrangement for pro by the resonance neutrons can be elimi duction of unmoderated fission neutrons nated, however, by surrounding the foils inside shield of the Brookhaven pile. The with protective layers of the element being foil inside cadmium is activated by fission irradiated, which remove the resonance neutrons, and the hare foil is a monitor of the thermal flux. neutrons (self-protection) but have a negligible effect on the fast neutrons. 2.23 Fast Neutrons from HJ(/,n)He4 Reaction in Pile. It is possible to obtain 14-Mev neutrons in the pile by utilizing the reaction of tritons (H') on deuterium (H1), the tritons being obtained from the (n,a) reaction on Li6. As an example, in some work of Novey," lithium nitrate was dissolved in DjO in the ratio of 1 lithium atom to 20 deuterons and 250 cm' of this solution was placed in the maximum flux of the Oak Ridge pile. The arrangement produced a 14-Mev neutron flux of 10* in a thermal flux of 10". This fast/thermal ratio of 10~* is somewhat smaller than a ratio of 10~* obtained by Anderson*0 at Argonne with a similar arrangement. The experimental ratios are smaller than the value 2 X 10~4 that can be calculated* from the reaction cross sections involved, because the actual conditions in the pile do not correspond to the ideal case assumed. The Li depresses the thermal flux (in a manner similar to the depression at a fuel rod), and the tritons collide with oxygen atoms (losing energy but producing no neutrons) as well as with deuterons. It is instructive to compare the experimental results for the triton-deuteron source with the fast flux obtained from a uranium receptacle slug in the pile. Assuming that the latter gives a fission flux equal to the thermal flux, and using the fission spectrum, it can be computed that for the fission source 5 X 10~4 of the neutrons will be above 14 Mev, hence the fast/thermal ratio will have the same numerical value. This flux is definitely higher than that attained experimentally with the tritons, although of the same order as the theoretical upper limit calculated for the triton method. The triton-deuteron neutrons have a very small energy spread compared with the fission neutrons, but moderation in the graphite of the pile will inevitably widen the spread greatly.

\

'//////

\\

/

''/////

\


WW \W

3

INTERMEDIATE -ENERGY NEUTRONS

For intermediate-energy neutrons, the cross section curves shown in Ref. 2 appear extremely complex because of the large number of resonances that are exhibited. * See Ref.

19, p. 117.

■

Sec. 5-2]


5-73

in spite of this complexity, the processes represented by the cross section much simpler than for fast neutrons. The complexity of the curves results from the fact that the resolving power of the instruments is sufficient to isolate the typically The simplicity of the reactions arises from the fact that closely spaced resonances. the excitation energy of the compound nucleus formed by absorption of the incoming neutron is insufficient for the various reactions to occur that are caused by fast neu In addition, as the energy is insufficient to cause appreciable reactions for trons. anything but / =» 0 collisions, the angular distribution in the center of mass system is However,

are

isotropic.

Even though the resonances in general represent only scattering and capture, properties are extremely important to basic nuclear theory and to reactor design. The theoretical implications arise because the neutron resonances represent excitation levels of the compound nucleus and enable measurement of the widths and spacings of these levels. The effect on reactor theory follows from the high absorption that may occur in resonances and the manner in which the absorption changes with thick ness of material and material temperature (via the Doppler broadening of the levels). A typical manifestation of the resonance absorption important to reactors is the Because of the many absorption lines present temperature coefficient of reactivity. in the cross-sectional curves and the close analogy to optics, the field of neutron reso nance study is usually referred to as "neutron spectroscopy." their


3.1

Velocity Selectors

Cross sections for intermediate-energy neutrons are usually measured with some type of high-resolution velocity selector, which selects neutrons in a narrow energy, or velocity, range. Below 10 kev there are at the present time two methods of neutron spectroscopy in general use, one based on Bragg reflection of neutrons by single The instruments used in these crystals and the other on time-of-flight measurements. methods, crystal monochromators and time-of-flight velocity selectors, respectively, have in common a neutron source produced by moderation of fast neutrons. This moderation, usually produced by elastic collisions with light nuclei, results in a neutron " so familiar in pile neutron flux inversely proportional to energy, the "dE/E spectrum physics. Furthermore, although the two techniques are based on entirely different methods of velocity selection, they both have resolutions (energy spreads AE) that In other respects vary with energy in the same way (&E/E proportional to E^). there are important differences; these determine the specific energy regions and cross sections best measured by instruments of the two types. 3.11 The crystal monochromator, or "spectrometer," as it is usually described, requires a well-collimated high-intensity neutron beam; thus a chain-reacting pile is Used with the first chain-reacting piles to be necessary for its successful operation. put into operation, crystal monochromators are now to be found at essentially all research

reactors.

The beam of neutrons for a crystal spectrometer is formed by opening a hole through the shield of the pile and into the "lattice" (uranium fuel plus moderator), usually to a point as near the center of the pile as possible. At this point a high flux of neutrons in the resonance The flux is distributed in energy according energy region is found. to the dE/E spectrum, thus with a magnitude that is the same in each energy decade II to 10 ev, 10 to 100 ev, etc.), this magnitude being 7 X 10" em'/sec in the Brookhaven pile, for example.* The flux incident on the monochromating crystal, placed just outside the pile shield, is, of course, much less, being reduced enormously (a factor of 10' to 10*) by the small solid angle subtended at the crystal by the source area (the base of the hole). The crystal is set at a small, adjustable angle to the incident collimated beam, and the diffracted neutrons are detected by a counter,

counter. diffracted for a particular incident glancing angle

usually an enriched BF> proportional

The neutron wavelength given by the Bragg formula • See Ref. 18, p. 64.

X,

nX = 2d sin 9

8, is

(12)

5-74

EXPERIMENTAL

TECHNIQUES

[Sec.

5

where n is the order of the reflection (the reflectivity being the strongest for n = 1) and d the spacing of the lattice planes causing the particular reflection. As the incident beam is not strictly parallel, there is a range of incident angles AO and a cor responding range of reflected wavelengths given by 2d AX = — cos 6 A9 n

or

— X

= cot 9 Afl

(13)

a typical crystal, LiF, the value of d for the (111) planes is 2.32 A; hence for neutrons of 1-ev energy the glancing angle will be .3.5°. Whereas this value of 8 is a practicable one, although small, 0 becomes 0.35° for 100-ev neutrons, of the same order of magnitude as the angular collimation of the best monochromators in use today. Even for a collimation of 0.1° the conditions just described would imply an energy resolution A£/£(~2AX/X) of about 50 per cent at 100 ev. The angular requirements just mentioned are important factors in reducing the range of utility of present crystal monochromators to energies well below 100 ev, in practice below about 20 ev. Another effect that restricts the instruments to low intensity with increasing energy. This energy is the rapid decrease in reflected decrease is a result not only of the decreasing incident flux (1/E) but of the reflec tivity of the crystal itself, which varies as \/E. The resulting flux at the neutron detector drops off rapidly with energy, with the decrease in actual counting rate even more rapid because of the usual 1/2?M response of the detector itself. The general characteristics of crystal monochromators and their use at the pile are described by Hughes,* and a description of a particular instrument at Brookhaven, which has an energy resolution of 2 per cent at 1 ev and 16 per cent at 50 ev, is given by Borst and


For

Sailor."

3.12 Time -of -flight Velocity Selectors. The second general method in use for low-energy neutron spectroscopy involves the production of a short burst of neutrons, of their time of of the highest possible intensity, together with the measurement arrival at a detector some 20 m distant. The underlying principles of the time-offlight techniques are the same whether the burst is produced by modulation of the output of a charged particle accelerator (cyclotron, betatron, or linear electron accel erator) or by mechanical interruption of a pile neutron beam. Of the sources named, The beta the cyclotron was the first to be applied to time-of-flight measurements. tron and linear electron accelerator, which have been used more recently, have some what better resolution, as do the pile mechanical shutters, or "fast choppers," which are now coming into extensive use. Because of their varying characteristics, these instruments complement each other and, taken together, constitute a potent means for unraveling the complex neutron "spectra" of nuclei. For all the time-of-flight instruments the neutron energy E, in electron volts; velocity v, in meters per second; and flight time per meter /, in microseconds per meter, are related as follows: v =

E

10« —

I

= 52.3 =

X

!j

(14)

723

(E)»

Thus for a typical path of 10 m, the flight times to be measured for 1-kev and 1-ev neutrons are 22.9 and 723 jisec, respectively, easily accomplished electronically. The spread in energy actually existing for a "single" flight time, i.e., for a single time channel of the electronic recording circuits, is the result of a number of contributions * See Ref. 18, pp. 159-165.

Sec. 5-2]


5-75

to the uncertainty in timing. The most important of these are the length of the electronic time channel itself, the neutron path length in the detector, the time delay in the action of the detector, the length of the neutron burst, and the moderation

of the fast neutrons (the last not occurring for the pile choppers). The timing uncertainties just enumerated combine to give a "resolution function," whose full width at half maximum, divided by the path length, is the "resolution" M in microseconds per meter. In the past few years resolutions have been improved about fivefold with the advent of the recent instruments, and at the present time values of about 0.05 /iscc/m are available. Corresponding to the resolution in micro seconds per meter we have the following uncertainties in neutron velocity and energy: time

Av =

E


AE

=

t1

= lO'v* At

^ t

(15)

= 0.028E'* At

Thus a resolution of 0.1 fisec/m gives an energy spread of 0.09 ev at 10 ev (0.9 per cent), rising to 88 ev at 1 kev (9 per cent). It is the energy uncertainty AE that is represented by the "resolution triangles" shown with the curves of Ref. 2. It is seen that the change in energy resolution with energy, AE <* E&, is just the same as for the crystal monochromator. However, an examination of the factors involved in the counting rate shows that for a dE/E flux, a l/v detector, and a con stant time resolution At, the counting rate will be constant with energy for the timeof-flight instruments. As a result, while the crystal has a very good intensity in the 0.1- to 10-ev region, it rapidly becomes inferior to the time-of-flight instruments as the energy is increased and is thus most useful in the lower energy range specified. 3.13 Pulsed Accelerators. For the time-of-flight methods in which the neutron source is pulsed, the accelerator itself (cyclotron, betatron, or linear accelerator) is

specifically designed for the purpose and only the sharp pulsing of the ion source represents a special technique. The pulsing of the conventional cyclotron, as used at Columbia" for several years, accomplished by modulation of the arc source, pro duces bursts of the order of 2 jusec, with sufficient intensity to allow a flight path of 6 m, and a resolution (including all timing uncertainties) of 0.5 ^sec/m. The betatron and linear electron accelerator are easily adapted to time-of-flight work. It is necessary only to install a neutron-producing target (utilizing the y-n reaction in a light or heavy element depending on the y energy), surrounded by a neutron moderator, and high-speed detection and timing equipment. At the present time the 100-Mev betatron at Schenectady and the 15-Mev linear accelerator at not

Harwell are being used as neutron velocity selectors, and in each case a resolution of about 0.1 /jsec/m has been reached. The high-energy frequency-modulated cyclo trons produce extremely short bursts of bombarding particles; as a result they can be for neutron spectroscopy with resolution better than 0.1 jisec/m. 3.14 Fast Choppers. Whereas the charged particle accelerators used as time-offlight sources are of conventional construction, the pile fast choppers are used only for neutron spectroscopy and were developed specifically for that purpose. In order to produce sharp bursts of neutrons, a shutter, necessarily heavy to stop neutrons of all used as sources

energies, must open and close quickly. The large size and high speed of the "fast" chopper (i.e., a chopper for fast neutrons) make it a much more complicated device than the thermal neutron "slow" chopper, for which thin cadmium sheets suffice to stop the slow neutron beam. Because of the limited energy range of the slow chopper '0 to 0.4 ev), very few neutron resonances can be studied with it.

The requirement of short burst length for the fast chopper — about 1 /usee — leads neutron beams of small area, hence low intensity; this latter tendency must be counteracted as much as possible to obtain a finite counting rate at the distant detector. These conflicting requirements result in a rather heavy, rapidly rotating shutter, containing shaped slits for the passage of neutrons, slits whose cross-sectional to

5-76

EXPERIMENTAL

TECHNIQUES

[Sec.


area is of the order of 0.01 by 1 in. As the slits in the moving shutter pass similar slits in a stationary collimator, neutron bursts of duration about 1 to 5 /isec, depending on the particular chopper, are produced. The first fast chopper was used at the Argonne laboratory. It is essentially a steel cylinder of 4-in. diameter and 16-in. length, rotating about its long axis, which is horizontal, and containing machined slits along its length near the periphery. This chopper is normally operated at 10,000 rpm, at which speed it produces 5-^sec bursts. The limitation in the shortness of burst arises from difficulties associated with the horizontal mounting on fixed bearings. The bursting strength of the rotor is not a limiting factor; it has been successfully spun to 40,000 rpm in spin tests (without bearings). Similar choppers, on horizontal axes with fixed bearings, are now in use at Oak Ridge and Harwell which give bursts about half the length of the Argonne machine. With the usual flight paths of about 10 m, the resolutions are of the order of 0.5 jusec/m when the other timing uncertainties are included (mainly channel time and detector delays).

Fig.

10. A photograph of the assembled rotor and its housing. The rotor spins about a vertical axis. Entrance and exit stators are not in place. The pile shield is in the back ground.

The Brookhaven chopper (Fig. 10), which has been in operation for about 5 years, has contributed much data on neutron resonance properties. The rotor is a hori zontal disk, 30 in. in diameter, suspended by a thin flexible shaft in an evacuated The narrow slits for burst formation are machined in plastic pieces, which chamber. are held in the rotor between aluminum forgings. Designed to reach an ultimate speed of 15,000 rpm, at which a burst of 0.6 jisec woidd be attained, it has operated With a scintillation neutron detector and a flight path of at 10,000 rpm for 5 years. 20 m, the measured resolution function (including all timing uncertainties) has a width of 1.0 ,usec; hence the resolution is 0.05 Aisec/m under these conditions. At Argonne laboratory, a fast chopper of the same general design has been constructed to take advantage of the high flux (about 20-fold greater than that at Brookhaven) of the Argonne research pile CP5. The Argonne fast chopper, with a 60-m flight path, has a resolution in the 0.04-yusec /m region. 3.16 Neutron Detectors for Time-of-flight Measurements. Although the usual enriched BFs proportional counters or ion chambers have been used as detectors for time-of-flight work, their low efficiency (arising from the l/v boron cross section), finite neutron path length, and long pulse collection time make it clear that better

Sec. 5-2]


5-77

The detectors mentioned can, of course, be improved to some for example, at Brookhaven a multiple BFj proportional counter is now being In this detector tested, containing 128 nested counters in a common BF3 atmosphere. the small counter diameter (Yi in.), large central wire, and high voltage reduce the time uncertainty (a total spread of 4 /jsec in the standard 2-in. tube) to 0.5 fisec. Scintillation counters, however, seem to hold the most promise. At the present time a boron-containing liquid scintillation counter is being used at Argonne, in which neutrons are moderated somewhat to increase the efficiency, although a time A scintillation counter incorporating uncertainty of about 1 /iscc is thus introduced. this counter has an ZnS grains in fused BiOi has been developed at Brookhaven; efficiency higher than BFa counters, although lower than the liquid scintillator, and a The Brookhaven chopper presents a more difficult detector speed of about 0.2 /isec. problem than other designs because of the larger beam area at the detector position and the higher 7-ray content of the beam. The detectors described thus far are used primarily for total cross-section measure ment by sample transmission. Measurement of the partial cross sections, primarily capture and scattering, requires more elaborate equipment; both of these processes involve more than a simple relative measurement, which is sufficient for transmission results. Data from partial cross-section measurements are very sparse at present relative to total cross-section results because of the increased technical difficulty. For a scattering measurement as a function of neutron energy a detector of scattered neutrons is placed close to the scattering sample and the scattering cross section is Neutron capture usually determined relative to a standard scatterer, such as carbon. may be detected by means of the capture y rays, but the measurement of the actual level parameters for capture is an involved process that is not on a routine basis as yet. A special partial cross section, but one of great practical importance, is fission, which is usually detected by means of an ionization chamber containing a thin foil of the fissionable isotope under study. The absolute fission cross section can hardly he obtained directly because of lack of knowledge of neutron flux, so the relative toss section only is determined using the l/v cross section of a standard material, *uch as B10, for comparison. Although fast choppers and the recent pulsed accelerators have approximately equal resolutions, there are several respects in which they differ, the principal differ ences pertaining to sample and counter sizes. The sample size is much less in the case of the fast chopper because of the inherently small slits. As a result samples as small as 10 mg have been used, and available supplies of separated isotopes are sufficient for investigation. On the other hand the counter area is usually smaller for the pulsed source machines, and they are more suitable for measurements utilizing complex detection instruments, such as those appropriate for scattering and capture cross sections. However, as for the high-energy region, by far the majority of the results pertain to total cross sections resulting from relatively simple transmission measurements. It is noteworthy that none of the time-of-flight methods can be used for activation cross sections, as can be done with the crystal monochromator; neutrons of all energies would activate the target, whereas in the latter instrument a beam is actually produced. monoenergetic detectors are needed.


extent ;

3.2

Measurement

of Resonances

The sharp neutron resonances observed in such large numbers in heavy nuclei are the manifestation of virtual levels in the compound nucleus at excitation energies of the order of 5 to 8 Mev. The observed parameters of the neutron resonances, such as peak heights, neutron widths, and radiation widths, give directly the char acteristic properties of the nuclear energy levels. The field of "neutron spectroscopy" concerned primarily with the determination of these level parameters, which are useful to the theory of nuclear structure and of neutron reactions as well. 3.21 The Breit-Wigner Formula. The "shape" of a neutron resonance, i.e., its toss section as a function of energy, is given in terms of the level parameters by the The single-level formula, in which the effect of Breit-Wigner dispersion formula. is

6-78

EXPERIMENTAL

TECHNIQUES

[Sec.

5

levels other than the one under investigation is neglected, is almost always sufficiently Further simplification results because only zero angular accurate in practice. momentum («) collisions are appreciable for low-energy neutrons, which are used to excite resonances in most nuclei. Under these conditions the cross sections for neutron absorption (a„) and scattering (a.) are given" by

i) **** where

o =

2

\

( 1 ±

V

- eS> ; (r/2).

r„/2

,

E-El>+iV/2

+

K

U

+

(16)

Ml

— J 2/ + 1/

- g)R*

(17)

lis

Here E is the neutron energy; 2jtXo the wavelength of a neutron of energy E0, the resonance energy; and r„, I\, and r( = r» + I\) are the neutron, radiation, and total widths of the level. * The target nucleus has radius R and spin /, with g the statistical factor giving the probability of formation of a compound nucleus of spin / + Although measurements of
ao

-

4wU*grn/T

(19)

As the total width of the level V is just the full width of the resonance at half maxi mum, it follows that a careful measurement of the height ao and width T of a total cross section resonance gives r and gT,. If the level spin (hence g) is known, then and hence (given by T — r») follow directly from the total cross section. Even though g is known exactly in only a few 1 for even-even cases, its approximate value ().£ in general, target nuclei) is often sufficiently accurate that the procedure just sketched is very useful in giving neutron and 7 widths for many resonances from total cross sections alone. A subsidiary measurement of
r

* Note that as the widths are values for the level, they are constants is far different from Eo.

I\

and do not change even when K

EXPERIMENTAL

Sec. 5-2]

NEUTRON PHYSICS

5-79

described briefly. We shall assume that the measured area of a trans dip (in electron volts) has been corrected for the effect of nearby resonances and for potential scattering, so that it refers to the resonance of interest alone. The analysis of the resonance is carried out in terms of the transmission curves alone with no conversion to actual total cross section in the process. The principle of area methods of analysis is that in the limiting cases of very thin and thick samples the dip area A is given" by and it can be

mission

Thin sample:

A =

Thick sample: A

=

%*0r

(20)

(xna^V

(21)

m

irith n the number of atoms per square centimeter in the sample. Measurement of for these two cases, of course, gives
areas

lOOi

,

1

1

,

1

,

1

,

1

60-


40-

Fio.

11.

The curves used in obtaining resonance parameters from measured areas of trans

mission dips. However, because of numerous experimental exigencies, it is usually true that only For these nonideal cases, the effect intermediate sample thicknesses can be used. of Doppler broadening, which is negligible in the ideal cases (see Fig. 15 of Seidl et a/.,'4 for example), greatly complicates the analysis. The analysis involves only the single set of curves shown in Fig. 11, in which A/ A is plotted against naa{V/A) for various All these quantities are dimensionless, and the input data A /A do not values of A/r. involve unknown parameters. The Doppler width A is given by

A =

2(^y*

(22)

with T the sample temperature and m/M the neutron-nucleus mass ratio (see Ref. 28 for small corrections to T). The use of the curves for the thick-thin sample case is very direct. Points are plotted for each sample at the correct ordinates A /A and a distance apart horizontally

5-80

EXPERIMENTAL

TECHNIQUES

[Sec.

5

corresponding to the ratio of the sample thicknesses n. If these two points are moved horizontally, they will fit only one of the curves, whose A/r value then gives r directly, and the abscissa gives an, hence gr„.* Another type of analysis is involved when only one sample thickness is available. For this situation, which occurs frequently, the usual procedure is to assume a value of rT for the resonance and thus obtain gr„ from the single sample area. The 1\ is usually known to about 20 per cent accuracy from other considerations, and their uncertainty produces an error in gYn that varies with sample thickness in a com plicated manner. The curves of Fig. 11, however, give a simple solution to this The measured area and the assumed ry give an problem also. and a A/r value that r„ (assuming at first that r = rT, that small), from which ruro(r/A), hence If the r„ obtained gVn, obtained. not small compared with rT, the A/r value used can be changed and revised gY„ obtained. Again, as for the multiple-sample method, no lengthy computations or use of table are necessary, the conversion of the rwo(r/A) value to gT„ being done by slide rule. The error in the final gV„ resulting from the experimental error in and that arising from the uncertainty in the assumed I'T are easily determined by simple geometrical considerations. In all the foregoing discussion, the statistical factor unknown. For most of the applications of resonance parameters to reactor calculations exact knowledge of the statistical factor not necessary. As measurement of gives the level spin, however, of interest for basic nuclear theory. measurement of one of the partial cross sections or even its ratio to the total cross section in the resonance gives value, but because of the difficulty of partial cross-section measurements only the few J'b have been determined. For fissionable materials the resonance analysis already discussed holds, but for measurement of the fission cross section complete determination of the parameters in the resonance also necessary, a measurement usually carried out with fission chamber. For these materials r„ usually so small that essentially given by is

is

g

J

is

A

a

is

r

is

a

is

IV +

Ty.

Integral Measurements

3.3

f"

is

it

a

a

is

a

is

is

is

is

The use of intermediate energy neutrons inside the pile limited by the difficulty of interpreting the results when such a wide range of neutron energy present. The main use of resonance neutrons inside the pile lattice relates to measurement of integral effects extending over large energy regions rather than the selection of par ticular resonance energies, which accomplished with neutron beams outside the pile. The integral cross sections, however, are important in reactor design and con stitute an important phase of neutron measurements. 3.31 Resonance Absorption Integral. When subjected to the 1/2? material spectrum, the rate of occurrence of particular nuclear reaction proportional to the integral of its cross section as function of neutron energy multiplied by the flux density; that is, proportional to Ja(E) dE/E: Reaction rate =

a(E)^

(23)

a

is

is

a

is

is

is

is

is

is

is

is

where the neutron slowing-down density and £Na, the slowing-down power of the well-known "resonance the moderator. The integral appearing in Eq. (23) usually considered to extend from the cadmium cutoff, about 0.4 ev, integral" and The form of the integral up to fission energies (denoted °o for convenience). based on the assumption that the dE/E spectrum not distorted by the absorbing isotope, an assumption that implies extreme dilution of the absorber in the moderator. If distortion of the spectrum usually true inside pile lattice, the present, as integral over the distorted spectrum called the "effective resonance integral." The cross section under the integral usually the absorption or the activation cross section, although the "resonance scattering integral" has definite meaning as well. Until recently, the only method of obtaining resonance integrals was by direct just (4»Xo'n /&)ar,. •The Bhscissa two( IV A) q


g

a

is

it

is

g

A

a

is

is,

A/\

Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

6-81

Determination of the neutron using the methods to be described. parameters has now given sufficient data, however, so that the integrals can be computed from the parameters directly. In one case, that of U"8, the value obtained from the parameters agrees with and is of higher accuracy than the directly measured value (activation method). 3.32 Measurement of Resonance Integral by Activation. Although absorption integrals can be obtained by oscillating samples in the pile inside cadmium, the simplest method of measuring the resonance integral is by means of activation, for then simple counting of foils can be used for the determination. If the ratio of the resonance to the thermal flux is known, the resonance integral follows directly from the measured cadmium ratio for activation 7>c
resonance

(24)

quantity (nv)tk/(q/iN. As usually measured, the resonance integral has as its lower limit the cadmium cutoff, here taken as 0.4 ev; hence no contribution is obtained from those neutrons between the practical upper limit of the thermal distribution (about 0.17 ev) and the cadmium cutoff. In other words, in the measurement of the cadmium ratio, the cadmium cover removes some of the resonance neutrons, those between thermal and 0.4 ev, in addition to the thermals, and the part of the resonance integral cor If it is desired responding to this energy band is not recorded in the measurement. to obtain the complete resonance integral down to thermal, as is necessary for pile calculation, then a correction (which can be calculated for a thin foil if the cross section is known) must be added to the measured value of the activation inside cadmium. The actual experimental method used in determining the resonance integral is It involves only the straightforward in the ideal case of an extremely thin foil. irradiation of a cadmium-covered and a bare foil, with subsequent measurement of the activities with a 0 counter. For accurate results, the foils should not be activated simultaneously because the Cd cover will lower the thermal flux and thus affect the bare-foil reading. This undesirable effect can be avoided in simultaneous activations by separating the foils about 4 in., but this procedure raises the possibility that the fluxes might not be identical at the two positions. In practice it is sometimes very difficult to ensure that the foils exposed to the resonance flux are thin enough so that no "self-protection" of the foil, i.e., distortion of the dE/E spectrum, will occur. The condition of no self-protection is quite severe Indium, for instance, because the peak cross section at a resonance can be quite high. has a peak cross section of 30,000 barns; an indium foil must therefore be less than 11.6 mg/cm1 thick in order that less than 1 per cent of the neutrons of exact resonance If the foils are sufficiently thin so that all the atoms are energy will be absorbed. exposed to the undistorted l/E spectrum, the counting rate will be proportional to the activation integral — the exact constant of proportionality usually being obtained by comparison with some standards, as already mentioned. Indium is a very convenient standard for determination of the resonance to thermal flux ratio because its resonance integral is determined almost completely from the large resonance at 1.44 ev, whose shape has been well determined by velocity selector Indium also has a high thermal cross section, is easily made into measurements. thin foils, and has a convenient half-life (54 min) for activation. The cross-section measurements give the total cross section, but that for activation of the 54-min period can be obtained by correcting for the relatively small scattering and the con tribution to the absorption of the periods other than the 54-min. The value of the resonance absorption integral for indium determined by integrating the velocity The


flux in a

EXPERIMENTAL

6-82

TECHNIQUES

[Sec.

5

curve in this way, with a lower limit of 0.4 ev, is 2,800 barns, accurate to about 5 per cent. The ratio of the resonance to thermal flux at a particular point in a pile, where the activation integrals are to be measured, can be determined by measuring the cadmium ratio for thin indium and using Eq. (24) with a thermal (2,200 m/sec) = 150 barns selector

barns.

The resonance

integral can then be obtained for

other materials from their cadmium ratios, and the flux ratio obtained from indium. The resonance to thermal flux ratio can also be obtained from the measured cadmium ratio for a \/v detector, because the ratio of the resonance integral to the thermal A thin foil of a sodium salt (15-hr halfcross section is 0.50 for a cross section. life) is a convenient detector for this purpose, and the resonance to thermal flux ratio follows directly from the cadmium ratio and the 0.50 factor by means of Eq. (24). The indium method of determining the flux ratio has an advantage over the 1 /» detector because its cadmium ratio is near unity while the resonance activation in the l/v case is only about 3 per cent of the thermal in a typical pile lattice. A dis advantage of the indium method arises from the necessity of extremely thin foils, although this difficulty can be avoided by use of secondary thick indium foils, cali brated in terms of thin foils. The "standard" indium foils, 0.1 gm/cm2 thick, used for resonance flux and thermal flux measurements are such secondary standards. Cadmium ratios measured with these foils are about five times the values for thin indium foils because of self-protection of resonance neutrons. In the case of an actual absorber in a flux of 3.33 Effective Resonance Integral. resonance neutrons the sample size is usually large enough so that its absorption modifies the neutron energy distribution appreciably. As a result of the absorption, the neutron spectrum is depleted at the resonance energies and the resonance absorp tion integral is smaller than it would be if the spectrum were unmodified by absorp The correct integral is then one in which the actual spectrum, depleted at the tion. resonance This corrected resonance integral is energies, replaces the l/E factor. called the "effective resonance integral," which is important in the calculation of p, the resonance escape probability. The effective resonance integral will depend in a rather complex way on the nature of the absorption resonances, the amount of scattering material present, and the physical shape of the absorber. The integral is an extremely sensitive function of the amount of scattering present. The value for normal uranium, which is 280 barns for infinite dilution, drops to 9.0 barns for uranium metal (no moderator present, the only scattering being that of the uranium nuclei themselves). Analysis of the slowing-down process, taking into account the flux modification by resonance absorption, actually shows that the effective integral should increase as the square root of the scattering cross section per uranium for low values of the effective integral, and experi mentally a 0.415 power variation is observed. The ratio of the effective integral to the true integral (that for an undistorted dE/E spectrum) is the measure of the flux at the resonance energies compared with that for the case of infinite dilution. The effective resonance integral for the homogeneous case as a function of dilution can be measured quite directly by comparison with indium. For normal uranium mixed with graphite, for instance, the measurement is made by placing a volume of powdered uranium oxide mixed with powdered graphite inside the pile lattice and irradiating thin foils of U"* and indium wrapped in cadmium inside the volume. The in the abundant U"8) given by the 23-min resonance absorption in uranium (that U™ activity produced in the U"8 foil relative to the indium activation. foil of normal uranium cannot be used because the fission product activities produced in would interfere with meastirement of the U***. The volume occupied by oxide and graphite need be only large enough to ensure that the resonance flux moving into the volume reaches equilibrium as far as the depletion at the resonance energies is con cerned, and volume of the order of 10* cm' sufficient for this purpose. For the more usual yet more complicated heterogeneous situation, for example, made by following graphite moderator, the measurement piece of uranium metal in the same procedure as for the homogeneous mixture. The experiment more cornis

it

A

is

is,

is

is

a

a

a


l/i

Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

5-83

plicated because the atoms in the thin surface layer of the uranium lump have a much The neutron spectrum higher resonance absorption than those inside the lump. incident on the lump shows very little depletion at the uranium resonances because equilibrium has been reached with the graphite atoms. As the neutrons move into

uranium lump, however, the spectrum quickly reaches an equilibrium depletion, which is quite constant throughout the interior of the lump. It follows that the resonance absorption integral is made up of two contributions, one proportional to the

the

surface area of the lump and the other to its volume.*


4

THERMAL NEUTRONS

An examination of cross-section curves in the thermal neutron region shows irregular variation with energy, but only rarely of the typical resonance shape familiar for neutrons of slightly higher energy. The thermal region represents such a small range in energy that the chance of resonances occurring in this limited region is quite However, as the wavelength of thermal neutrons is about equal to the inter small. atomic distances, the neutron waves scattered from various nuclei interfere and com plex diffraction effects occur. It is these diffraction effects that cause the abrupt The coherent changes in cross section characteristic of the thermal energy range. or optical effects of scattering of slow neutrons (neutron optics), while of great impor tance in the study of structure of matter, does not have so important a place in reactor design as the nuclear effects of thermal neutrons, such as activation cross sections and fission. The high intensity of thermal neutrons present in reactors has resulted in of high-precision instruments for their use; experiments are possible development with these instruments that could not be considered in the days when reactors were not available. 4.1

Thermal Neutron Sources

Before the advent of chain-reacting piles neutron experiments had been performed with a variety of sources in conjunction with a material in which the neutrons were thermalized (paraffin or graphite). However, all these sources give such low intensi

ties that accurate thermal neutron experiments can hardly be performed. A 1-curie Ra-a-Be source, for example, has a neutron emission rate of about 10' fast neutrons per second, and if this source is placed in a moderating medium, graphite, for example, the maximum thermal flux will be of the order of 10*. A particle accelerator such as a cyclotron, Van de Graaff, or transformer-rectifier accelerator can produce higher intensities of fast neutrons but usually of the order of at best 100 times that obtained from a Ra-a-Be source. The fluxes quoted refer to those found within the moderating material, of course, and if neutron beams are obtained from any of these arrangements, the intensity will be reduced greatly by geometrical factors, with a resulting beam flux that is extremely small. 4.11 The Maxwell Distribution. Of the wide energy range represented by pile neutrons, the highest density is for those in thermal equilibrium with matter at room temperature; the velocity of these neutrons exhibits the Maxwell distribution familiar from kinetic theory:

n(v) dv =

This equation, illustrated

in Fig.

12,

,

An

_

i)»e-*v.o«

dv

(25)

gives the density of neutrons per unit volume

n't;) as a function of velocity, where n is the total number per unit volume. The is the "most probable velocity," corresponding to a kinetic energy of velocity The *r(0.O253 ev); numerically it is 2,200 m/sec at room temperature (293.7°K). wavelength of the neutrons of most probable velocity is 1.80 A, of just the order of mag nitude of interatomic spacing in matter, hence the prevalence of "crystal effects" in thermal neutron scattering. The neutron flux is obtained by multiplying the density by * The fundamental measurements described in detail in Ref. 25.

and theory for the heterogeneous

case, performed

in 1941-1942,

are

EXPERIMENTAL

5-84

TECHNIQUES

[Sec.

5

the velocity; this quantity (Fig. 12) gives the number of neutrons crossing a unit area Neutrons of the wavelength required for a particular per second in a neutron beam.* experiment must be selected from the distribution of Fig. 12, which drops off rapidly


both above and below v0. Attempts to shift u0 by varying the temperature of the neutron source have not been fruitful, mainly because of the difficulty of changing the temperature of a sizable section of a chain-reacting pile. 4.12 Thermal Neutron Beams. The actual neutron flux available in a pile neutron beam is calculated directly from the geometry of the hole that is opened into the pile to form the beam. For example, let us assume that a hole of 10- by 10-cm cross section is opened into the pile to a point where the flux is 5 X 10'*. The flux in the beam formed by this hole will be given by the number of neutrons crossing a

2000

4000

6000

v- m/s

Fio.

12. The Maxwell distribution of thermal neutrons and the flux nv as a function of neutron velocity.

showing

the density of neutrons

n

l-cms area at the point of interest in the beam, which distance will be of the order of The beam flux will then bef

5 m.

5

X 10" X 4x(500)«

10»

1.6

X

10s

neutrons/(cm,)(sec)

(26)

The collimation (angular divergence of neutrons in the beam) in this particular case will be of the order of 10/500, or about 1°. The angular resolution needed for neutron diffraction work is higher than that just calculated by an order of magnitude, and for mirror experiments even greater collima tion is necessary (about 1 mini. Higher resolution is obtained by installation of diaphragms to limit the angular divergence of the beam, with an attendant reduction in intensity. Thin sheets of cadmium suffice to stop the thermal neutrons, but it is necessary also to reduce the fast neutrons that are always present as a background. As fast neutrons are much more penetrating than thermal neutrons, collimators usually consist of large blocks of material. In actual practice, combinations of materials, for example, plastic, iron, and lead, are used in order to stop the various The intensities in pile beams are so high that types of radiation that are present. it is of utmost importance to limit the stray radiation, not only from the standpoint of interference with the experiments but to eliminate the possibility of harm to the * The manner in which the fast neutrons are slowed down to thermal, or " moderated." as well as properties of the Maxwell distribution of slow neutrons, is discussed in Ref. 18. pile example The flux for this the value chosen is at the center of the Brookhaven pile at full power t (28 Mw).

various

Sec. 5-2]

EXPERIMENTAL

5-85

NEUTRON PHYSICS

The techniques for the formation of beams have been developed experimenters. sufficiently well so that neutron fluxes of the order of 10* thermal neutrons are avail able in beams sufficiently well collimated for diffraction experiments, while somewhat lower intensities with correspondingly higher collimation are used for neutron mirror research. *

T

T

800

700

600


500 FILTERED NEUTRON DISTRIBUTION

<

K

o

400

z z

3 o

300

UJ

200

100

J_ O

X

II

-A

Fio. 13. Wavelength distribution of neutrons after passage through a graphite filter. The distribution would show an abrupt drop at X« were it not for the finite resolution of the detecting instrument, the Brookhaven slow chopper. Filters. Whereas in some work the entire Maxwell distribution has been 4.13 used, at the present time there are very few experiments that are still performed with Several methods are available the wide energy spread represented by the Maxwellian. for selecting monoenergetic neutrons, or, more precisely, neutrons whose velocities Some of these methods (hence energies and wavelengths) lie within a narrow band. perform only a crude "monochromatization" but allow high intensity, whereas others give almost monoenergetic beams but, of course, at a correspondingly high loss * For details of these technique!, see Ref. 18, Sees. 9.1; 10.5; 11.1, 2, 3; and 12.4.

EXPERIMENTAL

5-86

TECHNIQUES

[Sec.

5

in intensity.

One of the first methods used for isolation of discrete energy groups involves the use of neutron filters; for special applications filters are still of great value. The type of filter that concerns us here is based on the fact that in a polycrystalline material no Bragg reflection occurs for which the wavelength is longer than twice the largest lattice spacing. For wavelengths longer than this "cutoff wavelength" no Bragg scattering occurs and the only remaining processes that remove neutrons from a beam are incoherent scattering and neutron reactions (usually neutron capture). For certain materials the capture and incoherent scattering are extremely low, and these materials are therefore suitable for neutron filters. The materials that have been used most extensively are graphite, beryllium, and beryllium oxide, with cutoff wavelengths of 6.7, 4.0, and 4.5 A. The neutrons of wavelength beyond the cutoff are transmitted through a block of filter material with practically no loss in intensity, whereas the shorter wavelength neutrons are removed by Bragg scattering. Although the resultant beam is not monoenergetic, it has an exceedingly sharp cutoff wave length, which for some purposes, particularly mirror reflection, is as useful as a monoFigure 13 shows the energy distribution of thermal neutrons after energetic beam. nitration through a graphite block 10 cm in length. 4.2

Although many valuable measurements have been made with thermal neutron containing the entire Maxwell distribution, they have been almost completely displaced by the more exact measurements in which neutrons of specific energies or velocities are isolated. The velocity selection can be performed either with a crystal monochromator or with time-of-flight techniques using a "slow chopper." Because of the serious interfering effect of high-order reflections when the crystal is used for slow neutrons, the latter method has been more used in the thermal region for crosssectional work. The crystal as a monochromator is extensively used, of course, for a particular application, neutron diffraction as a means of crystal structure analysis. 4.21 A mechanical neutron "chopper" operating in the The "Slow Chopper." thermal region is the same in principle as the "fast chopper" used in the intermediateenergy range, but the slow chopper is of much simpler mechanical design. The latter is of simpler construction because the bursts produced are of the order of 20 instead of 1 /isec and, in addition, the chopping of thermal neutrons can easily be accomplished with thin layers of cadmium instead of many inches of steel or plastic. A slow chopper was used by Fermi at Argonne as early as 1943, and his design was incorporated into similar instruments at many other laboratories. As an example, a cross-sectional view of the Brookhaven slow chopper is shown in Fig. 14. It is simply a rotating cylinder made up of parallel cadmium-plated nickel sheets, driven by an electric motor at speeds up to 15,000 rpm. The desired neutron velocities are selected by measuring electronically their time of flight to a detector, usually a BFi counter, several meters away. For cross-section measurements in the 5- to 25-A range, the background of epicadmium neutrons can be greatly reduced by insertion of neutron filters in the beam, which have a high stopping power for neutrons in and above the Maxwell distribution but are practically transparent for long wavelengths. The most useful filters in this The background counting rate, which is as large respect are Be, BeO, and graphite. as the timed neutrons at the long wavelength, is measured by inserting a thin (0.01-in. ) cadmium sheet in the beam at the chopper. A thick piece would stop some of the neutrons near the cadmium cutoff energy that penetrate the chopper, and the resulting cadmium difference would be larger than that appropriate to the true timed neutron rate corresponding to the cold neutrons. The effect of a thick cadmium shield in changing the background is negligible for a transmission measurement using a steady ("unchopped") beam, but for the chopper, where the duty cycle of the background is greater than that of the real counting rate by a factor of about 100, the effect is significant. For accurate velocity measurements it is essential that every component of the timing be carefully standardized. Fortunately, an excellent over-all check of the beams


Velocity Selectors for Thermal Neutrons

Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

5-87

furnished by the position of the cutoff breaks already described in connection neutron filters. It is necessary only to place a filter, for example, a piece of graphite, in the beam and observe the counting rate in the different channels or, more accurately, measure the graphite cross section by transmission for the different chan nels. Timing errors can be detected and their origins located by study of the crystal timing is


with

Fio. 14. The rotor of the Brookhaven slow chopper containing curved channels, which pro monochromatization of neutrons in the bursts. The channels are formed by curved nickel screens, plated with cadmium, and held in a hardened-steel cylindrical shell. duce some

such parameters as counter distance and rotational speed while In careful work observing any apparent shift of the cutoff wavelength that results. it is necessary to take into account the fact that the effective center of the BFj pro portional counter, hence the flight path, changes with neutron velocity as the boron cutoffs, by varying

absorption changes. The foregoing effects must be carefully studied for such impor tant cross-section measurements as those of Um and boron at 2,200 m/sec.

5-88

EXPERIMENTAL

TECHNIQUES

[Sec.

5

4.22 The Spiral Monochromator. A true mechanical monochromator is one in which neutrons of a narrow velocity band are produced by mechanical methods, with no time-of-flight measurement involved. A monochromator has certain advan tages over a time-of-flight instrument, advantages that follow from the fact that actual separation of the different velocities is not accomplished in the latter; i.e., no monochromatization occurs. For example, activation with monoenergetic neutrons is possible with the monochromator and impossible with the time-of-flight method. Because no measured flight path is involved with the monochromator, the conditions for scattering cross-section measurements are much more favorable for it than for the time-of-flight apparatus. The obvious advantages of a mechanical monochromator for activation and scatter ing have led to the construction of several models all of similar design; the principle of all of these is illustrated in Fig. 15. A device of this type consists of several disks of cadmium containing radial slits, with the disks mounted on the same axis, which is parallel to a neutron beam. As the disks rotate, the neutrons in a collimated beam pass through the slits if the neutron velocity bears the correct relationship to the speed of rotation. POSITION OF SLOT WHEN NEUTRONS REACH DISC


MAXWELL DISTRIBUTION

SINGLE VELOCITY

CADMIUM DISCS

Fio.

15.

The principle of the mechanical

monochromator

for slow neutrons.

The monochromator is analogous to the slow chopper, with the openings in the first disk corresponding to the rotor and those in the last disk to the electronically con trolled counting channels of the chopper apparatus. The resolution and intensity are determined by the size of the openings and the speed of rotation, and high resolu tion always dictates operation at the highest possible rotational speed. Because of the correspondence of the action of the first disk to the rotor of the slow chopper, it is expected that the resolution available with the monochromator will be about the same as that of the chopper for size of openings and rotational speeds of the same order of Of course, neutrons of only one velocity band at a time are available magnitude. as compared with the multichannel operation of the chopper. A rotating-disk monochromator has recently been built at Los Alamos, designed In order to get the highest possible counting specifically for use with cold neutrons. rate the resolution was made poor (about 30 per cent in energy) and the beam bole The high background of fast neutrons in the extended to the center of the pile. beam is effectively removed by disks of lucite placed between those of cadmium, which scatter the fast neutrons out of the beam. With this monochromator, neutrons have been obtained with wavelengths extending to 30 A. It is probable that the instru ment will be of definite use in this extremely long wavelength region. Although little in the way of experimental results has been obtained by the use of mechanical monochromators, it seems that continued effort should produce a device capable of yielding resolutions of a few per cent in the subthermal region. While the variation of the capture cross section is \/v for most materials in this region, the use of the monochromator seems logical for a few important materials whose crosssectional behavior is not known in this energy region. The monochromator may also

Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

5-89

prove to be a useful instrument in conjunction with other equipment, for example the crystal and mirrors, especially when these are used for low-energy neutrons.

4.3

Thermal Neutron Reactions

The great importance of thermal neutrons in the field of design and application of reactors is well exemplified not only by the variety of techniques used to exploit neutrons in the thermal region but by the great range of interactions with matter The cross sections are studied within the pile lattice using the entire they exhibit . Maxwell distribution and with beams of thermal neutrons, usually with velocity selection, to determine cross-sectional variation with neutron energy. 4.31

Activation

Cross Sections for Maxwell

Distribution.

For most materials


for which there are no neutron resonances at or near thermal energy, the activation rate in a neutron flux nv is given by the simple activation formula already given as Eq. (10) even though the entire Maxwell distribution is present. The use of the 2,200-m/sec cross section rests on the fact that the flux is practically always determined by a method that actually measures the neutron density n inde pendently of v (because the response of a 1 /v detector is proportional to n rather than The quoted result of a flux measurement is obtained by multiplying the measured nv). neutron density by an arbitrary v, which is almost universally taken as 2,200 m/sec. The actual flux may be, and probably is, different from the value obtained by assuming 2,200 m/sec for v, because the temperature of the distribution is usually higher than room temperature. This difference is unimportant, however, when the activation

material of approximately l/v cross section is concerned. The activation of the is, like the flux measurement, independent of v, but the cross section to be used in Eq. (10) must be the cross section at the velocity that was assumed in the quotation of the flux in the same equation (almost certainly 2,200 m/sec) if the correct activation is to be obtained. While this velocity independence of the activation holds strictly only for l/v sub stances, most elements are l/v within experimental error in the thermal region, and fortunately no further complications ensue for them. For those few materials that depart significantly from l/v, the calculation of activation in a Maxwell distribution must take into account the neutron temperature. The simplest way to correct for the non-l/» behavior is to calculate, from the known cross-sectional curve of the material, the ratio of the actual activation to that for al/v material of the same 2,200constitutes a correction factor for the m/sec cross-sectional value. This ratio non-1 /» effect that is easily calculated from the actual cross section as a function of velocity, for it is merely the ratio of the activation of the actual cross section to a l/v cross section, both integrated over the Maxwell flux distribution: of a

material

/

/ J

n(v)vax dv n(v)v

(27)

— dv

(28)

In these equations, /, is the correction factor for a material x whose cross section at 2,200 m/sec is fci/2,200 (thus kx/v is a l/v cross section equal to that of x at 2,200 From Eq. (28) we see that is just the cross-sectional ratio weighted accord m/sec). ing to the Maxwell density (not flux) distribution. The correction factor for a room-temperature distribution has been made for those isotopes that differ appreciably from l/v in the thermal cross-sectional table in the In use, the 2,200-m/sec cross section in the cross-sectional compilation of Ref. 2. table is simply multiplied by /before insertion into Eq. (2) to give the activation that changes As will be obtained with a Maxwell distribution at room temperature. extremely slowly with temperature of the neutron distribution, the factors listed can

/,

/

be used

for neutron temperatures several hundred degrees above room temperature.

5-90

EXPERIMENTAL

TECHNIQUES

[Sec.

5

/

As an example of the use of factors we can calculate the number of neutron captures per second in 1 g of Cd"s, hence formation rate of Cd"4 (actually not an activation The appropriate because the latter is stable) in the Brookhaven pile (nv = 5 X 10"). Thus the actual cross section given in Ref. 2 is "19,500 barns, not 1/v, X 1.3." 2,200-m/sec cross section is 19,500 barns, but because of the resonance near thermal energy, the effective value of the cross section for a Maxwell distribution is 19,500

and the reaction rate is 5

X 10" X

25,300 113


/

X X

1.3 = 25,300 barns 0.602

6.7

X

10u capturcs/sec

Strictly speaking, the value given is accurate for a room-temperature Maxwellian only, but the change in for a few hundred degrees centigrade is less than the crosssectional error in most cases. For a few important heavy elements the factor is given as a function of neutron energy in Ref. 2. Although most of the activation resulting from irradiation inside the pile lattice represents absorption of thermal neutrons, resonance neutrons do contribute a nonIt is therefore necessary to take into account the resonance negligible activation. activation in a thermal cross-sectional activation measurement made inside the pile lattice. Fortunately, the resonance contribution can usually be ascertained experi mentally by an additional irradiation inside cadmium. After elimination of the resonance activation by a cadmium difference in a thermal cross-sectional measure ment, there are very few materials that cannot be considered 1 /v. 4.32 Measurement of Thermal Activation Cross Sections. The thermal activa tion cross section (2,200-m/sec value) can be obtained directly from Eq. (2) from the measured disintegration rate produced by thermal neutrons, together with the con ventional (v = 2,200 m/sec) thermal flux. As the flux itself is measured with a l/v detector, the activation cross section resulting from Eq. (2) can, of course, be con sidered as one measured relative to the cross section of the l/v standard (assuming gold or boron, rather than a standard pile) of the flux measurement. However, no gain in accuracy usually results from this relative approach because the errors in the disintegration rate are larger than in the flux. The activated material is usually counted on a calibrated Geiger-Mtiller counter or some other absolute 0 counter, and the disintegration rate is obtained from the counting rate by means of the known efficiency of the counter. Although the principle of activation measurements is quite simple, it is exceedingly difficult to obtain reproducible, accurate results in practice, with the result that activation cross sections in general are known only to about 20 per cent. By far the most significant errors arise in connection with the conversion of the counting rate to the disintegration rate /. For a sample known to disintegrate by emission of f) particles alone, it is reasonably certain that each 0 entering the counter will be recorded. Even in this ideal case, however, significant error arises in the conversion of counting to disintegration rate, that is, in the estimation of the efficiency of the counter (fraction of disintegrations giving rise to counts). The uncertainty arises from lack of knowledge concerning the fraction of disintegration 0's that enter the counter. Some of the 0's are absorbed in the material of the sample, some in the window of the Geiger-Miiller tube, and still others are scattered from the surroundings. The efficiency of the counter is usually determined by counting a sample with a known disintegration rate, such as the RaE 0 standards" supplied by the Bureau of Standards. These standards consist of small amounts of Pbsl0(RaD) in PbO». electrolytically deposited as a layer about 1 mg/cm* thick on a palladium-coated silver disk, and are available in approximate strengths 100, 200, and 500 disintegra tions per second. The absorption of 0's in the Geiger-Mtiller window (usually about 3 mg/cm* thick) and in the sample itself can be studied for a particular geometrical arrangement byuse of aluminum absorbers. The absorption curve, measured by placing absorbers on an intense thin sample, is approximately exponential with an initial slope that has a

/

/

EXPERIMENTAL

Sec. 5-2]

NEUTRON PHYSICS

6-91

/S.

correspondence to the (end-point) energy of the Figure 16, for instance, illustrates the absorption of several 0's of different energies as detected by an endwindow counter, and curves of this type give an empirical relationship between the energy and the absorber half thickness (the absorber thickness that reduces the

0

rough

17.

of end-point

4

1—i-l 6

810s

4 .

0

TV

2 .

6 8

4

The half-thickness and mass absorption

102

6 . . I

ill.

© Flo.

600

emitters.

4

curves of several

2

4

6 8

2

10

500

MG/CM*

810

L^lmgm./em*)

coefficient

for

P

The absorption

_i .6 8 I

A

16.

400

300 THICKNESS

2

Fig.

2

absorption

as a

function

energy.

An empirical relationship between energy, absorption factor 2). counting rate by coefficient, and half thickness appears in Fig. 17, where the half thickness in milli In the disintegration-rate determination, called La. grams per square centimeter the observed counting rate must be corrected for the absorption of 0's in the sample For a sample of thickness small compared itself as well as in the counter window. made by adding the average thickness, with the half thickness, this correction that is, half the sample thickness, to the window thickness and correcting the counting is

is

0

a


200 ABSORBER

5-92

EXPERIMENTAL

[SEC. 5

TECHNIQUES

is

a

is,

rate to zero absorber by means of the slope of an actual absorption curve or that given by Fig. 17. 4.33 In the production of radioisotopes under condi Second -order Reactions. tions of high flux it is necessary to consider various possible neutron reactions of the The the "second-order reactions" must be investigated. radioisotope itself, that second radioactivity to be produced from the second-order reactions can cause It easy to see radioisotope itself, an activity that may be desirable or undesirable. that the amount of the second-order activity at saturation wall be proportional to the square of the neutron flux, because the amount of the primary radioisotope will be proportional to nv, and the secondary product proportional to this amount multiplied by nv: Disintegration rate of primary radioisotope = Nnvo by Eq. (2) Hence of atoms of primary

radioisotope (X

X

Number

Disintegration

= primary

disintegration Nnvo — -—

rate of secondary radioisotope

—

nvo'

the cross section for production

—

(29)

of the secondary radioisotope.

The absorption of neutrons by the primary radioisotope

nva'

and the saturation than Nnvo; i.e.,

= natural disintegration const of primary = disintegration const in flux nv

disintegration

radioisotope

rate of the primary radioisotope will be slightly less rate = Nnvo X

Saturation

+

(30)

,

X

X

is

is

causes some destruction almost always minor. It can be con of this radioisotope, although this effect sidered simply as a change in the effective disintegration constant of the radioisotope, a change that proportional to nv:

+

nva

a

it

a

is

a

is

4.34 Activation Analysis. The high thermal fluxes in present-day piles can give rise to appreciable radioactivity even for extremely small amounts of material, and It a this fact can be made the basis of method for analysis of trace elements. method whose sensitivity varies greatly with the thermal cross sections of the elements of interest, but in a well-chosen situation the approach powerful. Assuming a flux of 10", we obtain from Eq. (2) an activity at saturation of 600 disintegrations per second for an impurity of 10 barns thermal cross section, present in a concentration of 10~4 per cent (in terms of number of atoms, or "atom per cent") in 10"' mole of Without question, this high counting rate from such slight impurity target material. The general utility of this method of analysis, that is, represents great sensitivity. "activation analysis," however, involves consideration of the cross section of the elements under analysis, the ease of chemical isolation of the desired activities, and the amounts and cross sections of other constituents that would produce similar masking activities. The cross sections and decay characteristics" of practically all the elements are known sufficiently well so that little error arises from uncertainty on this score. The ease of production and isolation of the activities of interest can be decided only from examination of the particular material undergoing the analysis and of the nature of the impurities contains. The activity desired can often be separated from mask study of the decay ing activities by simple chemical separation, together with a


o' is

- N(n»)* where

constant)

Sec. 5-2]

5-93

and the energy of the particles emitted. As the energy measurement is used purposes, it can usually be performed by simple absorption. 4.35 Measurements of Absorption Cross Sections. One method of measuring the thermal absorption cross section by means of the effect on pile reactivity was devel oped" with the first chain-reacting pile because of the importance of the method for the determination of the purity of pile materials, such as graphite, uranium, and aluminum. As neutron-absorbing impurities in these pile components were par ticularly "dangerous" relative to the attainment of a chain reaction, the procedure " method of neutron absorption measurement. came to be called the "danger coefficient As the reproduction factor of a pile k, or the pile reactivity in inhours, depends on the thermal utilization /, the effect on the pile reactivity caused by a particular sample is thus a direct measure of its neutron absorption. Changes in the reactivity of the pile can be measured either in terms of the period of the pile or by means of the position of a control rod for exact criticality In actual practice, the reactivity (k = 1.000). is best obtained by a method combining both approaches, in which the control rod is first set so that the pile is almost at criticality, that is, so that the pile power is slowly rising or falling. The period of this slow rise or fall is then measured by timing the rate of change of pile power, usually with a sensitive galvanometer connected to a boron ionization chamber in the pile. The change in reactivity corresponding to a change in control-rod position is established by a separate calibration of the control rod, consisting of observations of pile periods for different rod positions, while pile periods are converted to reactivity by the "inhour formula." The change in pile reactivity caused by a certain sample of material is obtained from the reactivity measurements just described, made with and without the sample in the pile. The sample is placed at the center of the pile to secure the maximum sensitivity, for the effect of a neutron absorber on reactivity is proportional to the square of the neutron flux at the point of absorption. The "pile oscillator" follows the same principle as the danger coefficient method, although the pile power does not reach equilibrium because of the rapid oscillation of the sample. Even though the change in pile power is a complicated function of neutron absorption when short-period changes are considered, it is nevertheless true that the amplitude of the pile-power variation is strictly proportional to the neutron absorption of the sample. If a sample is moved periodically in and out of the pile or even from one point of the pile to another and the pile flux at a particular position recorded as a function of time, the oscillating component of the pile flux will be pro portional to the total neutron absorption cross section of the sample. The fact that the position and extent of motion of the sample, as well as the position of the neutron detector (for recording of the pile power), can all be varied as desired is utilized to make the oscillator technique sensitive to various parts of the neutron spectrum. The Oak Ridge oscillator, which is operated in a part of the pile where the flux is predominantly thermal (Fig. 18), moves the sample in and out of the neutron detector, which is a hollow ionization chamber. The detector is thus sensitive mainly to the local flux oscillation, and the over-all pile power varies only slightly because of the limited motion of the oscillating sample. The Oak Ridge apparatus can be used with the pile operating at full power, hence without interference with other experiments (because the sample is outside the pile lattice), as contrasted to the Argonne design, which necessitates pile operation at a level of a few watts. The pile oscillators are extremely accurate because the circuits are arranged to detect only that component of the pile power oscillating with the period of the mechani cal motion. The limit of detectability of the Argonne oscillator, for example, is about 0.0005 inhour, a sensitivity 20 times greater than the danger coefficient method. A reactivity change of this amount corresponds, in the Argonne heavy-water pile, for example, to the presence of only 25 pg of boron. Unfortunately, the increased sensitivity of the oscillator over the danger coefficient The most important of method is somewhat offset by several interfering effects. these has to do with scattering and is caused by the change in reactivity of the pile resulting from the variation in scattering effect of a sample with position in the pile. A sample at the edge of the pile, for instance, increases the reactivity by scattering period

only for identification



6-94

EXPERIMENTAL

TECHNIQUES

[Sec.

5

neutrons back into the pile that would otherwise escape from it through the oscillator hole. At the center of the pile, however, this increase in reactivity caused by scatter ing practically disappears. The reaction rate of thermal neutrons inside the pile for a material of high cross section is sufficient to cause a measurable change in the quantity of sample present. For example, if boron is irradiated in a flux of 10" for 1 year (3.16 X 107 sec), the fraction of the B10 isotope (a = 4 X 10* b) that would disappear is given by 4

X

10'

X 10-" X

3.16

X

107

X 10"

= 0.13


The effect of the neutron irradiation would thus be a 13 per cent change in the amount of B10, that is, in the isotopic ratio of B'° to B", for the latter isotope would be essen tially unchanged. The cross section for the reaction could be determined by a com parison of the isotopic ratio of boron after a long irradiation with the preirradiation

. '

Fio.

U 18.

. '

SECTION A-A

IONIZATION LEAD

CHAMBER

The pile oscillator in operation at the Oak Ridge pile.

As a result of the reaction Bl0(n,a)Li', the Li7 is formed in an amount sufficient to be measured with a mass spectrometer; hence the cross section could also be obtained from the Li7 produced. We thus see that mass spectrometric methods can be used to measure
nucleus formed. 4.36 Transmission Cross Sections for Maxwell Distribution. Some of the first cross-section measurements made with thermal neutrons involved transmissions of the complete Maxwell velocity distribution through samples. Total cross sections could easily be measured in this way because the well-collimated neutron beams ensured conditions of good geometry. The cross section obtained is a particular average over the Maxwell distribution, and while it can be measured with great accuracy, its interpretation as a cross section at a particular velocity is difficult. The development of velocity selectors, which allow cross-section measurements at discrete velocities, made the average cross sections of less value, but the fact that many cross sections have been determined as average values and that for some


Sec. 5-2]

5-95

materials the method is still useful makes a description of the factors involved worth while.

The relationship of the observed cross section to various averages involves a careful consideration of the shape of the neutron distribution and the type of counter used in the transmission measurement. For a l/c detector, the counting rate as a function of velocity can be identified with the Maxwell velocity distribution / as the integral of the Maxwellian,

and the total counting

rate

/

-

X

const

[ 0.4

/

ev

»V '<*»

(31)

The t>o is the most probable velocity, 2,200 m/sec at room temperature. counting rate is not the integral of the flux as it would be if the detector were "black," Because most detectors are i.e., if every neutron hitting the counter were detected. 1/t), and because the identification of counting rate with Maxwellian approximately is helpful in analysis, we shall assume a l/t> detector unless specifically stated other wise. Certainly thermal beams have such high intensity that there is no necessity for using a black detector to attain a high counting rate. The observed absorption cross section has a simple interpretation when the sample used is very thin. In that case (T almost unity), the transmission equation becomes where

simply 1

-T

= Nax

(32)

— T) observed obviously gives the average (total) cross section, weighted according to the neutron distribution. For all cases of interest, scattering is small and can be subtracted from the total to obtain the absorption cross section. This latter cross section we shall call the average cross section for a Maxwell distribution, — k/v), », and it is seen to be, for a l/c absorber (o-

9

=

-\a„ Vx

2_ k

jgptg- (./..)' dv

= 1.12SV,,

(33)

a

a

a

T

a

is

is 1

it

a

is

a

is

a

is

a

is

Thus for material (and detector), the average cross section that obtained thin sample transmission measurement (with scattering subtracted) 1.128 times the commonly listed value at the most probable velocity.* Although the interpretation of very direct for transmission measurement thin sample, in practice thicker samples must be used because the statistical accuracy of the cross section obtained from very poor. For transmission near unity example, obvious from Eq. (32) that of 0.95, measured with an accuracy of If thicker sample per cent, will give cross section accurate only to 20 per cent. used, the accuracy increases being given by from

In

l

a

1

dT

1

da

\$

1

if

T

= —1 (1/e transmission), a so that In per cent accuracy in transmission leads to per cent per cent accuracy in cross section, while for a (1/e)' transmission a accuracy would result. Although the cross section accuracy increases with sample distorted an increasing amount because of thickness, the Maxwell distribution necessary to preferential absorption of the slow neutrons by 1/ti capture, and consider the effect of this "hardening" on the observed cross section s. is

it

is

1

The increasing distortion of the spectrum with sample thickness implies that the cross section measured for a particular thickness will be an average over a distribution •

This

average

theory ssr

-

cross section not to be confused See Art. 4.3 of Sec. 6-2.

»,o/1.128.

is


and the (1

with the average cross section denned in reactor

EXPERIMENTAL

5-96

TECHNIQUES

[Sec.

5

that changes shape as it penetrates the absorber; hence
OVO",


I.OO|

0.90

LOG T

Flo. 19. The "Bethe hardening curve" used to correct for the hardening distribution in a transmission measurement.

of the Maxwell

that if transmissions close to 50 per cent are consistently used, the observed cross The sections are automatically the values at v0, the ones always listed in tables. correction of an observed cross section to
T

= 0.70 (Fig. 19).

Fig.

20. The

total

neutron

cross

section


of Be from

Ref.

2.

II 3

r

t—

'

i

T -a 3

»

1 I i t

3M •

I>-

sujoq- \0

6-08

i i i

«

i

_

■ ■

ii

-

«

II Ii 1 i

1

i

N

O

05

i

1

-

/J

1

9

ii

Normol U, Corrected for UtM

1

8 1

§

S

8


1 1 1 1 i

>

3 M i ■

a

f40

N

3

•

3

u H

j3


Sec. 5-2]

EXPERIMENTAL

NEUTRON PHYSICS

5-99

become very small because of decreasing neutron wavelength, and the constant cross section there observed is the true nuclear cross section for the "free atom." At neutron energies below a critical cutoff value, the neutron cross section decreases very rapidly, actually discontinuously, at a point where coherent crystal scattering becomes impossible because the neutron wavelength exceeds twice the largest lattice spacing. The cross-section curve for beryllium furnishes a good example of another type of neutron scattering that is directly observed with monoenergetic neutrons, namely, the inelastic scattering in which energy is exchanged with the lattice vibrations. For Be, because of the small capture, the cross section at energies below the crystal cutoff This scattering, (0.005 ev), a strongly temperature-dependent scattering is observed. which is proportional to 1/v, represents gain of energy by the neutrons from the lattice vibrations, that is, the absorption of "phonons" by the neutrons. For most materials the neutron capture, also proportional to 1/v, is much larger than the thermal inelastic scattering and is the major contribution to the cross section for energies below the crystal cutoff. For these materials the major use of cross sec tions of the type now being discussed is to measure the capture cross section. The capture cross section, except for the unlikely case of a neutron resonance in the thermal region, is proportional to 1/v and independent of temperature effects, which facts aid in the identification of the neutron capture. A good example of the use of monoenergetic cross sections for the determination of neutron capture is furnished by U'", as shown in Fig. 21. Here the neutron absorption cross section is 2.75 barns of the same order of magnitude as the thermal scattering (which is about 8 barns). The crystal effects are seen in the energy region just above the crystal cutoff, which is at about 3.5 X 10_* ev. Below this energy the 1/v cross section observed is practically independent of temperature because the capture is shown much larger than the inelas tic scattering. The value of the absorption cross section at 2,200 m/sec is accurately obtained by extrapolation from the straight line shown below the crystal cutoff, this being a more accurate method than to attempt to subtract the scattering from the total at 0.0253 ev. In other important cases, such as U"6 and gold, the same method has been used, that is, extrapolations from the energy region below the cutoff, to obtain accurate absorption values at 2,200 m/sec. In the extrapolation it is neces sary to take into account possible deviations from 1/v behavior caused by resonance levels in the neighborhood; in the cases mentioned these corrections are only of the order of 1 per cent and are easily made. The crystal effects are a manifestation of diffraction of neutrons. These effects can, in fact, be used to gain information on the structure of the particular crystal lattice; however, crystal-structure analysis by means of neutrons can be done much better by a direct study of the angular distribution of monoenergetic neutrons scat

from crystals. These latter studies constitute the very extensive modern field "neutron diffraction," which is being actively pursued at all existing research reactors. The crystal-structure information gained from neutron diffraction has very little direct effect on constants needed for reactor calculations but is one of the most fruitful applications of reactors to basic research. Another effect in the field of "neutron optics" is the refraction and reflection of neutrons. It is possible to reflect neutrons from highly polished mirrored surfaces at grazing angles, and the method is an extremely potent one for accurate determinations of fundamental nucleai interactions, such as the neutron-proton and the neutron-electron interaction. Again, as for neutron diffraction, these results are primarily applications of the high neutron fluxes available in research reactors to basic research and have little application to tered

of

reactor design.

REFERENCES B. T.: Article on The Neutron in Segre (ed.), "Experimental Nuclear Physics," vol. 2. John Wiley & Sons, Inc., New York, 1953. Hughes, D. J., and J. A. Harvey: " Neutron Cross Sections," Report BNL-325, Govern ment Printing Office, Washington, D.C., July, 1955. Rutherford, Chadwick, and Ellis: "Radiation from Radioactive Substances," pp. 8-30, Cambridge University Press, London, 1930. Hanson, Taschek. and Williams: Revs. Mod. Phya., 21: 035 (1949).

1. Feld. 2. 3. 4.

EXPERIMENTAL

5-100 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22.


23. 24. 25. 26. 27. 28.

TECHNIQUES

[Sec.

5

Barschall, H. H.: Revs. Mod. Phya., 24: 120 (1952). Hibdon, Langsdorf, and Holland: Phya. Rev., 86: 595 (1952). Henkel, Cranberg, Jarvis, Nobles, and Perry: Phya. Rev., 94: 141 (1954). Ncreson, N., and S. E. Darden: Phya. Rev., 89 : 775 (1953), 94: 1678 (1954). Walt, M., and H. H. Barschall : Phya. Rev., 93: 1062 (1954); Willard, Blair, and King ton: Phya. Rev., 98: 669 (1955); Walt, M., and J. R. Beyster : Phya. Rev., 98: 677 (1955). Whitehead, W. D., and S. C. Snowdon: Phya. Rev., 93: 114 (1953). Huber. P., and E. Baldinger: Helv. Phya. Acta, 25: 435 (1952); Fowler, J. L., and C. H. Johnson: Phya. Rev., 98: 728 (1955). Beyster, Henkel, Nobles, and Kister: Phya. Rev., 98: 1216 (1955). Rosen,

L.:

Nucleonica,

11(7): 32 (1953), 11(8): 38 (1953).

Kiehn, R. M., and C. Goodman: Phya. Rev., 95: 989 (1954). Barschall, Rosen, Taschek, and Williams: Reva. Mod. Phya., 24: 1 (1952). Burtt, B. P.: Nucleonica, 5(2): 28 (1949). Graves, E. R., and R. W. Davis : Phya. Rev., 97: 1205 (1955). Hughes, D. J.: "Pile Neutron Research," Addison- Wesley Publishing Company, Cambridge, Mass., 1953. Novey, T.: Unpublished Oak Ridge work, 1945. Anderson, H. L. : Unpublished Argonne work, 1944. Borst, L. B., and V. L. Sailor: Rev. Set. Instr., 24: 141 (1953). Melkonian, Havens, and Rainwater: Phya. Rev., 92: 702 (1953). Blatt, J. M., and V. F. Weisskopf: "Theoretical Nuclear Physics," pp. 328, 391-394, 426, John Wiley & Sons, Inc., New York, 1952. Seidl, Hughes, Palevsky, Levin, Kato, and Sjostrand: Phya. Rev., 95 : 476 (1954). Creutz, Jupnik, Snyder, and Wigner: /. Appl. Phya., 26 : 257 (1956). Nuclear Data, Natl. Bur. Standards Circ. 499, 1950, and supplements. Anderson, Fermi, Wattenberg, Weil, and Zinn: Phya. Rev., 72: 16 (1947). Bethe, H. A.: Reva. Mod. Phya., 9(69): 135 (1937).

6-3

MEASUREMENT OF REACTOR CONSTANTS BY


Herbert Kouts The physics program associated with reactor core design consists of closely coordi nated theoretical and experimental work. This is particularly true when novel features are being incorporated into the system. Separately neither calculation nor experiment is a good enough basis for design. The inadequacy of theory unassisted by measurements results from a combination of difficulty of mathematics and lack of physical data. The processes which take place in a chain-reacting fissioning system are well understood qualitatively, and the equa Yet these tions which describe the neutron cycle can be written down exactly. equations contain constants and functions which are not known well enough and which it is very difficult to measure with the necessary accuracy. Some examples are the energy dependence of inelastic and elastic scattering cross sections and the angular distributions of the scattered neutrons. Even if the detailed knowledge of the necessary cross sections, etc., were available, the problem of solving the equations describing the neutron cycle would be formidable, requiring an inordinately large amount of high-speed computer time for any suffi ciently accurate treatment. The existence of these two barriers to a purely theoretical program (complexity of equations and lack of detailed data) dictates the present way of attacking reactor On the one hand, the theoretical approach is simplified by approxi physics problems. On the other mations to the basic equations and the data which are used in them. hand, the experimental program is formulated in such a way as to lead to quantities of design interest which can be compared directly with the calculations. Each of these two approaches serves to improve the other, and agreement between them is The combined purpose is the understanding of sought as a confirmation of both. the physics of the reactor core in the region of engineering feasibility so that a precise design point may be chosen. It has not been unknown for a preliminary calculation of the critical mass of a reactor to be wrong by as much as a factor of 2 if the theoretical work could not be The theoretical migration compared directly with experiments on similar reactors. area may be wrong by as much as 10 to 20 per cent. Calculational errors of this sort are usually worst if the reactor does not consist of a single region of uniformly mixed fuel and moderator. Critical mass calculations are also less reliable if fc„ is not much greater than 1. The kind of experimental physics program which must accompany the design of a In connection with a relatively low-powered reactor depends on the circumstances. research reactor, it is generally sufficient to measure the critical mass, temperature Such a coefficient, and control-rod characteristics and to find a few detailed flux plots. program is simply carried out with a critical assembly mock-up of the proposed design. On the other hand, the physics program for a high-flux power reactor would necessarily be much more complex, including a number of measurements intended to provide such information as transient behavior, neutron economy, and the lifetime of the reactor core. In the following, discussion is limited to thermal reactor systems; in other words, to those in which most of the fissions are caused by neutrons with energies below 1 ev. 6-101

EXPERIMENTAL

6-102

TECHNIQUES

[Sec.

5

However, many of the techniques are easily modified to apply to intermediate and fast reactors.

EXPERIMENTAL

1

METHODS

Practically all reactor physics measurements involve determinations of neutron flux levels. Usually it is the thermal flux which is measured, but sometimes neutrons of other energies are studied. 1.1

Thermal Neutron Detectors

Thermal neutrons are readily detected by activation methods. All substances decay. capture thermal neutrons to some extent and most of them subsequently The amount of radioactivity so induced during a controlled exposure is a measure of the neutron flux intensity. The half-lives and activation cross sections of several elements commonly used for this purpose follow:

Element

Gold Dysprosium


Copper Manganese

Half-life

54. 14 min 2. 69 days 2.42 hr 12 .8 hr or 4. 34 min 2.59 hr

Activation cross section, barns

139 94 760 3.0

0.6 12

Foils of indium or gold are most commonly used, because these elements are readily available in high-purity metallic form, are easily rolled to a uniform thickness, and have relatively large cross sections. The last consideration is important because the flux level is often low, and so a sensitive detector is needed. Both gold and indium have strong low-energy capture resonances, indium at 1.41 ev and gold at 4.9 ev. Therefore resonance capture occurs as well as thermal capture. To separate out the thermal absorption requires making a cadmium difference measure ment. Two identical foils are exposed in the same place, one bare and the other shielded entirely by cadmium. Cadmium has a high-absorption cross section for neutrons with energies below about 0.4 ev and a very small cross section for higher energy neutrons (including those at the above-mentioned resonances). Therefore the activity induced in the bare foil is caused by the capture of both thermal and resonance neutrons, while the activity of the cadmium-covered foil is produced by resonance neutrons only. If the two foils are given identical exposures, the difference in their activities is to be attributed only to the thermal component of the flux. Dysprosium is also a very useful detector for thermal neutrons because of its rela tively high cross section and its low resonance activation. The most important resonances are at 1.7 and 5.5 ev, but they absorb weakly. Thus the radioactivity induced in a dysprosium foil is caused for all practical purposes by thermal neutrons In any particular case, this fact can be checked by a cadmium difference only. measurement. The major disadvantage of dysprosium is that it is not available in metallic form. The oxide may, however, be molded into plastic sheet (polyethylene or methyl methacrylate), and foils punched from this. Such foils are less durable than metal ones and so must be handled carefully. Moreover, they are not com pletely uniform in their dysprosium content and must be intercalibrated. Manganese and copper are used as detectors when the neutron flux is high. The activities measured are those of the emitted 0 rays or y rays. The former are usually detected by Geiger counters, cased in lead boxes so as to reduce background counts from cosmic rays and other sources. The lead must be on the order of 2 in. or more thick. Then the background seen is due mostly to natural radioactivity in the In order to reduce this, a special Japanese lead which is counter walls ar in the lead.

Sec. 5-3]

measurement of reactor constants

5-103

When well shielded, than most from radioactive decay products may be used. average Geiger counter will register about 20 to 40 background counts per minute, A Geiger counter has an insensitive time immediately after depending on its size. This time, response to a /S ray, during which period it cannot register another count. which is on the order of 1 msec, places an upper limit on the counting rates which may be usefully used. The counter used is Beta rays can also be detected by proportional counting. essentially the same as that for Geiger counting, but the high voltage applied is less. The pulse height (peak voltage of a pulse) is less from a proportional counter, and The insensitive time is only a few micro auxiliary amplifying equipment is needed. The sensitivity of a propor seconds, and so higher count rates can be tolerated. tional counter is usually more dependent on electronic stability than that of a Geiger counter. Use of the former therefore requires more care in selecting the components of the counting system and in guarding against the effects of changes in sensitivity. Gamma rays are usually detected with scintillation counters. These can be used only as proportional counters. Gamma counting has a number of advantages. Whereas 0-ray spectra have a continuous energy distribution, -y-ray spectra contain sharp lines each associated with a particular mode of decay of a single isotope of the substance whose decay is being used. Gamma rays of these precise energies can be isolated by means of a single-channel or a multichannel analyzer. One can there fore be sure that the decay being observed is the one desired, without interference from impurities or other isotopes of the substance used as a neutron detector. The disadvantages of -y-ray counting are again associated with the necessary stability of the electronics. The high voltage to the photomultiplier tube must be stable to about 1 part in 10* in order to limit counting variations to about 1 per cent. Thus instrumentation for this kind of counting is definitely more costly than that used for 0-ray counting. Detailed discussion of the design and use of radiation counters can be found in the works listed as Ref. 1. The activity A of a foil is loosely defined to be the count rate it produces with some counting system. Thus the activity depends on the type of counter and the counting The saturated activity A, is defined as geometry, as well as the decay rate of the foil. the activity which would have been observed immediately after an infinite exposure. The two are related by the expression freer

where X is the decay constant of the foil (half-life divided by log, 2), T is the period of exposure to the neutron flux, t is the time elapsed between exposure and the beginning of the measurement of A, and r is the duration of the counting.

The ratio of saturated thermal neutron induced activities

of two identical

foils

counted the same way in the same counter is equal to the ratio of the neutron fluxes which these foils were exposed to. Therefore the shape of the spatial neutron flux distribution over a region can be found from the saturated activities of identical foils exposed there.

Most reactor physics experiments depend on measuring relative fluxes rather than flux. Thus it is usually sufficient to reduce measured activities to saturated activities. The saturated activity depends on the thickness and area of the foil. The variation with the latter is essentially linear; that is, doubling the foil area also doubles the saturated activity. The variation of saturated activity with thickness however, much more complicated, depending on the blackness (per cent absorption) of the foil separately to thermal and resonance neutrons and also on the self-absorption of the radiation being detected. shows a variation of saturated activity as Figure measured with an indium foil and a Geiger counter.*'* It seen that as the foil thick ness increases, the curve passes through a maximum and then levels off. For relatively is,

absolute

is

1


the


refer to References


5-104

EXPERIMENTAL

TECHNIQUES

[Sec.

5

large thickness, the activity which is observed is caused only by neutrons which have struck the side nearest the counter. The existence of a maximum in the curve is important; it represents an optimum in the region of which no corrections need be made for difference in thickness of foils. In some cases, counters can be used directly for thermal neutron flux measurements. A measurement of the spatial variation of flux found with a counter usually takes less time than one made with foils, and the data obtained do not have to be reduced to saturated activity. The counters commonly used depend on fission or on the (n,a) decomposition of B10. Those of the former kind contain fissionable material (such as U*") as a coating on the walls or collector wire, and 100 300 *ne Pu'ses counted are caused by the MG/CM2 , -i fission fragments. Boron counters can be ^, , T ja ■ j •» , saturated activity vs. ,, Fio. 1. Indium-foil made in the same way, but usually iU the thickness boron is part of the filling gas, as BFS. The pulses are bursts of ionization caused by the Li' and He4 disintegration products of thermal neutron capture. Usually BFj-filled counters have more sensitivity than fission counters of the same The reason for this is the short path length of fission fragments through matter; size. only those from uranium within a few microns of the surface can get out. This is the origin of another feature of fission counters. The energies of the fission fragments in the gas depend on the amount of uranium they have passed through, and so there is a continuous range of pulse heights. Therefore fission chambers usually have poor On the other hand, the path length of fission fragments in the gas is small, plateaus. too, and this is conducive of small size. After 1.11 The absolute neutron flux can be measured with a gold foil as follows. exposure to the neutron flux, the foil has a saturated decay rate equal to its capture rate:


...

D.

= Naa4,

(2)

where N is the number of atoms in the foil, is the flux. The foil is simultaneously ^-counted and -y-counted, and the p-y coin The 7 ray used is the 0.411-Mev one, which follows cidence rate is also recorded. the p decay. If the three saturated activities are respectively Ap, Ay, and Apy, then

D.

=

(3)

can be found from Eqs. (2) and (3) and the known values of AT and is the thermal cross section divided by 2/ s/tt = 1.128. If significant epithermal flux is present, a cadmium difference measurement must be made. The analysis just described is strictly correct only for very thin foils (about 1 mil or less). Otherwise a correction must be applied to account for the local depression

of the thermal flux by the foil.' 1.2

Fast-neutron

Detectors

The fast-neutron flux (energy > 1 Mev) can be measured by means of a fission counter coated with highly depleted U2*8 or a similar fast-fissioning substance. Cross sections in this range are low, and so a fast-fission counter is relatively insensitive. Other instruments which serve similar purposes depend on the ionization caused by One such type is the protons which undergo recoil from fast neutron collision.


Sec. 5-3]

5-105

Another is the Hornyak hulton, which is a tablet of mixed methane-fdled gas counter. The recoil protons in the plastic cause the zinc sulfide to plastic and zinc sulfide.* fluoresce, and the light flashes are detected by a photomultiplier. 1.3

Catcher Methods

Some measurements involve finding the rate at which fissions take place in uranium. Inserting a fission counter would require leaving a void and thus disturbing the quantity to be measured. The fission rate can be found another way : by means of catcher foils. These are simply foils placed next to a clean uranium surface. The escaping fission products stick to the foil (actually penetrate into it), and their activity can be counted.

The catcher foils display a background activity due to neutron absorption, and this must be corrected for. With careful choice of material, the background correction can be kept low. In various cases, acceptable catcher foils have been made of highpurity aluminum,6 fluorcarbons,' and even some filter papers.' 2

SLOWING-DOWN AND DIFFUSION MEASUREMENTS

Design calculations involve the neutron slowing-down and diffusion properties of The determination of these forms an important group of Thermal Diffusion

2.1

Length

The Several methods have been used to measure the thermal diffusion length Li». two which have been most common and which are most reliable depend, respectively, on an observation of the spatial variation of neutron flux from a steady source8 and on the timewise variation of the flux following a pulse of neutrons.* The first method requires a , COLUMN FACE thermal column at a reactor. I ^THERMAL A thermal column is simply a large mass of

_

V AiJo t-1

(at

oo

=
-

-j

material with good neutron-moderating proper SAMPLE ties and with a very low absorption cross | | [ section. Most consist of graphite, are about 5 ft square and 5 ft thick, and penetrate the shield of the reactor. Pile leakage flux enters vJLj.PEDESTAL | on one side, and those neutrons which diffuse all the way through are so well moderated as to Fio. 2. Physical arrangement for measuring L. be almost in thermal equilibrium with the graphite. Figure 2 shows the physical arrangement used in a spacial-variation measurement. The sample shown is made of the material whose Lth is being measured. The thermal neutron flux in its interior can be written as

exp

(-tmz)

(4)

d is

is

.

Jo

^2.405

—I—

,)

ft* =

A

R

where A,- and m,- are constants, the a< are the successive roots of the zero-order Bessel the "extrapolation distance" (see the radius of the sample, and function, The purpose of the graphite pedestal later article on Thermal Diffusion Constant). Thus a little above the . . much smaller than Ai. to make all the At, Ai, pedestal-sample interface the flux has the form

is


materials used in reactors. measurements.

exp

(-mi*)

(5)

EXPERIMENTAL

&-106

TECHNIQUES

[Sec.

5

The spatial variation of neutrons

is measured along r and z separately, and the results The results of this analysis are are least squares fitted to the functions in Eq. (5). values of R + d and of mi. The diffusion length is found as

«-[(fS)'+-r Ai does not appear in Eq. (6), only relative neutron fluxes are used. Some precautions which must be taken in the measurement are: 1. The optimum value of R must be approximately La,. 2. Thermal neutrons must not be allowed to enter the side of the pedestal or the sample. It is best to cover these with cadmium. The correct 3. The axial flux variation must be shown to vary as the exponential. Since

expression

is more nearly like


sinh mi (A + d

—

z)

(7)

where h is the sample height. If there is any doubt, Eq. (7) should be used. 4. Graphite preferentially passes some neutrons of energies above about 5 Mev. Enough of these can penetrate the thermal column to be seen superimposed on neu The effects of such a high-energy component trons which enter at lower energies. can be eliminated by the measurement of two sets of spatial distributions, one taken with cadmium between the pedestal and the sample. The difference in the flux dis tributions is then caused solely by thermal neutrons which have passed through the Use of this technique also eliminates the need for cadmium pedestal-sample interface. difference measurements if the spatial flux variation is obtained with foils. Thus the 5. The analysis assumes that no neutrons are produced in the sample. measurement cannot be used with a material in which fissions could take place. The second method of measuring L(a is carried out with a pulsed neutron source Immediately after a burst of neutrons (such as a Cockcroft-VValton accelerator). into a substance, the time-dependence of the thermal flux is given by exp

with B,' the so-called geometrical buckling and lo the thermal neutron lifetime. The latter is (for \/v absorbers) the absorption mean free path for neutrons of velocity 1 cm/sec. The geometrical buckling is a function depending on the shape and size of the sample; its form for several simple shapes is given in Table 1. Table

Geometrical Buckling for Several Simple Shapes

1.

Sphere:

B„.=

(—;--)'

Rectangular .olid: Cylinder: R = radius; a, 6, t ■=lengths

/V

=

(~)'

- (±™)' \R + d/

+

+

(fffc)'

f-^Y lit \h

+

(^)'

+

of edgea; h = height;

d = extrapolation distance.

Thus if the decay constant m

1

+ L.h'R.' In

(9)

of Eq. (8) is measured for several values of Rg* and is plotted against B,1, the result is a straight line whose slope is Lth'/lo and whose intercept is (V1The ratio of these two numbers is the thermal diffusion area, which is L,^. It should be noted that the measurement also yields the thermal neutron lifetime and hence, in principle, the thermal neutron cross section of the material. The Russian and Swedish measurements listed in Ref. 9 do not show the linear


Sec. 5-3]

5-107

of Eq. (9). These workers have assumed that the deviations are caused The precise origin of the difference is phenomenon called "diffusion cooling." still uncertain. variation by a

2.2

Thermal Diffusion Constant

2.21 The thermal capture cross sections of most Found from Diffusion Length. The elements are now well known and are reproduced elsewhere in this handbook. thermal diffusion constants of such substances (or mixtures of them) can be found from the diffusion length by means of the relation

-

Da

(10)

cross section Z„ used in Eq. (10) is the value averaged distribution. This means (for l/v absorbers) the 2,200-m /second

The macroscopic absorption over

a

Maxwell

divided by the factor 2/y/r — 1.128. For substances whose cross sections do not vary as l/v, the Maxwell average can be taken by multiplying the 2,200-m/sec value by value

/

(11)

1.128 The so-called from

/ factor

l/v; it has

is a measure of the amount by which the cross section deviates been evaluated in a number of cases and is given in the cross-sectional


tables.

2.22 If 2« is not known well, A* can still be found from a Found from d or X. measurement of the diffusion length. As can be seen from Eq. (5), the radial flux variation does not vanish at the boundary of the material but extrapolates to zero at a greater distance R + d. The difference between this extrapolated radius and the true physical radius is called the extrapolation distance. Curve fitting the radial flux variation therefore leads to a value of d, which is approximately related to the

diffusion constant by Da.

-

0.47d

However, this measurement is not very accurate. obtained from the expression

D

=

(12)

Better values can generally be

HX,

(13)

where Xi is the transport mean free path found from cross-sectional data. 2.3

Neutron Age

The energy can be large.

losses which fast neutrons undergo on collision with very light nuclei Thus moderators containing hydrogen or deuterium do not slow down neutrons in anything like a continuous way, and strictly speaking the concept of neutron age does not apply to them. There is a related number, however, which has physical significance in every case and which reduces to the age when the latter is meaningful. If a point source emitting neutrons of energy Ea is placed in an

(essentially) infinite block of moderator, the spatial neutron distribution The quantity

all lower energies.

r(Eo, E{) is

-

^7T«

will contain

(14)

called the second moment of the neutrons with energy E\, and for heavier modera

it is identical with the age from E<, to Bi. The quantity r,1 is the mean square distance of neutrons with energy E\ from the point source. At the risk of causing It clearly depends on the moderator properties, confusion, we shall call r the age. as well as E<, and E\. If the point source emits neutrons of several energies or even a continuous spectrum, tors

ri* can still be measured. The second moment is then an average neutrons from birth to energy E\.

age of the source

EXPERIMENTAL

5-108

TECHNIQUES

[Sec.

5

The useful quantities in reactor design are the average ages from fission neutron to thermal for moderator materials. energies These are not easy to measure. What can be found, however, is the age from fission to the gold or the indium resonance energies. Such a measurement proceeds as follows.10 A large mass of the moderator material is placed on a thermal column neutron source. A small amount of fissionable material, approximately a point, is inserted in the moderator near the thermal column interface. The saturated activities of cadmium-covered foils are found as a function f(r) of the distance above the source. The quantity rr„s is then evaluated as

(15)

t,„ follows from Eq.

(14).

The contribution to the neutron age between the resonance and thermal energies can be estimated theoretically. It is always a small fraction of the total age, and so any error which it introduces is not very significant. The age to resonance of neutrons from other sources can be found by the same methods as above. For instance, there have been a number of measurements involv ing Ra-Be and Po-Be neutrons.11 Of course, a thermal column is unnecessary in


these cases.

Some necessary precautions in an age experiment are as follows: If the source is small, r can be taken as the distance from its center. Otherwise, the zero of r must be estimated from calculations too lengthy to include here. Also, if the source is massive, it may cause appreciable inelastic scattering of the neutrons which it emits. The measured value of t will then apply only to those neutrons which escape the surface. The measurement is not valid if fissionable material is distributed through out the moderator, because then the source is no longer a point. 3

EXPERIMENTS WITH REPRODUCING ASSEMBLIES

The possibility of establishing a chain reaction through thermal neutron fission into existence when fissionable material is mixed with a neutron moderator. In such an assembly, the neutrons from one fission will induce subsequent fissions The average number of such second-generation by being absorbed in uranium. fissions is called k,//. If k,tl is exactly 1, eacli fission has a probability of causing one other the next time around, a sustained chain reaction takes place, and the assembly If ktll is greater than 1, the assembly is supercritical and is then said to be critical. the fission rate and neutron flux increase with time. If k,f/ is less than 1, the assembly is subcritical. In any reproducing assembly there is always some background level of fissions. Some of these are spontaneous fissions, and some are induced by extraneous neutrons from such sources as cosmic rays. Each such event in a subcritical system leads on the average to a total number of fissions in all generations: comes

(16)

M

is called the neutron multiplication. The factors which determine the magnitude of kef/ are the concentrations of the substances in the assembly, their physical arrangement, their neutron cross sections, The size enters through the geometrical buckling Bff5 and the size of the structure. k,lr monotonically increases as B„s decreases (increasing size). The (see Table 1). value which kelf would have for B„2 = 0 is called kx (infinite size). If fc„ < 1, then the assembly could not be made critical in any size. If fc„ > 1, criticality can be achieved at some finite value of B,*. This is called the critical buckling B'.


Sec. 5-3]

5-109

Most reactor physics measurements are intended to provide information on the reproducing properties of pile cores and must therefore be carried out with reproducing assemblies. Critical experiments are made at k,// = 1. Exponential experiments are done with assemblies with k,ff < 1. 3.1 3.11

peater. for heat

Critical Experiments'*

Hazards.1* A critical assembly is a reactor core loaded to its critical size or It is distinguished from a full-scale reactor only in that it has no provision removal, is not shielded for high-power operation, and is usually not built for

permanence. CONTROL ROD POWER

VIBRATING REED GALVANOMETER

ION I CHAM kMBERf

METER RELAY

FLUX

LEVEL TRIP

RECORDING GALVANOMETER


ION CHAMBER

LOG

AMPLIFIER RECORDING GALVANOMETER

PREAMP

FISSION COUNTER

FISSION

PREAMP

COUNTER

LINEAR AMPLIFIER

3.

METER RELAY


COUNT RATE

CIRCUIT

SCALER


LINEAR AMPLIFIER

COUNT RATE CIRCUIT

SCALER


rtW Fio.

PERIOD

CIRCUIT

Typical instrumentation for

a

METER FLUX RELAY LEVEL TRIP

METER

RELAY

FLUX LEVEL TRIP

critical-assembly facility.

Since a critical assembly has potential hazards similar to those of a going reactor, it must have most of the same kinds of radiation instrumentation that a reactor has. There must be flux-level detectors, pile-period circuits, and control and safety rods. There must be provision for rapid automatic shutdown (scram) of the assembly on the receipt of danger signals from the instruments, and manual shutdown must also

be possible. All instruments must be designed to be as nearly "fail-safe" as possible; that is, under any foreseeable failure such as loss of electrical power or equip ment malfunction, the assembly must be taken subcritical quickly and automatically. A block diagram of a typical electronic system used with critical assemblies is given in Fig. 3.

All operations which involve criticality

hazards must be approved in advance by The procedures which must be followed in order Energy Commission. to obtain Commission review are given in Chap. 8401 (Reactor Safety Determination) of the AEC Manual. The principal topics which must be covered in the required hazards report include the nature of the operations, the physical arrangement, hazards and safety features, and the possible consequences of the worst conceivable accident. The initial approach to criticality must be made carefully, so as to avoid an acci the Atomic

EXPERIMENTAL

TECHNIQUES

[Sec.

5

The precautions taken depend on the kind of assembly, i.e., solid dental runaway. In any case, fuel with solid moderator, solid fuel with liquid moderator, or solution. the guiding principle for assuring safety of operation is that as many operations as The effect of each such step possible which add reactivity should be done remotely. Estimates of the critical size should be made at every stage should be measured. from the data which have accumulated. All, however, The safety rules for initial loadings vary from one location to another. include most of the following features: 1. The primary responsibility for safety is limited to at most two people. When one person is responsible, he must personally order or take all actions which can If more than one are responsible, they must increase the reactivity of the system. agree to take these steps. 2. The initial and second fuel loadings are low enough to be certainly safe. 3. After each step in the loading, count rates and/or ion chamber currents of the These are plotted on a "critical approach curve" flux-level detectors are measured. The curve is made the basis for (see below) to give an estimate of the critical size. deciding the amount of fuel to be added in the next step. 4. Above a predetermined value of k,// (usually about 0.99) the upper limit on the stepwise addition of fuel is set as some fraction of the control available in the safety rods. 5. A neutron source (e.g., Ra-Be) is kept in' the assembly as long as it is needed to provide reliable response from the flux level detectors. 6. Fuel is added only when at least one safety rod is withdrawn from the assembly and ready to trip. Thus an accidental overloading of fuel can still be compensated for. 3.2

Critical Approach Procedures

As an assembly is loaded above about 0.98, the interior flux varies with the size approximately as oc

B,'

--

(17)

B'

Thus a plot of B,' against This provides a prediction of

is a straight line which intersects the axis at B„* = B! the critical size. The critical approach procedure and the analysis of the derived data, of course, The practices for the three most com depend on the kind of assembly being used.

mon kinds of assembly are: The solution of fissionable material in light or heavy water 1. Solution Beaclor. is pumped into an assembly tank. If the tank is cylindrical (as is usually the case), only the solution height varies as fuel is added. Thus the variable part of the geo metrical buckling is [tr/(h + 2d)]', and

"

trs)'

(18)

The quantity to be plotted against

\h

1

(m-V

is

the critical height.

"

+

is

where

ft.

-

then

Id)

it

is

is

is

is,

is

a

it

ft.

d

The quantity must be estimated on theoretical grounds. The predicted 'value of however, insensitive to this choice. assumed that Solid Fuel, Liquid Moderator. For purposes of illustration in the form of rods and that the loading approximately cylindrical. the fuel Since would be very difficult to change the tank radius to match the fuel loading, an effectively the assembly generally takes place in very large tank. Thus there 2.


6-110

Sec. 5-3]


"infinite" liquid reflector

(see

later).

The variable part of B,1 is then

M»Y +s/

(R

(19)

The value of Rc indicated by the inter plotted against (R + «)"*. not very sensitive to the choice of the reflector savings s. For reasons of safety, fuel is added only while the tank contains no moderator. The assembly room is then cleared of people, and the moderator is pumped in slowly until its predetermined height is reached. The flux levels upon which the approach curve is based are then measured. This procedure avoids the chance that personnel will absorb excessive radiation because they have accidentally added too much fuel. 3. Solid Fuel, Solid Moderator. The re activity can be increased either by adding fuel to a fixed-size matrix of moderator or by adding complete lattice cells of fuel and The former can be moderator at once. illustrated by a loading of uranium rods q-i into a graphite cube. If the loading is kept nearly cylindrical, Eq. (19) applies and the critical approach curve is deter mined as in case 2 above. The form of the critical approach curve depends to some extent on the neutron source used and its location and on the position of the detectors used to determine flux levels. At low neutron multiplication a separate source is required (e.g., Ra-Be). (a) The best position for this is the center of the assembly. Neutron detectors must be placed where they can sample the over-all neutron flux of the assembly without hav ing too high a fixed background level from the source. They must be located in such a way that they are sensitive to changes in the neutron multiplication of the assembly. 9 If these conditions are not met, the critical approach curve will have the shape shown in Fig. 4a. Such a curve is unsafe, because at every point the extrapolated critical size is an overestimate, and this can lead to adding too much fuel in any one step. o, The other kind of shape an approach curve may have is shown in Fig. 46. The (b) associated detector positions are outside riu 4. Critical approach curves, the assembly and exposed to neutrons from one of its bare (unreflected) surfaces or inside the assembly, reasonably far from the source. A critical approach curve with this shape is conducive to safe loading intervals. An approach curve which is straight from low to high neutron multiplication is hard to achieve. The deviation from linearity of Fig. 46 is caused by the harmonic content of the flux in the assembly, and this can scarcely be avoided at loadings A linear curve is thus the result of a mean between reasonably far from critical. the conditions which lead to the curves Fig. 4a and 46. If the assembly is only a little bit subcritical, the increase in flux after an addition of reactivity rises toward an asymptotic steady level with a fairly long time constant. It is then necessary to wait before making any flux measurements until this rise has and 4>~l should be

cept is


6-111

5-112

EXPERIMENTAL

TECHNIQUES

[Sec.

5

is,

At any point the simplest and surest way to be certain that criticality leveled off. has or has not been reached is to remove the neutron source. If the flux continues to rise, criticality has been exceeded. If it decreases, the assembly is subcritical. If it remains level (after a slight initial dip), the assembly is just critical. The dip in the latter case is a transient effect caused by delayed neutrons not having been in equilibrium with the prompt neutrons emitted in fission. An assembly with a flux level which is steady upon removal of a strong neutron source is really a little bit The rate at which fission chains are dying out is exactly compensated subcritical. by additional spontaneous fissions, etc. The deviation from critical in this case however, so slight as to be completely insignificant. If, on the other hand, the assembly were precisely critical, spontaneous fissions, a

etc., would continually be starting new chains and the fission density (hence neutron linear rate equal to the source strength. flux) would rise at

Excess Reactivity and Reactor Period

-

(20)

fc..

k./f

1

3.3

The quantity

p

it

is

a

is

v

I

/.

is

a

is

v

is

is

I

is is

p

is

it

is

(2./A/

a

reactor containing only fuel and =

(21)

+WB,')

!

;

heterogeneous

I

A

(i>Z„)(l

moderator has

+ XmmAm)v(l + L*B0')

(22)

where 20/ and 2arl are the absorption cross sections in the fuel and moderator and A/, Am are, respectively, the fractions of the neutron density in the fuel and the moderator. reflected reactor has a longer neutron lifetime than an unreflected one with the same kind of core. Approximately 0.75 per cent of the neutrons emitted as a result of Um thermal fission are delayed, that This appear an appreciable time after the fission.14 different for different fissioning species. Six separate delayed neutron fraction delayed neutron groups from fission have been observed. The half-lives and yields for several fissionable materials are given in Table 2. Immediately after a sudden change in reactivity, the neutron flux in a reactor under goes an abrupt variation due to the altered rate of emission of prompt neutrons. then settles onto an exponential rise or fall (depending on the sign of the reactivity given by change) with a stable period

T

It

is is,

A


p

is

is

is

is

called the reactivity or excess reactivity. Clearly negative, zero, or positive subcritical, critical, or supercritical. depending on whether the assembly Any change in reactivity followed by changes in the neutron flux in the assembly. If decreased below zero, the flux level will drop toward the value should have If by virtue of the multiplication of the source strength. increased above zero, the flux level will continue to rise until by some means the reactivity again reduced to a nonpositive value. The nature of the flux change in either case determined by the neutron lifetime in the reactor and by the existence of delayed neutrons from fission. The neutron lifetime or prompt generation lime the average time neutron spends in slowing down and diffusing in the reactor core before lost by absorption or Therefore, except for leak leakage. Most of this time spent at thermal energies. = (w2„)_1, where age, a homogeneous reactor has the thermal neutron velocity. The precise value of not important, because 2„ has very nearly 1/v behavior. Neutron leakage reduces The corrected expression

Sec. 5-3]


5-113

Delayed Neutron Yields* X 10'

Table 2.

(Neutrons /fission)

Yield Group

1 2 3 4 5 6

Half-life, sec

54.2 21.6 5.4

U1"

Pu«»

JJMI

0.62

0.67

0.25

3.63

2. 14 1.77 2.47 0 57 0. 18

1.78 1.29 2.53

0.35 4.44 6.95

3.32 7.08

2. 1

0.55 0.20

2.34 0.31

17.95 10 47

0.80 0.05

3.83

Th»"

2.08

8.38 10.84 28.85 10.65 2.20

* Based on Ref. 14 and on G. R. Keepin, private communication X,- are, respectively, the effective delayed neutron fractions and decay for the several groups. The quantities J< are the relative importances of the indicated delayed neutrons compared with prompt neutrons. All are near 1 but differ from this value because delayed neutrons are not emitted at the same energies as prompt neutrons. Thus the part played in the chain reaction by the two kinds is not the same. The values of the vary from one reactor to another. Little attention has been pven to evaluating them in any instance. Sometimes they are guessed at on theoreti cal grounds, but mostly they are taken to have the value unity. Equation (23) is the basis for experimental determination of p. The measurement proceeds as follows: The time dependence of the count rate of a flux-level detector is measured as the flux rises on the stable period. The data are fitted to the appro priate rising or falling exponential function. The result of the fit is the stable period T, which is inserted into Eq. (23) to give p. The quantity fc,// on the right-hand side of Eq. (23) can be set equal to 1. The period can also be obtained with a period meter, an electronic circuit which takes the time derivative of the logarithm of the neutron flux. Period meters have, however, too poor an accuracy to be used for good reactivity measurements. Reactivity is generally given in one of three units: k,z, dollars, and inhours. From Eq. (20)

The

ft and


constants

(24)

An inhour is the amount of excess reactivity which will lead to a 1-hr reactor period. Although the difference in importance parameters £; causes the inhour to vary to some extent from one reactor to another, the change is not. great. It is almost always suffi cient to put P = 2.6 X 10"5 X inhours (25)

A reactor which is critical on prompt neutrons alone is said to have an excess reactivity of one dollar. From this definition, it can be shown that Reactivity in dollars

p

(26)

Of course, one-hundredth of a dollar is a cent. 3.4

Thermal reactors usually

Control-rod

Calibration

use control rods which contain strong neutron absorbers boron, cadmium, hafnium). Sometimes, however, they contain fissionable material instead. Reactivity is increased either by withdrawing a poison rod or by inserting a fuel control rod. The calibration procedures about to be described refer to poiaon rods, but almost identical methods are used with fuel control. (e.g.,

5-114

EXPERIMENTAL

TECHNIQUES

[Sec.

5

Calibration curves which give the reactivity added by rod removal as a function of the amount removed are always needed. The carrying out of such a calibration requires other control rods. The procedure is as follows: First, there must be enough available excess reactivity so that the assembly can be made critical when rod / (which is to be calibrated) is in any position. There must be enough control in the other rods to hold the reactivity down to zero no matter what position rod / has. The assembly is made just critical with rod / fully inserted. Rod / is then removed a small distance X\. The associated reactor period Ti is measured. The auxiliary rods are then used to bring the assembly back to critical. Rod / is again withdrawn to a new position x%, and the period Tt is measured. The process is iterated until rod / has been fully removed. The T, correspond to reactivities pit which can be calculated with the aid of Eq. (23). Thus the reactivity associated


Figure 5 gives a calibration obtained by the above means.

200

Fio.

with removing the control rod a distance x,

I

is

curve of a Brookhaven graphite reactor control rod, The shape of the curve is quite typical.

400 600 CM ROD INSERTION

800

5. Calibration curve of Rod No. Brookhaven graphite research reactor.

Fio.

1 at

6.

Flux

transient

in

a

rod-drop

experiment.

Usually the worth of any one control rod depends to some extent on the positions of others. This phenomenon is known as rod shadowing. Such effects are difficult to compensate for, although some approach to doing so can be made by carrying out calibrations with several configurations of auxiliary rods. The total worth in reactivity of a rod can be found by means of a rod-drop experi ment.™ If the assembly is just critical and rod / is dropped suddenly from some position x, the neutron multiplication undergoes a fast transient decrease and then continues on the stable period, as has been described earlier. The behavior of the neutron multiplication in a rod drop experiment is shown in Fig. 6. The analysis requires extrapolating the stable period section of the curve back to the time when the rod was dropped. Let the flux so obtained be 4>i, and let that just prior to the rod drop be ô. Then the reactivity worth of the length x of the rod is given by
1

dk

+ ^TT

In practice, it is not the multiplication which is It is therefore necessary for flux level instrument. its response is proportional to the total neutron flux It must be insensitive to after the rod is dropped.

(27) measured, but the response of a the detector to be located where in the assembly, both before and the neutron flux dip near the rod


Sec. 5-3]

5-115

surface and to associated effects on the neutron leakage out of the assembly. result* are obtained only by trial and error. 3.5

Good

Danger Coefficient

Critical experiments can be used to find the cost in reactivity of extraneous sub stances in a reactor core. Examples of materials which must be tested are fuel rod cladding, corrosion inhibitors, structural elements (such as fuel spacing plates), and The worth in reactivity is found by means of a danger-coefficient or instruments. measurement." The danger coefficient is the decrease in reactivity associated with inserting unit This quantity varies with the position in which mass of the substance in the reactor. the substance is placed, and for neutron absorbers with large cross sections it also depends on the thickness of the material with which the measurement is made. On the addition of the material being tested, a The procedure is straightforward. control rod must be moved in order to maintain the assembly at critical. This control rod is one which has previously been calibrated. Thus the net amount of rod motion corresponds to a reactivity change which may be read from the calibration curve. The assembly Safety usually dictates that the measurement be done in two steps. is made critical with and without the sample in the core. The difference in rod positions is then converted to reactivity as above. reactivity-coefficient


3.6

Temperature Coefficient17

The quantity and hence the reactivity are temperature-dependent. A hetero geneous system with low enrichment fuel has at least two coefficients, associated If there is a reflector, there separately with the fuel and moderator temperatures. is a temperature coefficient associated with it also. The uniform temperature coefficient is the quantity

dk'ff

l9v,

where all components having an effect on reactivity

have the same temperature T.

+50r

P OT »K35'

-lOOr 10

20

30

40

50

10

•C

Fio. 7. Change of reactivity with temper ature of the Brookhaven medical research reactor. (Re/. 2.)

Fia.

8.

20

Temperature

•C

30

coefficient

40

50

of

the

Brookhaven medical research reactor.

The quantity a, can be measured in a way similar to a danger coefficient. As T is changed, its effect on reactivity is balanced out with a calibrated control rod ; or„ is then the slope of the curve of kt// vs. T. Figure 7 gives the change in reactivity with temperature of the Brookhaven Medical Figure 8 is the associated tem Research Reactor (MRR), measured as above.' perature coefficient.

6-116

EXPERIMENTAL

TECHNIQUES

[SEC. 5

The experimental separation of the uniform temperature coefficient into separate moderator and fuel components requires changing the temperature of only one at a time. This is usually very hard to do because of rapid heat transfer. Some experi ments of this nature have been tried, but with uncertain results. The temperature of the reflector also has a separate effect on reactivity, but this should be noticeable only for very small reactors. Migration Area

3.7

The migration area is found from a set of period measurements at several small values of excess reactivity." These are all performed with control rods completely withdrawn, and the reactivity is changed through the size and therefore the geometri cal buckling. The measurement is most simply analyzed in terms of one-group theory. Here

*. k-»

-

For each value of S„*, from Eq. (31)

p

r+W' *.// —

(29)

- - M*Bt*

*-

1

r k.u

1

r

km

is found from a period measurement and Eq. (23).

(30) (31)

Then

This equation differs only to higher order from M*

dp

dB,*

Equation

1

M1

+ M'Ba'

(33)

AP:

(33) can then be solved for

- - dp/dBS BAdp/dB,') 1

A single typical value of B,* is used in Eq. (34). The way in which B,1 is varied depends on the kind of assembly studied.

(34)

It

is

easy to change the reactor core height of a cylindrical homogeneous solution or of a It is simplest to change the number of loaded liquid-moderated, solid-fuel system.

channels of a solid-fuel, solid-moderator assembly. If B* has been measured previously (as with an exponential experiment), a single measurement of p leads to a value of M *. This is obtained from

"

"

M>(B>

-

/?,»)

1+ M'B'

{6b>

which is the result of combining Eqs. (29) and (31). The above analysis leads only to the one-group value of M ». If one-group theory The is not applicable, the correct critical equation must be used from the start. For instance the age-diffusion critical analysis then does not lead directly to M1. equation

■*" An assumed value of t

= (1 =

+ L*B')erB'

r+%B7+T

can be used to get

a

km

J;.


p =

+ M'B*)

(1

value of L* and vice versa.

(36) (37)

Sec. 5-3]

MEASUREMENT OF REACTOR

CONSTANTS

6-117

EXPONENTIAL EXPERIMENTS19

4

;

a

is

is

is

it

a

is

is,

Exponential experiments are performed with less than critical sizes of reactor cores. Hence they are safer and more economical of fuel than are critical experiments. The real basis for choice between exponential and critical experiments in a reactor program not that great. however, neither safety nor cost because the difference Although number of quantities can be measured either way (particularly those concerned with the neutron economy), some parameters can be found much more easily in one way or the other. Critical experiments are much more suitable for measurements involv In fact, difficult to see how these could be done with an exponen ing reactivity. tial assembly. On the other hand, exponential experiments appear to be more useful for many measurements involving flux plots. Since the neutron multiplication of an exponential assembly quite low, high flux levels are possible only with large source strengths. The source always a thermal column, supplied with neutrons either by (7,Be) or (a,Be) sources (such as Ra-Be) or by the leakage flux from reactor. The latter can provide more source strength It also makes possible this feature leads to faster and more accurate measurements. some experiments which could not be done with sources based on radioactive decay.

One of the most important

quantities measured in an exponential experiment

is

Buckling

4.1

the

REACTOR CORE

THERMAL COLUMN

Fig.

9. Unroflected

exponential

The thermal neutron flux density in the

assembly.

core can be written as

d

sinh [mi{h

+

rf)

r

(en

R

AJt

=

- z)]

(38)

l

=

i

n

^

SO

AJ0

(2.405

sinh \m(h

d

=

+

)

1

z,

The terms with where all symbols have the same meaning as in Eqs. (4) and (7). so that after some distance from > decrease relatively rapidly with increasing the source plane (usually about 10 cm),

i


is

found from the observed spatial distribution of thermal neutrons. buckling, which The procedure given here applies to an unrcflected assembly with cylindrical sym metry, with a thermal column source at the bottom (see Fig. 9).

-

z)]

(39)

6-118

EXPERIMENTAL

TECHNIQUES

[Sec.

5

The reactor equation is =

-B.

(40)

Hence, inserting Eq. (39) into Eq. (40) leads to


The measurement is quite similar to the one of diffusion length which was described earlier. The radial and axial flux distributions are found with foils or counters and are fitted to a Jt> function and a sinh function, respectively. The values of R + d and in which result are then inserted in Eq. (41).

Fio.

10.

Thermal flux in reflected assembly.

If

the assembly has a reflector, the radial flux distribution is modified somewhatenough away from the core-reflector interface, the flux is still given by Eqs. (38) and (39), but with the extrapolation distance d replaced by the reflector savings 8. The expression for B1 is still Eq. (41), but s replaces d. Near the interface, the reflec tor modifies the neutron flux in a way typified by Fig. 10. The measurement when a reflector is used is obtained the same way as when it is absent, but the points fitted to the o and the sinh functions must be measured in the region where reflector corrections are small. A rule of thumb is that these effects are not noticeable more than a migra tion length from the edge of the core. If the reactor core is heterogeneous, the flux distributions in Eqs. (38) and (39) There is also an intracell represent only the macroscopic (cell-to-cell) variation. variation which is unimportant in finding B2. The buckling measurement is as described above, but all flux values must be found at corresponding points in different lattice cells. The above analysis is readily modified to fit other geometries.

Far

J

Migration

4.2

Area

If the exponential experiment is water-moderated, there is a technique for measuring the migration area.10 The procedure used is very different from that described earlier and is based on the following considerations: In the most general case, the fc„ of a thermal reactor can be written From Eq. (29),

K

=

tpfv =

If

*vh 1

+ M*B*

(42) (431

the moderator is poisoned by a neutron absorber (e.g., boron), /and B* are affected while the other quantities are unchanged (M2 varies slowly with the poison content").


Sec. 5-3] Then, from

Eq.

(43),

/ and

B* are linearly related, and =

If

the uranium

and B1

all change.

5-119

(44)

G)i>'.-oJr>

content of the fuel can be altered (different enrichments), Then

/,

ij,

The experiment is performed with several amounts of neutron absorber dissolved water. At each stage S* is measured as above, and is measured as discussed later. A straight line fitted to the sets of points (B*,/) yields M* through Eq. (44). If on the other hand the fuel enrichment is varied, (B2,f) can be measured but ij must be taken from cross-section data. Equation (45) is then used to get M*. Other forms Equations (44) and (45) are derived from the critical equation (29). of critical equations lead to more complicated analysis. Equation (36), for instance, has two free variables on the right-hand side; these cannot be determined separately by this method. If the measurements are restricted to the region where

/

in the

km


then

L* +

t

=

M*

1

+

(L«

+ t)B«

(46)

If, alternatively,

can be evaluated as above. (1

a similar

-

+ L'B»)

«***■

(47)

analysis leads to

«

TS?

Several variations of this experiment have been applied to heavy-water-moderated systems. 6

NEUTRON

ECONOMY

Among the most instructive measurements in a reactor physics program are those leading to information on the neutron economy, which can be defined as the account ing for all neutrons emitted in fission by finding out which fraction becomes involved in each possible reaction.

A number of kinds of measurements in this category have been performed. There not space to describe them all, and so only two (thermal utilization and fast effect) are discussed here. The most important omitted topics are resonance escape proba bility," epicadmium fission fraction, epicadmium 1/v absorption in Um, and breeding is

gain.

Neutron economy measurements can usually be made with either exponential or critical assemblies. 5.1

The thermal utilization

Thermal Utilization

of a heterogeneous

tural material is

2/>/

mixture of fuel, moderator, and struc

V*/ + 2„°$nV„ +

Z.'i.V.

(49)

where 2/a, 2m", 2," are, respectively, the macroscopic cross sections of the three sub are the average #„,, stances averaged over the thermal energy distribution, respective thermal neutron fluxes, and Vm and V, are, respectively, the ratios of volumes of moderator and structural components to the fuel volume.

5-120

EXPERIMENTAL

In a homogeneous system, ponents is the same, and

/ has

TECHNIQUES

a much simpler form.

[Sec.

5

Here the flux in all com

/-—

(50)

where now the 2's do not apply to the pure substances but are evaluated per cubic Here the evaluation of/ is quite simple; it involves centimeter of the reactor core. only calculating the atom fractions, multiplying by the cross sections per atom, and No measurements are called for. inserting the results into Eq. (50). As can be seen from Eq. (49), the thermal utilization of a heterogeneous system The way now used to find is based on measur depends also on the flux distribution. The $'s can then be evaluated and inserted along ing the intracell flux distribution. with the cross sections and volume fractions into Eq. (49)." The flux in a cell of a heterogeneous lattice can vary quite rapidly with position, These are usually made and so very small foils must be used to measure its shape. of dysprosium, because, as has been mentioned earlier, cadmium difference measure The foils are placed in milled depressions in a cross-cut ments are then not needed. section of a fuel element, and others are placed in an adjacent region of the moderator. The latter set must be positioned in a foil holder, the composition of which depends In light water, methyl methacrylate can be used, on the nature of the moderator. because it matches the slowing-down and absorbing properties of the moderator Sur quite well. In D20, the corresponding deuterated plastic should be used. prisingly large errors in the observed flux distributions can be introduced by very small amounts of a foil-holder material which does not match the moderator properties (e.g., aluminum in light water). An older way of measuring /, which has been ascribed to Fermi, avoids the need for Two sets of identical indium foils are successively cross sections as in Eq. (49). One exposed in the moderator in the same plane transverse to the fuel elements. The saturated activities set is exposed in cadmium covers, and the other left bare. are found, and from these are found the distribution in the moderator of the cadmium The average values of difference activity and the cadmium-covered foil activity. these, called Ad and Ar, respectively, are measures of the thermal neutron capture rate and the slowing-down density at indium resonance in the moderator. A calibration experiment is now carried out. This consists of placing a neutron source such as Ra-Be in an effectively infinite volume of the pure moderator and finding the spatial distribution of the cadmium difference and the cadmium-covered and 8,, If the space integrals of these are called, respectively, foil activities. then the thermal utilization of the lattice is given by


/

/ _

i

_i_'*£

This method was used to evaluate the thermal utilization of the Manhattan graphite piles. 6.2

(51)

Project

Fast Effect

The fast effect of a heterogeneous reactor can be found from the relative fission rates of U"* and U"6 (the measurement in other systems such as Th-U*" is precisely If this ratio is called /i, then similar).

.

-

1

+„

~ 1 ~

(52)

vis and vS5 are, respectively, the neutrons per fission of U"* and U,M, and arts is the Suitable capture-to-fission ratio of U"8, averaged over a fission neutron spectrum. assumptions of the values of these parameters lead to e

-

1

+ 0.557*.

(53)

Sec. 5-3]


6-121

Three methods have been used to measure the relative fission rates : 1. Small calibrated fission counters have been inserted into holes in a fuel rod. 2. Catcher foils have been used." The experimental arrangement is illustrated by Foils 1 and 6 give a measure of the total fission rate in the fuel element. Fig. 11. Foils 3 and 4 give a measure of the fission rate in a highly depleted U"8 foil inserted in the fuel. Foils 2 and 5 are used to find the background induced in 1,3, 4, and 6. After the activities of 3 and 4 are corrected to the same U"8 content as the fuel element, n is

calculated from (3, 4)corri;cW
_

^g^j

An additional correction for the difference in decay characteristics of fission products U"s is discussed in Ref. 236. The experimental setup is shown in 3. Sample uranium foils are 7-counted." Foil 2 is made of Fig. 12. Foil 1 consists of the same material as the fuel element.

of U»8 and


\y

Fio. 11. Measurement of t with catcher foils.

BUFFER

FOILS

Fio.

12. e measurement based on 7-eounting.

uranium very highly depleted in U"5. The aluminum buffer foils serve to keep the depleted uranium from being contaminated by U1" or its fission products. The foils are -y-counted with the discriminator bias set to reject energies below 500 kev. In this way the bremstrahlung from Np"* |3 rays is avoided and only fission-product y rays are observed. After the activity of foil 2 is corrected for U"8 content, n is

found

as

„

--

1

2°°""'"'

(55)

2porrcctod

A correction must also be made for the difference in decay characteristics of U,S8 U"1 fission products, and this is found the same way as that mentioned in method

and 2

above.

REFERENCES The following two publications are referred to in the list respectively

as

I

and

II:

I. " Proceedings of the International Conference in Geneva, August 1955," vol. V, United Nations Publications, New York, 1956. II. " Conference of the Academy of Sciences of the USSR on the Peaceful Uses of Atomic Energy, July 1-5, 1955, Session of the Division of Physical and Mathematical Sciences," Government Printing Office, Washington, D.C. la. Rossi, B. B., and H. H. Staub: "Ionization Chambers and Counters," McGraw-Hill Book Company, Inc., New York, 1949. b. Curran, S. C, and J. D. Craggs: "Counting Tubes," London: Butterworths Scientific Publications, London, 1949. e. Staub, H. H.: "Experimental Nuclear Physics" (E. Segre, ed.). Part I, John Wiley
6-122

EXPERIMENTAL

TECHNIQUES

3. Tittle, C. W.: Slow Neutron Detection by Foila, Nucleonics, 8(6): 5 (1950). 4. Hornyak, W. F.: Ret. Set. Instr.. S3: 264 (1952). 5. Price. G. A.: "Fast Effect Measurements," BNL-1690 (no date). 6. Sher, R.: Private communication. 7. Stolyorov, G. A., et al.: II, p. 161. 8. For typical measurements see Heintze, L. R.: Nucleonics, 14(5): 108 (May, 1956), also ORNL-933, CP-3364, CP-3195, ANL-4076, HW-21793, BNL-77. 9a. Manley, J. H., L. J. Haworth, and E. A. Luebke: Phys. Ret., 61: 152 (1942). 6. Von Dardel, G. F.: Trans. Roy Inst. Technol. Stockholm, No. 75 (1954). c. Antonov, A. V., et al.: I, p. 3. d. Ramanna, R., et al.\ Nuclear Energy, t: 145 (March, 1956). 10. For typical measurements see: ORNL-181, ORNL-187, ORNL-641, CP-3453, ANL4076. 11a. Amaldi, E., and E. Fermi: Phys. Ret., 60: 899 (1936). Research, A25 : 157 (1947). 6. Munn, J. I., and B. Pontecorvo: Can. 12a. Callihan, A. D., J. W. Morfitt, and J. T. Thomas: I, p. 145. 6. Horton, C. C., and J. D. McCullen: I, p. 156. c. Snyder, T. M.: I, p. 162. d. Krasik, S., and A. Radkowsky: I, p. 203. e. Flerov, G. N.: II, p. 133. /. White, R. H.: Nuclear Sci. Eng., 1: 53 (1956). g. Bebbs, E. H., R. H. Vogt, and K. W. Downes: NRL Rept. 4729, May 24, 1956. 13. See Refs. 12. Also "Meteorology and Atomic Energy," Government Printing Office, Washington, D.C., July, 1955. 14. Keepin, G. R.: "Progress in Nuclear Energy Series," vol. 1, p. 191, Pergamon Press, London, United States ed., McGraw-Hill Book Company, Inc., 1956. 15. Snyder, T. M.: I, p. 162. 16. Hughes, D. J.: "Pile Neutron Research," pp. 196-200, Addison- Wesley Publishing Company, Cambridge, Mass., 1953. 17a. Kaplan, I., and J. Chernick: I, p. 295. b. Krasik, S., and A. Radkowsky: I, p. 203. c. Callihan, A. D„ et al.: I, p. 145. d. Horowitz, J.: I, p. 256. 18a. Horowitz, J.: I, p. 256. b. Krasik, S.. and A. Radkowsky: I, p. 203. 19a. Groshev, L. V., et al.: II, p. 39. b. Kouts, H., etal.:l, p. 183. c. Cohen, E. R.: I, p. 268. d. Persson, R.: I, p. 239. e. Davenport, D. E.: I, p. 309. 20a. Kouts, H., et al.: I, p. 183. b. Cohen, E. R.: I, p. 268. 21a. Kouts, H.: BNL-C-7573, Oct. 30, 1953. b. Egiazorov, V. B., V. S. Dikarev, and V. G. Madeev: II, p. 59. c. Krasik, S., and A. Radkowsky: I, p. 203. 22a. Mummery, P. W.: I, p. 282. b. Krasik, S., and A. Radkowsky: I, p. 203. c. Kouts, H., et al.: I, p. 183. d. Cohen, E. R. : I, p. 268. 23a. Hill, D. L.: CP-1548.

J.

J.


[SEC. 5

Price, G. A.: BNL-1960. Stolyorov, G. A., et al.: II, p. 161. 24a. Kranz, A. Z., and G. G. Smith (WAPD): Private communication. 6. Hanna, G. C, and P. R. Tunnicliffe: CRRP-623, Jan. 4, 1956. 6. c.

6-4

ACCELERATORS BY G.

1

K. Green

INTRODUCTION

Types of Particle Accelerators

1.1

Particle accelerators are constructed primarily for use in research on nuclear particle physics. They are also used for isotope production (a major use prior to the development of the reactor) and as X-ray sources for radiography and radio Commercial use of intense beams of accelerated particles for sterilization therapy. and similar processes is just beginning. Table 1 classifies accelerators and indicates particles accelerated and highest energy built or in process. Sections on each type should be consulted for further qualifications too lengthy to include in a table. Note that beam currents and energies are subject to rapid obsolescence because the accelera tor art is evolving rapidly. High-energy electrons and protons are fundamental in research, and so are the most usual accelerated particles. However, deutcrons are often used, a particles less often, and heavier ions in a few machines. In principle, any charged particle can be accel erated, but limitations are imposed by ion sources and by the practical range of chargeto-mass ratio. 1.11


and

Table

1.

Types of Particle Accelerators Particles

Classification and type

accelerated

Straight-line accelerators : Direct- voltage types: Cockcroft-Walton*

R-F

acceleration

epdah epdah

Maximum energy built or in process

4 Mev 12 Mev

types:

Electron linear accelerator* Circular or magnetic accelerators: Constant-gradient types:

ph 0

600 Mev n 1 Bev

pdah e

25 Mev pd 300 Mev 5-10 Mev 700 Mev p 300 Mev* 10 Bev

t

pdah e P

Alternating-gradient types:

ep ep (other?) Particles: e, electron; p, proton; d, deuteron; a. a particle; k, heavier ions. Maximum energy stated is an approximation (1956). 'Commercially available (but probably not at maximum energy), t Limited commercial availability, or built on contract, t I Bev for electron racetrack.

6-123

6 Bev «, 30 Bev p Bev range

5-124

EXPERIMENTAL

TECHNIQUES

[Sec.

5

Units and Formulas

1.2

All accelerators operate by interactions between fields and charged particles. Since very high velocities are involved, it is usually necessary to use the mechanics of Units and symbols in this section are defined as follows: special relativity. e = charge on the m0 = rest mass of

electron or proton, esu. e = 4.80 X 10~10 esu Electron mass is 9.10 X 10~" g; proton mass particle.

1.67 X 10-" g m = total mass of particle c =

E, T,

velocity of light,

c — 3

X

1010

cm/sec

En = total, kinetic, and rest energy of particle respectively, ergs unless stated as Mev t; — velocity of particle, cm /sec 0 = v/e p = momentum V = electric potential difference, esu or volts if stated B = magnetic field, gauss R = radius of curvature, cm unless stated as ft = angular velocity, frequency, of revolution a, ev = electron volt. Energy gained by unit charge falling through potential difference of 1 volt. 1 ev = 1.6 X 10-11 erg Mev, Bev = million electron volts, billion (10*) electron volts Generated for wjivans (University of Florida) on 2015-09-23 02:50 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

f

The fundamental mass-energy relationships are

E

En =

— mc1

ergs, g, cm /sec

moc*

giving for the rest mass of the electron and proton m,

"

9.1

X 10-"(3 = X 1.6

1010)s

X 10"

An easily remembered approximation of Mass is related to velocity by

„

n = 0.51 Mev

mv = 940 Mev

rest energy is electron \^ Mev, proton 1 Bev.

-

m0

(1

leading to the energy relationships

E

"

/, ""^.u. — (1 ti*)>*

T

=

E

- E0

- /ST52 -

= m^»[(l

II

Reducing, energy and velocity are connected by

\E/ This gives

\T

+

Ej

the momentum in various forms:

c

Figure

1 shows velocity vs. kinetic energy in general units. an electric field t a force is exerted on a particle F = et. through potential difference V, it gains kinetic energy T = eV.

In

c

If the particle falls If V is in volts, « is

number of charge units, then T will be given in electron volts. In a magnetic field the force on a moving particle is orthogonal to both v and B and is F = (e/c)v X B. The particle will travel a curved path (p and B orthogonal)

5-125

ACCELERATORS

Sec. 5-4]

with radius of curvature given by

BR At low or

= 25 =

" nonrelativistic "

velocity of revolution 01 =

~ £o')H

-

{— + 2r£°)H

energies this converts to

BR

= (2TEa)Vl/e.

The angular

and its direction

-

B

B the frequency

About an axis parallel to of

(g'

revolution

B 2* (T + E») express kinetic and rest in million electron volts; then for singly charged particles For convenience,

energy

BR

=

)4

X lO'CT1 + 2TE0)H + 2 TEo) H

= 109 ( T* .


/=

1.43S ■=— -=

gauss-cm gauss-f t .

mc/sec

2

7 .6 8 4 .5 .3 T/E„(»I0 FOR CURVE 2 )

It should be noted that the energy of a Fio. 1. Ratio of velocity to velocity of light particle is not changed by transit through vs. kinetic energy in rest masses. a magnetic field. If the magnetic field is changing with time, electric fields, which must satisfy Maxwell's equations, are set up. These electric fields will accelerate a particle (see the betatron). 2

STRAIGHT-LINE ACCELERATORS

The straight-line accelerator is one in which the particle traverses the accelerating orbit only once. The usual straight-line orbit could have bends inserted by deflecting fields if there were a good reason. The direct-voltage types apply the full accelerating voltage continuously to a long vacuum tube and are thus limited in energy by voltage breakdown. The energy obtainable can be increased by applying many successive accelerations along the orbit with r-f fields, the method of the linear accelerator. 2.1

Direct-voltage

Accelerators

The Cockcroft- Walton and the Van de Graaff have a common element — the highThe only basic difference between the two is the method voltage accelerating tube.3-' of generating the high voltage, a transformer-multiplier or cascade generator in the former, and an electrostatic generator in the latter. (The common X-ray machine is a true accelerator in this class but will not be treated in this article.) 2.11 The Cockcroft-Walton was the original accelerator used for nuclear research.4 As outlined in Fig. 2, a transformer supplies a rectifier-capacitor multiplier to provide

The multiplier circuit, shown connected for positive required d-c potentials. polarity, is evidently extendable to as many stages as are required and as can be The rectifiers can be vacuum diodes, but these are being sup properly insulated. planted by stacks of dry-disk rectifiers.6 2.12 The Van de Graaff machine, Fig. 3, has a fast-moving belt of insulating material which is sprayed with charge at the ground end and carries the charge to the The terminal rises in potential until the belt charge current high-voltage terminal. equals the beam current plus leakage. Most of the leakage is down corona points on the corona rings, equalizing the potential gradient. Variation of the corona cur rent to points facing the terminal provides fine control of voltage.8 the

EXPERIMENTAL

5-126

TECHNIQUES

[Sec.

5

Both types can be built in air up to 1 to 2 Mev, but higher energies require pressurization in a controlled atmosphere. Pressurization is usually done with dry nitrogen Either vertical or hori plus carbon dioxide in a heavy steel tank at 100 to 300 psi. zontal construction can be used, the arrangements of Figs. 2 and 3 being arbitrarily chosen from the numerous possible combinations. The accelerating tube can be a simple structure of glass cylinders containing an axial array of cylindrical electrodes for moderate voltages. At high voltages the ION SOURCE

c 3

F-TUBE

RECTIFIER CONDENSER

ELECTRODE

I

MULTIPUER-r-

oj}o

TRANSFORMER 2. Simplified

-TARGET

Cockcroft-Walton accelerator and cascade multiplier circuit.

tube takes the form of Fig. 3. Insulating rings, usually ceramic, are sealed to elec trodes which can be disks with holes or more complex spun forms. Great care is All parts must be skill exercised to obtain a nearly uniform potential distribution. fully designed, and a very clean vacuum system must be employed to avoid severe Electrons limitations by secondary currents in the tube and voltage breakdown. can be introduced into the tube from a hot filament, or positive ions from an ion source, and accelerated down the tube. When the particles reach the end of the tube, they possess kinetic energy in electron volts equal to the product of their specific CORONA

RINGS

SUPPORT COLUMN

ACCELERATING TUBE

ION SOURCE

VACUUM

PUMP — SECTION OF ACCELERATING TUBE

FACEPLATESUPPLY6

Fig.

PRESSURE TANK-

3. Pressurized

Van dc Graaff machine.

a

is

is

is

4

a

is,

The tube electrodes, at successively higher poten charge and the applied voltage. tials, are weak electrostatic lenses and focus the particles, that apply lateral forces which direct the particles toward the axis of the tube.5 Mev.' Models at Cockcroft-Walton accelerators are being built up to about few hundred kev, with beam currents of milliamperes, are used extensively for neutron Any ion can be accelerated, but the energy production by the D-D or D-T reaction.' often When accelerating electrons this type small for heavy ions. per nucleon The energy readily adjustable, and beam "constant-potential X-ray set." called current limited only by ion source capability and tube breakdown induced by is


Fio.

secondary effects of high beam intensities. The Van tie Graaff has attained some 10 Mev at MIT, and many 4- to 5- Mev Compact, pressurized, 1- to 2-Mev models are convenient machines have been built.

ACCELERATORS

Sec. 5-4]

5-127

for either electrons (X rays) or positive ions. The Van de Graaff is useful for produc ing a beam of monoenergetic particles, whose energy can be controlled and stabilized to close limits. Currents of the order of 50 jua direct or a few milliamperes in short An analyzer is generally used with ions because a considerable pulses are normal. fraction of the beam consists of molecular ions which must be swept out for most research. 2.2

The Linear Accelerator10-"

The early linear accelerator of Sloan and Lawrence" operated by launching particles into a row of cylinders of successively greater axial length, alternately connected to an r-f line (Fig. 4). Thus, if the particles spent one half cycle in each "drift tube" they would be accelerated by the field across each gap, since the potential would

— IONS TO RF SOURCE


Fio.

ax 4. Early version of linear accelerator.

If X is the free-space reverse while the particle was shielded inside the drift tube. wavelength of the radio frequency then the length of each drift tube must be 0X/2. The tube length must be cut for the increase in 0, and this, in turn, requires that the injection energy and r-f voltage be known for the design. This type was used to accelerate heavy ions, and various modifications are possible for low energies. How ever, the simple form has unstable particle orbits, and refinements are necessary for long machines and high energy. DRIFT TUBE

- ION SOURCE

AND

PREACCELERATOR

Fio.

RF OSCILLATOR

-TANK

Q^vACUUM

PUMP

5. Sketch of proton linear accelerator.

2.21 A proton linear accelerator is diagrammed in Fig. 5'°. The copper-lined tank is excited with radio frequency in the axial electric mode. Drift tubes along the axis

shield the protons from the electric field while it is reversed and cause the particles to be accelerated only at the gaps. Since the drift tubes are in phase, the gap spacings are ff\. In order to eliminate the impractically short gap spacings required for small energy (low value of 0) at the entrance end, the protons are preaccelerated by a Cockcroft- Walton or Van de Graaff and in jected into the linear accelerator at a few hundred kev to a few Mev. The injector energy determines the first gap spacing; then the energy gain per gap, which fixes the increase of 0, determines the successive The energy gain per gap is gap spacings. a function of the voltage gradient down ^— PHASE STABLE the tank-drift-tube structure, which, in REGION turn, is very sensitive to small mechanical Fio. 6. Phase-stable region of the linear distortions and to minute variations in fre accelerator. A proton linac for, say, 30 Mev quency. would have about 50 drift tubes, while a higher energy machine might easily run over 200. Keeping a structure in the exact adjustment necessary to preserve the phase of the protons down a long row of gaps is impractical. There is an important region of phase stability™ which will automatically compensate for small errors. Con sider the wave of Fig. 6 to be the time-varying r-f voltage on an accelerating gap.

6-128

EXPERIMENTAL

TECHNIQUES

[Sec.

5

o

The stable phase angle defines the correct voltage to match the geometrical design. If a proton arrives late at the gap, as at a, it will receive an excess of energy, its velocity will increase, and it will arrive a little earlier at the next gap. After a number of but its momen gaps are crossed, the proton will have returned to the proper phase tum will be too high, owing to the "catching up" process, and so it will cross the next The energy gain is then too small, the velocity begins to diminish, and gap early. at point b the advance in phase will cease and the too low velocity will begin to make the proton cross the gaps at later phase position — working back toward a. In this fashion a phase displaced proton oscillates in phase about the stable phase angle, provided the peak r-f voltage is greater than that necessary to match the change of 0X built into the drift tube lengths. Errors in amplitude of the radio frequency are also compensated. The protons are phase unstable on the falling side of the r-f wave. The frequency of phase oscillations is dependent on geometry and voltage distribution In a proton linac of, say, 50 Mev and has been calculated for a few specific designs. the phase oscillation will span a few gaps at the start and a few tens of gaps at the

o

end.


In a long machine the radial stability is also important. For cylindrical drift tubes, the field pattern (Fig. 7a) indicates that a proton off axis is exposed to a radial If the voltage is force toward the axis as it enters the gap and away as it leaves. increasing with time, to ensure phase stability, the radial defocusing force predomi Simultaneous phase and radial nates and the proton beam will blow up radially. stability can be achieved only by operating near the peak of the r-f wave, a very critical and unsatisfactory condition. The difficulty is fundamental and can be solved only by application of focusing lenses or a means of introducing charge into the beam area. The latter method is applied with grids across an end of the drift tube (Fig. 76). The radial forces (t>) are now always directed inward so the (a) proton will oscillate radially about the Fio. 7. Electric field shape in linac gap. axis. The grids can be reduced to a few (a) Without grid, (b) with grid. very thin slats so that they intercept little of the beam, but in a linac of many gaps the intensity reduction by absorption and scattering in the grids is serious. The strong radial defocusing forces can be over come by lenses inside the drift tube, concealed from the main r-f fields. Solenoids or "strong-focusing" quadrupoles give practical radial focusing for proton linacs of any desired length, with phase stability and good beam intensity. Large quantities of r-f power are needed for a linear accelerator. Proton machines favor the 200-Mc region where special triodes, resnatrons, and klystrons are usable with some difficulty. The attainable voltage V of a linear accelerator, length /, r-f power P, and free-space r-f wavelength X is T.

KPW'

K is a geometrical constant." The Berkeley proton linear accelerator has constants 1.5 m, I = 12 m. quoted as about V = 30 Mev, P = 2 Mw (pulsed), X This gives K at about 7. Numerous r-f structures other than the one illustrated can be used, and various efficiencies are attainable, but K will not vary markedly, and power in megawatts is required for energies much in excess of 15 Mev. Proton linacs produce a beam which emerges directly from the machine with small size, energy spread, and angular divergence. "Conventional" types give average beam currents of microamperes, which should increase, with improved focusing, to milliamperes. Energy output is limited only by availability of powerful r-f genera tors and cost considerations. Machines with large drift tube aperture, strong mag netic focusing, and very high r-f power can be built to produce output currents of the order of an ampere.11 A useful modification of the proton linac principles gives the heavy-particle linear

-

ACCELERATORS

Sec. 5-4] accelerator.

5-129

Multiple ionized atoms of such elements as C, N, O have readily usable "strippers" can be inserted along the machine axis. These strippers

t/m values, and

thin foils or transverse gas jets which remove electrons so as to change the e/m Designs can be evolved which will conform with following machine geometry. accelerate a considerable variety of "heavy" ionB — useful in the study of nuclear structure and of nuclear chemistry. 2.22 Electron Linear Accelerator.10 The electron linear accelerator has one advantage denied the proton linac. Electrons are highly relativistic at moderate energies; that is, their velocity is almost constant — reaching 0.99c at about 2.5 Mev. With constant velocity there is no phase stability, but neither is there disturbance of the velocity by structural and voltage errors, so precise control of frequency and Since ft geometry enable long electron machines to operate without phase stability. is nearly 1, even a few gaps after injection, a loaded waveguide can be designed with A simple and useful loaded guide is phase velocity equal to the electron velocity. shown in Fig. 8a, a cylinder loaded with spaced washers or irises and a favorable are


to

RF FROM MASTER OSCILLATOR (b)

Flo.

waveguide,

8. (a) Loaded

(b)

Sketch

of electron linear accelerator.

structure for the very precise control of dimensions necessary in the absence of phase An electron linac consists of sections of accelerating guide, each supplied stability. radio frequency by a powerful amplifier (capacity of numerous megawatts), the amplifiers in turn excited by a master oscillator through phase shift adjustments for A wavelength of 10 cm (3,000 Mc) is commonly synchronizing the sections (Fig. 86). used. The electron linac at Stanford University is 220 ft long, utilizes 10- to 20-Mw klystrons and should eventually give about 1 Bev. The radial stability problem is relieved by extreme relativistic operation. At 0 nearly 1 the magnetic fields of the r-f wave compensate the radial defocusing component of the electric fields and an essentially neutral radial equilibrium results. Electron linear accelerators have emergent beam characteristics similar to those However, owing to lack of phase focusing, the electron described for proton linacs. Electron linacs of 10 to 20 Mev type has greater energy spread in its emergent beam. are useful as X-ray sources and for concentrated external beams. High energies, for research, are limited only by r-f generators and cost. 3

MAGNETIC ACCELERATORS

The curved path of a charged particle in a magnetic field makes it possible to send the particle around repeatedly and to accelerate it with an electric field applied many times in succession. It is then possible to attain high energies with moderate voltages. It is necessary to provide focusing, or restoring, forces which will confine the particles to the accelerating vacuum tube or chamber and to synchronize the accelerating electric field with the recurrence of the particle orbits.14'10 Launch a particle in the r,S plane of Fig. 9, and apply a magnetic field which is per pendicular to this plane at z .= 0. Let the field have the form B, which the field index can be defined r dB,

—

Bo(r0/r)n, from

6-130

EXPERIMENTAL

TECHNIQUES

[Sec.

5

If the particle has velocity perpendicular to r and momentum p ■» eBR/c, it will move in a circle of radius R unless disturbed by electric fields or collisions with gas molecules. The equations of motion are given by the force equations, setting rate of change of momentum equal to the unbalance of centripetal and magnetic force for the radial relation, and equal to the magnetic force for the vertical one:

( — -rim dt\ dlJ) d ( dz\ dr\

d

=

?nv2

e.

t

c

n vB,

dl\mdt)

If radial and vertical excursions are small, we can replace r by R + x and restrict z to small values. On the equilibrium orbit the angular velocity uo = v/R, mv = RB
35. dr

_

0

true for a cylindrical

field with small gradients.

FORCE ON

PARTICLE -

Fio.

9.

Coordinate system and general magnetic field shape for constant-gradient

accelerators.

Using the relationship Bt = Bo(R/r)n and integrating, B, Substituting, the reduced equations of motion become .

...

+ wo'(l 4-;

at*

at1

- n)x .

+ «o*n*

-nzB./r ~

magnetic

- nzB,/R.

0 =

0

d*x —

Mr =

(1

a

a

simple harmonic oscillator, showing that particle dis These are the equations of turbed from the equilibrium orbit will execute free, or "betatron," oscillations about that orbit with angular velocity related to angular velocity of revolution by —

n)ûo

ui = (n)^«o

a

0,

>

1;

For the oscillations to be stable the roots must be real and the first equation requires that n < physically this means that with increasing radius the magnetic force In the vertical direction stability must fall off more slowly than the centripetal force. which corresponds to the line of force curvature shown in Fig. 9. requires n If the magnetic field increases with radius, vertically displaced particle will be forced away from the median plane and lost. S.l

Constant Gradient Magnetic Accelerators

is

a

1

0

These have their magnetic field and its gradient constant at any constant radius. condition of magnetic field < n < as Stability of particle motion requires The radial and vert ical restoring forces on the particles are not very strong, shaping.*0 Figure 10 shows the BR necessary. so relatively large aperture for acceleration product as a function of energy for electrons and protons. A 3.11 The cyclotron'4-18 was the first of the family of magnetic accelerators. large electromagnet produces a cylindrical region of magnetic field, at the center of a


and this relation is approximately

_

ACCELERATORS

Sec. 5-4]

5-131

The ions encounter an r-f electric field between the dees are accelerated synchronously if the angular velocity of the radio frequency At "low" energies the mass m equals the angular velocity of the particles, eB/mc. does not increase appreciably, so u is almost independent of velocity and radius — the important "cyclotron condition." However, the field must diminish with increasing radius to provide the restoring forces necessary to keep the ions from striking the dees. Sufficient focusing is obtained with a very small radial gradient, and the particles spiral out (Fig. 11) as their energy increases, with only a small drop in rota tion frequency. There is no phase stability, so the number of revolutions must be kept small to limit the cumulative phase error and continue the synchronous accelera tion to the edge of the dee where the orbit can clear a thin separator, forming one side of the deflector channel. High voltage on the deflector produces a radial elec tric field which extracts the beam from the dee space for external use. which is an ion source.4' and

K>7

10*

u.

l

l0"


m

10* I

10 MEV

100

I

10 BEV

100

-T Field-radius product for electron and proton in a magnetic field. KINETIC ENERGY

Fig.

10.

Protons, deuterons, or a particles are usually accelerated. For special purposes highly ionized heavy ions (such as C'+) can be produced in ion sources and accel erated." Energy limits for the conventional cyclotron depend on the attainable r-f dee voltage, since phase errors accumulate from mass increase as well as from radial field decrease. 14'4' A crude approximation of the energy limit in million electron volts for deuterons is fn« = 2VM, where V is the dee voltage in kilovolts. Cyclo trons produce internal circulating beams of many milliamperes with 0. 1 to 1 ma avail able externally. Sizes range to about 65-in.-diameter poles, producing 25-Mev deuterons at high working fields (~ 19,000 gauss), while larger diameters have been used with long gaps at lower fields to obtain high beam currents. The radio frequency is 1.52 X 10_,B mc/sec for protons and about half that for deuterons. The cyclotron is not useful for electrons because the frequency changes rapidly with energy at constant B. 3.12 The betatron was the first magnetic electron accelerator. It is the only such type that does not use r-f acceleration. Where the cyclotron accelerates par ticles at increasing radii in a field constant in time, the betatron uses an orbit of constant radius in a magnetic field rapidly increasing with time. If the orbit radius remains constant, the momentum must be increased by acceleration as the field increases, since p = eBR/c. If the orbit is threaded by magnetic flux * there is an electric field c along the orbit given by 2kR* = di>/dt. This produces a force on the Integrate and set equal to the radius electron F = dp/dl = e*/c = (e/2*-/?c) dQ/dt. condition -fro) = eBR
-

so

that

*

2tRc

- *o - 2*R*B

c

6-132

EXPERIMENTAL

TECHNIQUES

[Sec.

5

" necessary for a stable equilibrium orbit. It is the "betatron condition requires that the flux change through the orbit be twice what it would be if the field were uni form. The betatron (Fig. 11) has a vacuum doughnut which surrounds the magnet The core is laminated, to diminish losses and eddy-current field distortions, core. and is excited with alternating current in the coils. During the a-c cycle when the field rises above zero, a stream of electrons is emitted from the gun with energy of kilovolts or tens of kilovolts, timed such that the injection energy matches the orbit This

-COIL

-YOKE

CCHL-i

r-YOKE

r DOUGHNUT


ORBIT

Diagrams

of the constant-gradient

(low-n)

magnetic

accelerators.

field. The electrons are then accelerated by the flux change (which must be in the same direction as the guide field), and the betatron stability condition is maintained When the peak of the by adjusting the air gap for the necessary ratio of flux to field. a-c cycle is reached, the electrons are forced into a target by deliberate mistracking of the flux. This is done by letting the core saturate or by pulsing auxiliary coils. The electric field is continuously applied, so there is no problem of synchronization However, the electrons travel hundreds of kilometers while as in the cyclotron. being accelerated, and a good vacuum is necessary to avoid serious losses from gas In the simple form of Fig. 1 1 the maximum orbital field is limited by the scattering. flux requirement to a value well less than half the saturation induction of the iron Since the required condition applies to change of flux, the constant of integration *o

Sec. 5-4]

ACCELERATORS

5-133

can be chosen negative and a betatron flux core back-biased with direct current to obtain about twice the energy from a given radius and magnet weight. The long path requires a high degree of orbital stability that would be difficult to achieve with the low-n value used in cyclotrons. In the betatron n is set within the region Y± to are a little slower than the revolution frequency. % and the "betatron oscillations" A more refined treatment of the equations of motion for varying magnetic field shows that amplitude of the oscillations (set up by injection errors, gas scattering, and mag net inaccuracies) decreases as the field rises, being proportional to B~^. The betatron is used mostly as a source of high-energy X rays, although external beams of electrons have been obtained.1* Machines in the 20-Mev range are common (commercially available), and one has been built for 340 Mev at the University of Illinois. X-ray output ranges from tens to thousands of roentgens per minute at a meter from the target.

In common with all circular electron accelerators, the betatron is plagued by radia The electrons lose in radiation

tion loss from the electrons at high energies.41

Kev /turn

=

—— H


with E in billion electron volts and R in meters. A 300-Mev betatron, radius 1 m, is subject to radiation loss of 0.7 kev per revolution, which is large enough to require correction of the flux condition. Parenthetically, the betatron method is not, in principle, confined to acceleration of electrons. 3.13 The microtron was made possible by the development of high-power micro wave techniques. A constant, essentially uniform magnetic field in the vacuum chamber bends electron paths into circles. A microwave resonator is placed near one edge of the field and excited with many kilowatts by a magnetron or other r-f source, The most obvious operating mode is given operating at, perhaps, 3,000 mc/sec." by a resonator pulsed to a peak r-f voltage of 510 kv (Fig. 11). Electrons will be produced by field emission, and some will emerge from the resonator with 510 kev of kinetic energy so that their masses will be 2Af 0. The period of revolution is

_

2rmc

and the value of B can be adjusted to make t equal 2 cycles of the radio frequency. On the next passage, the electrons will gain another rest mass and t will equal 3 cycles of the radio frequency, and so on, until the final orbit strikes a target or enters an ejection channel (magnetic shield). There Other modes of operation can be used.13 is a region of phase stability for the microtron, but radial and vertical orbital stability

The small number of revolutions for acceleration makes approximately neutral. to operate the microtron with careful adjustment in lieu of vertical and radial restoring forces. The microtron has not been used extensively as an X-ray source but has been proposed as an injector for high-energy electron synchrotrons, since the wide spacing of the orbits facilitates beam extraction. 3.14 Phase Stability. The synchrocyclotron and the various forms of synchrotron attain very high energy by accelerating particles with a relatively small r-f voltage each revolution for an enormous number of revolutions. Vertical and radial restoring The resulting free, or beta forces are maintained by using a field with 0 < n < 1. tron, oscillations can be contained in an aperture of practical size for path lengths which exceed 10' miles in some machines. This orbital stability solves the guidance problem, but there must be a mechanism for keeping the radio frequency and the particle in phase to high precision." In Fig. 12 a particle is shown traveling an orbit in a cylindrically symmetrical field of indicated radial falloff and accelerated by the gap field of an r-f cavity. If the particle velocity is v and the orbit length is (, the time of a revolution is t = l/v and, differentiating, are

it possible

At

Al

Av

t

I

V

EXPERIMENTAL

6-134 Substituting

TECHNIQUES

[Sec.

5

the field relation and the relations of the first section, At t

=

f \1

-l_

n

_ £V\

dp

EW

p

Since n lies between 0 and 1, the first term of the parenthesis is greater than 1, while Therefore an increase in the second is less than 1, so the parenthesis is positive. momentum will cause an increased transit time. Although the velocity of the par ticle increases, it moves on a larger radius and the increase in path length is propor If the particle must cross the gap at phase fa (Fig. 12) to obtain tionately greater. the required energy gain per turn, assume that it arrives at a and obtains excess The transit time will lengthen as it travels on a larger radius, and it momentum. Oscillation of the will cross the gap at b, obtaining less energy the next transit, etc.

Fio.

12.

Coordinate system

and

phase-stable

region

of

magnetic

constant-gradient

accelerator.

o

will occur at a frequency much slower (two to three orders of mag Note that this phase stability is found on the nitude) than the revolution frequency. decreasing slope of the r-f voltage wave whereas the linac stability is on the increasing slope. 3.16

The synchrocyclotron or f-m cyclotron uses a solid pole magnet with poles shaped to give a radially decreasing field for focusing forces. n~n The radio frequency is varied by a rotating condenser (or tuning-fork capacitor). One dee is used for convenience in r-f design. Particles emitted by the ion source will be accelerated twice each revolution by the r-f field of the dee. Suppose the frequency fixed, then as a particle spirals out and gains momentum, its mass will increase and its increasing = eB/2rmr radius carries it into smaller magnetic field. The frequency of revolution will diminish, the particle will arrive later in r-f phase each revolution (Fig. 12) and will eventually reach the phase where the r-f voltage is zero as it crosses the dee gap. Further retardation in phase will decelerate the particle, making it spiral back to = 180°. smaller radius, so it will oscillate in phase about a The radius is

/

o

is

A

is

is

a

is,

Now suppose the frequency to be lowered slightly ; the r-f phase will retard with respect to the particle phase so the particle will cross the gap at 4> < 180° and will spiral out to a larger radius and higher energy. In the synchrocyclotron the frequency is lowered continuously each cycle of the rotating condenser and a bunch of particles is accelerated from center to rim each cycle. The variation of frequency with time is not critical, but the maximum rate is set by the dee voltage; that the variation must not be so rapid that the energy gain per turn exceeds the peak r-f dee voltage. The f-m cyclotron can be designed for energies involving large mass increase with reasonable dee voltages (the order of 50 kv). The University of California 184-in. machine has a magnet weighing over 4,000 tons. It will deliver 700- Mev protons with The energy limited only by economics of the frequency ranging from 35 to 19 Mc. huge magnet and difficulties of r-f design. Beam not emitted continuously, as in the conventional cyclotron, but in bursts at the rate of the modulating condenser. Time The change in radius per turn at high average currents are about a microampere. few per cent of the beam can be energies too small to use the splitter-deflector. extracted by perturbing the magnetic field near the rim," but the beam normally is


phase about

Sec. 5-4]

ACCELERATORS

6-135


collected on an internal target, just inside the n => 0.2 radius. Since the magnetic field falls off rapidly at the edge of the pole, the rapid increase in gradient is then If n = 0.2, the radial betatron accompanied by a rapid increase in field index n. oscillations are twice the frequency of - the vertical ones, and at n = ^ the vertical oscillation Irregularities in the magnetic frequency is half revolution frequency. field will couple the oscillations at n = 0.2 and excite the vertical ones at n — Unless the field is precisely uniform as a function of angle 8, these "resonances" will blow up the oscillations and lose the beam." Resonances at n = 1 and 0 are especially bad, since vertical and radial frequencies, respectively, are equal to revolution fre quency and small irregularities will destroy the beam in a few turns. 3.16 The electron synchrotron takes the general form of Fig. 11, although con " siderable variation in design has been used.'0-33 Acceleration is at "constant radius in a rising magnetic field. Electrons injected from the gun at tens of kilovolts are The small flux bars given betatron acceleration to a couple of million electron volts. begin to saturate, and, if the process continued, the electrons would spiral into the inner wall of the doughnut. However, r-f accelerating voltage is then applied each turn by the high-Q cavity at constant frequency, and the electrons continue to accelerate as the magnetic field rises. The phase stability keeps the electrons in synchronism with the radio frequency, and as the electron velocity increases, the orbit radius increases to match. Since the velocity at r-f turn-on is only about 1 per cent less than c, the resulting 1 per cent change in radius is easily accommodated in the doughnut. It is necessary that the peak r-f voltage exceed the energy gain per turn (in electron volts) required to keep the radius constant in the rising magnet field. Energy gain per turn, in electron volts, with R in feet, is

A£

= 5.85

X lO-'ft3

~ at

Radiation loss must be supplied in addition. The guide field is shaped as in the for orbital stability. The electron synchrotron needs only a ring of guide field and does not require the Used mostly massive solid pole magnet of the cyclotron or flux core of the betatron. for research on y rays, this design is built up to energies of some 300 Mev. Above this energy the form is modified to that of the next section. Smaller versions for 50 to 100 Mev are quite compact and are useful X-ray generators. This type of electron synchrotron also lends itself to construction with air-core coils, that is, con ductors shaped to produce proper magnetic fields without iron cores. Another type of electron synchrotron is the Racetrack with "straight sections" left between magnet sectors. Its operation is identical with that of the proton syn chrotron next described, but the r-f change is much less. Energy is limited only by the excessive radiation loss in the billion-electron-volt region. The California Institute of Technology synchrotron will produce electrons over 1 Bev. 3.17 Proton synchrotrons are practical means for accelerating protons to billion electron volts.34 38 These complex and massive machines are the largest of the They are used for research in particle physics, which requires very accelerators. high energies. The proton synchrotron involves a composite of principles developed in the previous sections. Referring to Fig. 1 1, the magnet is in ring form with straight sections left for insertion of equipment and the protons are accelerated at approxi mately constant radius. The magnet is excited with a triangular current pulse. During the current rise a bunch of protons is injected from a "straight-line accelera tor" when the injector energy matches radius and field. The injected beam is elec trostatically deflected so it is launched tangent to a stable orbit. Radio-frequency voltage is applied to an accelerating cavity and increases the proton energy as the At a desired energy the beam is shrunk into a target, or a large magnetic field rises. Stability is maintained by using a field index n about fraction of it can be ejected. With N straight sections of length L the betatron oscillation frequencies are 0.6. Injection is at a few Mev and a (3 of the increased in the ratio (1 + NL/4rR).3i order of 0.1. At a few Bev 0 is almost 1, and, since the radius is fixed, the radio betatron

EXPERIMENTAL

5-136

TECHNIQUES

[Sec.

5

frequency must vary some factor of 10, tracking the rising magnetic field. Phase stability maintains the proper stable phase angle provided the radio frequency is free from transients and noise. However, the frequency must match the magnetic field accurately or radial errors and particle loss will ensue. This is a difficult engineer The phase oscillations, when V is the peak r-f gap voltage and 4>ois ing problem." the stable phase angle, have frequency V.

R + NL/2*

{_

eV cos %rE

o

I"

L(T

n)(l

+

NL/2rR)

-m*

which is of the order of 10~3 revolution frequency." Three machines," at Birming ham University (England), Brookhaven National Laboratory, and the University of California, have radii of 12, 30, and 50 ft and produce energies of 1, 3, and 6 Bev, One at 10 Bev has been undertaken in Russia. The beam is accelerated respectively. for 1 to 2 sec and contains about 1010 protons. The over-all average intensity is low because the acceleration cycle is repeated only after several seconds owing to the high power involved in charging and discharging the stored energy of the magnet. The Brookhaven Cosmotron magnet weighs over 2,000 tons, the Berkeley Bevatron 10,000 tons, and the Russian machine 35,000 tons.


3.2

Alternating-gradient

Magnetic Accelerators

Alternating-gradient Synchrotrons, Strong-focusing. 3.21 Further scaling up of the "conventional" proton synchrotron becomes impractical because energy scales as radius while magnet weight scales as the radius cubed, although operating expe rience permits some economies by reducing aperture. The alternating-gradient, or strong-focusing, synchrotron was invented to make use of a small vacuum chamber in a magnet of small cross section but large radius."-"

f <

Fig.

u

DFDFDFOF AZIMUTH — -

+

--

+

-- +N

SIGN OF n

"focusing OR DEFOCUSING

Typical particle orbits in alternating-gradient field, ning of similar orbit in constant-gradient field. 13.

Dotted line indicates begin-

The magnetic accelerators previously described are stable only if the field index n lies between 0 and 1. If n ^ 1, there are strong vertical restoring forces, but there is also radial instability or defocusing and the inverse for n 0. However, these restrictions are imposed by the requirement that the field and its gradient be azimuthally uniform — a requirement that is not fundamental. Suppose a series of magnets of high-ra value (large radial gradient) set up with positive and negative Orbits of typical disturbed particles (and all particles in an gradients alternating. accelerator are disturbed from the central orbit to some degree) are shown in Fig. 13 A view of the vertical oscillations with the radial dimension expanded for clarity. would appear the same if the plus and minus labels on n were inverted. The oscilla tions are no longer sinusoidal but are increased in frequency and decreased in ampli tude as compared to the low-n case. The beginning of a low-n oscillation is dotted in An alternating-gradient synchrotron has N magnet periods, and each period Fig. 13. contains a focusing and defocusing field (or a plus and a minus). The betatron oscillation phase shift per period /i must be in a stable region and is selected near ir/4 for large machines. The number of betatron oscillations per revolution r is v = Nii/2t. The n value of the fields is approximately ^N*/** or N*/16 for p = Another important attribute is the small radial displacement of the equilibrium orbit

«

r/i.

ACCELERATORS

Sec. 5-4]

5-137

This "momentum compaction" is roughly 50 times over the low-n machine and is as important as the amplitude reduction in betatron oscillations for reduction of aperture. Phase stability in the alternatinggradient synchrotron begins on the rising side of the r-f gap voltage, as in the linac, but at a transition energy Tt ^ (v — l)Eo there is instantaneously no phase stability, and above this the phase stability is on the falling slope of the radio frequency, as in Precision r-f control circuits will carry the beam through the the low-n machine. Alternating-gradient phase transition by shifting the r-f phase at the proper time. proton synchrotrons are being built at CERN (Switzerland) and Brookhaven National The Brookhaven alternating-gradient synchrotron will have an energy Laboratory. of 25 to 30 Bev, diameter 840 ft, N «= 60, v — 8J£, magnet of cross section 33 by 39 in., and vacuum-chamber aperture about 3 by 6 in. Electron accelerators using One at Cornell University gives 1 Bev, and strong focusing are quite compact. larger ones are limited only by radiation loss. FFAG — Fixed-field alternating-gradient accelerators apply the alternating3.22 gradient principle to d-c magnetic fields.4'-'4 The increase in duty cycle over the The pulsed accelerators makes it possible to obtain large increases in beam current. Thomas cyclotron, a precursor of the FFAG, uses a scalloped pole to compensate for relativistic effects and can be built for steady output at high energies. The Midwestern Universities Research Association is intensively developing the FFAG.44 Because of momentum compaction the orbits of particles over a large At the same time radial and range of energy can be contained in a d-c ring magnet. vertical stability can be maintained, and the particles accelerated by either radio In the "Mark type, the field increases with frequency or a betatron flux core. increasing radius in all sectors but the field is reversed in alternate sectors so the gradient alternates. Particles injected near the inner periphery will follow scalloped orbits which expand as the particles are accelerated. The Mark I has a large circum A spiral-ridge (or Mark V) type has cir ference factor due to the field reversals. cumference vs. energy comparable to more conventional machines and produces alternating gradients because the field is rising as a particle approaches a ridge and is falling after the particle passes the ridge. The FFAG basic principles are applicable to betatrons, cyclotrons, and synchrotrons. Although only one small working model is extant (1956), the FFAG provides another large extension of accelerator design. It is evident that particles can be accelerated in a great many different configurations of electromagnetic fields. As the demand arises, more new types will no doubt be with particle energy change.

improvement


I"

invented.

REFERENCES These references are selected for descriptive content and lists of additional references. Much of the detailed information on accelerators is found only in privately circulated reports, some of which are in AEC Depository Libraries. Detailed constructional data are obtainable only by paying an extended visit to a laboratory where accelerator building is practiced. Four extensive bibliographies list the accelerator literature through June, 1954. These also contain lists of accelerator installations. Brookhaven National Laboratory Report BNL-L-101.

University of California t'CRL-2672.

Radiation

Laboratory

Reports UCRL-1238,

UCRL-1951,

General 1. 2.

Livingston, M. S.: "High Energy Accelerators," Interscience Publishers, Inc., New York, 1954. Sturrock, P. A.: "Static and Dynamic Electron Optics," Cambridge University Press, London,

1955.

Direct-voltage 3. Buechner, 754 (1947).

W. W.,

et

Accelerators

al.: Electrostatic Accelerator for Electrons,

Rev.

Sci. Inttr., 18:

EXPERIMENTAL TECHNIQUES

6-138

[SEC. 5

4. Cockcroft, J. D., and E. T. S. Walton: Experiments with High Velocity Positive Ions, Proc. Roy. Soc. (London), A136: 019 (1932). 5. Elkind, M. M.: Ion Optics in a High Voltage Accelerating Tube, Rev. Sci. Instr. 24: 129 (1953). 6. Heilpern, W.: Cascade Generator for Acceleration of Particles to 4 Mev, Helv. Phys. Acta, 28: 485 (1955). 7. Peck, It. A., and H. P. Eubank: High-current Cockcroft-Walton Accelerator for Neutron Production, Rev. Sci. Instr., 26: 444 (1955). 8. Turner, and E. J. Rogers: Van de Graaff Accelerator, Rev. Sci. Instr., 24: 805 (1953). 9. Tuve, M. A., L. R. Hafstad, and O. Dahl: High Voltage Technique for Nuclear Physics Studies, Phyt. Rev., 48: 315 (1935).

CM.,

Linear Accelerators 10. 11. 12. 13.

Linear Accelerator Issue, lire. Sci. Instr., vol. 26, February, 1955. (Describes both proton and electron types.) Lawrence, E. 0.: High-current Accelerators, Science, 122: 1127 (1955). Slater, J. C: Design of Linear Accelerators, Revs. Mod. Phys., 20: 473 (1948). Sloan, D. H., and E. O. Lawrence: The Production of Heavy High Speed Ions, Phys. Rev., 38: 2021 (1931). Cyclotron

14.

Cohen, B. L.: Theory of the Fixed Frequency Cyclotron, Rev. Sci. Instr., 24: 589


(1953).

16. 16. 17.

Lawrence, E. O., and M. S. Livingston: The Production of High Speed Ions without Use of High Voltages, Phys. Rev., 40: 19 (1932), 45: 008 (1934). Livingston, M. S.: The Cyclotron, J. Appl. Phys., 15: 2, 128 (1944). Livingston, R. S.: Acceleration of Partially Stripped Ions, Nature, 173: 54 (1954), 170: 221 (1952).

18.

Nahmias, M. E. Paris, 1945.

:

"Le cyclotron"

and "Machines

atomiques," editions of lire, opt.,

Betatron Foote, R. S., and Ben Petree: A Pulsed Magnetic Extractor for a Betatron, Rev. Sci. Instr., 26: 694 (1954). 20. Herat, D. W.: Acceleration of Electrons by Magnetic Induction, Phys. Rev., 60: 47 (1941); with R. Berber, Electronic Orbits in the Induction Accelerator, Phys. Rev., 60: 53 (1941); A 20-Mev Betatron, Rev. Sci. Instr., 13: 387 (1942). 21. Schwinger, J.: Classical Radiation of Accelerated Electrons, Phys. Rev., 78: 1912 19.

(1949). 22. Westendorp, W.

J.

F.,

Appl. Phys., 16:

and E. E. Charlton:

581

100

Mev Induction

Electron Accelerator,

(1945).

Microtron 23. Kaiser, H. F., and W. T. Mayes: General Purpose X-band Laboratory Microtron, Rev. Sci. Instr., 26: 565 (1955). 24. Redhead, P. A., H. Le Caine, and W. J. Henderson: The Electron Cyclotron, Con. Research, A28: 73 (1950); Nucleonics, 6(4): 00 (1949).

J.

Synchrocyclotron 25. Anderson, H. L., et al.x Synchrocyclotron for 450-Mev Protons, Rev. Sci. Instr., IS: 707 (1952). 20. Bohm, D., and L. L. Foldy : Theory of the Synchrocyclotron, Phys. Rev., 72 : 649 (1947). 27. Le Couteur, E. J., A. V. Crewe, and J. W. G. Gregory: Extraction of the Beam from the Liverpool Synchrocyclotron, Proc. Roy. Soc. (London), A232: 236, 242 (1955). 28. Henrich, L. R., D. C. Sewell, and J. Vale: Operation of the 184-in. Cyclotron, Ret. Sci. Instr., 20: 887 (1949). 29. Richardson, J. It., etal.: Development of the FM Cyclotron, Phys. Rev., 73:424 (10-1S).

ACCELERATORS

Sec. 5-4]

6-139

Electron Synchrotron

J.

Appl. Phya., 18: 810 (1947); also Radia F. R., et al.: A 70-Mev Synchrotron, tion from Electrons in a Synchrotron, Phya. Rev., 74: 52 (1948). 31. Frank, N. H.: Stability of Electron Orbits in the Synchrotron, Phya. Rev., 70: 177 (1946). 32. Lawson, J. L., el al.: 300 Mev Nonferromagnctic Electron Synchrotron, Rev. Sci. Inatr.. 26: 809 (1955). 33. Thomas, J. E., W. L. Kraushaar, and I. Halpern: Synchrotrons, Ann. Rev. Nuclear Sci., 1: 175 (1952).

30. Elder,

Proton Synchrotron 34. 35.

36. 37. 38.

Cosmotron Staff: Cosmotron Issue, Rev. Sci. Inatr., 24, September, 1953. (The most complete description of an accelerator in the accessible literature.) Blackman, N. M., and E. D. Courant: Dynamics of a Synchrotron with Straight Sections, Rev. Sci. Inatr., SO: 596 (1949); also Scattering of Particles by Gas in a Synchrotron, Phya. Rev., 75: 315 (1949); Rev. Sci. Inatr., 24: 836 (1953). Blewett, J. P.: Recent Developments in Proton Synchrotrons, Ann. Rev. Nuclear Sci., 4: 1 (1954). Hibbard, L. V.: The Birmingham Proton Synchrotron, Nucleonica, 7(4): 30 (1950); See also Nature, 172: 704 (1953). Twiss, R. Q., and N. H. Frank: Orbital Stability in a Proton Synchrotron, Rev. Sci. Inatr., 20: 1 (1949).


Alternating-gradient 39. 40.

41. 42. 43. 44.

Types

Adams, J. B., M. G. N. Hine, and J. D. Lawson: Effect of Magnet Inhomogeneities in the Strong Focusing Synchrotron, Nature, 171: 926 (1953). Adams, J. B. : The Alternating Gradient Proton Synchrotron; also G. Luders: Theory of Particle Orbits in the AG Synchrotron, Supplement to vol. II, ser. X, Nuovo cimento, No. 1, 1955. Blewett, J. P.: Radial Focusing in the Linear Accelerator, Phya. Rev., 88: 1197 (1952). Courant, E. D., M. S. Livingston, and H. S. Snyder: The Strong-focusing Synchrotron, Phya. Rev., 88: 1190 (1952). Thomas, L. H.: Paths of Ions in the Cyclotron, Phya. Rev., 54: 580 (1938). See also Ref. 11. Symon, K. R., el al.: Fixed Field Alternating Gradient Accelerators, Phya. Rev., 98: 1152 (1955).

(Abstracts only.)

Ion Sources 45. Gow,

J.

D., and

606 (1953).

J.

S. Foster:

A High Intensity

Pulsed

Ion Source, Rev. Sci. Inatr., 24:

J. Jones: High Intensity Ion Source for Cyclotrons, Rev. Sci. Inatr., 25 : 552 (1954). 47. Moak, C. D., H. Reese, and W. M. Good: Radio Frequency Ion Source for Particle Accelerators, Nucleonica, 9(3): 18 (1951). 46.

Livingston, R. S., and R.

5-5

ENGINEERING TESTING IN REACTORS BY


John R. Huffman and F. L. McMillan Reactors, being very large sources of neutrons and 7 rays, in many cases of high intensities, are used for studies of the effect of these particles on matter and of the There is no other major source of neutrons.* effects of matter on these particles. Many studies of fundamental importance to reactor engineering are carried out in reactors, such as properties of particles and attenuation of neutrons and y rays in In design matter. This type of experiment is discussed in Sec. 6-2 of the •handbook. ing and constructing reactors, the nuclear engineer may be forced to proceed without the data and experience obtained from pilot-plant or semiworks experimentation. These are generally too costly and too close in size to the ultimate reactor to warrant their use. To establish feasibility, he is forced to resort to testing and confirming his While these tests are design by conducting experiments in an operating reactor. costly, they are orders of magnitude cheaper than the construction and test of the reactor. Engineering testa are more concerned with changes in mechanical and physical Such experiments are properties, dimensional stability, and corrosion of materials. usually performed on components of a new design. They may consist of testing small samples, segments, dynamically similar models or full-scale models, of fuel-bearing, moderating, or structural materials. 1

1.1

STATIC TESTS

Before and After Measurements

1.11 Component testing can take several paths. Tests at Ambient Temperature. Depending on the reactor used and hence the restrictions placed on the experiments, valuable information can be obtained by testing under the environment of the reactor. For an air-cooled test reactor, air cooling can be used; for a low-temperature watercooled test reactor, water cooling can be used- — in both instances up to temperatures, heat-transfer and hydraulic conditions imposed by the reactor. The simplest type of test is to insert a small specimen directly in the test reactor. Generally it is preferred to encapsulate the specimen in the material of construction of the reactor to avoid problems in the reactor and to divorce the specimen from the coolant used in the reactor. The properties of material under test are determined before and after irradiation. Many specimens can be incorporated in a single capsule, or a series of capsules can be installed in a single test hole. If the environment (temperature, pressure, coolant, etc.) is not important, full-sized fuel assemblies of new design can be substituted for those of the test reactor. An important type of test facility is the shuttle. Specimens up to 10 to 100 g In large can be sent into a reactor by pneumatic, mechanical, or hydraulic forces. reactors an endless chain or train can be used to carry large numbers of specimens through the reactor. For short-time irradiations, pneumatic systems are provided. * Accelerators smaller volume.

do produce

higher

neutron

intensities

5-140

than are available in reactors,

although

over a

Sec. 5-5]

ENGINEERING

TESTING IN REACTORS

5-141

In water-cooled reactors, hydraulic devices are available. The advantage of these systems is that a large number of irradiations can be performed independent of reactor Most other types of tests require the reactor to be shut down when operation. experiments are inserted. Mechanical systems are generally slow moving and are best suited for relatively long time irradiations. Hydraulic systems are perhaps next and are used for irradiations lasting 5 min to several days or weeks. Pneumatic systems, which handle the smallest quantities, are excellent for irradiations required to last from less than a second to several minutes. They can also be built to deliver samples rapidly to the chemistry or physics laboratory remote from the reactor. Depending on the half-life of the mate rial being studied, these shuttle systems each have their place. They are a great con venience to the staff operating a test reactor. 1.12 Tests at Elevated Temperatures and Pressure. To establish a static


environment different from that of the test reactor, specimens can be inserted in capsules containing the desired coolant fluid or some other fluid for heat transfer. Often, to fit the temperature conditions of the test reactor and at the same time provide higher temperatures in the test specimen, heat sources and thermal insulation are designed into the capsule. When more exact knowledge of test conditions is desired, leads are brought out from the capsule so that temperature and pressure conditions can be recorded. Hence, to some extent controlled conditions for the experiment can be maintained and safety conditions for the reactor can be satisfied. Converters. To provide a flux of fast neutrons, sleeves of fissionable material 1.13 can be designed into a container and the specimen placed inside this sleeve.

1.2

Static Systems — In-pile Measurement

An important type of experiment is one attempting to measure the properties of in situ in a reactor. This is intended to determine whether mechanical properties are the same during neutron bombardment as they are after bombardment. In such instances remotely controlled load-applying equipment and remotely con trolled recording-measurement instruments must be part of the experiment. Exam ples of this type of experiment involve measurements of tensile strength and creep. One problem involved is to establish the effects of irradiation on the load-applying equipment and the detection instruments themselves.

materials

2 2.1

DYNAMIC TESTS Cycling Systems

serve to test various fuel elements or materials under conditions or level reactor operation. Some interest may be expressed on the effect of reactor startup or shutdown on the stability of proposed fuel assemblies. Testing fuel under these conditions is somewhat more complicated than static burnup and Test reactors and production reactors which provide experimental stability tests. facilities must operate at constant full power during most of the operating cycle. The experiment itself must then be cycled into and out of the reactor flux zone on some prearranged schedule. In order not to affect neighboring experiments or the stability of the reactor itself, such experiments must be compensated. Com pensation results in no change in reactivity as the experiment is cycled into and out of the flux zone. Temperature monitors in the form of thermocouples may be used to record the number of cycles experienced by the sample as well as the temperature Static irradiations

of full

excursions.

Another possible method, such as a poison sleeve, could be used to shade the test Two similar test assembly, but this method is much more difficult to compensate. sections, one of which replaces the other as the experiment cycles to and fro, is by far the simpler method to use.

6-142

EXPERIMENTAL

TECHNIQUES

[Sec.

5

The principal objection to tests outlined above lies in the fact that test operating conditions do not necessarily duplicate the desired operating conditions of the test specimen.


2.2

Loops

The developing interest in power generation by nuclear power has produced a demand for test facilities more nearly duplicating proposed reactor operating con ditions. To satisfy the demand for testing operation under conditions of high tem perature and pressure in those reactors using a primary water loop, high-pressure loops have been developed. Questions of fuel stability, heat transfer, water chemistry, radiation-accelerated corrosion, fission-product leakage, and fuel stability under desired operating conditions may be answered in high-pressure water loops. Basically, a high-pressure water loop consists of an in-pile test assembly which carries the test element, a thermally controlled pressuring surge tank, and some method of circulating the water. Heaters may be provided in the line to maintain Flow can be controlled by means of automatic flowloop operating conditions. Auxiliary control valves or variable-frequency power supply to pump motors. systems such as water make-up, water purification, sampling, gas injection, and Figure 1 radiation monitoring may be necessary to maintain such an experiment. illustrates a flow sheet for a typical high-pressure, high-temperature water loop. An external test facility may be provided in order to obtain comparisons between effects under identical operating conditions with the exception of irradiation. The system is pressurized by means of a heated surge tank, the temperature of the water in the surge tank being maintained at the saturation temperature corresponding The connection between the loop and the to the desired loop operating pressure. surge tank is small enough that pressure alone is transmitted and no appreciable heat Circulation is maintained is transferred from the surge tank to the primary system. by means of the circulating pumps, two pumps being provided to ensure circulation in the event of failure of a pump. The control circuitry should be such that automatic The heat take-over by the standby pump results from primary pump failure. exchanger is provided to remove heat from the recirculating water in the loop. This heat removal is compensated for by the line heaters and the heat generation in the fuel Conditions of inlet and outlet temperature to the sample section under irradiation. In case of trouble in the loop, the heat may be closely maintained in this fashion. exchanger also serves as a thermal sink for cooling the circulating water and thus reducing the pressure on the loop. The radiation monitor maintains close supervision on the activity of the loop water, indicating possible fission breaks or the build-up of corrosion products being carried by the water. A water-purification system, consisting of a side stream from the Ion-exchange primary loop, continuously maintains the desired water chemistry. resins maintain the desired pH and conductivity and remove corrosion products and possibly fission products. An auxiliary power supply is furnished in order to provide pump power and super visory circuit power in the event of plant power failure. Many of the experiment controls must, of necessity, be connected to the reactor safety circuits in order to provide protection for the reactor and experiment in case Automatic control is provided for hazardous conditions develop during operation. such important factors as surge tank liquid level, flow, pressure drop in test section, loop pressure, water temperature, surge tank pressure, surge tank temperature, and The more important parameters, those likely to result in the develop water activity. ment of hazardous conditions, are connected to the reactor control circuit and may initiate automatic reactor power reductions at a rate consistent with the hazard. These may range from an immediate scram to much slower rates of power reduction. Figure 2 shows the flow sheets of a make-up system and a pump cooling system which might be used with such an experiment. Experimental loops, in which high levels of activity are anticipated, are provided with auxiliary circulating systems to be used to provide cooling for the in-pile assembly


p s

I H 5

en

o:

ft -1 < en

H bj

a 5

111 0^

rr UJ

1

5-143

6-144

EXPERIMENTAL

TECHNIQUES

[Sec.

5

during periods in which the main loop is being decontaminated. In general, these loops operate at lower temperature and pressure than the primary loop and are used when excessive activity develops or mechanical failure occurs in the primary loop. The purification system receives a small side-stream flow from the loop. The water passes through filters and ion exchangers to remove both solid and ionic particles. Sample taps are usually provided. The decontamination system is designed to receive and process the loop contents after shutdown. Contaminated water is received by catch tanks and either stored or pumped to hot-waste storage facilities. The make-up system processes demineralized water for addition to the loop. Figure 3 shows in block diagram the type of control system which might be used on a loop of this sort. Troubles such as low flow or low differentia! pressure across the in-pile tube, which could result in serious consequences, automatically scram the

DEMINERALIZED


WATER

ION EXCHANGE

CONDUCTIVITY CELL

MAKE UP SYSTEM

Fio.

2.

PUMP COOLING SYSTEM

Flow sheets of make-up system and pump cooling system.

reactor. Other troubles of less serious nature will call for reduction of power at a slower rate. When conditions requiring the attention of the operator develop, annunciators provide warning signals to this effect. Some difficulties are auto matically corrected by means of the experiment control circuits. Instrument trouble can result in erroneous calls for power reduction, and some means should be provided to release the reactor from these false calls for power reduction. A switch or test block arrangement is provided to disconnect from the reactor controls all experi ment automatic power-reduction circuits except those which cannot be handled manually, because of short time factor, by an operator in continuous attendance at the experiment. Loop tests have been performed in MTR, at Hanford, Chalk River, and Oak Ridge. Considerable care beforehand should be given to methods of inserting involved experiments into a reactor, removing them after irradiation, shipping them when they are quite radioactive, and examining them by remote-control means. Insertion in a reactor generally requires shielding around insertion points. Upon removal much

Sec. 5-5]

ENGINEERING

TESTING IN REACTORS

5-145

NQI OIL PUMP

FAILURE

LOW OIL FLOW

NO.2 OIL PUMP START

LOW

LOW LOOP FLOW _ BOTH LOOP PUMPS FAIL HIGH LOOP TEMPERATURE

fv

LOOP ^P

NQI LOOP

PUMP FAIL

LOW LOOP FLOW

N02 LOOP PUMP START

REACTOR SCRAM MANUAL

STOP LOOP HEA'TER

PUSHBUTTON

LOW LOOP PRESSURE HIGH LOOP PRESSURE HIGH OIL TEMPERATURE

INCREASE COOLING WATER

REACTOR SLOW

SETBACK

HIGH GAMMA

ACTIVITY

HIGH PUMP


REACTOR ANNUNCIATOR

TEMPERATURE HIGH SURGE TANK LEVEL LOW SURGE TANK LEVEL REACTOR CONTROL POWER OFF

STOP MAKEUP HIGH PRESSURE

PUMP

SWITCH

PRESSURE CONTROLLER

COMMERCIAL POWER OFF

STOP SURGE TANK HEATER START

MAKEUP

PUMP

LOW AIR

PRESSURE EMERGENCY POWER ON

Fia.

3.

Block diagram of control system on loop.

more elaborate shielding is required, both around the reactor and around the experi mental equipment. The equipment must be transported to a storage place or a shielded inspection place in heavy lead containers. Hot cells with remote handling devices are required to disassemble the equipment. Containers with Bureau of Explosives approval are required for shipment. Fully equipped hot cells are required to examine and test the specimens.


SECTION

6

REACTOR PHYSICS BY JOHN W. WEIL, B.S., Ph.D., Manager, Power Reactor Physics, Atomic Power

General Electric Company. J. R. DIETRICH, Ph.D., Vice President, General Nuclear Engineering Corporation, formerly Associate Director, Reactor Engineering Division, Argonne National Laboratory. Equipment

Department,


CONTENTS 6-1

REACTOR THEORY BY JOHN W. WEIL

1 Neutron Transport 2 The Spherical Harmonics Method and the Diffusion Equation 3 Other Techniques for Transport Prob lems in Neu 4 The Treatment of Boundaries tron Transport 5 Reactor Equations 6 The Four-factor Formula 7 Special Reactor Problems References

6-»

PACE

6-3 0-10 6-12

7 Reflected Reactors 8 Matrix Solutions for Reflected Reactors, by Otto Schulse 9 Evaluation of Material Constants of by Otto Schulze and J. the Reactor,

6-15 6-18 6-21 6-25 6-27

R. Dietrich

10 Multigroup

Calculations,

by

David

Okrent

11 Sample Calculations 12 Lumped Thermal Absorbers

REACTOR CALCULATIONS BY J. R. DIETRICH

1 Introduction 2 Diffusion of "Monoenergetic" 3 Slowing Down of Neutrons

4 Energy Distribution of Thermal Neu trons 5 Critical Reactor Equations for Thermal and Near-thermal Reactors 6 Solutions of the Wave Equation for Bare Reactors Homogeneous

Neutrons 6-3o

Reactor 13 Noncritical Reactors 14 Perturbation Relations References

6-42

6-1

in

PAOE 6—48 6—51 6-57

6-00 6-70

6-75 6-88 6-05

the 6-105

6-110 6-117

6-120

6-1

REACTOR THEORY BY

John W. Weilt


The principal object of reactor theory is to provide useful techniques which will give an accurate description of the spatial and energy distributions of neutrons within a chain-reacting system. In a system operating in the steady state it is required that for a given source of neutrons born in fission the slowing down and eventual capture or loss of these neutrons shall be correctly described. The resulting distribution of fissions induced by these neutron captures must agree both in strength and in dis tribution with the given source. Only if this condition is satisfied will the reactor be critical and operating in an equilibrium condition. In order to be of use to engineers, reactor theory must obtain this correct description of the neutron distribution for reactor geometries of engineering interest and must include a realistic representation of the complex nuclear cross sections of the materials of which the system is composed. It often turns out, however, that these practical considerations prevent the accurate mathematical descriptions from being brought to a solution with any reasonable amount of effort. Consequently, the subject of reactor theory is largely composed of methods which may be used to obtain approxi mate solutions to a detailed problem for various special cases of engineering interest. This chapter is devoted, then, to a discussion of these approximations and their validity in order to present an over-all viewpoint to the nuclear engineer. Detailed mathematical treatment is left to the references. In choosing among the various available methods, one should consider several specific questions: Is it possible to use some rough but convenient approximation to provide a general This often allows effort in other areas to proceed and, in addi idea of the situation? tion, allows for a more effective concentration of refined calculation in the areas of most importance. Are neutrons of all energies equally important to the behavior of the chain-reacting In this latter system, or are certain neutron energies of particular importance? case perhaps a treatment of a limited number of neutron energies will suffice and will help to reduce the mathematical complexity. Is the majority of the reactor volume distant from boundaries by a mean free path or more? If so, certain valuable simplifications can be made in the form of the equa tions to be used. Or, in the other extreme, is the reactor composed of a structure so fine that all detailed description can be neglected? It may then be possible to treat the entire medium as homogeneous if suitable care is exercised in formulating the equivalent homogeneous material. Will the system behavior be sensitive to details of geometrical shape? If it will not, then simplified geometries may be used for calculation with great reduction in the effort because of a reduction in the number of dimensions to be handled or because of the occurrence of simpler mathematical functions. It Are important variations in composition of the system of a gradual nature? may be entirely satisfactory to represent these variations by choosing several regions of uniform composition and connecting them with appropriate boundary conditions. t The assistance of Drs. J. Sampson and T. M. Snyder in the preparation of portions of this materia! is gratefully acknowledged.

The material

was reviewed

6-2

by Drs. P. Zweifel and T. M. Snyder.

REACTOR

Sec. 6-1]

6-3

THEORY

effort can be achieved with little reduction However, cases will occur where very few approximations can be made with accuracy, and the system will, in the end, have to be studied in an experimental mock-up or crit ical experiment in order to establish sufficient confidence in the prediction of final system performance. Great reduction in over-all calculational

in accuracy

by judicious selection of approximate methods.

NEUTRON TRANSPORT

1

which may occur to a neutron in a system com of scattering and absorbing materials is called neutron transport. In neutron transport no attempt is made to obtain a chain reaction or, for that matter, to treat any other processes except those which cause a change in the space or energy coordi nates of the neutron. The combination of neutron transport with other physical phenomena to form a power-producing, chain-reacting system is treated in Art. 5. The study of the physical processes

posed

1.1

The Boltzmann Equation

It is possible to derive an equation which rigorously describes the behavior of neutrons in any scattering or slowing-down material. This equation is based on the fundamental physical principle of neutron conservation within a closed system. Consider the vector neutron flux

is

a

is,

which is a vector whose magnitude > is the number of neutrons of speed v which in one second cross a unit of area perpendicular to the vector direction Q at the position given by the vector r. The equation for the conservation of neutrons then, composed of terms which represent the various possible sources and sinks in system. The net number of neutrons leaving a unit volume per unit time in the speed interval dv and the direction interval da V • (!>,a,r) dv da The number of neutrons removed from this unit volume and this speed interval and direction interval by all forms of interaction with other nuclei, including slowing-down, is

N(r(v)(v,a,r)

dv

da

.V = atomic density of the medium, atoms/cm' = cross section for all removal interactions, em'/atom (For an isotropic medium with no crystal effects this independent of Q) The number of neutrons introduced into the unit volume and into this speed and and /or a direction interval by slowing-down collisions involving a change in speed change in direction Q will be equal to the number of these collisions integrated over all possible original speeds and directions. This source is, then,

Jj

v

is

where

dvda

4>(v',a',T)Na.(v,v',a,a') dv' da'

/a' the primed quantities represent the speed and direction of the neutron before a form of differential cross interaction and where the quantity
where the

is

is


4>(t>,0,r)

S(v,a,r)

dv

da

This term can be used to represent neutrons born in fission or introduced by some external means.

6-4

REACTOR

PHYSICS

[Sec.

6

Clearly if the rate of change of neutrons within the unit volume and this velocity (speed and direction) interval is to be zero (that is, the system is in steady state), then the number of neutrons entering the unit volume and this velocity interval must equal the number of neutrons leaving the same volume and velocity interval. Equat ing these two quantities yields

V-+(»,0,r) + Nv(v)4,(v,a,t)

=

S(v,a,i) +

JJ 4>(v',a',i)N
dv'

da'

This

is the steady-state Boltzmann equation and is the basic equation governing the behavior of neutrons in a system containing arbitrary scattering and absorbing To be completely general the atomic density and the cross sections must materials. be considered to be functions of position, but it will suffice to treat the case for a

1.2

The Space-independent

Integral Equation

For cases where the neutron distribution is space independent or where the flux is slowly varying, it is possible to make a reduction in the complexity of the Boltzmann For a region of constant flux the divergence term is zero. Then integrat equation. ing over the various directions of the velocity vector yields

N(v)

= S(v) +

fv,[Jf *(»',0')AM»y,0,0')

da

aa

rfO'J

dv'

This space-independent Boltzmann equation

is then rearranged into an equation for the collision density ty. The collision density is equal to the neutron flux times the total probability of a collision (the cross section). Consequently, the space-inde pendent equat ion can be made to take the form of an integral equation (in the energy variable E instead of the speed v)

HE)

=

}r^TpjS&,E')ME')dE' +SW

is

A

Superscript numbers refer to References


a

is

is, a

where the kernel g(E,E') is now the probability that a neutron of energy E' will be slowed down to energy E as the result of a scattering collision which it undergoes with probability a.(E'). Solutions of this equation depend upon the choice of the kernel g(E,E'). Because this kernel will have energy cutoffs for most elements (that neutron can lose only a fixed maximum fraction of its incident energy upon collision with an atom of mass greater than the neutron mass) both limits of the integral will become functions of the energy E. This situation does not lead to a form of the equation convenient for Thus, the variable limit on the integral solution. usually avoided by using one of number of synthetic kernels. These kernels do not give an exact description of the slowing-down process but are approximate functions which do not differ greatly from the correct function and which will yield a convenient reduction of the integral. discussion of these synthetic kernels included in several references. '-f

t


uniform medium. If all the cross sections and sources involved were known experimentally, and if the equation could be solved, a rigorous description of the neutron flux would result. Unfortunately, the Boltzmann equation is an integrodifferential equation involving source and cross section functions of an arbitrary nature. As a consequence, it cannot be solved exactly for any but a few highly restricted cases. For most engineer ing purposes, approximations to this equation or approximate reformulations of it are of most use. To obtain equations of manageable difficulty it is usually necessary to make approximations as to both the space-wise and the energy-wise behavior of the neutrons. Below are treated briefly a number of such approximations to the Boltzmann equation which have been of use in practical problems.

REACTOR

Sec. 6-1] 1.3

6-5

THEORY

The Hydrogen Equation

For the case of hydrogen a neutron may lose all its energy slowing-down kernel may be shown to have the form

=i

g(E,E')

The

if E < E' if E

= 0 and only

in one collision.

>

E'

the lower limit on the integral is a function of the energy E. equation reduces directly to the differential equation

Consequently,

the integral

(E) dE which

^{E}
a,(E)

may be solved explicitly.

m - Nvwtw

=

1.4

1

E

_

dS(E) dE

_

The solution is

/; sffl «(r) e,p [ - //

The One-velocity

«•

+

am

Boltzmann Equation

Let us return now to the original Boltzmann equation (see Art. 1.1) and seek methods which do not require the assumption of a space-independent neutron distri bution. Quite obviously there will be many engineering cases where such an approxi mation will be useless. However, there are often cases where the energy dependence of the neutron flux can be neglected. In these cases the Boltzmann equation reduces immediately to


of solution

V

• <,(r,0)

+ AW(r,Q)

=

f

JO'

JW.(0,0')*(r,Q')

da' + S(t,Q)

This equation forms the basis for a great many engineering calculations and is of particular value in treating the distribution of thermal neutrons in a reactor. The most common method for treating this equation is the spherical harmonics method which is discussed in Art. 2. The principal result of such treatment is the approximate reduction of this integrodifferential equation to a differential equation called the diffusion equation:

D V»* (r)

- N
= 0

D is the diffusion constant, a property of the medium. This equation can be explicitly for a number of cases of particular interest. It can also be shown that this equation yields solutions which are the asymptotic solutions of the onevelocity Boltzmann equation at points distant from boundaries and in low-absorption For this reason it forms the basis for many special methods discussed later media. on and has been used extensively for obtaining a first, reasonable estimate of the characteristics of many engineering systems. A special form of the one-velocity Boltzmann equation can be obtained from con sideration of the collision density
solved

*,(r) where

=

£

S(rOK(|r-r'|)
.S(r') is an isotropic source which may vary with position and where the kernel

is given

by

6-6

PHYSICS

REACTOR

[Sec.

C

=

g'(r)

+

/(r)

+

1

/

1

+

If

is

a

r'|

K(\t — r'|) is the product of the inverse-square-law attenuation which is characteristic of a point source times the exponential probability that the neutron will arrive at the without having point r, having traveled the path length |r — collision, times the probability of its having a collision at r. The latter simply the total collision cross section at r. all scattering and absorption events are now regarded as absorptions, with neutrons being emitted as the result of the absorption, then all collisions By definition can be regarded as first collisions. w'(r)

o-i (r)

v the number of neutrons emitted per fission. The equation for the collision density then becomes (where all collisions are regarded as first collisions)

=

f

*(r)

KQt

- 1-1)15(0

+

[1

is

where

+/(rO]*(r')l

dr>

2.

1.6

The Fermi Age Equation

- !)',_ -

1

(A

A

,

1

t

a

is

Let us examine now several techniques for obtaining approximate solutions to the Boltzmann equation where both space and energy variations in the neutron flux must be treated. Of these methods the simplest, and perhaps most direct, is the Fermi age technique. The age equation based upon the continuous slowing-down neutron loses an model, which states that in each collision with a moderating atom amount of energy characterized by the mean logarithmic energy decrement

is

£.

A

is

sufficiently large), the slowing down and that for sufficiently heavy nuclei (where of the neutron can be approximated by a continuous history with a parameter This picture holds only very roughly for light elements where a neutron may lose a For heavier elements the continuous slow large amount of energy in one collision. One important result of the con ing-down model is, however, very satisfactory. the identification of the slowing-down density tinuous slowing-down model q(t,E)

= ZN
which the number of neutrons slowing down below a given energy per unit volume. The diffusion equation, including a slowing-down source, written

E

per

unit time

Introducing

- NaMr,E)

'lq (T'E)

on

S(t,E)

=

0

V* (t,E)

+

D

+

is

is


is

is

Note, This the integral formulation of the one-velocity Boltzmann equation. however, that in the conversion of all collisions to first collisions the correlation between incident and scattered neutron directions has been lost. To this extent, this equation less general than the integrodifferential formulation of the one-velocity Boltzmann equation. Integration of the point kernel K(\i — r'|) over various geometries results in the derivation of simpler forms of kernels for use in restricted geometries. The forms of the line, plane, and cylindrical and spherical shell kernels are given in Art. 2.25 of Sec. 6-2 and in Rcfs.

the continuous slowing-down model and defining the Fermi age

the age

6-7

THEORY

REACTOR

Sec. 6-1]

equation with absorption results:

V*(r,r)-^,(r,r)-^+fl'(rIr)-0 U or If the resonance

escape probability

is defined as

(-//>.f?) then the

redefinition of the slowing-down density as q =

gives

p(E,E')q'

immediately the result that

vV'

(r,T)

-

dr

+

1 S'(r,r) p(E,E>)

= 0


This shows that the solution with absorption, upon proper treatment of the external source term, is simply the solution of the Fermi age equation without absorption Finally, since many cases of interest multiplied by the resonance escape probability. have a source term only at high energy and are source-free at all lower energies, the equation reduces simply to the most familiar form of the age equation:

vlg

(r,T)

_ aJj?A dr

= o

The similarity of this equation to that for the diffusion of heat should be noted. The quantity t plays the role of time in the heat equation. Because of this it is called the "Fermi age" in the present application. It is, however, not a time variable but rather has the units of length squared. It can be identified with one sixth of the second moment of the neutron distribution and is thus one-sixth of the mean square distance traveled by the neutron in being slowed down to energy E. 1.6

The Selengut-Goertzel

Equation

A very useful approximate solution to the Boltzmann equation has been given by Recognizing the basic difference between slowing down in Selengut and Goertzel.' hydrogen and slowing down in all other materials, these authors have treated the processes separately. The Selengut-Goertzel equation, in its usual form, is an exten sion of the diffusion equation in which that part of the source which is due to slowing down in hydrogen is treated exactly while the remainder of the source is treated with the continuous slowing-down model. Any correlation between the deflection of the neutron and the degradation resulting from a collision with a hydrogen atom is ignored. The resulting equation has the form

Dv»* (r,E)

- (Nhv.b

+ N^)4>(t,E) + S(t,E)

The Selengut-Goertzel equation may be solved numerically by incorporation into a multigroup formulation or by other numerical methods. It has proved to be reason ably convenient for a number of different kinds of problems. Note that the hydrogen Nivr,n appears added to the absorption AVa. scattering This method, while clearly approximate, nevertheless avoids treating hydrogen hy continuous slowing-down theory and thus avoids the major approximation incor porated in the otherwise useful age theory. The Selengut-Goertzel method has met with surprising (and perhaps fortuitous) success in the description of processes of several types in water-moderated systems. More exact methods, which are much

6-8

REACTOR PHYSICS

[SEC. 6

more tedious, sometimes give results which agree less well with experiment than does the Selengut-Goertzel method. For this reason the Selengut-Goertzel method occupies a special position because of its unexpected accuracy. 1.7

The Multigroup Equations

Another very useful approach to the energy- and space-variation problem is obtained the continuous neutron energy spectrum by a finite number of Within the rth group, which extends from £i_i to Et, neutrons are energy groups. assumed to diffuse according to the one-velocity diffusion equation until they have suffered a number of collisions (l/£) In {Et-i/Ei), after which they move on to the next group. The group equation may be written in the form

by approximating

D

V*

(r)

- iWM

+ S(r) +

^-^

= 0

The source term S(t) includes neutrons intro Here q is the slowing-down density. In the latter case, a* duced into the system from fission or by inelastic scattering. Integrating this equation includes inelastic scattering out of the group in question. over the energy width of the tth group yields the multigroup equation


OiV*

(r)

- Naaiiib)

+ Si(t) + q'(t)

- q°(t)

=- 0

q°(t) and g'(r) are, respectively, q "out" and q "in," the slowing-down densities at The constants Z); and AV„the low-energy and high-energy limits of the group. are now averages over the width of the group, and the source S,(r) is integrated over the group width. For a small number of groups, usually two or three, the equation is formulated as In particular, for two groups given above.

fli VVi (r) DtV'ti

- #(»„, + (r) - AV.2*,(r)

£,«.,)

i(*) + S,(r) + £,AV.i*,(r)

= 0

-0

where the subscript 1 indicates the so-called "fast" neutron group and the subscript 2 indicates the "thermal" group. However, for a large number of groups (over 40 have been used on occasion) the multigroup equation is usually written as an equation for the slowing-down density, using the continuous slowing-down model

yfe) ** w - (to)

*(r)

+ s<(r) + ?,(r)

- «°w

=

°

where the bracketed quantities are again averages over the group width and where the slowing-down density, q,(i), and the source are integrated over the lethargy width The lethargy variable is defined as of the group.

"

=

,

'"

pOMcvl

L^MeVjJ

need not be chosen to be 10 Mev; however, it has become customary to do so. For the highest energy group (the lowest lethargy), i = 1, and if q'(r) and Si(r) are known, then the equation for this group may be solved for g°(r), provided that some relation among
The reference energy

?. = w,q'

+ wtcf

REACTOR

Sec. 6-1]

6-9

THEORY

A number of ways of choosing these and Wi are constants to be chosen. is discussed in Ref. 4. Care is required in choosing the cross sections for each medium in each group. The correct formulation for these constants can be obtained from the averaging process For a very large number of groups the group constants are usually indicated above. For two-group work, however, calculated from the cross-sectional curves directly. the constants needed are usually taken from experimental data where possible, using where w\ constants

the relations

Dt =

Di

L*iV<7„,

= rA'(ff«i + £,
L* is the thermal diffusion area and r is the Fermi age of neutrons entering the energy group. Finally, it should be noted that the techniques of multigroup theory, while usually applied to the form of the diffusion equation discussed above, are not limited to this particular application and can be used with various other forms of the Boltzmann In particular, the Selengut-Goertzel equation has been used, as has the equation. form of the diffusion equation with a variable diffusion constant. In this latter case, the Laplacian term becomes div Di(t) grad
thermal

The Greuling or Diffusion Kernel Equation

An alternative formulation of the group picture stems from the integral form of the This formulation is most simply illustrated by one-velocity Boltzmann equation. If the medium is considering a point, isotropic, unit source of neutrons at position r0. infinite and nonabsorbing, then the solution of the one-velocity integral equation for the slowing-down density is 9i(r) = /fi(r,r0)

K\ is the point diffusion kernel and is denned as the probability that a neutron which is born at r0 will be removed from the group by an elastic scattering collision at r. The point diffusion kernel is similar in its effect to the point transport kernel discussed earlier (see Art. 1.4) except for the change in definition of the units involved. The slowing-down density and the collision density ^ are related by where

If the energy interval, Au, described by this equation for the slowing-down density is wide compared with the energy loss in one collision, so that the continuous slowingdown model applies, then the slowing-down density resulting from this first group may be used as a source for the next lower energy group of neutrons, for which the solution for the slowing-down density is immediately ?>«

-

/r,

^(r.rOS^r'.r,,) dr'

is

a

This process may be continued for any number of groups, the solution for the slowingdown density for the nth group being merely the convolution of the n slowing-down kernels which apply to the higher energy groups. In an absorbing medium very similar result can be obtained. The solution for qn as convolution of kernels valid for other geometric diffusion kernels provided that the material cross sections are independent of position in the medium. This particular integral form of the group equations uses the continuous slowingdown picture. Consequently, the Fermi age equation can be shown to be the limiting case of this group formulation as the number of groups goes to infinity. Also, as in the case of the space-independent integral equation, various forms of the kernels can be used for different physical models in order to give alternative, approximate approaches to the description of the energy and space variations of the neutron flux. a


1.8

6-10

PHYSICS

REACTOR 2

[Sec.

6

THE SPHERICAL HARMONICS METHOD AND THE DIFFUSION EQUATION

The one-velocity Boltzmann equation forms the basis for many calculations in engineering practice. The most common technique for treating this equation is the spherical harmonics method. In this article a number of approximations based upon this method will be discussed with emphasis on their relative validities in physical problems. The Spherical Harmonics Method

2.1

Basically the spherical harmonics technique consists of expanding the source and flux distributions in spherical harmonics of the cosine of the angle between the direction a and some principal coordinate direction. The source and flux distributions are then considered to be rotationally invariant with respect to this coordinate direction. If p is the cosine of the angle between the direction tt and the reference coordinate, then the flux is given by 30

m =-0

j_l

-

*»(r)

(T,n)P„(ii)

dp

/i

is

is

is

The source term treated in a similar fashion. The quantity Pm(*i) the Legendre polynomial of order m. The cross section
a.m

-

l)AW.O.)
P„Gx')*(r,/»')
■

0')P«(0 Q')d(a •

where

(2m +

f*t

£

H

r.

■

Q')

1

I

1.

is

is

is

The spherical harmonics method then assumes that all terms containing harmonics higher than Pn(/i) contribute negligibly to the result. This approximation to the The resulting approximate equation solution known as the P„ approximation. < multiplied in turn by each spherical harmonic Pniii) and integrated over — < Because of the orthogonality of the spherical harmonics, this results in a set of n + first-order differential equations.

n

The Diffusion Equation

2.2

B

=

+

A

is

it

In the Pi approximation assumed that, in the expansions, contributions from terms beyond the first two are negligible. It can be shown that this assumption implies that the cross section angular distribution can be expressed in the form cos (Q • O')

is

Also, because the higher and that the medium involved only weakly absorbing. harmonics express transient effects near boundaries, the Pt approximation assumes that the region in question several mean free paths from any boundary. The two resulting equations can be combined into a single equation for the isotropic component of the flux. is


where

REACTOR

Sec. 6-1]

D

V* (r)

THEORY

- 2a*(r)

where

+ S(r)

•

the average value of cos (O

Q')-

0

- a)

3z.a

where jt is

-

6-11

The source here has been assumed to be

isotropic.

This equation is the so-called diffusion equation and is of particular use in reactor it can be solved analytically for many geometries. It provides satis factory accuracy as long as the conditions for validity of the approximations mentioned above are met. In particular this equation finds use in describing the distribution of flux of any monoenergetic neutron group within a large medium such as a reactor The diffusion equation is also often used as the basis of the derivation of the core. The flux disadvantage factors within a unit cell of a heterogeneous reactor lattice. however, open to question, since most lattice cells validity of this last application contain strongly absorbing material and because the regions involved are not large however, the diffusion mean free path. Because of its convenience, compared with first approximation to these flux disadvantage very often used to derive equation factors. a

is

a

is,

physics because

2.3

The

P2 Approximation

It

can be shown that the P» approximation yields the same differential equation as that certain correction terms occur Pi approximation. The only difference to the constants in the equation. In particular, the diffusion equation resulting from the Pi approximation is

V'4> (r)

- 2„0(r)

+ S(r) =

0

is

D with

is

is

While this equation yields improved values of the constants, the generality of its application not greatly improved and the remarks in the previous section apply. Again, the approximation valid only for the case of weak absorption where

2.4

The P3 Approximation

is

is

is

is

a

is

a

is

is

it

If assumed that all terms beyond the fourth contribute negligibly to the result, distinct improvement in the generality of the solution obtained. In particular the equation which results for the flux now of fourth order and has solutions contain ing terms which affect the flux particularly in the region of boundaries. Experience has shown that the P3 approximation provides much more accurate description of the neutron flux within a reactor lattice cell. Generally the P3 flux dis tribution entirely satisfactory within the fuel element and fairly good in the sur rounding moderator. Comparison with higher harmonics (P6) Pi, etc.) has shown that in most cases of interest the P> approximation satisfactory and provides most of the needed improvement over diffusion theory. The Pj approximation involves the solution of large algebraic systems and usuaUy carried out on digital computing machines. should be recognized that although the Pj approximation effectively provides a correct description, for most physical cases, of the neutron flux distribution for a onevelocity group of neutrons, many cases of interest are not correctly described by such a one-velocity model. Because of its complexity the P3 approximation has not been widely applied to two-or-more-velocity approximations although some work has been done on two-group approach.6 a

It


does the

6-12

REACTOR 3

PHYSICS

[Sec.

6

OTHER TECHNIQUES FOR TRANSPORT PROBLEMS

In this section several techniques of a somewhat unconventional nature will be Historically, the solutions to most reactor problems in neutron transport considered. have been obtained with the methods which have been described in the previous sections. Recently, however, two suggestions have been made concerning other approaches to the above equations. These two ideas are the hemispherical harmonics method and the Sn method. In addition, an entirely different technique of great general applicability has been developed. The Monte Carlo method, as it is called, may become important in the future for the solution of complex problems of all kinds if the abilities of computing machinery continue to increase at their present rapid rate. 3.1

Hemispherical

Harmonics


The method of spherical harmonics, which has been discussed in some detail earlier, has proved of great use in many applications. However, the rate of convergence of the expansion in spherical harmonics is not known, and the success or failure of the method has been determined largely by numerical computations and by comparison with experiment. The complexity of solving the higher approximations in spherical harmonics has limited the use of the method, in most cases, to the P3 approximation. It would be desirable to have a method which would retain the advantages of the spherical harmonics method but which would converge more rapidly, thus yielding equivalent accuracy with less algebraic complexity. It has been suggested that in most cases of interest the majority of neutrons flow in the directions of the principal coordinate axes. That is, the boundaries between regions are often surfaces of constant radius r or displacement x. Consequently, the gradient of flux will often be directed perpendicularly across these boundaries. The spherical harmonics method, however, expands the flux in a series of harmonics containing some components which are largely in a direction parallel to the boundaries. Improved convergence might be obtained if an expansion were used which eliminated This is what the components of importance which were parallel to the boundary. hemispherical harmonics method attempts to do. Instead of a single expansion in p, the cosine of the angle between the velocity vector and the principal coordinate direction, the flux is expanded in two different ways. That part of the flux with p between —1 and 0 is expanded in harmonies of the variable (2ji -f 1), while that part of the flux with n between 0 and 1 is expanded in harmonics of the variable (2/x — 1). In this way two complete expansions are derived, the sum of which represents the neutron flux as a function of angle at the space point in question. As with the spherical harmonics method, these expansions are trun cated arbitrarily after a given number of terms to achieve an approximation. For consistency, the two expansions are cut off after the same number of terms. Because the functions involved are still Legendre polynomials, the approximations are called P„,„ for the case where each expansion is cut off after n + 1 terms. Although not much work using this approximation has appeared in print, it is characteristics, with a expected to have considerably improved convergence approximation yielding accuracy comparable to a P3 approximation in the spherical harmonics method, but with considerably reduced algebraic complexity. Unfortu nately, however, the hemispherical harmonic equations are not analytically solvable in cylindrical geometry. Because of the great interest in this type of configuration, however, numerical solution of the Pn,« equations will probably be attempted for this case in the near future. An additional improvement is to be expected from the hemispherical harmonies method in the treatment of black boundaries. Because the hemispherical harmonics do not contain flux components parallel to most boundaries, a more correct descrip tion is to be expected in the vicinity of absorbing boundaries. The method of hemispherical harmonics is sometimes referred to as the method of

fj.j

Yvon.'

Sec. 6-1]

REACTOR 3.2

THEORY

6-13

The Sn Method

The Sn method of Carlson7 is still another approach to the solution of the steadyBoltzmann equation. It is usually applied to the one-velocity equations which are used in multigroup theory. However, it does not require the diffusion approxi mation but is applied to the case where, for each energy group, the one-velocity Boltz mann equation holds with but two restrictions. The source term is taken to be iso tropic, and energy degradation is taken to be independent of the deflection angle of the scattered particle. The SH method then consists of approximating the flux as a function of direction, 0(r,/i), by a series of straight-line segments state

♦
-

"


where

~

w

*(r,M,) +

i

-w-\ 0,

< m 1, 2

~* *(r,w-,)

m n

The order of the approximation is characterized by the number of line segments n This approximation is- then used to reduce the one-velocity Boltzmann which is used. An additional equa equation to a set of n equations in the n + I variables 4>{x,m). tion is obtained by setting m m — 1 directly in the one-velocity Boltzmann equation. The resulting set of equations is then solved numerically for the fluxes, an initial The equations for the various energy groups guess for the sources having been made. are solved in order of decreasing energy, and a new approximation to the fission source is obtained. The process is continued until an acceptably small change is obtained between two successive source calculations. The Sn method has been applied to time-dependent problems as well as to the steady-state problem outlined above. For the stationary case the anisotropic problem (which includes the correlation between scattering angle and energy degradation) has been solved for spherical geometry. The chief value of the S„ method would appear to be its ease of adaptation to cal culation on digital machines. It has been applied to a large number of varied prob lems, chiefly at the Los Alamos Scientific Laboratory. One over-all comment is applicable to the 5„ method. Although it is usual to choose the n intervals in n as of equal size, it is possible to choose unequal intervals. For a given value of n the most efficient choice of intervals would appear to be that given by the Gaussian integration formula,8 and it can be shown that this choice of Gaussian intervals will yield values for the isotropic flux exactly equal in accuracy to those obtained with the spherical harmonics approximation with the same n. (The angular distributions will, however, not be the same as given by the spherical harmonics Therefore, it would appear that the S„ method would be less efficient than method.) However, it may have advan the corresponding spherical harmonics approximation. tages in ease of application which would outweigh the reduced efficiency in convergence

of the approximation. Such a tech Sykes9 has reported a method using a double Gaussian integration. nique would appear to be a numerical equivalent of the hemispherical harmonics method of Yvon. 3.3

The Monte Carlo Method

A new technique, which is of a different nature from any discussed so far, is gaining importance for use in transport problems as more powerful calculating machines become available. This technique applies the statistical methods of random sampling to those physical and mathematical problems to which a probability analysis is applicable either directly or as an analogue. Because of the random-sampling approach, in which random numbers are actually used, the technique has acquired the name of the "Monte Carlo" method.

6-14

PHYSICS

REACTOR

[Sec. 6

is

is

a

is

is a

a

a

is

A

if

is

is

P

is

a

it

is

is

P

of the number of case histories necessary to calculate the probability The assumption made that ordinary sampling process to within the relative error e. procedures are used. desired to know the prob Consider now a problem in reactor shielding, where shield of some arbitrary composition and ability of a given radiation penetrating may he of the order of 10~10, so that .V For cases of engineering interest, shape. Physically, 100 per cent error may be tolerable. prohibitively large even though being made to get statistical accuracy in the num the difficulty that an attempt ber of radiations penetrating the shield. But a very large number of radiations must be followed in order to find just one which succeeds in getting all the way through. The situation made still more difficult, the composition or geometry of the shield complex, since individual particle histories then become tedious to calculate. considerable amount of work has gone into obtaining improved mathematical tech Basically, these niques for reducing the number N of histories which are required.10 improvements deal with sampling methods which sample more efficiently the particles having larger probability of penetrating the shield. The improvements also attempt to obtain more information from given history or to perform a suitable transformation of the problem into a related one in which the probability of success Here iV

P

considerably larger. As a second example, consider the calculation of the resonance escape probability short, Here each history in a heterogeneous lattice moderated with light water. small number of collisions are required to carry neutron completely since only about through the resonance region. Furthermore, the probability of success 0.9, so that 10' to 10' histories will determine the resonance escape probability to an is

P

a

a

is

is


it

if

4

if

it

is

is

is

a

is

is is

a a

is,

The history of a neutron in a chain reactor may be represented accurately as a A cross section of statistical events. by definition, the probability that neutron will suffer a collision in traversing a given distance of material. Thus, neutron may be introduced into a system at some random point with an energy chosen at random from the distribution of fission neutrons and with random orientation. random selections are then made to determine, with appropriate dis Successive tributions of cross sections and other material constants, when and where the neutron scattered or slowed down or absorbed. The process continued until the neutron captured or escapes from the system. The average characteristics of the system, such as neutron diffusion, aging, and attenuation, can then be obtained from a statis tical average of many such histories. In general, the Monte Carlo method may be regarded as technique for evaluating Its chief virtue becomes integrals or for solving integral or differential equations. evident for problems involving many dimensions in that the difficulty of solution increases roughly with the number of dimensions rather than with the nth power, the number of dimensions, which where n the case for a numerical technique such In practice, turns out that Monte Carlo superior to other as Simpson's rule. numerical approaches the dimensionality of the problem exceeds or 5. With the Monte Carlo method, in principle, the fundamental probabilities of the events are known (cross sections, etc.), any problem can be solved to any desired degree of accuracy. However, very time-consuming sampling and statistical cal culations are often required. Consequently, Monte Carlo has been applied primarily to problems which have been prohibitively difficult to handle by other methods. In may still be necessary to restrict the Monte Carlo analysis to even these cases certain aspects of the problem or to perform supplementary approximate calculations or experiments on aspects amenable to simpler treatment. It illustrative to consider two problems to which the Monte Carlo method has been applied. The very different requirements of these problems are indicated by the following formula, which derivable from the binomial probability distribution: sequence

THEORY

REACTOR

Sec. 6-1]

6-15

4

THE TREATMENT OF BOUNDARIES IN NEUTRON TRANSPORT is

a

it

a

Once solutions have been obtained for specific material by any of the analytical methods discussed above, necessary to be able to join solutions for different media and to include geometrical restrictions on the size of the system. Although very simple methods can be used to achieve these definitions of the geometrical problem, there are more subtle techniques which can materially improve the accuracy of calculation. 4.1

Simple Boundary Conditions

is

a

The fundamental condition for matching solutions across boundary between regions that of physical continuity. In particular, for the Boltzmann equation this condition may be stated that the flux, (v,Q,i), must be continuous across a boundary for all Q and v. For the more practical reductions of the Boltzmann equation, this condition extends directly. Consider, as an illustration, the spherical The condition of continuity requires that the components of harmonics equations. the flux, into which the total flux has been expanded, shall each be continuous across In the Pi and Pi approximations, the one-velocity Boltzmann equation boundary. reduces to two first-order differential equations for the flux components o(r) and where

a

i(i),

is

it

is

r

i(t)

is

is

a constant, The ô(r) component merely the integrated isotropic flux, since Po(n) component of the independent of m- Similarly, since Pi(n) equal to ji, the Con flux direction. directly interpretable as the net neutron current in the sequently, the boundary conditions for the spherical harmonics method in the diffusion approximation amount to requiring continuity of flux and neutron current across a In practice, sufficient for criticality calculations to reduce these two boundary. conditions to requiring that the diffusion constant times the logarithmic derivative of is


is

a

is

it

in

is

A

is,

is,

This problem lias been carried out successfully using ordinary accuracy of 1 per cent. random sampling." It should be noted that although this is a case peculiarly suited to Monte Carlo calculation, several hours of Univac time are required to complete a single computation of resonance escape. One further comment on the Monte Carlo method is appropriate. Suppose that a By randomcalculation is to be performed of the reactivity of a nuclear reactor. in principle, possible to determine the average number of sampling techniques it neutrons produced by each neutron introduced into the system (that the reactivity). But the spatial and energy distributions of the neutrons will not be determined accu rately by this calculation. very much larger number of histories would have to be followed in order to determine, to satisfactory accuracy, the flux of neutrons of each On the other hand, an analytical or conventional energy at each point in the reactor. numerical solution of the Boltzmann equation will give not only the over-all char acteristics of the system but the distributions of the neutron flux. This because, the analytical case, the over-all characteristics are computed from the solutions for the fluxes. From this comparison may be seen that the Monte Carlo method useful primarily where the desired results are limited number of specific system characteristics rather than the complete solutions for the system. The present time-consuming Monte Carlo method will, in the future, become prac tical as better computing machines develop and as improved sampling techniques provide greater calculational efficiency. With the rising cost of experiments and with the inevitable increase in the complexity of engineering nuclear systems, the Monte Carlo method can be expected to be brought increasingly into use. This will be true even though the first wave of enthusiasm for the method now over and the practical difficulties are clearly realized.

6-16

REACTOR

PHYSICS

[Sec.

6

the flux

D- V* be continuous across an interface. Of course, the condition of continuity of flux and current (or of higher components of the flux in other approximations than the diffusion approximation) assumes that the interface is passive. In the event of an active interface, which may be either an

absorber of neutrons or a source, an obvious conservation relation replaces the above continuity conditions.


4.2

Extrapolated End Points

Special treatment is required for the case of a black boundary or its equivalent, a boundary with an infinite region of vacuum. The former case may arise in the treatment of strongly absorbing media (such as control rods), and the latter may arise during the treatment of bare reactors where the active lattice has an outer boundary in common with a region assumed to be of infinite extent and which has the properties of a vacuum. The obvious boundary condition which may be imposed is that the flux shall go to zero at the boundary of such media. However, in the adjoining active region the flux does not physically go to zero at the boundary, since there will be a streaming of neutrons toward the infinite sink represented by the black or vacuum region. The solution to this seeming paradox has been obtained, in practice, from an examination of the desired quantities in the finished calculation and takes the form of the so-called extrapolated end point. What is desired in calculations of criticality and in most flux distributions is a In such correct treatment of flux at relatively large distances from a boundary. regions, the transient terms in the solution to the equations will have disappeared and only the asymptotic solutions will remain. It would be desirable, thus, to require that the boundary condition imposed produce a correct asymptotic flux distribution. In the commonly used diffusion approximation, the solution to the flux equation contains a limited number of transient terms and gives a physically correct description only at points far from boundaries. Put another way, the diffusion approximation can be shown to be the asymptotic solution of the transport equation in weakly absorbing media and away from boundaries. Consequently, it is desired to achieve a boundary condition such that the diffusion-theory solution will approach the correct transport solution at interior points in the region. Extrapolation of the correct asymptotic or diffusion solutions through the boundary and into the adjoining medium shows that if the flux is required to go to zero at this extrapolated boundary, then a This treatment, however, will not correct asymptotic description will be obtained. yield a correct description of the neutron flux in the vicinity of the boundary. If the physically meaningful boundary condition is imposed that the current return ing from the black or vacuum region should be zero, then solutions may be obtained which yield quantitative estimates of the size of the extrapolation distance which is A linear extrapolation of the diffu to be used with a zero-flux boundary condition. sion solutions shows that, for a plane boundary, the extrapolation distance is ?^X,r. However, consideration of more accurate approximations to the Boltzmann equation and its solutions shows that a more correct extrapolation distance for the asymptotic The quantity \,r is the transport mean free path of the active solution is 0.71X(r. medium and is equal to three times the diffusion constant D. Thus for a bare reactor, the flux is required to go to zero at an extrapolated boundary which is 0.71Xir into the adjoining vacuum medium. The above values for the extrapolation distance apply for a plane boundary and for the case of small capture and isotropic scattering. Similar results for plane boundaries, with other conditions of capture or scattering, are given in Refs. 12. For most cases, however, the value of 0.71Xir is adequate. For curved boundaries the extrapolation distance varies as a function of the radius of curvature. For a black sphere, the extrapolation distance increases monotonically This is physically because a to %)*r, in diffusion theory, as the radius goes to zero.

THEORY

REACTOR

Sec. 6-1]

6-17

black region (such as a sphere) depresses the flux in the adjoining medium than a boundary which is very large (such as a plane). The use of diffusion theory with the appropriate extrapolation distances for the boundaries involved will give rigorously correct values for the flux a few mean free For a very large class of cases, this treatment will yield paths from the boundary. satisfactory accuracy. restricted less

4.3

Serber-Wilson

Boundary

Although the Serber- Wilson method is a technique for performing criticality cal it is included in this section because it is primarily a special case of the conditions to be applied to transport problems. The Serber-Wilson method applied to the calculation of small, reflected reactors with considerable Although this has been its major use, it has also been applied to other success." problems, notubly the calculation of the thermal flux distribution in a unit lattice cell of a heterogeneous reactor.14 The Serber-Wilson method is primarily for one-velocity problems and deals with the The asymptotic forms of the solu asymptotic solutions to the transport equation. tions are assumed in each of the media concerned, and special boundary conditions are used to assemble these individual solutions into a single description of the physical Because conservation of neutrons is implied by the continuity of net cur system. rent across a boundary, this condition is retained from the usual diffusion-theory However, for the simplest case of a spherical reactor with an infinite approach. reflector, the usual continuity of flux is not required, since, with a true solution to the transport equations, the asymptotic or diffusion solutions are not continuous at the boundary. For this simple example, only one additional equation is needed to The so-called "Serber condition" replace the discarded flux continuity condition. which is used requires that the one-velocity Boltzmann equation be satisfied exactly at the center of symmetry of the system, which is the origin in this case. This con dition then allows the system to be completely described. For this simple case, with a single infinite or finite reflector, numerical results are easy to obtain and are given in Ref . 13 in convenient form for the solution of problems. For systems involving more than one intermediate boundary it is necessary to obtain such an additional equation for each boundary. This is accomplished by In the simple spherical case, satisfaction satisfying the Serber condition "in detail." of the Serber condition in detail results in the so-called "Wilson conditions," which require that the flux in the one-velocity Boltzmann equation (r,ii) be continuous at all boundaries for the value >i = —1. These radially inward fluxes are, however, to be computed as if the two adjoining media were infinite in extent. Although the Serber-Wilson method has been useful for calculation of reflected Consequently, systems, it has not generally been extended to the many-velocity case. it can be helpful only where the nature of the system makes a one-velocity approach a reasonable approximation to reality.


culations, boundary has been

Albedo

4.4

The albedo is a concept which has been of use for those systems where one-velocity diffusion theory is adequate. It is simply defined as the ratio of the current density out of the medium in question to the current density into the medium in question. The albedo is thus the fraction of neutrons which are scattered back across the bound ary after any number of collisions and is, to this extent, a "reflection" coefficient. For the diffusion-theory case, the albedo can be shown to be a property of the reflecting medium only and, for this reason, allows certain simplifications to be made in the treatment of boundaries between a reactor core and a reflector. Inserting the defini tions of neutron current into the definition of the albedo gives immediately Albedo =

/3 =

1 1

+ (2D/4>)(d4>/dz)

-

(2D/4>)(d4,/dx)

6-18

PHYSICS

REACTOR

[Sec.

6

2x1) coth xa

~ 2xD

2xD

+

1

-

1

g

a is

where the slab thickness, including the extrapolation distance, and x rocal of the diffusion length. For an infinite slab reflector this becomes

It

is

1

- 2x1) coth xa

+

_

1

a

ij>

Because of the required continuity of flux and current in diffusion theory, the quanti ties and D d/dx may be chosen for either medium. For finite slab reflector the solutions of the diffusion equation can be inserted directly to give

the recip

is

will be noted that the albedo of a finite slab less than that for an infinite slab of the loss of neutrons by escape from the outer boundary. When the reflector more than about two diffusion lengths thick, a finite reflector essentially indistinguishable from an infinite reflector. For spherical boundaries the expression for the albedo will be different, with the This albedo of a medium being less when surrounds because spherical source. neutrons which get into the reflector an appreciable distance have a smaller solid Thus the angle into which they can be scattered and still return to the source region. albedo will depend both on the properties of the medium and on its geometrical shape. For other shapes see Art. 2.3 of Sec. 6-2. good reflecting medium will have a diffusion coefficient small compared with its The albedo diffusion length, as may be seen from the above expressions. 90 per cent or over for infinite slabs of good reflecting materials such as heavy water or Because of its absorption, an infinite, light-water slab has an albedo of graphite. only 0.821 for thermal neutrons. For The albedo can be used to replace other forms of boundary conditions. plane boundary is

is

p

J

a

and the extrapolation distance at the boundary

+ is

L

2D

1

dx=

For the

is

5

it

is

is a

it

is

zero and this case of a nonseattering absorber or of vacuum, the albedo equation reduces to the extrapolated end point treated earlier. limited in its reactor conceptually pleasing quantity, the albedo While Because adequate. applications to cases where the one-velocity diffusion picture has not found wide application to the more recent reactor systems, par of this, ticularly those containing light water where important boundary effects like the rise in thermal flux at a core-reflector interface can be described only on a multivelocity The albedo has, however, been useful in shielding work." basis.

REACTOR

EQUATIONS

a

a

is

is,

Many problems in reactor engineering may be solved as transport theory problems they take no cognizance of the multiplying nature of the system. That directly. necessary in performing criticality calculations to include this important However, The modification and extension of the methods of transport theory characteristic. valuable tool for to include these problems will be treated in this section, along with chain-reacting system. predicting the effects of small disturbances on it

5.1

The Reactor Equations and Their Relation to Transport Theory

The methods of transport theory have always included a provision for an arbitrary of neutrons. In the reactor equations, this source assumed to come from

source

is


is

A

a

it

is

because

THEORY

REACTOR

Sec. 6-1]

6-19

born in fission. External sources arc normally not included in critical In this section, although they are often used during the approach to critical. however, only systems which are critical will be discussed, so that no provision for an external neutron source will be made. In any reactor, the number of neutrons introduced into the system is proportional to the number of fissions, which, in turn, is proportional to the flux of neutrons which For a uniform thermal reactor, then, the source of neutrons will be produce fission. proportional to the thermal flux at each point. Consider the simple case of a one-group diffusion equation. The source of thermal neutrons will be proportional to the flux, the factor of proportionality being the absorp tion cross section times the number of neutrons produced for each one absorbed at the The one-group reactor diffusion equation is then point in question, or kx. neutrons

systems,

D

V* (r)

+ JW.(A„

-

l)*(r)

= 0

earlier, however, the one-group picture does not provide an description for most engineering reactors. In a two-group picture, the source of fast neutrons is proportional to the thermal-neutron flux, the constant of Here p is the resonance escape proba proportionality being equal to (A;„/p).Vo-„2. bility. Assuming that resonance capture occurs between the fast and slow groups, the source of thermal neutrons is proportional to the fast flux with a constant of of pA{i

adequate

C V'i

-

(r)

AT(«r«,

+ *,
-

Dt VVs(r)

^

AVo2*2(r)

P AV..*2(r) + p.V|,
= 0 = 0

equation and the homogeneous part of the two-group equations written in the form of a wave equation:

The one-group may be

V«4>(r)

+

BVW

= 0

Solutions of this equation for various geometries are well known.18 homogeneous solution in the two-group equations leads directly to

-lDtB'

+ N(*ai + £i
-

pAr£,<7„0i

^=

P

No.rti

(Z)2B» -(- AVa2)tf,

Use of this

= 0

-0

the constant BJ is taken to be the same for both groups. By Cramer's rule, condition for a nontrival solution leads immediately to the two-group critical equation (for small absorption in the fast group) where the

fc. = (1 +

rB»)(l + L'B»)

The corresponding one-group equation is kK

-

1

+ M-B*

In both of the above critical equations the following relations have been used r =

-J^-

L> =

M» =

:

IS + t

Obviously, almost any transport theory approach can be used to produce solutions The approach can be made equations and to give a critioality equation. either through the integrodifferential formulations or through the integral equation formulations of the Boltzmann equation. One further form of the critical equation is worth noting here. On the basis of age theory (or from the integral equations, using Fermi age or Gaussian kernels) the rigorous critical equation can be shown to be (no to the flux

6-20

PHYSICS

REACTOR

"group" approximations having

[SEC. 6

been made)

k.e-B^ =

1

+ L*B*

For a large reactor where the buckling B* is small, all the above equations can be reduced to the form of the one-group equation = 1

+ M*B*

In general, the remarks concerning the validity of the diffusion equation which were made earlier are still pertinent for criticality work. The diffusion equation is It will give correct results only an asymptotic solution of the Boltzinann equation. for a bare reactor if the flux is required to go to zero at a correctly chosen extrapolated Since the solutions for the various energy groups of neutrons must be boundary. required to go to zero at the same boundary, the above equations assume that the extrapolation distance is independent of neutron energy. This assumption is adequate for large reactors but may cause difficulty for the case of reactors which are not very


large compared with the extrapolation distance. Finally, it should be noted that the constant B*, as a property of a medium, is called the "material buckling." However, as a solution to the wave equation for a given geometry, this constant is referred to as the "geometrical buckling." In order that a reactor be just critical and that the properties of the medium be just that required by the geometry, the material buckling and the geometric buckling must be equal. In this particular case, they are both referred to as the "critical buckling." This condition forms a simple, alternative statement of the criticality equation B'cri. 6.2

£„' =

=

Bm*

The Adjoint Equation

Although it is a purely mathematical artifice, the concept of an adjoint flux is very useful in evaluating the effect of small disturbances on a reactor by means of per turbation theory. The adjoint equations are usually based upon the group reactor equations discussed above. If the group fluxes are represented by the vector then the reactor equations may be written in the matrix notation M«(.

-0

The adjoint of the matrix M may be formed by taking the complex conjugate of each matrix element and interchanging rows and columns. The resulting adjoint matrix M* will then have eigenfunctions analogous to the neutron flux vector These eigenfunctions are defined by an eigenvalue equation similar to the reactor equation

.

M**

= 0

The eigenfunctions of the adjoint matrix are referred to as the "adjoint fluxes." equations for the two-group adjoint fluxes arc thus

-

Z>iVVi*(r) 2>s

N(
V*s*(r)

These equations may be solved The neutron flux equations. physical significance. For reactor work, the most adjoint matrix may be shown to conjugate or adjoint)

fr

+

i,,*(r)

- AV„20s»(r)

The

+ pA'£,
^ P

tfa„2*i*(r)

= 0

in a manner entirely similar to that employed for the resulting adjoint fluxes, however, have little direct

important property of the adjoint fluxes and of the be (coming directly from the definition of a Hermitian

dV

=

fy 4»M*V

dV

REACTOR

Sec. 6-1] where

t

and

two particular

<(»** are

THEORY

6-21

eigenfunctionB of their respective eigenvalue

equations.

This property makes possible the derivation of perturbation theory. Perturbation

5.3

Theory

Let the reactor equation matrix be perturbed by some small change AM in such a that the reactor is just maintained critical. The reactor equation under this

way

new

condition is then

(M +

'*,

AM)4>'

= 0

the fluxes are those corresponding to the perturbed condition. Multiplying equation by subtracting the product of the perturbed neutron flux and the adjoint equation, and integrating the result over the volume of the reactor, the result is where this

jr

dV

- jv

$'M*$*dV

+

Jr

* AJAQ'dV

= 0

The first two terms are equal in magnitude because of the properties of the adjoint fluxes and the adjoint matrix. Consequently, the perturbation equation is «J)*AM

dV

-0

This equation provides the basis for evaluating the changes which must be made, for example, in a control-rod setting to offset a change in some other reactor parameter. In essence, the perturbation relation gives the relative weighting of small effects in the reactor. If this small effect does not change the neutron flux appreciably, the unperturbed flux $ may be used as an approximation to the perturbed flux. The relative weighting of perturbations to a reactor may thus be stated to be the integral of the product of the adjoint flux times the neutron flux over the region of the per turbation, or

f

*AM

dV

= 0

For the special case of a one-group approximation, the reactor matrix M is selfadjoint, and perturbations are to be weighted as the square of the neutron flux, except for changes in the diffusion coefficient. These latter are to be weighted as the square of the flux gradient. By the same approach it can be shown that if the reactor is large (or k — 1 small enough), regions of different buckling in a reactor are to be weighted as the square of the flux integrated over each region of constant buckling. 6

THE FOUR-FACTOR FORMULA

a

is,

The four-factor formula provides a very useful conceptional approach to neutron multiplication in a chain-reacting system. Standard formulas for calculation of the four quantities involved are well known and are presented in Art. 9 of Sec. 6-2. It however, necessary to realize that this physical picture is, in many cases, only rough approximation, so that the formulas involved are sometimes inadequate for cases of engineering interest. In this section the four-factor formula will be discussed from the point of view of its validity and its application to cases where the usual approximations fail. 6.1

The Physical Concept a

In the four-factor formula the physical processes within reactor are pictured as follows: All fissions in the primary fuel are caused by neutrons of thermal energy. 2. The neutrons produced in fission are all fast. The number of neutrons entering the resonance absorption region of fast neutrons produced equal to the number tj

is

1.


Jr

6-22

REACTOR

PHYSICS

[Sec.

6

per thermal absorption in fuel, times the fast fission factor t. The fast fission factor is the number of fast neutrons slowed down below the fast fission threshold per fast

neutron introduced into the system. 3. Of the neutrons entering the resonance region, a fraction p escapes resonance absorption. 4. The resulting thermal neutrons are then absorbed in fuel with a probability /, as opposed to their being absorbed in all other materials. This neutron cycle was conceived primarily for use in graphite or heavy-water reactors and is very often adequate in reactors of these types. The neutron mul tiplication k„ for an infinite region of the lattice in question is the product of these four factors = Vtpf fc» and is a useful quantity reactors.

as a first step in proceeding to calculations


6.2

of finite-sized

The Fast Effect

A fraction of the neutrons produced in fission are at energies sufficiently high to These neutrons will also be able produce fission in materials such as U"8 or ThS3S. to produce fission in isotopes such as U2", but the relative effect of these latter mate rials will be small, since their relative abundance in low enrichment reactors is small and since their microscopic fission cross sections at high energies are only comparable to the cross sections of other materials. In the conventional formulas for the fast effect, it is assumed that a fission neutron will produce fissions only in the lump of fuel in which it is born. If it escapes from this lump without interaction, it is considered to be lost from the high-energy region. Corrections may be made for neutrons which leave a lump but are immediately scattered back into it. This formulation of the fast effect has been developed exten sively for a number of fuel geometries. As long as a fission neutron has a negligible chance of striking some lump of fuel besides the one in which it was born, the traditional approach to the fast effect holds satisfactorily. However, some reactors, notably those moderated with light water, have their fuel elements very close together, so that a very significant number of fissions in the high-energy region is produced by neutrons from one lump entering an Where this effect is small, a correction may be derived for the inter adjacent. lump. However, action fast effect and may be used to modify the conventional approach. in many reactors of engineering interest the fuel lumps themselves are small enough and close enough together that the majority of the fast effect is produced by this interaction type of phenomenon. In this case, it has been found more successful to approach the fast effect from another standpoint. The majority of the fast effect in a close-packed lattice is essentially that effect which would have been produced in a homogeneous mixture of fuel and moderator Hence, a homogeneous fast-effect calculation can of the same average composition. be made and can then be corrected for the alight increase in fast effect due to the heter This approximation has been found to give good agreement ogeneity of the lattice. with measurements.17 Fast-effect measurements for light-water uranium lattices are much larger than would be expected from the conventional formulas.11 6.3

Resonance Escape Probability

The normal formulation of the resonance escape probability p depends almost entirely upon the evaluation of an effective absorption cross section for the fuel in its actual geometrical configuration. This effective cross section has been measured for several specific cases,19 and these measurements have given rise to the empirical This empirical surface-and-mass formula conventionally used in such calculations. approach to the calculation of resonance absorption relies upon a formula which breaks the effective resonance cross section into a strongly self-shielded surface term Unfortunately, the problem of arriving and a weakly self-shielded volume term.

Sec. 6-1]

REACTOK THEORY

6-23

at the constants in this formula, starting from fundamental cross sections, is very difficult because of the very largo self-shielding of the absorption resonances. Care must be taken in the use of the effective resonance integral formulation for If epithermal absorption in Um is neglected, then a compensating resonance escape. In correction must be made in the epithermal absorption properties used for U,,s. particular, the relative cross sections used for these two isotopes in the thermal region

adequately account for the competition between the l/v portions of their absorp sections even in the resonance region. Thus, as a minimum, if epithermal fission in U2" is ignored, the l/v component of the Ui3* cross section should be sub This treatment still ignores absorption in the tracted from the resonance integral. lT,3S resonances which are, however, not very pronounced. Note, however, that when explicit account is taken of epithermal U*3' absorption, the l/v correction to the U"' mass absorption must not be made. For the case of a close-packed lattice, where a neutron has a reasonable probability of traveling from one fuel lump to another fuel lump without encountering a moderator In these lattices, atom, the concept of surface absorption is subject to correction. the uranium atoms at the surface of a fuel lump are partially shielded by the uranium atoms in adjoining fuel lumps. That is, the neutron spectrum impinging on the lump is not that produced by full moderation but is partially depressed in the energy region of the strong absorption resonances by the presence of other lumps. Such an inter action correction," similar in mathematical formulation to the correction for inter action fast effect, should be applied to all lattices which are sufficiently tightly packed. The justification for the use of an empirical formula for resonance absorption is simply that for a heterogeneous lattice there is no other adequate way of evaluating such a quantity. In a difficult geometry the mathematical complexity is just too Some progress is being made, however, with the use of high-speed digital great. computers in the calculation of resonance escape from fundamental cross sections by use of the Monte Carlo technique." It is to be hoped that a sound theoretical basis for investigating resonance escape in geometries of engineering interest will result from this work. It has already been reported*' that an energy-dependent cell calculation, using Selengut-Goertzcl theory in the moderator and an appropriate boundary condition at the surface of the fuel rod, gives excellent agreement with the results of the Monte Carlo calculation. With 'he aid of digital computers such an approach may well provide the engineering tool will


tion cross

that is needed. 6.4

Thermal Utilization

The fraction of thermal neutrons captured in fuel material (usually uranium) is thermal utilization. Calculations of the thermal utilization by conventional methods arc approximate chiefly because of the use of inadequate methods for com puting thermal neutron flux distributions and because of the approximate nature of the description of the energy distribution of thermal neutrons. In many cases diffusion-theory formulas are used to calculate the disadvantage factors with which to weight cross sections. The disadvantage factor is an expression of the relative weighted average fluxes in the different materials and is dependent essentially upon the ability to calculate correctly the detailed thermal flux distribution within the reactor lattice. For many cases of interest, diffusion theory will not give this flux distribution to adequate accuracy. Various other approaches have been used to obtain adequately correct disadvantage factors. In particular, the Pt spherical harmonics approach has been found to be extremely useful." The Serber-Wilson This method appears to yield technique has also been used for lattice calculations.14 improvements over diffusion theory but it is not expected to yield as accurate results a« it docs in crit icality calculations because of the much greater importance of higher order terms in the solutions for the flux in a reactor lattice cell. The second difficulty with calculations of the thermal utilization arises from the energy distribution of thermal neutrons. To begin with, this distribution is affected by moderator temperature and by the absorption in the lattice. If all cross sections c alled the

6-24

KEACTOH

PHYSICS

[Sec.

6

/


varied with energy in the same way (usually l/t>), these spectrum effects on would be However, the cross sections of many fuel materials are decidedly relatively minor. not l/v and, to this extent, provide a need for the calculation of the neutron flux spectrum in some detail. This is particularly important for reactors containing appreciable amounts of plutonium because of the pronounced resonance in the cross sections of this material just above the thermal-energy region. For such materials, in order to choose cross sections correctly, it is necessary to average the energydependent cross-section curves over the neutron distribution. The calculation of thermal-neutron spectra is exceedingly complex because of the effect of molecular binding on the scattering atoms. To date most calculations have dealt with gaseous moderators and no adequate treatment is available for the cases of real interest. Methods for getting a temperature characteristic of some equivalent Maxwellian distribution are often used. This effective temperature problem is dis cussed in Art. 7, Special Reactor Problems. A further complication in the determination of neutron spectra for use in thermal utilization calculations is the fact that the spectrum is not constant throughout the lattice but will be of somewhat higher energy (usually referred to as "hardened") in the regions of low moderation and high absorption. Both for the choice of cross sections and for the calculation of the flux distribution, it is the neutron spectrum locally incident upon the atoms in question that is desired. Hence, it is necessary, ideally, to know the spectrum at each point in the lattice. Calculations of these hardening effects have been made, but generally the variations of the neutron spec trum throughout a lattice are not of first-order importance and are usually neglected in reactor calculations, except for reactors of unusual type. 6.6

Fast Neutrons per Thermal Absorption

The calculation of the number of fast neutrons produced per thermal neutron absorbed in fuel is usually straightforward. It is made complex only by the fact that the ratio of capture to fission in fissionable materials varies slightly with energy. Thus, r) is somewhat dependent upon the neutron spectrum. For materials other than plutonium this variation is relatively minor. For plutonium the presence of the 0.3 ev resonance, with its accompanying change in the value of capture-to-fission, again demands a considerable knowledge of the thermal neutron spectrum in order to obtain a satisfactory value for i). The value of tj for Pus", for example, is given as However, because tj is lower in the large 2.03 at 2,200-m/sec neutron velocity." resonance at 0.3 ev, the value of i; for Pu"' in a typical power reactor will be about 1.85. 6.6

Inadequacies in the Formulation

The above discussion has primarily involved improved methods for calculating the various components of the four-factor formula. It is important to realize, however, that the four-factor formula itself contains some important inadequacies arising To begin with, the fast effect will directly from the physical picture involved. probably be dependent upon reactor size for small, close-packed reactors. The fast neutron leakage correction which is made to fc. thus probably does not adequately give the effect on fc„ caused by the loss of very fast neutrons. This effect, however, should be relatively minor. The major inadequacy in the four-factor formula is the omission of nonthermal In many reactors an appreciable fraction of the fissions is produced by fissions. This will be particularly true in close-packed lightneutrons of epithermal energies. water lattices and is even true to some extent in the more diffuse graphite or heavywater lattices. It should be noted, however, that the addition of resonance or epi thermal fission terms to the conventional reactor formulations must be accompanied For reactors by associated corrections to resonance absorption in U"8 (Art. 6.3). containing a small percentage of epithermal fission, the inclusion of resonance fission However, will cause relatively minor changes in calculated reactor characteristics.

reactor theory

Sec. 6-1]

6-25

if epithermal fissions are a significant fraction of the total number of fissions, then the conversion ratio and the reactivity may be appreciably affected by the different com petition between materials in the epithermal and resonance regions and the fact that the epithermal value of ij for U"s (and for the plutonium isotopes as well) is much less favorable. 7

SPECIAL REACTOR PROBLEMS

In the application of many of the methods discussed in this handbook there are special effects which may have important consequences in actual reactor operation. A few of these effects will be discussed in this section in an effort to provide a better understanding of the behavior of the nuclear systems involved.


7.1

Thermal

Spectrum Effects

In many reactor calculations the low-energy neutrons are treated as a single energy Cross sections for this group must be obtained by averag group of thermal neutrons. ing cross sections over the thermal-energy spectrum in order to arrive at physically significant results. However, in such a treatment of the thermal group it is usually necessary to take the characteristic temperature of the neutron distribution as that of the moderator molecules with which the neutrons are in energy equilibrium. For fairly dilute systems which are geometrically homogeneous this approximation is entirely adequate. There are important cases where this treatment fails, however. In heterogeneous geometries, where the lumps of fuel are of appreciable size compared with the mean free path of a neutron within the lump, the effective neutron spectrum impinging on the fuel atoms varies with position. The neutrons at lower energies are more strongly absorbed, with the result that the remaining spectrum is composed of neutrons with a higher effective "temperature." This phenomenon is usually described as "absorption hardening" and may be of importance in describing the characteristics of heterogeneous reactors with large fuel elements. However, the present trend toward small fuel elements (for engineering reasons) will avoid some of the corrections due to this effect. Of course, clumped, small fuel elements (for example, in a water-graphite or sodium-graphite reactor) will again exhibit appreci able neutron spectral hardening in the absorption in the fuel. Perhaps a more important aspect of the absorption hardening problem arises in both heterogeneous and homogeneous reactor lattices which contain strong thermal This applies particularly to light-water lattices, where the relative absorption. amount of thermal absorption is high because of the large amount of fuel present. In these cases the absorption hardening effect is a characteristic of the entire lattice rather than of a restricted geometrical region. The increase in the effective tempera ture of neutrons for a homogeneous reactor moderated with gaseous hydrogen has been studied by Wigner and Wilkins" and for a reactor with a moderator of heavy A useful approximation to the results of these calculations elements by Wilkins." has been suggested by Cohen", who found that the results were roughly of the form

—

vtr

= 1.128

+ 1.36^— £2.

where 8 = average velocity of hardened "Maxwellian" vtr = velocity corresponding to thermal temperature So"- = absorption cross section at vkT (assumed to be ~l/w) {2, = scattering properties of the medium (assumed independent of velocity) The resulting average velocity may then be used to obtain an effective kT for use in the usual cross section averaging techniques. This empirical formula has been checked roughly by the work of various experimenters. More recent work has been concentrated on treating the problem of nongaseous moderators." To date results of engineering significance have not been achieved for liquid moderators.

6-26

KEACTOH 7.2

Self-absorbed

PHYSICS

[Sec.

6

Cross Sections

The depression in neutron flux in the region of a high-absorption cross section has discussed elsewhere. One aspect of this problem, however, deserves special In the calculation of thermal utilization in a heterogeneous reactor the mention. various absorption cross sections are weighted by their geometrical average fluxes to take into account the depression of the flux, for example, in a fuel rod. The effective cross section used may be written (where is the average flux in material i) been


In many calculations, however, it is inconvenient to perform such a space-wise flux This is particularly true in multigroup weighting as a part of reactivity calculations. work. Consequently, it is often desirable to regroup the above expression:

The quantity 2,// may be thought of as the effective absorption cross section in a material having the same over-all nuclear properties as the actual homogeneous heterogeneous The cross section
The Doppler Effect

Most neutron interaction calculations picture the collision process as being made up of a neutron of some given velocity impinging upon a scattering or absorption atom which is at rest. In actuality this picture is not correct, since the target atom is in thermal motion in the atomic or molecular system of which it is a part. Because there are components of this thermal motion in the direction of the incident neutron, the effective energy of the neutron becomes a spread-out distribution about the actual neutron velocity with respect to the average position of the target, atom. This phenomenon is entirely analogous to the Doppler broadening of lines in optical spec troscopy or to the shift of acoustic frequency due to a motion of the source of sound with respect to the observer. For this reason it is referred to as a Doppler effect . It can be shown, for a Maxwellian distribution of target atom velocities and a 1 /r cross section, that the motion of the target nucleus makes no difference in the effective Although this condition covers a very large fraction cross section of the interaction. of the cases of interest, certain cross sections which depart widely from l/v will have a In particular, for resonances the effect noticeable change due to the Doppler effect. It may be shown that the area under a Breit-Wigner resonance can be important.

Sec. 6-1]

REACTOR

THEORY

6-27

not changed by this target atom motion, so that in very dilute systems there is no reactivity change due to Doppler effect. However, in many cases of interest the resonance peaks may be strongly self-shielded. The vibration of the target atom (which lowers the effective resonance peak height but spreads it out, thus maintaining the same area under the peak) cannot change the effective cross section inside the However, the widening resonance itself, since the absorption is already saturated. of the resonance will cause a widening of the region of strong capture, so that absorp tion in the skirts of the resonance will increase. This effect is referred to as "Doppler It is of importance to reactor operation. broadening" of the resonance. In low-enrichment reactors and, indeed, in many others, resonance absorption is less desirable than thermal absorption because of a lower probability of fission. Hence Doppler broadening will reduce reactivity with increased temperature. That In most reactors Doppler broadening produces negative temperature coefficient. this negative coefficient in the range of to This a fairly lO~s(Ak/k)/°C. small temperature coefficient in most systems. It still important, however, since In reactor power excursion the removal of reactivity due to this prompt. no necessity for temperature coefficient does not lag, since there heat-transfer process to take place. Since fission energy largely dissipated directly in the fuel material by the passage of the fission fragments, the temperature of the fuel rises immediately and the Doppler broadening takes effect promptly. Because of this, although other temperature coefficients may be larger, the Doppler broadening important to reactor dynamics and safety. is

is

3. 4. 5.

6a. 6. 7.

1,

p. 367, AECD-3645, March, 1955. For example, "The Reactor Handbook," vol. p. 376, AECD-3645, March, 1955. "The Reactor Handbook," vol. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," p. 127, D. Van Nostrand Company, Inc., Princeton, N.J., 1952. Goertzel, G.: "Criticality of Hydrogen Moderated Reactors," TAB-53, 1950. Ehrlich, R., and H. Hurwitz, Jr.: Nucleonics. 12(2): 23 (1954). Mandl, M. E. "The Spherical Harmonics Method in Plane and Spherically Sym metric Geometry in Multi- Velocity-Group Theory and Its Application in the TwoVelocity-Group PJ Approximation," AERE T/M 1295, Ministry of Supply, Harwell, Berks, 1953. Kourganoff, V.: "Basic Methods in Transfer Problems; Radiative Equilibrium, and Neutron Diffusion," p. 101, Oxford University Press, London, 1952. Yvon, J.: La Diffusion Macroscopique des Neutrons — une Methode d'Approximation, 1,

2a.

:

1.

REFERENCES

b.

Journal

of Nuclear

Energy, 4(3): 305 (1957).

Carlson, B. G.: "Solution of the Transport Equation by

Sn

Approximations," LA-1891,

1955.

12. 13. 14a. h.

1, 1,

10. 11.

Wick, G. C: Z. Physik, 121: 702 (1943). Sykes. J. B.: If on. Not. Roy. Astron. Soc. 111: 377 (1951). For example, Kahn, H: "Applications of Monte Carlo," RM-1237-AEC, April, 1954. Richtmeycr, R. D. "Monte Carlo Study of Resonance Escape in Hexagonal Reactor Lattices," NYO-6479, May, 1955. p. 390, AECD-3645, March, 1955. "The Reactor Handbook," vol. p. 434, AECD-3645. March, 1955. "The Reactor Handbook," vol. Bunemann, O.: Private communication, referred to in Geneva Conf. Paper A/Conf.:

8. 9.

8/P/433. A. Henry, private

"The "The

communication,

October,

1953.

el

el

1,

I,

p. 695, AECD-3645, March, 1955. Reactor Handbook," vol. p. 443, AECD-3645, March, 1955. Reactor Handbook," vol. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," p. 214, D. Van Nostrand Company, Inc., Princeton, N.J., 1952. 17. Weil. J. W.: Unpublished. al.: Exponential Experiments with Slightly Enriched Uranium Rods in 18. Kouts, H., Ordinary Water, Geneva Conf. Paper A/Conf.8/P/600, June, 1955. sea. (1954). al.: J. Appl. Phys., 26: 257 19a. Creutz, E. C, b. Muehlhause, C. O., el al.: Unpublished. c. Untermyer, S.: Unpublished. 15. 16a. b.

el


is

is

a

is

a

is

it

is

2

X

1

is

a

is,

again

6-28

REACTOR

PHYSICS

[Sec.

6

J. R., G. B. Arfken, T. R. Cuykendall, R. J. Stephenson, and D. O. Caldwell: "Surface to Mass Dependence of Effective Resonance Integrals for Uranium-238 Cylinders," ORNL-958, March, 1950. 20. Dancoff, S. M., and M. Ginsberg: Unpublished. October, 1955. 21. Stein, S., and R. L. Hellens: Private communication, 22a. Fleck, J.: Unpublished. b. Haeffncr, R.: Unpublished. c. Weil, J. W.: Unpublished. 23. "Neutron Cross Sections," BNL-325, July, 1955. 24. Wigner, E. P., and J. E. Wilkins, Jr.: "Effect of the Temperature of the Moderator on the Velocity Distribution of Neutrons with Numerical Calculations for H as Moderator," AECD-2275, September, 1944. 25a. Wilkens, J. E.: CP-2481, 1944. b. Hurwitz, H., Jr., M. S. Nelkin, and G. J. Habetler: Neutron Thermalization I, Nuclear Science and Engineering, 1: 280 (1956). 26a. Cohen, K.: Unpublished. b. Deutsch, R. W.: Computing 3-Group Constants for Neutron Diffusion, Nucleonics, d.

Risser,

15(1): 47 (1957). 27a. Cohen, E. R.: The Neutron Velocity Spectrum in a Heavy Moderator, Nuclear Science and Engineering, 2: 227 (1957). 6. Nelkin, M. S.: Neutron Thermalization II, Nuclear Science and Engineering, 2:199 (1957).

BIBLIOGRAPHY Generated for wjivans (University of Florida) on 2015-09-23 02:50 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Davison, B.: "Some Remarks on Applications of the Spherical

Case of Complex Geometries," T/R 700, 1951. Davison, B., and J. B. Sykes: "Neutron Transport Theory,"

London,

Harmonics Method in the Oxford

University Press,

1957.

S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952. Hurwitz, H., Jr., and P. F. Zweifel: Slowing Down of Neutrons by Hydrogenous Modera tors, /. Appl. Phys., 26: 923 (1955). Kahn, H. : Random Sampling Techniques in Neutron Attenuation Problems, Nucleonics.

Glasstone,

6(5): 27 (1950); 6(6): 60 (1950).

Kahn, H.: "Applications of Monte Carlo," RM-1237-AEC, 1954. Mark, J. C: "The Spherical Harmonic Method I," CRT-340, November, 1944. Marshak, R. E., H. Brooks, and II. Hurwitz, Jr.: Introduction to the Theory of Diffusion and Slowing Down of Neutrons, Nucleonics, 4(5): 10 (1949); 4(6): 43 (1949); 6(1): 53 (1949); 5(2): 59 (1949).

Marshak, R. E.: Theory of Slowing Down of Neutrons by Elastic Collision with Atomic Nuclei, Revs. Mod.'Phys., 19: 185 (1947). Weinberg, A. M., and L. C. Noderer: "Theory of Neutron Chain Reactions," extracts May, 1951. from vol. I, chaps. 1 to 3, CF-51-5-98, Tait, J. H.: The Calculation of the Fine Structure of the Thermal Neutron Flux in a. Pile. by the Spherical Harmonics Method, Proc. Intern. Conf. Geneva, August, 1955, vol. 5. United Nations Publications, 1956. "The Reactor Handbook," vol. 1, AECD-3645, March, 1955.

REACTOR

6-2

CALCULATIONS BY

J.

R. Dietrichf Nomenclature

(See also

Table

6 for special symbols used in

Arts.

and 8)

Complete definitions of the symbols may be found in the sections referred to. A — mass number B' = buckling. The geometrical buckling of a medium is that which satisfies the equation V'Q + Bl* = 0 in the medium with proper The material buckling is that which satisfies boundary conditions. 1 + (k/p)Pm(B.,B*) the equation -L'B* 0 (Art. 5.3) B'% = in the two-group representation, the "transient" buckling (see

-


7

D

E

F(E)

/

G(x,x')

Table 6) = diffusion coefficient = neutron energy —

'-

(Art. 2.13)

collision density per unit energy (Art. 3.2)

= thermal utilization (Art. 9.2) = diffusion kernel for thermal neutrons

= flux at x due to

unit source

at x' (Art. 2.25) »,°' i-1' v' v' I / o, ' 1 1 An, A.i,

J

~ Bessel functions = neutron current density

(Art. 2.11) constant k — multiplication constant (infinite) (Art. 9.1) k,n = effective multiplication constant in a finite reactor (Art. 13.1) k.x = excess multiplication factor (Art. 13.1) k =• Boltzmann

L L/

I

£

-

= diffusion length \/fl/2« (Art. 2.15) = slowing-down length (Art. 5.6) —

prompt neutron lifetime (Art. 13.3) of neutrons produced which leak from reactor

= neutron leakage = fraction

M

= migration length = -y/L* + t (Art. 5.7) n = neutron density, neutrons/cm' .V = number of nuclei per cm* N — Avogadro number = number of atoms per gram atomic weight p = resonance escape probability (Art. 9.4)

P(E,r,r')

P*(E,'r

—

/]) FX(E,B')

= finite slowing-down kernel (Art. 5.1) = infinite slowing-down kernel (Art. 5.2) = three-dimensional Fourier transform of the infinite

slowing-down kernel (Art. 5.3) q(E) = slowing-down density (Art. 3.2) Q, q = neutron source strength, neutrons/sec or neutrons/(cm*)(sec) R = radius of a spherical or cylindrical medium

f The author acknowledges by D. H. Shaftman.

with thanks valuable suggestions

6-29

and review of parts of the manuscript

6-30

REACTOR

s

= in the

PHYSICS

two-group representation, the ratio

t = time variable T = t hickness of slab medium = absolute temperature

f/,

[SEC. 6 (see Table 6)

T

(S

= =

ft

-

y

=

e

=

1

_

=

H = = *« = = =

r

=

t

Ao

= = = = = =

T

= =

£.

= =

20/

=

6)

.

V —

= = = = =

1

/

E

6,

2/;

1.1

INTRODUCTION The Fission Reaction

if

is

the fission of heavy The basic process which makes possible the nuclear reactor nuclei by neutrons. The products of a given single fission cannot be predicted, but many fissions are considered, an average equation for the reaction can be written for given fissionable isotope; for example, a

neutrons other radiation + energy

+

v

fission product nuclei

+

neutron

►

1

fission

U"s

+


X. X

*/

is

a

neutron velocity volume most probable neutron velocity in a Maxwell-Boltzmann dis tribution (Art. 4.1) albedo = ratio of current density out of medium to current density into medium net current into medium (Art. 2.3)] fraction of fission neutrons originating from delaved emitters (Art. 13.2) fraction of fission neutrons originating from delayed emitters of ith species (Art. 13.2) extrapolation distance measured in transport mean free paths (Art, 2.22) fast fission factor (Art. 9.5) regeneration factor = average number of fission neutrons pro duced per neutron absorbed by fissionable material (Art. 9.2) reciprocal of diffusion length (l/L) reciprocal of diffusion length reciprocal of slowing-down length (l/L/) mean free path of neutron = 1/2 decay constant of j'th species of delayed-neutron emitter (Art, 13.2) average cosine of scattering angle per collision of neutron with scattering nucleus (laboratory coordinate system) (Art. 2.13) average number of neutrons emitted per fission average logarithmic change in energy per collision (Art. 3.1) reactivity = k.z/k,rl (Art. 13.1) density, g/cm' microscopic cross section, cm'/nucleus microscopic absorption cross section microscopic scattering cross section cross section corresponding to most probable velocity in MaxwellBoltzmann distribution at temperature (Art. 4.2) macroscopic cross section, cm-1 macroscopic absorption cross section macroscopic scattering cross section macroscopic absorption cross section for thermal group of neu trons in two-group representation (see Table macroscopic slowing-down cross section for fast group of neutrons in two-group representation (see Table also Art. 5.6) Fermi age from source energy to energy (Art. 3.4) neutron flux density (Art. 1.3) flux density of thermal neutrons flux density of fast or epithcrmal neutrons adjoint function (Art, 14.1) [if

V = —

V

(1)

Sec. 6-2]

REACTOR

CALCULATIONS

6-31

The significant point for the present discussion is that the average number v of new neutrons produced per fission is considerably greater than unity (actually about 2.5). The fission chain reaction in a nuclear reactor is maintained at a steady rate if the materials of the reactor are so arranged that, exactly one of the v neutrons produced by each typical fission interacts with another fissionable nucleus to produce still If, on the average, less than one of the fission neutrons produces a another fission. further fission, the neutron population of the reactor will decrease continuously and If, on the other hand, more than one of the the chain reaction will eventually die out. neutrons from the average fission produces further fissions, the neutron population of the reactor will grow from generation to generation and the level of fission power production in the reactor will increase continuously with time. Most problems in reactor physics involve either the determination of the necessary conditions for a chain reaction (critieality conditions) or the determination of the rate steady-state of growth or decay of the chain reaction (reactor kinetics) under given reactor con In either case, the statics or kinetics of the chain reaction is determined by ditions. the statics or kinetics of the neutron population of the reactor. The problem is usually solved by investigating the life histories and reproductive powers of "average" Evidently, then, the subject of reactor physics consists mainly of neutron neutrons. physics or, more specifically, of the physics of neutron transport and neutron inter actions with matter.


1.2

Fundamental

Neutron Processes

The individual neutron can exist for only a short time as a free particle. It ceases to exist as a particle when it enters the nucleus of an atom and becomes integrated into the structure of that nucleus. When such an event occurs, the neutron is said The resulting modified nucleus may be stable to have been absorbed by the nucleus. or may be unstable and emit other particles, including neutrons (as in the case of fission), but in any case the original neutron is considered to have ceased to exist. Aside from absorption, which ends its life, the neutron can interact with matter only These by making scattering collisions with the nuclei which compose the matter. interactions are called collisions because the free neutrons in a material medium are The veloci normally traveling about, in the spaces between nuclei, at high velocity. ties may have been imparted to the neutrons when they were formed or may be the In a scattering collision the velocity of the colliding velocities of thermal agitation. If the neutron neutron is usually changed, both in magnitude and in direction. imparts only kinetic energy to the body with which it collides, the collision is said to be elastic. If it changes also the internal energy of the target body, the collision is said to be inelastic. 1.3

Quantitative Specification of Interaction Rate: Cross Sections

it is found that the number of interactions of any given type Experimentally (scattering, absorption, fission, etc.) which occur per second per unit volume of a material containing neutrons depends, for a material of given nuclear species, only on the number of nuclei per unit volume, the number of neutrons per unit volume, and the effective neutron velocity. This relationship is formulated quantitatively by supposing that each nucleus of a given species presents a target for a given type of interaction of area cm1 to any neutron traveling with velocity v (the neutron itself being considered a dimensionless point). On this basis a given neutron, in the time interval At, will travel a distance v At cm and will experience interactions of the specified type with all nuclei which lie within the volume SI cm'. If there is a uniform density of N nuclei per cubic centimeter, the number within this volume is just Na(v)v At. If the neutron density within the medium is n neutrons per cubic centimeter, then the rate at which interactions of the specified type are occurring per unit volume is

)

No. of interactions/(cm*)(sec)

The quantity

nv is called the neutron

flux.

It

= nvNo(v)

(2)

is often represented by the single

e-32

REACTOR

.

PHYSICS

[Sec. 6

The Its dimensions are neutrons per square centimeter per second. symbol quantity a is called the microscopic cross section for the type of interaction under Its dimensions are square centimeters per nucleus. Microscopic consideration. The barn is a unit of area equal to 10~" cross sections are usually quoted in barns. cm'. However, in the equations presented here the units of a are square centimeters The quantity No is called the macroscopic cross section. unless otherwise specified. It is the total target area for a given interaction which is presented to a neutron by all the nuclei in a cubic centimeter of material; its dimensions are cm' per cm", or cm-1. It is often represented by the single symbol 2. The average number of interactions made by a neutron in traveling 1 cm is just N

to be so.

Evidently the concept of the cross section is not a particularly useful one unless the cross section is a property of the nucleus alone, unaffected by such considerations as the molecular or crystal structure of the material in question. This condition is met in most cases which arise in elementary reactor theory. When it is met, any material may be regarded simply as a mixture of the various nuclear species present, and the macroscopic cross section for any given type of neutron interaction is just the sum of the macroscopic cross sections of all the nuclear species for that type of interaction; i.e., (3)

where

A',- is the number of nuclei of the ith species per cubic centimeter of the medium, pi is the number of grams of the ith species per cubic centimeter of medium, Ai is the atomic weight of the ?'th species, and N is the Avogadro number.

1.4

Sizes of Cross Sections and Other Important Dimensions

Scattering cross sections for neutrons of energies important in reactor physics from about 2 X 10"" to about 20 X 10~" cm*. Absorption cross sections cover a much wider range of values. For neutrons of thermal velocities the absorption cross section of hydrogen is about 0.3 X 10~" cm', that of iron is about 2 X 10~" cm1, and that of the boron isotope of mass 10 is about 4 X 10~" cm'. If the cross sect ion is visualized as circular in shape (i.e., as the cross section of a sphere), the radius cor responding to a cross section of, say, 10 X 10~" cm' is about 2 X 10~" cm. The average distance between nuclei in typical solids or liquids is very much larger, of the order 10~8 cm. That is to say, even though many nuclei are packed into each cubic centimeter of material, the targets which they present to a neutron for interaction are so small that the neutron may, in typical cases, travel distances of the order of centi meters between interactions. Consequently, one of the problems in maintaining a neutron chain reaction is simply to contain a large fraction of the neutrons in the reactor long enough for them to interact with the fissionable material. The wandering away of neutrons before they can be absorbed is generally referred to as leakage from the reactor. Despite the relatively long distances traveled between interactions, typical neutrons have such high velocities that they exist for only a short time before being absorbed. The mean lifetime of a neutron in a reactor depends upon the reactor design but is seldom more than 10_s sec. It is not possible to maintain a high density of neutrons range

CALCULATIONS

REACTOR

Sec. 6-2]

6-33

in the face of such a short lifetime unless the rate of production of neutrons per unit volume by fission is very high, i.e., unless the power production per unit volume is In the reactors built to date it has not been possible to achieve power very high. densities which give neutron densities greater than about 10' neutrons/cm'. Since the atomic density in typical solids or liquids is of the order 10" atoms/cm', it is evident that the probability that a neutron will encounter an atomic nucleus is enor mously greater than the probability that it will encounter another neutron. liecause of this circumstance, that every free neutron in the medium acts independently of all other free neutrons, the equations which describe the flow of neutrons in a medium are linear. Some of the consequences of this linearity are discussed in Art. 2.25.

Conditions for the Chain Reaction

1.6

As stated in Art. 1.1, a steady self-sustaining chain reaction is maintained in a nuclear reactor if the materials of the reactor are so arranged that exactly one of the » neutrons produced by each average fission reacts with an atom of fissionable material to produce another fission. The subsequent articles have indicated that two processes compete with the fissionable nuclei for the free neutrons: absorption by nonfissionablc nuclei and leakage from the reactor. The condition for criticality of the reactor can be written in terms of these two processes if two new symbols are defined: £ = fraction of neutrons produced which leak from the reactor

_

No. of second-generation neutrons produced No. of first-generation neutrons absorbed (by all materials present, including

fissionable material) then,

k(l

- £)

=

(4)

1

criticality

is,

The equation for

=

ri

k,

ij

is

is

k

v

y

is

k

if

The quantity another symbol can be stated in a more significant way introduced. The quantity the average number of fission neutrons produced per The expression neutron absorbed in fissionable material. In general less than v. for becomes

No. of neutrons absorbed in fissionable — material total No. of neutrons absorbed

a

is

is

a

is

if

if

is

is

k

£

must be made relatively must be made relatively large and According to Eq. (4), small the chain reaction to be self-sustaining. The maximum value of attained reactor of little the reactor Such made of pure, fissionable isotope. use as power source, since no effective means supplied for removing the heat which generated and converting to useful work. Practical power reactors must contain it

is

£

is

is

k

materials other than the fissionable isotope. These materials serve the purposes, for example, of coolants, structural materials, and cladding to prevent the escape of fission products from the fissionable material and, in some cases, to protect the fission able material from chemical attack by other materials. Furthermore, in many cases fertile materials — isotopes which can be converted to fissionable isotopes by the absorption of neutrons — arc included in the reactor. Consequently, in many cases the practically attainable value of must be not much greater than unity, and made quite small in order to achieve criticality. particularly This requirement is

demanding the reactor to be fueled with the naturally occurring mixture of the fissionable isotope U"' with the fertile isotope U"8. One way of decreasing £— of decreasing the probability that neutron will wander out of the reactor — more to make the reactor large. Another method, which economical, to include in the reactor a good moderating material: a material of low mass number, low absorption cross section, and relatively high scattering cross The neutrons, in elastic collisions with the moderator nuclei, are slowed -section. down to relatively low velocities. Since absorption and fission cross sections are much larger for slow than for fast neutrons, the moderation, or slowing down, of the is

is

is

a

if


^

6-34

REACTOR

PHYSICS

[Sec.

6

neutrons increases their probability of absorption and hence decreases the leakage Most frequently, in a moderated reactor, the amount of moderator probability. employed is so large that most neutrons come into energy equilibrium with the thermal agitation of the moderator nuclei before they are absorbed. Such a reactor is called a thermal reactor. Reduction of leakage is not, of course, the only way in which neutron moderation affects the criticality condition. Since fission and capture cross sections vary with neutron energy in different ways, the value of k which applies for a given mixture of absorbing and fissionable isotopes depends upon the energy spectrum of the neutrons This dependence is strikingly illustrated by the fact that whereas it is present. quite feasible to construct a critical thermal reactor which employs as fuel the natu rally occurring mixture of U23' and U"8, the value of k for a large block of pure natural uranium (no moderator added) is considerably less than unity. The methods outlined in this subsection apply primarily to thermal reactors, although much of the basic material, as well as the specific methods of Art. 10, are more generally applicable.


1.6

Outline of Subsequent Treatment of Reactor Physics

To summarize the foregoing, the behavior of a nuclear reactor is determined by the migrations of neutrons, their slowing down, and the relative rates at which they are absorbed by fissionable and nonfissionable isotopes. When these processes are such that the neutron population of the reactor remains constant in time, the reactor is said to be critical; the study of the conditions for criticality constitutes the subject of reactor statics. The behavior of the neutron population when the criticality con dition is not fulfilled constitutes the subject of reactor kinetics. The remainder of the subsection is devoted principally to reactor statics; further material on kinetics is contained in Sec. 8-1. The material in the remainder of this subsection may he summarized as follows: Article 2 treats the flow of neutrons of uniform energy in a medium. Article 3 treats the slowing down of neutrons to thermal energy, and Art. 4 describes their distribution in energy after they have reached thermal equilibrium in the moderator. In Art. 5 the criticality condition for a reactor is stated quantitatively as a partial differential equation in terms of the fundamental processes covered in the preceding sections. Article 5 amounts to a quantitative statement for the leakage £ of Eq. (4). Article 6 gives solutions for the critical reactor equation in simple geometries, and Arts. 7 and 8 outline methods of solution for the more complex cases of reflected reactors. Article 9 covers the evaluation of the constants which enter the reactor equation, including the multiplication constant k [Eq. (4)]. This article covers both homogeneous cases and the more complex "lumped" cases. Article 10 outlines briefly the multigroup technique which can be used for both thermal and nonthermal reactors. In Art. 11 Article 12 treats typical computation procedures are illustrated by specific examples. the effects of lumped thermal-neutron absorbers, such as control rods or experimental samples, in a reactor. Article 13, on noncritical reactors, is a very brief treatment of reactor kinetics and concepts related to reactor kinetics. In Art. 14 relations are given by which the reactivity effects of small changes in a previously critical reactor may be evaluated. . Throughout the subsection an attempt has been made to limit the coverage to those items appropriate to handbook treatment. Individual computations which would normally take more than an hour or two of work on a desk-type calculating machine or a slide rule for their execution are considered to be beyond the scope of a handlxiok. The treatment of neutron migration has been limited to the diffusion approximation. In general, where specific approximations are used, they are noted. A complete discussion of the implications of such approximations and of their range of appli cability is evidently beyond the scope of the subsection. Finally, although an effort has been made to arrange the topics in a logical sequence and to impart as much physical understanding as can be done in a brief space, the presentation is obviously not intended to approximate a logical and integrated development of reactor theory.

REACTOR

Sec. 6-2]

6-35

CALCULATIONS

rather, that the subsection constitute a convenient reference for those formulas and procedures which occur frequently in the course of reactor

It is intended, elementary

computation. 2

DIFFUSION OF "MONOENERGETIC"

NEUTRONS

the material of this article applies to the diffusion of monoenergetic Conceptually, neutrons; practically, it is applied to groups of neutrons whose energy spread is narrow values of properties of the medium enough and predictable enough that averaged The most common case is that of thermal neutrons. The averaging may be used. of diffusion properties over the thermal-energy spectrum is treated in Art. 4. The Diffusion Equation

2.1

J

2.11 The current density Current Density. is the net number of neutrons flowing in unit time through a unit area normal to the direction of flow.

or, in

J

= —D grad

(5)

'--l>(
(6)

Cartesian coordinates,

D is the diffusion coefficient and i, j, and k are, respectively, the cosines of the between the x, y, and z directions and the normal to the plane across which current density is being measured. That is to say, the net current density across the given by plane has directional components Jx, Jv, and where


angles

J,

J„ /, - -D**

-fll* dx

=

J,

dy

=

-D^ dz

Each component (e.g., Jx) of the net current density is the difference between a for ward current density component Jx + and a reverse current density component Jx~, given by r

J'+

"

~ 4

D

d
2

Yx

,D

J--l+2 T

a

(7)

0l

with similar expressions for the other directions. 2.12 Neutron Leakage Rate. The rate at which neutrons leak from unit volume in a diffusion medium is Neutron leakage /(unit vol) (sec) =

—Dv*

--!,(** + ?* + **) \dx* dy*

dz* >

(8)

in cartesian coordinates where Vs is the Laplacian operator (see Sec. 3-2 for other coordi nate systems). 2.13 Evaluation of Diffusion Coefficient D. If experimentally determined values of D are not available, the following relation (from transport theory) may be used:

D

32(1-

A.)

7~-1at 2b

(l \

51

+

r

:

2.1—

/io

+

• •

\

(9)

•) '

where 2 and 2. are the total and absorption macroscopic cross sections, respectively, and >2ois the average cosine of the scattering angle per collision (stationary coordinate system). If the medium absorbs only very weakly, the expression becomes

D whfre

1

32.(1

Ai, is the transport mean free path.

- £„)

= hz 3

(10)

REACTOR PHYSICS

6-36

[SEC. 6

In reactor systems, scattering of neutrons is essentially isotropic in the center of mass coordinate system, and for this condition — mo = cos ^ =

-

(11)

SA

where ^ is the scattering angle in the stationary (laboratory) coordinate system and A is the mass number of the diffusion medium. The rate of 2.14 Neutron Balance per Unit Volume—The Diffusion Equation. change of neutron density n ( = number of neutrons per cubic centimeter) is given by

— at

= rate of leakage into unit vol — rate of absorption

=

D

W

- 2.*

/unit vol + rate of product ion /unit vol

+ S(r)

(12)

where S(r) is the effective source strength, or effective rate of production of neutrons The quantity S(r) is smaller than the actual source strength by the per unit volume. factor [1 — J
D

VV

generally written Generated for wjivans (University of Florida) on 2015-09-23 02:50 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

-

V*4> —

£„

*V

■=0

= 0

(13)

where x* = 20/D. The reciprocal of x?, D/Xa, is called the diffusion area and is represented by the symbol L1, which can be shown to be one-sixth the mean square distance (crow flight) traveled by a neutron in a diffusion medium between the time The square root L of the diffusion area of its birth and the time of its absorption. is called the diffusion length. The diffusion equation, when written in the form of Eq. (13), is frequently referred to as the wave equation. 2.2

Solution of the Diffusion Equation

2.21 The solution of the steady-state diffusion equation General Considerations. in an infinite medium can bo written out formally, for the general case of an arbitrary distribution of sources, by use of the diffusion kernels (see below). The solution for a finite medium is difficult unless the source distribution and the boundaries display a A few useful cases of relative simplicity are treated reasonable degree of symmetry. in following paragraphs. Other cases are covered later (Art. 12) in connection with Solutions for a great many cases are given in a lumped absorbers in the reactor. paper by Wallace, '-f Before the solutions can be given, it is necessary to consider the boundary conditions which apply at the edges of finite media. The diffusion equation does not describe If, however, appropriate boundary accurately the flux behavior near boundaries. conditions, derived by transport theory, are applied, the solutions of the diffusion equation will describe the distribution of flux adequately in all parts of a finite medium The conditions for a except those within about one mean free path of the boundary. number of cases are given below. 2.22 "Black" Boundaries. A boundary is called black if no neutrons which cross the boundary from the medium return to the medium. External boundaries between the medium and vacuum are black. Boundaries between the medium and very strong absorbers, cither internal or external, may often be considered black for practical purposes, the criteria being that the absorber be thick enough to reduce the neutron flux practically to zero in its interior and that 20 Evidently, 2. in the absorber. the neutron flux in the medium will decrease as the boundary is approached. The boundary condition depends on the geometry of the boundary and on the transport

»

t Superscript

numbers

refer to References


Sec. 6-2]

REACTOR

CALCULATIONS

6-37

It is generally stated either as the value of the free path in the medium. logarithmic derivative of the flux at the boundary mean

\im L

OX Jbound.ry

(14)

y\,r

where x is the coordinate normal to the boundary (the positive direction being from medium to boundary) and 7 is a number characteristic of the shape of the boundary, or as the extrapolation (or augmentation) distance e: « = 7Xir

(15)

The extrapolation distance is defined as the distance beyond the boundary at which the flux would become zero if its normal derivative at the boundary were extrapolated The two methods of statement of the boundary condition are equivalent. linearly. A convenient way of applying the condition of Eq. (15) is simply to set up a fictitious boundary (the extrapolated boundary) at a distance c beyond the true boundary and

0

,1 RADIUS

2

4

3

This boundary

5

OF SPHERE OR CYLINDER, IN MEAN FREE PATHS

length for "black" spheres and cylinders. 1. Estimated values of linear extrapolation (Reproduced from: B. Davison and S. Kushneriuk, Linear Extrapolation Length for a Black Cylinder, iVational Research Council of Canada, Division of Atomic Energy, MT-214, March 30, 1946.)

Fio.

is

I

is

it

A

7

1

4
condition is not strictly equivalent to that of Eq. (14) (since the solution of the diffusion equation is not, in general, linear between the real and extrapolated bound Values of 7 for various geom aries), but it is adequate for almost all practical cases. etries are given below. These apply for media of low absorption (strictly, for nonabsorbing media). y = 0.71. Plane Boundary, The extrapolation distance varies Internal Cylindrical and Spherical Boundaries. for vanishingly small radius to 0.71 for infinite with the curvature; y ranges from on radius of black absorber radius. shows the probable dependence of Figure as estimated by Davison and Kushneriuk. "Gray" Boundaries. 2.23 strong absorber which yet does not absorb all the Diffusion theory often referred to as a "gray" absorber. neutrons incident upon does not describe the neutron flux behavior accurately in the vicinity of such an Kushneriuk and McKay have derived extrapolation distances, by approxi absorber. mations to transport theory, for "gray" cylinders embedded in nonabsorbing, scatter These extrapolation distances may be used with diffusion theory, ing media (Fig. 2). in the same way as the black boundary extrapolation distances, to determine the flux distributions in the scattering medium. The Media. 2.24 Boundaries between Nonabsorbing (or Weakly Absorbing boundary conditions between two diffusion media are The neutron flux continuous across the boundary. 1.


require that the neutron flux vanish at this extrapolated boundary.

6-38

HEACTOIt

PHYSICS

[Sec.

6

2. The neutron current density in a direction normal to the boundary is continuous across the boundary. If the two media are nonabsorbing, the presence of the interface does not affect

the adequacy of the diffusion approximation. If either or both of the media absorb, higher order approximations to transport theory are needed to describe the neutron distribution adequately.* Nevertheless, the diffusion approximation, with the above boundary conditions, is used in many re 32, actor calculations when the absorption is not extremely large. 2.26 Solutions of the Steady-state Diffusion Equation in Infinite Medium — The Diffusion Kernels. The steady-state diffusion equation is


DV*

=1

0

1 0.4

1

I

I

0 8 12 16 2 0 2.4 2.8 32 3.6 RADIUS OF CYLINDER I'm. 2. Linear extrapolation length for a "gray" purely absorbing cylinder in a noncapturing medium. The unit of length for both radius of cylinder and extrapolation length is the mean free path in the sur rounding medium, a is the radius of the cylinder, and a is the ratio of macroscopic absorption cross section in cylinder to scattering cross section in the surrounding (Reproduced from S. A. Kushmedium. ncriuk and C. McKay, Neutron Density in an

- 2„«

+ S(r) = 0

(16)

The equation is linear. Physically, this is because the mutual interactions be tween free neutrons are completely negli gible at practical flux densities. Because of the linearity property, any superposi tion of solutions is a solution of the equa tion, and the flux distribution resulting from any arbitrary distribution of sources is just the superposition of the flux distri butions due to the individual sources. It is therefore useful to tabulate the flux distributions due to unit delta-func tion sources of various geometries (point sources, line sources, etc.) in an infinite medium. The flux distribution for such a unit source is called the kernel for the source. It will be denoted by G(u,u'). The quantity G(u,u') is just the flux at u due to unit source at u'. The flux (u) at u due to an arbitrary distribution of sources S(u'), defined in the volume V in an infinite medium, is

Infinite Non-Capturing Medium Surround ing a Long Cylindrical Body which Scatters

(")

=

JYS(u')G(u,u')dV

(17)

where the differential (IV signifies thai the volume integral is to be taken over the source distribution. The kernels G(u,u') are the solutions of the steady-state source-free diffusion equation and Captures Neutrons, Atomic Energy Canada Limited, C'Sr-566, July, 1954.)

of

DVV

- S„0

= 0

in an infinite medium, the single source being considered a separate region of infini tesimal extent. The solutions apply (within the limitations of diffusion theory) throughout the infinite medium except infinitesimally near the source. Kernels for a number of source geometries are given in Table 1. Solutions of the Steady-state Diffusion Equation in Finite Media with 2.26 Localized Sources. If the geometry of the medium (and of any sources which may exist in the medium or at its boundaries) can be described by a single coordinate, the solutions are simple. Table 2 gives the general solutions for such one-dimensional cases in the usual geometries, as well as the solutions of some typical specific problems. The three-dimensional case will be illustrated by a problem in rectangular eoordi-

Sec. 6-2]

CALCULATIONS

REACTOR Table

Diffusion Kernels in Infinite Media

1.

Symbol for

Source geometry

6-39

Kernel = neutron flux at r (or at x, as the case may be)

Source strength and location

kernel

r-*|r-r'|

-

Point

OiM(r,rO

I neutron/sec at r'

Plane

Qpi(x,x')

I neutron/tcm'Maec) over a plane of infinite extent at x'

Line of infinite length

G/(r.r')

I neutron/(cm)(sec) over a line of infinite length, in z direc tion, at r'

4*D\r r'l e-*l*-*'[

IxD

Thp vectors r and r' are both taken from the z axis, in a plane normal

Q.iry)

Spherical shell

1 neutron /sec per shell of radius with center at origin

/,

I neutron /sec per shell of radius r' and unit length, with center at axis

J

(.-.l,-r
Tin

|r — r'l denotes the absolute value of the vector difference of r and r\ Bessel functions of the first and second kinds, respectively, of zero order.

-

ô(»cr)/o(*er')r > r*

Jo and Ko are the modified

nates, for which the diffusion equation in the source-free regions is

dy*

dx*

—

H

dz1

x'o

0

(18)

If the medium consists of a single material, the equation may be assumed separable and to have a general solution of the form 4, =

X is a function of x alone, solutions of the equations

Y

where

dy* are constants chosen <*s

0s

X, Y, and Z

are then

dz* to fit the boundary conditions plus the addi

+

y* + x* =

(19)

0

fi,

where a, and tional condition

7

ax*

to be

X(x)Y(y)Z(z)

a function of y alone, etc.

+

the

*2

exp

if

6;

0,

z

— (

sin

nirV

/

abDym 7mn

(20)

1

1

ntrx

2Q

— - sin —

n ■»

+

and

=

mn(z) sin

)

>

= m

(z)m«

y

x

is

y

Q

•t>(x,y,z)

where

0, is

is

is

The rectangular prism a typical problem, and the arrangement frequently used for the measurement of diffusion length. If the prism infinite in the direction and = = a; and the source bounded in the x and directions by the planes of effective strength located at the point (x',y',z'), the above proce neutrons/sec dure leads to the solution

is


Cylindrical shell of infi nite length

to the axis

(wnr\* ~~a)

\b)

ym„\z

—

«'|)

z

? a

AB

Jr

y

J

;

B + il/o(*r)

Q

b;

-

r

b;

+

7

6

+

r

J

*2> Ko(xa)Ti(xb)

i4

_7

B

6

K\ are the modified

x) for x

x cosh «(a —

x(a —

— x)

/o(*a)Ki(x&)

+

Bessel functions

of first order.

/»(xa)iYi(xb)

7o(«a)

+ x/>/Co(wa)/i(jr6)

*(a

K*(xa)

BXo(xr)

M

w(a —

fsinh

sinh

sinh x(a

sinh x(a

'

and are arbitrary constants, to be chosen to fit the boundary conditions. = the diffusion coefficient for the medium; x* = 2*fD area for the medium. ,/La = /diffusion lo and Kn are the modified Bessel functions of first and second kinds, respectively, of sero order; and

0

-

["sinh

A/o(*r)

_

4«-/>sinh

«a

xD cosh ita

2xa

x'

J I

A

D

-

*(r) where

*(r)

sinh

L

+ BKa(xr)

0;

b)

$(r)

is

outer shell, of Spherical extrapolated diffusion radius = a, inner radius ■current of uniform neutrons/ density = (cm*) (sec) flowing into shell at outer shell, of extrapolated Cylindrical diffusion radius = a, inner radius = current of uniform neutrons/ density (cm1) (sec) flowing into shell at_r =*

*
*(*> -

xD

Q

r

dr

Net diffusion current of uniform density neutrons/(cmI)(sec) flowing into slab at x = slab extends to extrapolated « o boundary x Point source of strength neutrons/sec at center of sphere of extrapolated radius =>

«

sinh x(a — x) for x

~] r)

Cosh

*(x)

*') 2xa sinh x(a — x') sinh

sinh x(a

xD

problem

Sources

of specific

Localized

Solution

with

6)

dr*

-

+

+

. sinh xr

*(r)

— —a slab boundaries at plane source of a; infinite uniform neutrons/tcra^tsec) strength at * - X*

Extrapolated and z ™

Q

long cylinder

=

sinh « cosh « x

J

dr

x'«

4>(z) ■»

Media

Q

"

*»«

+

<

Infinitely

77 dr1

-

One-dimensional

specific Typical problem

in Finite Equation (For x

Q

J

Sphere

and

General solution

Diffusion

-f-

.

— O

in

of Steady-state

Expanded form of diffusion equation

Solutions >

Slab, infinite directions

2. 0)

Geometry of medium

Table


x* >

~|

./

1

I

1

B

is

reactor calculations

T

sm

T

z

in the x and

the solution

abDy„„

.

smh y„„(c

sinh 7„»c

smh >„„c

2.3

The Albedo}

a

—

.

.

for

>

z,

mrj/'sinh —— -— —ym,,z'

z

.

sin

z)

a

,

mux'

sin

b

AQ—

abDy„n

b

=

5

*(z)m»

,

is

z

in

is

(21)

wy'

For this case the quantities jr'/a* and ir'/o1 are called the buckling* directions, respectively. = direction, hounded by the planes finite the the prism again that given by Eq. (20), but withf

If

2'|]

wx'

sm

-

0,

.

UV^

|z

+

2Q

+ <,•/«') +

£

[

(*'

-

exp

.

"

sin

+

whprc

"

sin

^

=

A.

,

*(x,y,i)

A

n

1

z'\

those for which m =

j/

all terms of the summation except For large values of |z — = become negligible, and the solution becomes

1,

6-41

c,

Sec. 6-2]

(22)

and 7„„ as before.

is

a

0 is

reflection coefficient which characterizes this process quantitatively. It as the ratio of current density out of the medium (negative) to current density the medium (positive)

measured along

-

+

(*/4)

LO/4)

(JQ/2)(a*/ai)

(D/2)(^
boundary

vector normal to the boundary and directed into the is

Within the limitations of diffusion theory, the albedo geometry and the material characteristics of the medium. geometries are given below.

characteristic only of the Albedos for a number of

Infinite Medium with Plane Boundary:

- +- 2"D 2hD

0=1-

4xD

(25)

is

the scattering isotropic also, xD = (l/y/%) \/'Za/'Z. But 2/Zo just the total number of collisions (JV) made by the average neutron the medium before absorption, and

This relationship

=

1

P

in

is

If

1,

«

For the case of xD

(24)

1

»

1

2.31

(23)

1

[

boundary

a

is

~T

where x medium.

=

-J 1

- — JL I"

B

into

:

defined

- -±= -7=5

aids in the physical understanding

<26)

of the neutron

"reflection"

process. is

Within the framework of diffusion theory only an isotropic distribution of velocities of incident neutrons can be considered. In many practical cases the distribution

z

is

if

is

is

t

is

U

t

dimension, The solution valid regardless of which dimension of the prism taken as the regardless of the dimension in which the sinh term However, used to describe the distribution. vergence of the series good only the longest dimension dimension. taken as the See Ref. 3. See also Art. 4.4 of Sec. 6-1.

J


of

a

a

If

neutrons are incident on the boundary of source-free medium, large fraction the neutrons which cross the boundary into the medium may, after one or more The albedo collisions, be scattered back across the boundary and out of the medium.

i.e., con

REACTOR

PHYSICS

[Sec.

6

7=

1

_

i

(3

=

g

is,

not isotropic (e.g., a neutron beam incident on the medium). The albedo as derived from more detailed considerations of neutron transport for the case of low absorp tion and isotropic scattering,

_

for glancing incidence

(27)

normal incidence

(28)

2.9078

apply for isotropic incidence only.

lll^^L1^

Infinite Medium Surrounding Sphere of Radius

-

Spherical Shell, Inner Radius a, Outer Radius

3

1 1

_

- 2D[x coth

x(b

+ 2D\x coth «(6

SLOWING

.

2D[x + (l/a)| + 2D[x + (l/o)]

-

a) a)

2.35 :

Neutrons Incident from

+ (l/o)] + (l/o)]

DOWN OF NEUTRONS

a

a it is

The kinetic energy of fission neutrons varies over a wide range, but in all cases In the course of their lives in reactor such high compared with thermal energies. neutrons will, in general, lose energy by nonabsorbing collisions with nuclei of the reactor materials. The energy spectrum of neutrons in typical thermal reactor will consist of slowing-down spectrum which describes the distribution of energies thermal spectrum of neutrons which among the neutrons being slowed down, plus have come into equilibrium with the thermal energy distribution of the reactor atoms. The slowing-down distribution will be described in this section, and the thermal dis tribution in Art. 4. For very fast neutrons the slowing-down collisions may be either elastic or inelastic. The elastic process by far the more important one in determining the effective thermal reactor and the behavior of neutrons over the entire energy spectrum in The effects of inelastic slowing down can, process which will be considered here. however, be included in the constants, such as r (Art. 3.4), which describe the slowingdown distribution in the reactor. 3.1

Loss of Energy in

a

a

is

is

a

a

Single Elastic Collision

f

C

is

is

is

9

The loss of energy in a single collision with a stationary nucleus depends upon the mass number of the scattering nucleus and upon the angle between the original direction of the neutron velocity and the direction after the collision. It useful in describing the mechanics of elastic collisions to employ the center of mass coordi at the center of mass of the neutron plus nate system. The origin for this system Thus the center of mass (or C) system moves colliding. the nucleus with which at constant velocity with respect to the laboratory (or L) coordinate system, and the system. The scattering angle collision always takes place at the origin of the it


1

m

Inside

- (l/xa)\ - (l/*o)]

xa

+ 2xD[coth xa

1

2.34

- 2*D[coth

a :

1

_

1

:

Sphere of Radius a

g

2.33

(2.,)

+ 2xD coth xa

b;

=

1

p

:

The albedos below, for other geometries, 2.32 Slab of Finite Thickness a

in the

CALCULATIONS

REACTOR

Sec. 6-2]

L system

6-43

is related to the scattering angle > of the C system by

— -—1

cos 8 =

VI

+ A cos

+

<4!

—

.

.

(33)

+ 2.4 cos

A is the mass number of the scattering nucleus. The ratio of the neutron energy after collision to the energy before collision is

where

E, where

——

In

(36)

r

-

1

- - In

+

t

-f-

= 180° (head-on collision) and is equal to r. The minimum value of E/Et, occurs for For neutrons in the energy ranges important for thermal reactors, the assumption is usually made that all values of cos > are equally probable (the scattering is spherically symmetric in the C system) and hence all values of E/E0 from 1 to r are equally probable. The average logarithmic change in energy per collision £ is then

0.

1

A

1

A

is

£

n

t

= i=±

(37)

it

is

is

convenient to Frequently, in treating the slowing down of neutrons Lethargy. employ a dimensionless variable, called the lethargy, in place of the energy variable. The lethargy u denned by:

E

,

En u = In —

per

£,

is

the energy at the instant in the neutron and E just the average change in lethargy of a neutron

is

the initial energy of then, The quantity

is

where E<> question.

collision. 3.2

Collision and Slowing-down

Densities

q

Q

(E)

is

is

E

is

Q

is

a

is

is

E

is

defined as the number of neutrons The slowing-down density in a medium In the general per cubic centimeter per second. down past a given energy In an infinite medium which absorbs case, a function of energy and position. uniform source of fast neutrons only thermal neutrons and which supplied by constant per cubic centimeter per second of energy E, the slowing-down density and equal to at all energies from thermal to E. 3.22 The collision density per unit energy, F(E), simply the number of collisions made by neutrons with atoms of the medium per cubic centimeter per second per If the neutron flux per unit energy interval unit energy interval at the energy E. S,(£), then the collision density at energy and the scattering cross section 3.21

that, slow q


if

If

{

= = oo are, respectively, and The limiting values of for (hydrogen) and the moderator contains n different species of nuclides, of scattering cross section 2,i and average logarithmic energy decrement per collision £„ and all the scattering with energy, way cross sections are either independent of energy or vary in the_same then the average logarithmic energy decrement per collision for the mixture

F(E)

= -L.{E)4>{E)

(38)

6-44

PHYSICS

REACTOR

[Sec.

In an infinite medium which absorbs only thermal neutrons, the collision density function of energy is given by

F(E) if the medium is hydrogen.

|

=

6

as a

(39)

For moderators other than hydrogen the collision

density

is

F(E)

=

~

(40)

for all energies E below about r*E0 [see Eq. (35)]. For higher energies the variation of F(E) is more complicated; it is discontinuous at the energy rEe because of the cir cumstance that above this energy neutrons are present which have suffered only one collision (as well as those that have had multiple collisions) whereas below rEo all neutrons have had at least two collisions.4 Equation (40) applies for a moderator composed of a mixture of atomic species provided it is possible to write an average logarithmic energy decrement | (see Eq. The neutron flux can be obtained from Eq. (39) or (40) (37)] for the mixture. by use of Eq. (38).

(E)

_ F(E) _

Q

1

*(E)-^)-Evm

.... m


Thus, over any energy range for which 2, is constant, the flux is just proportional to 1/E. This condition holds approximately for many moderators over the resonance energy region. 3.3

Spatial Distribution

of Slowed Neutrons

If there is a point source of fast neutrons in a uniform moderating medium of infinite extent, the slowing-down density at any energy below the source energy will have some spatial distribution which is spherically symmetrical about the source and which is otherwise characteristic of the energy and 010 of the moderator. If we are concerned with the calculation of a thermal reactor, we are interested in the slowing-down den sity at some energy just above thermal. Each fission in such a reactor is a source of fast neutrons. If we know the char acteristic distribution of slowing-down density into the thermal-energy region from such a fast source, then we can treat each fission as a distributed source of t her mal neutrons which diffuse according to the laws of Art. 2 until they produce fur ther fissions to maintain the chain reac tion. By the principle of superposition CENTIMETERS FROM SOURCE Fig. 3. Flux of 1.44 ev neutrons from a point (sec Art . 2.25), the effectsof all the sources source of 1 fission neutron per second in an can be added up independently to give the infinite medium of HjO. Fo(r) is the flux in reactor equation. The method by which neutrons/(cm!) (sec) (ev). (Reproduced from this is done is treated in Art. 5. We con J. E. Wilkin*, R. L. Hellene, and P. F. sider here the methods of describing the Zweifel, Status of Experimental and Theo slowing-down distribution. retical Information on Neutron Slowing Figure 3 is a plot of the measured Down Distributions in Hydrogenous Media, Geneva Conf. Paper A/Conf.8/P/597, June slowing-down density distribution, at the indium resonance energy, from a point 30, 1955.) fission source in water (the indium nance energy is a convenient one at which to make the measurement; the distribution This curve is actually the kernel of is nearly the same as that just above thermal). the slowing-down density (see Art. 2.25) for the case in question. If such kernels are

6-45

CALCULATIONS

REACTOR

Sec. 6-2]

incorporated into a reactor equation, it is evidently desirable to describe them by relatively siraplcequation. Furthermore, it is desirable to be able to compute the distributions from microscopic data. It is by no means obvious that the distribution should be describable by a simple mathematical function; indeed, in the general case, it is not. The Fermi approximation, discussed below, gives a simple description which is satisfactory for many cases. For an extensive treatment of the slowingdown distribution by more rigorous methods, see Refs. 5 and 6. to be

some

3.4

The Fermi Age Approximation

This approximation is valid only if the scattering mean free path does not vary appreciably within any energy interval E to E', where E' = rE [see Eq. (35)], and if the number of collisions experienced by the neutron in reaching the energy of interest

The approximation

is therefore quite poor for media in which the moderation by hydrogen or deuterium, but it is widely used for moderators of higher atomic weight than these. The approximation does not describe properly the slowingdown distribution at distances very far from the source, but at such distances the slowing-down density is too low to be important, in most reactor problems. 3.41 The Fermi Age. If we have source neutrons produced at energy E0 and are interested in the distribution as a function of E for all energies E less than EQ, we define a new variable, the Fermi age r, such that is large.


is mainly

is the diffusion coefficient (see Art. 2.13), 2, is the macroscopic scattering and £ is the logarithmic energy decrement per collision. The slowingdown density q then becomes a function of space and of r, and its distribution is given by the Fermi age equation. 3.42 Fermi Age Equation : where

D(E)

cross section,

VHM for all

values of

t

=

^

(43)

less than Ea),

> 0 (i.e., for all energies

lim V?(r,T) =

dj^-

+ 5(r)

The initial condition is (44)

S(i) is the source strength of neutrons of r = 0 (i.e., energy Eo); Vs is, of course, Laplacian operator. Note that r, although referred to as the age, has the dimension (length)*. It can be shown to be one-sixth the mean square (crow flight) distance traveled by a neutron from the time of its emission by the source until it reaches the energy E, which cor responds to the value of r. An alternate statement is that r is one-sixth the second where the

spatial moment of the slowing-down is defined by

density where the second spatial moment r'(r) /

H(r)

=

^ / JO

r'q(r,T)4rr' dr (45) <3(r,T)4*T!

dr

Equation the slowing-down density from a point source at the origin. to compute t for a mixture of elements provided the average value |Eq. (37)] is used for £. 3.43 Solutions of the Age Equation. The solution The Slowing-down Kernels. of the age equation for a unit point source of monoenergetic neutrons at r0 in an infinite medium, the neutrons being emitted at an energy corresponding to t = 0, is
(42) can be used

,,(r,r)

,

(4«t)W

'
( Ir4/"1')

(46)

6-46

HEACTOR

PHYSICS

[Sec.

6

This solution is the point kernel for the slowing-down density. It is a displacement kernel; the slowing-down kernels for a distribution of sources may be superimposed to give the net slowing-down density, as discussed for the diffusion kernels in Art. 2.25. Note that the slowing-down density has the same spatial distribution as the Gauss error curve. In a given moderator containing a single point source, the slowing-down density is always highest near the source ; it falls off more or less rapidly with distance according as t is small (high energy) or large (low energy). The kernels for the slowing-down density in an infinite medium with various source geometries are given in Table 3. Table 3. Gaussian Slowing-down Kernels in Infinite Media The kernel gives the slowing-down density, at point r (or z as the case may be) and age r, produced hy unit sources of neutrons of age t' located at point r* (or x'). The flux 0(r,r) can be obtained by dividing power |2,(t), provided the value of t is one which corresponds to the the kernel hy the slowing-down slowing-down distribution (i.e., provided the neutrons are not thermalised). Source geometry

Point

Symbol for kernel

P(r,r;rV)

Piano

Source strength and location

1 neutron/sec

at

r', t'

1 neutron

Kernel

of age

over a plane of infinite extent at X 1 neutron of age over T'/(cni)(8et') a line of infinite length, at r', *'

Line

Spherical shell

P.(r,T;r'.T>)

1 neutron of age t'/scc per shell of radius r' , with center at origin

Cylindrical

P.(r,T,rV')

1 neutron of age r'/sec per cm h-ngth of infi nitely long shell, of radius r', with axis at r = 0

shell

4
-

L

4(,

- r')J

r')

J

- O]'' [4»(r - r'l'l T —I* ■ exp 1 Vl 4(r - r') J [4,(t - r01* Mr where p1

- r')

»

r* + r'1 — 2rr' oos («» —

««>[-;,_,,] 4r(r-r')

/

Mr

„•

n

- r>\J

If the number of neutrons Slowing-down Distribution from Fission Source. 3.44 emitted per second between energy E' and E' + dE' by a point source at the origin, in an infinite medium, is F(,E') dE', then the slowing-down density from the source is "[r'T(E)]

"

[">

Je

e-rV4[r(g)-r(«')lF(g')

|4,[r(£)

....

dE'

(4']

^rjEWr

is,

Similar equations will hold for the other slowing-down kernels. Evidently, the slowing-down density at any energy E, as given by Eq. (47), is no longer (iaussian. Nevertheless, in reactor theory, it is often the practice to define an effective age f, which can be used with the point kernel for a monoenergetic source however, [Eq. (45)], to describe the neutron balance in the reactor. The proper f For very a function of the size of the reactor (or, more precisely, of the buckling B1). large reactors (small buckling) the effective value of t for the slowing-down density just above thermal energy given by is


r'/fcm') (sec)

L

Aft

.

(E',E„.)F(E') dE'

(48)

REACTOR

Sec. 6-2]

CALCULATIONS

6-47

In other words, for this case, one uses just the S is the total source strength. arithmetic mean of the ages of all the fission neutrons, from their source energy to thermal.

where

3.5

Slowing Down with Absorption

In an infinite medium containing a uniform Resonance Escape Probability. of high-energy neutrons of strength Q(E„) neutrons/(cm3)(sec), the slowingdown density q(E) at some lower energy E will be just equal to Q(Eo) if there is no If there is absorption, the slowing-down density at E will be some absorption. This fraction is called the resonance escape fraction p(E) of the source strength. 3.61

source

probability:

q(E) = p(E)Q(E0)

(49)

For a homogeneous medium of total macroscopic scattering cross section X,(E), total and effective logarithmic energy decrement ?(£), the absorption cross section resonance escape probability is calculated by


P(E)

= exp

{

-

i ^ ^}

jE

(50

It is valid for modera The expression is correct if all the moderation is by hydrogen. tors of mass number greater than 1 if all the absorption is by a single narrow resonance or by a number of narrow resonances provided in every case the energy of a resonance is higher than that of the next lower resonance by a ratio which is at least as great as (l/r)»or (l/r)« |Eq. (35)]. If the moderator is of mass number greater than 1, and if the absorption cross section and the scattering collision density vary only slowly with energy, the resonance escape probability is given approximately by7

*»— >(-//' fdhF.f) -

wbm

,

and

« =

l-'-"-"V2 1 —

In

-

r

—

rt

T

<5" (52) (53)

Although many cases arise in which Eqs. (49) and (50) do not apply rigorously, they are used extensively in practice and give reasonably accurate results for most cases, particularly if the resonance absorption is not large. It should be pointed out that in those cases where the absorption is by resonances, the above expressions will give very poor results unless the absorber is distributed in a manner. For resonance absorption in heterogeneous very nearly homogeneous media see Art. 9.4. The spatial distribution of slowing-down density from a localized source of 3.52 fast neutrons in an infinite medium will, of course, be modified from that discussed in Arts. 3.3 and 3.4 if the medium absorbs neutrons. If the Fermi model is used, and if the medium absorbs only weakly, it can be shown that the spatial distribution of slowing-down density q'{x,E) from any source distribution in the absorbing medium is related to the slowing-down distribution q(r,E), which would result from the same source distribution if there were no absorption, by

q'(T,E) = p(E)q(i,E)

(54)

It is generally where p(F.) is the resonance escape probability discussed above. assumed in reactor calculations that the same result applies regardless of the slowingdown kernel used and, indeed, that the result applies not only for an infinite medium but for the finite reactor as well. 3.53 The Resonance Integral. If the microscopic absorption cross section ofv*54) material, as a function of energy, is denoted by oa(E), then the resonance inter

6-48

REACTOR

E

for the material between the energies

PHYSICS

[SEC. 6

and En is

fEt o<,(E') fig' — JE E'

Resonance integral =

/

(55)

Note that since for typical moderators the flux is proportional to l/E in the resonance region [cf. Eq. (41)], the resonance integral is just the integral of the absorption cross section weighted by a factor which is proportional to the flux in such a moderator. 3.64 Use of Resonance Integral to Calculate Resonance Escape Probability. For the case in which the macroscopic absorption cross section is very much smaller than the macroscopic scattering cross section of the medium, the resonance escape prob ability becomes

p(E)

= exp

- -)

^- jg

(-

= exp

— -)

jg

(56)

where Na is the number of absorber atoms per cubic centimeter. Again, if the scatter ing cross section is constant over the range of important absorption, the resonance escape probability can be written


p(E)

= exp

(

^

~ jg

= exp

£

-i^

(resonance

integral)

J

(57)

Note that the ratio Na/2, is the reciprocal of the macroscopic scattering cross section of the medium per atom of absorber in the medium. Measured values of the resonance integral are usually quoted for the energy interval from fission (En) to the cadmium cutoff. The upper limit of the energy is usually unimportant as long as it is high, since absorption cross sections are low in the millionelectron-volt range of energies. 3.65 The effective resonance integral is so defined that an equation similar to Eq. 2, does not hold : (57) will apply even if the inequality 2„

«

p(E)

= exp

|

—

i

—

and Effective resonance

(effective

resonance

f integral = /

-

~

Eo

JE

2

-

E'

1 1

+

(58)

— o-„ dE' -=—

s. + 2.

fE°

)e

integral)]

(iW2.)

dE' ~W

nun (W)

Note that for a given absorber the effective resonance integral will depend only on the quantity t,/Na, the scattering cross section per absorber atom. Thus, a few measure The ments of the effective integral as a function of this ratio will define its variation. curve of effective resonance integral vs. 2,AVo will, of course, extrapolate to the true resonance integral for very large values of 2,/JVo. 4

ENERGY DISTRIBUTION OF THERMAL NEUTRONS

Neutrons slowing down in a moderating medium, if they are not captured first, will eventually reach such low energies that upon collision with nuclei of the medium they may either lose energy or gain energy from the thermal motions of the nuclei. Ulti mately they come into equilibrium with the nuclei of the medium, gaining, on the The neutrons in such a case are referred to as average, as much energy as they lose. thermal neutrons. 4.1 J

The Maxwell -Boltzmann Distribution

If the absorption is quite low, the neutron energy distribution follows the Afaj-ipW/iltzmann law, the same as that which describes the distribution of energy of thermal If n(E) is the number of neutrons per :tation among the nuclei of the moderator.

REACTOR CALCULATIONS

Sec. 6-2]

6-49

unit volume per unit energy interval at the energy E, if n is the total number of thermal neutrons per unit volume, and if T is the absolute temperature of the moderator, the Maxwell-Boltzmann distribution of neutron energies is

_

n(E)

B* r_K/lT

2

(60)

where it is the Boltzmann constant and is equal to 8.61 X 10-s ev/°K. The velocity distribution of the neutrons is given, in terms of n(»), the number of neutrons per unit volume per unit energy interval, by n(i>)

!T

=

/

v;te)\H 4

m

/ mv'\ exp(-2kf)

v

.

(61)

where m is the mass of the neutron.

The most probable velocity a of the neutrons in a Maxwell-Boltzmann

hkT "V

distribution

(62)

m

The energy corresponding to this most probable velocity is

E

-

3^ ma'

- kT

(63)

4.2

Specification of Neutron Energy: kT Neutrons

The energy of a thermal-neutron distribution is ordinarily characterized by giving energy kT corresponding to the most probable velocity in the distribution. Neutrons in a thermal distribution of temperature T are sometimes referred to as kT neutrons. "Thermal cross sections" are usually quoted as the cross sections for This velocity corresponds to the monoenergetic neutrons of velocity 2,200 m/sec. kT value 0.0253 ev and is the most probable velocity for the temperature 20°C. For computing the number of processes occurring in a medium containing a thermal dis tribution of neutrons, an average cross section is usually derived, as discussed in the

Art.

4.3. 4.3

Averaging of Thermal Cross Sections

In a diffusion medium, if any process is characterized by an energy-dependent cross section Z(i?), then the number of such processes occurring per unit volume per second is the neutron flux per unit energy per unit energy interval is
jj^y

*(£) dE

[J

=

J*

dE

-Z(E)n(E)v(E) dE n(E)v(E) dE

J

-Z(E)4>(E)

J

Jo~

(E)

JQ~


but the most probable energy is \$kT.

2(B) Ee-WT dE

m)

6-50

REACTOR

PHYSICS

[Sec.

6

The averaging

can be done numerically for any measured variation of the cross section. Obviously, either the macroscopic cross section 2(£) or microscopic cross section r is

l

and the average cross section for the absolute temperature cross section for the absolute temperature To by


(65)

l2g

T

is related to the average

For certain important

cross sections which do not follow the l/v variation, the averag ing indicated by Eq. (64) has already been done over a range of values of kT and factors have been tabulated which correct for the departure from l/v dependence. Thus, in the cross-sectional compilation of Hughes and Harvey* a correction factor/ is given as a function of energy for a number of nuclides. If the kT cross section of any of these nuclides, at energy E, is multiplied by the value of/ corresponding to the same energy E, an effective kT cross section, o,//(kT), is obtained. The use of this effective kT cross section in Eq. (65) will yield the correct value of 9 averaged over the Maxwellian distribution which has its most probable velocity at the energy E. 4.4

Deviations from the Maxwell-Boltzmann

Distribution

If absorption in the thermal region is large, the energy distribution of neutrons is modified by the resultant loss of neutrons. The effect for practically occurring absorbers is to shift the energy distribution toward higher energy values. The energy distribution in a medium of atomic hydrogen (constant scattering cross section) with l/v absorption has been investigated by Wigner and Wilkins.9,10 These studies indicate that the absorption does not change the shape of the distribution drastically but shifts it toward higher energies. Experiments on existing reactors indicate that the effective neutron temperatures exceed the moderator temperatures by the order of 50°C. Even in the case of weak absorption the Maxwell-Boltzmann equation cannot give the true energy distribution of neutrons in the energy range several times kT, for in this region the thermal distribution must merge into the 1/E distribution which char acterizes the slowing-down process (Art. 3). These two regions may be joined approximately by equating the slowing-down density just above thermal energy to the total absorption of thermal neutrons *

(thermal) £„ = q(E,h)

=
where tjiiE) is the flux per unit energy interval in the energy range just above thermal and is thus given in terms of the total thermal flux by

*(E) (thermal)

_

a. E(Z.(E)

(67)


Sec. 6-2]

6-51

other hand, 4>(E), the flux per unit energy interval, is given in the thermal in terms of (thermal) by the Maxwell-Boltzmann distribution

On the region

»(g) (thermal)

"_

m

__2_ vim

= —

v

—

(*r)M

*e_E/kT

«-*/*»"

(68)

The boundary between the thermal and the slowing-down energy regions can arbitrarily be defined for a given medium as the energy at which the two distributions It is to be understood that this is a (67) and (68) give the same value for 4>(E). and that neither describes the true purely formal joining of the two distributions neutron energy distribution well in the transition region. In most practical problems a precise knowledge of the energy distribution in this region is not required.

6

CRITICAL REACTOR EQUATIONS FOR THERMAL AND NEAR-THERMAL REACTORS

The thermal-neutron

diffusion equation expresses the balance between production of thermal neutrons in a diffusion medium. In a thermal or near-thermal reactor the absorption of thermal neutrons by fissionable material leads to fissions which provide further neutrons, which slow down and ultimately become thermal. When the characteristics of the reactor materials and their geometry are so adjusted that the production of thermal neutrons by fission followed by slowing down (during which process some of the epithermal neutrons may leak from the reactor) is just pqual to the loss of thermal neutrons by absorption and leakage, the neutron popula tion of the reactor will remain constant in time and the reactor is said to be critical. The equation which satisfies this condition of criticality can be derived from the diffusion equation [Eq. (12)] by setting the rate of change of neutron density to zero and relating the source strength S(r) of thermal neutrons to the fission process. For the latter purpose the multiplication factor k will be defined. In defining k, the medium (reactor) is imagined to be infinite in extent but to contain the same energy distribution of neutrons as the finite medium under consideration. The quantity k is defined as the number of thermal neutrons which would be produced in such an infinite medium, by fission followed by slowing down, per thermal neutron absorbed. When the procedure described above is applied, a general reactor equation is derived, the specific form of which depends upon the neutron slowing-down law By the application of approximations for the reactor composition. appropriate which are valid for most practical cases, the formal solution of the general equation can be written in terms of the wave equation, which specifies the spatial distribution of neutrons, plus a characteristic equation, which specifies the conditions for criticality in terms of t he buckling of the wave equation and the constants of the reactor materials. The remainder of this section will present the reactor equation in detail, its general solution, and the form of the equation and its solutions for various common slowingdown laws. Detailed solutions for various geometries will be treated in Arts. 6, 7,


and loss

and 8. 6.1

The Finite, Steady-state, Reactor Equation

If we apply the "monoenergetic" diffusion equation [Eq. (12)] to the thermal group of neutrons, using appropriately averaged values for the thermal diffusion properties of the medium (Art. 4) and identifying the local source strength of thermal neutrons »S(r) with the local value of the thermal slowing-down density q{E,,t) we have D

V*4>.

(D

- Z.*(r)

+

j(«.,r)

=

where the subscript a designates the thermal (i.e., slow) group of neutrons.

(69)

e-52

PHYSICS

REACTOR

[Sec.

6

t

r',

In a chain-reacting system, the slowing-down density is that resulting from the We now designate by P(E„t,t') the finite thermal slowing down of fission neutrons. slowing-down kernel, i.e., the probability that a fission neutron created at inside the reactor, will become a thermal neutron in unit volume at r. The slowing-down density then becomes

-

Jf

reactor

(r)

- Z.*.(r)

+

-

f

W.

taken over the entire reactor volume.

p

D

is

where the integral tion then becomes

Z„0.(r')P(£.,r,r') dt'

(70)

volume

jrnactor

The finite reactor equa

Xa.(T')P(E.,T,t') dt'

=

^1

(71

)

=

p

q(E.,t)

dt

volume

or, substituting L2 for £>/£a and applying the equation to a critical reactor (&n,/dt = 0),

This

t

reactor volume

(72)

0

J[

- *(r) -

p

v«*. (r)

+

L'

is

is

5.2

The Asymptotic Reactor Equation

all space

*.(t')P„(E.,\t

- r'|) dt'

=

0

- *.(r) + -

J

V«*. (r)

p

L>

f„

a

is

a

r'|

|r

is

Far from reactor boundaries the finite slowing-down kernel can evidently be replaced by the corresponding infinite kernel P„(E„\r — r'|), which has the characteristic — that its value determined only by the absolute distance between the point of measurement and the point at which the fission neutrons were born. Such kernel called displacement kernel. The reactor equation for this condition becomes (73)

is

a

is

a

if if

the extrapolation distance can be considered independent of It can be shown that the finite and infinite slowing-down kernels satisfy the same neutron energy, and linear equation (true for the Fermi, group, and transport kernels and convolutions of them), then the asymptotic solution holds everywhere in a critical reactor, except within a distance of the order of mean free path from the boundary, and the slowingdown density in the reactor identical with the infinite slowing-down density except The asymptotic within distance of the order of a mean free path from the boundary. this equation can, therefore, be used for most practical reactor problems. It equation for which solutions will be given in the following sections. 6.3

General Solution of the Asymptotic Equation

The solutions of Eq. (73) are also solutions of the wave equation: + B**.

-

(74)

0

V»
is

is

is

where B* a constant, which generally referred to as the burkling of the reactor. Until further restrictions are put upon B', the solutions of Eq. (74) specify only a necessary family of possible spatial distributions of neutron flux in the reactor. It to consider further restrictions on the solution which take into account the neutron k

Historically, both and P(E».t,t') have been so defined as to take into account resonance absorption. The same definitions have been followed here. Hence the appearance of the resonance escape proba bility in the denominator of Eq. (70). p

t


the finite reactor equation. Evaluation of the final term evidently quite difficult, since P(E„t,t') will depend upon the leakage probability of fission neutrons formed at r'. In the following section approximations are made which simplify the evaluation of P(E„t,t').

6-53

CALCULATIONS

REACTOR

Sec. 6-2]

eP„(E.,\T - r'|)d(|r' - r|)

-

-P„(B.,Bl)

+

=

then (76)

V

-L»B>

(75)

0

equation of the transformed reactor equation 1

The characteristic

=

is

f\(£.,B>)

J_"_

Evidently, once a formal specification of the flux dis balance within the reactor. tribution has been made, as by Eq. (74), a formal evaluation can be made of the space In practice this is done integral of the slowing-down density [last term of Eq. (73)]. by taking the three-dimensional Fourier transform P„(E.,B2) of the slowing-down density with respect to B*:t

is

the complete statement of the general solution as follows: The asymptotic reactor equation [Eq. (73)] satisfied by any solution of the wave equation [Eq. (74)] a root of the characteristic equation [Eq. (76)]. Furthermore, the provided B*

q.(E,t)

given by

- Z„P.(£,B')0.(r)

(77)

V

=

is

slowing-down density q„(E,i) at any energy

E

is

is

and

is

if

a

is

is

is

a function (scalar or vector) defined by the equation *({.».?> =

j"m

in x, y,

dx

z

J

space

is

t £.

If tix.y.z) C

q.

space, its Fourier transform

fZ„

<*!/

^(f ,»?.i*>in the new

** «"** *",**rV'(*.lM)

-

*(r)e«*»-rdr

*M

=

~

J(o>).

f

J(u)

J

A

is

compressed notation generally used, in which (z,y,z) =* r; ({.ij.f) = oj; dx dy dz -~ dr; d£ dij df ^ du. In this notation the transform and its inverse are

if>(ta)depends

only on w

-

|o>|. and

f

r

if-fr) and ^(u) are either both scalara or both vectors. If the function >f>(r}depends only on = [r|, then its transform the transformations simplify to 00 4ir sin ur dr

that developed

by Weinberg

and described

in Ref.

vol.

part

1,

outlined here

ul(w) sin wr dr 2,

treatment

90

u

2,

1

5.

is

The

chap.

J If

^,(r) » _ — 2r*r

J


is

Equations (74) and (76) give two specifications for the buckling Bs of the reactor. Equation (76) specifies Bs in terms of the properties of the reactor materials. The sometimes referred to as the material buckling B„s. On buckling when so specified the other hand Eq. (74), when solved with appropriate boundary conditions, specifies B! in terms of the geometry of the reactor. When so specified, B* often referred to as the geometrical buckling Bg*. the reactor geometry and the characteristics Only of the reactor materials are such that Bm* = B„* will there be a solution for the steady-state reactor equation that satisfies the boundary conditions. This condition referred to as the critical condition. (B„» = B0l) The above treatment of the general reactor equation obviously little more than More detailed treatments will be helpful in fundamental applications conceptual one. of the theory. However, for the practical purpose of calculating critical reactors for which the usual slowing-down distributions apply, values of P„(E„B*) have been tabulated (see following sections). For these cases the appropriate value of P„(B„BS) can be substituted into Eq. (76), and solution of the reactor equation involves only the finding of a consistent value of B* which satisfies both the algebraic equation (76) and the partial differential equation (74).

6-54

PHYSICS

REACTOR 5.4

Gaussian Slowing-down

Distribution

[SEC. 6

(Fermi Age Theory) f

The infinite slowing-down kernel and the resulting characteristic equations are

P.(W)

Kernel:

= P

^

^ T/'^^

(TO

Fourier transform: Pm{B„B*) = pe^*"**' 1 + L'fl' - ke'3^^

(79)

Characteristic equation:

(80)

In these equations, t(E.) is the Fermi age from some "effective" value of the fission Rigorously, this effective value depends upon B„', but the value energy to thermal. for a medium of infinite extent can be used in most practical cases (see Art. 3.44). It will be recalled that the Gaussian slowing-down distribution results from the

assumption of "continuous" slowing down (Art. 3). The corresponding character istic equation (80) therefore is used most successfully for moderators which approxi mate this condition, i.e., for relatively heavy atoms such as carbon. Zero Slowing-down

6.5

Length (One-group Theory)

Kernel: P„(E.,t,i') = pi(r Fourier transform: P„(£'.,fls) = p Characteristic equation: 1 + L2B* = k 5.6

Exponential Slowing-down

- r')t

(81) (82) (83)

(Two-group Theory)

Distribution

It is assumed that a fission neutron diffuses without change of energy and with a diffusion coefficient D/ until a single (fictitious) slowing-down event occurs which reduces its energy to thermal in one jump. A cross sect ion Xa/ is assigned to specify the probability of the fictitious event per centimeter of travel, and a characteristic slowing-down length Lj is defined in analogy to the thermal diffusion length L:

PJE.,B*) L,'B')

|

r

(fB)

=

(86)

Lf^B*

(87)

k

+ L>B*)(l

+

(1

Characteristic equation:

—

-

-

<■ 1

Fourier transform:

(-l/L^lr

exp = P 47rL/!|r

p

Kernel: P„(E.,t,t>)

r»l

apply to this picture (when multiplied by 2„/ to give

2

-f"

The diffusion kernels of Art. slowing-down density) :

+

J/

0 is

such that

is

(89)

who* value whose

dx =

sero at all values of

i except r

—

0

*

pXa/4>/ = is

See Art. 3. the Dirac delta function, . defined as a function

and whose value at x =

(88)

P

-

+

W.

1.

D.

=

0

- 2°/*/ - ^..

Df V'*/

0

it

convenient to write For the most useful applications of the two-group model, the two coupled partial differential equations which, for this model, are equivalent to the asymptotic reactor equation (73):

t t 4 is


In this representation it is assumed that the fission neutrons are born as thermal It is a poor approximation for most cases; the "modified" one-group neutrons. theory (see below) is a much better approximation of equal simplicity.

REACTOR

Sec. 6-2]

CALCULATIONS

6-55

In these equations the subscript s designates the thermal group of neutrons and the designates the fast group. subscript These two equations are simply the diffusion equations for the thermal flux and the fast flux They are both satisfied by solutions of the wave equation

,

/

/.

Bi

/ ,

V*> +

= 0

and are so chosen as to preserve the relations between provided the solutions for the fast and thermal fluxes as specified by Eq. (88) or (89); e.g.,

,

*£ =

*e

+ D'B'

(90)

V'Z'I

Art. 7). Computation of the mean square slowing-down distance from the kernel |Eq. (85)] shows that it is just 6L/1. Hence L/' has the same significance in the two-group Obviously, the two representation as the age t has in the Fermi age representation. characteristic equations (80) and (81) will not predict the same criticality conditions for a given reactor if identical values are used for r and Lj1. Experience has shown that the two-group picture is the better approximation for hydrogen-moderated reactors and that reasonable agreement between experimental and calculated critical sizes for such reactors results if the measured value of the "age" or slowing-down area is used as the value for L/1. For reactors moderated by relatively heavy materials such as graphite, the Fermi age picture is much the better approximation, and good results can be obtained in two-group calculations only if some adjustment is made on the value of L/1. Despite its limitations the two-group approximation is used extensively for multiregion calculations (e.g., reflected reactors) on many types of reactors because it is the simplest approximation which will give reasonably accurate results for such problems. When moderators other than those rich in hydrogen are involved, L/1 is often considered as an adjustable quantity whose magnitude can be adjusted to compensate for the relatively poor description which the diffusion kernel gives of the For example, if a reflected reactor is to be solved, a slowing-down distribution. (ruess may first be made of the equivalent bare dimensions of the reactor (Art. 7.6), and the equivalent bare reactor may be solved by some approximation better than the two group. An adjustment may then be made in L/* to force the two-group calculation to give the same result for the equivalent bare reactor, and this adjusted Lf1 may be used in the calculation of the reflected reactor.


face

6.7

Modified

One-group Theory

If the left-hand side of Eq. (87) is expanded and the term in B* is neglected, obtain the characteristic equation for the modified one-group approximation:

I +

(L» +

L/')B*

= 1 +

= k

we

(91)

A similar result is obtained if the Fermi age equation (70) is expanded to terms of The quantity 3/1 = L' + L/s (or L* + r) is called the migration

first order in B*. area.

5.8

The Multigroup

Approximation

In the multigroup representation treated here, the epithermal energy range is divided into a number of smaller energy intervals and all the epithermal neutrons are sorted into the same number of "groups" according to the energy interval into which they fall at the instant considered. A neutron of a given group is considered to diffuse at constant energy until a slowing-down event occurs which slows it into the next lower group in one jump. Every neutron, during its lifetime, is considered to pass through every energy interval (for a more general multigroup representation, see Sec. 10). Evidently the two-group representation is the special case of the n-group

6-56

PHYSICS

REACTOR

[SEC. 6

representation in which n = 2. For the n-group case we define n — 1 fictitious slowing-down cross sections 2„i which describe the probability of slowing from one group to the next and (n — 1) L.'s and x,'s which are related to the 2„; and the average diffusion constant Z>,-for each group: (92)

di,

dr2

^

din^'

—

exp — x;|r3 — r2| m

4irL2!|r,

-

-

xi]r8

exp —

—

rt|

r2|

—

x„-i\u

4»L»_i,|r»

r»_i| - r»_,|

(93)

+Li»B»)(l

and the characteristic equation

+L2'B*)

-I- L„_,»B«)

is

= (1

P„ (£.,«»)

(1

The Fourier transform of the kernel

is

x,- = 1/L<.

where

i-l is

it

will be remembered, L„' just the thermal diffusion area L*. The characteristic equation thus an algebraic equation of degree n.

where,

1,

k

If

a

1

is

> there are always n — complex or negative roots in addition to the one real, positive value for B*. In the case of bare reactor, the complex and negative roots are eliminated by the boundary conditions.

- r,|) - /rfr2/rfr,

a

is

an example of Equation (93) has the general form

P(|r„

Kernels; Synthetic Kernels

Convolution of Slowing-down

6.9

of slowing-down kernels.

convolution

-

r,|)][P,(|r, r,|)] X [P,(|r4

/dr„_,[P,(|r2

-

-

r2|)]

[P„-,(|r„

■• •

If the P._,

,

— r'|/4r

,-|r-f|/i +xZ.«|r

Transport

Transform

- r'|

e-|r-r1A 4xX|r

-

r*|«

1

Diffusion

.— (Fermi)

-i-

AO

If

a kernel

- r._,|)]

+ L>B« tan

(XB)

(94)

Fourier transthen the

(B»),

the mean free path)

'

Gaussian

Kernel

f„(£,r,r')

above;

1

Type

as in the discussion

X is

Kernels and Their Transforms

have the same meaning

|r

r

Table (L* and

4.

.

,

a

is

said to be convolution of the kernels Pi, Pi, . . . P» forms of these kernels are, respectively, Pi(Bl), P2(B'),

it


cxp

■■■

r,|

= p

_ _

P.^.r^r!)

J_a

If we let i = 1 refer to the highest energy group, the slowing-down kernel into the nth (thermal) group is

HE ACTOR CALCULATIONS

Sec. 6-2]

transform of the convolution

P(B»)

kernel

P

6-57

is simply the product ■■•

=

[Pn-,(B')1

(95)

Convolutions of kernels of the same or different types may be used to approximate The three types of kernels slowing-down distributions in various moderators. which have been employed, with their transforms, are listed in Table 4. If a convolution kernel is arbitrarily constructed to fit the measured slowing-down distribution in a moderator without regard to the physical aspects of the slowing-down For example, the slowing-down process, the result is often called a synthetic kernel. distribution in HtO has been described by a convolution of four diffusion kernels whose characteristic lengths [Li, Eq. (92)] were arbitrarily chosen to fit the experi mental measurements. the

SOLUTIONS OF THE WAVE EQUATION FOR HOMOGENEOUS

6

BARE REACTORS

pointed out in Art. 5, solution of the critical reactor equation consists of finding of B1 which satisfies both the wave equation (74) and the characteristic equation (76). We consider here the solutions of the wave equation for homogeneous bare reactors. A one-dimensional (slab) and a three-dimensional (rectangular parallelepiped) case will be treated as examples. As

& value


6.1

For

a

chain-reacting

slab,

Infinite Slab

infinite in the y and z directions, the wave equation

becomes

If

we take

the center plane of the slab as x = 0, the solution is = A cos Bx + C sin Bx

where must

A and C are constants, to be chosen to fit the boundary conditions. Since be symrfletricnl about x = 0, the constant C must be zero, and the solution a

Fia.

4. Solution of the wave equation

contains only the cosine term (Fig. extrapolated boundary of the slab. slab

4).

for infinite bare slab.

= 0 on the The boundary condition is is met if the half thickness of the

This condition (a/2) is equal to ir/2B or 3t/2B or 5r/2B . ,

. , etc.

6-58 Table

PHYSICS

REACTOR 5.

Reactor shape

Extrapo lated bound-

Form of 7'0 +

BV

= 0

^ + B>* - 0

«* i4 cos —

£».

r = It Planes

dr"

Critical

Solution

-

:

(¥0

t dr

ft =

2.405

(2.405)'

j»

0'4>

I

*•»

r ar

d'» a7~

+

«

.

-0

b>

- _+ »«

,

/2.405

\ H>

(2.405;'

R>

For

critical

H>

B


B

Rectangular paral lelepiped of sides = a, b, and c; origin of coordinatesat center

Planes

ay

ay

a*»

ai/>

ay a7» + bj« = o

■4

toe

1 mi

icy • A. COBrx — COB— COBwz — ■5* + o'

^+ ^

f

o«

t»

c'

For minimum critical volumr ^

c>

V3, B

,4i Sphereof radius R

6

Solutions of the Wave Equation for Homogeneous Bare Reactors

Slab; infinite in y Planes and z direction, thickness *=a in x direction ; ecu tral plane at x 0 Infinitely long cylinder of radius R

Cylinder of radius R, tenth H; ori gin of coordi nates at center

[SEC

~ _

5.441 B

I6I\

H>)

T= R dr"

A . * _ _ „„

r dr

r

«i =

irr

R~B

ft 4

ft'

4x«

130

3

When the condition is added that the flux shall not be negative anywhere in the slab, nil conditions are satisfied only by a slab of extrapolated half thickness a/2 = x/2B. Thus the multiplying slab is critical if its extrapolated thickness a is just equal to */B, where B is the solution of the characteristic equation (76). t The neutron flux will be constant in time, and its spatial distribution will be given by 4> = 4n>cos

Bx =

where
o cos -a x »= 0, the

center plane of the slab.

Rectangular Parallelepiped

The expanded form of the wave equation is

^

dx'

av dy'

£^ dz>

t The physically meaningful expressions for PîE.B*) have such form that the characteristic equation In Negative roots may, however, be present. has a single solution for which is real and positive. slab geometry they lead to solutions of the wave equation involving sinh and cosh (or exponential) In the case of the single-region reactor these solutions are eliminated by the boundary condi terms. tions, cither because they cannot be made to go to tero at the reactor boundary or because they give discontinuities in neutron current at the center of the reactor. These arguments do not hold, however, in the case of a multiregion reactor (see, for example, Art. 7).


Sec. 6-2]

If the origin of coordinates is taken at the center metric solution is where

(cf.

Art.

6-59

of the parallelepiped, the sym

= 4>i> cos KiX cos Kuy cos
2.26)

= B»

*** + «„' +

Note that the solution is just a product of three slab solutions. This possibility of separating the solution into independent geometrical components is characteristic of bare reactors. If the extrapolated dimensions of the parallelepiped are a, b, c in the x, y, and z directions, respectively, K »

and the

- l'a'

,

» =

**

-

1 ■

b2

ll c2

reactor is critical if the dimensions are such that

*IT2 o2

+

IT2 IT2 I. + L 62

c2

= B2

Thus, the reactor can be made critical by proper adjustment of any one of the dimen the other two being fixed arbitrarily provided they exceed a certain minimum value (the value for which the third must become infinite).

sions,


6.3

Other Shapes

Table 5 gives the expanded form of the wave equation, its solution, and the critical dimensions for bare homogeneous reactors of various shapes. 6.4

The Absolute Value of the Thermal-neutron

Flux

This value in a thermal reactor is given by
X lO"-^,

Hjhj

(96)

Mf

$ = average thermal flux over the reactor core volume A = atomic weight of fissionable material in reactor over 9/ — microscopic fission cross section of fissionable material, averaged thermal energy spectrum, barns E = average energy liberation per fission, Mev P/Mf = ratio of operating power of reactor to fissionable material content of the reactor, watts /g or kw/kg If the fissionable isotope is Um, and if 9/ is taken, roughly, as 500 burns and E as

where

200

Mev,f

} = 2.4

X 10"

(97)

Mf

For an effectively homogeneous reactor the above expressions give the average flux over the core volume. For a lumped reactor they give the average thermal flux in the

fuel. 6.6

The Absolute Value of the Fast-neutron

Flux

This value in a reactor is not often a very useful concept, since the total fast flux includes neutrons of such widely varying energies. For rough estimates the value of the two-group fast-neutron flux may be used. In a bare reactor (or far from the If an accurate value is required, it must be evaluated for one. t This value is only an approximate the specific reactor in question, taking into account the leakage of neutrinos and some y and neutron energy from the reactor, as well as the production of extra energy by the nonfission absorption of Typical values lie between 190 and 200 Mev. neutrons.

6-60

REACTOR

PHYSICS

[Sec.

6

reflector in a large reflected reactor) the two-group fast flux is everywhere propor tional to the thermal flux in the ratio

*l „ * ?S 4>.

p

2./

1

1

(98)

+ L,*B*

where 2, is the macroscopic thermal absorption cross section, L/* is the slowing-down area, B1 is the geometric buckling, and 2„/ is the fictitious slowing-down cross section

/

Although this equation is convenient to use, since the absolute value of is known, a more straightforward expression for

*' _

v

,

is usually

X No. fissions/(cm3)(sec) Z.,(l + L,W)

Thus, although the thermal flux depends on the specific power (power per unit mass of fissionable isotope), the fast flux depends only upon the number of fission neutrons produced per unit volume per second, the slowing-down cross section, and the fast leakage 1/(1 + L,*B'). The Fractional Leakage (£) of Neutrons

6.6


This value from

a critical reactor is given by

2

_

No. neutrons leaking/sec No. fission neutrons born/sec

k

— 1 A:

For a hare thermal reactor, if it is assumed that the leakage of neutrons in the energy interval between resonance absorption and thermal can be neglected, the leakage can be divided into fast and thermal components (see Art. 5): No. fast neutrons leaking/sec No. fission neutrons born /sec

_

^

P„ p

^

= 1 — e~rH'

for Fermi slowing down

L,'B'

for "two-group" slowing down + V#! &B1 No. thermal neutrons leaking/sec _ /»„(£. ,B>) 1 + L'B1 No. fission neutrons born/sec 1

A further useful

leakage

( 100 1

(101)

relation is

No. thermal neutrons leaking/sec No. thermal neutrons absorbed/sec 7

L*B*

(102

REFLECTED REACTORS

A reflector is ordinarily a mass of material having k <1, which is placed around the outside of the multiplying core of the reactor to reduce the leakage of neutrons out of the core. This reduction of leakage has the two advantageous effects of reducing the quantity of fissionable material required for criticality and of causing more nearly uniform flux and power distributions in the reactor core. The most effective reflector material will be that which has the largest albedo for neutrons of the The reflector is often energy-distribution characteristic of the reactor in question. In this case it returns neutrons to the reactor core at made of moderating material. considerably reduced energies and modifies the energy spectrum near the corereflector boundary. Except in the special case for which the diffusion properties of the core and reflector are identical for all epithermal energies, reflected reactors can be solved only by one


Sec. 6-2] Table Dimensions: T, R\, Rt. a, b Variable*;

6-61

Definitions and Symbols Used in Arts.

6.

Dimensions

of reactor

regions.

See Figs.

7

and 8

4 to 9.

Dimensional variable of flux distributions; measured from inner dimension of each region in a slab, measured from center of core for cylinder and sphere = Veetor whose four components are, in order, fast flux, thermal flux, negative fast f current, and negative thermal current at an internal boundary M, N, V, W, X — Functional values of flux distributions A, B, C, E = Numerical coefficients Characteridtic constants of the reactor: k = Multiplication constant in the core material u* = For any region, the contribution of the leakage from iinreflected boundaries of the region to the total buckling = (it /a)2 for an unrejlected slab of thickness a = (2.405//f)2 for an unrejlected cylinder of radius R xf* = Reciprocal of flowing -down area = l/Z./1 x<* =• Reciprocal of diffusion area = 1 P

/LJ

I + u*

Macroscopic thermal absorption cross section Macroscopic slowina-doiun cross section = D//L/ Diffusion coefficient for thermal neutrons Diffusion coefficient for fast neutrons Material buckling of any region "Fundamental" buckling in the core,


2.

+ H V(*/' + : :,*)» + 4* ->*<*/' + "Transient" buckling in tlie core. I

-

* VBl«

+ •>.')

1) I)

V-Bi'» + u' Coupling coefficient between fast and thermal fluxes = (2* + D,Bl)/Z/ in general S (S. + D.Bi')/Z/ 1 . the core 5- = (2. +

Mrt/I/J"

"

S' Matrices:

4X2 4X4 2X4

Q

Y.

!

>2 V(*/» + «.')' + 4»/>*,'(t

-

u'

-

Subscripts:

-

H*

(S. + />.*/») /Z/}in matrix matrix matrix

= Denotes Denotes

constructed constructed constructed

the "Hector

for the outer reflector for the nth intermediate for the core

region

fast group of neutrons thermal group of neutrons

of the group methods. This is because other descriptions of the slowing-down dis tribution provide no means for matching the neutron fluxes and currents at. the coreIn this and the following reflector boundary over the epithermal energy distribution. sections the two-group formulation will be used. The treatment for larger numbers of groups is similar but usually becomes too cumbersome for hand computation. The mathematical treatment is similar for the various geometrical cases of reactors One of these will be presented in some detail (Art. 7.1). reflected in one dimension. In the other cases only the final equations will be given (Arts. 7.2 through 7.4). 7.1

Cylinder of Finite Length with Radial Reflector (Fig. 5)

This case will be worked out in some detail to illustrate the method.

The reflector

has k = 0.

The diffusion equations for the fast and thermal-neutron fluxes have been given To minimize the complexity of the notation in the [Eqs. (88) and 89)]. following treatment, some of the subscripts will be omitted and Eqs. (88) and (89) will be written as (103) Z/tfv + kZ,. = 0 Df previously

W/

D, VV.

-

-

-

S/4>f

= 0

(104)

6-G2

REACTOR

PHYSICS

[Sec.

6

It is to be understood that 2, in these equations designates the thermal absorption cross section and 2/ the fictitious fast absorption cross section (see Table 6 for a complete list of the notation used in Sees. 7 and 8). It has been assumed in Eqs. (103) and (104) that resonance absorption is negligible. If this is not the case p can be included as indicated in Eqs. (88) and (89). In the core, k > 1 for a critical reactor; in the reflector, k = 0. In either case, the solutions are the solutions of the wave equations

W/

V*.

+ +

B*4>f

B'.

= 0 = 0

(105) (106)

the solutions being coupled by the condi tion that the relation between fast and thermal flux, as defined by Eq. (104) [or, alternatively, by Eq. (103)], must be preserved :

S-+L*.

*- + DJit 2,

(107)


and the solutions being further restricted by the requirement that the characteristic equation be satisfied: Fin.

5.

This characteristic equation is a quadratic in B*, which defines two values of B* (B' B'*) in each region. In the core, which will be characterized by the subscript 1, -(»■/' + «i.») +

B,«

B.'i

V(*,/'

+ *>.s)' + 4»,A,.»(fc 2

_

-(»■/' +

»■■')

~

V(*
+ 4».A..Mfc

2

In the reflector, which will

-

1)

-

1)

and

(109)

(n0)

be characterized by the subscript 2,

B,'

*,,« =

-X!/i

(111) (112)

If

it is assumed that the wave equation is separable in both core and reflector, so that the solutions are the products of independent functions of z and r, then the solution for each flux (fast and thermal) in each region (core and reflector) is of the form = cos

— F (r) a

(113)

where F(r) is the solution of the equation d*F

ldF

rfr5

r dr

[b.-@>-o

(114)

Since there are two values of B* in both core and reflector [Eqs. (109) to (112)], a total of four equations of this type must be satisfied to define each F completely in both regions, thus: r

f/r2

d*Fit dr*

1

r

(115)

dr

dFu dr

m'Ftf

(116)


Sec. 6-2]

^'+r£-"v*-°)

fl

6-63

™

t

^+i^_^2/=0jrefleCtOr

(118)

In the above equations the signs of the equations for Fu and F-... m*, /»/*, and it,1 are so chosen that the constants themselves are inherently

Z',

with identical constants

P

positive:

-m«

-

-

-B.2

- B," -

-/./*

=

(119)

(^)*

-

-*./'

(120)

(^)8

(121)

(^)*

d)'

(122)

r

In the core, the complete solutions for the fast and thermal fluxes are the sums of those = 0: solutions of Eqs. (115) and (116) which are finite for

- AJa(lr)

F F

,,

=

+ C/0(mr)

ASJodr)

(123) (124)

+ CS,70(ror)

In the reflector the solutions for fast and thermal fluxes are the sums of the solutions of Eqs. (117) and (118).

P„

HI,(ji/r) + M K0(M/r) + Sr'Hh^r) + S-'MKob/r)

=

+ GK^.r) + - Ehin.r) SiEhM + SjGKoM

= S2' H „(n,r)

(125)

I

Fu

+ S,'MK0(n/r)

(126)

In these equations, A, E, G, H, and M are constants to be determined by the boundary conditions and Si, Si', Si, and St' are the coupling coefficients [Eq. (107)] appropriate to the region under consideration and the value of B*: C,

S- + D"B''

Sl>

=

z"

+ *>'•*'"

S,

=

*u

S,'

=

*»

~ Z>**»' = s»/ ~

(127)

(128) (129)

0

S, =

2?""/'

(130)

Ft.(Rt)

i/(Ri) F,.(Ri)

-0

=

0

Ftf(R,)

Ftf(R,)

- Ft.(R,)

D./^(«.) -D./^(«>) -Dto^*-"(B,) Di.^-'(Bi) dr dr

E,

G,

C,

Equations (123) to (126) contain six undetermined coefficients (A, H, and M). Five of these, plus the criticality condition, are determined by the following six boundary conditions:

f


u

(131) (132) (133) (134) (135) (136)

6-64

REACTOR

PHYSICS

[Sec.

6

Applying conditions (131) and (132) to Eqs. (125) and (126) there result EI0(n.R2)

Hh(mRt) + MKa(»,R,) St'HI<,(vRt) + SSMK.b/R,)

+ GK0(m.Ri) +

= 0 = 0

£=-G^

(137)

(138)

Using these relations, and applying conditions the four simultaneous equations result:

AJSRi)

+ C/.(mJ?,) + G

r*°k'ft'> /o0,,fi,) +

ASMWi)


D,.AUi(lRi)

(133) to (136) to Eqs. (123) to (126),

_

M \

+ CS,7,(mft,) + S.'Af

K0(M.R,)1 T7o(m/«.)

[~--^-

- DuCmltfaRi) - Dt.*jG - ^>w/W [tT^T

DtfAS^xilRt)

- Dt/CSi'mldmRi) - Si'DttufM

These four linear homogeneous

/o(m/«.) *■(*■"■)

- Ko(m/«.)1 - A'„(M/R.)1

= 0

(139)

= 0

(140)

1

-0

(141)

+ K.(m/Ki)1

= 0

(142)

+ «•(/».«■) + #.(*/«.)

\K.'}"f™ /.&»/«.)

1

equations, of the form

+ kuC + ki/J + kltM = 0 0 + fc22C + fc21G + fc»M k»A + k»C + k„G + k„M = 0 0 ktlA + ktiC + k»G + kuM

-

Jfc,w4

will have

a nontrivial

solution for A, C, G, and

M

(143>

provided the determinant

kiikukijkxi D =

kiikzikzsku kt\k*ik22ku

= 0

(144)

ktikiiktiku Physically, this condition is the condition of criticality for the reactor. In general, for an arbitrary choice of reactor characteristics, the determinant will have some nonzero value. A practical method for finding a condition of criticality is to compute the value of the determinant D for a number of different values of one of the reactor characteristics (usually R\, H, k, or uranium density), to plot the value of D as a function of the value of the variable characteristic, and thus graphically to determine the value of the characteristic for which D = 0. Once the condition of criticality has been established, the set of equations (143) defines the relationships among the constants A, C,G, and M. If one of the constants (most conveniently, .4) is set arbitrarily, the remaining three may be determined in terms of it by three of the equations (143), and the constants E and H can be specified in the same terms by Eqs. (137) and (138). Substitution of these values into Eqs. (123) to (126) specifies the thermal and cpithermal flux in the core and reflector, This multiplier, which specifies the absolute except for an arbitrary multiplier. level of the flux, can be determined if the power at which the reactor is operating is known (sec Art. 6.4).

The fluxes are given by

F, and F /

6-65

Finite Cylinder Reflected on the Ends (Fig. 6)

7.2

where

CALCULATIONS

REACTOR

Sec. 6-2]

. f

=

„ , , r 2.405r —r-- F.(z) Jo

(145)

=

J,

(146)

2.405r

are given by equations of the form (147)

dz* which have the solutions

Fi,

= 4 cos iz + C cosh mz = 5i-4 cos /z + Si'C cosh mz F„ Fj, = E cosh /x,f + G cosh /i/f F2/ = Si'G cosh M/f

where


C-f5)' The coefficients

A,

C,

A

E, and G

SiA -IDUA

sin

IH —

+ mDuC sinh

-IDifSA

sin

^

----- -fe"

(149)

are related by the equations

+ C cosh

cos

(148)

cos 7tiH —

^

—

+ Si'C

E

cosh ju.r — G cosh

cosh

^-

+ titD2tE sinh

+ mDtfSi'C sinh

^

S2'G cosh

n/T

0

M/r

0 (150)

+ n/D2.G sinh + n/Ds,S2'G sinh

0

„/T

0

REGJ0N2_

■Li REGION I

r-

REGION I

j_ _R£5iPN_2_

Fig.

6.

Fio.

The methods used for Eqs. (139) through (142) can he applied to these equations to find the condition of criticality and to determine three of the four coefficients A, C,

E, and G.

6-66

REACTOK PHYSICS

6

Rectangular Parallelepiped Reflected on Two Opposite Faces (Fig. 7)

7.3

The fluxes are given by

.

= cos

<*./ = cos

where

[Sec.

F,

and

F/

tx — a

cos

— cos a

ttu . . — r ,(z)

(151)

t f(z)

(152)

b

-— b

are given by equations of the form

(153)

The solutions for F are the same as those given in Eqs. (148) above, and the equations which determine the coefficients and the criticality condition are Eqs. (150) above, with the following changes in the definition of symbols:


'-*■-©■-©•

7.4

The 0i« = =

(154)

Reflected Sphere (Fig. 8)

fluxes are given by

r

(A sin Bir + C sinh

I (S,A sin B,r r

- [/?(cosh

tf

S —

=

-—*•-©" -©'

S/C

sinh B,'r) (155)

x.r

—

'

r

+

fi/r)

coth x,/i2 sinh x.r) + H (cosh x/r

//(cosh x/r

—

Fio.

8.

—

coth x//£» sinh x/r)]

coth x/rf2 sinh x/r)

Fig.

9.


Sec. 6-2]

coth x,R2 sinh x.Ri

Ri

(s^

B,R,

cos

~Y8' CD,/

(Si'BS

^S>'x/

(156)

coth X/R2 sinh x/Ri

BlKl) cosh

Ri

Ri

cosh

x/Ri

=

=

- ^-S,' sinh B/B,)

B.'R,

coth xfRt cosh x/Ri ^-

HDj/

sin

— —

—

Si'xf sinh x/Ri

coth x/R2 sinh x/Ri +

Ri

x/Bi

cosh

these equations may be used to determine the criticality condition and to of the coefficients, as in the case of Eqs. (139) through (142).

fix three

7.6

Bare Cylindrical

Reactor Containing Central "Reflector" or Absorber (Fig.

9)

and again,

.

= cos

and

Ff

,(r) (157)

coB^F/(r)

are given by equations of the form

;S

F.

+

where

-

-"

h

is

is

This case occurs in computing the effects of centrally located control elements or The sometimes in computing the effects of experiments inserted in the reactor. assumed to be <1, but for the sake of generality not taken to be quantity fcj necessarily zero. The fluxes are given by

+

[B,-(5)>-°

(158)

-(*!/'

+ *..*)

- V(*l/' _

4x„W(A-,

-

4x,/'x,.'(fc,

-

+

.

:

0

+ x,,5)2 +

(159) 1)

+ K|.«)

+

„ Bl

VW

-(x,/s

+

D

B»

1)

and B* has two values for each geometrical region:

,


+

+

ADU

x/sinhx/Ri 0

Ri

x.Ri

0

—

—

B.'R.)

sinh x.Ri

-tr- cosh

+ HD21 ^x/cothx/Rj coshx/Ri —

— x,

sinh

/J

Ri

coth x.R2 cosh x.Ri

-

B,'R,

cosh

= 0

/)

^-

J_

—

CDU

i-

B,R,) +

x/R,)

cosh

/)

sin

+

-

-

x/R2sinh x/Ri

(b,'

B,R,

^x,

cos

+

AD,,

-

cosh x.Ri) + £(coth x.R-. sinh x.Ri + B(coth KfRt sinh x/Ri — cosh x/Ri) = 0

BiRi + HSt' {coth

B,Ri + OS/ sinh

(b,

.4S,sin

B/Ri

+ C sinh

are related by the equations

+

BiRi

i-

A sin

H

coefficients A, C, E,

and the

6-67

6-68

REACTOR PHYSICS

_ -(««/'+««.')

R 2,

_ -(«»/'

flj,2

/2,

The quantities

and

I.

m*,

-

1)

+ »».')' + 4»»/W(fct

-

1)

it,1 arc defined

-

xi.1)1 +

2

-

+ »,.')

4*„W(*!

VW +

+

-

Bl.

"

*' - (*)'

(J)*

[SeC.

6

(160)

as

-

ft"

-«* -

Bi"

(§)' -

(161)

(*)'

The solutions of the four equations of the form (158) are, then,

2,. +

„

•3l =

8,

+

+

„ Ol

- /)2,B2»

„

—

2,. + ft.ft'» S„

-s„ft.B2's

(163)

2j/

Xif

and G, respectively:

_g*.(mft)

(165)

/.(mft)

- /r,(?ft)l

-

- ft.[fti./i(ji.ft) m

J (166)

7,(mft)l

m

-

-

0

L

-

=

/„(mft)l /t(mft) S^r/,0i.ft) S,'M/,0i/r) =

-

-mft (mft)

/,(mft)l

0

io(m/fi)

- ff/o(/i.ft) - Mh(n,R,)

S.'G

-mft(mft)

-

J

O

U«(mft)

L

+

+

JI

dr

+ Afp,/,(^/ft)l

°(mf /0(mft)

- ft/tS2fti./.(M.ft)

/,(mft)l

S2'MM//i(A./ft)l

-

0

+

^ (ft)

-

0

y,(/ft)

f

ft,G

(164)

and Af are related by the equations

J

M?!?

ft.

dr

f

f,

s,ft,c

!%

ft

y0(/ft) + K„(/ft)l

+

f°(/^ Jo(llti)

+

f

-

L

[

S.C

'•<««)

=

^ (ft)

g.

_cMft) •/„(/«,)

=

L

C

\

^S-! Jo(/'m)

ft.(ft)

can be evaluated in terras of

and the remaining constants

-

ft.

dr

G,

£

t

and

^ (ft)

ft.(ft)

J

DV

0

=

= =

C

(ft)

F,.(ft)

'}

ft,(ft)

ar

A

the constants

=

C,

ft,

=

F,AR,)

+

F./(ft)

0

Applying the boundary conditions

L


ft.ft'

z„

Z,.

(162)

MI0(n,r) S2'Mh(n,r)

,

HUM

= = S2HI0(n,r)

Eh(mr) + Gft(mr) SAEIo(mr) + Gft(mr)]

,

Flf

where

AJ„(lr) + CY„(lr)

SMJoilr) + CY„(h)) +

Fu

=

-

+

ft.

ft/

CALCULATIONS

REACTOR

Sec. 6-2]

6-69

The criticality condition can be determined and three of the constants can be evaluated from these four relations as explained in the case of Eqs. (139) through (142). If the central region is a control rod, it is usually black to thermal neutrons, and the boundary condition on the thermal flux in region 1 is 1 0»i

(ft)

dr

(ft)

=

i

« is the extrapolation distance for thermal neutrons (see Art. 2.7). Often the material is a poor moderator, and a suitable boundary condition on the fast flux is

where rod

^ dr

(*«) = o

When these conditions hold, the two constants C and G, which must be evaluated to describe the flux distribution in region 1, are determined by

« '

c\~ L

/l(/ft)

tS

Jo(t/ti)

-

y,«ft)

1

+ ™*»>

- Gm

/,(mft)

f

^(^ w J]~G\I -rrnrr /o(otKi)

+

ft(mft)l

-

J1

(167)


and

J

LJo(f/ti) 7.6

L

/o(mft)

J

= 0

The Reflector Saving

Given a reflected reactor known to be critical (either by calculation or experiment) core has dimensions a, b, and c in some coordinate system whose directions are specified by a, 0, y, we remove the reflector which extends in, say, the a direction and determine (by either calculation or experiment) how far the bare core would have to be extended in the a direction to achieve criticality. If this critical dimension of the bare core is a', then we define the reflector saving in the a direction as o' — o. The dimension a' is called the equivalent bare dimension of the reactor. If the reflector saving for a given core-reflector combination is known, the criticality calculations for the reactor can be made as though the reactor were bare, using the reflector saving in place of the usual extrapolation distance. The value of the reflector saving is often relatively insensitive to changes in some of the important characteristics of the core. Furthermore, in many practical cases the reflector saving is a relatively small fraction of the equivalent bare dimension. These considerations make the reflector-saving concept a very useful one in practical reactor calculations. For example, if calculations are to be made over a wide range of reactor variables for design optimization, reflector savings can be calculated for only a few conditions and values for the remainder can be interpolated. whose

7.7

Reactors Reflected in More than One Dimension

There is no analytical solution for the reactor reflected in more than one dimension, although numerical methods (e.g., the relaxation methods) are possible. The usual method for practical calculations is to assume that the reflector saving in any given dimension is the same as that which would result if the reactor were bare and of the equivalent bare size in the other dimensions. Thus, for a three-dimensional problem one guesses first an equivalent bare value for each of two dimensions and solves the actual reflected case in the third dimension, thus determining the equivalent bare value for that dimension. The same process is applied in turn to each dimension, using the "correct" values for the equivalent bare dimensions as they are deter

6-70

REACTOR

PHYSICS

[Sec.

6

mined and guessed values when necessary. After a calculated result has been obtained for the equivalent bare value in each dimension, criticality can be determined by the If, as the calculation proceeds, it is evident that some of the bare reactor solutions. guesses made were very poor, it may be advisable to repeat the calculations using the improved values for the equivalent bare dimensions. Experience has shown that, as would be expected, this method gives accurate results for reasonably large reactors having reflector savings which are not large fractions of the equivalent bare dimensions. An alternate method can be used if the core is of a compact shape and is identically reflected on all sides. The core and reflector are replaced by spherical regions of the same materials and of roughly the same geometrical buckling. The reflector saving is computed for this spherical assembly, and it is then assumed that the same reflector saving will apply to each dimension of the actual core-reflector assembly. This method involves less computation than the preceding and may give more accurate results if the reflector saving is a large fraction of the equivalent bare dimension. 8

MATRIX, SOLUTIONS

FOR REFLECTED REACTORS

By Otto Schulze


The problem of the two-group reactor reflected in one dimension, which was treated in Art. 7, can also be solved by the application of matrix algebra. The results obtained can also be applied approximately to reactors reflected in more than one dimension by the same schemes outlined in Art. 7.

SLAB GEOMETRY

CYLINDRICAL OR SPHERICAL GEOMETRY

Fia.

10.

Notation for multiple-reflector

problems.

The matrix solutions are particularly useful for reflectors which consist of more They than one region (Fig. 10), all of which are here assumed to be nonmultiplying. and were were originally described by Garabedian and Householder (MonP-246) reduced to a straightforward computational procedure by Spinrad and Kurath.t The symbols used here are the same as those of Art. 7 (see Table 6). 8.1

General Procedure for Calculating Criticality

For each region there is a matrix whose elements arc determined by the two-group constants and the dimensions of the region. For the outermost reflector the matrix is called the Q matrix. It has a simple form because the fluxes vanish at the outer For each intermediate region n there is a matrix desig (extrapolated) boundary. nated F„, while the matrix for the core region is symbolized by Fcor». t This flection is a condensation of Ref. 11. if extensive computations are to be made.

The report contains

form sheets which

are very useful

CALCULATIONS

REACTOR

Sec. 6-2]

6-71

After all the elements of all the matrices have been determined, one proceeds from outer reflector through the intermediate regions in consecutive order to the core region, forming the product first of the outer reflector Q matrix mult iplied into the Y„ matrix of the adjacent intermediate region. This product is then multiplied into the Yn matrix of the next intermediate region, and so on, until finally a product is formed with the yCor« matrix. This final product is a 2 X 2 matrix. The critical condition is that the determinant of this final matrix be zero. Criticality can be achieved either by varying a particular parameter (for instance, the dimension of the core) until the value of the determinant is reduced to zero or (as is more convenient if the nuclear constants, i.e., fuel loading, are to be varied) by setting the determinant equal to zero and solving for one of the functions I tan (It), l\Jt(lr)/Ji(lr)], or I cot (Ir) (depending upon whether the geometry is a slab, cylinder, or sphere). In this notation I is the square root of the partial buckling. Knowing If B,* is not the same as the /, one can obtain the value of the total buckling Bi*. test value, one iterates, using the new solution as a test value, repeating this process The procedure is demonstrated in Art. 11.7. until convergence is obtained. the

8.2

The Q Matrix (Outer Reflector) 1

0

0

1


0

^-

Q =

—

q
?(/».) I

where

Slab

(GO

-

Cylinder

Sphere

tanh pT 1

!

?(m/,m.) =

8.3

- Ri)

+ uRi coth Mitt

and

1

S'D,

m

[?(m/)

Ko(vRi)

Ki(jiR,

-

1

- tfoOiK»)/o(/ifli)"

1

+

Ki(Mgi)/o(j.gt)J

?(m.)]

The Yn Matrix (for Any Intermediate Region n) 1

X«0,)

0

Xdnf)

0

Xi
Xi(ii,)

Xiiiif,?.) Xl{liJ,H.)

—I Xiiit.)

D,X,(j,,)

0

Xt
0

X3(ji/,n.)

D.Xtin.)

D,

Xtifif,?,)

A^Gi.)

where

Slab

Sphere

nT

cosh n{Ri

go

sinh nT

sinh n(Ri

Ri)

= cosh

A\M

=

A,to

= n

sinh ptT

l

?R,R:

MA

= cosh

fiT

cosh /i(g* —

Ri)

-

-

4- sinh

R,) cosh ^(B2

+ (n'RiR*. A",rM)

+

1) 1

uRi

Cylinder ,.g,[/o(Mg»)tfi(M«i) + Ko(Mg.)/iO»gi)]

-

g,[/oOig,)Xo(Mgi) KoOigO/oOigi)]

- g,)

sinh *.(g2 sinh ft(Rt

SRAIdnRJKApRi)

- g,)] gi)

- X.OigO/iOigi)]

+ K,(j.g,)/o(*.gi)]

6-72

REACTOR

PHYSICS

[Sec.

6

and Xi(/l/,M.)

=

jp

[Xi(jtf)

D. S'D, 8.4

—

Xi(nJ]

- X4(„.)l

[XtM

The Kwrc Matrix (for the Core Region) 1

1

1

J_

S

S'

-D,XC(»

D,Xe(m) D,Xe(m)

-D,XC{1)

S' where


Slab

X.(Q

I tan

Xc(m)

tanh

8.6

Cylinder

Sphere

IT —

—

I cot IRi

2

mT

m

,JxVRx)

1

—

Ri —

coth mRi

— R\

mh(mR1) /o(mft,)

General Procedure for Determining Flux Distributions

After the critical condition has been achieved, the distribution of the fast and Starting with the core the constant A' is calculated thermal fluxes may be obtained. from the ratio of the elements of the critical determinant; the other coefficients are stated in terms of A'. A vector f which represents, in order, the fast and thermal fluxes and the negative These values of the fluxes currents at the boundary of the core is then obtained. The super and currents are used in calculating the coefficients of the next region. script t — 1 designates that the outer edge values from the previous region are to be used. One proceeds

through the various intermediate regions in order and finally to the outer reflector. Here the boundary values from the previous region form an overdetermined system for obtaining the fluxes and afford a check on the final result. Distribution Core:

-Mp)

M

=

MUp) + AN(mp)

=

^

Slab MUp) N(mp) A

COS Ip

cosh mp

A'

(IT/2) (mT/2)

cos

cosh

-J

dp) +

8'

(mp)

Sphere sin Ip

Mb)

p

sinh A

'

Cylinder

mp

h(mp)

P

sin \R\ sinh mR\

A'J0(IR>) 7o(mft,)

Sec.

6-2]

where

A'


6-73

is obtained from

la A'

Willi

The outer edge

61

d

6

c = — a

A

eg-,

- -i

c

(check)

values of the fluxes and currents are obtained from the matrix product-

*/ *•

UK.

Intermediate

j.

Region:

AV(Mp) + B\V(p,p)

4>f{p) =

.(p) Generated for wjivans (University of Florida) on 2015-09-23 02:50 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

JA

AN(mIii)

=

^,

*/(p) + CV(ji#) + EW(».p)

Slab

Sphere

P meas ured from

cosh MP ] > ainh ppj

inner edge of re gion

A

-

coah p(p p ainh p(p

BO R,)

Cylinder

p measured from center of reactor

p

ft.*/'-'

Ri [»/Xi(j./S,)#/f-i

B

Rl

p/

«/£>/ C

'o(pp) 1 >P measured from axis of reactor Ko(pp) J

-

Ri

J;

R,

♦/'-')

[./flfp/SO^/f-l

£

P.O.

-

SW>/

J;

*/<-'

-

V

y

/

-

Ri [p./i(p.R1)

s'fl/

/„(„,«,)

S'O/

P.

P.B. V

tfo(p/R,)]

S'D,

1

llfll (previous region) =

||f||

.

D.

J

#,<-i)

J

The outer values of fluxes and currents are obtained from the matrix product

II I'll

]

_ 1

[p./C,(p,R,) V

i£l

+

D.

J

A

(i/so*/'-'

B

p.D, cosh p.71 (l/ft,)

(I/Ä.)

£ p.(R,

(-R,/D.)J.'-> sinh

Sphere

05 */W

AXijifp) BX(w)

-

fti)

+

fti)

m. cosh p,(Ri

(R,/S'D,)J/<> -

-{H,/D,)J,'-' - ft.) + m/ cosh u/(ft> -

(I/S')*/'"'

,i/(«i

— fti)

sinli M.(ft« -

-

sinh

pf(Ki

fti*/'-'

*i(ft; —

«,'-'

(l/fti)

sinh

sinli

-

=

fti)

fti)

-

-

p.DM,(p.R,)

/o(
ji/D//i(wfti)

/.(d/fti)

/o(t>Ri)

-

Cylinder

(IAS')*/'-1

[/o(M/*i)/«<(M/fi!)|A"i(wRi)

•//'-'

/K,(p,R,)]K,(p,Rx)

y.'-i

(D./S'D,)J/<-' [/.(^.ft1)//C»(M.R.)]Äi(c.ftOl

-

[/o((..ft.)/A-o(M.ft.)]Xo(M.ftl)

*.*-'

~ .

♦/'"' [/.(M/ft.)

Ai(fint)

+

sinh m-7

-

|

*.•-'

*1/jT

is p

M/^>/ cosh

-//*"!

p/T

from inner p)

sinh

measured

Slab

.(»)

/(p)

,i

(check)

p)

sinh ^(T — edge of region

Reflector:

,

(check)

A'(W)

Outer


+

,

+

+

Sec. 6-2] 9

CALCULATIONS

REACTOR

EVALUATION OF MATERIAL CONSTANTS By Otto Schulze and

J.

6-75 OF THE REACTOR

R. Dietrich

In order to set up the reactor equations for a giveu thermal reactor (cf. Arts. 5 and to evaluate a number of constants which are characteristic of the materials of the reactor and their disposal in the reactor. In the reactor core, these constants are L* and D„ which apply to the diffusion of thermal neutrons; k, the infinite multiplication constant, which is usually broken down as the product of four other constants i\, t , p, and ; and constants characterizing the behavior of fast neutrons in the reactor. It will be assumed here that one of the simpler descriptions of the distribution will be employed and that the behavior of fast neutrons slowing-down can be characterized by two constants, a slowing-down area t (or L/') and a diffusion constant D/. If a reflector is provided on the core, the same constants must be evaluated in the reflector also. Usually, k is zero in the reflector. If the structure of the reactor is so fine-grained that none of its parts exceeds a small fraction of a diffusion length in its smallest dimension, the reactor may be con sidered to be effectively homogeneous and computation of the material constants is If this condition is not met, the reactor is generally referred considerably simplified. to as a lumped reactor, f The particular type of lumping, in which the fissionable material (fuel) is arranged in discrete lumps throughout the moderator — generally in a regular lattice pattern — is of frequent occurrence in reactor design. This type of arrangement reduces the absorption of resonance neutrons by U"8 and thereby allows criticality to be attained with lower U"*/U"8 ratios (lower enrichment) than would be possible with a homogeneous arrangement . In the following articles, evaluation of the material constants will be discussed both for the effectively homogeneous case and for the lumped case described above. 10), it is necessary


/

The Infinite Multiplication

9.1

Constant k

The infinite multiplication constant is arrived at in the following way: It is first assumed that the region for which k is to be evaluated extends in all directions to infinity. In this infinite medium, k is defined as the number of secondary neutrons reaching thermal energy per primary thermal neutron absorbed in the medium and can be broken down into the product of four terms: k

- vtpf

(168)

where ij is called the regeneration factor, t the fast fission factor, p the resonance escape probability, and the thermal utilization. Equation (168) is often referred to as the four-factor formula. Evaluation of these factors is discussed in the four

/

immediately following articles. 9.2

The Regeneration Factor

y

The quantity i/ is defined as the number of fission neutrons produced per neutron absorbed in the fuel. The definition of fuel is to some extent arbitrary. In a reactor which is effectively homogeneous, the term fuel is usually applied to the fissionable In a lumped reactor the term usually refers to the fissionable isotope plus isotope. any isotopes which are mixed with the fissionable isotope or are geometrically closely associated with it. If the reactor is effectively homogeneous and the fuel is con sidered to consist of the fissionable isotope only, n is a constant characteristic of the fissionable isotope. For thermal neutrons, n is given for a number of fissionable isotopes in Table 7. If the fuel is lumped, it may consist of a number of isotopes, some of which are fissionable and some not. If the various isotopes are intimately "f The reactor might also be referred to, io terms of its neutron properties, as a heterogeneous reactor. terminology leads to some confusion, however, as the terms homogeneous and heterogeneous have used to characterize the state of actual physical homogeneity of the reactor.

This

l«n

6-76

REACTOR

mixed,

they are all exposed to the same average neutron flux and

PHYSICS

'"I

[Sec. 17is

6

given by

JV«r..

Ni

is the number of atoms of the ith isotope per unit volume,
scopic

Table

7.

Number of Fission Neutrons Emitted per Thermal Neutron Absorbed (ij) by Thermally Fissionable Isotopes* Itotope

q

H»' U»'

2.08± 0.02 2.31 ± 0 0} 2.03 + 0.03

Pu>»


♦From "Neutron Cross Sections,"

McGraw-Hill

Book Company,

Inc., 1955.

In such isotope. An example is the material used for the cladding of fuel elements. a case the average flux may not be the same in all isotopes considered as part of the fuel. The quantity ij is then given by

V

- -U

(170)

i flux over the z'th isotope. For symmetrical cases, computed by methods similar to those discussed in Art. 9.3. where $ is the average

9.3

The Thermal Utilization

& may

be

/

The thermal utilization is defined as the ratio of the number of neutrons absorbed per unit time in fuel to the total number of neutrons absorbed per unit time in the reactor. For an effectively homogeneous reactor it is given by

/

M^l

2„(fuel) + 2„(all other materials)

I

1

+ (S„(all other materials)/2„(fuel)]

(l71)

where 2„ is the macroscopic absorption cross section. In a lumped reactor the absorption cross sections of the various materials must be weighted with the average thermal neutron flux over the materials. In comput ing for such a case the reactor core material is usually considered to be infinite in extent. such an infinite system is written as a function of some If the thermal neutron flux

/

space coordinate r, then

in

/ is given by

_

[Xai(r)4,(r)dV

the integration is taken over a typical volume of reactor material. quantity 2„i is the macroscopic absorption cross section of the fuel material and Z0 is the total macroscopic cross section, including both fuel and any materials present. where

Thr onlv

othe(

6-77


Sec. 6-2]

If

the reactor core is made up of regularly spaced regions of fuel and moderator, of which can be considered individually as homogeneous, and if the macroscopic absorption cross section is designated by 2„i in the fuel and by SOJ in the moderator, then is given by each

/

/-

!.i

l

,

/fuel

/fuel

4>(r)dV

*(r)dV + 2.2

(173)

Jfmoderator *(r)dV .

are taken as indicated over a typical fuel region and a typical is convenient to work with the average fluxes in the fuel and respectively:

where the integrations

moderator region. moderator, î and

5"

It

(174)

-±[ Yi J

and

V\ and Vj are the volumes of

where

/

=

a typical

fuel region and a typical moderator

V,2.,*, ViS„i#i + VtS.fi,

1 1

+

/by

...

,

Disadvantage

, factor

(176)

(Z../2.i)(Vs/Vi)(*s/#i)

The ratio #s/i?i is called the disadvantage factor. of

(175)

Then

region, respectively.


+{r)dV

mod-

vi

2.iî -—— 2«i V i

/l I-

\/

—

,\ II /

in terms

(177)

For

/ is

a fuel-moderator lattice of the type considered here, the first step in calculating the division of the lattice into a number of identical unit cells in such a way that

Thus, if the fuel is dis the unit cell possesses the maximum possible symmetry. posed in equally spaced slabs, the cell breakdown is usually made as in Fig. 11.

FUEL-

-TYPICAL CELL

Flo.

11.

Section

of

reactor

with fuel in

Fig.

Ltypical 12. Section

cell

of reactor with fuel in long

round rods.

slabs.

Figure

12 is

a typical breakdown

for a reactor using fuel rods in the form of long

cylinders, disposed in a square array. The next step is to approximate the true shape of the cell by a shape which can be described by a single dimension. This must be done in such a way as to maintain the true volume of the cell: thus, the cell having the shape of a long square prism

with side length

as in

Fig.

12 is approximated

by a long circular cylinder of radius R2 = a/s/ir (Fig. 13). W'ith the problems reduced to one in a single dimension by a suitablecell approxima tion, may be computed by diffusion theory or by higher order approximations to

/

6-78

REACTOR PHYSICS

[Sec.

6

Only diffusion theory is treated here, although it cannot be con transport theory. sidered a good approximation for the cases of small dimensions and strong absorption which often occur in the cell treatment. The case of the cylindrical cell will be treated as typical (Fig. 13), and solutions will be given for other cases (Table 8). The cell problem is treated as a problem in the diffusion of thermal neutrons; the slowing-down density is assumed to be uniform over the region of the cell occupied by moderator (region 2) and to be zero in the fuel (region 1). There is thus a source of q neutrons/(cm')(sec) uniformly throughout the moderator, and the diffu sion equation is

fVi

fl, D, V*(h

- 2ol0, - 0

- 2„j0s

in the fuel = 0

(178)

in the moderator

(179)

+9

where D is the diffusion constant and 2. the macroscopic absorption cross section. Writing x = 2„/D, the solutions of Eqs. (177) and (178) are, respectively,


0,

- Ah(x,r)

(180)

+ C7u(*,r) + -£-

4>2 = £/o(x2r)

(181)

where the Ko solution in region 1 has been precluded by the requirement that 4> be finite at r = 0. Since all the cells are identical and are assumed to constitute an infinite array, there must be no neutron current from cell to cell; hence d((ii(lti)/dr = 0. Application of this boundary condition to Eq. (181) gives the relation between the constants B and C:

Fio.

13.

Square

cell of Fig.

12

cylindri

cized.

/■(*7«*) C =

Ki(xiB.)

B

(182)

and Eq. (181) can be rewritten as
ClK,(*,tf,)/o(*.r) + /.(xsftOtfotxir)] +.

(183)

where G is a new constant G = Table 8.

Values of

E

and

F

Cell

Slab Sphere

(184)

for Cylindrical, Slab, and Spherical Geometry* E

geometry

Cylinder

B A',(x2/?,)

xjh

/o(«ifli)

2

/i(«.fti)

xifti cuth *\R\ tanh »ifii m'fti' 3 *ifti — tanh mli\

- «■') T 7o(«,Ri)Ki(«.ff;) 4- K.(«ifii)/i(»iJ»t) "1 L /,(»,«,)# ■(«,«,) - JCi(«.«,)/,(xift,)J fti i coth ooth »,(«, - fto xi(/2i - fti) » - *i") I - xiRi coth «,(R, - fti) [" U1 - «iftifti - xi(fti - fti) coth «,(«, ifti ,(ft.'

2ft,

172

For sphere and cylinder, H\ is the radius of fuel, Hi the cell radius. For slab. Hi is the half t\m\ Subscripts I refer to fuel and 2 to moderator in all i Hi the half thickness of the cell. M. Weinberg, "Science and Kncinecring of Nuclear Power," vol. II, chap. 6, Addis*1 ^ OtW* Publishing Company, Heading, Mass., 1949.

of fuel. * A.

1

reactok calculations

Sec. 6-2] Applying between

andG: 1

A

"

6-79

the conditions 4>i(#i) = fc(Ri) and £>i*i'(#i) = Dit'(Ri) at the interface regions 1 and 2 to Eqs. (180) and (183) yields the values of the constants A

2„2

/),*, Il(x,R,)[K,(x2Rt)h(xiRi) + I1(x,Ri)K0(xlR

[hixiR l

and

-

^)

Dm Kl(x2Ri)Ii(x1Rl) I^xiRJKiixiRx) A D,x, /i(x.fii) 1

a

(185)

nm

although, for the evaluation of /, the value of G is not required. To evaluate the basic definition of is recalled:

/

/

,

_

absorption rate in fuel total absorption rate

Absorption

- 2vR,

2*RiJ(R,)

by fuel per unit length =

Di

(Ri)

— I,(x,Ri)

2rR,A

=

-j1

These equations will be evaluated for unit length of cell. The absorption rate in fuel per unit length is just the net neutron current into region 1 per unit length, or

just the total source strength per unit

- b,«)

(7?,'

,

/,(«,«,)

J

L

2

F,Z.,

/

i

_ va* r».î /o(*ifii)1

1/A,

2i?i

-

—

Ri*)q

ff,»)

1_

=

22o1R,/1(x1R1)

g»,(g,«

A

2xK,S„1/1(x1fi1)

and, inserting the value for =

- /?,')

= ir(/?s*

A

xqxi(Rt*

2

1

/

=

_1

Total absorption per unit length whence

,18_.

r/.(«,g1)y,(»,«,) + g0(»iR1)/1(x>R,)i K-1(«/?,)/,(«B,)J L/1(xI«s)X,(x,/i1)

-

(188)

Thus 1// consists of two terms, one of which involves x of the fuel only and the other x This form of expression for 1// can be derived for other onedimensional geometries, the customary form of the equation being of the moderator only.

E

1

F

where the values of

and

1

=

+

E

1

1

//

is

If

8,

F

is

is

F

is

F,

actually the ratio of the flux density at the Here a function of *i and Ri only, fuel-moderator interface to the average flux density in the fuel. The quantity (J'sSaj/KiZai) The quantity E — often called the relative absorption. ••ailed the excess absorption. of the higher average flux It accounts for the effect and The values of density in the moderator relative to that at the interface. for the case of cylindrical symmetry are obvious by comparison of Eqs. (188) and They are repeated in Tabic along with the values for the cases of spherical (189). geometry and slab geometry. the fuel surrounded by a thinf layer of nonfissionable material of volume V, per unit length of cell and macroscopic absorption cross section Saa, its effect can be included by modifying the equation for to

(YÂJj^fF

{B-1)

(190)

the effect of the layer on the flux distribution — whether the effect be due to diffusion constant, or slowing-down properties — negligible.

"Thin" implying that

absorption,

+

remain unchanged. is

t


per unit length of cell

:

The total absorption length

is

Xl

6-80

REACTOR

PHYSICS

[Sec.

6

The Resonance Escape Probability pt

9.4

In the expression for k of a thermal reactor the resonance escape probability of interest is that over the neutron energy range from the upper energy limit of important resonance absorption to thermal energy. If the upper energy limit is designated £<>, then the resonance escape probability to thermal energy p(th) for an effectively homogeneous reactor containing a resonance absorber is

where Za(E') is the macroscopic absorption cross section, 'S.(E') the macroscopic scattering cross section, and l(E') the average logarithmic energy decrement per collision (Art. 3.1) in the medium. These quantities are seldom known in sufficient detail to permit evaluation of Eq. (191); an experimentally determined effective dE' fEo — - is usually used for evaluation of p. If £ and 2, can resonance integral / ff0
J Elk

E

be considered constant with energy

(usually true in the resonance range), p(lh) is given

by

pW-exp(-|f /*''."§')

(192)

given by

dE'

[E.

dE'

1

(B°

is

is

is a

and function only of the species of absorber and the ratio iV0/2,. Figure 14 gives experimentally determined curves of the effective resonance integrals as functions of Na/2, for uranium and thorium, the two most important resonance absorbers which occur in reactor problems. The procedure for computing p(lh) for these cases as follows: (1) Compute AT./Z,, where 2, the total macroscopic scattering cross section in the medium,' including the contributions of all scat terers present. (2) Find the fEl>

J/

corresponding value of

Elk

dE'

aa.,t —— from the curve Fig. 14.

E

(3)

Use this value in

A(E) Of. Art. 3.5.

=

is

itWNiVMB)

+

is

is

is

a

is

a

is

If

|

a

Eq. (192) to compute p{lh), employing value of consistent with the scattering cross sections previously used (Eq. (37)]. lumped resonance absorbers are present, the resonance absorption density inside much lower than that at the surface because of the very strong absorpt ion lump by the resonances. If the material of the lump has very low moderating power, the energy regions depleted of flux by the strong resonance absorption near the surface are not replenished by moderation, and this "self-protection" of the absorber against resonance neutron absorption It this effect which makes particularly great. possible the construction of a critical reactor from lumps of natural uranium in a moderator, despite the high resonance cross sections of U"8 which, in a uniform mix ture of uranium and moderator, would absorb too many neutrons to permit criticality. Unless the lumped resonance absorber has a simple and well-known resonance structure, its absorption must be determined by semiempirical method. The case of uranium typical of the latter approach. It predicted theoretically and found experimentally to be true that the rate of resonance absorption by lumps of uranium of simple shape (no reentrant surfaces! can be represented by an expression of the form

t


is,

where JV« is the number of absorber atoms per unit volume. Although the effective integral is an experimentally determined quantity, it resonance in principle,

UE)N,VME)

M

(W4)

Sec. 6-2]

REACTOR CALCUL.WTIONS

6-81

where A (E) is the number of resonance absorptions per second per unit energy interval, 4>i(E) is the average flux per unit energy interval in the interior of the lump, 4>.(E) is the average flux per unit energy interval over the surface of the lump, Ni is the density of absorbing atoms in the lump, and Vi, S, and M are the volume, surface area, and mass of the lump, respectively; a(E) and b(E) are to be determined experimentally. If the lumped resonance absorbers are arranged in a regular lattice, some volume of moderator Vi will be associated with each fuel lump, the absorber plus moderator It is assumed making up a cell similar to those previously discussed (Figs. 11 to 13).

Mill 200

1 1 1 1 11

1

A URANIUM A THORIUM o URANIUM • THORIUM

METAL METAL

(URANIUM)

OXIDE, U02

OXIDE, Th02

G GRAPHITE S SUCROSE

L LIGHT WATER H20 H HEAVY

80

S

WATER

L,

r""' (T

HC R UM) •

s G


40

G

20

r.

r

'

I

r,

J^*

^

8

4

-tf

"

4

20

8

40

80

400

200

800

2000

4000

SCATTERING CROSS SECTION PER ABSORBING ATOM (lO'^M2)

Fio. 14. Effective resonance-capture integrals of UM* and Th"2 diluted with neutronscattering materials. (Reproduced from R. L. Macklin and H. S. Pomerance, Resonance Capture Integrals, Geneva Con/. Paper A/Conf.S/P.S'S'i/RevA.) that there is no resonance absorption in the moderator. The decrease with decreasing energy of the number of neutrons slowing down in the cell past any energy E is then due to the absorption in the lump; i.e.,

A{E)

=

dQ(E) tlE

(195)

and Q{E) is given, very nearly [Eq. (41)], by

Q(E) whence, combining Eqs. (194),

ld-2 QdE

=iT

El

(195),

=

V^2(E)hÊ

(196)

and (196),

4>,(E)NlVla(E)

+

^.(E)NIVME) S/M'

(197)

and integrating between the limits En, and Eo, p(lh) is given as

Q{E„)

y\

K.^2.2

I J En

E'

4,2(E')

+

M

cim^'fw

>™

6-82

REACTOR

PHYSICS

[SeC.

6

4,(E)/fa(E) and that the ratio $i(E)/fa(E) is If it is assumed that $.(E)/$t(E) independent of energy in the energy range of important resonance absorption, then =

The above argument, makes it appear reasonable that the effective resonance integral for a lumped resonance absorber in a lattice arrangement can be represented, for various geometries of absorber, by an expression of the form

J/ resonance

/£'

-A

+/»-£ J[J

(200)

Where the subscript "resonance" on the integral sign denotes that the integration is taken over that energy range which includes all important resonances, from thermal to fission energy.

Values of A and n which have been used for natural uranium and natural uranium compounds are given in Table 9. Resonance Constants for Natural Uranium Metal and Some of Its Compounds, for Use with Eq. (200) *


Table 9.

Material

A. barns

«, barn g/cm'

U U.O. UOi

9.25 12 11.51 14 6

24.7 20 22. 1 16.3

UF. •From

"The Reactor Handbook,"

table for sources of information.

vol.

1, Table

1.5.26.

AECD

3645. March,

1955.

See origins!

In order to compute the resonance escape probability, it is necessary to evaluate the ratio 0i/«f2 |Eq. (199)]. The computation of this ratio is formally identical with the computation of the disadvantage factor (Art. 9.3). If a resonance disadvantage factor /, is denned in a manner completely analogous to the disadvantage factor of Art. 9.3, then [cf. Eq. (177)1 1

resonance

disadvantage factor I

F [see

p(th)

+

(S.,V,/Z.tF,)(J?

-

1//,

— 1

(201)

1)

Eq. (189)], and = exp

2.i?lS.»(«

N,Vi

f

./resonance

In this

*.
Z..JV,

f

./resonance

-

1)

(202)

r.tE'HdB' /£')

expression the symbol 2„s denotes a fictitious "absorption" cross section (macroscopic) which is really a slowing-down cross section, characterizing the prob ability that a neutron will be slowed out of the energy range occupied by the reso It is equal to 2., the number of scattering collisions per unit volume per nances. second per unit flux divided by l/{ In Ei/Ei, the average number of collisions required


Sec. 6-2]

neutron through the resonance =

2a2

./resonance

L J

resonance

"
to E2: (203)

by definition,

a.(E'WE'/E') 11

J[

AT, —

2.,

El

{l/h)\n\Eim

absorption cross section in the absorber 2ol

f

The average

region covering the energy range

is,

to slow a

6-83

resonance

(204)

In {Ex/Et)

dE'/E'

Jit

From Eqs. (203) and (204),

2„j

£2,2 es„

resonance

a

(F'\

(lE'

(205)

Eq. (202) becomes = cxp

+

p(th)

resonance

-

(206)

,.t/f(E')(dE'/E')

is

it

In order to apply Eq. (206), necessary to know the effective values of x for and the moderator (for evaluation of and F) and an effective value of {2, for the moderator. Experimentally determined values of xi are given for uranium and uranium oxide in Table 10, and values of x» and J2282 are given for various modera tors in Table 11. The procedure for calculating the resonance escape probability is, then, the follow ing. cell system appropriate to the lattice geometry (cf. Figs. 11 to 13). (1) Set up Designate the fuel as region and the moderator as region (2) Take values of xi and xi from Tables 10 and 11 and use them with the cell dimensions to calculate

F

1

2.

a

E

the absorber

p

U

xi = 0.0222p «i = 0.025,>

UaO»

m density, g/cm3.

In,

xj/p, cmVg

barns per atom or per molecule

38.5 28 1.26 1.76 76

HK>

0.583 0. Ml 0. 128 0.0690 0.0672

if

•From "The Reactor Handbook,"

vol.

I.

0

5

D.O Be BeO Graphite

IV E. Values of xt/p are for use with U; ■»density, g/cm*.

cm" cm-1

Moderator Resonance Constants*

Table 11. Material

1

Values of x, at Resonance Energy in Natural Uranium Metal and UjOs

Table 10.

Table 1.5.27.

(.See Ref.

17.)

Original source CL-697,

fuel is oxide, multiply by 0.88.

effective resonance

integral

resonance

a*

(E')(dE'/E')

by Eq. (200).

J

compute the

9,

by the appropriate equations of Table 8. (3) Compute the surface /mass use and from Table to with values of

(S/M) of the fuel lump, and

p

—

A

(J!

it,

ratio

/

and

1)

p


(E

1)

and

(1) Take the effective value of {52,2 for the moderator from Table 11, and combine with the other quantities determined above to find p(th) by Eq. (206).

6-82

REACTOR PHYSICS V^tS

If

it is assumed that $,(E)/fa(E) = î(E)/d> independent of energy in the energy ranp(lh) = exp

v

r-

i*!&

.«>

*y %&■

Theab

^l^*t>*u * *£«■** ..v •v*'

e\v

>'**^> «:->::
\*

tak to &

-•'

•a

w.A

^Va;>* ..*•>»

M

&sf*

aVP

0«»

^>volV^

compo,

,*<

*

to*

*#

•^to.*1

.*•

■■■'■

Vai

^Ja*^N.*°

*?5 '^>?>^>v

*SX5"fr«

\\

iv* .^v

V^V*"",»*.

,«#*:
H%

for

1&*

df*»

gP^°£ôÂ *>*

V«v^


Tablt

V

;>< '

v"'

1*

*■ .•»

.,"' "M' * From

"

The Reactor of inforu.

table for sources

^-&»<*

»««*' »»»

In order to compute the ratio &/& [Eq. (199 the computation of the dL factor f, is defined in a ma Art. 9.3, then (cf. Eq. (177)| *1

resonance

pi

*" |see

Eq.

(189)),

+

(2.iF,A

and

p((A) = exp

KifiZ.,/? -V,E

J.

resonance

cr.JE'HdE'/E')

2„5 denotes i (macroscopic) which is really a slowing-down . ability that a neutron will be slowed out of the nances. It is equal to 2,, the number of scutti second per unit flux divided by l/£ln E,/Ei, thei

In this expression the symbol

■&*&*■■

v-

\VW

n.WtvV

0
6-85


_>ec. 6-2]

consideration are distributed in the fuel lump according to the thermal distribution in the lump, whereas in computing P' it is assumed that the sources Computed curves of values of P for Stributed uniformly within the lump. and geometries Note that P, since it involves given Fig. it 15. sizes .lump <.lump sizes and geometries are given in Fig.are 15. Note thatin P, since involves is a of fissions, of thermal function both x, the reciprocal of the thermal ibution ^"'«s in the lump, and XtR, the dimension of lump measured in total mean ..-"* -^v^-Jength ' ■ *;. '. ."jj j£y The quantity f" for a given geometry is a function of Sir? only; it is given * .a - .tf^Lîy the curves for which x/2( = 0. the other constants needed for Eq. (207), for natural uranium metal, <8 under

„
^

^"Table

12.

Table

12.

Fast Effect in Uranium

mputation when there is no faat-neutron interaction between fuel lumpa:* 0. 29 barna 1.5

" <0.04 " 4.3 <2. 55 (dimensionleas) f faat effect («) for 0.600-in. -diameter rods of slightly enriched ttron interaction between fuel lumps, t c. », p

uranium in light


£

227± 105 + I . 072 ± 1.061 + 1.047 ± 1.043 ± 1. 109+ I . 074 ± 1. 042 ± vol.

1. Table 1.5.20. AECD-3645,

0.011

0.002 0.001 0 001 0 001 0 001

0.002 0.001 0.001

March, 1955.

Original source

R. Sher, and V. Walsh, Exponential Experiments with Slightly Vater, Geneva Conf. Paper A/Conf. 8/P/600. June 30. 1955. tfiose which occur in some water-moderated reactors, , and there is a substantial increase of < over that as a result of the mutual fast-neutron interactions

sured values of

1

Diffusion

e

for some H20-uranium

lattices

Coefficient Aa

thermal diffusion coefficient can be im absorbs strongly, by Eq. (9). Measured '.ilable. If they are not, the scattering cross ■i^y'^jh a value of /j0 computed from Eq. (11). If the » ■> '*» ôr 80me materials of importance. copic transport cross section (Zir = 1/Xir) ,\ total macroscopic transport cross section mm

i

,y\*,Ol 3

V-'

•

. .

moderator volume to fuel volume is as the ,vbondition does not hold, an approximate >n can be computed by taking a fluxJ1 p. over a lattice cell, exactly as is done for

/diffusion coefficient can be used

m if

t/fft

6-84

REACTOR PHYSICS 9.6

The Fast Fission Effect

[Sec. 6 c

The fast fission effect applies to the fission of U"8 nuclei by fast neutrons in reactors containing relatively large proportions of U"8 to Um. It is usually a negligible effect in effectively homogeneous thermal reactors because of the usually low ratio of U"8 to moderator in such reactors. Only neutrons having energies above about 1 Mev can cause fission of U"8. For this reason the following two assumptions can often be made in calculating < for lumped reactors: 1. Any inelastic scattering collision within the fuel reduces the neutron energy below the TJ"8 fission threshold. 2. Any neutron which escapes the fuel lump is degraded to an energy below the U"8 fission threshold by the moderator before it reaches a new fuel lump or is scattered back into the original lump. The quantity t is defined as the number of neutrons slowing down past the fission threshold of U"8 per neutron produced by thermal fission. With the above assump tions it is given by

l~

[v

- - (2./Z/)](2//X,)P 1

,

_

(rX, + S.,)P7^T~

(207)

where Se, 2/, 2d, and 2, arc, respectively, appropriate average values of the capture, fission, elastic scattering, and total cross sections in the fuel lump and v is the average number of fast neutrons produced per fission of U"8 by fast neutrons. Generated for wjivans (University of Florida) on 2015-09-23 02:51 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

0 9

Fig. 15. Probability of first-flight collision within a fuel lump. 2 is the macroscopic total cross section, x the reciprocal of thermal diffusion length. For spheres and cylinders R is the radius; for slabs it is the half-thickness. P' is given by the curves for x/2 =0. (Repro duced from Fig. 15.14, The Reactor Handbook, vol. 1, AECD 3645, March, 1955; original from CD-644.)

In Kq. (207) P is the probability that a neutron produced by thermal fission in the fuel lump will make some kind of a collision within the lump before escaping. The neutrons produced within the lump by fast fission may, in turn, produce further fast fissions within the same lump, and on occasion a sequence of such fast fissions may last through several fast-neutron generations. The denominator of Eq. (207) takes account of these fission sequences. The factor P' is the probability that a neutron produced by fast fission in the fuel lump will make some kind of collision within the lump before escaping. In computing P it is assumed that the sources of fission neu

Sec. 6-2]

6-85


irons under consideration are distributed in the fuel lump according to the thermal flux distribution in the lump, whereas in computing P' it is assumed that the sources are distributed uniformly within the lump. Computed curves of values of P for various lump sizes and geometries are given in Fig. 15. Note that P, since it involves the distribution of thermal fissions, is a function of both x, the reciprocal of the thermal diffusion length in the lump, and 2|/?, the dimension of lump measured in total mean The quantity P' for a given geometry is a function of S,ff only; it is given free paths. in Fig. 15 by the curves for which x/St = 0. Values of the other constants needed for Eq. (207), for natural uranium metal, are given in Table 12.

Table

Fast Effect in Uranium

12.

Constanta for computation when there is no fast-neutron interaction between fuel lumps:* Constant

aj

0 . 29 barns 1.5

a,i

" <0.04 " 4.3 w <2. 55 (dimensionless) Experimental values of fast effect (e) for 0.600-in. -diameter rods of slightly enriched There is fast-neutron interaction between fuel lumps. t water.

r. a,

Volume water


Volumo uranium

0. 1785 1 1.5 2 3 4 1 1.5 4

%

uranium in light

Um in uranium (by weight)

1.299 1.299 1.299 1.299 1.299 1.299 1.143 1. 143 1. 143

l.227± 1. 105 1.072 1.061 1. 047 1.043 1. 109 1.074 1.042

0.011 + 0.002 + 0.001 + 0.001 ± 0.001 ± 0.001 ± 0.002 ± 0.001 ± 0.001

* From "The Reactor Handbook," vol. I. Table 1.5.20. AECD-3645. March, 1955. Original source CL-697. t H. Kouts, G. Price, K. Downes, R. Sher, and V. Walsh, Exponential Experiments with Slightly Enriched Uranium Rods in Ordinary Water, Geneva Conf. Paper A/Conf. 8/P/600. June 30. 1955.

In close-packed lattices such as those which occur in some water-moderated reactors, assumption 2 above does not hold, and there is a substantial increase of t over that calculated by the foregoing method as a result of the mutual fast-neutron interactions Measured values of « for some HjO-uranium lattices between separate fuel lumps. of this type are given in Table 12. 9.6

The Thermal Diffusion Coefficient A*

For an effectively homogeneous medium the thermal diffusion coefficient can be Measured computed from Eq. (10) or, if the medium absorbs strongly, by Eq. (9). If they are not, the scattering cross values of \,, should be used if they are available. sections must be used in conjunction with a value of £o computed from Eq. (11). If the Table 13 gives transport mean free paths for some materials of importance. medium is a mixture of materials, a macroscopic transport cross section (2(r = 1/Xir) can be obtained for each component and the total macroscopic transport cross section of the medium can be taken as their sum. In the case of lumped reactors, the ratio of moderator volume to fuel volume is often sufficiently large that the moderator diffusion coefficient can be used as the If this condition does not hold, an approximate diffusion coefficient of the medium. value for an effective transport cross section can be computed by taking a fluxweighted average of the transport cross section over a lattice cell, exactly as is done for

6-8G

REACTOR PHYSICS

[Sec. 6

the absorption cross section in Eq. (209). The quantity D, can then be computed as for the homogeneous case above. The resulting expression for the effective diffusion coefficient is

-

D

„

1

l + (JVVjKWji)

*

where 1

-/2al

The approximation is a poor one for use in matching neutron currents at a reactor boundary, since the cell approximation does not hold at the boundaries. Such defects are inherent in the cell method and are the price paid for the simplicity of the method. Table 13.

Thermal

Diffusion Length and Transport Various Materials*

2.85± 0.03

HtO

D:0 (0.16% H.O)

-


Be (p BeO (p

116

GBF

to p — 1.60) :

±0.5 ±2

54.4

±0.5

1. 43 ± 0. 05 (using
(ffnba = 4. 4 millibarns)t 52 ± 1 4. 8millibarns)t (»nh«

AGOT Th(p ThO. U (p UjO.

0.48+ 0.01 2.65± 0.15

± 4

29

20.8

-I.8S) 2.69)

Graphite (corrected

Transport mean free path (thermal), cm

Thermal diffusion length L, cm

Substance

Mean Free Path of

-

- 11.2) (p = 6) - 18.9) (p - 6.0)

±0.3 ±0.4 l.55± 0.05 3.7 ±0.4 2.7

4. 1

• From "The Reactor Handbook." Table 1.5.4. AKCn-3645. March, 1955. See original table for sources of information. t 2,200-m/sec values; obtained by using a calculated value of 4.70 barns for the Maxwellian average of the total cross section.

9.7

The Thermal Diffusion Area i»(= 1/*')

The thermal diffusion area

is given, in all cases, by the equation L>

=

A.

(20S',

-ol*

where 2a(* is the total thermal macroscopic cross section of the medium. In the case of a lumped reactor an effective absorption cross section 2„ must be used in the reactor core. The effect ive value is taken as the volume average of the true absorption cross section, weighted by the thermal flux density; i.e., for a cell (sec Figs. 11 to 13) where the fuel is designated by the subscript 1 and the moderator by the subscript 2 Vs + V'i(«,/0,)

„

_r«? 1

-fV,

M

+ V, (*,/&)

(200/1'

where $i/$i is the reciprocal of the disadvantage factor [Eq. (177)] and/is the thermal utilization [Eq. (189)]. Usually the term Fi(0i/^) in Eq. (209) can be neglected in comparison with V%. For this case

2.

1-/

(210)

Sec.


6-2]

and if the

6-87

diffusion coefficient of the medium is taken to be that of the moderator, area becomes, approximately,

the diffusion

L' where L»s is the

Experimental in Table 13.

L,»(l

=

-/)

(211)

diffusion area of the moderator. values of thermal diffusion length for a number of materials are given

The Fast-diffusion

9.8

Coefficient Df

An effective fast-diffusion coefficient covering a range of energies can be defined only the formalism of the group treatment of slowing down. effectively homogeneous with respect to fast Most reactors can be considered neutrons, for even in lumped reactors the slowing-down lengths and absorption mean free paths are sufficiently long that large variations in fast flux do not occur over a Hence, at a given energy, a total transport cross section Z„(E) typical lattice cell. can be evaluated which is just the sum of the macroscopic transport cross sections of to be uniformly mixed together. all the reactor core materials, considered Thus a diffusion coefficient D(E) = l/32(r(£) can be evaluated as a function of energy over the energy range E\ to B2 covered by the fast-neutron group in question. Usually D{E) will not vary rapidly with neutron energy, and it is therefore permissible to make the assumption that the variation in energy spectrum from point to point in An effective diffusion coefficient Df can then be the medium can be neglected.

for the group simply by averaging Df(E) over the energy distribution

Df

-

D(E)(E) dE «(£) dE

If

JEi

the energy distribution becomes

by the 1/E distribution

of flux can be represented

3.2) the expression

D>

9.9

'

"

of

(212)

(Art.

HdE/E) (213)

in EWE,

The Slowing-down

Area

(t

(E):

[*'

flux

/

evaluated

or L/1)

is

The computation of slowing-down area quite difficult except in those cases for applies (the approximation applies fairly well for not beryllium and moderators of higher mass number provided inelastic scattering important). When the approximation does apply, r can be calculated by the equa For other cases, see Ref. tions of Art. 3.4. In all cases measured value of should be used Measured one available. values are given for number of moderators in Table 14. The table contains measured values for fission neutrons slowed down to the indium resonance energy and cal culated values from the indium resonance energy to thermal. The total t from fission to thermal energy the sum of the two. As pointed out above (Art. 9.8), even lumped reactors can usually be considered in computing fast neutron constants. effectively homogeneous In many lumped very nearly the reactors the slowing-down area for the uranium-moderator lattice same as that for the moderator alone; the high inelastic scattering cross section of the uranium compensates for its low elastic slowing-down power. In two-group calculations the slowing-down area designated by L/2 instead of and treated as a constant which may be adjusted to compensate for lie deficiencies of the exponential description of slowing down distribution. These adjustments are discussed in Art. 5.6. is

if

t

t

is

is

is

a

t

a

5.

is

which the Fermi age approximation

is


within

6-88

REACTOR PHYSICS

[Sec.

6

Age of Fission Neutrons in Various Moderators*

Table 14.

(Room temperature) Age to indium resonance, cm3

Moderator

H.0 U,0 (0.16% HjO)

-

Be (p 1.85) BeO (p 3.0) = (p C 1.60) HiO, 66.7% 33.3% 50% IM), 50% Al 80% H.O. 20% Zr 50% HiO, 50% Zr

Calculated age indium resonance to thermal, cm*

30. 4 ± 0.4 100 ± 5 80 ± 2

-

+ 3

311

All [ >

cm1

31.4

1

125

97.2

17.2

105 ± 10 364

53 2 3

49.6 76.8 35.7

b>y vo1,

Age to

thermal,

51.6 79.8

6lt

(from reactor analysis)

-

(p I.I g/cm') D.O + l.4gThOi/g DiO DiO + 2.8 g ThOi/g D.O 9.69 g/cm«) (ThOi density 96.7% DiO, 3.3% U, by vol

DiO

127)

I32icalc.t 140 J


-

107 ± 5{

Values of r for IliO-U mixtures can be inferred from M1 measurements in the following reports: S. Krasik and A. Rudkowski, Pressurized Water Reactor (PWR) Critical Experiments, Geneva Confer ence Paper A/Conf. 8/P/60I. June 30, 1955; H. Kouts el of., Exponential Experiments with Slightly Enriched Uranium Rods in Ordinary Water, Geneva Conf. Paper A/Conf. 8/P/600. June 30. 1955. •The first eight values are from "The Reactor Handbook," Tables 1.5.1 and 1.5.2. AECD-3645. March, 1955. See original table for sources of information. t J. Ernest Wilkins, Jr., Robert L. Hellens, and Paul F. Zweifel, Status of Experimental and Theoreti cal Information on Neutron Slowing Down Distributions in Hydrogenous Media, Geneva Conf. Paper A/Conf. 8/P/597. June, 1955. t M. Tobias, Certain Nuclear Data and Physical Properties to Be Used in the Study of Thorium Breeders, CF-54-8-179. Aug. 26. 1954. 5 E. Richard Cohen, Exponential Experiments on DjO — Uranium Lattices, Geneva Conf. Paper A/Conf. 8/P/605. June 30, 1955. 10

MULTIGROUP CALCULATIONS By David Okrent

In Art. 5.8 the multigroup formulation was given for the case in which fission occurs only in the thermal group and all fission neutrons pass through all the epithermal A more general formulation is neces energy groups before becoming thermalized. sary for the treatment of fast reactors or reactors in which an important fraction of the fissions occur at epithermal energies. Applications and methods of solution of multigroup calculations are varied and are often too complicated to be handled by manual computation. In the following article the method is applied to the case of a bare reactor, which requires only a modest amount of computation but suffices to illustrate the application of the method. 10.1

If those neutrons

Ej)

The Multigroup Equations

whose energy lies between two arbitrary limits Ef and Ej-\ (2?,_i > j, the time-independent diffusion

are considered to make up the flux in group equation for this group can be written in the form

Dj v>/(r)

-

(2,.,-

+ 2,., +

2,„„-

+ 2.,moJ.y)*y(r)

-(-

St(r)

(214)

where Dj TVjfr) = usual leakage term 2C,/ = macroscopic capture cross section for group 2/,, = macroscopic fission cross section for group 2i„,, = macroscopic cross section for scattering inelastically out of group 2,i „„,(,, = macroscopic cross section for elastic moderation out of group 2„m„,„, = 2,., + 2/., + 2,-„., + 2,i „,„,)./ = the total macroscopic cross sec

j

j

j

tion for processes

removing neutrons from group

j

j

reactor calculations all neutrons born into group j at position

Sec. 6-2]

6-89

r. These neutrons S, (r) includes the result of fission by neutrons of any and all energies or the result of an energy-degradation process from some higher energy into the band between E,- and (Herein the possibility of an external source will be omitted.) The entire energy range of interest will be divided into N groups or energy bands. Furthermore, 7/ will be defined as the fraction of all fission neutrons born into group j, and vt as the total number of fission neutrons born per fission in any group k. Then Eq. (214) may be rewritten in the form and where could be

D,

v'*,(r)

- 2„m„„./0,(r)

N

+7/

j-l + Of course, 2;„(J —* into group j. Thus

j)

^ (-1

Y

«*Z/.*<*»t(r)

k=\

j-1

2i-(/->i)*i(r)

+

^

2"

=0

">«*(m

(215)

m-l

is the macroscopic cross section for scattering from group I (+ i

Y tUl-* J)

y-i

= 2.W

(216)

similar expression holds for the elastic moderation cross sections. The elastic moderation source, in principle, could have contributions from all higher energy groups; hence it has been summed from the highest energy group 1 to that just above However, for materials other than hydrogen, such contributions are possible group j. only from relatively nearby energy bands and frequently only from the next higher


and a

group. Equation (215) then describes the diffusion of neutrons in group within a particular A set of cross sections for all N groups is required for this homogeneous region. region. The result is N simultaneous equations which describe the problem over the entire range. Of course, if more than one region needs to be considered, a crossThe usual boundary conditions of con sectional set is required for each region. tinuity of flux and current are applied to each group, and the multiple sets of simul taneous equations solved. This latter problem of multigroup theory in more than one region cannot generally be solved analytically. A numerical solution of an iterative type is required. This solution is of such magnitude and needs such great accuracy that only modern high speed computing machinery is really practical for its achievement. (in energy)

j

10.2

Simplifications for Single-region

Computation

If multigroup diffusion theory is being applied to a single region (specifically, a bare reactor core) a simplifying assumption can be made which makes possible a semianalytic solution with only a few hours work at a desk computer. The basic hypothesis is that the flux has the same shape in every group. Within the framework of diffusion theory, this would be rigorously true, independent of geometry, if and only if the extrapolation distance were the same for all groups. This is not usually the case. Furthermore, there is direct experimental evidence that the Nevertheless, the neutron energy spectrum is not space independent in a bare core. assumption of a space-independent neutron energy spectrum in a bare core is not a very bad one and gives results which are rather good away from the boundaries. Thus, it is assumed that =

k(r)

= kF(r) = 4>iF(r)

*,(r)

In

tjFir)

4>,(r)

other words, F(r) is a spatial shape factor independent of energy.

It

is further

6-90 assumed

REACTOR

PHYSICS

VV(r)

B»F(r)

[SEC.

6

that (217)

where B* is the so-called material buckling. If a bare core of this material were built in any shape, its extrapolated dimensions would have to be such that the geometric buckling equaled this material buckling at critieality. With these assumptions Eq. (215) becomes — DiB*4>j

—

2„move.jj +

Ji V

"&f,k
k-1

J-l

S-l

+

y 2j)
+

y

2.,m„j(m-+ j)*m

=0

(218)

m-1

1= 1

The spatial dependence has been completely removed, since it is now common every term. It is convenient to rewrite Eq. (218) in another form, namely,

y-i

at


i

yj

£ = ——

vi,2/.k4k +

y

2,-.(/-»

j)i

y-i +

— 10.3

to

y

2rf

„„i(m -»

j)4>m

—

(219)

Procedure for Solution

Equation (219) really comprises a set of N simultaneous equations, one for each group. In practice, this is usually solved by an iterative procedure. For the highest energy group (j = 1) the numerator consists only of the fission source, no contribu tions being possible from inelastic or elastic moderation processes. If it is now N assumed

that the total fission source

number, say unity,

i

for all groups

^

vtS/,*^*

is some

arbitrary

is given by (220)

D,B> +

i

The quantity still remains a function of B'. The procedure for solution is to guess some reasonable value for B1, keeping in mind that it is this quantity which will be iterated on. When the solution converges, the critical value of the material buckling will have been determined. The quantity 4>i is computed from Eq. (220} with the trial value of B2; it is then possible to evaluate

2:

02

. T. + 2..'-"^+ + 2„^-^, •D2B5

(22n

2„in„,.„2

Since this group gets source neutrons other than fission only from group I, it is com The process is continued through all the pletely specified and >2 can be calculated. groups in numerical order in this fashion. At this point a set of <£,'s has been obtained, and the convergence test is to be applied. AT

This

is done by calculating a value for the total fission source

V t=i

vk2f.tk

and coro-

If the two agree within the desired paring it with the original assignment of unity. accuracy, the equations have been solved. Otherwise, a new guess on B* is made and a second iteration performed. The fission source is again computed, using the new set

CALCULATIONS

REACTOR

Sec. 6-2]

6-91

and compared with unity. This process is repeated until convergence, which quickly with some practice. When the solution is complete, not only the material buckling B1 but also the neutron energy spectrum have been determined. The spectrum of course, given the relative values of the total flux in each energy band. of 0,'s,

by

is,

comes

Applications

The single-region calculation has

It

variety of applications, particularly

for inter

can be used in parameter studies to give the neutron spectra and critical masses for cores of varying composition (assuming energy previous knowledge of reflector savings). In connection with the latter, some caution must be exercised in applying the results to reflected assemblies, since the reflec tor savings may differ sufficiently to alter the apparent trends of the one-region calculations. The neutron energy spectrum can be used in predicting various performance criteria, also in establishing set of one or two group cross sections for performing multiregion calculations. Caution must again be exercised in that this spectrum quite accurate performing energy averages for the core but will probably not be representative of the reflector spectrum. This latter will probably change quite rapidly with dis tance from the core. in

is

a

a

mediate and fast reactors.

a

10.4

Example

it

it

1

is,

a

is

it

is a

is

.

is

a

is

a

is

is

it

desired to determine the critical mass and various other performance Suppose data of a certain fast power reactor. It sodium-cooled, uses plate-type fuel ele ment having a partly enriched uranium meat and stainless-steel jacket. It assumed that previous experience indicates a reasonable reflector saving for this core and the type of blanket contemplated. Then the material buckling required for the assignment of critical mass. First, a group energy structure must be chosen so that cross sections can be assigned The choice arbitrary, and the structure function of personal taste and previous experience. However, the technical facts do suggest that certain boundaries are For example, the fission of U"8 will be an important contributor to reasonable. an important performance criterion. reactivity; also, Furthermore, U"a has fission threshold in the neighborhood of does not fission below this Mev; that Above, Hence, a logical place for a group energy. rises rapidly to a rough plateau.

is

a

if it

a

is

this threshold. boundary Again, suppose the fission or capture cross section of the fuel U"5 exhibited marked change below some energy. It might be quite flat above 100 kev but rise sharply below this energy. This, too, would then be a logical place for a group significant amount of the total flux was expected that boundary. Indeed, present was to be found below 100 kev, several groups might be used below 100 kev in order to describe these events accurately. Thus, some idea of the neutron energy spectrum to be expected helpful in assigning group structure. one of the materials had a large resonance in a region of considerable flux, Again, a group might be assigned to bracket it. Finally, the manner in which certain important cross-sectional measurements were made might be suggestive of group structure. For example, to find where in the energy scale inelastically scattered neutrons go, threshold detectors like U"8 nnd Np"7 are used. The neptunium threshold around 400 kev. If a significant amount of the inelastic scattering data based on this detector, this energy becomes a natural position for a group boundary. The above only illustrative, not all-inclusive. Depending on the particular subject of interest, one or another group structure may be chosen. In the present example, there no special side interest which requires bunching total of six groups selected as a reasonable number to work with. of groups, etc. Since most of the neutrons are expected in the neighborhood of 100 to 300 kev, a

a

is

A

is

is

is

is

if


10.6

6-92

REACTOR

PHYSICS

[Sec.

Thus, the following group greater density of groups is used in that region. structure is chosen, and the -y/s are assigned accordingly. Group No.

Energy interval, Mev

7

»-l.4

0.47 0.31 0. II 0.07 0.04 0

1 Z

1.4-0.7

J

0. 7-0. 4

0.4-0.2 0.2-0.05 0.05-0.0

4 S 6

6

energy

The 7,'s are obtained by integrating the normalized fission spectrum with respect n

to energy between

the limits of the various bands.

Of course,

V

y-i

7,- = 1.

it

is

is

a

is

x

j

a

if

is

is

{E)(E) dE HE similar expression applies for other cross sections of interest. For inelastic scattering, frequently convenient to divide the description of this First, the average inelastic scattering out of each group process into two parts. obtained. Then a matrix of coefficients assigned to describe where, in energy these neutrons go. We may write is

is

is

it

A

2,-„,r*

the average macroscopic inelastic scattering cross section out of group

and n

over all groups lower in energy than group

I.

The summation

is

1-1

+ 1

I

for material

I

where

- Z,„.i*C,—

,

2,„*(/-»i) x is


is,

One now needs to assign group cross sections. The technique used depends on the accuracy desired and the information available on spectrum and cross sections. One Some assumption must does not, a priori, know the shape of the flux in each group. In principle, if the cross sections be made, and average cross sections computed. are well known as a function of energy, one might get greater accuracy by using a large number of groups and iterating on the cross sections. That assume some flux shape in each group, find average cross sections, and solve the particular "bare core," Now use this spectrum to reassign average cross getting a neutron energy spectrum. sections in each group, and again solve the bare core problem. In practice, cross sections may not be too well known, and a reasonable assumption on flux shape reasonable flux assumption sufficient. For a fast power reactor, for our particular group structure that the flux constant with energy in every resembles the fission spectrum in this energy region. group except the first while It usually convenient to assign average group cross sections to each material, then combine them in the appropriate proportions. This technique accurate for there are wide all cross sections except transport, where some difficulties arise fluctuations within group. However, for a mixture of elements, such fluctuations are usually minimized, so that the transport cross section can be given the same averaging process as the others. as Thus we may write the average fission cross section in group for material

CALCULATIONS

KEACTOIt

Sec. 6-2]

6-93

For the materials of the example and the particular group energy structure chosen, moderation from any group will contribute only to the group next lower The group cross section of course, a function of the average loss in the in energy. the scattering cross section 2K, and the shape of the flux logarithm of the energy within the group. Again, possible to iterate on cross sections or to use approxima tions thereon. However, the previous assumption of a flat flux in every group but the first quite reasonable for this example. One possible method of procedure then to compute flux-weighted average energy for each group. One may then write for material is

Ej

x a

is

is

it

£,

is,

the elastic

—

j

mod,

In

~ Ei/Ei+1

J

0.35

6

0.41

Si.

0.0010 0.0017 0.0025 0.0037 0.0059 0.0154

0.050 020 0.012 0.007 0.001

2.1 mod

0.012

077 0377 0.0365 0.0307 0.0319 0.0414

0 0

0.014 0.010 0.010 011 0.016 0.026

St

0.006 0.012 0.009 0.009

0

4 3 2 1

0. 16 0. 19 27 0. 28

0

2,r

0

No. group

Furthermore, for simplicity, vj will be taken as 2.5 for all groups. will be assumed that the matrix of coefficients which prescribes the destination of We take these to be as the same for all materials. inelastically scattered neutrons folio W8:

0.7 0.2 0.

3

4 0.

1

0. 0.05

0 2 8

0.05

0.2

3

0.65

0. 15 0.

1

2

0.4 0.3

1

6 S 4 } 2

^\

1

\^ From To

1

is

it

The quantity

a

If

is

is

As previously mentioned, the elastic moderation for this example always into the next lower energy group, so that no corresponding matrix required. no such back One makes a first guess on B1 usually from previous experience. ground exists, a rough one-group set of cross sections, averaged over composition, will give reasonable first guess, using the formula fis

_

vX,

- Z.

In

is

is

A

is

in

,'s

is

As the present example, B1 = 0.01 will be tried for the first iteration. shown Hence, 1.03414. Table 15, the fission source calculated using the first set of the result second guess of B* = 0.0105 3.414 per cent away from convergence. If only 0.95 per cent away from convergence. made, and this time the result this were not satisfactory, a third iteration would be made.

is


a

If

is

of

A

is

final note on cross sections that the transport cross section used in the definition the diffusion coefficient (D = l/3Zir) the sum of the elastic transport 2„(1 — jio) and the removal cross sections. one goes through the process of obtaining group cross sections and then averaging them appropriately in accordance with composition, one obtains a set of multigroup cross sections for the particular core under consideration. For purposes of illustra tion, set of cross sections representative of a small, fast power reactor will be assumed and two iterations on buckling B* will be made.

« £?

+

4.7535 8.2342 8284 6.4326 4.5475 1.6580

±

8041 8.3942 6.9924 6.6249 4.6908 1.7156

iH

mn-*o*n^-Q

•/>—

oooooo oooooo

ooooc>o

6

4

ay (N eo-3 Mfl (D N

moor-.

oo-*n5» »>OQ>^-0>

- ir-i ff.

oooooo

tV

(N N IS O IN **Ra*g

**SS2:1

oooooo

II

oooooo

□Ou"*

c o o e o o o o

s

oo ooo ooooo

1

•

(10)

NNCCO* -irt»v»

— N l*\ (îM iftirt — S

W-N

7

I

Col.

col.

9

«

7

§

a

oooooo

S

S

8

3 ** 8 "i

§

o o ooooo

o oo

T

■BO

3 — QO*
S

e w

7

KIT

— u-lQOu->
oooooo

OOOOOO

H

I

5? '

T

*

+

o

b

1

£85 boo b

X* jS

oooooo 6-94

^ 6>o

a a

!lisi-^j.j bdebe

g

•

s -

-

§

s

oo

1

5

in

II

O O C

«

■

H

°

J

;isi

b

oo bob

s:

ds

■s ■ !

-o

o ooo

St

■5

d

1 a5

*

i

§

o>■ u-i00 HI r-. so u"»—

•c

•

T

vfi(N — «}ff. r»»r>to
>is

k

■

r

^>

r


■5

i

7

■&

=

ooo oooo o

Poooooo

The following

6-95

SAMPLE CALCULATIONS

11

the procedures involved in calculating the a typical lumped reactor. Since the purpose to arrive at very accurate results, the numbers

calculations illustrate

material constants and reactivity for is to illustrate procedures rather than involved are carried out to slide-rule will usually be found desirable to use more

CALCULATIONS

REACTOH

Sec. 6-2]

significant figures, particularly

In making practical calculations it accuracy. a desk-type calculating machine and to carry in calculations of the type covered in Arts. 11.5

and 11.6.

Again, to illustrate procedures, values of some of the reactor constants are cal It should be remembered culated even though applicable measured values exist. that the methods used here are only approximate, and when great accuracy is required, measured values must be sought or higher order approximations to transport theory must be used. Description of the Reactor


11.1

The sample reactor is taken to be one moderated by D2O, using as fuel natural uranium in the form of circular rods 2 cm in diameter, canned in aluminum of 0.05-cm thickness. The rods are arranged in a triangular lattice, with uniform spacing of 16 cm (Figs. 16 and 17). The core of the reactor is a circular cylinder, 200 cm in diameter and 200 cm high, and is surrounded on all sides by a reflector of pure D20, 30 cm thick (Figs. 18 and 19). The reactor is assumed to be at room temperature. The area of the triangle drawn in Fig. 17 is The moderator contained in a tri110.9 cm1.

Fia.

16. Section

of reactor for sample calculations.

Fio.

17.

Cell arrangement for sample

calculations.

[

y

IOO CM

I30CM

co RE

|

/

J

260 CM

/

CORE

DjO REFLECTOR

Fio.

18.

reactor.

Horizontal cross section of sample

200CM

-D2O REFLECTOR

Fio.

19.

reactor.

Vertical cross section

of sample

6-96

REACTOR

PHYSICS

[Sec.

6

angular prism of this cross section and unit height is associated with the volume of uranium contained in half a fuel-rod section of unit length (1.571 cm3) and with the volume of aluminum making up half a can of unit length (0.16 cm3). From these numbers the volume ratio of moderator to fuel and the radius of the equivalent circular cell It*, can be computed. These quantities, the atomic density in the uranium region Ni and in the aluminum region A^, and the molecular density in the moderator region .V2 arc tabulated below. The density of uranium is taken as 18.5 g/cm' and that of D20 as 1.10 g/cm3. Vol.

D,0

110.9

-

Vol. U

1.571

- o.iot

Vol. Al

(r) (2) (0.05)

Vol. U

t(1)«

n,

J(2)

0.10

(110.9)

X 10")

m (18.5H0.603

_

8.41 cm

=

1QU

238


v Ni

= (1.10)—(0.603 18.0

X

= (2.70) (0.603 27 0

X 10")

, , 10") = nmro„„ni , , 0.0308 X 10" molecules/cm3

11.21 Regeneration Factor r/.t number 28; then

__

v

nnr„„

+ 2„(28)

.

.

Constant k

byEq.(169)

y(25)Ar3t
AW„(25) +

.

atoms/cm3

Designate U"5 by the number 25 and U"" by the

,(25)21,(25) Z,,(25)

=

Infinite Multiplication

11.2

69.4

= 1.00 cm

R, _

,. N,

_

1.571

A,8aa(28)

>(25V/(25) + (AS8/Ass)
(2.46)(0.981)(580) (0.981) (687)

Thermal Utilization

11.22

2„i =

+ (0.993/0.007 15) (2.75)

/:

V,[0.00715
= 0.0169

X

10»«[(0.00715)(0.981)(687)(10

")

+ (0.993)(2.75)(10-»)]

1

1.128

= 0.314 [See Eq. (65).] X,,

3LJ(D,0) 2„, = Jv>„(Al)

2.65

-

(3) (116)'

= 6.57

X 10"'

cm"1

(0.0603) ( 10") (0.230) (10"»)

from Table

(—!—) V 1 . 128 '

13

= 0.0123 cm"'

t An inconsistent number of significant figures is carried in the equations to aid the reader in recog nizing the numbers. Since only ratios of cross sections are involved in computing % All cross sections are from BNL-325. 17,the 2,200-m/sec cross sections can be used instead of values averaged over the Maxwell distribution. However, it is necessary to include the correction (0.981) for the departure of the UMi cross sections from a l/v variation (see BNL-325).

Sec. 6-2]

1

18.5

1

= 0.632

1.55 18.9

l Xl = — =

= (0.00862) (1)

E

=

xtjRi*

1

= 0.00862

116

= 0.632

»,/?, = (0.632)0)

x.Ri

2ft,

1

/

=

7„(x,7?,)

2

7,(x,7J1)

FJ-i


j"

11.23

+

_

2

yJiy X

(69.4) (6.57

(1)»

0.601

from Table 8

+ K0(0.00862)/i (0.0725)

-

7C,(0.0725)71(0.00862) 1.0067

(13.70) (0.00431)

0.632 7,(0.632)

r i2ai

1

/=

-

(0.0363)(116)

kiR,

1+

- Kl(xiR1)Il(x1Rl)

(1.000)(13.70) + (4.87) (0.0363)

= 0.301

-

(1)'

70(x.7?,)Ki(x,g,) + KtixtRÎiixtR,)

7,(0.0725)^,(0.00862)

2

= 0.0725

(0.00802K8.41)'

B,

/^.iftlK.M,)

13

= (0.00862X8.41)

R,')

xARr

= 0.00862

- ff.')

from Table

x«R<

0.601 /o(0.00862)A'i(0.0725)

m

6-97


=

/, (0.632)

-

(*

+

1.1024

'

0.332

_

Q49

from Eq. (190)

1)

10-') + (0.10) (0.0123)

1 (1.049)

+ 0.0067

= 1.0201

0.9746

Resonance Escape Probability (p): x, = (0.0222) (18.5) x, = (0.141)(1.10)

x.fi,

E

=

2

-

2

-

(0.801) (6.26)

7,(0.411) 7,(0.411)

=

1 *v/*'>

*

tRi2p

4 +

m

h.

|2.

~

=

= 2.13

02Q

^

2tt7?i

U

dE>

0.210

2

= 11.92

Nu

- £,(1.304)7,(0.155)

(0.3703) (0.0777)

0411 L043

= 0.155

+ fl,,(0.155)7,(1.304)

+ (2.00) (0.801)

(1.006) (0.370)

mass of

x,R,

= 1.304

7,(1.304)iC,(0.155)

surface area of U

M

from Table 10 from Table 1 1

cm-'

1) 7o(0. 155) A', (1.304)

1

04H F _

x,R,

= 0.411

0.155 (70.7

- 5.40

= 0.411 cm"1

- 0.155

R\p

= 0.108 cm »/g

(1)(18.5)

= [9.25 + (24.7)(0.108)](10-»«)

X 10"" cm'/l'"8 atom

by Table 9

= -V, (0.993) = (0.0469 X 10") (0.993) = 0.0466 X 10" = 5.28 X 10"" cm»/D20 molecule by Tabic 11 (5.28 X 10-") (0.0368 X 10") = 0.194 cm"'

-

1

exp V, exp

-

&.F

Nnfv..„(E')(.dE'/E')

+ (E

-

by Eq. (206) \)

1

(69.4) (0.1 04) (1.020) (0.0406

X

10s1) (11. 92

X 10"")

atoms/cm'

+ 2.13

-

,.-0.0]87 1

_

0.902

6-98

REACTOR PHYSICS

[SEC. 6

Fast Fission Factor i:

11.24

=

Xl (thermal)

—J

— (

L

from Table

= 0.632

1.55 V18.9/

13

2, = (0.0466) (4.3) = 0.200 (0.0466) (0.29) = 0.0135 2,, = (0.0466) (1.5) = 0.070 0 2„ v 2.55

=

2,

0.200

1

~

(2.55

= 0.21

-

2,

+0.070]

k

= 0.025

J

21)

k:

L

0.200

The Diffusion

11.3

(2Q7)

Eq<

/n onn mti on 1)(0.0135/0.200)(0.21)

r(2.55)(0.0135)

1

by

15

P r

-

= Vtpf = (1.326)(1.025)(0.9G2)(0.975)

1.276

Coefficients Df and /'. per cent of the total neglected, and D, for

is

is

Since the volume of metal in the reactor core less than volume, the effect of the metal on the diffusion coofTioients both core and reflector taken from Table 13:

_

£

D.

o

_ ?|8 o

m

0883

9.(1))

- 3.4

X

10~"

9.(0)

-

3.6

X

2

O,

is

evident, however, from the cross section D/ may bee omputed by Eq. (213). It curves for D and that the average values for the scattering cross sections between, Mev and thermal energy are very nenrly say, 10

"

L>

t

0.958

[1

+

X 10"") +

=

L,»(l

-/)

(0.958) (0.0368

By Eq.

= (116)'(0.0254)

X

10") (3.6

10

")

J

(2) (3.4

X

X 10")

The Diffusion Area IS and Age t.

11.81

- /io(0) -

1

(0.666) (0.0368 = 1.14 cm

age

= 0.666

- Ao(0)]2.(0)J

by Eq. (11) 1

1

»-\[

- jio(D)

_ 2

1(1

- jio(D)]2.(l>)

/j„ =

where

The

3

3

using these average values:

3 L


from Fig.

+ T 2.,) ""'

Infinite Multiplication Constant

11.26

3.16

(2./Z/)1(Z//Z,)f

(1/2/ - '-'

1

=

-

P

= 0.21 l>

_

1

.

P'

.

™?2

is 2

'-1

2/

(211)

- 342 cm1

for the metal-D.O mixture can be taken to be the same as that for

DiO.


Sec. 6-2] From Table

14,

t

= 125 cm

The Material Buckling B„«

11.4

The material buckling

6-99

is given by the modified one-group equation as

B"

=

^ST M*

^T 467

=

589 X

=

cm"2

10~4

This is a relatively large value for which the modified one-group equation can be The characteristic equation based on the expected to give only a fair approximation. Fermi age equation is a much closer approximation: (1 +

This equation

can be solved by trial and error or otherwise to give

B„*

X lOr* cm"'

= 5.50

Critical Size of Bare Reactor

11.5

This quantity

is not required for the solution of the problem under consideration, If the reactor is the computation is included for the sake of completeness. assumed cylindrical in shape, the ratio of length H to radius R must be specified arbitrarily. Assume here that it is 2.

but

'

_

£

B,'

=

B„*

= 5.50

5.50

X

R*

£406)..

=

R*

H>

X

10"'

[-

lO"4 L 4

R 11.6

-

_4_ (2ft)» +

+

cm''

by the criticality condition

(2.405)»1

= 1.500

J H

122.5 cm

from Table 5

(2^51 R*

X

10'

= 245 cm

Reactivity of Reflected Reactor

In

the following, the method of Art. 7.1 will be employed. Since the two-group reactors, is less accurate than that of the Fermi age for DjO-moderated the first step will be to determine an effective slowing-down area L/ which will make the two-group material buckling equal to the Fermi age material buckling, i.e.,

formulation

(1

L/* '

+ L'B')(1 + L,*B') =

=■>

B*

5.50

= (1 +

X

10"*

L'B')e""

= 134.5 cm'

Since the problem must be reduced to a one-dimensional

problem,

the choice is

arbitrarily made to examine first the radial solution, using an estimated equivalent A guess is made for the axial reflector bare reactor solution in the axial direction. Because DjO is a good reflector, a reasonable guess for the reflector saving saving. on each end of the core would be just slightly less than the actual reflector thickness — say 25 cm. The first estimate (which will be reexamined later) for the partial buckling then, in the axial direction is,

Axial partial buckling

-

+2(25)1'

=

158°

*


= k

L'S'Je'-B'

10"4 Cm_i

6-100

REACTOR

PHYSICS

[Sec.

6

The remainder of the problem consists of adjusting one or more of the reactor constants to a value which will make the critical determinant [Eq. (144)] vanish. In the following, k, the infinite multiplication constant, will be varied arbitrarily to achieve this condition. Although such a variation of k alone is not a physically possible procedure (e.g., if k were changed by changing fuel enrichment, L* would also change), the determination of the critical value of k is a useful operation, since a comparison of k (critical) with k (actual) yields a determination of the reactivity of the system. To proceed with the calculation, a trial value of k must be arbitrarily chosen. Since DsO is known to be an effective reflector and the reflector is relatively thin, a reasonable first choice is that value of k which would make the reactor critical if the reflector saving were equal to the reflector thickness. This value of k should be near the true critical value, but almost certainly a little too small. Thus the direction to go in choosing a second trial value of k will be known. The specified first trial value is k = 1.245. In Table 16 are listed constants which will be used in the calculation; these have been evaluated in the preceding sections or are derived from constants evaluated The notation is that of Art. 7. The slight variations from previous notation there. are necessary to avoid an excessive number of subscripts in writing the complex equations involved. Table 16


Core constant! (region I) k = 1.245 (trial value) R\ = 100 cm 342 cm' Lu 134.5 cm' L\i

- 130 cm (Fir. 5) - (116)' = 13.450 cm' Li/' - 134.5 cm" Rt

-

0i.

Reflector constants (region 2)

Lu'

-

= 0.883 cm 1.14 cm Si. = D,/L.' = 0.00258 cm-' 2i/ Dt/Lt' = 0.00848 cm ' xi.» l/L.' = 0.00292 cm-« xi/« = \/L/> = 0.00744 cm >

Du =- 0.883 cm

Di/ »

Di/

Bi'

-

xj/> = 0.00744 cm-' = -0.744 Bi> =

-xi.'

4.88 X I0~< cm"'

[Eq. (109)]

Bi'« = -0.01085 cm-« [Eq. (110)]

I'

=

Bi'«

B,'

1.580 X I0"« >H + 2R = 3.30 X 10 ' 1.580 X 10 ' = 0.01 101 -m> Bi'»

-

-

Zi. + Di.fli'

Si

Si/ 2„ + D,.B,''

S,

iKi

-

= 1.14 cm

Zi. = 0.656 X 10"' cm-i 0.00848 cm"' 2>/ xi.' = 0.744 X 10 « cm-»

-

-

(Table 13)

- -xi/' -

-,.,'

= Bi«

-f./'

-

-

-0.00848 cm-'

1.580 X 10 '

-

B,'>

X !0-«cm->

-

1.580 X 10-'

-2.324

X

10"'

0.00864

0.355

2i, — Bi.xi/'

Si'

0.826

2./

2l/

-0.767

n.Ri = 1.525 p/Ki = 9.30 ,i.R, 1.983 M/Kj = 12.10

1.818

mfl, = 10.50

-

The following Bessel functions will be needed to evaluate the critical determinant:

JoilR,)

= y0(1.818) = 0.3295 = /„(10.50) = 4,500 /0(1.525) = 1.672 = /o(1.983) = 2.253 /•0».Hi) = /o(9.30) = 1,447 /o(a«//i>i) IointRz) =» /0(12.10) = 2.078 X 10' KoinMt) = A'o(I.525) = 0.2070 K„(n.Rt) = K0( 1.983) = 0.1163 AVu/fti) = A'0(9.30) = 0.372 X 10"* K„(M/ft,) K„(12. 10) = 1.990 X 10"

/„(»"«,) /0(p,ft,)

-

-

JAIR,)

/i(mft,)

= 0.5818

= 4,280

/,(/!./?.)

-

1.0067

/i(m/«i)

-

1,366

A'.Oi.fti)

= 0.2676

A.U/ft,)

= 0.392

X 10-«


OS

OS

+

as

OS

a

q

c

"3 a; g

5

£

*> 0!

3*

S

05

ii

c

6-101 05

e I"*

3

OS oj

I

-T OS fc

5

£

7i

a

5 =5

-5 Qi 13

6-102

REACTOR

Putting in the numerical values yields

-

0.3295 0.1170 0.00934 0.00428

4,500 3,715 397 423

-

PHYSICS

[Sec.

-

0.1207 0 0.004304 0

0.371 0.285 0.0322 0.0319

X

10"'

X 10"' X 10"4 X 104

The determinant may be evaluated by expanding by minors.

D

X

= 0.0825

6

The result is

10-'

g,

X,.') + = -(»■/» +

-

+ Kl.8)' + 4*,/'»,.«[fc(0)

1]

2

= 5.20

X

10~4

cm'

The axial buckling of the assumed reactor (ir/250)' is 1.58 X radial buckling of the critical reactor must be Radial buckling = 5.20 X 10"4

-

1.58

10~4 cm*;

10"4 = 3.02

X

X

hence the

10"1

and the equivalent bare radius of the reflected reactor is

R (equiv)

V3-62 X

„

126 cm

10"4

To obtain an accurate specification of the criticality condition, one must solve the problem of the cylindrical reactor, bare radially and reflected on the ends (Art. 7.2), using the above value for the equivalent bare radius. If reasonable guesses were made originally for the axial reflector saving, the two computations (one radial, one axial) should suffice to determine the critical value of k. If a very poor initial guess was made, a second radial computation may be necessary, since the two components of the buckling are not completely independent. A reasonably accurate value for the critical k can be obtained without resorting to the axial calculation by assuming that the axial reflector saving (on each end of the This is a rather good assump core) is equal to the computed radial reflector saving. tion if the reflector saving is considerably less than the core dimensions (provided, of The computed course, the materials of axial and radial reflector are identical). radial saving is =

R (equiv)

- R (actual)

The equivalent bare height of the reactor

H (equiv)

B* = 3.62

= 126

then,

= 200 + 2(26)

X

lO"4

+

The corresponding buckling

is,

RS

-

100 = 26 cm

- 252 cm

is


Since D is not zero, the assumed value of k is not the critical value. A larger value is chosen for the second trial. This is arbitrarily taken to be the actual k of the reactor core material (k = 1.275). Note that only the elements of the first two columns of the determinant must be recomputed. The new value of the determinant is D' = —0.095 X 10-4. Linear interpolation between D and D' yields, for the value of k which will make the determinant zero, k(0) = 1.259. If an extremely accurate value of k(Q) is required, a third trial computation should be made, using the value The third trial will not be made here. 1.259 for the trial value of k. The value of k(0) is not an accurate value for the critical k of the actual reactor but is the critical value for the reactor which has the assumed equivalent bare dimen sion in the axial direction [H (equiv) = 250 cm]. The useful quantity which can be evaluated with some precision is the radial component of the buckling. The critical material buckling of the assumed reactor is given by

(2^2)'

= 5.17

X

10"4 cm'

and the

CALCULATIONS

REACTOR

Sec. 6-2]

6-103

critical value of k is given by

(1

=

k (crit)

+

—J (l

+

—,) -

1.259

Because the original guess for the axial reflector saving was quite close, the value of Obvi fc(crit) agrees with that of fc(0) to within the accuracy of the computation. ously, a change of a centimeter or two in reflector saving has no very great effect on the reactivity of a large reactor. The original problem undertaken was to calculate the reactivity of the specified This is most conveniently expressed in terms of the excess reactivity reactor. ik.ii/k.n, which is given by Eq. (261):

_

Sktff = Sk

k.fl

k

k (actual)

- k (crit)

k (actual)

_~

-

1.275

1.259

1.275

"

= Q Q125

If it is necessary to make the reactor exactly critical, one or more of its character In making istic properties (e.g., size, enrichment, lattice spacing) must be altered. such adjustments, if they do not change fc,// by more than 2 or 3 per cent, and if they do not alter the moderator density, it is usually safe to assume that they will not affect reflector savings appreciably.

Solution by Matrix Method


11.7

To illustrate the use of the matrix method of solution outlined in Sec. 8, the problem of the preceding section will be re-solved by that method. The same first guess will be made for £(1.245), and all the constants listed in Table 16 will be applicable. In order to construct the Q matrix for the reflector, the following elements arc evaluated, using the constants of Table 16 in the expressions listed in Art. 8.2:

_L n( qW>) Dtf

=

j

-w K.G./B,)

K^Ri)

_

Ko(iifRt)Io(iifRi)

KtbfRJhb/R,) KvinfRM^R,)

_

/C,(M/Bi)/oGi/fii) 1

Du

7

/

%=

Kobi.Ri)

J_

v.

j _

KA^Ri) j ?(*■/,*<.) =

The Q matrix is (Art. 8.2): 11 O * = [ 0

Kt>(,ii,Ri)It>(nMRi)

,

Kobijii)!

wjw 1

_

„g

Qg

Kib.R,)Un.Ri)

O! D'.f 0

o(n.Ri)

Ko(M.R>)li(».Ri)

90»/)

- !W

= 16.87

0 28.06

8.79 16.87

Since there arc only two regions involved in the problem, the Y„ matrices for inter mediate regions are not required. The following elements are needed for the core matrix Ym„ (Art. 8.4): — = -i— = 2.818 0.355 8i DuXe(m)

fa Si

=

1.0.)

—

S,'

=

Dlfm S

=

/o(m/ti)

\MXM)

DUX&)

=

%rX.(l)

= 2.49A-.(0

=

-

—

-0.826 01138 -0.1067

l.UlJ,UR,)

JoVRi)

=

-1.210

6-104

REACTOR PHYSICS

[SEC. 6 is,

Note that although quantities are available in Table 16 for evaluating Xr(l), we then, to retain it as a variable for the time being. The core matrix 1

1

choose

-1.210

2.818

-1.UX.Q)

0.1138

-0.1067

—

I 1 |

- 89.0X.fJ)

2.013

that the determinant

-

-

WA4XAD

2.818

89.0A%(2)

-2.300 of the QYeart matrix be zero;

2.013

1

The condition for criticality i.e.,

10.14X,(0

n 2.818

-2.300

|

QV oora

giving the matrix

0

t

formed,

|

QFCOr«

is

The product

is

-2A9Xe(f)

arbitrarily to set the determinant here for determining criticality This specifies a value of which will make the equal to zero and solve for Xc(l). determinant zero. If the value so determined agrees with the trial value (Table 16), the condition of criticality has been determined. If does not agree with the trial value, criticality has not been found, since no value of other than the trial value consistent with the other quantities used in the Fcoro matrix. Expanding the critical determinant yields = 0.0394

Xc(l) The above equation

can be solved quickly

=

Mfil

by trial and error, giving

I

= 0.01902

B,'« = mi

-

(tfTlRs)'

-(B,« -B,'«

■

x 10"

1580

+

+

++

=

p

*''

*

is

to use this value which differs from the trial value. The easiest procedure now (0.01902) as a new trial value, recompute all the elements of the Ken, matrix, take the product QYcon, and test again for criticality. To determine the new elements consistent with the new trial value, recall that

from Eqs. (109) and (110)

Xl/» + *„») 1.580 X 10~«

1

1

The new core matrix

is

etc.

-1.206

2.79 =

-1.14X.(J)

i

Kcore

0.1138 0.1064

-2A5XAD

zero value of the determinant give 10"4

I

bis

I

bn bn

the number of columna

of the first

I I |

(anbu + aiibn + anbit) au6« aii6u) (anbu anbti 4- anbu) (anbu +

anbu) anbu) anbu)

bit 6:i bit

is

can be multiplied only rows of the second.

Two matrices

if

I

4- +

aitbn anbu (rtn&ii -f- anbu

= 1|

and

as

(anbu (anbu

X

\

i

an an an

I

ait att an

b

an ait

+■4-

I

On

defined

= 3.61

for example.

| |

The product of two matrices,

= 0.01900

I*

= 0.0392

+ + +

t

Xr(l)

<"

a

Formation of the product QYcon and determination of the value of XAI) correspond ing to

is


I

is

it

I

is

The course chosen

equal

to the number of

CALCULATIONS

REACTOR

Sec. 6-2]

6-105

A;

k

?*

is,

This value of I is sufficiently close to the trial value to make further iteration unprofitable unless the computation is carried to larger numbers of significant figures; it incidentally, in satisfactory agreement with the value obtained in the previous calculation (Art. 11.6). The critical value of the value of the axial may be determined by adding to in Eq. (109). buckling (1.580 X 10"4), to obtain B,», and solving for 12

LUMPED THERMAL ABSORBERS IN THE REACTOR is

it

necessary to know the effect of an absorbing Many practical coses arise in which object of macroscopic size in the reactor. Typical cases are slugs for the production of radioactive isotopes, experimental apparatus for investigation of radiation effects,

control rods. the rate of absorption of neutrons in the absorber Usually the information desired for given level of average neutron flux in the reactor. Often the effect of the always true that absorber on reactivity also desired. It the absorber has the thermal neutron flux macroscopic thermal absorption cross section Z„, and at some point in the absorber, then the rate of absorption per unit volume at that point Rate of absorption = 2«, neutrons/ (cm1) (sec) (222)

if

is

is

t<

if

is

is

a

is

and

by

J

if

is

is

A a

Absorbers in Infinite Medium

12.1

is

A

2,

§

is

it

is

2.

is

1

2,

2.

1

is a

1,

2,

is

is

The computation of absorbers of simple shapes in infinite diffusion media rela For these cases the absorption rate evaluated in terms of the thermal tively simple. neutron flux in the medium infinitely far from the absorber. In many cases this computation can be used as an approximation for the more difficult case of the absorber in a finite reactor by methods discussed below. To evaluate the absorption we consider the two-region problem, the absorber the infinite diffusion medium being region and the origin of coordi being region nates being at the center of the absorber. uniform source of thermal There in region neutrons of strength Si neutrons/(cm')(sec) and a uniform source of strength S2 neutrons/(cm')(sec) in region Practically, Si and Si are just the slowing-down densities into the thermal-neutron group in region and region respec tively. For the infinite medium, since all neutrons slowed down are eventually the value of the thermal flux infinitely far from absorbed, Sz = $.Zsj, where 0„ the absorber and 2sj It the macroscopic absorption cross section in region small enough that does not produce a significant assumed that the absorber If, then, for example, the slowing-down perturbation in the fast neutron density. properties of the absorbing material are the same as those in region Si = St. is

if

I

c

is

t

is

comparable to an absorption mean free very poor the absorber thickness The assumption path. If the thickness equal to a small fraction of an absorption mean free path, then the assumption of no perturbation of flux by the absorber will lead to a fractional error of the same order of magnitude as «. Many cases of thermally black cylindrical absorbers are treated in Ref. 12. is

a small fraction of a slowing-down length (\/r) in the absorber That is. its smallest dimension If this condition not met, the fast flux distribution must be taken into account by, for a two-group calculation (see Art. 11.22).

material. example,

is

S


if

f

is

is

it

both small and only weakly absorbing, cannot be Unless, however, the absorber that the flux at the absorber position the same as that which would assumed exist at that position the absorber were absent. In general the absorber will depress the thermal flux in its vicinity, and the true rate of absorption can be determined only computation which evaluates the effect of this flux distribution. impractical here, for on the one general treatment of the absorber prohlem hand, diffusion theory inadequate for the treatment of very small (but highly absorbing) absorbers, while on the other hand, the treatment of large absorbers can become very complex the geometry of the absorber-reactor combination lacks Some relatively simple but useful specific eases are treated below. symmetry.

6-106

REACTOR PHYSICS

[Sec. 6

case of frequent occurrence is that in which the absorber is a material of much higher atomic weight than that of region 2. In such a case, Si may be taken to be zero.

Under the conditions outlined above, the thermal-neutron flux in both regions satisfies the equation (the group notation of Sec. 7 is used here)

D,

V'd>.

-

2.0. + 5

w. - «v.

+

S

= 0 0

D.

(223)

where D, and 2, are the diffusion constant and the absorption cross section, respec tively, for thermal neutrons in the appropriate region and x* — 2.//),. For an absorber in the shape of an infinite slab of half thickness T or an infinitely long cylinder of radius J? or of a sphere of radius R, the thermal fluxes ,» in regions 1 and 2, respectively, are given by equations of the form

,i

=

4>.i

- AiY

A\X + +

—

i

(224) (225)

2.,

X

Table 17 gives the values of the constants A\ and At and the functions the three geometrical cases. Table 17. Generated for wjivans (University of Florida) on 2015-09-23 02:51 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Neutrons

Region 1

Type

*,,

Slab Cylinder Sphere

-AiX

+

Region 2

+,-A,r + £

— 2.1

X cosh xu

Y cosh xix — ainh xu

/•(«ir) sinh xir

Ktt(xir) cosh xjr — sinh xir

r

T

A:

.4i

Cylinder Sphere

Y for

Flux Distributions for Absorbers in an Infinite Medium with Uniform Source of Thermal

Slab

and

Ai/^xi

cosh kiT+ (D.i*i/D.,«)

&/Z.,

- Si/S.,

/).jxa/Ci(xiR)

- Si/2.,

(sinh x,R)/K + [C,,(Rxi cosh x,R

- sinh «iD

.1iD.,xl/i(xiR)

/»(«,«) + (»„*i//J.,*,)[Ko(x.ft)//ii(x>K)I/,(x>B)

S./2.,

ainh xiT

/).,x,(cosh x,T

ainh x\T

- sinh xiR)/fl.,R(l

.4tD.,(Rx.

+ x*R)]

cosh xiR

- sinh xiff)

/>.,(! + XiR)(cosh x.R

- sinh xjfi)

The relation for the flux in the absorber having been established from Table 18, the rate of absorption at any point in the absorber can be computed by Eq. (222). If, however, only the total rate of absorption by the absorber is desired, it may be evaluated by taking the derivative of the flux at the absorber surface and computing the density of neutron current into the absorber by Eq. (5):

J

= —D grad d>

ncutron8/(cm,)(sec)

is just /
18.

Infinite slab Half thickness Origin of coordinates

Table

Distributions

T

H

at center

R *(r)

-

StR

2.,

-

cosh *ir

+ «».X.(««fi)

— sinh mr .iff - sinh St

I

(#/«))«-*.*

S.,

+

e-'tr

or

Z.,

- »inh xtT)

(«/K)l(cosh

I

Z.,r

[««,

+

[<«]

K r

5> Z,,

l)
or

*;T

with

+

=

e +

2., KoUjK)

Sj

Z., (.»:

. 1)(cosh

— xtx

Z.i (c*t

^ 3t

Medium

Thermal neutron flux, neutrons/(cm*) (sec)

in an Infinite

2.,

Uniform

J(R)

Current

Source

Neutrons

«.[.«!

«■(•*,

s,

surface,

(#*,/!

Si ft.,)]

[A'o(».ff)//Ci(«IR)H

s,

density into absorber neutrons/(craJ) (sec)

of Thermal

+

*(r)

*(r)

=

-

Absorbers

*(«> -

Black

4-

Sphere Radius ■» Origin of coordinates

around

+

long cylinder Infinitely Radius = at center of coordinates Origin

at center

Geometry of absorber

Flux


I) +

+

2,,| +

+

+

6-108

REACTOR

PHYSICS

i /i

[Sec.

6

is,

however, (boundary) =« 1/e. Table 18 rigorous boundary condition with this condition for the three absorber shapes gives the flux distribution in region The table also gives the current density into the absorber surface. considered. It evident from Table 17 that the effect of the absorber on the flux in medium the diffusion length) and, even falls off rapidly with increasing mr (or r/L, where for black absorbers, becomes small at about three diffusion lengths from the absorber. The infinite-medium case may be used as a reasonable approximation to the case of an absorber in a finite reactor provided the undisturbed flux distribution in the reactor (i.e., the flux distribution before the absorber installed) does not amount to a large flux variation over the region in which the absorber, when installed, will influence the The infinite case often used as a rough approximation flux distribution strongly. used, even when this condition not fulfilled. When the finite approximation common practice to assume that the equivalent uniform source strength Si for the infinite case just equal to the undisturbed slowing-down density at the point is

is

is

is

it

is

is

is

L

2

is

2

more

a

is a

if a

is a

is if

A

1.

is

X

*

2.

(vol. of absorber over which a

computed) o

is

3.

is

the average thermal flux in the undisturbed reactor in the region where computed. The absorber can then be treated, by perturbation theory, as an absorption cross section of magnitude 2„ distributed in the reactor over the volume occupied by the and

absorber. 12.2

Absorbers in Regular Array

A

the case, the relation can be written

Absorption by all other materials (including fuel) Absorption by "lumped" absorbers

^

When this

1

of the other reactor materials.

is

is

/

is

is

is

is

a

a

Occasionally the case occurs in which identical absorbers are arranged in a regular For such or nearly regular array over the reactor or over some region of the reactor. case the absorbers can be treated by the same methods as those used for calculation of the thermal utilization in lumped reactor in Art. 9. First the region occupied divided into cells symmetrical about the absorbers; the approximaby the absorbers " thermal ion infinite in extent. then made that the region occupied by the cells utilization" /„(,, then computed for the absorbers in exactly the same way as the Usually for these cases thermal utilization was calculated for fuel lumps (Art. 9). the total absorption of thermal neutrons by the absorbers small compared with that

t

fat*

if

is

and an equivalent absorption cross section 2e can be assigned to the lumped absorber. The quantity 2, that cross section which, treated as uniformly distributed, will result the same fractional neutron absorption as that of the lumped absorbers. in


A

a

is

a

is

is

is

is

An alternate procedure, in the finite reactor where the absorber to be installed. which makes some allowance for the leakage of thermal neutrons out of the reactor, the undis to evaluate an effective source strength by setting Si =

Sec. 6-2] It

is given

by 2« =

Z„„to, (

-

1

6-109

)

(227)

the total absorption cross section (or, in the case of "lumped" reactors, absorption cross section) of all materials except the absorbers in question. After the effective cross section is determined, the absorbers may be treated as uni formly distributed, with absorption cross section 2„ for computation of both their total absorption and their effect on reactivity. Frequently the region occupied by the array of absorbers does not extend over the located in the core. In such a case, the entire reactor core but is symmetrically properties of the region occupied by the array can be computed as indicated above, and the reactor may be computed as a symmetrical, multiregion problem, using the methods illustrated in Art. 7. Often the use of reflector savings will reduce such a problem to a two-region problem. In the case of absorbers "black" to thermal neutrons the cell problem of most In this frequent occurrence is one which is symmetrical in cylindrical coordinates. case, the problem and its solution are stated as follows: The absorber is a cylinder of radius R\\ the surrounding "cell" has radius Rt. In the surrounding cell, there is a source density of thermal neutrons, assumed to This source is just equal to the be of uniform strength = q neutrons/(cm»)(sec). density into the thermal energy group. The source strength is taken slowing-down to be zero in the absorber. The boundary condition at the absorber surface is is the thermal flux density and c is the extrapolation (\/)(d

the effective

V!4>

-

x'4>

+ 1 = 0

The solution is

*(r)

GlK^xRJIoixr)

=

+

J,(»S,)K,(«r)l +

|

28

where 2, is the macroscopic absorption cross section for thermal neutrons in the material surrounding the absorber and G is a constant whose value is given by G

«[K,(*R5)/.(*«.)

- Ii(xR,)K, (*«,)]?/2.- [K^xRJloixRJ - /,(xK2)Ko(xK,)]

The current density into the absorber is

t

2.

-

1

IAxR^KAxR,)] Ex[KdxR~)I,(xR,) K^xRÛixR,) + hixRJKoixRt)

_

}

where D is the thermal-neutron diffusion constant in the material outside the absorber. The "thermal utilization" fab. of the absorbers is given by /obi =

12.3

2t«,J(R,)

Central Absorber in Bare Reactor

If the absorber is so shaped that the geometry of the absorber-reactor combination exhibits a high degree of symmetry, the problem can often be solved in as much detail as is desired by use of the group methods. The problem of this type which occurs most frequently is that of a central cylindrical absorber passing axially through a cylindrical reactor. The solution for such a system, treated by the two-group approximation, is given in Art. 7.5. The case of a bare reactor is considered. If a

6-110

HEACTOR PHYSICS

[SEC. 6

reflected system is to be computed, the reflector saving can be computed first without the absorber and can be assumed to be unchanged by the presence of the absorber. This assumption is adequate unless the absorber diameter is an appreciable fraction of the reactor diameter. The procedure is to solve the criticality problem for the reactor containing the The rate of absorption in the absorber can then be evaluated, if desired, absorber. from the thermal flux distribution which is obtained in arriving at the solution. The effect of the absorber on reactivity can be found by comparing the criticality condi tion (e.g., critical £•«,) of the reactor containing the absorber with that of the reactor without the absorber. 13

NONCRITICAL REACTORS

If the neutron balance in a reactor is such that exactly one of the neutrons emitted in a typical fission reacts with a fissionable atom to produce a subsequent fission, the reactor is said to be critical. If in a critical reactor there are no extraneous sources of neutrons, the neutron population will remain constant in time. This section deals with the behavior of reactors which are not critical. In general, the neutron popula tion of such a reactor, if not zero, will vary with time (exception — subcritical reactor with source; see below). General Concepts and Definitions


13.1

If the reactor, although not critical, is nearly so, the spatial distribution of neu trons in it will be very nearly the same as in the critical reactor. The general sig nificance of quantities used to specify the characteristics of noncritical assemblies can best be appreciated in terms of such cases, which are also the ones occurring most These quantities! A;,//, k.„ and p are alternative frequently in practical problems. descriptions of the relation between the number of neutrons in a given generation of the chain reaction and the number in the immediately succeeding generation. The effective multiplication factor k,f/ is just the ratio of these two numbers: ,

The

_

excess multiplication

k„

k,n

=

No. neutrons born in (n + l)st generation No. neutrons born in nth generation factor

— 1 =

k„

is neutrons born l)st generation

in (n + [No.

1

J

_ /

No. born in generation/

\nth

\

No. neutrons born in nth generation

(229)

No. extra neutrons produced per generation No. neutrons starting the generation

The reactivity

p is Ktx

(230)

k,/t The value of k,/f in terms of the characteristic reactor quantities

*'"

„<1

(Sec. 5) is

(231)

+ IAB>)

where Rl corresponds to the fundamental mode in the solution of the wave equation V1* +

B*4> = 0

for the reactor with proper boundary conditions. Equation (231) may be regarded of k,n for reactors which can be described by an equation of that type, irrespective of whether or not the reactor is near criticality.

as the definition

t The

extant.

use of these symbols follows that of Weinberg and of Gladstone

and Edlund.

Other usages are


Sec. 6-2]

6-111

by Eq. (229), if an average time between generations of neutrons i«v« average generation time) could be specified (and if the spatial distribution of neutrons in the reactor were independent of time), the time dependence of neutron density n in the reactor would be given by Evidently,

(the

dn

_

krx

U"

dl

n = no exp -—

(232) (

•

•

•)

S

+ *•//* +

+

+

No. neutrons born per sec = 5(1

- k.fl S

1

=

(233)

=

located well inside will hold to within factor of

the source

13.2

Kinetic Behavior

of

a

2

a

if

is

if

the reactor not much subcritical, and However, the core, the above expression for the multiplication or 3.

is

B*

0

W

+

is

if

is

is a

is

In other words, the source production increased by factor 1/(1 — fc,//) by the — k,//) noncritical chain reaction. This factor 1/(1 sometimes called the mul tiplication M of the reactor. Strictly speaking, the source multiplied by this exact factor only the source distributed throughout the reactor in the same way as the fissions are distributed in the critical reactor (i.e., according to the solution of the wave equation

Bare.t Near-critical

Reactor with Delayed Neutrons /3

The emission of delayed neutrons accompanies the decay of a certain small group of fission products. The rate of emission by each member of the group of delayed emitters decreases exponentially with time after fission. There are six species of fission products which emit delayed neutrons in sufficient quantity to have an impor

C, and the rate

of

emission

=

ATfte-*'

0,

(

if

is

is

tant effect on reactor kinetics. The fraction of total fission neutrons emitted by the ith member of the group of delayed emitters designated by the symbol ft, the decay constant of the emitter designated by X,-, and the concentration of the emitter per unit volume of reactor by d. Thus, an instantaneous burst of N fission neutrons = per unit volume of reactor were to occur at the concentration C,- of the ith emitter species resulting from the burst would vary with time as (234)

of delayed neutrons per unit volume by that emitter would is

is

Although the following discussion based on bare reactor concepts, the results are applicable simple reflected reactors premded an appropriate value used for the effective neutron lifetime.

t


is,

Because a fraction of the neutrons are delayed (i.e., emitted in the process of radio active decay of fission products), it is possible to specify an average generation time only if k.x is very small (about 0.0004 or less) and if k.t has been constant for some time. In general, the time behavior of the neutron population is a complicated function of reactivity, the delayed-neutron characteristics, and the prompt neutron lifetime (see Art. 13.2 and Sec. 8-1). If an extraneous source of neutrons, S neutrons/sec, is introduced into a subcritical reactor, the neutron flux in the reactor will, after a time, reach some steady value. In this condition the production of neutrons by the source plus fissions is just balanced by the loss of neutrons by leakage plus absorption. If the reactor is nearly critical, the number of neutrons produced per second by relation (228), directly from the source plus Sk.// from the immediately preceding generation plus S(k.//)1 from the generation before that, etc.; i.e.,

to

6-112

REACTOR

PHYSICS

[Sec.

6

vary as (No. delayed neutrons of ith species emitted)/(sec) (unit volume) = XiCj =

.VX,/3,e~x''

(235)

Values of Xi and 0< are tabulated for thermal-neutron fission of U"' in Tables 1 to 4 of Sec. 8-1 and Table 39 of Sec. 1-1. f The number of prompt fission neutrons produced per unit volume per second in a reactor operating with a thermal-neutron flux density is {k/p)Za(\ — p), and if the concentrations C< of all the delayed emitters arc known, the rate of emission of delayed neutrons per unit volume can be specified

as

V

CiX,-.

If

the

reactor

is

i sufficiently near criticality that the spatial distribution of flux does not differ sub stantially from that which would exist in the critical reactor, the leakage of neutrons can be expressed in terms of the geometrical buckling li* of the critical reactor. % If the assumption is made that the effective energy of the delayed neutrons is the same as that of the prompt ones,§ the slowing-down density in the reactor will be q =

[- 2,0(1

- p)

+

P„(£\,B») ^ CiX,]

(236)


where P*(E„Bi) is the Fourier transform of the infinite slowing-down kernel (sec Art. 5.3). The balance equation which holds for thermal neutrons at every point in the reactor is

-DB'4

- Za*

+

2.0(1

-«

+

£ C.X.] P„(E.,B*)

=

^

= 1

^

(237)

where v is the average velocity of thermal neutrons. Applying the definition of k./i [Eq. (231)] and the relation LJ = Z)/20, Eq. (237) reduces to

[*.„(!

- A - 1), +Pglc*

=

{l+^)vX^

{

(238)

The quantity 1/[(1 + L'B^tiZo] is the effective lifetime of thermal neutrons in the finite reactor (see Art. 13.3) and is designated by the symbol I. To specify completely with time, Eq. (238) must be coupled with the six differential the variation of equations which specify the variation of the concentration d of the six delayedneutron emitters. The complete set of equations becomes

[*.„(i

-« -no+^y — = 7T

—

at

X.Ci

H

* p

c.x..

=/i£ (239)

PiZa4>

It

is quite easy to obtain the solution of Eqs. (239) in the asymptotic case for which the reactor has been in a state of constant positive excess reactivity for an effectively infinite time. For this case and all the C, are increasing exponentially with a con stant period If T, whose value is to be det ermined.

Assuming

= A
iii

t

r

d dt

- OiT t

See also Rpf. 14. For delay«»d-neutron data on other isotopes, sao Hefs. 15 and 16. Table 5 of Sec. 1-1 lists other data contained in this handbook. % Many practical reactor kinetics problems concern reactors which are not far from criticality, for For the more general case, see Uef. 13. which this approximation is valid. $ This assumption should be examined critically for cases in which extreme accuracy is necessary, as, for example, when reactor periods are used for the measurement of reactivity changes. % This result is to be expected on physical grounds; mathematically it results from the < solution of Eqs. (239).


Sec. 6-2]

6-113

in Eqs. (239) yields

substitution

*-[(i/r?+J^-

a-,)*,,-^*,^^—-!

and

(240)

t

recognizing that

&

»

> 0,-,

Eq.

(240) can be reduced to

i

k.i,

Tk.f/

L

(1/D

(241)

+ Xi

This equation, often called the inhour equation, relates the asymptotic or stiihlr reactor It is a simple period to the reactivity [note that (k,// — l)/k,// = p = reactivity). relation which is much used in experimental reactor physics for evaluating reactivity

from the measured reactor period. Figure 20 gives T as a function of p for two values Note that curves for other of the lifetime /. Also plotted is the curve for I = 0.


10-

r^

^

Xs

10'

mr=

nr^ o z o

10

\

>>

\

V =*

o o

V >

e o

ft

{=I0"*SE<

m\

< Id or at

u N.

<= 10"4SE

10"

[ -r

H\

f =C

10"

—

— V

\

,

I0-3 10"

3

4

8

I0"3

3

4

6

8

''•if

Fio.

20.

The Inhour relation (delayed-ncutron data).

10"

^

6-114

REACTOR

PHYSICS

6

[Sec.

values of / can be easily constructed from the latter curve simply by adding the term l/Tk,//. For values of p which are greater than 0, the period becomes very short and the term (\/T) becomes large compared with \(. When X,- can be neglected, Eq. (241) reduces

to

Tk.„ T

Tk.„ =

I

(242)

k
T

(/3

This approximation is usually good for p > + 0.0025). In addition to the single positive value of which will satisfy Eq.

9

8

7

6

5

4

is a

=

0,

constant after assuming some initial value at time

(

X

1

I

p

is

the case in which sum of exponentials:

p

SECONDS

Fig. 21. Time variation of neutron flux (t) after instantaneous application of reactivity — to critical reactor operating at steady flux
(243)

A

needed.

is

/

/

is

p

p

is

For positive, To and .4o are positive and T0 equal to the asymptotic value dis For negative, To and all the cussed above; all the other A'a and 7"s are negative. discussed other 7"s are negative and all the .4's are positive. This general solution The at length in Sec. 8-1 and in texts on reactor theory (e.g., Glasstone and Edlund). results for a particular value of (10~4 sec) are plotted in Fig. 21. involves considerable Since the general solution of Eqs. (239) for any particular useful to find approximations which can be used when only rough values are labor, it


(n is

a

(241) (for positive p), there are n negative values which will also satisfy the equation the number of delayed-neutron groups). Thus the complete solution of Eqs. (239), for

very crude approximation

to

the

solution

for positive instantaneous

Sec. 6-2]

CALCULATIONS

REACTOR

6-115

can be made in the following way: The problem is assumed to that of determining the variation of flux in a reactor which has been operating for an effectively infinite time at constant flux t> prior to the time when the effective increased instantaneously from the value to the value multiplication constant maintained at that value for all time later than to. k,u > and Just prior to the reactivity increase delayed neutrons were being formed at some was ij>oP. steady rate such that their direct (unmultiplied) contribution to the flux The value of the -prompt effective multiplication constant was (jfc,//),m>ropt = 1-/3. The prompt multiplication (Afprompt) was 1/[1 — (fc,//) prompt] = 1/0. Thus the flux changes in reactivity

1

is

1

is

to,

be

20

10

0

30

SECONDS

1

Time variation of neutron flux (t) after instantaneous = to critical reactor operating at steady flux 4>„.

o

10-1 sec.

of the delayed-neutron

is

can be considered to arise from the prompt multiplication This concept source. illustrated by the identity

application of negative X

22.

reactivity

I

Fio.

p

flux due directly

-(

)(pTp; Vmultiphcation/ )-+*\-*

/

p

to delayed sources/

is

For a short time after the effective multiplication constant increased to the new value k,n, the flux due directly to delayed sources will remain nearly at the old value
p)

- *,//(! -

uP

o(3

=

p)

i

Pk,n

- k„

a

1

1

- k,„(\ -

Consequently, the total flux will be expected to rise rapidly to

1


o

new value <£i, given by (244)

6-116

REACTOR PHYSICS

[SEC. 6

For smaller values of does not apply if k„ > 0. to the kinetic behavior can be made by assuming that the flux rises instantaneously to the value given by Eq. (244) and thereafter follows the asymptotic variation, as given by Eq. (241). Such a curve is drawn in Fig. 21 for comparison with the correct curve. The kinetic behavior of the neutron flux for negative values of reactivity is not strongly dependent on / after the first few seconds. Thus the curves for negative /> in Fig. 22 can be used for other values of I except during the period of rapid flux decrease. Quite evidently the approximation k.x a rough approximation

The Effective Neutron Lifetime

13.3

In the development of Eq. (238) the time required for neutrons to slow down was neglected; consequently the effective lifetime I which appears in the equation is really the effective lifetime of thermal neutrons In,. The physical interpretation of the expression for la, is as follows: The number of neutrons absorbed per unit volume per second, if the steady neutron density is n, is = nvXa

Absorption/(cm*)(sec)

The

loss of thermal neutrons by leakage is [Eq. (102)]

No. thermal neutrons leaking from


Effective thermal-neutron

=

lifetime

1

cm*/sec = nvXa(L*B')

—

(loss of neutrons) /(cm5) (sec) n

1

nvXa(l + L*B')

t>2,,(l

-f-L'B1)

(245)

The true effective neutron lifetime is made up of the effective thermal lifetime plus the effective fast lifetime. Equation (254) is the correct expression for the effective lifetime in the two-group approximation. If the reactor is bare and if its properties are uniform, Eq. (254) reduces to ,

_

1

*»*

+

*/

1

d (252),

♦/_*+**

.*

and

2, + D,B*

_

If the small difference between k and k, is neglected and the reactor is near enough criticality that the characteristic equation (87) may be assumed to hold, the equation for / becomes

*

/ =

».2„(1

+

L.W)

I

:

»/2o/(l + L/»fl»)

(246)

where v, is the average velocity of the thermal neutrons, vf is the average velocity of the fast neutrons, 2„, is the macroscopic thermal absorption cross section, and 2„/ is the macroscopic "slowing-down cross section." If the reactor is reflected by a material having a thermal-neutron lifetime much different from that of the core, the effective lifetime for the reactor may be significantly different from that for the equivalent bare reactor. Equation (254) may be used to determine the effective lifetime for this case. Since, however, the integration of the flux and adjoint functions over the reactor volume may be quite tedious, an alternate method is sometimes useful. The effective fast lifetime, which is usually a small fraction of the total lifetime, is taken to be that of the equivalent bare reactor, as given by the last term of Eq. (246), and the effective thermal lifetime is computed by a scheme devised by R. P. Feynrnan, which proceeds as follows: It is assumed that the criticality condition for the reactor has already been determined by the methods of


Sec. 6-2]

6-117

A small amount of l/v absorber is then added Art. 7 or 8 or by some other method. The average macroscopic uniformly throughout the reactor (core plus reflector). The poisoned absorption cross section due to this added poison is designated 2op. reactor is then solved by some convenient method (Art. 7 or 8), and its effective The quantity &,// will, of course, be multiplication constant (fc«//) is determined. Since the prompt neutron lifetime I is to bo determined, it is assumed less than unity. Then, by Eq. (232), the rate of change of flux that all fission neutrons are prompt. is at any point in the poisoned reactor where the instantaneous value of flux is d4>

dt

"_

-

4>(k„,

1)

I

Recalling, however, that the reactor would be critical (and flux constant) except for the presence of the added poison, one concludes that the rate of decrease of neutron density is just equal to the rate of absorption by the added poison and the rate of change of flux is . dn „ «,

d dt

dt

Equating the two rates of change of flux above yields

*(*.//

-

1)


I

= — V2ap

I

k,ff

Or, more generally, if the uniform addition of thermal-neutron absorber of average macroscopic cross section 2„p results in a reactivity change Afc,//, the prompt thermalneutron lifetime /,» is Im

- - A|^

(247)

Note that the above reasoning involves the tacit assumption that the addition of the This thermal absorber does not change the flux distribution to an important extent. can be true only if the amount of absorber added is quite small. 14

PERTURBATION RELATIONS

It is often useful to evaluate the effect on reactivity of small changes in the proper ties of an initially critical reactor. Perturbation theories have been developed which give approximate evaluations of such small changes, whether the changes be localized in space or distributed over the entire reactor. Perturbation treatments can be Only specific based on any of the group formulations of the reactor equation. For general treatments and formulas from the simpler treatments are given here. for the theory of the method other sources should be consulted (e.g., Rcfs. 7, 17, and

18). 14.1

First-order, Two-group

Perturbation

Formulas

The first-order, two-group perturbation theory was first published by L. W. Nordbeim (CP-2824). This form of the theory is the most widely used in thermal reactor calculations. In the development of the formulas certain terms involving second orders of small quantities are neglected, but the major assumption is that the perturba tion does not change significantly the flux distribution in the reactor. This condi

tion can generally In deriving the

be used as a criterion of the applicability of the method. expression which evaluates the perturbation in terms of reactivity, the period of the perturbed reactor is first determined formally, and the period is Thus the then related to reactivity by means of the inhour equation (Eq. (241)]. starting point of the development is the time-dependent, two-group reactor equations

6-118

REACTOR PHYSICS

- 2.0.

4-

+ Jt„2.0. +

y

D. V'*, V»0/

- 2/0/

dt

-

C.Ki =

(248)

+

ft

1

k,

=

.

i

factor

>

C{,<

1

the prompt infinite multiplication

-

(

is

where k,

ft/lyr.*

k

=

at

\

^

i

D,

- r,- ^

2/0/

^-'

(the notation of Art. 7.1 is used here)

[SEC. 6

is

It assumed that the solutions for the fluxes are separable into space-dependent and time-dependent terms, the latter being exponentials of reciprocal period v. Equations (248) can then be written 2.0. + 2/0/

- 2/0/

=

+ £,2.0. =

- 0. v

\p- +

(249) fcp2.

0.

-——)

1

D, V'0/

-

)

D. V*0,

D, V'0/*

- 2/0/*

= v

(^

-(-

fcy2.;

0/* — ^—

)

2.0.* + fc„2.0/*

(250)

t

-

+

D, V'0.*

are defined by the equations

2/0.*

=

rî-

No discussion of the general significance and properties of the adjoint functions will will be taken that they are simply functions useful in the It can be shown that identical values of satisfy evaluation of perturbation effects. Eqs. (249) and (250) and that both equations define the same criticality condition = 0). The procedure for determining the effects of small perturbations on the as follows: critical reactor Write out the two-group equations for the fast and slow fluxes (0/ and 0.) in the critical reactor:

D. V»0. Df V»0/

- 2.0. + 2/0/ - 2/0/ + *2.0.

-=

0

1.

is

(»

v

be given here; the approach

=0

D. V'0.*

D,

V«0/*

- 2.0.*

- 2/0/* ++fc2.0/* 2,0.*

=

(0/* and 0.*) in the same

0

and the equations for the fast and slow adjoint functions critical reactor:

=0

Solve the flux equations (251) for the critical values of the buckling B* by the or and write out the solutions for the fast and slow fluxes as methods of Arts. indicated in those articles. 3. Using the critical value of the buckling determined in above, determine also Note that Eqs. (252) the solutions for the fast and slow adjoint functions 0/* and 0,*. are identical in form with Eqs. (251), the only difference being the changes in coeffi cients of the final terms. The methods of solution for 0/* and 0,* are therefore the same as those used in the solutions for 0/ and 0,. 4. The values for the fluxes and adjoint functions having been obtained as func tions of position over the entire volume of the reactor (core plus reflector), the effect on A-,// of local changes in any of the reactor properties can be evaluated in terms of these functions. The total effect on k,// can be evaluated by integrating all the The expression for the local perturbations over the entire volume of the reactor. 2

8,

7

2.


The adjoint functions 0/* and 0.*


Sec. 6-2] total

fractional change in k.//, &k,///k,//,

*!» „ fc,//

1 ,

/

/ ycore volume

„

kPt.Mf*

-

+

.

dv

{I

6-1

19

is

I

J reactor

frfr*) i2,

[Mf*S(kX.)

volume

+ V*. V*.* SD. +

- *.*.*

V*/ V*/*

«2,

(253)

SD,]dv}

The quantity S(A-2.) in the above equation may be written in either of two ways:

- «(ij2„)

S(kZ.)

Ik +

a(*X.) = 2,

or

=

+

2«

i? aZjs

k «2.

Note that since each term in both numerator and denominator of Eq. (253) contains product of a flux and an adjoint function, the absolute normalizations of both the flux and the adjoint are immaterial. The proper relationships must, of course, be maintained between and *, and between *.* and a single

/*.

The Neutron Lifetime

14.2

theory gives the following expression

First-order, two-group perturbation

for the

neutron lifetime I* in the finite reactor:

,.L

[(1/9.)*.*.* + (1/8/)*/*/*! dV

volume

/

/core volume

(254)

kp2..f* dV

where v, is the average velocity of a typical thermal neutron and vj is the average The quantity 2, velocity of a typical neutron during its lifetime as a fast neutron. of course, the average macroscopic absorption cross section for thermal neutrons. is,

Nonuniform

Perturbations

of

a

14.3

Homogeneous Bare Reactor

V** + B2*

=

0

The spatial distributions of both the fluxes and the adjoint functions in a homo geneous bare reactor are given by the solutions of the wave equation

=

*,*(r)

=

/?i£>'

+

2/

If

the

*,*(r),

*.0)

(255)

^

and

*/(r)

S-

4

is

,(r)

5.

Solutions of the equation are given for the usual reactor shapes in Table solution from column of that table taken as the solution for both and the solutions for 4>/(r) and */*(r) are simply

j^t5f *,*(r)

(256)

14.4

Uniformly Distributed

Perturbation

of

a

With these expressions for the fluxes and adjoints the effects of local or distributed perturbations can be evaluated by Eq. (253). Homogeneous Bare Reactor

For the case of perturbations distributed uniformly in a homogeneous Eq. (253) reduces to (D,/Z,)B*

«2.

IT,

+ (Z>//Z,)B* 2/ SD. (D./T.)B*

2.

+ (Z>./2,)BJ D,

1

(i>./2.)£»

(D,/?,)B* +

_

bare reactor,

1

ofrZn) t;2„

1

"_

1

*.//

1

»fc.//

+


reactor

(D,/2f)B*

SDf

D,

(257)

6-120

PHYSICS

REACTOR

The expression equation

[Sec.

can, of course, he obtained also by differentiation

k.„

= [1

+ (D./2.)B»J[1

6

of the characteristic (258)

+ (D,/Z,)B>]

Frequently, for uniform perturbations, it is more convenient to use the expression derived from the more usual statement of the characteristic equation: k">

=

(259)

The perturbation expression in terms of these quantities is

tk.ff

Sk

*:.//

k

_

LHP

I

6L*

+ L»B> A1

1

r/i' t +- rfl»

Vl +

1

+

tBV

Bs

(2(10)


The above expressions can be used to evaluate the effects of uniform changes of core properties in reflected reactors provided these changes do not have a significant effect on reflector savings. It is usually safe to assume that changes in k alone will not produce important changes in reflector savings. Changes in B1 will not cause large changes in reflector savings provided the reactor is already fairly large (say Changes in L* or r will, in general, have important effect* km < 1.2 for criticality). on reflector savings if they are both important in determining neutron leakage. If either L* or r is small compared with (L* + t), small changes in it will not have an important effect on reflector savings. 14.5

Perturbations

in Lumped Reactors

The effects of perturbations in lumped reactors can be evaluated by the methods described in the foregoing parts of this section provided effective values for the per turbations over a typical cell of the reactor are used. These effective values may be determined by methods analogous to those described in Art. 9. Often in lumped reactors the effects of perturbations on neutron leakage are not important, and the effect on k.// can be evaluated in terms of changes in k alone, where k is expressed in terms of the four-factor formula k

- vpf

When effects on leakage are unimportant, Sk./f = *^=i? + k t) k.tt The effects of the perturbation on ij, in Sec. 9.

e, p, and

«e (

+ ?P + */

/ are

p

f

(261)

determined by the methods applied

REFERENCES Philip R.: Neutron Distribution in Elementary Diffusion Theory, I and II, Nucleonics, vol. 4, February and March, 1949. Weinberg, Alvin M., and L. C. Noderer: Diffusion and Slowing down of Neutrons, in "Theory of Neutron Chain Reactors," vol. 1, pp. 1-43, OUNL-CF-51-5, May 15, 1951. Placzek, G.: The Concept of Albedo in Elementary Diffusion Theory, in C. Goodmau of Nuclear Power," vol. 2, chap. 7, Addison(ed.), "The Science and Engineering Wesley Publishing Company, Reading, Mass., 1949. Placzek, G.: On the Theory of the Slowing down of Neutrons in Heavy Substances, Phye. Rev., 69: 423 (May 1 and 15, 1946). Marshak, Robert E. : Theory of the Slowing down of Neutrons by Elastic Collision with Atomic Nuclei, Reve. Mod. Phys., 19(3): 185-238 (July, 1947). Wilkins, Jr., J. Ernest, Robert L. Hellens, and Paul Zweifel: Status of Experimental and Theoretical Information on Neutron Slowing down Distributions in Hydrogenous Media, Geneva Conf. Rept. A/Conf.8/P/597, June 30, 1955.

1. Wallace, 2. 3.

4. 5. 6.

REACTOR

Sec. 6-2]

CALCULATIONS

6-121

7. Gladstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," chap. N.J., 1952. 6, D. Van Nostrand Company, Inc., Princeton, 8. Hughes, D. J., and A. Harvey: "Neutron Cross Sections," McGraw-Hill Book Com pany, Inc., 1955. Also published as BNL-325 by Superintendent of Documents,

J.

Washington.

J.

E. Wilkins, Jr.: "Effect 9. Wigner, E. P., and on the Velocity Distribution of Neutrons with

of the Temperature

of the

Numerical Calculations for H

Moderator as Moder

ator," AECD-2275, Sept. 14. 1944. E. Richard: A Survey of Neutron Thermalization Theory, Geneva Conf. Paper A/Conf.8/P/611, June 30, 1955. Spinrad, B. I., and Dieter Kurath: "Computation Forms for Solution of Critical Problems by Two-group Diffusion Theory," ANL-4352. Garabedian, H. L.: "Control Rod Theory for a Cylindrical Reactor," AECD-3666,

10. Cohen, 11. 12.

August, 1950. 13. Weinberg, A.

M., and H. C. Schweinler: Pkyi. Rev., 74 : 851 (1948). D. J., J. Dabbs, A. Cahn, and D. Hall: Delayed Neutrons from Fission of Um, Phys. Rev., 73: 111-124 (1948). Keepin, G. R., and T. F. Winnett: Delayed Neutrons, Geneva Conf. Paper A/Conf.8/P/831, June 29, 1955. Brunson, G. S., E. N. Pettitt, and R. D. McCurty: " Measurement of Delayed Neutron Yields in Pu, U»3, U"\ and Th Relative to Yield in U»»," ANL-5480, August, 1955. "The Reactor Handbook: Physics," McGraw-Hill Book Company, Inc., 1955. Also published as AECD-3645 by U.S. Atomic Energy Commission, Technical Information

14. Hughes, 15. 16. 17.

Service.

H. L. : "Theory WAPD-19, September, 1950.


18. Garabedian,

of Homogeneous

Control of

a

Cylindrical

Reactor,"


SECTION

7

RADIATION AND RADIOLOGICAL PROTECTION BY JOHN M. WEST, B.S., M.S., Vice President, General Nuclear Engineering Corpora formerly Associate National Laboratory.

tion,

Director,

Reactor

Engineering

KARL ZIEGLER MORGAN, A.B., M.A., Ph.D., Director Oak Ridge National

Division,

Argonne

of Health Physics Division,

Laboratory.

EVERITT p. BLIZARD, B.A., M.A., Director, Applied Nuclear Physics Division, Oak Ridge National Laboratory. C. GOERTZ, B.S., Director, Argonne National Laboratory.


RAYMOND

R. FERGUSON, Laboratory.

KENNETH WILLIAM

A.B.,

Remote Control

Engineering

M.S., Associate Physicist,

Division,

Argonne National

B. DOE, B.S., Associate Chemical Engineer, Argonne National Labora

tory.

CONTENTS CALCULATION OF NUCLEAR RADIATION

7-1

7-3

BY JOHN M. WEST 1 2 3 4

Nature

of Radioactivity

Transmutation by Neutron* Fission Products Reactor Coolant Systems References

BY EVERITT P. BLIZARD PAGE

7-2 7-7

7-14 7-17 7-21

1 2 3 4 5 6

7-2 HEALTH PHYSICS BY KARL ZIEGLER MORGAN Ionising Radiation Radiation Monitoring Decontamination Radioactive Effluents and Wastes Shipping and Handling of Radioactive Materials Handling of Fuel, by Thomas J. 6 The Safe Burnett Toxicity, by Myron F. Fair. . . 7 Beryllium 1 2 3 4 5

References

NUCLEAR RADIATION SHIELDING

7-22

7 Ray Attenuation Sources of t Rays and

7-50

X Rays

Neutron Attenuation Principles Neutron Sources

Geometry Shield Materials References

7-4

7-45 7-48

PAGE

MECHANICAL HANDLING RADIOACTIVE MATERIALS

7-fil

7-72 7-78 7-90 7-93 7-109 7-1 15

OF

BY RAYMOND C. GOERTZ, KENNETH R. FERGUSON, and WILLIAM B. DOE

7-55

1 General Solutions to Handling Problems 7-117 2 Typical Equipment for Handling Radio active Materials 7-125 7-139 References

7-55 7-57 7-57

7-1

7-1

CALCULATION OF NUCLEAR RADIATION Y

15

John M. West 1

NATURE OF RADIOACTIVITY


1.1

Radioactivity

Nuclei which spontaneously undergo atomic disintegration by emitting particles or electromagnetic radiation are said to be radioactive. 1.11 Natural Radioactivity. Unstable nuclei which exist in nature are said to be naturally radioactive. In addition to certain individual radioactive isotopes such The isotope of uranium having a mass as K40 there are three major decay chains. of 238 (U238) is the first member of a chain which, after emitting eight a particles Similar chains and six /S particles, terminates as a stable isotope of lead (Pb206). The three begin with Um and thorium (Th232) and terminate as Pb206 and Pb208. natural radioactive chains are referred to as the uranium-radium, uranium-actinium, and thorium families,1* respectively. 1.12 Induced Radioactivity. When a nucleus is made to react with a particle or electromagnetic radiation to form an unstable nucleus which did not previously exist in nature, the new atom is said to be artificially radioactive. 1.13 In their approach to a stable state, radioactive atoms Decay Mechanisms. may emit charged particles of various types (a particles, 0 particles, positrons), uncharged particles (neutrons), or electromagnetic radiation (y rays, X rays). Alpha Decay. The o particle is identical with the nucleus of a helium atom. The emission of an a particle creates a new nucleus with an atomic number reduced by 2 and an atomic weight decreased by four mass units. Alpha decay is common in the three natural radioactive chains. The mechanism by which U238 decays to Th2" follows:* ,2J8

"0.05

-4 25Mev

Mev

It should be noted that the a ray may bo emitted in either of two ways. The 4.20-Mev a is emitted followed by a 0.05-Mev y ray about 22 per cent of the time. In the remaining 78 per cent of the transitions, the full energy difference between U238 and Th23J is carried away by the 4.25-Mev particle. Beta Decay. In fi decay a negative electron is emitted from the nucleus of an atom. Since atomic number is defined by the net positive charge of the nucleus, ejection of a 0 ray increases the atomic number by unity. Gamma rays are usually emitted also to carry away excess energy of the new nucleus. An example follows :J * Superscript numbers t Sec Ref. 2, p. 007. t See Ref. 2, p. 482.

refer to References

at end of subsection,

7-2

Sec. 7-1]

CALCULATION OF NUCLEAR RADIATION izMg27

7-3

2.64 Mev 1.65 Mev

0.84 Mev

ISAI"

0

In a large number of Mg" disintegrations, 80 per cent of the cases will involve a 8~ In the remaining maximum energy 1.80 Mev followed by a 0.84-Mov y ray. cases (20 per cent) a 8 ray of 0.79-Mev maximum energy is followed by two y rays in cascade of energies 1.01 and 0.84 Mev. The energy listed for 8 particles is always the maximum energy unless specifically The 8-ray energy spectrum curve for a given disintegration indicated otherwise. The average 8 energy is about 40 per cent process approximates a probability curve. of the maximum 8 energy. The difference between the energy of a given 8 ray and the maximum energy is carried away in the form of a low-reaction type of radiation failed the neutrino. Positron Decay (8+). The positron is a particle whose mass is the same as that of an electron and whose charge is equal in magnitude but opposite in sign. Positron emission represents a mechanism whereby a nucleus can decrease its atomic number by unity : Gamma-ray Emission. The y ray is electromagnetic radiation originating in the nucleus, as distinguished from X rays, which usually are less energetic and which arise from energy adjustments as electrons move between orbital shells outside the nucleus. Gamma rays are usually emitted instantaneously when a neutron is cap tured in a nucleus (n,y reaction). Gamma rays are frequently emitted following In a few cases (isomeric transitions), ejection of a particle from a radioactive nucleus. radioactive nuclei emit y rays alone. Neutron Decay (n). A few cases are known in which unstable atoms emit neutrons. The fission products Kr" and Xe187 and the isotope ()" are examples. An unstable atom may decay by capturing an orbital Electron Capture (EC). This electron in the nucleus, resulting in a decrease of atomic number by unity. capture produces a vacancy in the orbital shell which is rilled by an electron moving from an outer shell, giving rise to X rays characteristic of the daughter nucleus. The nucleus may be in an excited state following orbital electron capture, in which case the ground state is reached by emitting a y ray. processes may produce an unstable Isomeric Transition (IT). Transmutation


ray of

Isotope which has one or more metastable states in addition to its ground state. The ground and metastable states are called isomers. The nuclide may, in its ground If the ground state is unstable, an excited isomer state, be either stable or unstable. may decay to its daughter directly or by first emitting a y ray (isomeric transition) to become the unexcited isomer which then decays. Each of the two isomers decays In the example which follows, the metastable with its own characteristic half-life. state of Co*0 decays with a half-life of 10 min, sometimes going directly to Ni*° by emitting a 8 ray and a y ray but much more frequently (99.7 per cent) emitting a y ray to reach the ground state of Co*0. The latter decays with a half-life of 5.3 years by emitting a weak 8 ray and two y rays in cascade.*

1.33 Mev Ni60 • See Ref. 2. p. 490.

0

7-4

RADIATION AND RADIOLOGICAL PROTECTION

[SEC. 7

The energy of transition from an excited state to the Internal Conversion (IC). ground state may, instead of appearing as a 7 ray, be imparted to an orbital electron That this process ejected from the atom. This process is called internal conversion. occurs is evident from the ejected electrons of discrete energy levels as well as the characteristic X rays resulting from refilling of the electron shells. It should be noted that neither y emission nor ejection of electrons by internal conversion changes the identity of the atom. These processes are secondary effects of particle excitation of the nucleus. Radioactive Decay

1.2

Radioactive decay may take a variety of forms. In the simplest case an unstable Frequently, however, the nucleus decays to a stable nucleus in a single process. stable state is not reached until two or more transitions, each with a characteristic In some isomeric transitions two different half-lives char half-life, have taken place. acterize a single nuclear change. 1.21 Single-step Decay. During a given interval of time there is a definite prob ability that a particular radioactive atom will disintegrate. Let this probability If the number of parent atoms A is very large, the per second be designated by X. number of atoms disintegrating per second is \A and the rate of decrease in the number of parent atoms is given by dA/dt = — XA, which integrates to


A = A„e-*' where A0 is the number of parent atoms at time t — 0. 1.22 Half-life. The number of atoms living for a length of time between time I and I + dt is

\A

t

and decaying

dt = XAoe-x' dt

The mean lifetime is therefore given by \Aoe~Kt dt

A0

_

1 X

mean life is seen to be the time required for A to decrease by a factor 1/e. The half-life is obtained by setting N/No = \§ = e~x< and taking the logarithm both sides. Then

This

x„ X7V.

= ,In o 2

or

X

Ta Chains. If

XX

1" 2 0.693 rr -= 7>, =

or

= 0.693

of

(mean life)

1.23 nucleus A decays to nucleus B which is also radioactive, Decay the abundance of B as a function of time can be expressed as

5 dt

B

and

=

Xu4° X2 — Xi

= X,A

(e'^'

-

- UB e~x><) +

B^-^

The last term accounts for any of isotope B which may have been present at t = 0. Note that if the daughter product B is stable, then Xj = 0. Nucleus B may decay into radioactive nucleus C, in which case the abundance of C is given by

-f

C = C»e-X'' .

.

Bo

/

— —- (e-^

A3

Xl

A2

X2

-

e~x'')

_x .

.

Xl

Xs

* .

.

Xi

Xj

calculation of nuclear radiation

Similarly if C decays to D, the abundance of the latter

-

Doe-x<'

+ Bo .

.

+

Co

^ (— V\| —

(

^—

Xj

X2 — X. Xi

-x

Xa

.

Xl

\t

x»

c-x,<

X3 X« — X|

— e"^«)/

—^

— X2 — X< X3 Xi ^3

^s

^1

Xi — X2 X3 — Xj

*1_ —

+

e"*'<

— X3

1

Xi Xj — Xi X4 — X| Xl — X3

—

—-*

e"*" +

Xi

Xi —

—

_|_

\Xj

x«

X-.' X,

by

- e"x'')

— ^— (e-^ —

x«

H

D

is given

7-5

\\

xI_

—

—1J

Xi

x'

x>

Xi — Xi Xj — X4 Xj — X4

e-x4A

/

Sec. 7-1]

Ionization

1.3

0

A

is

(J

0

a

fj

(3

is

0

is

is

0

2

where

R

R

„

E d

0.543£ _

- 0.16

= range, cm = energy, Mev = density, in g/cm! is

is

is

approximately true for all kinds of absorbing material, Although the Feather rule the accuracy somewhat better when the absorber aluminum. The rate of energy loss, and hence the specific ionization, by positrons in passing very similar to that of rays. through matter Ionization by a Rays. Alpha particles passing near neutral atoms exert 1.32 strong attractive forces which strip off orbital electrons and leave the atoms as posi tively charged ions. The specific ionization much higher and the range cor The specific ionization respondingly shorter for a particles than for particles. increases as the energy of the a ray reduced. The range of a 4-Mev a particle only about 2.5 cm in air, as compared with a range of about 1,600 cm for particle of the same energy. 1.33 Ionization by Neutrons. Unlike charged particles, neutrons cannot ionize atoms through electric or magnetic forces. Nuclei are knocked forward by neutrons as they traverse matter. These charged particles produce dense local ionization. Molecules can also be disrupted. is

a p

is

is

0

is

0

is


is

is

the process by which normally neutral particles become electrically Ionization Atoms or groups of atoms which have lost valence electrons are positive charged. ions, while negative ions are free electrons or particles which have acquired additional The number of ion pairs produced per centimeter of path termed the electrons. Values of specific ionization vary with the kind of radiation, specific ionization. energy of radiation, and nature of the material being traversed. Ions may be produced in many ways. The ions which are normally present in the atmosphere to the extent of about one thousand per cubic centimeter are due to cosmic rays and to natural radioactivity. Both particles and electromagnetic radia tion can produce ions. Ionization by Rays and Positrons. 1.31 particle passing near an atom exerts a strong electrostatic force on orbital electrons, which may separate an electron from the atom, thereby producing an ion pair. The average energy loss per ion pair about 34 ev.3 Thus, 1-Mev formed in air particle will produce about 30,000 ion pairs in air before being stopped. particle increases as the energy of the The specific ionization by a particle In air the number of ion pairs per centimeter increases from about 25 at degraded. Mev to 200 at 0.04 Mev. The linear ranges of particles of the same energy vary considerably because of the deflections resulting from electrostatic interaction with orbital electrons. The fairly definite, however, and represents the basis for a common maximum range Feather's rule states experimental method of determining maximum 0-ray energy. that the maximum range of radiation given by

7-6


[SEC.

7

In traversing matter, neutrons can be inelastically scattered, giving rise to y rays, which, in turn, produce ionization. 1.34 Ionization by y and X Rays. Electromagnetic radiation can produce ionization by three mechanisms, namely, photoelectric effect, Compton effect, and The rate of formation of ions by these three processes depends upon pair formation. the energy of the photon as well as the nature of the material being traversed by the Figure 1 shows the trends for low-atomic-number materials (air). photons. The absorption of photon energy by the photoelectric and Compton effects decreases with an increase in photon energy for any given material being traversed. The absorption of energy by pair formation increases with photon energy. These opposite trends determine a minimum absorption value for each element. Pair formation also increases with atomic number, and for high-energy photons in heavy elements this is the dominant effect.


ABSORPTION

.01

02

COEFFICIENT p(CM'')

.04 .06

Fig.

.1

.2

4

.6 .8 I

2

4

6

8 10

ENERGY, MEV 1.

Absorption of y radiation in air.

In the photoelectric effect all the energy of the y ray is given to an orbital electron. The electron is ejected from the nucleus with a kinetic energy equal to the energy

of the photon less the binding energy of the electron. Since the binding energy of the electron is usually very small compared with the energy of a y ray, essentially all the y energy shows up as kinetic energy of the electron. The elastic collision between a photon and a free or loosely bound electron in which only a part of the energy of the photon is given up is called the Compton effect. For a given scattering angle of the photon there are a corresponding energy and direction for the recoil electron. In pair formation a photon having an energy in excess of 1 Mev can, as it passes through matter, convert its energy to mass to form an electron and a positron. Any excess energy over that required to produce the masses of the two particles (1.02 Mev) appears as kinetic energies of the particles. Pair formation is much more probable in heavy materials than in light ones. 1.35 Measurement of Radiation. Charged particles such as 3 particles, a par ticles, and positrons are detected and measured by their direct ionization effects. Gamma rays and neutrons are detected by the secondary charged particles which they produce. In air a 1-Mcv charged particle will produce about 30,000 ion pairs,


Sec. 7-1]

7-7

number per centimeter of path varying from 25 for 2-Mev /3 rays and positrons values for lower energy 0 particles and positrons and very much higher These high values of specific ionization make measurement of values for a particles. radiation in the range of intensities of interest from the health protection standpoint the

to higher

relatively easy. A curie is the amount of a given radioactive substance which undergoes 3.7 X 1010 radioactive transformations per second. It should be noted that the curie is not a measure of specific ionization or biological effectiveness, inasmuch as 1 curie of a given isotope may be many times more dangerous than 1 curie of another substance. A roentgen is the amount of electromagnetic radiation which will produce one electrostatic unit of positive or negative ions per cubic centimeter of air under standard conditions. The intensity of y rays, measured in roentgens (abbreviated r), is deter mined in ion chambers or electrometers which collect the ions in a definite volume of air or other suitably calibrated gas. The intensity from a y source of known strength can be calculated by reference to Fig. 1 and remembering that about 34 ev of energy is required to produce one ion pair. For example, suppose it is desired to calculate the y intensity at a distance of 100 cm from a 1-curie point source of 1-Mev y rays. The number of y rays per square centimeter on the surface of a sphere 100 cm in radius with the point source at its center is


*

-

^flux

-

mmk*

- 2-9

x

10V8ec

Reference to Fig. 1 shows that the fraction of energy absorbed per centimeter of path (ft) is 3.5 X 10~'. Now since 34 ev is required per ion pair and since the charge of an ion (electron) is 4.8 X 10~10 esu, the number of electrostatic units produced per

cubic centimeter per second is

j _

4>Ene

34

(energy /photon) (absorption coefficient) (charge of electron) = (7 flux) energy required to produce ion pair in air

_

(2.9

= 0.50

X

10') (1

X

10') (3.5

X 10-')

(4.8

X

10"'°)

34

_ j

4

lQ_t r/ge(J

r/hr 2

TRANSMUTATION BY NEUTRONS

All elements except helium capture neutrons. The new atom resulting from capture of a neutron may be either stable or radioactive. If the target material contains more than one isotope, each of these interacts separately, with its own individual characteristics. Thus, several new isotopes may be produced, some of which are radioactive and others stable. 2.1

Cross Sections

For a particular nuclear reaction

each atom can be imagined to have a sphere of associated with it such that if a neutron penetrates the sphere an interaction with the atom takes place. This sphere offers the same cross-sectional area in every The magnitude of the area is called the cross section of the atom and is direction. usually represented by the symbol a. The cross section varies greatly with the nuclear reaction, the neutron energy, and other variables of the system. Consider a beam which contains n neutrons per unit volume moving with a velocity If v through a slab of material of cross-sectional dimensions x and 1/ and depth 2. the slab contains jV0 atoms per unit volume, the fractional area blocked out to neu trons is (Noiyz/xy)o. The total number of neutrons entering the slab per unit time is nvxy, so the number of interactions per unit time is influence

nvxy

—

xy

a

-

nvNoxyza

where Ff Ls the total number of atoms in the slab.

= nvNo

0.015 7.5 92.5 100 81.2

H" Li' Li' Be« B"

1

1.

0

75.4 75.4 24.6 0.37 99.6

Cl» CI" CI" A" A"

do ±

±

0

2

6 ± ±

2

±

3

1

I

2.

1 n

6

±

± 0.03 0.9

4

±

13

±

12 0.14

Sc" Ti" V" Cr" Mn"

—

#-(0.17). 7(none) #(0.7), 7(none) #-(4.81. 2.77. 1.11). 7(2.15. E.C.. 7(none) 1.2). 7(1.37) #(2.5.

#-(1.2. 0.36), 7(0.89. 1.12) #(2.2. I.9).7(0.32) #(2.6). 7(1.5) E.C., 7(0.32) #~(2.BI, 1.04, 0.65). 7(0.82,

85 days min 3.9 min 26. days

1.77. 2.06)

1.60)

6.1. 7.1) 6.1. 7.1)

#-(1.3). 7(1-46). F..C. (3.58. 2.04). 7(1.51) #(0.25). 7(nonc) #-(~2.7). 7(hard) I.T., 7(0.142)

5

±

5 ±4 4 II

5

83 4

2.6hr

Mevf

#-(2.86). 7(1.78) #-(1.48), 7(weak or absent) #-(1.70). 7(none) #-(0.17). 7(none) #-(4.3. 1.6). 7(2.7)

#(4.5. 2.9). 7(1.6) *" #-(5.4). t(I.64) #-(4.2. 1.2).7(~3) #-(1.4), 7(1.38. 2.76) #"(1.8. 0.9). >(l.0l. 0.84)

(3(0.155). 7(none) #-(0.155). 7(nonc) 4.3. 3(10.3, 3.8), 7(6.0. #-(10.3. 4.3. 3.8). 7(6.0. #(3.7). n(~l)

10s years I2.4hr 152 days min 20 sec

6

±03

(n.7) (n.y) (n.y) (n.y) (n.y)

38 min 34 days 1.8 br

87 days

8.

100

0

5

100 S3 99

+

2

X

0. 18 100

K« K« Ca" Ca" ScMn> (n.y) (n.y) (n.y) (n,>) (n.y)

0

X 10» years

4

#

8c" Ti'° V" Cr" Mn"

3

5

1.0 ±0.2 0.63 0. 12 I.I 10

min 2.7hr 14. days 87 days min

3

93. 6.9

sec sec sec hr rain

2.

0.53

29 12 40 15

5.800 years 5.800 years sec sec 4.14 Bee

Radiations.

nuclides

#-(0.018), -r(none) #-(0.018), T(none) 3). >(weak or absent) #"(13. #-(0.56). 7(none) 0-
Radioactive

years years 0.9 sec 2.7 I0« years 0.03 sec

Half-life

by Neutrons

3

02

S" CI" CI" A" A"

(n.p) (n.y) (n.y) C».7) (n.y)

0.04 0. 17 30 20 0.56 12

Al" Si" P»> S» S"

Mr"

O" F» Ne" Na"

C'« C'< N» N" N"

H» H» Li» Be" B'>

Identity

Transmutations

6

9.

K" K" Ca" Ca" 8c«

100 4.2 0.017

± ± ± ± ± (n.y) (n.y) (n.y) (n,7> (n.y)

000

04 01 05 0.05 04

0.21 0. II 0.23 0.26 0. 14

±

05

9 ±

±

100

(»,7) (n.7) (".>) (n.7) (n.y)

±

0.21

±

mb

• 0.04 2mb 36 15 mb 0.56 + 0.03 50 10 mb

(".>) (n.a) (n.7) (».>) (n.7)

Reaction

(n.7> (»,P) (n.y) (n.p) (n.p)

mb

barns*

Primary

0.9 0.3mb 1.75 05 24 8,1b 0. 040 mg for virgin fission neutrons 0. 78 mb for virgin fission neutrons ±

1

Al" Si" p., 8" S"

9 ±

0

Mg»

±

7 7.

0.204 100 8.82 100 11.3

5 4. 4

0" F" Ne" Na»

0.01

cross section,

Table

X

99.6 37 99.8 0.037

±

mb 3mb <50 mb

0.57 945 33

Activation

nuclides

2 12 1

C" N» N" 0" 0"

Per cent abundance

Identity

Parent


1. 6.

8

5

0.31 100 100 12

49.8 49.8 9.2 9. 50

50.5 49. 0.35 56.9 56.9

17.4 72 27.8 9.9 82

100 17.4 2.8 100 15

Se" So" Se" 8e« Br"

Br" Br" Kr" Kr" Kr"

Kr" Rb" Rb" Sr" Sr"

Y" Zr" Zr" Nb" Mo"

19

7.7 7.7 100 0.87 9.0

±

Go" Ge" A." Se" Se"

0 33 0.4

1

0.63 60 39.8 20.55 36.7

0.5 0. y»«™ 47 days 10. min years hr

I.T.. 7(0.103) #"(1.38). (none) (3.4). 7(?) (1.5). 7(1.1. 0.37. 0.17) I.T.. 7(0.049, 0.037)

59 min 17 min 67 sec 25 rain 4.4 hr

±

± ±1 4 ± 33

1 20 mb

±

30 0.2

±

18 00

63 ± ± 2 ±± 4 7 (n.y) (n.y) (n.y) (n.y) (n.y)

1.4 8.5 3.5 0.5 2.0 ±0.5 10 0.03 60 20 mb

± ±

±

4 ±

25 ± ±

±

0 + ±

0

±

0 5 1.2 0.3 0. 0.05 0.1 0. 1.0 0.5 <6 mb

5 20 mb 0. 15 03 0.4 mb

±

72 0. 12 1.3

2

6

9

±

1

±

Y" Zr" Zr" Nb" Mo"

min daya min hr days 61 hr 65 days 17 hr 6. min hr

(2.18). 7(none) (0.37. 0.84). 7(0.72. 0.23) (1.91). 7(0.75) (1.3). I.T.. 7(0.042) T.. 7(0.26. 0.69. 1.51)

(3.63. 1.27). 7(2.3. 1.89. 0.41) (1.82. 0.72). 7(1.08) (5.30. 3.6. 2.5). 7(2.8, 1.86. 0.90) fl(nonc), I.T., 7(0.39) (1.46), 7(none)

####

78 19.5 17. 2.7 53

#-(1.99. 1.1). fl*(~l). 7(?) (0.46). 7(0.55-1.31) (0.6). #*(?). 7(0.26, 0.044) (0.85). 7(0.15). I.T., 7(0.31) (0.69. 0.15). -,(0.54)

#

min hr hr hr years

18 36 34 4.4 10

###

5

(n.y) (n.y) (n.y) (n.y) (n.y)

Kr" Rb" Rb" Sr" Sr"

Br" Br" Kr" Kr«m Kr"

# ## #

60

±

2

2.9

# #

(n.y) (n.y) (n.y) (n.y) (n.y)

(n.y) (n.y) (■:y) (n.y) (n.y)

30 0.5 50

10 mb 0.1 25 mb mb 0.5

26

1 1 # #

Se""> Se" Se"» Se" Br"i»

#"(2.8). I.T.. 7(0.38) (2.2. 1.38. 0.71). 7(0.04-1.75) (3.04. 2.49. 1.29), 7(0.56-2.06) E.C.. 7(0.067-0.405) I.T.. 7(0.162)

59 see 12 hr 27 hr 115 days 18 sec

Ge'"° Ge" As" Se" Se"

±

2 2

0.45

4.

20 mb 0.3 0.7

min 250 daya 14 hr 52 min

3

85 1.4

(n.y) (n.y) (n.y) (n.y) (n.y)

7(1.34)

0.08

#-(0.57). #+(0.66), E.C., #(2.63. 1-5). 7(1.04) E.C., 7(1.12), #'(0.32) I.T., 7(0.44) (0.90). 7(none)

0-(2.l). y(?) 0-(l.t). 70) #-(0.6 to 3.15). 7(0.63 to 2.51) K.C.. 7(none) #"(1.14). 7(0.26. 0.42. 0.57)

I2.9hr

E.C.. >(DUDUJ #-(0.46. 0.26). 7(1-30. 1.10) I.T., 7(0.06) (0.3 1). t(I.33. 1.17) (2.10. 1.01. 0.60). 7(1.49. 1.12. 0.37)

min 20 min 14 hr daya 82 min

± ± ± Zn" Ga" Ga" Ge" Ge"

9 3 (n.y) (n.y) (n.y) (n.y) (n,y)

0 (».7> (n.y)

4

Cu" Cu" Zn" Zn""> Zn"

Fe" Fe" Co""> Co" Ni"

#

0.8 1.8 OS 0.1 0. 10 0.03 1.0 ±0.2 .>) (n.y)

(".>) (n.y) (*.7) (n.y) (n.y)

736 . 52

Zn"> Ga" Ga" Ge" Ge"

±

9 ± ±

16 20 2.6

9

69. 30 48.9 18.6 18.6

111

2

Cu" Cu" Zn" Zn" Zn"

Fe" Fe" Co" Co" Ni"


/S # 7

I

67

± + ± ±

12

4

±

± ±

±

3

1

±

±

9

0.5 23 0.5 0. 15 mb

±

±

± ± ± ±

1

i 0. 14 0.16 1.0

4

33.0 33.0 4.7 4.7

0 6

8n'» Sn'» So'" Sn'» 8n>«

±

± ±

±

±

±

6

± 0.3 mb mb

±

26

1

±

±

6 10

±

±

Sb'" 8b'«™« gb'iuni 8b'" Tc'»

days hr min days days days 21 min min 60 days 100 days

#-(1.94. 1.36). 7(0.56-1.25) (?). I.T., 7(0.018) #-(3.2). I.T., 7(0.012) (0.50-2.29). 7(0.12 2.04) I.T., 7(0.088. 0.159)

#-(0.42). 7(?) #-(0.38), 7(none) (1.26). 7(0.15) #-(1.4). 7(none) #-(2.37. 0.40). 7(1.90)

#-(1.00. 0.87. 0.60). 7(0.14-2.09) #-(2.95), 7(none) E.C., 7(0.393) I.T., 7(0.159. 0.162) I.T., 7(0.024. 0.065)

(1.61. 0.7. 0.3). 7(0.46-1.28) #-(1.11. 0.58). 7(0.33-0.52) I.T., 7(-1.2) I.T., 7(0.19) #-(1.98), E.C., 7(0.55-1.30)

#~(I.T.), 7(complcx) #-(2.24. 2.82). 7(0.66. 0.9) #*(0.32). E.C., 7(0.85. 0.094) I.T., 7(0.15. 0.25) #(0.59). 7(7)

#

8

2.

± ± ±

5 1

2 5 2 I.I

(-.7) (n.7) (n.7) (n.7>

1.5 IS mb mb 0.5 0.3

± ±

0 1

6.8 30 30

>400 27 40 26

1

57 42.8 42 42 11

Sn>«'"« Sn'" 8n>»"> 8n'» Sn"»

54 min 13 sec days 14. days 245 days

Mevf

I.T., 7(0.052) #-(- 2.6). 7(0.04. 0.18. 0.94) (0.96). 7(0.087) (2.15). 7(0.38. 0.56. 0.65. 0.73) 0-( 1.5), 7(0.45. 0.66)

#

Bb»> Hb'" Hb'M Mb'" Ti>'»

20 1

(n.7) (n.7) (n.7>

± 0.03 0.04 0.5mb 2mb

In""" In"1 Sn'" Sn'" Sn"»

2 15 1

(-.7)

(n.7) (n.y) (n.7) (n,7) (n,7)

37

15

2. 2.

43 days days hr 50 days 72 sec

Radiations,

nuclides

(1.23. 0.45), 7(0.04-0.78) (1. 2.1). 7(0.19, 0.96) E.C., 7(0.22) (0.22. 0.70). 7(0.50) #-(1.15). 7(0.73. 0.13)

#

145 52 1.3

Cd""° Cd"« Cd'" Inium In'"

5

(n.y) (n.y) (n.y) (n.y) (n.y)

days sec hr min years

5

0. 14 0.03 1.1 0.3 1.5 0.3 56 12 2.0 0.6

270 24. 6.7 49

2.

Ag"° Cd"" Cd'" Cd"»

Agioi

min 4. 44 sec 13 hr 26 min min

3

(n,7) (n,7) (n,7> (n.7) (n.y)

Rh'»"° Rh'°« Pd">» Pd'»

67 hr 15 min days 42 days hr

3

2.8 110 1.0 0.2 30

(»,>) (n,7) (n.7> (n.7) t>,7)

Mo" Mo'« Ru" Ru>" Ru'°«

Radioactive

(Continued)

Half-life

4

12 12 140 30 12 0.3 0. 44

(1.7) (n.7) (n,T) (n.7> («.>)

Identity

by Neutrons.

8

95.8 95.8 0.95 14. 24

Ag"» Ag'M Cd"» CM"" Cd>»

0.05 0.05 mb 0.3 0.2

Reaction

Transmutations

2.

# /S

In»» In>'» 8n"« Sn'" Sn'»

48.65 48.65 1.22 12.4 24.1

Rh"» Rh"» Pd>°« Pdno Ag">'

0. 13 0. 20 10 1.2 0.7 ± ±

3

28.9 28.9 7.6 4.2 4.2

ICO 100 26.7 13.3 51.35

Mo" Mo"» Ru" Ru"» Ru'«

Primary

cross section, barns*

Table

3

Cd'" Cd"« Cd'" In'" In»«

23.75 9.6 5.7 31.3 18.3

Idcntity

Activation

nuclides

/3

O

Per cent abundance

Parent


1. 2,

#

3. 1

t4

8 8

10 1.8 3.7 <2 140

100 17.2 5.7 3. 16 26

22.5 47 23 24 21

Pr"> Nd'« Nd'" Sm' " Sm'"

Sm1" Eu'" Gd'" Gd'" Gd'"

677

5 ± ± ±

3 ± 0.1 0.2

2 0. 1.7

mb

±4 5 8

± ±

± ±

5.3day» 9.2hr 3.9 min hr years days 20 years

Xe'» C'wm Cs'" Ba'" Ba'"

6

8

0 4

99 0.3

min hr days hr min

I.T.. y(0. 10. 0.035) I.T.. 7(0.089) (none) 0-(O,7), I.T., -H0.I06) 0-(l.8). >(0.3. 0.8)

7(0.56-1.36)

7(0.16, 1.05) 0-(2.3). (1.32. 1.67. 2.261. 7(0.093-3.0) E.C., 7(0.17. 0.28) (0.58. 0.44). 7(0.14) 1.09. 0.71). 7(0.04-0.72) 0-(l.39,

y(?) 0"(~4). I.T., 7(0.128) 0.64. 0.08). jS"(0.68. E.C., 7(0.12-0.50) 0.08) E.C., 7(0.32.

I.T., y(0,I77) 0-(2.O. 1.4). 7(0.16. 0.7) E.C., 0*(T) 7(0.43) 0(2.02). 0-(O.34). 7(0.081) 0-(O.9l). 7(0.25)

0.42). 7(0.10-0.76) 7(0.08. 1.38) 0.67). 7(0.1 1-0.42)

0.40). 7(0.09-0.96)

0-(l.8). 7(0.11. 0.25) E.C.. 7(0.12. 0.34. 1.0) 0-(l.88). E.C., 7(0.10) 0-(O.9). 7(0.055. 0.38) 0-( 1.5). 7(0.37)

0.64), 7(0.14. 1.59) 0-(2.l5. 0-(O.83. 0.60. 0.38). 7(0.09. 0.32. 0.53) #-(1.5. 1.1, 0.95). 7(0.03-0.65) E.C., 7(0.061) 0-(O.68. 0.80). 7(0.07-0.70)

0-(O.97. 0.88), 7(0.084) E.C., 7(0.02-0.31) 0-(O.5O, 0.13), 7(0.14-0.40) 0-0.3). 7(0.15) 1.2). 7(0.089) 0-(l.l.

30 130 1.000 3.000 40 60 1.0 5.5 15 35

< ±

129 days 33 days days 1.8 hr 3.7 hr

>22 2600 300 < 1000 60 12

1

± Tm'» Yb'» Yb'» Yb'" Lu'» (n.y) (n.y) (n.y) (n.y) (n.y)

±

2

3.

Dy'" Ho'" Er"'

± ±

6

0-(O.86. 0.52. I.T., 7(0.109) (1. 25. 0.88. 0.55). 0-(l.84. 1.05. 0-0.49. Tb'»

(n.y) (n,y) (n.y) (n.y) (n.y)

± ±

0.8

25 9. 225 18

2

73 days min 2.4 hr 27 hr hr

Sm'» Eu'» Gd'" Gd'» Gd>«'

(n.y) (n.y) (n.y) (n.y) (n.y)

19 hr days 1.7 hr 60 days 47 hr

Pr'" Nd"7 Nd'" Sm'« Sm'»

1.1 300

± 40

3 0.6 1.2

min hr days days hr

86 40 40 28 33

Ba'» La'« Ce'» Ce'«' Ce>"

1 1

5.5 1400 25

daya daya hr daya min

30 hr 25 min 27 min

58 90 9.3 32 72

1

2

100 0. 14 31.8 12.7 97.4

0. 15 17 26 24

5 1 ± ± ± >

Tm'" Yb'" Yb'» Yb'» Lu"'

8 ±

± 2.

0

100 28.2 28. 100 14.9

±

11 332 1

0

Tb>" Dy'" Dy'" Ho'» Er"»

±

±

5 Te1"1" Te'" I>« Xc>» Xe'»

To'»» Te'"™ Te"' Te'"» Te'» ->

(n.y) (n.y) (n.y) (n.y) (n.y)

(n.y) (n.y) (n.y) (n.y) (n.y)

0.5 8.4 <0.4 0.3 1.0

71.7 99.9 0.26 88.5 III

Ba'" La'" Ce>" Ce'" Ce'"

(n.y) (n.y) (n.y) (n.y) (n.y)

1

90 0.

0.08 mb

8.96 100 100 0. 101 0.097

Xe'» C»"> C»1" Ba'" Ba'»»

5 (n.y) (».t) (n.y) (a.T) (n.y)

±

<8 mb 0. 22 0.05 5.5 0.5 0. 0.2 0. 0.2

34 34.5 100 26.9 10.4

Te1" Te"» I'" Xe>" Xe'»

3 (n.y) (n.y) (".7> (n.y) (n.y)

2

20 mb 0. mb 0. 13 0.03

4. 18. 18. 31.8 31.8

To'" Te'" To'" Tc'" Te>"


1

3

3. 1

7

±

±

2

9 +

4

+

±

1 ±

±

+

Per cent abundance

0

2.60 35.4 100 100 14

30.6 28.4 37.1 62.9 0.018

26.4 41.0 38.5 38 61.5

0.78 25.4

Identity

Ln"« Hfino Ta'" Ta"" W'M

W>" W>« Re'" Re>»' OS'"

Os>" Os'" lrn. Ir>" Ir>»

Pt>" ptIM Pt>" Au'"

3

8 ±

5

to

0.4 100 200 30

±

±

2

7 ±

±

±

±

8 0. 10 0.6 19

York,

I.T.. E.C., 7(0.13-1.5) (0.67), 7(0.08. 0.19) fl-(1.8).7(?) (0.96. 0.29. 1.37). 7(0.41. (0.21). 7(0.28)

4. 18 31 2. 44

±

2

±

days min

5

5

3

*

For neutrons of 2.200 meters/sec. Beta energies are maximum Average energy about 40 per cent of maximum. energies. Cross Sections," Source: D. J. Hughes it al., "Neutron McGraw-Hill Book Company, BNL-325, Inc., New 469-651 (1953). Natl. But. Standard! Circ. 499. Seaborg. Table of Isotopes, Revt. Mod. Phv%., SI(2):

23.

years min 3.3hr

min

days days

days hr min

1955.

/J-(l.23).

J.

M.

7(T)

Hollander,

0-(l.8>. 7(f) 0-(O.76). E.C., 7(none) 0-(~l.6). 7(none) 0-(O.64). 7(none) #-(1.17). 7(none)

Perlman,

and O. T.

0.68. 1.09)

0-(O.I4), 7(0.13.0.04) 0(1.1). 7(0.065) I.T.. 7(0.057) fl(0.66), E.C., 7(0.14-0.61) 0-(2.2). 7(0.29. 0.33, 1.51)

0.21. 0.32)

Mevf

days 32 hr inin 70 days 19 hr

0-(O.5O. 0.47. 0.17). 7(0.11. /9-(0.4l). T(0.14-0.6I) I.T., 7(0.18) 0-(O.6). 0-(O.53). 7(0.07-1.22) E.C.. 7(0.03. 0.60. 0.80)

Radiations,

0"(O.43). 7(0.13) /S-(l.33. 0.63). 7(0.07-0.78) 0.93). E.C., 7(0.12-0.76) 0-(!.O7. (J-(2.1). 7(0.15-1.3) E.C.. 7(0.65. 0.88. 0.23)

days days

days days tniu

nuclides

77 days 25 hr 90 br .18 hr 97 days

6.7 46 16.4 22 20

Radioactive

(Continued)

Half-life

5. 2. 4.

Th"'

Hg»" Tl»o» T1"M Pb»" BiJio

Pt'" Pt'"' Pt'" Au1" Hg«"

Os'»> Os'» Ir. Sim Ir'" Ir'"

W>" W>" Re1" Re>" Os'»>

Lu>" Hf"> Ta1"™ Ta>" W"

Identity

by Neutrons.

572

7.7

(n.7) (n,7)

(»,7) (n.7)

(«,T) (n.7)

(n,T) (n.7)

(n.7> (n,7) (n,7> (n,T)

(n,7) (n.T) (n.7> (n.T) (n.7)

Reaction

(n.7)

± 0.4

3 0.03 0.2 mb mb

± (n.7) (n,7) (n.7) (n,7) (n,7)

barns*

Transmutations

7

0.43

cross section,

Primary

3

00

0.10

90 40 1.1 0.3 3.9 ±0.8 96 10 3.0 0.8

± ± ± ±

100

±

Tht"

±

6.8 29.5 70.5 52 100

±

1.6 260 700 130

7

HE'" Tl«« Tli°i Pb»" Bi»"

i

2.1 ±0.6 34 100 20 75 15 <200

10

7

100 29.8

± ± ±

5 1

4. 1

Hg'»'

' 3 10 mb

800

Activation

nuclide*

4000 10 30 19 10

Parent

Table


1.

11

0

I.

t


Sec. 7-1]

7-13

The combined cross section for all processes leading to disappearance of neutrons This given material is termed the absorption cross section of that material. quantity is most commonly used in calculating the total number of neutrons dis appearing in a target or the total number of atoms transformed, without regard for the nature of the new atoms. The probability of interaction with neutrons to form radioactive isotopes is referred to as the activation cross section. This quantity may be given for the normal mixture of isotopes in the element, in which case it is referred to as the atomic activation cross For individual target isotopes, it is called the isotopic activation cross section. section. It follows that if a given element contains two isotopes x and y, with abundances of 20 and 80 per cent, respectively, and x interacts with neutrons to form a radioactive isotope while y interacts to form a stable isotope, with cross sections of
in a

Continuous Irradiation

2.2

2.21 Production of Stable Isotopes. If JVio is the original number of atoms of target isotope in a constant neutron flux nv and the capture cross section for this isotope is
—— at

nvaiNi

- Nue-""'

N,

and

-= —

If iVi is the number of new atoms resulting from interaction of target atoms #1 with neutrons and the cross section of Nt is tr2, then at

Nt

and

= niWiJVi — =

nwiNi

"

"'

(«""« at — ffi

-

— nv{aiNufi~n"<-i

—

«-*•»■«)

The target element may contain several isotopes, in which case each isotope and its daughter products can be treated separately by the foregoing formulas. 2.22 Production of Radioactive Isotopes. If the daughter isotope A': is radio active, the number of atoms of N2 as a function of time varies as at

which integrates

to AT,

=

nv
(nwi +

— nvo\

\>)

[e-„,ffli

_

e-(n.»,+x,)<]

Usually the number of target atoms A\ irradiation

period, in which case

N,

=

-n

Ni

does not decrease = Nltl and

nvaiNl nvci + Xi

>

[1

- e-(-.+x.)«]

materially during the

■

7-14


[SEC.

7

When isomers are produced by capture of neutrons in a given isotope, each isomer can be treated as a separate daughter product, each with its own cross section for The rate of formation of the secondary end formation and its own decay constant. product from decay of the two isomers would be the sum of the rates of formation from the two isomers. Intermittent Activation and Decay

2.3

It sometimes happens that the target atoms are in the neutron flux only a fraction of the time. Decay proceeds both during and between irradiation intervals. As an example, circulating coolants for reactors are activated as they pass through the core and decay in the external equipment as well as in the core. Let the number of radioactive atoms formed in a unit volume of coolant as a result of the first pass through the reactor core be represented by k. If the neutron flux is constant and the time spent in the core is t\, then k is given by

_

_ nvNv

e_,

X

As the unit volume of coolant passes through the external circuit with a transit time of ti, this activity decays by a factor e~Xl', and the number of radioactive atoms after one complete cycle is A, fee"*''

In making a second complete cycle, the number of new radioactive atoms formed is again ke~M', but the number created in the first cycle decays by a factor e~X(,i+<»). After n passes through core: Activity at exit from core,

I _

e-n\ «!+
to core,

_

I

= ke

c~»X(li-K,)

=

A„(cntry)

=

X«,

-** X(i,

+U)1 *** „

tt)

-

«-«x«1+«>)]

«-»>•«.+<>>]

k/\(ti +

<-)

+

Note that after an infinite number of cycles the saturation value at exit and at entry ke~Ut/\(ti tt).

is

.i„(exit)

[1

small compared with the mean life 1/X

(1

and

<*)

is

the cycle time

+

If

(
An (entry)

+

Activity at entry

FISSION PRODUCTS

3

is

About 60 radioactive nuclides result from fissioning of uranium and plutonium. On the average each of these original radioactive isotopes decays through two addi tional radioactive stages before reaching a stable terminus. 3.1

Decay of Fission Products

Data on individual fission products and their descendents are given in a number of in the literature.4"' Formulas for the amount of a given fission product as a function of irradiation time and decay time are very similar to those already presented for neutron activation and decay. In particular

— Xi

=

(1

places

A


-

-

e-x.'.)e-x.<>


Sec. 7-1]

7-15

A

= number of atoms of first member of fission product chain = fission yield = fraction of fissions giving isotope A = number of fissions per second = (3.1 X 10'°) (power level in watts) U •« irradiation time, sec U = decay time after termination of irradiation, sec For the next member B in the chain, the following applies:*

where

Y F

Xj(Xs

- X,)

L\

\

/

X,

/

x8

J

In some cases involving The foregoing equation assumes simple decay only. fission products with high neutron cross sections (such as Xe'*s), removal by neutron capture is very significant and must be taken into account. Heat Generation from Fission Products

3.2

Characteristics of those fission products which contribute significantly to the gross shutdown from operation for 10 hr in Table 2. By statistical methods, Way and Wigner8 derived a handy rule-of-thumb expression This has been modified to give which gives the 0 or y energy from fission products. the shutdown power as a function of time after shutdown for either 0 or y rays relative to the total power generation during the short irradiation. 7 activity between 30 min and 1,000 days after or more have been reproduced from NDA-27-39

where

io-'ri

X

s

P'

= power generation due to 0 or 7 rays Po' ~ total power generation during irradiation t = time after the irradiation, sec

The applicability of the foregoing formula to either 0 or y rays stems from the fact that the total j3 energy (exclusive of neutrino energy) is roughly equivalent to the total 7 energy. This formula gives values which are correct within a factor of 2 for decay times between 10 sec and 100 days. To obtain the total 7 or 0 energy from fission products created at a constant rate over an extended period of time T 0 followed by a decay period / in which no fissions occur, the Way-Wigner formula must be integrated over the irradiation period. In the following formula the 0 and 7 rays have been combined to give the total decay

-2

£

= 6.22

rT'

/

X 10-»(r + t)~l*dT

6.22

1

X 10-s[r°

- (To + 0"°

0

P

or

P

■*]

energy.

Untermyer and

Weills9 developed an empirical formula to fit experimental data accurately than the simple statistical equation by Way and Wigner. The empirical formula for heat from irradiated natural uranium, including contributions from the decay of U*" and Np"', follows:

-

+ To +

2

l

X 10')-»

(<

» 2

0.87

+

-

10)-»

1(1

T„

+

-

+

+ 10)-»-'

(t

~ = 0.1

«

more

{

X 10')-»']|

± ±

t

1

In comparing the foregoing formula with existing experimental data for heat generation as a function of time after shutdown, Untermyer and Weills estimated the following accuracy: under sec large error to 10' sec 50% 101 to 104 sec + 30% 10* to 10' sec 10% 10« to 10" sec + 50% 1


~- 6.22

Pa

7-16

RADIATION

AND RADIOLOGICAL

Table 2. Isotope

Br"

Half-life

Parent nuclide

Parent half-Ufe

32 min

Kr"»

4. 36 lir

Kr"

2. 77 hr

Rb"

17. 7 min

8r"

Gamma-emitting

PROTECTION

Fission Products Yield 0 0065 0.01

Branching ratio 0.35 0.76 0.15 0 85

Kr"

2.77hr

0.031

0.80

0 031

u u» 0.01

0.08 0.21

9.7hr

0.05

0.07 0.33 0.15 0. 14

Y'lm

Y"

8r»

8r"

9.7hr 9.7hr

0.26 0.00235

1.00 0 001 0.001

u

0.059

10 hr 16.5 min 65 days

Sr"

7 min

Y»

0.06 0.05

10. 5 min

0 06

Nb"

90 hr 35 days 60 sec 72. 1 min

Zr" Zr" Zr" Zr"

Ru1"

39. 8 days

Y»»

Y"

Zr"

Nb"» Nb" Generated for wjivans (University of Florida) on 2015-09-23 02:51 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

51 min 61 days

Nb"">

Mo" Ru1M

Rh'"

Te""" Te'»

4.5 hr 30 sec

days min hr

T«'«

Iiii

8. 14 days

Te1"

I'"

65 65 17 17

days days

hr hr

68.3 lir

33. 5 72 30 24. 8

Te1,lm

Y"

Ru»"

8b'» 8b'»

1 year

4.2hr 4 2hr

77.7 hr 2.4 hr

Te>"

24. 8 min

77.7hr

0.0019

0.028

52. 5 min

tiii

6.68 hr

Xe"» C»'»

9. 13 hr 13.7 days 32 min

J1IS

Xe"»

17 min

85 min 12. 8 days

Cs'»

9.5 min

Te1"

44 min

6.68 hr

0.94 1.00

0.24 0. 12 0.01 0 02 0.01 u 1.00 u 1 00

0.45 0.028 0.093 0.85 0.022 0.007 0.88

0.059 0.000062 0 06

0.063 0.0617

1.8

2.18 0.38 2.8 0.9 1.86 1.41 1.025

0.66 0.747 0.64 0.551

0.2 1.2 0.7 1.4

0 235 0.231 0 745

u

0.059

0.3 0. IS

0 02 1.00 1 00 1.00 1.00 0 13 1 00

0.044

0.057

1.89 0 89

0.725 0.758

0 034

0 046

I>«

Ba'"

0.0019

Mev

0.49 0.49

0. 10 0 02 20 8 hr

Ba"»

0.037 0.009 0.0052

u

1.00

Te»"

I1"

Cb'm

0.0006 0.06 0.062 0.062 0.062

0.0044 0 028

min

[SEC. 7

0.94 0.05 0.01 u

0.30 0.02 0.04 u 1.00 u 1.00 0 43 0 33 0 26 0 70

0.30 0.10

0.75 0 67 0 728

0.1412 0 494

0.726 0.513 0.624 0.87 1.045 1.55 2.41 0 106

0.3 0.17 0.7 0.16 0 722

0.637 0.364 0 284 0 163

0.231 0.69 1.41

2.0 0.53 0.85 1.4 2.2 1.2 2.4 1.8 1.3 0.25 1.2 1.44

0.98 0.463 0. 16 0. 162

0.54 0.304


Sec. 7-1]

Table 2. Half-life

Isotope

La'"

Gamma-emitting

Fission Products.

Parent nuclide

Parent half-life

Ba'"

12. 8 day.

40.2hr

Branching ratio

0.05 0. 94 0 26 0 36 0.050 0.67 0.30 0. 40 0.005 0 005 0 002 0. 25 0.05 0. 10 0 33 0.0007 0.0003 0.60

0.0617

32. 5 daya 34 hr

La'*1

La1"

3.7 hr 19 min

Pr'"

17. 5 min

Ce'"

273 daya

Nd'«

11.6 daya

0 026

8m'"

47 hr

0.0015

Eu"*

IS. 4 daya

0.057 0.054 0 0464

10 hr

Sm>"

(Continued)

Yield

Ce'« Ce'"

7-17

0.00013

Mev

2. 5 1.6

0.82 0.49 0.33 0. 145 0.70 0.28 2 18 0 695 1 48 0. 52 0 30 0 39 0. 104 0. 53

0.60 2.0


* u ia uncertain.

The heat generated from U"* fission products alone is obtained by subtracting from the foregoing empirical formula the following approximate expressions for heat due to U»" and Np"« decay: -K~ — >

Pn

P(Np"9)

- e->/i.o»i>]

-

0.0025[e-',/»

_

o.0013[e""/,,0'ol>,)) — e_(,+I'ii>/,,0 000l

040'

reactor coolant systems

4

4.1

Water Coolant

4.11 Pure Water (H*, N", 0", N"). The hydrogen in HiO produces no important radioactive products when irradiated with either slow or fast neutrons. Of academic interest only is the capture of a neutron to form deuterium which can then capture another neutron to form tritium (H'( 12.5-year half-life). The amount of tritium produced by this secondary reaction is too small to constitute a problem in reactors The amount of tritium formed from the 1 part in 7,000 of cooled with light water. deuterium already present in natural water is also insignificant. In heavy-water reactors the H,(n,7)H* reaction produces enough tritium to be of some concern in those cases where intimate contact is a possibility. The very weak (0.018-Mev) 0 ray is absorbed by pipes or other container walls. Thus, if The cross section of H' for thermal neutrons is about 0.6 millibarns. the average neutron flux were 1 X 10" neutrons/(cm,)(sec), the tritium concentration in 1 cm* of DjO after 1 year would be

»*'a

e-M)

A

-

1.2

X

„

(10")(6-6

X 10")(6 X 10-) 1.8

or

10'» atoms/cm*

X

.

10-» C1 1

V IfliMCl 3

fX

_ 8

^...x,.-.,,,.,*,..,, V

10->1

f0'°

= 6-8

X 10"4curie/cm'

The oxygen constituent of both light and heavy water captures neutrons to form radioactive nuclei by the following reactions:

O" O17 O1*

+ fast neutron = N" + proton + fast neutron = N" + proton + slow neutron = O1' + y

7-18

RADIATION

AND RADIOLOGICAL

PROTECTION

[SEC. 7

The N18 isotope decays with a half-life of 7.4 sec, emitting very energetic (6- to 7-Mev) 7 rays. Neutrons must have an energy greater than 9 Mcv to produce the

reaction 016(n,p)N16. Only one in several thousand fission neutrons has an energy in excess of 9 Mev. Rather than use the actual fission neutron spectrum and the variation of the O" cross section with energy, it is convenient to use the measured cross section value of 4 X 10-" cm' for virgin fission neutrons as a whole. Since a single scattering in HsO is likely to reduce the energy of virgin neutrons below the 9-Mev threshold, only the virgin flux need be considered. This virgin flux is cal culated from the relation a,

_ JL _

X 10")(2.5)(P)

(3.1

Sin

n = number of virgin

where


iVii
neutrons released per second in 1 cm* of the reactor core Si„ = total scattering cross section of hydrogen for fission neutrons above 9 Mev /cm3 of the core P = power level of core, watts/cm3 3.1 X 1010 = fissions/(sec) (watt) 2.5 = number of neutrons per fission JVh = number of hydrogen atoms/cm' of core o-H = scattering cross section of hydrogen for neutrons above 9 Mev » 1 barn In some reactors the attenuation of neutrons by core materials other than water must be taken into account in calculating the virgin neutron flux. As an example, consider a reactor in which 50 kw of heat is produced per liter of water in the core. Then

_

(3 1 (6.6

The saturated

,Nc

-

(5.9

X

10") (2.5) (50)

X 10")

N" activity

(1

X 10"")

_

ncutrons/(cm8)(sec)

is therefore

X 10")(3.3 X 10")(4 X

lO"8*)

= 7.8

X

10'

disintegrations/(cm»)(sec)

Ordinarily the coolant would pass through the core quickly and the N'6 activity would not reach saturation. If the transit time were 1 sec, the N" activity would be only 9 per cent of its saturated value, or 7.0 X 10* disintegrations/(cm*)(sec). In a heavy-water reactor the method of calculation is the same as illustrated for The smaller energy loss per collision is partially compensated by the light water. slightly higher scattering cross section of deuterium relative to hydrogen. According to current data the N" activity in a heavy-water reactor might exceed that in light water by about 30 per cent. The O" isotope has a cross section of about 8 X 10~88 cm' for fast neutrons to form N" via the reaction 0"(n,p)N". The N" isotope decays with a 4.1-sec half-life by emitting a p ray of 3. 7-Mev energy to form an excited level of O17, which promptly emits a neutron to form O".10 The O" isotope is only 0.038 per cent abundant. When the lower abundance and higher cross section are taken into account, the saturated concentration of N" in water coolant is about 0.007 times that of N". The O" isotope is formed by the reaction 018(n,y)0" with thermal neutrons. The cross section is about 2.1 X 10-88 cm8. The O18 isotope is 0.20 per cent abundant. Thus in a thermal neutron flux (nv) of 1 X 10" neutrons/(cm8)(sec) the saturated activity of O" is (1

or

X

1018)(3.3

X 10»)(2.0 X 1' 4

3.7

10~8)(2.1

V *

10'

X

10'°

X

lO"88) = 1.4

X

10' di8integrations/(cm8)(sec)

= 3.8 X 10-8 curie/cm8

Activation cross sections for elements which might 4.12 Impurities in Water. exist as impurities in water are shown in Table 1. The activity level resulting from


Sec. 7-1]

7-19

will depend upon the concentrations. For example, the water coolant For a thermal flux of 1 X 10u the saturated contain 0.01 ppra of manganese. MnM activity would be

impurities might

(1

x 10")

((T) 01

X

10~«1'

55

(6

X 10»)(12 X 10"")

- 1.3 X 104disintegrations/(cm3)(sec)

The N" activity in a relatively pure recirculating water coolant system will ordinarily be several orders of magnitude more intense than the radioactivity due to impurities. The longer-lived impurities constitute the major problem during a decay period such as a reactor shutdown, however. 4.13 In a direct-cycle boiling reactor, the concentration Boiling-water Reactor. of radioactive impurities in the steam is far less than that of the water in the reactor. This is because the distillation process leaves most of the impurities behind. The The volatile N™, N17, and O" activities will distill off with the steam, however. relative concentrations of these gaseous activities in the steam and the water are

known at present. Boiling reactor designs usually employ a bypass ion-exchange circuit to keep the In the formulas which follow, concentration of impurities in the reactor water low. the fraction of water entering in the bypass circuit per unit of time is represented by fi. Let ft = fraction of reactor water leaving reactor as steam per sec C =/, +/* a = entrainment factor for water (and impurities) in the steam leaving the reactor Si = moles of isotope i entering the reactor per sec as a result of corrosionerosion of external system parts m = mass of water in reactor

not

^

Then

at

= B<

Mi

or

Let

Ai

=

- (/, + afJMt - e-c')

u

(1

= number of gram molecules of radioactive

second.

-

Then

Ai

=

-

(1

J^-

-

e~«)

BVCM*

-

atoms formed from

Mi

per

(X, + C)A{

(X,.

+ X.,

+

The number of gram molecules of radioactive isotope Ai per unit weight of reactor is Ai/m. Likewise the number of gram molecules in a unit mass of steam leaving the reactor is a(Ai/m). water

4.2

Gas Coolant

Air (A41, N16, O", C14). The major activity resulting from exposure of air neutrons is the A41 isotope formed from the argon in air by the reaction A40 (71,7) A41. A41 decays with a half-life of 1.8 hr by emitting both 0 and y rays. The cross section of A40 for thermal neutrons is about 0.53 barns. Air contains 0.94 per cent by volume of argon or 2.5 X 1017 atoms of argon per cubic centimeter Thus, irradiation of 1 cm3 of air in a thermal flux of 1 X 1013 for 1 sec will of air. ''reate A41 having a disintegration rate of 4.21

to

(1

X 10l')(2.5 X

10l7)(0.53

X

10"14)(1

-

e~' •«xio-«)

= 1.33

X

10s

disintegrations/seC

7-20


[SEC.

7

If air coolant were swept through the reactor continuously with an average transit time of 1 sec and an average flux of 1 X 10" and then exhausted, the amount of A" per cubic meter would be -(1.33 X 10»)(10«) — 3.7 X 10l°

= 3.6

X

, . . . 10 ' curie/meter'

Other radioactive isotopes of minor importance compared with the formed in air coolant by the following mechanisms: O16

A41

would

be

+ fast neutron — N" (7.4-sec half-life) + proton slow neutron =0" (29-sec half-life) + y + fast neutron = C14 (5,500-year half-life) 4- proton

O" + N14

Commercial helium 4.22 Helium. Pure helium does not become radioactive. usually contains enough air impurity to give a detectable concentrat ion of A41 activity. 4.23 Carbon Dioxide. The following reactions produce radioactive isotopes in carbon dioxide:

O" + fast O" + slow


C"

neutron = N 18(7.4-sec half-life) 4- proton neutron O" (29-sec half -life) 4- y + slow neutron = CI4(5,500-year half-life) + y

-

The The reactions with oxygen have already been described for water coolant. concentration of oxygen per cubic centimeter is less in CO2 than in water, and the Nls and 0'» activities would be reduced in proportion. Carbon contains 1.1 per cent of the isotope C", which has a cross section of 1 milliIrradiation of 1 cm3 of C02 at 0°C and 1 atm pressure barn for thermal neutrons. in a flux of 1 X 10" neutrons/(cm*)(sec) for 1 sec gives (1

X 10")

(2.7

X 10")(0.011)(1 X 10-") X 10""

4.1

(1

_

e_4 1X10-11)

- 3.0

X

10»

atoms of

C14

per cm'

If this COs were recirculated for 1 year with half the time being spent outside the neutron flux, the activity would build up to about 5.0 X 10~11 curie /cm'. 4.3

Alkali Liquid-metal

Coolants

The alkali metals are used in liquid form as Thermal vs. Fast Reactors. 4.31 coolants for both thermal and fast reactors. In calculating the induced radioactivity in fast reactors the formulas are the same as for thermal reactors but the appropriate The neutron spectrum varies considerably from cross sections are quite different. one fast reactor to another and from one region to another within the same reactor. The Much less is known about activation cross sections due to fast neutrons. activation cross sections are very much lower for fast neutrons than for a thermal spectrum. For a fast reactor in which the average neutron energy is 200 kev, the activation cross section of sodium for the Na"(n,7) reaction is about 1 millibarn. In general, the cross section increases with atomic weight, varying between about Lead and bismuth 0.1 and 100 millibarns for lithium and mercury, respectively. are exceptions to the general rule, having much lower cross sections than other elements of comparable atomic weight. Although the cross sections tend to be 100 to 1,000 times as high in thermal reactors as in fast reactors, the neutron flux may be several orders of magnitude higher in a The fast reactor. The transit time would normally be shorter in a fast reactor. net result, is normally a somewhat lower activity per unit volume of coolant in the fast reactor, but this could be reversed in particular cases. Because of the paucity of data on activation cross sections due to fast neutrons, The the following treatment, of liquid metals will be restricted to thermal reactors. reader who must calculate radioactivation of materials in fast reactors is advised to investigate the status of these data at the time the problem arises.


Sec. 7-1]

7-21

Natural sodium consists entirely of the isotope Na". 4.32 Sodium. This The isotope has a cross section of about 0.6 barn for the reaction Na"(n,7)Na". Thus irradiation of 1 cm* of sodium for density of liquid sodium is about 0.9 g/cm». 1 sec in a thermal neutron flux of 1 X 10" gives n 10 ' X io») 0 v

™ 23

(6

X 10") (6 x io-») (1.2 X 10-')

(

_

0-t)'

6

= 1.4 (1.4

X

10") (1-2 3.7

Na" decays with

X

X

10-»)

1010

= 4.5

X 10" atoms of Na" per cm'

X 10"' curie/cm3

a half-life of 15 hr by emitting a 0 ray and two y rays per dis

integration. 4.33 Potassium.

The naturally radioactive potassium isotope K40 has such half-life (10* years) that its radioactivity is no problem. Irradiation of potassium with thermal neutrons produces K" (12.4-hr half -life) The parent K41 isotope is 6.9 per cent abundant and by the reaction K41(n,7)K41. has an activation cross section of about 1.0 barn, corresponding to an atomic activa tion cross section of 0.060 barn. Because of its lower cross section, potassium becomes only about one-tenth as radioactive as sodium. 4.34 Lithium. The atomic activation cross section of lithium for the reaction Li'(n,7)Li« is about 0.03 barn. Li8 decays to helium with a half-life of only 0.9 sec, Because of its low cross section and short life, this emitting energetic 0 and a rays. activity in lithium is a very minor problem compared with sodium and potassium. The Li* isotope reacts with thermal neutrons to form helium and tritium


a long

(Li« + n

- He4

+ H«)

H* decays with a half-life of 12.5 years by emitting very weak 0 rays. 4.4

Heavy-metal

Coolants

The radioactivation of pure lead is very minor. The cross section 4.41 Lead. for formation of Pb*" by the reaction Pb,08(n,7)PbM» is only 0.0006 barn. The halflife of Pb»°9 is 3.3 hr. 4.42 Bismuth. The principal radioactive isotope of bismuth is Bi110, formed by Bi'10 neutron capture in Bi20S. The cross section for this reaction is 0.017 barn. decays with a half-life of 5 days by emitting 0 rays to form Po"°, which in turn decays with a half-life of 140 days by emitting a rays to form Pb,M.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

Cork, J. M.: "Radioactivity and Nuclear Physics," pp. 25-27, D. Van Nostrand Company, Inc., Princeton, N.J., 1947. Hollander, J. M., I. Perlman, and G. T. Seaborg: Table of Isotopes, Revs. Mod. Phys., 26(2): 607 (1953). Curtis, R. B.: Phys. Rev., 53: 98C (1938). Gillette, P. R.: "Activity of Fission Products," HW-17415, Apr. 14, 1950. Faller, I. L., T. S. Chapman, and J. M. West: "Calculation of Fission Product Decay Chains," ANL-4807, May, 1952. Clark, F. H.: "Decay of Fission Product Gammas," NDA-27-39, Dec. 30, 1954. Moteoff, John: " Fission Product Decay Gamma Energy Spectrum," APEX-134. Way, K., and E. P. Wigner: Phys. Ret., 70: 1318 (1948). Untermver, S., and J. T. Weills: "Heat Generated in Irradiated Uranium," ANL-4790, Feb. 25," 1952. Bur. Standards

10. STatl.

Circ. 499, Sept. 1, 1950.

7-2

HEALTH PHYSICS BY

Karl Ziegler Morgan Health physics is a branch of physics relating to biology, chemistry, engineering, medicine, and industrial hygiene, the purpose of which is to prevent damage to Its organization in the present form persons or property from ionizing radiation. began in 1942, and in 1955 there were over a thousand persons in the United States devoting full time to health physics.


1

1.1

IONIZING RADIATION Effects of Ionizing Radiation

1.11 The Cause of Injury. The exact mechanism of damage from ionizing Usually it is assumed that the damage is proportional to the radiation is unknown. The mechanism of ionization may be ionization which radiation produces in tissue. direct in the case of irradiation by charged particles. Indirect ionization is caused by electrons released in tissue as a result of interaction with incident X or gamma radia tion; or by electrons, protons, alpha particles, or heavier charged particles that dissi pate their kinetic energy in tissue as a result of interaction with incident neutrons. The number and energies of these various ionizing particles depend upon the primary These particles produce ion pairs along radiation and the composition of the tissue. their paths, and the density of this ionization along the tracks depends roughly on the square of the ratio of the charge to the velocity of the moving particle. The specific ionization of a 0 particle or a secondary electron increases from about 64 to 10* ion pairs per centimeter of air as it slows down from 1 Mev to 100 ev, and the specific ionization of an a particle increases from about 2 X 104 to 7 X 10* ion pairs The energy w per centimeter of air in slowing down from 8 to 1 Mev (see Table 3). required to produce each of these ion pairs is an insensitive function of the type of ion, The average value commonly its energy, and the medium through which it passes. assumed for w in air is 32.5 ev per ion pair for a 0 particle or secondary electron and Recent measurements of Bortner and Hurst*-1 36 ev per ion pair for an a particle. and Jesse2 give w = 35 ev per ion pair for or particles in air, and data of Jesse* seem to indicate that the value of w for electrons in air is about 34 ev per ion pair. When the velocity of the moving ion is less than the velocity of the electrons in the outer shell of atoms in the surrounding medium, much of the energy is lost in elastic collisions in which momentum and kinetic energy of the ion are transferred to the translatory motion of the stopping atom as a whole3 and the efficiency of ionization becomes very low. For example, the value of w for the heavy recoil ion of ThO'-212 (E = 168 kev) is about 100 ev per ion pair.4 The ratio of the velocity V of the moving ion to the velocity v of the orbital electron is

(1) ' Superscript numbers

refer to References


7-22

HEALTH PHYSICS

Sec. 7-2]

7-23

where E = Mev of moving ion E0 = Mev of electrons in outer shell of surrounding medium A = atomic weight of moving ion It may be useful to remember that V/v = 1 for moving ions in air (Eo = 15 X 10_s In the high-velocity range (where F/t)» 1) w is relatively Mev) when E = 0.03 A. constant and on the average only about half the energy lost in the production of an ion pair is actually used in ionization. The remainder of this energy produces excitation which leads to various molecular transitions and the formation of free radicals. The

importance of these factors depends upon the initial velocity of the moving the type of damage under consideration. Whereas some biological damage results from the passage of an ionizing particle through or close to the nucleus of a cell where the molecular composition of the genes may be changed or chromosomes broken, other damage is the result of the formation of radicals — H, OH, H»Os, II()2, etc. — which may migrate considerable distances in the fluid within and between the cells before communicating damage to the cell. relative

ions and


1.2

Units of Radiation Exposure As Applied to Dosimetry

1.21 The most common unit for measuring exposure dose of ionizing Roentgen. radiation is the roentgen. It is defined as that quantity of X or 7 radiation such that the associated corpuscular emission per 0.001293 g of air (1 cm3 of dry air at 0°C and 760 mm Hg) produces in air ions carrying 1 esu of quantity of electricity of either sign. It should be noted that the roentgen is defined only with respect to X or y radiation and, strictly speaking, cannot be used in connection with other types of radiation such as a, g, and neutron radiation. The number of electrostatic units per cubic centimeter of air corresponding to a roentgen is converted to ion pairs per cubic centimeter by dividing by the electronic charge, e = 4.80 X 10_,° esu. Since the energy of X or y radiation in the common energy region below 3 Mev is used in the production of sec ondary electrons and these in turn produce ion pairs, it follows that the roentgen is a measure of the number of ion pairs produced in 1 cm3 of air (at 0°C and 760 mm Hg) by The roent secondary electrons originating in air from the primary X or 7 radiation. gen is not an energy absorption unit and is related directly to absorbed energy only when in is constant. We have the relation

1 r

(corresponding to

1

esu/cm3 air)

_

1.602 4.8

2
= 83.9 ergs/g air if w = 32.5 ev/ion pair*

(2)

In the above equation each term is defined in parentheses. The roentgen ordinarily not include the electrostatic units contributed by Compton scattered photons, and since the roentgen is intended to relate the X or 7 flux at the point in question in air to the ion pairs produced per unit volume of air at that point, the roentgen cannot be applied to high-energy (> about 3 Mev) X or 7 radiation where a considerable fraction of the ion pairs collected at the point of measurement is produced br scattered photons and electrons that originate at considerable distance. 1.22 The roentgen equivalent physical, rep, is an absorbed dose unit tka Rep. has had several definitions. Originally it corresponded to 83.9 ergs/g of soft tissue. The most widely accepted definition is: The rep is a unit of absorbed dose correspond ing to 93 ergs/g of soft tissue (of approximate composition oxygen 73 per cent, carbon 12 per cent, hydrogen 10 per cent, and nitrogen 4 per cent, by weight). The rep applies to any type of radiation directly or indirectly capable of producing ionization in tissue. The value of 93 was arrived at by assuming that X or 7 radiation loses energy to tissue only by the Compton effect where the energy loss is proportional to the electron Then the ergs per gram of tissue equivalent to a roentgen become density. does

*

It should

trons in air.

be pointed out that recent experiments8 have indicated that w = 34 ev/ion pair for elec If this value of w is used in Eq. (2), one roentgen corresponds to 87.7 erRs/R air.

7-24 83.9

X


7

Y(fZ/A)tim.

=!

___

83.9[(10 m

[SEC.

X

1)/1 + (12 X 6)/12 + (4 X 7)/14 + (73 X 8)/16 + (0.2 X 15)/31 + (0.2 X 16)/32 + (0.37 X 19)/39] (75.5

X

X

8)/16 + (1.29 X 18)/40 ergs/g tissue if w = 32.5 ev/ion pair 98 ergs/g air if w = 34 ev/ion pair

7)/14 + (23.2

o> 93

»

or

(3)

/, Z, and A are the fraction present by weight, the atomic number, and the atomic weight, respectively, of the elements in tissue or air. It is apparent that this value of 93 would not be equivalent to the roentgen absorbed in bone and could be in considerable error for soft tissue in the low-energy region where the photoelectric absorption varies as the third or fourth power of Z. However, in the interest of simpli fication the rep was given a fixed value of 93 ergs/g for all tissue and for any type and If the old value of energy of radiation that results in the production of ionization. u>, 32.5 ev per ion pair, is replaced by the more recent value* of to, 34 ev per ion pair, in Eqs. (2) and (3), the rep has a magnitude of approximately 100 ergs/g of tissue. The rep unit is becoming obsolete and is being replaced by the rad. 1.23 Rad. The radiation unit, rad, is an absorbed dose unit corresponding to an energy absorption of 100 ergs/g of any medium. It can be applied to any type and Since the roentgen is energy of radiation that leads to the production of ionization. approximately equivalent to 100 ergs/g of tissue and can be equivalent to 90 to 150 ergs/g of tissues — depending on the energy of the X and 7 radiation and type of tissue— the rad, like the rep, is closely related to the roentgen. The rad differs from the rep in that it can be applied to any medium, and thus in soft tissue,


where

number of rad

— 0.93

X

(number of rep)

(4)

1.24 Rem. The roentgen equivalent man, the rem, is an RBE dose unit that has had several definitions, but the presently accepted definition is

number of rem =

RBE X

(number of rad)

(5)

in which RBE is the relative biological effectiveness of the radiation. This unit is useful in establishing permissible exposures to mixed radiations and in compiling Sometimes it is applied to exposure of an individual exposure records of individuals. organ, such as the eye or the bone, when expressing the portion of the permissible exposure rate of 0.3 rem/week attributable to various components of mixed radia It tions. Commonly it is applied to total body exposure to mixed radiations. should never be used in experimental work where it is necessary to maintain a separate account of each kind of ionizing radiation in rad or rep units. 1.26 The gram-roentgen is the roentgen Gram-roentgen, Gram-rep, Gram-rad. In some cases body damage dose per unit volume integrated over the entire body. is more closely related to the gram-roentgen absorbed dose than to the roentgen absorbed dose in any local part (or parts) of the body. The gram-rep and the gramrad units are similarly the absorbed dose integrated over the body and correspond to the total energy delivered to the body by radiation which results in the production of ionization. We have gram-rad =

J

(rad

X density of

tissue)

dv

(6)

1.26 Absorbed Dose. The absorbed dose of ionizing radiation is the amount of energy imparted to matter by ionizing particles (e.g., secondary electrons, protons, can and heavy ions) per unit mass of irradiated material at the place of interest. be expressed in ergs per gram, rep, rad, etc. The roentgen is not a unit of absorbed dose even in air. 1.27 Exposure Dose. The exposure dose at a certain place is a measure of the Exposure dose is expressed in radiation based upon its ability to produce ionization. roentgens of X or y radiation.

It

HEALTH PHYSICS

Sec. 7-2]

7-25

1.28 Air Dose and Tissue Dose. As used in radiology, air dose refers to the roentgen exposure dose at the desired point in the beam of X or 7 radiation measured in air in the absence of other material. If a patient is placed with the surface of the skin nearest to the source at the same point in the beam, the measurement of the roent The tissue dose at the sur gen dose at this same point is the tissue dose at that point. face of the body is frequently 10 to 50 per cent greater than the air dose, the per cent increase depending primarily upon the width of the beam and the energy of the radiation. 1.29 RBE Dose. The RBE dose, or relative biological effective dose, is the absorbed dose in rad multiplied by the RBE of the radiation with respect to the stand ard radiation. (Refer to the report of the International Commission on Radiological Units' for detailed information on units.)

1.3

Equation and Its Application to Dosimetry

Bragg-Gray

In a few cases calorimeters have been used to make direct measurement of the energy per gram imparted to matter by ionizing radiation, but usually the sensitivity of available instruments is insufficient for direct measurement. It is customary to estimate the absorbed dose by measuring the ionization produced in a known volume of gas. In order to do this use is made of the Bragg-Gray* equation


(f) \dm/ \ where (dE/dm)b is the energy loss from ionizing radiation absorbed per unit mass of medium b, Ph is the mass stopping power of medium b relative to the gas g, w„ is the average energy required to produce an ion pair in the gas, and the quantity usually measured experimentally, is the number of ion pairs produced per unit mass of gas in the cavity surrounded by medium 6. In order for this equation to apply, the following conditions must be met: 1. The gas cavity must be small compared with the range of the ionizing particles. 2. The gas cavity must not contribute an appreciable number of secondary ionizing particles. 3. The intensity and composition of the radiation near the cavity must be uniform. 4. The cavity must not disturb the radiation flux by changing the direction and velocity of the radiation crossing its boundaries. 5. The stopping power Pb must be independent of the energy and physical state of the medium or must be given for the solid medium b with respect to the gas g for the energy range considered. 6. Appropriate average values of wt must be available for the energy range con

sidered.

listed above is often the most difficult because in some cases, in the measurement of absorbed dose of fast neutrons, the diameter of the ion chamber should be less than 0.1 mm. Often this difficulty is circumvented in prac tice by constructing the chamber wall of tissue-equivalent material and filling it with tissue-equivalent gas. In this special case the requirements of the Bragg-Gray principle do not limit the size of the chamber. The first limitation

e.g.,

1.4

Relative Biological Effectiveness

There are many definitions of relative biological effectiveness, the best definition

RBE

RBE,

but perhaps

is

=

dose, rads, of

7

radiation from a

Co80

source

to produce a biological change

dose, rads, of the radiation under comparison to produce the same biological change (8)

Frequently

200- or 250-kv

X

radiation is used in the above equation instead of the

7-26


[Sec.

7

y radiation from Co*0, but this practice is less desirable because of the wide spectral distribution of X radiation. Also the X or 7 radiation is sometimes measured in roentgens instead of rads; this is undesirable, since RBE should be a dimensionless unit. The RBE depends on the type of biological damage —damage to the bloodproducing system, cataract formation, tumor incidence, etc. — and the kind and energy of the ionizing particle. Probably the most significant characteristic of the passage of radiation through matter relating to biological damage is the stopping power of the medium or linear energy transfer (LET) expressing the rate of energy loss per centimeter of particle track. The relationship between LET and specific ionization S is given by the equation


LET

= wS

(9)

The LET is proportional to S only when w is constant. Unfortunately, neither S nor w is known except for gases, but satisfactory values of LET or dE/dx can be obtained from a theoretical consideration. Probably there is an LET corresponding to a maximum RBE for each type of biological damage. For example, if in a par ticular case the region of damage in a cell is 1 ju, a 5-Mev a particle with a specific ionization of 2.7 X 103 ion pairs per micron of tissue (2.9 X 10' ion pairs per centi meter of air) would be inefficient in producing damage if it required only one ion This a particle would have pair in the region considered to bring about the damage. an RBE < 1 compared with a 1-Mev 0 particle with a specific ionization of 5.6 ion If, however, it required pairs per micron of tissue (64 ion pairs per centimeter of air). 10* ion pairs in this region to produce the damage, the RBE of the 5-Mev a would be greater than one relative to the 1-Mev 0 particle because of the low probability of In this case one would expect coincident 0-particle tracks in the region of damage. the maximum RBE to occur for a particle for which S » 10s ion pairs per micron of tissue. If this value of £ corresponds to a 20-Mev a particle, the RBE for an a particle with an energy of 30 Mev would increase as the « particle lost energy in the medium until the energy reached 20 Mev ; over the remainder of the track the RBE would decrease. There are wide variations in the experimental values reported for RBE; for example, the RBE for the mean lethal dose to bacterium E. Coli decreases from 1 for 0 rays (LET = 0.2 kev/,i water) to 0.67 for X rays (LET = 2.1 kev/V water), to 0.17 for a rays (LET = 130 kev//i water).7 For most biological changes of common interest, the RBE of a particles is greater than 1 and the RBE increases as the LET or S increases from the value for secondary electrons associated with 200-kv X radiation to the values associated with 1-Mev protons, 5-Mev a particles, heavy ion As a consequence it has been customary to make the simplified assump recoils, etc. tion that RBE = 1 for X, 7, p+, e~, and (3 radiation; RBE = 10 for protons, fast neutrons, and a radiation ; and RBE = 20 for heavy recoil ions of mass greater than 4. These values of RBE are intended to indicate relative chronic over-all damage to man from ionizing radiations in t he common energy range in which they are most frequently available — 50- to 1,000-kv X rays, 0.01- to 3-Mev 0 and 7 rays, 0.1- to 14-Mev pro Obviously, since one could choose tons and fast neutrons, and 4- to 8-Mev a rays. these ionizing particles with energies such that they all had the same LET or S, the above values of RBE should be applied as approximations only for the aboveAlso the values of RBE are different for each specific indicated energy regions. type of radiation damage such as cataractogenesis, leukemia incidence, genetic changes, etc., which are different from the general RBE values listed above. Handbook 59, "Permissible Dose from External Sources of Ionizing Radiation,"8 gives RBE as a function of LET and S. These values are duplicated in Table 1 for convenient reference. As seen from the above discussion the fixed values of RBE for over-all damage to man with reference to type of radiation or the values given in Table 1 with respect to LET and S are oversimplifications in that RBE is different for each biological Also, there effect and there may be maxima in the curves of RBE vs. LET or S. may be some synergism in that irradiation from a mixture of two types of radiation and/or exposure of two or more body organs produce in some cases damage greater than the sum of the individual components of damage when received separately.

HEALTH PHYSICS

Sec. 7-2]

7-27

Recommended Values8 of the Relative Biological Effectiveness (RBE) Radiation of Different Specific Ionization Applicable to Exposure to Radiation from External Sources*

Table 1. of

Present knowledge of the biological effectiveness of radiations of different specific ionizations does not Therefore, ranges rather than individual figures are given in this table. warrant fine distinctions. For any range of specific ionization it is safer to use the higher of the two values of RBE given for that range, but a value obtained by linear interpolation is acceptable. X rays, electrons, and positrons of any specific ionization: RBE = I


Heavy Ionizing Particles

Average specific ionization, t ion pairs per it of water

RBEJ

100 or less 100-200 200-650 650-1.500 1.500-5.000

1 1-2 2-5 5-10 10-20

Average

linear energy transfer

(LET)

to water,!

kev per m of water

3. 5 or less

3.5-7.0 7.0-23

23-53 53-175

* The critical organs and biological damage considered are skin with respect to cancer, blood-forming organs with respect to leukemia, gonads with respect to impairment of fertility, and lenses of the eyes with respect to cataracts. t Specific ionization is expressed in ion pairs per micron of water in terms of its air equivalent. t RBE is in terms of the pertinent biological effectiveness of ordinary X rays for which the average specific ionization in the tissue of interest is assumed to be 100 ion pairs per micron of water. Permissi".le dose in rads -> (permissible dose in rems) /RBE. ^ Linear energy transfer is given in thousand electron volts per micron of water, using 35 ev per ion pair.

There is a slight indication

that there may be a rhythmic effect in which repeated of radiation at certain intervals in the cell recuperation and development may lead to greater damage than the same dose delivered continuously or with other rest intervals between doses. Also, as will be seen in the next section, RBE for certain types of biological damage is a function of the dose rate and of the accumulated dose. doses

1.5

Linear and Threshold Types

of Radiation

Damage

To a very rough approximation one may think in terms of two types of radiation damage: (1) the type which increases linearly with the accumulated dose and (2) the type which does not make its appearance until a certain threshold dose has been accumulated. Most types of radiation damage decrease as the dose rate is decreased until the rate of repair is equal to or greater than the rate of damage. In these cases, it is supposed thjat there is a threshold dose rate and dose below which radiation dam age is not and a saturation dose rate beyond which this damage is less depend observed ent on dose ralf and is related primarily to the accumulated dose. The threshold type of damage is illustrated in Fig. 1 by the single exposure curves for the production of radiation sickness and fatalities. It is supposed that there is no trace of radiation sickness if thje single dose is less than 20 r and that there are always severe symptoms of radiation /sickness for single doses above 300 r. In the cases of radiation sickness and fatal exposure the dose rate is very important. Four hundred roentgens of y radiation delivered to the total body of man in an hour or less would be expected to result in a j50 per cent probability of a fatality. However, 400 r delivered in a week One might ask about would be t^xpected to produce a negligible number of fatalities. the possibility of a dose rate so high that there is recombination of the ions in tissue and less .damage per roentgen. Ritchie9 has shown that there is little room for encouragement in this direction, since it would require a dose rate of 2 X 10" r/sec reduce the tissue damage by 50 per cent. Figur0 1 also illustrates the linear type of radiation damage for which there is little or1 no repair and for which the damage is more or less independent of the dose f»te. tf animal data can be extrapolated to man, it is observed that from 30 to 80 r is required to double the spontaneous mutation rate10 and each roentgen would to

[SEC. 7


7-28

TOTAL BODY EXPOSURE

TO X OR GAMMA

Fio.

RADIATION

( ROENTGENS)

\

1

r

7

7

5

a

f

5

2

1

1

r

ir}

and it is difficult to determine if there is a slight increase in this type\of dtmage due a few minutes Animal experiments indicate that 20,000 to 30,000 r to the radiation. probably would tipset man's nervous system, resulting almost immediately in uncon sciousness and possibly death shortly afterward. Occupational exposure of radiolo gists to long-continued intermediate levels of exposure, perhaps as low as 600 received at the rate of per week, has increased appreciably the incidence of leukemia." Small fractions of these threshold exposures may be harmful but are not expected to result in this type of damage. Biological effects of radiation such as those illustrated in Fig. for X and radiation depend upon the type of radiation, or the LET value, and upon the area and volume of the body exposed. Neutrons to for acute exposure are from times as damaging as 200-kv X rays; a nys on the surface of the body produce no damage because they cannot penetrate the rotective minimum depth of epidermal layer of skin that has mg/cm*, but they are con sidered to be to 20 times as damaging as 200-kv X rays when the a emittii? radio Many types of ndiation isotope deposited in critical body tissue such as bone. damage such as radiation sickness and leukemia are more a function of th< gramparticuhr body roentgen or gram-rad dose than the surface dose or average dose to organ. On the other hand cataracts and sterility are examples of damap tbftt a

is


shorten the life span" by about 10~4. There is some evidence that the slope of the "shortening of life span" curve is greater for single exposure than for continuous exposure, so that large single exposures may reduce the life span by as much as reduces it by 10~4 per 5 X 10-4 per roentgen while low-level continuous exposure roentgen. On this basis 100 r received over a long period of time would be expected to shorten the average life span by about 8}i months, whereas this exposure received in a single dose would be expected to shorten the average life of man by about 31^ years. Radiation cataracts are of special concern in that damage from neutrons is more or less linear with dose with very little or exceedingly slow repair whereas the cataracts produced by y radiation appear to be of the threshold type in that they do not make their appearance until the rate of damage exceeds the rate of repair long As a result the RBE of fast neutrons with enough to accumulate a threshold dose. respect to 200-kv X rays for cataractogenesis increases with the time over which the Other biological effects of ionizing radiation such as immediate radiation is received. to the nervous system, leukemia, tumor induction, and skin erythema seem damage to be of the threshold type, although at very low dose rates the statistics become poor

HEALTH PHYSICS

Sec. 7-2] depend

7-29

only on the dose in very localized tissue with a few square centimeter cross-

sectional area.

Blair" has offered another solution

to this problem by giving an equation which that possibly no type of radiation damage can be classed simply as "linear" or "threshold" but rather each type of radiation damage contains two components, His equation is one which is reparable, the other irreparable. suggests

dl/dt in which are

/

is the

constants.

- Ay - 0(7 - ayt)

injury, y is the dose rate, t For a single dose of duration

I

-

(y/MA

is the time of exposure,

(10)

and A, 0, and a

t,

- a)(l - e-»')

+ ayt

(11)

Blair has applied this equation to the reduction of life span and to the mid-lethal (LD-50) for y and neutron radiation and to both external and internal exposure. It seems likely that with the proper choice of constants this equation could be applied to other types of biological damage. dose

1.6

Permissible

Dose from External Sources

Maximum Permissible External Dose. The principal authoritative sources information on permissible dose are the publications of the National Committee on Radiation Protection814 and of the International Commission on Radiological Protection.16 Figures 2 and 3 summarize graphically some of the information given in the publications of the two above-mentioned committees. The basic maximum permissible occupational exposure rates apply primarily to the blood-forming organs which, as indicated in Fig. 3, are considered to be at a depth in the body greater than 5 cm. Here it is observed (1) that the basic maximum permissible exposure rate for * the adult man is 0.3 rem /week, (2) that the exposure rate, if averaged over a period of several years, is one-third the basic rate, i.e., 0.1 rem /week, and (3) that the exposure rates for minors under 18 and for others living in the neighborhood of the controlled area must be Ho of the above values, i.e., the basic rate is 0.03 rem/wcek and the This population is assumed to be a small portion long-time average is 0.01 rem/week. of the population at large for whom it is recommended the average accumulated exposure from all sources of man-made ionizing radiation (i.e., occupational, medical, A weapons testing, etc.) must not exceed about twice natural background exposure. controlled area is defined as an area in which the occupational exposure of personnel to ionizing radiation is under the supervision of a health physicist. The basic exposure rate may be increased by a factor of 5 (to 1.5 rem/week for the adult) when limited to the skin of the head, neck, hands, forearms, feet, and ankles. This exposure rate of 1.5 rem/week also applies to the skin of the entire body if the radiation is of very low penetrating power — with a half-value absorption layer of less than 1 mm of soft tissue. In Fig. 3 it is observed that the basic exposure rates (0.03 rem/week for age <18 and 0.3 rem/week for age >18) increase exponentially between the 5-cm depth and the surface of the body so that they are doubled at the surface of the body (0.06 rem/week for age <18 and 0.6 rem/week for age >18). These higher values at the surface of the body are the maximum permissible dose rates as determined approximately by personnel monitoring meters such as film badges, pocket meters, and dosimeters. The critical body organs for chronic occupational external exposure at low absorbed dose rates are considered to be the blood-forming organs, the reproductive organs, and the lenses of the eyes. The principal forms of damage one seeks to avoid in these cases are leukemia from exposure of the blood-forming organs, sterility and genetic effects from exposure of the gonads (testes or ovaries), and cataracts from exposure of the lenses of the eyes. The ovaries are considered to be at a depth of 7 cm, so are not as critical as the blood-forming organs, which are considered to be at a depth >5 cm. The testes, however, are considered to be at a minimum dept h of 2 cm as indicated in 1.61


of

* There is some indication that the NCRP and the ICRP may change the emphasis from a weekly to a quarterly or yearly dose; e.g., the limit may be set at 3 rem/13 weeks and 15 rem/year, provided the accumulated dose to age N does not exceed that determined by the equation 5(N — 18) rem.

7-30

RADIATION

AND RADIOLOGICAL

PROTECTION

[SEC. 7

Fig. 3. This may limit the exposure rate for men or require local shielding in order not to exceed the maximum permissible rate (0.3 rem/week for the adult of age > 18). Exposure of the gonads is not an important consideration after age 45, since few chil dren are conceived after this age. The lens of the eye is considered to be at a depth of 3 mm and often may be the organ that limits the exposure rate (0.03 rem/week for age < 18 and 0.3 rem/week for age > 18). For practical survey operations the maximum In the case of X permissible RBE dose rate as measured in air is 0.3 rem/week. or 7 radiation this measurement is made with a chamber having an air-equivalent wall thickness equal to or greater than the range of the secondary electrons; otherwise if the secondary radiation has not reached equilibrium with the primary X and y ',2'5 TO LENS OF EYE. (LENS OF EYE CONSIDERED TO BE AT DEPTH OF 3MM) SIGNIFICANT VOLUME IS VOLUME OF ENTIRE LENS.

(A) 300 MREM/WK


LIMIT TO ENTIRE BODY OF 1500 MREM IF THE HVL IN TISSUE IS
(D) 1500 MREM/WK TO SKIN OF HEAD, NECK, HANDS, FOREARMS, FEET, AND ANKLES PROVIDED EXPOSURE TO EYES IS NOTOVER 300MREM/WK IN THE CASE OF X OR T DOSE THE PERMISSIBLE 1500 MR/WK INCLUDES BACKSCATTERSIGNIFICANTAREA IS 1 CM2 IN REGION OF HIGHEST OOSE RATE IE ) 300MREM/WK1'2'3 TO PORTIONS OF BODY INSIOE 5 CM

(C) eOOMREM/WK2^ TO SKIN4 OF TOTAL BODY BASAL LAYER OF EPIDERMIS CONSIDERED TO BE AT A DEPTH OF 007MM

(F] 300 MREM/WK1^

(0) 1500 MREM/WK TO SKIN4 OF HEAD, NECK,HANDS, FOREARMS, FEET AND ANKLES PROVIDED EXPOSURE TO EYES IS NOT OVER 300 MREM,/ WK IN THE CASE OF X OR r DOSE THE PERMISSIBLE 1500 MR/WK INCLUDES BACKSCATTER SIGNIFICANT AREA IS I CM2 IN REGION OF HIGHEST DOSE RATE.

(HI DOSE MEASURED IN AIR MAXIMUM PERMISSIBLE WEEKLY DOSE OF X OR Y(E< 3MEV) RAOIATION MAY BE 300MR'.2 MEASURED IN OCCUPIED AREA.

TO BLOOD FORM ING ORGANS THESE ORGANS CONSID ERED TO BE AT DEPTH OF 5 M M SIGNIFICANT VOLUME ICM3 IN REGION OF HIGHEST DOSE RATE

(1) THE 300 MR/WK OF X OR T RAOIATION MEASURED IN AIR (WHEN E<3MEV) MAY BE INCREASED TO 450 MR/WK TO THE LENS OF THE EYE, AND ABOUT 400MR/WK TO THE BONE, AND TO OTHER TISSUE MORE THAN 5 CM INSIDE THE BODY WITH EXCEPTION OF THE GONADS WHICH ARE LIMITED TO 300 MREM/WK UNTIL AFTER AGE 45.

( 2 ) WHEN THE EXPOSURES ARE CONTINUED OVER A PERIOD OF SEVERAL YEARS THE INTEGRATED DOSE SHOULD NOT EXCEED 1/3 OF THE VALUE INDICATED BY THESE RATES (l E /TRATE DT =(BASIC PERMISSIBLE EXPOSURE RATE) x TIME)

(3) AN ALTERNATE RULE IS APPLICATION OF RULE (H) TO MEASUREMENT OF X OR T DOSE IN AIR.

(4) SEE FIGURE 3 FOR DOSE DISTRIBUTION

Fio. tions.

2.

Basic permissible

BELOW SKIN.

occupational exposure rates to external sources of ionizing radia

radiation, the tissue dose might be underestimated. Under these conditions of measurement it is considered that the X or y tissue dose may correspond (under the least favorable conditions) to a permissible exposure rate of 0.4 r/week to the blood-forming organs (at a depth of 5 cm) and 0.45 r/week to the lenses of the eyes (at a depth of 3 mm). In this case local shielding of the testes (of men age 18 to 45) may be required to reduce the tissue dose rate to 0.3 r/week. The absorbed dose from a and 0 radiation is measured with thin wall chambers. For neutrons the absorbed dose is measured using a chamber with a tissue-equivalent wall and tissue-equivalent gas in the cavity and applying the Bragg-Gray principle or it is calculated from a flux measurement." The maximum permissible occupational exposure rates given above are based on average weekly exposures. As explained previously, some types of radiation damage

Sec. 7-2]

HEALTH

PHYSICS

7-31

are rate dependent, and it would not be satisfactory to exceed appreciably these weekly even though the accumulated RBE dose for the year did not exceed 52 times the permissible weekly rate. Handbook 59 states that it is not prudent to extend the To exposure period without making some allowances for possible increase in risk. meet the requirements of some practical cases it is recommended in this handbook that in exceptional cases the exposure period may be increased to 13 weeks provided the RBE dose in the 13-week period does not exceed 3 rem. This RBE dose may be received at any rate and the 13-week period may begin at the time the weekly RBE Thus, in exceptional cases, the basic weekly occupational dose of 0.3 rem is exceeded. exposure rates given in Figs. 2 and 3 may be increased provided the accumulated RBE dose in 13 consecutive weeks is not more than 1%^, or 77 per cent, of the RBE dose

limits

1000


3 100

125REMSINGLEEXPOSURE (THISINCLUDESTHE 23REM LISTEDBELOWFORTOTAL BODY.) -62.5 REMSINGLEEXPOSURE (THIS INCLUDESTHE25REM LISTEDBELOWFORTOTAL BODY.) 25REM SINGLEEXPOSURE

planned emergency exposure of. Ihands.forearms.feet. ankles >h jofadultiwomen under 45 not/i t \perm!tte0 to take this exposure) -12.5 REMSINGLEEXPOSURE accidental exposure of total body of adult

it

!i

t4

PLANNEDEMERGENCYEXPOSURE TOTOTALBODYOFAOULT[WOMEN UNDER45 NOTPERM!TTEOTO TAKE THIS EXPOSURE)

10

5S

■<»* . 5

2 B

go, uj

°-F 3

|X

hit.

1**9

■.:, ■■-

8i8*8s ;3oi HIS? Si

I 5 REM/WK 1.2REM/WK ' EXPOSURETOENTIREBODYFROM RADIATIONWITHHVLOMM 0.6REM/WK THEMAXIMUMPERMISSI-045REM/WK") BLE AIR EXPOSUREOF 040REM/WK \ 03 REM/WKMAYINCREA03REM/WK I SET0O45REM/WKAT3 MMOR0.4 REM/WKAT 5 EYES(THE EXPOSURETOEYES MAY BE - CM DEPTHIN BODYIN INCREASEDTO0.43REM/WKIF ENTIRELY CASE OF X ORERADIAT FROMX OR T ION.

-

0.

Or:

-006 REM/WK LIMIT FORMINORStAGE < 13)

<

2 0.01

OVARIESI j BLOODFORMINGORGANS TESTES 8

Fio.

12 15 12

3. Distribution of maximum human body.

I I I - 0 03 REM/WK-

4 8 0 DEPTH IN HUMAN BODY (CM)

permissible

occupational dose rate and emergency dose in

13-week period. For example, the RBE dose to the bloodforming organs might be 2 rem in one week provided the RBE dose during the next 12 weeks did not exceed 1 rem. The top portion of Fig. 3 indicates permissible high-level single exposures. These may be received as a result of accidental or planned emergencies and should be limited to once in a lifetime. These exposures are assumed to have no effect on the weekly status of a person. However, if conditions of occupational radiation-tolerance exposure are such that there is the possibility of recurrence of a high-level exposure, the person may be required to modify his employment and it may be desirable to institute appropriate compensatory measures. A high-level exposure may be con sidered to extend over any period up to one month, and although only a portion of the emergency dose, e.g., 15 rem, may be received in this period, only one such epi sode should be permitted in a lifetime. The ordinary radiation exposures for medical

ordinarily permitted in this


[SEC.

7

purposes are not considered to have an effect on the radiation-tolerance status of a person, but in some special cases when large volumes of tissue receive fairly large doses repeatedly for medical reasons, such exposures should be taken into account in reckoning the radiation-tolerance status of a person. It is to be observed that if a person were exposed at the maximum permissible external exposure rates indicated in Figs. 2 and 3 and these were reduced by a factor of three as is required for prolonged exposure, the integrated dose to age 70 would be in excess of 270 rem to the blood-forming organs, 540 rem to the surface of the body, and 1,300 rem to the skin of the hands, forearms, feet, ankles, head, and neck or to the skin of the entire body for radiation of low penetrating power (HVL* < 1 mm However, it must be emphasized that such accumulations of RBE dose skin). might be harmful to the individual and would result in serious genetic consequences if they involved a large segment of the population. Handbooks 52 and 59 point out repeatedly that all unnecessary radiation exposure should be avoided. The report of the International Commission on Radiological Protection16 states that no particular harm is to be expected if the maximum permissible RBE dose rates are exceeded for a short period of time; however, it is expected that some detectable biological changes would be observed from a study of a large group of people that had maintained these exposure rates over a lifetime, and it is strongly recommended that every effort be made to reduce exposures to all types of ionizing radiation to the lowest possible values. 1.62 Maximum Permissible Flux. In some problems of shield design, instrument development, and absorbed dose calculation, it is helpful to make use of maximum permissible flux values as given in Table 2. The values for fast and thermal neu trons are those calculated by W. S. Snyder.17 " Table 2.

X

Flux for Various Types

Maximum Permissible

Type of radiation

Basic exposure rate,*

RBE

mrad/week

■

300

0-rays and electrons

i

300

Thermal neutrons

2. 5

120

and 7 rays

Fast neutrons

10

30

a particles

10

30

Protons

10

30

Heavy ions

20

15

of Ionizing Radiations

Approximate flux to give a maximum permissible exposure in an 8-hr dayf 4,200/B photons/(cm')(sec) in free air at 0°C 0.07 to 2 Mev) (error < 13?; for B 1.3 X 10s Y electrons or fi rays/(cm«)(oee) incident on tissue [^68 electrons or 45 fi per (emJ)(sec) of l-Mev energy] t 2.000 thermal neutrons/(cm*)(scc) incident

-

on tissue 58 neutrons of 2-Mev energy /(cni)»(sec) incident on tissue 1.3 X I01 K a particles/(cmJ)(sec) incident on tissue [~0.014 a of 5 Mev/(cm>)(sec)l 1.3 X 10» Y protons /(cm') (sec) incident on tissue [~0.I7 protons of 5 Mcv/(cm*) (sec)) 1.3 X I0» Y heavy ions/(cm")(sec)[ -O.OOOo oxygen ions of 5 Mev/(cm')(sec)l

* Basic exposure rate permissible to blood-forming organs, gonads, and eyes of persons of age > I S. t Maximum permissible exposure rate based on a 60-mrem RBE dose delivered to tissue in an B-hr day (7.5/RBE mrad/hr) and assuming w =» 35 for a, and w •= 34 for fi and secondary electrons. In this example the electrons have an instantaneous energy of I Mev. The 0 rays have an instan rays with a maximum energy of 1 Mer. taneous energy corresponding to the Fermi distribution of

J

j3,

Y

is

2

the average

HVL

-

g/(ev)(cm,)

(12)

dose measured to a depth in tissue equal to the

absorbed

or half value layer

is

For

Y

The product (RBE) Y is the inverse of stopping power or of linear energy transfer the ionizing particles. If it is desired to obtain the absorbed dose at the as used surface of the body from a normal beam of ionizing particles (a, p, c, etc.) in Table given by the equation

(LET) of

*


7-32

the thickness

of a shield required

to reduce

the exposure

maximum doee by H.

HEALTH PHYSICS

Sec. 7-2] range of the ionizing particles

In

Y

7-33

is given by the equation

these equations p, = density of tissue (^1 for soft tissue) p. = density of air ( = 0.001293 g/cm> at 0°C and 76 cm Hg) So = specific ionization ( «= ion pairs/cm of air at 0°C) to = ev/ion pair Pi = mass stopping power of tissue relative to air Ri = range in tissue, cm E = average initial energy of the particles, Mev

Only a small fraction of the ionizing particles reach the maximum range R, in tissue This is especially true for 8 particles and electrons where the because of scattering. The absorbed dose from 8 resulting attenuation is approximately exponential. particles and electrons at any depth of tissue d can be obtained approximately using Y as given by the equation Pi


(RBE)10V<2?e-»"''

g/(ev)(cm»)

(14)

The value of Pt for 8 particles ^1.14 in soft tissue and ~1.04 in bone. For a particles P, =^ 1.21 in soft tissue and ^1.03 in bone. The average value of w for 8 particles and electrons is 34 ev per ion pair and for a particles it is 35 ev per ion pair. Differential values of specific ionization in air S„ (ion pairs per centimeter of air), as used in Eq. (12), and values of range R, (centimeter of tissue), as used in Eq. (13), are given in Table 3. Approximate values of S„ for a particles, electrons, and B particles and coefficient of absorption mi (cm-1 of tissue), for 8 particles of maximum energy, E (in million electron volts), may be obtained from the empirical equations S„

=s=0.5

S,

^

~

10*

J?-°'4

9

55 4- —

E

Sm ==: 33

m

X

+ 63£~0

ion pairs/cm of air for a particles {
ion pairs/cm of air for electrons

' ion pairs/cm of air for

16K"1 ■•cm"1 of tissue for B particles (

( < 10 %

(15)

error from

E = 0.006 to 2 Mev) particles (<10% error from E = 0.05 to 2 Mev) <20 % error from E = 0.1 to 2 Mev)

(16)

B

(17) (18)

At best the representation of B attenuation by an exponential decay is a poor approxi mation and this should be kept in mind when using Eqs. (14) and (18). Equation

(18) is intended to give the average "absorbed dose attenuation," which differs The value of in corresponds to the somewhat from the count rate attenuation. average value for jS particles with an initial maximum energy E. Figure 4 is a plot of values of stopping power of tissue for various ionizing particles as a function of energy. Term (RBE)F as used in Table 2 and in Eqs. (12) and (14) is the inverse of stopping power as given in Fig. 4. The values of average energy E of 8 particles and positrons may be calculated from the empirical equation

£ in

which E

=

0.33ff(l

is the maximum energy

-g)(l+^)

(19)

(in million electron volts) of the B radiation and

Z is the atomic number of the ^-emitting radioisotope. In the case of positron emitters, Z = 0 in Eq. (19). With few exceptions this equation gives values of E with errors of less than 5 per cent.

Table

4 lists values of maximum

permissible flux for neutrons as calculated by

Values are given also Snyder17 and as rounded off in the international handbook.14 for the depth in the body of the point of maximum RBE dose and of the dis

RADIATION AND RADIOLOGICAL

7-34

PROTECTION

[SEC.

7

Specific Ionization, Range, and Average Energy of Ionizing Particles

Table 3.

Range in air and soft tissue 8m differential

specific ionization in air

a 1(article

energy of particle,

Mev o particle, ip/cmk

ff particle,

ip/cm*

In air,

In tissue.

cm*

era*

In air.

In tissue. cm'-'''

0 109 2.70

1 36 X 10 « 3 41 X 10-'

cm'^

10' 2 7 X 10'

0.0001 0.001

I. 1 X 10' 235

2. 1 X 900

4.0 X 10' 6.2 X I0<

150 75

540 175

0. 10 0.26

9.33 83. 1

1 18 X 10-' 0 105

0.6 0.8

6.9 X 10' 7 2 X I0<

68 65

130 110

0.38 0.43

148 219

0 168 0 250

10

7 2 X I0« 7.0 X 10'

64 64

97 89

0. 52

1.2

0.60

7 2 X io-« 7.9 X 10'

292 364

0 33S 0 428

1.5 2.0

6.3 X I0« 5.3 X I0<

64 65

80 73

0.74 101

1.1 X 10 ' 1.4 X 10"

476 659

0 155 0 776

3.0 4.0

4.0 X I0« 3 3 X I0<

66 68

69 70

1.67 2.50

3. 1 X

2.2 X 10' 10-'

1020 1370

1 24 1 67

4.5 5.0

3. 1 X I0« 2.9 X 10'

70 70

71 71

3.00 3. 52

3.6 X 10-' 4.4 X io-«

1530 1690

1 90 2 II

5.5

6.0

2.7 X I0< 2.5 X 10'

71 71

71 71

4.07 4.67

4.8 X io-» 5.5 X io->

1870 2050

2 34 2 56

6.5 7.0

2.4 X 10' 2.3 X I0«

72 72

72 72

5.28 5.96

6. 1 X 10-' 7.0 X io->

2210 2380

2 78 3 00

7.5 8.0

2. 1 X 10' 2.0 X 10'

73 74

73 74

6.64 7 36

7.8 X 10' 8.6 X 10'

2540 2710

3 22 3 43

10 20

1.7 X 10' 1.0 X 10«

75 80

75 80

10 5 34

1 2 X 3.9 X

io-»

3350 6540

4 29 8. 7J

170 590

0. 19 0 66

0.01


Electron, ip/cm'»

Electron or & particle

0.05

3. 1 X

0. 10

0.4

I0<

50 100

10' 0.06

io->

" H. A. Bethc and J. Ashkin, Passage of Radiation through Matter, in E. Segri (ed.)( "Experimental Nuclear Physics," vol. 1. John Wiley & Sons, Inc., New York, 1953. b M. S. Livingston and H. A. Bethe, Nuclear Dynamics, Experimental, Rev*. Mod. Phyw., 9: 270 ( 1937). ip = 35 ev/ip was assumed in calculating these values. « W. S. Snyder and J. Neufeld, "On the Energy Dissipation of Moving Ions in Tiasue."1 d W. S. Snvder, Health Phvsics Division. Oak Ridge National Laboratory, unpublished data. •L. Katz and A. S. Penfold, Revs. Mod. Phv>.. it: 28 (1952). ' R. M. Sternhcimer, Phys. Rev., 88: 851 (1952), and Ann. Rep. Nuclear Sci.. 8: 258(1953). oW. Bothe, "Handbuch der Physik," vol. 22.2, p. 56. Edward Brothers, Ann Arbor. Mich., 1943. * Calculated on basis of electron values by R. D. Birkhoff, Health Physics Division, ORNL. In this case Sa is given for the mixture of 0 particles of energies given by the Fermi distribution with B e*)ual to the maximum

energy.

tribution of absorbed dose in rads per neutron among protons (p), electrons (g), and heavy recoil ions (H ) at the surface, at the point of maximum dose, and at a depth of 5 cm in the body. Curves by W. S. Snyder.1' Figures 5 to 13 show some of Snyder's data. Figure 5 gives the "first collision dose" or dose in a small volume element of tissue exposed

HEALTH

Sec. 7-2]

7-35

PHYSICS

Table 4. Maximum Permissible Flux of a Beam of Neutrons Normal to the Trunk of the Body That Corresponds to a Dose of 300 Mrem in a 40-hr Week Distribution of dose in tissue phantom, rad /neutron X 10*

Maximum permisa ble flux Loca tion of

Calculated value,* n /(cm1) (sec)

Neutron

national dose, For RBE value.* cm For RBE constant n/(cm') for each (aec) surface function type of

Mcv

LET"

of

10 5 4 3 2 1 0 5 0. 1 0.01


I0»

I

P

H

5-cm depth

i

P

At maximum of rem dose

H

P

1

//

particle17

50 55

27 27

57t 58t

32t

sat

43t

36f

6lt

55 90 250

2 5 X 10 •

At

At surface

mum

86

230t 1.050t 1.60(11 1.900

l.200t l.600t 2.000

30 30 30 30 40 60 80 200 1.000 2.000 2.000

3 3 0 0 0 0 0 0 0 2 0 3

5.7 4 9 4 4 4 0

0.22 1. 1 0. 30 0.52

0.33 0.85 0.47 0.45

0 49 0 0.50 0 0.49 0. 3.5 3.0 1.0 0 46 0. 1.7 0.32 0. 17 0.55 0 53 0 0 70 0.35 0 023 0. 10 0 50 0 054 0 52 0. 10 0 35 0 040 0.40 0 040 0.35 0 063 0 25 0 009 0. 16 0 0 0 0

30 30 30 30

0 40

5.5 4.2 3.5 3.0 2.0

0.30 0.25 0.52

35 27 17 14 037

5 6 4 7

4.4 4 0

3.5 3 0 1.7 0 70 0. 10 0 040 0 075

0.30 0.95 0 40 0.50 0.30 0.40 0.30 0 30 0.30 0.25 0.30 0.52 0.32 0. 17 0.35 0 023 0.35 0.40 0.25

* The values of flux (or RBE as a function of LET make use of the functional relationship between and LET given in "National Bureau of Standards Handbook 59."" (Data supplied by W. S. Snyder.1*) The values of flux for RBE constant for each type of particle assume RBE is I for secondary The international values11 were rounded off electrons, 10 for protons, and 20 for heavy recoil ions.17 from the calculated values where RBE was assumed to be constant for each type of particle at a time before the more refined data (for which RBE is given as a function of LET) were available. t Values marked thus are interpolated from the calculated values.

RBE

Table 5.

Exposure of Man to Natural Background

Type of Exposure

External exposure isotope

to U, Th, and Ac

Exposure, mrep/day

mrem /day

Exposure,

0 13

0. 13

0.07

0.07

0 013

0 13

0 15

0. 15

0 36

0 48

series

to cosmic rays at sea level and 50° Geo. Mag. latitude

External exposure

Internal exposure products

to

Ra

and

its

Radiation Remarks

Attributable mostly to Rn and Tn products in air and on earth's sur Frequently less by factor of 5 face. atmosphere when no inversion exists Increases 75% in going to 2,000 meters elevation. Decreases 10% in going to 0° Geo. Mag. latitude Assumes 1.18 X 10~10g Ra in body. This may vary by factor of 5 either 245 g K or 0.18 mc K*° in body, 75% of which is in muscle

Assumes

exposure

by ±50%

to vary from these values

to a monoenergetic beam of neutrons of intensity 1 neutron per square centimeter. curves labeled D* — y, H, 0, C, N, Nu(n,p)C14 show the energy produced in the volume element of tissue by the principal interactions of the neutrons with the atoms of tissue. The first collision dose curve represents that portion of the energy produced which is absorbed in the volume element. Essentially this merely omits the energy of the y rays which will be almost entirely unabsorbed if the volume ele-

The


7-36

[SEC. 7

O.Oi

0.1

1.0

10

ENERGY (MEV)

Flo. 4. Stopping power of tissue for various ionizing particles us a function of energy. (O-ryyn ion, alpha particle, and proton curves: W. S. Snyder and J. Ncufeld, " On the Energy Dissipation of MovinQ Ions in Tissue," O/2ArL-1083; eleclron curve: W. Bothe, " Handbuch Also, R. M, Sternheimer [quoted by L. D, Marinelli in Ann. der Physik," vol. 22.2, p. 56. Rev. Nuclear Sci., 3: 259].)

is

6

5

4

7

1

oscillations near Mev are not entirely statistical but correspond to peaks in the cross-sectional curves. Figure shows the ratio of rem dose to rad dose at various depths within the slab. The change in energy spectrum of the radiation with depth in the slab may produce considerable variation in the RBE of the radiation. Figures to 13 show some of the data on which Table based. 1.63 Natural Background and X-ray Exposures. shows, for purposes of Table comparison, the exposure of man to natural background radiation. shows Table some common X-ray exposures. 8


6

is

is,

however, considered to be large compared ment is small. The volume element with the range of the protons or recoil atoms produced, and the dose curve includes a weighted sum of the all the energy of these particles. The RBE dose curve absorbed energy with each portion of the dose weighted with the RBE assigned by Handbook 59 of the National Committee on Radiation Protection,* page 48. shows the neutron flux which delivers a maximum RBE dose of 0.3 rem per Figure The 40 hr or for prolonged exposure, 0.1 rem per 40 hr, in a tissue slab 30 cm thick.

HEALTH PHYSICS

Sec. 7-2]

Table 6.

7-37

Common X-ray Exposures

Chest X rays at Oak Ridge National Laboratory using full size X-ray film in direct contact with intensifying screen Average chest X-ray exposure found from a survey in England Average chest X ray of majority in United States Chest X ray taking photograph on fluorescent screen Dental X rays Lower GI examination Upper GI examination Fluoroscopic examinations Average routine X-ray examination during 1st half of pregnancy* Average routine X-ray examinations during 2nd half of pregnancy* Treatment of malignant tumors (per 20 sec exposure) Fluoroscopy in shoe-fitting

J.

S. Laughlin and I. Pullman, "Gonadal Dose Produced By the Medical Use of X-Rays," Section of Report, The Genetically Significant Radiation Dose Received by the Population of the United States. Preliminary Edition, National Academy of Sciences — National Research Council, Washington, D.C., 1957. * W. E. Nolan and II. W. Patterson, "Radiation Hazards from the Use of Dental X-Ray Units." UCRL-1882. November 26. 1952; or Radiology, 61: 625-629 (October, 1953). * Memo to J. C. Hart from D. M. Davis and E. D. Gupton, ' Radiation Dose in Diagnostic X-Ray March 26. 1957. Procedures." d R. W. Stanford and J. Vance. The Quantity of Radiation Received by the Reproductive Organs of Patients During Routine Diagnostic X-Ray Examinations, Brit. of Radiol., 28 (329) : 266-273 (May. 1955). ' Dade W. Moeller, James G. Terrill, and Samuel C. Ingrahain. Radiation Exposure in the United States, Public Health Repts. (U.S.), 68 (1): 57-59 (January, 1953). / A. L. L. Bell, X-Ray Therapy in Fluoroscopy, Radiology, 40: 139-144 (February, 1943). * Precautions should be taken to reduce these exposures as they are sometimes excessive. See " Radiation Biology," Liane B. Russell, The Effects of Radiation on Mammalian Prenatal Development, A. Hollaender, editor. Vol. I, Part 2. Chapter 13, pp. 861-918, McGraw-Hill Book Co., 1954. *

III

J.


20 mr* 160 mr4* 200 mr° 100- 1,000 mrd t 1.5 300 r->.« 27 r" 35 r* 5-270 r*-**' 46 r° 56 r° 3.000 -7.000 r« 7-14 r*

Fia.

5.

First collision dose curve.


7-38

[SEC. 7

10,000 rr 5,000 —

2,000$ 1,000 500 J?

z

£ K

UJ

2

200 100 50

20 10 THERMAL


Fig.

6.

0.2

05

IKEV

2

5

IOKEV 20 50 I00KEV200 WFIITRON ENERGY FNFRrtY NEUTRON

Flux of neutrons to deliver the stated dose rate

15 DISTANCE (CM)

Flo.

7.

RBE 1.7

as function

as a

500 IMEV

2

5

I0MEV

function of neutron energy.

J

20

of depth and neutron

25

I

30

energy.

Permissible Internal Dose

The maximum permissible concentration in air (MPC)« 1.71 The Standard Man. and in water (MPC)„ and the maximum permissible body burden q of radioisotopes depend not only upon the chemical and physical properties of the radioactive material but also on the characteristics of man. In order to compare theoretical and experi mental values it has proved advantageous to relate all such data to the so-called "standard man." It is recognized that there are variations not only in individuals but in groups of people because of race, climate, food habits, and occupation. Never

HEALTH PHYSICS

Sec. 7-2]

7-39


theless, there are distinct advantages in relating the MPC and q values to a standard man for easier intercomparison and in order to make their application more general. Modifications and qualifications can be applied to the general values in adjusting The national and international hand them to individual and group requirements. books14'15 give values of the average composition of the body, concentration of the elements in some of the body organs and the daily consumption of these elements, the weight and effective diameter of the body organs, water consumption per day, urine and fecal excretion per day, and the deposition and disposition of inhaled radio active material, the fraction going from the gastrointestinal tract to the blood, the

2

4

6

8

10

12

14

16 CM

18

20

22

24

26

28

30

32

Fig. 8. Absorbed dose (rads) from 10-Mev neutron beam. fraction going from the blood to the critical organ, the fraction in the critical organ of that in the total body, the biological half-life in the critical body organ, the critical body organ, and other pertinent characteristics of the standard man. The critical body organ is that body organ receiving the radioisotope that results in the greatest body damage; almost without exception it is that body organ that receives the greatest absorbed dose from the ionizing radiation. 1.72 Calculation of Values of (MPC),, (MPC),,, and q for the Standard Man. Con tinuous Exposure. All the official handbook values —national14 and international16 — are based on continuous exposure throughout the lifetime (70 years)* of the indi vidual. The permissible body burden q and the corresponding concentrations in *

In the

revised handbooks14-1* (scheduled

of the daughter products of

Ra'"

are used.

for publication in 1958) more recent data for the retention This leads to a value of /tVfi(HBE)JV =110 and the con

stant in the numerator of Eq. (21) becomes 11 instead of 16. The period of occupational exposure in will be taken as 50 years rather than 70 years. The maximum permissible the revised handbooks exposure rates will be 0.6 rem/week to the skin and thyroid, 0.56 rem/week to bone, 0.1 rem /week to total body and gonads, and 0.1 rem/week to all other body organs.


7-40

[Sec.

7


air (MPC)o and water (MPC)» for occupational exposure are those which are esti mated to result in an exposure rate of 0.3 rem /week * to the critical body organ. These values are to be reduced by a factor of 10 for environmental exposure or for exposure If exposures are to a large fraction of the population, genetic considera to minors. tions demand that they be reduced to the order of magnitude of natural background exposure to the gonads.

0

2

4

Fio. The values of

q

9.

6

UBE

8

10

12

14

16 CM

18

20

dose (rems) from 10-Mev

22

24

26

28

30

32

neutron beam.

(in nc) are given by the equation, Q

=

(20)

/,££(RBE)tf

in which m = mass of critical organ ft — fraction of radioactive material in critical organ of that in total body E = average energy delivered to critical organ per disintegration RUE = relative biological effectiveness N = nonuniform distribution factor which is set equal to 1 for all tissue except bone, where it is set equal to 5 (with the exception of Ram, P", and radioisotopes that emit only X or 7 radiation) In the case of o-emitting radioisotopes deposited in the bone the value selected for q is based on a comparison with the long-standing permissible value of q =0.1 fic for Itam. The equation used in making this comparison is 9

* Sw footnote on p. 7-39.

-

10

/,£E(RBE)Ar

^c

(21)

health physics

Sec. 7-2] where

/2£i?(RBE).W

= 162* for

7-41

Ram and appropriate values for other radioisotopes

given in the handbooks of Refs. 14 and 15. The values of (MPC)„ and (MPC)„ are determined in the general case from the following equations: are

3.5

(MPC). and

(MPC).

=

X

10~Wi

- 10~Vt "*"T)

3iLX 7'/„(l

e-o

/ml

/

(22) }

(23)


In these equations, t is the period of exposure in days (taken as 70* years for con tinuous exposure), /„ and /„ are fractions arriving in critical body organ following

inhalation and ingestion, respectively, and equation

T

=

T

is the effective

half-life given by the

JFbT' T, Tt +

in which Tt is the biological half-life and TT is the radioactive half-life. half-life is the time required for half of the material in question to be the body from the critical organ. It applies only in those cases where elimination can be represented approximately by a single exponential,

gfi = (ftMo«(-0 «"""r') * See footnote on page 7-39.

(24)

The biological eliminated by the biological that is


7-42

The average weighted energy

££(RBE)JV

=

£ >''Xm'n

[SEC. 7

(in million electron volts) is given by the equation

[/*.#,.(!

-

*~°'d)

+ UfaM, +

10fa„EakNah

+fÊimNim+70fr,Er.Nr.]

(26)

in which fn,ftt,faufim, and/r„ are the fractions of the disintegrations emitting photons of energy Ey„ 0s of average energy J5/j„ as of energy Eai, electrons of energy E;m, and recoil nuclei of energy E,n, respectively. The term e~°

IP'7,

i

I

Fio.

11.

,

RBE

■

dose (rems) from 2.5-Mev neutron

beam.

In many cases the fraction of the radioactive material absorbed into the body is small, and as a result a portion of the gastrointestinal tract is the critical tissue. In most practical cases (when 7V > 0.5 days), the equations given in the international handbook14 for finding the MPC values can be simplified by the assumption that the lower large intestine is the critical organ and the radioactive material remains fixed in this part of the gastrointestinal tract for 18 hr, during which time there is radio active decay but negligible biological decay. In this case the equations become (MPC)„C' and

(MPC)„<"

2.7

= (1

-/,)£2?(RBE) 1.5

(1

X lQ-H? X IP-**?

-/O^(RBE)

lie/ml

(26)

/ic/ml

(27)

Sec. 7-2]

HEALTH PHYSICS

7-43

in which /i is the fraction of the radioactive material swallowed that is absorbed by the body from the gastrointestinal tract. In many cases (where T, > 1 week) the term G can be set equal to 1. This term corrects for the radioactive decay of the material in passing through the gastrointestinal tract. The equation for G is 0.52

(e-o.3»/7-f

_ e-o.»o/rr)yr

(28)


Ordinarily in the case of radioactive noble gases the exposure from the cloud of radio active material surrounding the body is greater than that from the small volume of

Fio.

12. Absorbed

dose (rads) from thermal

neutron

beam.

If a large radioactive cloud completely surrounds the gas contained in the body. body, the (MPC)„ value is given by the equation (MPC).

This

=

9

*

*°~7

Mc/ml

(29)

equation is usually conservative in that it assumes irradiation from a 4x rather

than a 2ir geometry. The reader is referred to the handbooks of Refs. 14 and 15 for specific MPC and q values. However, if the level of radioactive contamination is low, it may not be necessary to determine specific MPC values or even identify the radioisotopes for short periods of operation (a few months) provided the concentrations are less than the general values given in Table 7. No official MPC values have been selected for a single 1.73 Single Exposure. exposure

to radioactive material.

However, one can calculate from Eq.

(30)

the


7^44 Table

7.

General Maximum Permissible Medium in which

dose /omc

(ic/ml 5 X

io-> 10-'

10"

10'

D in rem in time I days to the critical organ of mass m g from a single intake of is deposited in the critical organ. of which the fraction

/

74//0r)'£(RBE)A'(l

D In most absorbed


a emitter,

(ic/ml

Air Water

7

Concentrations for Any Radioisotopes

ff or y emitter,

contained

[SEC.

- tr*'

(30)

cases of single exposure, the gastrointestinal tract receives the greatest dose.19 When the lower large intestine is the critical portion of the gastro-

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

CM

Fig.

13.

RBE

dose (rems) from thermal

intestinal tract (when Tr > 0.5 days), the given by the equation

RBE

neutron

beam.

dose to the gastrointestinal

tract

is

0.13//oY.E'(RBE) (31)

a where

/

= (1 —

/t) for ingestion and

0.62(1

—

/,)

for inhalation.

Sec. 7-2]

HEALTH PHYSICS

7-45

In using Eqs. (30) and (31) to find permissible values of Jo for a single intake of a A common radioisotope, one has to decide what integrated dose D is permissible. assumption — and a very conservative one — is that D = 0.3/7 = 0.043 rem. Per haps, if the single exposures do not recur frequently ( < 1 per month), it is satisfactory D = 0.3 rem. In the interest of national

to set

defense and /or to save a life, emergency exposures both external and internal may be much higher, and in such cases the hazard accepted from radiation damage ideally should be of the same order of magnitude as that from other associated hazards such as fire and explosion. The largest value of absorbed dose D that can be received over a single day without leading to detectable body Maximum changes, i.e., drop in blood count, drowsiness, etc., is about 25 rem. permissible concentration in air and water for single exposure can be calculated from Eqs. (30) and (31) by placing (32) in which C = 107 1,100

cm'/8-hr workday or 2 X 10' cm'/24-hr day for inhalation and cm'/8-hr workday or 2,200 cm»/24-hr day for ingestion of water.


2

RADIATION MONITORING

Radiation monitoring instruments may be classed in two groups40 — fixed and portable — and the portable instruments comprise two subgroups — area survey and personnel monitoring meters. In addition, the use of radiation monitoring instru ments requires a considerable amount of auxiliary equipment. 2.1

The

fixed monitoring

Fixed Monitoring

Instruments

instruments consist of radiation-detection devices, i.e., counters, proportional counters, etc., located in a fixed position where they normally operate continuously. Most instruments of this type maintain a written record on a moving chart of the dose rate and/or the integrated dose. Frequently a number of radiation-detection instruments located in various parts of a building or area are wired into a central recording system ; sometimes they are tied in by radio. Usually they are provided with an alarm system which may be set to sound a warning whenever any preset dose rate or integrated dose is reached. In an area where neutron exposure is possible, the sensitive element of the fixed monitoring meter often consists of a BFa counter surrounded by a few inches of paraffin. Such instruments are designed to give only a qualitative indication of the At best absorbed dose and to sound an alarm when a radiation hazard develops. they can be expected to give only a poor estimate of the absorbed dose an individual receives as he moves from one location to another. However, the constant vigil of fixed monitoring instruments is a source of comfort and often provides the first indica tion of pending radiation danger. One of the most useful fixed (or semiportable) monitoring instruments is the air monitor. This instrument usually consists of a filter collector, precipitator, impi nger, or dust-settling device. The air filter collectors have a Geiger-Muller tube or scintilla tion counter mounted close to filter paper through which the air is drawn and filtered, and when the buildup of radioactive dust particles on filter paper reaches the per missible level in a given interval, an alarm is sounded. These instruments serve to indicate the integrated radioactive dust hazard. Some air monitors draw air through a moving filter paper which, in turn, passes over a Geiger-Muller tube, ion chamber, scintillation counter, or proportional counter; either of the latter two are used where a-emitting contamination is present. The moving filter instrument serves to indicate more directly the differential air activity and the immediate hazard rather than the Also, if the instrument is provided with a second counter over integrated dose. which the filter paper passes, it provides a rough indication of the average half-life of the radioactive material that is collected. ion chambers, Geiger-Muller

"

7-16

RADIATION AND RADIOLOGICAL PROTECTION 2.2

[SEC.

7

Portable Survey Instruments

Fio.

14. Oak

7

Kidge National Laboratory badge

a

7

is

in

is

7

is

is

A

7

a

is

is is

A

a

is

is

is

This instrument such that the absorbed dose proportional to the count rate. widely used as neutron survey meter. It has portable and easy to operate and nondirectional response, however, which undesirable for some applications. in common use in research programs operates directional but less portable model that in accordance with the Bragg-Gray principle. The counter of this model has a tissueequivalent wall and gas, and the energy of the pulses integrated by feeding the output of the counter into an ordinary binary scaling circuit in such a way that the meter can be made to read absorbed dose directly and independently of the energy of the neutrons, even in the presence of radiation. Some of the instruments have built-in a source that can be readily exposed inside the counter, providing an absolute second method21 of neutron dosimetry that sometimes used calibration. research programs to make two measurements — one of the total absorbed dose, the other of the absorbed dose — and subtract the two readings. The total absorbed dose measured with a tissue-equivalent ion chamber, and the dose sometimes estimated from a measurement with a graphite ion chamber filled with COj. This practice should be discouraged, however, because the graphite chamber has a carbon and oxygen recoil ion response to neutrons that a function of neutron energy which is


7

a

is

a

7

0,

is

(3,

It would take many pages to list the varieties of portable survey instruments. However, most of them can be characterized as Geiger-Muller counters, proportional counters, scintillation counters, ion-chamber electronic meters, and fiber electrom eters. Some of the meters have thick walls so that they are suitable only for 7-ray radiation; and or measurements; others have thin walls so they will measure a, some have sliding windows so that a single instrument suitable for measuring and radiat ion. separately a, Neutron survey instruments20 are available for thermal- and fast-neutron measure ments. The thermal-neutron flux may be measured by using two fiber electrometers — one with graphite walls, the other innerlined with boron — and calculating the single reading neutron flux from the difference of the two readings or by means of with a BF3 counter. There are three common methods used for measuring fastneutron absorbed dose. The simplest method by means of the portable fastneutron survey meter.11 This meter consists of a proportional counter biased to cut out pulses from radiation. One model of the instrument uses chamber design

HEALTH PHYSICS

Sec. 7-2]

7-47

it impossible to measure in this manner the neutron dose for unknown mixtures and y rays. Unlike the proportional counters mentioned above, these double chambers are not available commercially; the tissue-equivalent chamber is difficult to maintain in calibration, and it is difficult sometimes to make certain they are exposed to identical geometry. The y component of absorbed dose can be meas ured using chemical dosimeters23 that are insensitive to fast neutrons. This y absorbed dose may be subtracted from the total dose as measured by the tissueOne of the best methods of measur equivalent chamber to obtain the neutron dose. of neutron ing the neutron absorbed dose is to determine the spectral distribution flux by means of a series of foils*4 — fission detectors and neutron capture foils — using the proper thermal-neutron shield. The flux spectra data can be multiplied by the first collision cross-sectional data to determine the absorbed dose. This method of neutron dosimetry has the advantage of being independent of dose rate and lends makes


of neutrons

VICTOREEN A LOW ATOMIC

POCKET

WALL PAPER SHELL C ALUMINUM TERMINAL HEAD a ALUMINUM TERMINAL SLEEVE El POLYSTYRENE SUPPORT BUSHING E2 CENTRAL ELECTRODE, GRAPHITE COATED NUMBER

B GRAPHITE -COATED

Fig.

15. Pocket

ER, MODEL 352 E3 POLYETHYLENE INSULATING WASHER E4 POLYSTYRENE FIXED BUSHING E5 ELECTRODE CONTACT F RETAINING RING G, ALUMINUM

BASE CAP

G2 POLYETHYLENE

FRICTION

BUSHING

fil er electrometer.

itself to a convenient means of estimating the average stopping power — or linear energy transfer — of the radiation. 2.21 All persons subject to radiation exposure Personnel Monitoring Instruments. should wear one or more appropriate personnel monitoring instruments. The prin cipal types in use are film badges (Fig. 14), pocket condenser chambers (Fig. 15), pocket fiber electrometers, and chemical dosimeters. The film badge should be pro vided with an open window for 0 detection and several filters to aid in estimating the y The film badges are provided with sensi absorbed dose independently of energy. tive dental film for 7- and 0-ray detection and may contain particle emulsion film for neutron monitoring. If one of the filters of the badge is made of cadmium, the portion of this film behind the cadmium will respond only to fast neutrons and the

Several types of pocket ion chambers remainder to both fast and thermal neutrons. and Geiger-Muller electronic circuits have been developed which sound a warning These, together with alarm if a predetermined radiation dose is accumulated. the direct-reading, direct-charging fiber dosimeters, are especially useful in "hot" operations.

7-48


[SEC.

7

Auxiliary Equipment

2.3

Many types of auxiliary equipment must be provided in order to conduct a satis factory radiation-protection program. Unless the film badge service is provided by an outside vendor, it is necessary to have a dark room with film-developing tanks and chemicals, densitometers, and film-marking equipment. In addition micro scopes, counting room with counting equipment (Geiger-Muller counters, scintilla tion counters, proportional counters, sample changers, integrating circuits, pressure chambers, range analyzers, etc.), body-fluid-analysis laboratory, decontamination laundry or equivalent, calibration facility with a selection of a, 0, y, and neutron standard sources, etc., are required. 3 3.1

DECONTAMINATION Prevention of Contamination

The best solution to the decontamination problem is to prevent contamination by providing proper facilities in the way of buildings, instruments, and equipment and by training personnel to employ acceptable techniques and for the persons involved to have a proper respect for radioactive contamination and the hazards associated with it.


3.2

This problem begins with

Building Facilities

the selection of an appropriate building site for reactors, chemical laboratories, and production facilities. If possible, one should avoid densely populated areas in locating the more hazardous types of reactors and asso ciated chemical operations. In such cases consideration should be given to meteoro logical, topographical, geological, and hydrological conditions of the proposed site; i.e., (1) the prevailing winds preferably should be in a direction away from densely populated areas, (2) hills and valleys may offer shielding protection to the surrounding communities in case of an accident, (3) certain types of soil formations are more suit able for the disposal of radioactive waste than others, and (4) large bodies of water which collect the drainage from the proposed site (both surface and underground drainage) may furnish dilution for the direct and indirect (air-borne particulates set tling on ground and later washing into streams) discharge of radioactive contaminants. The buildings themselves should be arranged so that the hot and y operations, the operations with a emitters, and the low background operations are isolated from one another as much as possible (Fig. 16). For example, it may be convenient to the chemists to have the counting rooms close by the hot chemical operations, but much saving in building costs can be made by isolating them and thus avoiding the use of thick concrete walls for shielding. The wall surface of the rooms and hallways should be made of such materials and provided with paint and other coverings that do not tend to collect radioactive con tamination and can be decontaminated easily. The building arrangement should be such that one cannot pass from a hot to a "cold" area without going through a clothing-change facility, and the flow of heating or cooling air and pressure differential should be in the order (1) counting rooms, lunch rooms, medical facility; (2) offices and hallways; (3) laboratory and reactor facilities; and (4) hot operations. Dust serves as a carrier of radioactive contamination and should be kept to a The air into buildings and into the hoods and the cooling air into nuclear minimum. reactors should be filtered on the inflow and in many cases on the outflow as well. There are usually economic reasons for keeping to a minimum the fresh air brought into a building and the contaminated air sent through filters and up the stacks. This practice of keeping the air changes to a minimum is compatible with the operation of hot hoods (where large volumes of air flow are sometimes specified) if "closed systems" are used to contain the hot operations. In such case it may be necessary to filter and send to the stack only the small volume of air that leaks into the closed system

"hot"

HEALTH PHYSICS

Sec. 7-2]

7-49

drawn into the hot gas and vacuum lines. In the small operations the hot conducted in glove boxes contained in hoods where little if any of the air escapes from the glove box to the hood, so that the hood air flow can be kept to a minimum and may not require filtration. In the larger operations an airtight cell must be constructed and provided with remote-control devices for the operation of the This cell must be kept at a negative pressure with respect to equipment inside. the operating area, and the cell ventilating air as well as the off -gas air from the equip ment usually requires filtering. A convenient means must be provided for the change of hot niters to prevent danger to personnel or the spread of contamination. The and is


work can be

CHIEF ACCESS Via air from

16. General

relationship of location for the various levels of radioactivity.

hot operations after passing through filters should be sent up a stack of

The reentrainment of proper height — 30 to 500 ft, depending upon the operation. radioactive dust particles in the neighborhood of the operation can be reduced by grassing the area, building hard-surface roads, and using road- and roof-washing equipment in dry weather; however, it is always best not to permit radioactive dust to fall out in the area in the first place. Easy access must be provided to the several facilities — especially the hot liquid waste and hot off-gas lines. This can be accomplished by arranging the laboratories in a row with a service corridor or tunnel back of or below the laboratories. The hot lines should run in the direction from the cold to the hotter areas, and precautions must be taken to prevent the possibility

of a reverse flow.

7-50


[SEC.

7

Every effort should be made in the design of buildings to localize and confine radioactive contamination. Space should be provided at the entrance to the hot areas where special "protective" clothing is provided —coveralls, shoe covers, caps, etc. These protective clothing should have distinctive markings so they will not be worn off limits. Complete change-room facilities with showers and toilets are required for work where there is the likelihood of clothing contamination. The larger operations should provide a special decontamination laundry to remove danger ous levels of radioactive contamination from protective garments, while the smaller operations may be able to handle the problem with a family-size washer and drier in a corner of one of the laboratories. A decontamination facility for cleaning tools and other equipment is a necessary adjunct to many hot chemical operations. It may consist of a shed with drain pits, tubs with various solutions, sandblasting equip ment, steam jets, hood facilities, etc.


3.3

Personnel

Some laboratories have special decontamination squads available to clean up con tamination in case of a spill of radioactive material. While this offers certain advan tages, it is probably best in most situations to have the person responsible for the This person should be familiar with the hazards and Bpill to look after the cleanup. problems of cleanup of materials with which he works, and the cleanup experience may aid in reducing the number of spills. All radioactive sources, hoods, and rooms in which work is being done with radioactive material should be properly labeled to indicate whom to contact and what precautions should be taken in case of fire or other accident. Plans and arrangements should be made with the local fire depart ment before rather than during an accident requiring their assistance in a contaminated area.

RADIOACTIVE EFFLUENTS AND WASTES

4 4.1

Air-borne

Radioactive Contaminants from Reactors

Ordinarily there are no serious problems associated with the discharge of radio active wastes from a nuclear reactor. The gaseous waste consists of cooling air or graphite "breathing" air and/or helium, and this may carry with it many types of radioactive gas and particulates. By far the greatest amount of radioactive con taminate of this type is the A4'(1.78h) produced by the A40(n,7)A41 reaction with the argon normally contained in the atmosphere. Radioactive isotopes of Xe and Kr and C14, HJ, N", O", etc., are produced in small quantities, but with the possible exception of C'*Oj none of the gases is ordinarily of consequence. Fortunately none of these radioactive gases is considered to be among the more dangerous radioisotopes, and the noble gases, especially, are less dangerous because they fail to concentrate in the body organs. If the fuel units are air-cooled, there is always the impending possi bility of a cladding rupture and of mixed fission products, uranium and transuranium elements, leaking into the gas stream and being blown up the stack with the cooling air. The radionuclides of greatest concern in this case depend somewhat on the type of reactor, but several of the more dangerous materials that may present a problem are Sr»°, Sr", I"1, Ba,4°, La140, Ce144, Prl4\ Pu"', and enriched uranium. One of the more worrisome problems associated with the operation of a nuclear reactor is the so-called particle problem, i.e., the dispersal of radioactive particles. These may consist of fission products carried on dust particles in the cooling air or on dust from the graphite and uranium. 4.2

Air-borne

Radioactive Contaminants Discharged from Chemical and Physical Operations

The chemical operations associated with production and research involving radio active materials can lead to the escape of air-borne radioactive contamination from the hoods into the laboratories or up the ventilation stacks and down into laboratory

HEALTH PHYSICS

Sec. 7-2]

7-51

I1", P", H*, CM, Sr"> are typical radionuclides that present some of the more areas. Also, improperly sealed sources — Ram, common air-borne contamination problems. Pol,°, Sr*°, etc. — or sources that are treated too roughly (e.g., a radium source operated by a pneumatic device where the source is brought into operating position when it is banged against a metal stop) may lead to widespread contamination and exceedingly expensive cleanup operations. 4.3

The RBE Dose from

a

Radioactive Particle

previously radioactive particles may become air-borne and some of activated uranium or dust from an uncased Co'0 source, may have a very The RBE dose rate at a distance r cm from high specific activity, i.e., /ic/particle. the particle in tissue is given by the equation As indicated

these, e.g.,

,s D =

-- —

,. , . w .. rem/(week) (radioactive particle)

0.02205,,)^, (RBE)e-w r*

.„.,,

(33)

Q = /ic per particle r = cm from particle D = RBE dose rate, rem /week The The other terms have the same meaning as defined in Eqs. (12), (13), and (14). average RBE dose rate in a sphere of radius r about the particle in tissue is given by in which


the equation

A = D

—

0.066QSaU)P,(RBE)(l —— r'/i,

-

e-c,')

..

,.

.

. . .

rem/(weck) (radioactive particle)

....

(34)

If the radioactive particle emits a or 0 radiation, Eq. (34) can be applied only when r < lit, in which R, is the maximum range of the radiation in tissue. If mr is small, Eq. (34) reduces

to

i) _ 0-OeeqfldwPigBE)

rcm/(week) (radioactive particle)

(34a)

In the above equations it may be convenient to set SawPt = pa(dE/dm) and obtain values of dE/dm from the graphs in Fig. 4 or to set S*wPt = WpatiiXE for Eqs. (34) and (31a). The average RBE dose rate in a sphere of radius r > Rt is given by the

equation

£>

- 85QZ£(RBE)

rem/(week) (radioactive particle)

(35)

A case of special interest is that of fallout particles from the testing of nuclear weapons. If the particle contains the normal mixture of fission products, it can be assumed that the radioactivity at any time is inversely proportioned to the 1.2 power of the time since the atomic explosion." In this case the integrated dose resulting from a particle fixed in body tissue for a time ti — to weeks is given by the equation

D

= 5tot)

—

\JJ

J

rem/radioactive

particle

(36)

In the above equation, D is the RBE dose rate at time to since the atomic explosion when the particle became fixed in the body tissue and is given by Eq. (33), (34), or The term ti — (o is the number of weeks the particle remains fixed in the body (35). tissue.

It is apparent from the above equations that the RBE dose close to a radio For example, the average dose active particle deposited in tissue may be very large. D is 9 X 10' rem in a sphere of 10-4 cm about a l-n/ic particle deposited in tissue 1 day following a nuclear explosion if it remains for 10 days and if the particle emits f) It is certain that such doses would radiation with a maximum energy of 0.6 Mev.

7-52

RADIATION

AND RADIOLOGICAL

PROTECTION

[SEC. 7

destroy some cells close to the particle and damage others farther away. What this means in terms of over-all body damage is unknown, but until biological information is available, it is wise to continue to take precautions to maintain the radioactive dust both from weapons tests and laboratory operations to a minimum. 4.4

Remedial

Measures for Air-borne Contamination


There are many ways to reduce the air-borne radioactive contamination; a few of them have been indicated. Some of these methods may be summarized as follows: 1. Install air filters for the air entering buildings and on the intake air to reactors and hot cells. 2. Install air filters for exit air from reactors, hot cells, and hoods. 3. Design and construct reactors so that there is a minimum of graphite and other kinds of dust that is loose in the reactor. 4. Prevent reactor fuel elements from rupturing by proper design, construction, and operation and frequent inspection. 5. Minimize reentrainment of radioactive dust particles in the area of the operation by paving the roads with a hard surface and by planting grass and other vegetation. 6. Construct inside surface of buildings with material having a smooth finish. Avoid exposed pipes, ledges, and other dust collectors. Keep floors and walls clean and polished. 7. Prohibit smoking, eating, and drinking in laboratories where the more dangerous radioactive materials are being used. 8. Instruct personnel in proper use of sources; for example: o. 6. c. d. e.

/. g. h.

i.

to other parts of the building following a spill. a spill. Wear protective clothing if contamination is possible. Know what to do about the source in case of a fire. Move calibration sources into place by tongs, pulleys, and wires or by pneu matic suction, not by pneumatic pressure. Have cushion stops at each end of the pneumatic suction system. Place filters in the pump exhaust of the pneumatic suction system. Inspect sources frequently and carefully account for each source. Use solid metal sources to replace Ra"" if possible. Often Co'0 is a good substitute for Ra"«. Do not track contamination

Turn off ventilators following

4.6

Radioactive

Liquid Waste from

a Reactor

Radioactive liquid waste problems can be subdivided as was done with airborne contamination, i.e., reactor waste and waste from chemical and physical operations. reactors that use water Unless one is concerned with large plutonium-production for cooling, there are few problems of liquid waste from reactors. One of the greatest problems is that of cleansing the water in the canal* that is usually a part of the reactor facility. This could and may be used to catch the fuel assemblies discharged from the reactor and as a convenient storage of various radioactive materials dis Inevitably the canal water becomes contaminated with charged from the reactor. such radionuclides as Co60, H3, P", U238 fission products, etc. If a large body of water is close by which can be used for dilution, the contamination of the canal water can be kept at an acceptable level by constantly draining off and replenishing a portion of it. A more acceptable solution, however, is constantly to scavenge out the radioactive contaminants by the proper use of filters — sand, anion, cation, etc. A large plutonium-producing reactor requires copious quantities of water from a The radioactive contamination problem depends pri reliable source for cooling. marily upon the purity of the water or the efficiency of the demineralization equip ment. Obviously, it is desirable to avoid pumping water loaded with minerals * In a "swimming-pool" moderation.

reactor the water provides the shielding and some or all of the neutron

Sec. 7-2]

HEALTH PHYSICS

7-53

through a reactor not only because of corrosion and plugging of cooling channels but also because every neutron lost to form Na", CI"'", Fe", Com, Ni".".« Cr", Ca", S", P", C14, etc., is one lost that might have been used to form a useful Pu"9 atom. reactor is to operate efficiently by Therefore, if the plutonium-producing dissipating the heat in a large volume of pure water, the concentration of induced radioactive contaminants in the water probably of necessity will be at a safe level from the standpoint of radiation danger.


Most types of power reactors presently conceived would not discharge much radio active liquid waste. Probably the loops of highly active fluid would be carefully shielded and connected only through heat exchanges to other fluid loops. Of course, any water leaving the high-flux region of a reactor contains a large concentration of the short-lived N" (7.35 sec) and traces of O19 (29.4 sec), H» (12.5 years), C14 (5,720 years), etc., but these are expected to lead to only minor problems of contamination. If the water circulates through the reactor in aluminum tubes, it is likely to contain a This large concentration of Na*4 (14.9 hr) from the (n,a) reaction with aluminum. Na" activity may require holdup basins for the water before it is discharged from the reactor control area. One of the problems that concerns those responsible for radioactive contamination from a reactor is the provision for a possible accident such as the one that occurred at Chalk River, Canada, on Dec. 12, 1952. This can be provided for ordinarily by building sumps and dump tanks that will receive the contaminated liquids by gravity flow and by containing the reactor in a large, airtight structure that is capable of containing the airborne waste in case of an accident. One solution is to build a pumping system that will pump the air of the building through filters and up a high stack following a reactor accident. This would maintain the air in the building at a slightly negative pressure without the expensive requirement of a completely airtight structure.

4.6

Radioactive Liquid Waste from Chemical and Physical Operations

This waste is likely to present much more difficult and long-range problems than the other types of fluid waste discussed above. Not only are the curies to be disposed of daily from a large chemical processing plant millions of times greater than from the associated reactor, but the average half-life of the waste material is usually much longer because of the holdup or cooling time allowed between the discharge of fuel from the reactor and the recovery of the metal and fission products. Many methods of disposal of this radioactive waste have been tried. Some methods are as follows: 1. Storage in underground concrete or steel tanks 2. Disposal in underground cribs, bins, gravel pits, etc. 3. Disposal in open earthen pits — both lined and unlined 4. Disposal at sea 5. Disposal in deep wells or in abandoned salt mines

Each of the above methods has certain advantages and disadvantages. The tanks the advantage that as long as they remain intact the radioactive liquid is available for future processing. This is especially important in the storage of uranium solutions from which the uranium can be recovered at some convenient time. The great disadvantage in tanks is that they are very expensive to build and maintain and that they are vulnerable to leaks and sudden failure as a result of explosions, fire, Some of the radioactive waste, e.g., Pu833, presents essentially a earthquakes, etc. permanent disposal problem. One may rely on the integrity of a tank for 100 years but probably place little dependence on it for a period of thousands of years. The type of tank used and its lining are determined in part by the kind of liquid waste, i.e., acid, base, salts present, turbidity, etc. One of the most promising methods of disposal of radioactive liquid waste is by means of various types of pits dug into the soil. Such pits are safe, however, only under certain qualifying conditions, but in a favorable locality they are not only have


7-54


[SEC.

7

much cheaper to construct and maintain than tanks but probably much safer because as the water seeps through the soil the radioactive material is deposited and firmly fixed to the soil particles near the pit so that earthquake, fire, and explosion cannot suddenly dislodge large quantities of radioactive material from the pit area. Some of the requirements for satisfactory use of pits for disposal of radioactive liquid waste are as follows: 1. The underlying geology and hydrology of the disposal area must be completely There must be no possibility of seepage into underground explored and understood. streams which vent into springs and wells used for human consumption, and, of course, such wells and springs should not be used to water livestock or for irrigation. 2. The soil should be a clay or shale formation instead of sand; if sand, it should contain some clay and humus material. Limestone soil is risky in that it is likely to be cavernous. The soil should be sufficiently soft that it can be excavated cheaply with power equipment. 3. The site should be on the side of a sloping hill that has a known drainage pattern to a large body of water or a stream flowing into a large body of water nearby. This provides dilution and permits constant monitoring of the seepage from the pit. 4. The sides of the pit should not be steeper than 45° to minimize caving in of soil The pit should be ditched completely around on all sides to prevent from the sides. surface water inflow, and if the fill-up due to rainfall is excessive, the pit should be covered with a roof to carry away rain water. The liquid waste can be brought 5. The pit should be accessible for maintenance. in by shielded tank trucks or by a pipeline. However, the pit should be sufficiently isolated, enclosed with fences, and marked with warning signs that it cannot be used for swimming or as a source of water for grazing stock. It should be covered with wire netting — if not covered with a rain shed — to prevent birds from using it as a It should be isolated from other occupied areas (probably regular stopover site. about a mile) so that radioactive mist will not cause an undesirable increase in the level of background radiation. 6. Under some circumstances there may be advantages in filling the pit with gravel and sand or lining it with various impervious linings and /or adsorbent materials. If the pit is to receive liquid waste from a large power reactor, some provision must This may mean the be made to take care of the heat generated by the radioisotopes. adding of cooling water to replace that which boils away or a provision to retain the fumes as the pit boils to dryness and perhaps melts some of its contents and the neighboring soil. Various ceramic materials may be added to the pits to facilitate the formation of a molten mass and the fixation of the radioactive materials at rela tively low temperatures, e.g., at less than 600°C. 7. There are certain advantages in adjusting the pH of the radioactive waste in a pit. This advantage must be balanced by considering the cost of the chemicals as against the risks of a less efficient pit system. Other types of disposal of radioactive waste, such as cribs and bins, may be con sidered as modifications of the pits. Direct sea disposal is probably not satisfactory However, in some situations the best method of disposal except for low-level waste. may be by means of a series of seepage pits and pumping from the last pit through a long line into the sea.

4.7

Disposal of Solid Radioactive Waste

Much of the above discussion relative to liquid waste applies also to the disposal of solid waste. Solid waste can be buried in pits or trenches that meet the above requirements, after which it should be covered with dirt and its site permanently marked and guarded. If the solid waste is contaminated with the more dangerous, long-lived radioactive materials, such as Pu,M, it should be covered with a foot or Other methods of more of concrete mix before the pit or trench is filled with earth. disposal of solid waste consist of mixing with concrete in a box or steel drum and Some of the burial at sea or storage in abandoned salt mines or in desert areas.

HEALTH PHYSICS

Sec. 7-2] solid waste can be burned

if incinerators containing suitable filters

7-55 are provided in the

off-gas system.


5

SHIPPING AND HANDLING OF RADIOACTIVE MATERIALS

There is no one source of information relative to the proper procedures for the A number of agencies, i.e., Interstate Commerce shipment of radioactive materials. Commission, Bureau of Explosives, Civil Aeronautics Board, Coast Guard, U.S. Post Office, and the Shipping Committee of the National Research Council, have pre There is some general consistency pared publications'* dealing with this problem. in the shipping regulations of the various United States agencies and with the regula tions of the various countries, but there arc enough differences to add to the already The shipping regulations complicated problems of shipping radioactive material. are designed to prevent radiation damage to persons under ordinary circumstances, but they do not guarantee no radiation damage to man in case of an accident. They assure protection of sensitive X-ray films under average conditions of shipment but would most certainly permit some damage to such films when there are delays during shipment and stopovers while the distance between film and sources is the minimum that the regulations require. A few of the more important general requirements for the shipment of radioactive material are as follows: 1. The package must not be less than 4 in. in its smallest outside dimension. This is to discourage a person from placing it in his pocket. 2. The package must be sturdy and of a type approved by the Bureau of Explosives that does not leak or spread radioactive contamination during shipment. 3. A single package must not contain more than 2 curies of radium, polonium, or other members of the radium family, and not more than 2.7 curies of any other radio active substance; a specific provision allows the shipment of not more than 300 curies of solid cesium-137, cobalt-60, gold-198, or iridium-192 in one outside container, the container to be fire- and shock-resistant, and approved by the Bureau of Explosives. 4. The surface of the package must contain no significant radioactive contamination, i.e., <0.1 mrad/hr of 0 or y radiation and <500 d/min of a radiation measured over 100 cm1 of surface. 5. The dose rate at any accessible surface must not exceed 200 mr/hr or equivalent in mrem/hr. 6. The dose rate at 1 meter must not exceed 10 mr/hr. 7. Shipments may be made by rail and motor express, air, ship, and private carrier. All these shipments fall into four classes, Groups I, II, III, and exempt. There are Such shipments are intended to very limiting restrictions for exempt shipments. permit the transportation of such items as clock dials, instruments, ores, etc., with a Only exempt shipments may be made by mail. minimum of restriction. Group I radioactive materials emit 7 radiation or both 7 and electrically charged particles. emits neutrons or both neutrons and other types of radiation. Group emits only electrically charged particles, or the 7, X, and the BremsstrahGroup lung radiation are so shielded that the absorbed dose at the surface of the package does not exceed 10 mr/24 hr et any time during shipment .

III

II

6

THE SAFE HANDLING OF FUEL By Thomas

J.

Burnett*

The pyrophoric nature of fuel metals can lead to incidents in which their oxides are dispersed in air. Such dispersions, as well as those produced in normal handling operations, are of concern because of potential exposure hazards. No detailed treat ment of the methods to be employed in preventing fuel fires or for bringing them under control will be given here, although these methods are quite important." There are special hazards of fuel handling and storage associated with the possible However, creation of critical assemblies which will not be considered in detail. standard control measures must be employed such as batch size limitation, inter•Oak Ridge National Laboratory.

7-56

RADIATION

AND RADIOLOGICAL PROTECTION

[SEC. 7

action spacing, and container geometry, usually supplemented by the use of neutron


absorbers.*8

Ordinarily external exposures are not a problem with U"s, U"s, and Pu"', as the radiations originating from these isotopes are primarily Where nonpenetrating. recycled fuel is handled, external radiation may become significant, depending on the degree of removal of residual fission products and on the build-up of 0— r-emitting In this case, exposure rates must be measured isotopes of uranium and plutonium. and personnel exposures monitored appropriately." Exposure-Control measures may include limitation of working time wit h rotation of personnel and /or the use of shielding and remote-handling equipment. Internal exposures are more serious than external, since reactor fuels are longlived a-emitting substances, which, when deposited in the body, confine their dis sipation of energy to a small volume of tissue and produce continuous damage until removed. Hence the precautions used in handling fuel are largely concerned with the Control measures must adequately limit, or where prevention of internal exposure. possible entirely prevent, the entry of radioactive substances into the body by inhala tion, ingestion, or through the skin. Ingestion of fuel usually results from putting contaminated fingers or other con taminated objects into the mouth. Enforcement of proper operating regulations can readily control ingestion. For example, smoking and eating should be prohibited in areas where fuel is handled. Before leaving an area of contamination, adequate cleanup is required. Special clothing may be issued for use in handling fuel to aid in confining contamination within working-area boundaries. Facilities for changing clothing, washing up, and checking for contamination should be located at the bound ary of the work area, and these facilities should be used by personnel as the principal point of access to the work area. Inhalation of fuel particles may be controlled by limiting the release and dispersal of fuel materials so that the contamination of air in the breathing zone of the workers remains within acceptable limits.14 If air-borne dispersal of fuel cannot be effec tively controlled, the wearing of respiratory protection devices will be necessary. The use of separate exhaust ducts (the air from which is filtered before venting) close to operations of high-dispersal risk is frequently very advantageous. For a par ticular combination of operating process and control measures, the quantity (j»g) of respirable fuel particles becoming air-borne is fairly constant. Hence the inhala tion hazard from different fuels will depend very strongly on their specific activity For some fuel materials, such as plutonium, the specific activity (curies per gram). may be so great that special control methods are required, such as total enclosure of the operating processes. Enclosure is necessary when the fuel is handled as a fine powder. Surveys to detect the presence of a contamination in the work areas and on the skin and clothing of personnel should be made frequently. The results of such surveys can be used to evaluate the effectiveness of housekeeping and other control measures. Continuous monitoring of the air for air-borne long-lived a emitters is necessary, as the results may indicate the need for modification of the Transient rises in air-borne contamination, while processing methods employed. significant for detecting improper operating conditions, are of lesser concern than the maintenance of average values within permissible limits. Personnel should be monitored for internal exposure on a routine schedule. No concern need be given to the exposure to neutrons which result from the spon taneous fission rate of fuels in normal use. Such neutron exposure, when detectable, is of much lesser significance than other external exposure. Fuel contamination may present serious problems other than those directly related to exposure as such contamination may interfere with the measuring of the health hazards potent ially present. Hence, special efforts are required to control the spread of contamination to counting rooms and analytical laboratories as well as to avoid the contamination of detection instruments and of repair or calibration facilities. Detailed precautions beyond the above outline will require study by a competent health physicist whose recommendations can be based on specific circumstances, quantities, methods, and materials.

HEALTH PHYSICS

Sec. 7-2]

7-57

BERYLLIUM TOXICITY2'

7

By Myron F. Fair* 7.1

Physiological

Effects of Beryllium

Beryllium poisoning manifests itself by inflammatory attacks on the skin and on the mucous membranes of the body. The skin manifestations are contact dermatitis and ulcers. The mucous-membrane manifestations are observed as nasopharyngitis, Pneumonitis is the most severe form of beryl tracheobronchitis, and pneumonitis. lium poisoning and has resulted in a number of fatalities. Symptoms appear in the acute form of the disease from a few hours to several days after exposure. The subject becomes acutely ill with cough, chest pain, occasionally fever, and marked prostration. Chest X-ray examination reveals lung findings indicative of acute pneumonia. The subject may recover completely within a few months, or he may continue on to the chronic form of the disease.

In the chronic form of the

disease, the latent period may last from 1 to 5 years the last known exposure and the onset of the illness. The symptoms are limited to the lungs showing shortness of breath, malaise, and progressive weight loss in spite of normal food intake. Chest X rays show scattered densities suggestive of silicosis. An experienced radiologist, however, can distinguish between the two conditions. There is no known treatment. Cortisone has been used with varying degrees of success for the relief of symptoms, especially shortage of breath. between


7.2

General Considerations

Berylliosis has occurred in association with most industrial processes in which beryllium or its salts contaminate the atmosphere. It has not been limited to workers but has occurred in people living in the vicinity of the plant and also in members of the workers' families. It has been observed by its pulmonary and dermatologic manifestations. Certain preexisting physical conditions may predispose one to, or Early diagnosis is adversely affect the course of, the chronic form of the disease. helpful in preventing complications, in reducing severity, and in shortening con valescence. The respiratory tract is the chief site of damage for beryllium poisoning. Inhalation of beryllium compounds may produce acute or chronic pneumonitis. There is evidence to show that pneumonitis will not occur if the beryllium concen tration is kept below 25 fig/m3. 7.3 1.

Rules for Handling

The foremost consideration is cleanliness and personal hygiene. All operations involving dust or fumes must be done in carefully

designed shops exhaust facilities are available. 3. Well-designed storage facilities must be available. 4. The in-plant atmospheric concentration of beryllium should not exceed an average of 2 jig/meter' for an 8-hr day. The out-plant breathing concentration should not exceed 0.01 ftg/m'. 5. One should not under any circumstances be exposed to concentrations greater than 25 jig/m3 for any period of time, however short. 6. There should be an adequate medical program, supervised by a physician fami liar with beryllium poisoning, periodically to examine workers exposed to beryllium. 7. Any complaint of throat or chest difficulties or of skin rashes should be reported to the physician immediately. If evidence of beryllium poisoning is found, the individual should be removed from any further exposure to beryllium. 2.

where

REFERENCES 1.

Bortner, T. E., and G. S. Hurst: Ionization of Pure Gases and Mixtures of Gases by 6 Mev Alpha Particles. Phys. Rev., 93(0): 1236-1241 (March 15, 1954).

* Oak Ridge National LaVxiratory.


7-58 2.

3.

5.

6.

[SEC.

7

William P., and John Sadauskis: Ionization in Pure Gases and the Average Energy to Make an Ion Pair for Alpha and Beta Particles, Phys. Ren., 97(6) : 1668 -1670 (March 15, 1955). Bohr, N.: The Penetration of Atomic Particles Through Matter, Kgl. Danske Videnskab. Selskab Mat. Fys. Medd., 18(8): 1-144 (1948). Snyder, W. S., and J. Neufcld: On the Passage of Heavy Particles Through Tissue, Radiation Research, 6(1): 67-68 (January, 1957). Snyder, W. S., and J. Neufeld: On the Energy Dissipation of Moving Ions in Tissue, Jesse,

ORNL

4.

"

1083,

November

14, 1951.

W. G., and L. W. Cochran: Ionization of Gases by Recoil Atoms, Phys. Rrr., 107(3): 702-704 (August 1, 1957). Report of the International Commission on Radiological Units and Measurements (ICRU), National Bureau of Standards Handbook 62, 1956, for sale by the Superin tendent of Documents, U.S. Government Printing Office, Washington 25, D.C. Bragg, W., and L. H. Gray: "Studies in Radioactivity," Macmillan and Co., Ltd.. London, 1912. Gray, L. H.: The Absorption of Penetrating Radiation, Proc. Roy. Soc. London, 122A: Stone,

647-668

(1929).

Gray, L. H.: An Ionization Method for the Absolute Measurement of Gamma Ray Energy, Proc. Roy. Soc. London, 156A: 578-596 (193G). Gray, L. H.: Radiation Dosimetry, part I, Brit. J. Radiol., 10: 600-612 (1937). Gray, L. H.: Radiation Dosimetry, part II, Brit. J. Radiol., 10(1 18) : 721-742 (October, 1937).


7.

8.

9.

Gray, L. H.: The Ionization Method of Measuring Neutron Energy, Proc. Cambridge Phil. Soc, 40 : 72-102 (1944). Boag, John W.: The Relative Biological Efficiency of Different Ionizing Radiations, National Bureau of Standards Report 2946, December 30, 1953, for sale by the Super intendent of Documents, U.S. Government Printing Office, Washington 25, D.C. Lea, D. E., R. B. Haines, and E. Bretscher: The Bactericidal Action of X-Rays, Neu trons and Radioactive Radiations, J. Hyg., 41(1): 1-16 (January, 1941). Permissible Dose from External Sources of Ionizing Radiation, National Bureau of Standards Handbook 59, for sale by the Superintendent of Documents, U.S. Govern ment Printing Office, Washington 25, D.C. Ritchie, R. H.: Health Physics Semiannual Progress Report, ORNL 1684, January, 1954.

Plough, H. H.: Radiation Tolerances and Genetic Effects, Nucleonics, 10(8): 16-20 (August, 1952). 11. Boche, R. D.: Effects of Chronic Exposure to X Radiation on Growth and Survival, chap. 10, "Biological Effects of External Radiation," National Nuclear Science Series. Div. 6, vol. 2, H. A. Blair, editor, McGraw-Hill Book Company, Inc., New York, 1954. Nat. 12. Henshaw, P. S., and O. W. Hawkins: Incidence of Leukemia in Physicians, Cancer Inst., 4(4): 339-346 (February, 1944). March, Herman C: Leukemia in Radiologists in a 20 Year Period, Am. Med. Sci.. 220(3) : 282-286 (September, 1950). Peller, S., and S. Pick: Leukemia and Other Malignancies in Physicians, Am. Med. Sci.. 224(2): 154-159 (August, 1952). March, Herman C. : Leukemia in Radiologists, Radiology, 43(3): 275-278 (September. 1944). Warren, Shields: Longevity and Causes of Death from Irradiation in Physicians. Am. Med. Assoc., 62(5): 464-468 (September 29, 1956). 13. Blair, H. A.: A Formulation of the Injury, Life Span, Dose Relations for Ionizing Radiations. I. Application to the Mouse, Unclassified University of Rochester Report UR 206 May 13, 1952. Blair, H. A.: A Formulation of the Injury, Life Span, Dose Relations for Ionizing Radiations. II. Application to the Guinea Pig, Rat and Dog, Unclassified University of Rochester Report, UR 207, July 3, 1952. Blair, H. A.: Shortening of Life Span by Ingested Radium, Polonium, and Plutonium, Unclassified University of Rochester Report, UR 274, September, 1953. Blair, H. A.: Recovery from Radiation Injury in Mice and Its Effects on LD-50 for Durations of Exposure up to Several Weeks, Unclassified University of Rochester Report. 10.

J.

J.

J.

J.

14.

UR 312, February, 1954. Maximum Permissible Amounts of Radioisotopes in the Human Body and Maximum Permissible Concentrations in Air and Water, National Bureau of Standards Handbook Printing Office, 52, for sale by the Superintendent of Documents, U.S. Government Washington 25, D.C.

HEALTH PHYSICS

Sec. 7-2]

7-59

J.

of the International Commission on Radiological Protection, Brit. Radiol., supplement 6 (1955). 16. Hurst, G. S., W. A. Mills, F. P. Conte, and A. C. Upton: Principles and Techniques of Application to Acute Lethality Studies of Mice with the Mixed Radiation Dosimetry. Cyclotron, Radiation Research, 4(1): 49-64 (January, 1956). 17. Snyder, W. S., and J. Neufeld: Calculated Depth Dose Curves in Tissue for Broad Ra/liol., 28(331): 342-350 (July, 1955). Beams of Fast Neutrons, Brit. 18. Handbook on Neutron Protection up to 30 Mev, National Bureau of Standards Hand 15. Recommendations

J.

book, 63,

1957.

K. Z., and M. R. Ford: Developments in Internal Dose Determinations, Nucleonics, 12(6): 32-39 (June, 1954). Morgan, K. Z.: Radiation Protection, chap. 4, "Nuclear Reactors for Industry and Universities," Ernest H. Wakefield, editor, pp. 38-50, Instruments Publishing Com pany, Pittsburgh, Pa., 1954. Morgan, K. Z. : Instruments for Measuring Radiations, "Encyclopedia of Instrumenta tion for Industrial Hygiene," pp. 937-947, University of Michigan, Ann Arbor, Mich., September, 1955. Davis, D. M.: Survey Monitoring Instrument Manual, ORNL 332, for sale by the Office of Technical Services, Department of Commerce, Washington 25, D.C. (1955). Hurst, G. S., R. H. Ritchie, and H. N. Wilson: A Count-Rate Method of Measuring Fast Neutron Tissue Dose, Rev. Sci. Instr., 22(12): 981-986 (December, 1951). Hurst, G. S.: An Absolute Tissue Dosimeter for Fast Neutrons, Brit. Radiol., 27(318): 353-357 (June, 1954). Glass, F. M., and G. S. Hurst: A Method of Pulse Integration Using the Binary Scaling Unit, Rev. Sci. Instr., 23(2): 67-72 (February, 1952). Rossi, H. H., and G. Failla: Neutrons: Dosimetry, "Medical Physics," O. Glasger, editor, vol. 2, pp. 603-607, Year Book Publishers, Inc., Chicago, 111., 1950. Roentgenol. Radium Rossi, H. H., and G. Failla: Dosimetry of Ionizing Particles, Am. Therapy, 64(3): 489-491 (September, 1950). Sigoloff, Sanford C: Fast-Neutron Insensitive Chemical Gamma- Ray Dosimeter, Nucleonics, 14(10): 54-56 (October, 1956). Hurst, G. 8., J. A. Harter, P. N. Hensley, W. A. Mills, M. Slater, P. W. Reinhardt: Techniques of Measuring Neutron Spectra with Threshold Detection-Tissue Dose Determination, Rev. Sci. Instr., 27(3): 153-156 (March, 1956). Way, K., and E. P. Wigner: The Rate of Decay of Fission Products, Phys. Rev., 73(1 1) : 1318-1330 (June 1, 1948). Evans, Robley D.: Physical, Biological, and Administrative Probloms Associated with the Transportation of Radioactive Substances, National Academy of Sciences, Nuclear Science Series, Preliminary Report No. 11, Publication 205, 1951. Transportation of Restricted Articles by Air, C. A. B. 16 — Official Air Transport Restricted Articles Tariff No. 6-B, Emery F. Johnson, Agent, National Airport, Washington, D.C, March, 1957. Regulations of U.S. Coast Guard, Federal Register, 21(183): p. 7108 (September, 1956), for sale by Superintendent of Documents, Government Printing Office, Washington 25,

19. Morgan, 20.

21.

J.


22.

J.

23. 24.

25. 26.

D.C. Regulations Governing Transportation of Radioactive Materials by U.S. Mail Service, U.S. Official Postal Guide, 5(1), chap. IV, art. 37, p. 53 (July, 1953), for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. "Postal Manual," Chaps. 1 and 2, 125.2, U.S. Post Office Department, December 1, 1954, for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. Campbell, H. A.: Tariff No. 10, "ICC Regulations for Transportation of Explosives and other Dangerous Articles by Land and Water in Rail Freight Service and by for Shipping Con Motor Vehicle (highway) and Water Including Specifications tainers," Bureau of Explosives, Office of Chief Inspector, 30 Vesey St., New York 7,

N.Y.

"Transportation of Radioactive Materials," Handbook of Federal Regulations, for Printing Office, Washington, D.C. (1955). sale by U.S. Government 27. Smith, R. B.: Pyrophoricity — A Technical Mystery under Vigorous Attack, Nucleonics, 14(12): 28-33 (December, 1956). Reactor Fuel Solutions, Nucleonics, 28. Callihan, A. D.: Nuclear Safety in Processing 14(7): 39-11 (July, 1956). — A Literature 29. Sachs, Frances L., and Katheryn D. Ballentinc: Beryllium Toxicology Search, Unclassified report Y 975, July 17, 1955. There is a classified search on the same subject, Y 974.

7-3

NUCLEAR RADIATION SHIELDING BY

it,

is

1

6

is

5,

3

2

a

is

is

is

a

is

a

The primary purpose of shields is to protect personnel and equipment outside a reactor from the radiations that are produced inside. In addition, of course, other radioactive sources must be shielded as well. The shield which protects personnel is referred to as the biological shield. Many layers of material are usually placed between the reactive core itself and the outside of the shield, and all these attenuate the radiation and must be taken account of in a shield design. Accordingly, the shield actually consists of all the material between the source of radiation and the occupied regions. Of these layers, several might be named. The first is the reflector, the prime purpose of which is to "reflect" or scatter neutrons back into the reactive Next might be the first part of the shield proper. At the inside of the shield core. there is often considerable heat deposited by the radiation. In order to ensure that this heat is removed and does not cause damage to other regions beyond special section, called a thermal shield, This shield might typically be usually installed. slab of iron cooled by water or gas. Following the thermal shield might be a pressure shell, to contain radioactive fission products, coolant, etc. Finally, the biological shield proper would follow, which in many stationary installations consists of concrete, either ordinary or of some special aggregate such as barytes or limonite. This section discusses the whole complex of layers outside the reactive core and the means by which the required radiation attenuation produced. The section divided into several articles, the first of which deals with that informa tion which best known, namely, 7-ray attenuation theory. Methods are described for calculating the attenuation of radiation from a point source as measured by point detector at some distance in an infinite homogeneous medium. In Art. the 7-ray sources are listed and described. Article discusses the neutron attenuation theory, again for a point source and point detector in infinite homogeneous media, with the neutron sources described in Art. 4. In Art. on geometry, the extension described, from point sources to actual large distributions of sources such as reactors and practical problems are treated in terms of the simple attenuation calculations of and 3. Article deals with shield materials, giving information on materials Arts. which have been of common interest in attenuation problems. The theory of 7-ray attenuation in single media well understood, and answers are available for all energies and thicknesses of interest. For multiple regions the For the case of neutron attenuation in hydrogenous theory considerably less clear. shields, such as water, there theory which agrees reasonably well with experiment. Inhomogeneities in the shield, as with layers of different materials, make the problems of calculating either neutron or 7-ray attenuation so difficult as to be essentially unsolved by the more sophisticated techniques. Even mixtures of materials present problems that cannot be simply solved at this time. As a consequence of this, much of the shielding lore in this handbook appears as rules of thumb or laws which have been developed simply because they describe reasonably well the phenomena which have been observed in measurements. The solutions are not satisfactory in the sense that they give exact answers. Nevertheless, the accuracy which required shield design not great, and possible to design shields with sufficient accuracy for most conditions. is

it

is

7-«0

in

is

is a

is


Everitt P. Blizard

Sec. 7-3]

NUCLEAR RADIATION SHIELDING 1

7-61

GAMMA-RAY ATTENUATION* '"' 1.1

Attenuation Processes

The attenuation of 7 rays in matter is attributable primarily to three processes which their energy is given to electrons. These electrons slow down quickly, The processes are described below. producing heat in the immediate vicinity. 1.11 The 7-ray photon gives all its energy to an atom, Photoelectric Effect. The causing the ejection of an electron from an orbit (usually an inner orbit). ensuing X ray caused by readjustment of electrons in the atom is usually replaced As a result the photoelectric effect can by emission of a second "Auger" electron. be accurately treated as an absorption The photoelectric-effect cross process. section varies from element to element approximately as (Z is the atomic number). It is the dominant process for low-energy photons and materials of high atomic number Z. (It dominates for energies less than 150 kev with copper and for energies less than 500 kev for lead.) The photon creates a positron-electron pair and is absorbed 1.12 Pair Production. in the process. The positron subsequently is annihilated nearby, giving two photons of 0.51 Mev. Since these are not very penetrating in comparison with the original photon, they are usually assumed to be absorbed nearby, so that the pair-production It does not occur for photons process is treated effectively as an absorption process. of energy below 1.02 Mev, and it increases in cross section with increasing photon It varies from element to element as Z1. It is the dominant effect for energy. high-energy photons and high-atomic-number materials (i.e., it is dominant for E > 80, 15, or 5 Mev, with materials of Z = 1, 13, or 82, respectively). 1.13 The photon interacts with an atomic electron, which is Compton Effect. loosely bound in the sense that its binding energy is small compared with that of the photon, giving some energy to the electron and proceeding with lower energy and altered course. The probability for scattering through any angle varies with primary photon energy, the scattering being more forward for the higher photon energies. The energy E' of the scattered photon is uniquely determined by the initial energy E and the scattering angle 8 as follows:


by

E'

= 1

+

(1

-

E cos 0)(£/O.51

Mev)

(1)

are expressed in million electron volts. For small scattering angles The cross section is also greatest for this case. degradation in energy is least. For 90° scattering, the scattered photon can have no more than 0.51 Mev regardless of the initial energy. For 180° scattering (back scattering), the scattered photon can have no more than 0.255 Mev. The cross section for the Compton effect varies from clement to element directly with Z. It decreases with increasing photon energy but is the dominant effect in the intermediate energy region. Because of the Compton-scattered photons the attenuation in matter is effectively decreased, a fact which is taken account of by the so-called "build-up factor" in attenuation calculations. where all energies the

1.2

Attenuation Coefficients

The attenuation coefficients are quantities which indicate the attenuation by all three of the foregoing processes. 1.21 The total linear attenuation coefficient ji is the sum of the macroscopic cross sections for all three processes. Thus the current density of uneollided photons at a distance from a point source of photons is

Y"{R,t) * Superscript numbers

- ^r- photons/(cm«)(sec)

refer to References


(2)

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7-64 where

RADIATION AND RADIOLOGICAL PROTECTION S R

— source

[SEC.

7

strength, photons/sec

= distance from source, cm = probability per unit distance for interaction, or the macroscopic ft cross section, for all processes in the material, cm-1 (Table 1 gives values of ii/p, where p = density, g/cm', for the material and for

the photon energy En, of the source) + fc + Mj>J> t = thickness of attenuating material, cm lift, He, and npP = macroscopic cross sections for photoelectric effect, Compton effect, and pair production, respectively The uncollided flux from a collimated plane source incident on a shield of thickness — Itp'

x is

r°(x)

=

r°(0)e-'«

(2a)


where r°(0) is the incident flux. 1.22 The energy-absorption linear attenuation coefficient p.a is the probability for interaction, per unit distance traversed, multiplied by the average fractional In common usage the energy loss on Compton scatterings energy loss on interaction. is assumed to be the difference in photon energy pre- and postscattering, but all energy is assumed lost on photoelectric and pair-production interactions.

1.23 Mass attenuation coefficients are the foregoing quantities n, na divided by the material density p. Since the linear attenuation coefficient is clearly proportional to the quantity of material traversed in a given distance, the mass attenuation coeffi cients are independent of the material density, being dependent only upon the energy of the 7 rays and the type of material. It is for this reason that the mass attenuation Table 1 gives the total coefficients for the different elements are given in the tables. mass attenuation coefficients, and Table 2 gives the energy absorption mass attenuation Figure 1 gives the differential Compton cross section for scattering of coefficients. photons through any angle. Figure 2 gives the same quantity multiplied by the fractional energy carried away by the scattered photon.*

nuclear radiation shielding


Sec. 7-3]

7-65

140°

130°

120°

110° IOO°9Q°80°

70*

60°

50°

40"

140°

130°

120°

110° 100° 90° 80"

70°

60°

50°

40"

Fio. 2. Differential Compton cross section for scattering by electrons of photons of energy anioc* (moc* = 0.51 Mev). through any angle 6, multiplied by the ratio of scattered to incident photon energy, in units of 10-2« cm'/electron/sterad. (Ref. 5, p. 83.) 1.3

Gamma-ray

Dose Rate and Heating Data

1.31 Units of Radiation Dose. Physical radiation dose is measured in terms of the energy deposition. The unit is the "rad," which is the quantity of radiation which deposits 100 ergs per gram of material. Physical radiation dose has been measured also in rep (roentgens equivalent physical), where 1 rep = 0.93 rad for the case in which the rads are measured in air. Biological radiation dose is measured in terms of the biological effect of X rays The factor by which the physical radiation (usually with those produced at 250 kv). dose in tissue must be multiplied to obtain the biological quantity is called the rela tive biological The RBE will vary with the biological effect effectiveness (RBE). The unit of biological dose is the rem (rad equivalent, man). The being studied. RBE is currently taken to be 1.0 for all X rays and y rays. This factor may change as better information is obtained. 1.32 Calculation of 7-ray Dose. The current density of unscattered photons from a single isotropic point source is given by Eq. (2). The dose rale which the unscattered radiation induces is given by

D°(R)

= 5.767

X 10-* ^ E0r°(R,l) rads/hr p

(3)

where Eo = photon energy of radiation source, Mev Ma/p = energy-absorption mass attenuation coefficient for the source energy and the material in which dose is to be calculated, cm!/g (Table 2). Unless otherwise specified, dose in air is usually implied, although dose in The two are about the same tissue is more nearly the desired quantity. r°(/?,<) = current density of unscattered photons, summed or integrated over all sources, photons/(cm2)(sec), as calculated by Eq. (2) Since most dose-rate calculations arc for tissue, Table 3 is supplied which gives the conversion factors in Eq. (3) as a function of photon energy in a convenient form. The dose rate from scattered plus unscattered photons due to a point source is given by Dr(R,t) = BA„t)D0(R,t) rads/hr (4)

7-66

RADIATION AND RADIOLOGICAL PROTECTION TableS.

[Sec.

7

Flux for Unit Tissue Dose*

B; photon energy, Mev

Photon flux for 1 rad/h r, cm"1 aec-1

5.90 3.88

0. 10 0. 15

0.20

2.91 1.847 1 357 1. 085 9.065

0 30

0.40 0.50 0 60

0.90

6.98

1.0 1.25 1.5

5.77 4 77

2.0 3.0 4.0 5.0 6.0

3.405 2 63 2. 10 1 80 1.577 1.278 1. 067

4.18

8 0 10.0

X X X X X X X X X X

X X X X X X

X X

10' I0« I0« 10« I0« I0« 10' 10' 10« 1010' 10' 10' 10' 10' 10' 10' I0>

Energy flux for 1 ra d/hr. Mev/(cmJ)(see) 5.90 5.82 5.82 5.54 5.43 5.425

5.44 5.58 5.77 5.96 6.27 6.81

7.89 8.41

9.00

X 10' X 10'

X X X

X X X

X X X X

X X X

9 46 X 1.022 X 1.067 X

10> 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10> 10' 10' 10* I0«

where Br(nl) is the dose-rate build-up factor chosen from Tables 4 and 5 for source energy Ed per photon, material of shield, thickness of shield t, attenuation coefficient m for source energy Ed, and geometry (point isotropic or plane collimated source — see below). For more complicated source configurations it is necessary to integrate this expres sion over the sources, since doses from different sources must be added to obtain total dose at a point. The build-up factors in Tables 4 and 5 are calculated on the assumption that the They are adequate for dose in tissue, since the mass dose is to be determined in air. attenuation coefficients are very nearly the same. For convenience, Table 6 is given, which shows the number of curies of Co'0 and Cs1" 7-ray sources which are adequately shielded in lead shields of different thicknesses. 1.33 Calculations of Heating Due to 7 Rays. The rate of heat deposition in the shield from scattered plus unscattered y radiation at a distance R from the source with t cm of shield intervening is given by

H(R,l)

=

1.002

X

10-«^J5o«„(^)r0(/1,,f) p

ergs/(sec)(g)

(5)

2)

7)

\xl

where BApl) = energy absorption build-up factor, chosen for photon energy Ed, material, and number of relaxation lengths in the shield (Table Ha/p = energy absorption mass attenuation coefficient for the shield material, cmVgm (from Table 1.4

Factors Affecting Calculational

Method

is

is

6)

4)

is

The use of the foregoing article and the accompanying tables dependent upon the conditions of the problem. In most instances a judicious choice must be made with some compromise in accuracy. The recommendations which follow are designed to give conservative estimates of dose rate with a minimum of complication. More sophisticated approaches yielding more accurate results can be found in the references. 1.41 Choice of "Point Isotropic" (Table or "Plane Collimated" (Table Build the source embedded in or immediately adjacent to the shield, the up Factors. "point isotropic" build-up factor to be used (Fig. 3a). If a small fin linear extent)

If


* Calculated from Ref. 4.

Sec. 7-3]

NUCLEAR RADIATION

7-67

SHIELDING

4. Dose Build-up Factor, B, (Point isotropic source, Ref. 2)

Table

lU So, Mev

Material

2

3.09

7.14

23.0

2 52 2. 13 1.83 1.69 1.58 1.46 1.38 1.33

5. 14 3.71

14.3 7 68

2.77 2.42

4.88

0 5 1.0

2.37 2.02

4.24

2.0 3.0 4.0

1 75 1.64 1.53 1.42 1.34 1.28

0.2S5 0.5

Water

1.0

2.0 3.0 4.0 6 0

8.0 10.0

Aluminum.

.

6 0

8.0 10.0


0 5 1.0

2.0 3.0 4.0 6 0 8 0 10.0

Tin

0.5

10

2 0

3.0 4.0 6.0 8.0 10.0

Tungsten . .

0 5 1.0

2.0 3 0

4.0 6.0 8.0 10 0

0.5

Lead

1.0 2 0 3 0

4.0 5.1

6.0 8.0 10.0

Uranium.

0.5

. .

1.0

2.0 3.0 4.0 6.0 8.0 10.0

* Extrapolated

4

1

values.

2. 17 1.91 1.74 1.63

3.31 2.61

2.32 2.08 1.85 1.68 1 55

3.91 3 34

2.76 2.40 2.19 9.47 6.57 4.62 3 78

3.22 2.70 2.37 2.12

1.98 1.87 1.76 1.55 1.45 1.34 1 27 1.20

3.09

1.56 1.64 1.57 1.46 1 38 1.26 1.19 1. 14

2.08 2.30

3.09 3.74

2. 17 1.96 1.81 1.57 1.42 1.31

3 53 3. 13

1.28 1.44 1.42 1.36 1.29 1.20 1. 14 1 II

1.50 1.83 1.85 1.74 1.62 1.43 1.32 1.25

1.84 2 57

1.24 1.37 1.39 1.34 1.27 1.21 1.18 1.14 1. II

1 42 1.69 1.76 1.68 1.56 1.46 1.40 1.30 1.23

1. 17 1.31 1.33 1.29 1.24

1.30 1.56 1.64 1.58 1.50 1.36 1.27 1.20

1.16 1.12 1.09

2 89

2.43 2. 15 1.94 1.72 1.56 1.42

5.98 5.39 4. 13 3.51

3.03 2 58

2.23 1.95

2.82 2 37

2.05 1.79

2.72 2.59 2.41

2.07 1.81 1.64 1.69

2.26 2.51 2 43

2.25 2.08 1.97 1.74 1.58 1.48 1.98 2.23 2.21

2.09 1.85 1.66 1.51

7

72.9 38.8 16.2

8.46 6.23 5. 13

3.99 3.34 2.97 21.5 13. 1

8.05 6. 14 5.01

4.06 3.45 3.01 11.7 10.2

7.25 5 85 4.91 4. 14

3.49 2.99 4.57 6. 17 5 87

5.28 4.82 4.17 3.57 2.99 2.24 3.62 4.09 4.00 4.03 3.60 3.05 2.62 2.00 3.02 3.66 3 75 3.61

3.44 3 34 2 89

2.52

10

166

77.6 27.1 12.4

8.63 6.94 5.18 4.25

3.72 38.9 21.2 11.9 8 65

6.88 5 49

4.58 3.96 19.2 16.2 10.9 8.51 7.11

6.02 5.07 4.35 6.04 8.85 8.53 7.91 7.41

6.94 6. 19 5.21

19.5 12.8

9.97 7.09 5.66 4.90 80 8

37.9 18.7 13.0 10 1

7.97 6.56 5.63

112 9 89

8.50 7.54

8.85 6.95 5.98 141

58.5 26.3 17.7 13.4 10.4

8.52 7.32

8.64 13.7 13.6 13.3 13.2 14.8 15. 1 12.5

12.0 15.7 15.2 14.0

2.65 4.81

4.84 5.30 5.44 5.55 5.69 5.07 4.34

6.87 8.44 9.80

4.66 4.60 4.36 3.78

17.0 12.9

25. 1 19. 1 16.0 14.7 13.0 12.4

2.27

4.51

82.2 27.7

17.6 13.5

3 74

3 27 3.21

982 334

55.6 42.7

6.25 8.07 9.66

1.85

20

35.4 28.3

3. 12

2.97 3.95

2.26

50.4

2.61

1.67

2.61

456 178

4.64 5.27 5.92 6.27 6.29 5.40 4.65

2.50 3.09 2.96

15

11.7 13.8 14.1 12.5

2.08 3.67 5.36 6.97 8.01 10 8 11.2 10.5

18.8 19.3 20. 1

21.2 29.1

34.0 33.4

(7.35)* (10.6)* 14. 1

20.9 36.3 41.9 39.3 (2.73)* 5.86 9.00 12.3 16.3

23.6 32.7 44.6 39.2

(6.48)* 9.88 12.7

23.0 28.0 28.5

Table 5.

Dose Build-up Factor, BT

(Plane collimated source, Ref. 2) \>x

Material

Ea. Mev 1

Water

2.26

4.29 3.39

1.84 1.69 1.58 1.45 1.36

2.63 2.31 2. 10 1.86 1.69

2.94 2.74 2.35 2.13

10.0

2 07 1.92 1.69 1.58 1 48 1.35 1.27 1.22

1.0

I.6S

2.24 2.13

1.0

2.0 3.0 4.0 6.0 8.0

2.0 4.0 6.0

1.58 1.39 1.27 1. 16


10.0

0.5 1.0

2.0 3.0 4.0 6.0 8.0 10.0

Uranium

0.5 1.0

2.0 3.0 4.0 6.0 8.0

*

111(I

Table 6.

9.05 6.27 4.28 3.57

2.63

1.0

0.5

Lead

4

0.5 2.0 3.0 4.0 6.0 SO

Tin

2

1.17 1.30 1.33 1.29 1.25 1. 18 1.13 1.10

1.28 1.53 1.62 1.57 1.49 1.37 1.27 1.21

15

20.0

35.9

11.5

18.0

74.9 30.8

6.96

9.87

5.51

7 48 6. 19

4.63 3.76

2.63 2.30

12.4 11.6

8.78

5.47

7.41

20.6 18.9 13.7 11.4

6.46 5.35 4.58 4.07

3.27 2.89

I.9S

8.54 6 78

4 00

8.31 7.81 6.11 5 26 4.61 3.81

2. 17

14.4 10.8

4.86

3. 16

4.87 4.57 3.76 3.32 2.95 2.48

1.80 1.57 1.33 1.39 1.68 1.76 1.71 1.56 1.40 1.30 1.24

10

3. 12

1.90 1.71 1.55 1.44

1.24 1.38 1.40 1.36 1.28 1.19 1.14 1. II

7

9.92 8.39 7.33

6.70

3.40 3.27 2.69 2.27

5.18 5.12

1.77

2.81

4.53

1.63

1.87

2.18

2.80 3.36 3.55 3.29 2.97

2 08 3.40

4.35 4.82 4.69 4.69

2.61

4. 18

2.27

7.70 9.53 9.08

3.54

7 70

4.31

3.72

2.41

2.42 2. 18 1.87 1.69 1.54 1.45 1 90 2.15 2. 13

1.60

1.73

2.32 2.87

2.70

Co80

and

10.5 11.0

II. 0

9.68 4.20 5.94 7. 18

3.60 4.89 5.94 6.47 7.79 7.36 6.58

3 56

3.99 4.06

2.94 2.74 2.39 2.12

1.82 1.61 1.48

for

5.77

3 02

2.02

Lead Shield Thicknesses

7. 19

7.13 6.30

4. 12 3 65 3.21

Cs'"

Sources

Activity, curies Lead thickness. in.

1 2 3 4 5 6 8 10 12 14 16 18

Co" 1.4 1.9 1.7 1.3 9 6 2. 3 8 3 8

Cs'»

X 10 '

10 < X 10 • X 10 ■ X I0"»

X

X 10'

X 10' X I" ' X I0« X 10' 2.3 X 10' 7 X

1 3 X 7.5 X 2 5 X 6.4 1.6 X 3 7 X

2 X 8 X

IO-<

10' 10-' 10' 10" 10" 10'

10"

These shields are presumed to be spherical with the source near the inside surface, outer surface will be 7.5 mrcm/hr, which is laboratory tolerance.

7-68

The doge at the

Sec. 7-3]

NUCLEAR RADIATION

7-69

SHIELDING

it,

source is far from the shield, so that the radiation is nearly collimated, use the "plane collimated" build-up factor (Fig. 36). In case the sources are spread over a large an integration must be made over the surface surface near the shield but not on

it

Build-up Factor, B*

Energy-absorption

(Point isotropic source, Ref.

2)

Table 7.

is

is

(Fig. 3c). In order to decide whether, in this integration over the array of sources, the point to be used, necessary to define isotropic or plane collimated build-up factor

W

10.0

0,5

II

1.51 1.76 1.58 1.37 1.24 1.19 14 10 1.08

2 0

1.0

3.0 4.0

II

1. 1.

6.0 8.0 10.0

9.97 7.00 4.53

20.4

3

2.38

73

1.99 1.78

2.99

5.51

8.58

2.61

5.47

9.31

7.48 5.62 4.47

35

2.03

3.30

1.74 1.45 1.31 1.24

2.67 1.74 1.58

2.38

1.80

2.19

2.61

2.37 10 1.73 1.49 1.39 1.29 1.21 1.16

2.06

47 2.73

4.74 4.70 3.85 3.25 2.96

39 14 2.50

2.09 1.89 1.70 1.50 1.38

2.63 2.21 1.92

28.3 18.2 13.5

7.62

6. 10

5

185 69.1

13.4 10.2

27

5.22

34.2 21.9

64.3 38.8

11.9

19.3 14.0

III

8.82 10 35 23 69

8.67 6.89 6.17

17.0 21.3 17.7 15.2 13.2 10.9

6.80 5.38 4.38 3.78

8.83 7.00 5.76

20.2

4.28 3.70

11.6 13.6 11.0 43

0

9 1

104 44.

8.94 6.93

3 4 5 7.

3.61 03

70 73

9.70 8.81

3.52 7.78 9.05 8.65 8.46 9.38 9.46 8.60 7.15

3.01 6.01 32 5.43

4.79 4.61

4.20 3.50 2.92

9.92 7.88 6.74 101

58.5 27.4 19.8 16.7 12.7 10.4

9.89

22

5

23 12.8

7.90 6.05 4.90

17.9 13

28.7 24.1

20.2 19.

1

49.2

15.0

13.6

13.4 10.

7.07

2

26.9 8.63 6.35 5.05 3.92 3.25 2.83

83.8 28.6

3.60

7.40 4.93 3.88 3.24 2.63 2.26 2.02

11.4

20.0

4 5

3.98 3.36 2.88

12.9 9. 12 7.01 5. 18 28

906 306

1

5.

2.76 2.42 2.13

18

28.2

4

1 1

7

4 3

37

3

3

8.67 6.57

421 163 51

20

20.8 20.8 20.3 9.70 12.0 12.6 13.9 18.5 21 23

0 5

1. 1.

1.73 1.49 1.34 1.21 14

18 54

154 71.5

8

2.0 3.0 4.0 6.0 8.0

3.38

68. 36. 16.8

15

6

2.25 2.12

1.0

2.21 1.95 1.63 1.45 1.35

10

3

0.5

2.64

7.85 4.99 4.10

4

10.0

58

13.4

3. 3

1.

2.0 3.0 4.0 6.0 8.0

5.

4.84

1.81 1.62 1.51

2 3

1.0

Lead

80 2. 19 1.78 1.58 1.45 1.30 1.21 16

2

0.5

1.89 1.74 1.60

4.92 3.64 2.74 2.37 2.09

2.

10.0

2.52 2.18

2.61 15 1.80 1.66 54 1.40 1.31 1.25

1

2.0 3.0 4.0 6.0 8.0

3.71 82

21.8

20.4

I

if

I

1,

-f-

id

is

of

at the outer shield cone radiation. This cone, the apex of which surface, includes all rays through the shield for which ui < being the shield Plane thickness, the attenuation coefficient, and the ray length in the shield. all the radiation incident on the shield collimated build-up factors are then used within the base of this cone arrives at an angle to the shield normal no greater than

the -principal ix


1.0

Tin

2

10.0

1.85 1.74 1.59 1.46 1.38 1.31

2

2.13

0.5

.

6.88 4.93

46

10

2.0 3.0 4.0 6.0 8.0 Aluminum.

3.03

2

0.255 0.5

2

Water

1

Eo, Mev

Material

7-70


USE

POINT

[SEC.

7

ISOTROPIC

BUILDUP FACTOR


USE

REACTOR

Flo.

CORE

PLANE COLLIMATED BUILDUP FACTOR

USE

POINT

ISOTROPIC

BUILDUP FACTOR AND INTEGRATE

3. a. Geometry for ease of source in shield, b. Geometry for case of source far away from shield, c. Geometry for case of large source near shield.


Sec. 7-3] the cone

In other words, if

half angle.

\%l

-

a d and d > I, factors are where d = a = I =

7-71

+T

pi

t

use plane collimated build-up factors. (Otherwise point isotropic build-up to be used.) distance from source array to shield maximum radius of source array, measured perpendicular to d

shield thickness Attenuation Coefficient for Single Element Not Listed in Tables. Since the commonly used in shields, this should not often be a tables cover most elements However, the quantities Ap/p and Apa/p (A = atomic weight) are smooth problem. Plot the values of this quantity for neighboring functions of the atomic number Z. elements against Z, and interpolate. 1.43 Build-up Factor for Element Not Listed in Tables. Build-up factors are smooth functions of atomic number Z. Interpolate. 1.44 Attenuation Coefficient for Homogeneous or Nearly Homogeneous Shield of Many Elements. Number the elements in the shield 1, 2, 3, . . . , i, . . . , n. the fractional

densities pi for each element

^shield

density

p =

V

shield

pH (?) \p/>

(either kind)

(6)

m

+

0.11

0.0636

X

0.89)

= 0.0706 cm«/g

1

- - (0.126

X

Mev

p

-, H20,

1

Example

:

i

p 2,

The attenuation coefficient

is

=

y

^

2,

a

is

,

1,

if

is

E

a

of

is

If

is

1.46 Build-up Factor for Homogeneous or Nearly Homogeneous Shield of Many Elements. There no simple way to choose the build-up factor for this case from a conservative but not necessarily accurate among the tables available here. answer required, choose the build-up factor for the component which would give the largest dose rate. Calculate the number of mean free paths pi, using the method the foregoing paragraph. If more accurate result needed, plot nip for the shield material vs. photon energy for all energies below the source energy and choose an element for which the same plot similar in shape. Use the build-up factors as the shield were wholly of this element. 1.46 Number the layers Attenuation Coefficient for Laminated Shield. ■ ■■,], . . . m. Determine the attenuation coefficients (p/p)i for these layers and their thicknesses
(-)

p^,

(7)

is

Note that empty regions in a shield are taken account of in this expression, since the density will be zero or nearly so, nullifying their contribution to the exponential attenuation, although the geometric attenuation l/7i* will be unaffected. 1.47 Build-up Factor for Laminated Shield. Build-up factors have not been calculated for laminated shields. However, the build-up factor largely determined char the total number of mean free paths (pi, from previous paragraph) and this at least two or three mean acteristic of the material in the outermost region free paths in thickness \(p/p)tp > 2]. Choose the build-up factor for the material of the outermost thick region and for the number of mean free paths in the whole shield (pi). In case the outermost single region not thick, use the method of Art. 1.45 above, for the materials constituting the outermost two or three mean free paths. is

if

is

is

by


and the attenuation coefficients (p/p)i. determined by summation as follows:

is

i

Determine

p?j

1.42

7-72

RADIATION

AND RADIOLOGICAL

SOURCES

2

OF

PROTECTION

[SEC.

7

RAYS AND X RAYS

y

X rays come from atomic rear Gamma rays are produced in nuclear transitions. rangement or from electron acceleration in the atomic coulomb field. Except for their origins they are indistinguishable. Sources of y rays in reactor installations are discussed below. 2.1

Prompt-fission

7 Rays

At the moment of fission, 7 rays are given off from the fissioning nucleus. The total energy given off is 7.46 Mev per fission'-7 on the average. Similarly there are Table 8. 7-ray energy,

Mev

Energy Spectrum of Prompt 7 Rays from Fission of TJm* 7 rays per 0.1 Mev

0.2 0.3 0.4

0.815 0.697 0.661

0.622 0.553 0.474 0.408 0.353 0.307 0.272 0.240 0.205 0.180


O.S

0.6 0.7 0.8 0.9 1.0 t. 1

1.2 1.3 1.4 1.5

16

1.7 1.8 1.9

2.2 2.3 2.4 2.5 2.6

~ZT

0.0409

2.8 2.9 3.0

4. 1

4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

0.0651

0.0579 0.0512 0.0457

0.0181

6. 1

6.2 6.3 6.4 6.5 6.6 6.7

0.0268 0.0244 0 0219

0.0150

0.0136 0.0126 0.0116 0.0106 0.00985 0.00908 0.00830 0.00764 0.00704 0.00649 0.00604 0 00556 0 00519 0.00470 0.00430

5.2

53

0.0165

0.0298

5 1

Mev

0.0198

0.0330

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

7-ray energy,

5.4 5.5 5.6 5.7 5.8 5.9 6.0

0.0369

3. 1

0.139 0.125 0.113 0.0907 0.0818 0.0726

2.1

Mev

7 raya per 0.1 Mev

0. I5B

0 102

2.0

7-ray energy.

6 8

6.9 7.0 7. 1

7.2 7.3 7.4 7 5

7.6

7 raya

per 0.1 Mev

0.00399 0.00371 0.00341 0.00314 0.00290 0.00264 0.00243 0.00223 0.00204 0.00188 0 00172 0 00157 0.00139 0 00128 0.00115 0 00103 0.000916 0 000833 0 000731 0 000629 0 000547 0 000467 0 000388 0.000308

* R. L. Gamble, "Prompt Fission Gamma Rays from Uranium 235," Dissertation, University Texas. Austin, Tex., June, 1955.

of

7.51 photons per fission on the average. The energy per photon is distributed accord ing to Table 8. A simple analytic form for this distribution, good to +30 per cent in the region from 0.2 to 7 Mev, is

Y(E)

7.5
= number of photons given off per Mev energy interval at energy = photon energy, Mev is to be noted that no 7 rays arc reported of energy greater than 7.6 Mev.

where

It

=

Y(E)

E

2.2

(8)

E

Decay 7 Rays from Fission Products

Most fission products arc radioactive, emitting 0 and 7 rays. Both the fission products and their emanations are tabulated elsewhere.8 The radiations from gross fission products vary widely in energy and the decays are of various half-lives. A quite complete analysis by Clark86 has been used* to obtain plots of 7-ray emanation


Sec. 7-3] ra million

7-73

volts per second per watt, for seven energy groups. Some of reproduced as Fig. 4. The figure gives the energy release, in million electron volts per second per watt, produced by U"4 fission products for infinitely long periods of operation. Thus, if Ti(T<,,Td) is the energy release rate in million electron volts per second per watt for the ith energy group at Td days after shutdown, following To days of reactor operation or fissioning, the ordinate of the figure gives electron


this data is

time after

shutdown

Fio. 4. Decay of fission products I

Mowing

For any other operation time relation: Tt{Ta,Ti) 2.3

tdoys)

after infinite operation.

the energy

= r,-( «,7\,)

- r<(«,

release rate can be found by the

To + T„)

(9)

Decay y Rays from Activated Materials

Decay y rays originate in activated reactor structure, shielding, housing, etc. 'M«iy elements, upon absorbing neutrons, become radioactive and decay with the The data (cross sections, decay rates, and energies) I *
1

V l-L

1

j

U 1 9

0

1

7 »

00?

3 \*

142

17

2 0.

1'

V V

00 7 7

? 77 ^

30 »•<•« 41

>27

0 II

380 2.4 4.8

00

25

93

7 0

no

50

22

7

1

o3

\ 71

6.446 15 997

«^9.2I6 7. 261

7JV^

7.0 0.4 10. 16

1.6

.7

.78

.99 .64

3.0

2.6 3.5

0 0

V

7

±

>59

04

24

0

0.059 12.6

► Q.95

L 3.3

References

14

26 14, 17. 20-23. 28. 29. 37 36 16 15, 21., 22. 28/29, 31.32, !<27 V7- 21. 22. 27 27

26 II. 32 14 14

16 19. 27

27. 38 27. 33

14 13. 17. 27 25. 30 13

7. 2. 5.

Mercury M'>!yt.
? ?

T

J

Many

7

4

3

43

95

6.1

7 36

33

3<

02000

38

67/ 3.

I

Magnesium Munganeso

\*

? 0

ioo

6

2

3'

73 67 2.8

4

23 66

8 914 63 78 494 230 86

767 625

42

97

22 35

>2

5 2

80

716 486

0.

4.

930

212 69

80 0. 93 26

0. 0.

2.6 3.68 3.3

78

0.

4

Lead' Lithium-6

7 18 49

748

5.

Iodine

f 1

>23

0

7.

Iron

0 T

0

J

*

3.59 0.009 36.300 94 0.330 190

1 1 4 4-

Copper Fluorine Gadolinium Gold HydrogenIndium

>

1

3 1

21

76796409 83 95 56

2.5

S(£.)

0.

18

110001 2.4

~2

1 4

9

12 36

16

>37

000 101

724 80 30 23 814 17 478 046

E. Mev 0. 3. 8. 3.

V

00 17

35

>

2. 6.

2.9 34.8

60 100 13

>44 50 <30 20

123

21 12 22 14 75

5-7

photons/ capture*1

Capture

Fraction of energy emitted S(£„)»«

Neutron

Average No.

Thermal

Highest energy ray, Mev

from

1

25.000 0.406 0.0045 32

73

77 36 47 75 50 100

000 20

>I3 ~80 ~80 -80

0

20

?

I

Chromium Cobalt

1 9

13 CadmiumCalcium Carbon- 12 Chlorine

4. 3*

V

7

1.17 0.009 015 3.990 3.500

1 3
1-3

in designated energy intervals per 100 captures, Mev

Rays

,

Beryllium' Bismuth' Boron- 10' Cadmium

0.215 6.4

0-1

Photons

5 3

Antimony Arsenic Barium

typo

Spec tral

Gamma

7

Aluminum

Target

(n,7) cross section, barns

Thermal

Table


9.

2 1313

\ 98

*T

1

uncertain

Complicated Complicated,

100(1.17).

9.58- min Mg": 2.59- hr Mn": 15(2.13)

15(1.49).

100(1.33)

10(0.84). 2(1.01) 100(0.845). 25(1.81).

54.3-min In'": 18(2.09). 54(1.27). 39(1.09)

12-sec F>°: 100(1.63)

5.3-year Co":

Fast B-l l»: 94(.fJ 478)

100(1.80)

2.3-min Al»:

Decay >afc.'

—- ■

-■

1.

0

4

8.

1

2 2 ~ 100

7 7 7

2 7

at

» 3)

? ?

2

4 1 13

10 0.5 18 17 2.6 2.8

2.4

2.7

2.5

0.82 0.60

1.02

0.76

1.44

0.48

14 13 13 14

12. 16

14 13 14 13 13 14 14

7 7

4 14

27

15. 18.27

15. 17.37

16

7.

7

7

2

is

7.

8

8

4

??

j ?J 7

is

1.

V": 100(1.46) 250-day Zn": 25(1.12) Zr«: 65-day ~98(0.750), Nb»: 100(0.764)

3.74-min

Complicated

35-day

270-day Ag'» 100(2.758). 100(1.380)

Complicated 14.9-hr Na»:

100(0.89) 100(1.12).

4(1.58)

ytfi*

85-day Sc":

19-hr Pr»": Exist (34)

Complicated

Decay

* Fraction Em the minimum has no deep of available y-niy energy observed in the measurement; by rays with energy greater than Em. energy emitted significance. of isotopes and/or the complexity of the decay schemes render esti means that decay 78 above 750 kev are claimed to exist but the multiplicity "Complicated" See Ref. 33 for data. mations of intensities difficult. for decay as follows: Half-life of decay-7 emitter Notation given firat, then number of decay rays of each energy >750 kev per 100 neutron captures. rays " 9.58-min Mg17: 10(0.84), 2(1.01)" means isotope emits 10 0.84- and 1.01-Mev decay in parentheses. Thus rays per 100 captures. Energy For elements thus marked, number of unresolved photons has been determined; for all others only the intensity of resolved lines given. For lead, in addition, the seven to 7-Mev For these elements, the highest energy the only rays ray ray present in the highest occupied energy interval. are all 6.73 Mev; for beryllium, in addition, the fifty 3- to 5-Mev 7s are all 3.37 Mev. Boron-10 from Li7 following a decay of B11. decay intensities for zirconium may be off by 50 per cent. Capture-7-ray

2.5

2.9 <2

3.00

0.66

13

5.

y

16 0.5

5.6

2.6

S(Em)

References

(Continued)

4.

(2-3)4

7 100

7

>50

14

80 04

2* 3* M

2 62 33 99 14.5 54 29 35

12

19 10 7.94 7.920 9.28 5.83 6.792 7.89 8.85 10.483 10.55 7.27 6.41 9.22 8.64 6.07 6.54 9.35 9.39 7.42 7.305 9.51 66

Mev

SiEmV"

Fraction of energy emitted

Capture.

3

V

29 27 41 17 29 49 91

II

39

0

V V

>

100 ~90 >50 ~140 >19 ~50 ~100

10

>7

Average No. photons/ capture2*

Neutron

2.

(2-

J

7 7

1*

14 90 43 15 32

54 <35 115 45 36 34 38 45 63 65 229 70 61 62 80 26 76 139 33 53 24 48 113

5

V

? ?

1* 3*

V 7 7

7

V

5-7

5 8

II 2.

51

2 ~120 36 ~80 ~70 ~150

? 7 7

00 1 5.

Tungsten VanadiumZinc Zirconium8

2.

? 1

1.89 11.2 150 10.600 22 11.8 0. 160 60 0.47 16 0.49 21.3 3.3 0.65 5.8 19.2 4.7 1.06 0. 18

<5

j

y

1-3

Highest energy ray, Mev

Thermal

5 2 3.

Sulphur Talantum Thallium Tin Titanium

11 193

V

0-1

0

0.

type

in designated ent'rpy intervals per 100 captures, Mev

trom

Photons

3

Niobium NitroRenPhosphorus Platinum Potussium Praseody mium Rhodium Samarium Scandium Selenium Silicon Silver Sodium Strontium

section, barns

Spec tral

\

cross

Thermal

Rays

Table

Gamma

9.

Target

■ "


\r 1

83 7

7

it

5-

is

is

7

-,

is

7

7

7

*

e

is

S

is

d

'

/

*


7,

.

1

(

:

9.

B. B. B. B. G. G. G. B. G. B. B. G. G. B. T. T. H. H. R. R. W. W. W. C. R. F. M. A. A. B. B. B.

Table Gamma Thermal Neutron Rays from Capture. (References) B. Kinsey, G. A. Bartholomew, and W. H. Walker, Phyt. Rev., 82: 380 (1950). B. Kinaey, G. A. Bartholomew, and W. H. Walker, Phyt. Ren.. 83: 519 (1951). and W. H. Walker, Phut. Rev., 85: 1012 (1952). B. Kinsey, G. A. Bartholomew, B. Kinaey and G. A. Bartholomew. Phyt. Rev., 89: 375 (1953). and B. B. Kinsey, Phya. Rev.. 89: 386 (1953). A. Bartholomew A. Bartholomew and B. B. Kinsey, Phyt. Rev., 90: 355A (1953). A. Bartholomew and B. B. Kinsey, Phyt. Rev., 93: 1434(E) (1954). B. Kinsey, G. A. Bartholomew, and W. H. Walker, Can. J. Phyt., 29: (1951). A. Bartholomew and B. B. Kinsey, Can. J. Phyt., 31: 49 (1953). 10. B. Kinsey and G. A. Bartholomew, Can. J. Phyt.. 31: 537 (1953). B. Kinsey and G. A. Bartholomew. Con. J. Phyt., 31: 901 (1953). 12. A. Bartholomew and B. B. Kinsey. Can. J. Phyt., 31: 927 (1953). 13. A. Bartholomew and B. B. Kinsey, Can. J. Phyt., 31: 1025 (1953). 14. B. Kinsey and G. A. Bartholomew, Con. J. Phyt.. 81: 1051 (1953). 15. H. Braid. Nuclear Sci. Abttr., No. 6B, New Nuclear Data (1953); Phyt. Ren., 90: 355A (1953). 16. H. Braid, Phyt. Rev., 91: 442 (1953). 17. T. Mots, Phyt. Rev., 90: 355A (1953), Nuclear Science Abttractt, No. 6B, New Nuclear Data (1953). 18. T. Mots, Phyt. Rev., 93: 925A (1954). 19. W. Prinirle and G. Isford, Phyt. Rev., 83: 467 (1951). 20. W. PrinEle, Phyt. Rev., 87: 1016 (1952). 21. A. Reardon, R. W. Krone, and R. Stump, Phyt. Rev.. 91: 334 (1953). 22. A. Reardon, R. W. Krone, and R. Stump, Phyt. Rev., 91: 442A (1953). 23. Thornton, et al.. Phyt. Rev., 86: 604A (1952). 24. O. Muchlhause, Phyt. Rev., 79: 277 (1950). 25. E. Bell. L. G. Elliot, Phyt. Rev., 79: 282 (1950). 26. Ajaenberg and T. Lauritsen, Revt. Mod. Phyt., 27: 77 (1955). 27. Rcier and M. H. Shamos, Phyt. Rev., 95: 636A (1954), also Phyt. Rev.. 100: 1302 (1955) 28. Recksiedler and B. Hamcrmesh, Phyt. Rev., 95: 650 (1954). Recksiedler and B. Hamermesh. 29. Phyt. Rev., 96: 109 (1954). 30. Hamermesh and R. J. Culp. Phyt. Rev., 92: 21 IL (1953). 31. Hamermesh and V. Hummel, Phyt. Rec, 88: 916 (1952). 32. Hamermesh and V. Hummel, Phyt. Rev., 83 663 951) 33. M. Hollander, Perlman, and G. T. Seaborg, Rat. Mod. Phyt., 25: 469 (1953). 34. P. Chudom and C. O. Muchlhause, Plutonium Project Report CP-3801, p. 25. Cross Section," AECU-2040. 35. "Neutrun 36. H. T. MoU. 1955). Phyt. Rev., 99: 656A Can. J. Rettarch, A28: 475 (1950). 37. C. H. Millur, A. G. W. Cameron, and M. Glicksimm. 38. H. E. Kuhitschek and S. M. Dancoff. Phyt. Rev., 76: 531 (1949). P. S. Mittleman and R. A. Licdtke, Nucleonics, 13(5): 50 (1955), revised and brought up to date as of April, 1956. source:

1

7,

1. 2. 3. 4. 5. 6. 7. 8. 9.

1.

1

(

I.

J.

NUCLEAK

Sec. 7-3]

2.4

off

7-77

RADIATION SHIELDING Capture y Rays

At the instant of neutron capture (within ~10-14 sec), the capturing nucleus gives y rays of total energy equal to the neutron binding energy. (Subsequently,

Since all neutrons which decay y rays may be emitted, often much later.) fission or other nuclear reactions are captured, this source is very strong — the most important single source of y rays from the point of view of shielding. Table 9 gives the yields for capture by various elements.11 Elements may be classed as capture 7-ray emitters according to the distribution of energy among the emitted For all (71,7) reactions, the total energy of all photons emitted for one photons. The types are as follows: capture is just equal to the binding energy. Type 1: Essentially all energy in a single photon of maximum or near-maximum radioactive

do not cause

energy.

Type 2: Spectrum shows many 7-ray energies possible, with distinct line structure evident.

shows many 7-ray energies possible, with no line structure Generally softer radiation than from Type 2. Type 4: Decay by charged-particle emission, e.g., (n,a) or (n,p) (see Table 10). Li8 reacts wholly by (n,a), giving no 7 ray. B'° gives a single 0.48-Mev 7 ray in 93 per cent of neutron absorptions. In 94 per cent of absorptions N14 gives no 7 rays but releases a proton instead; in 5.6 per cent of absorptions, the proton plus recoiling C"(n + NM -> p + CM) have 0.624 Mev. Type 3: Spectrum

Table

10.

Nuclei Useful in Suppression of Capture 7 Rays Thermal-neutron cross

Target nucleus

section,

barns

Energy release, Q, Mev

Reaction

Li' B»

Li'(n,<0H1

N'<

N«(n,p)C"

B">(n,a)Li! B'°(n,a)Li; + 0.48-Mev y

Natural element

Isotope

70 50 700 1.68 0. 1

930 266 3.720 1.68 0. 1

4.78 2.792 2.792 0.624 10.8

Suppression of capture 7 rays can be accomplished by introduction of class 4 nuclei, for which capture 7 rays are nonexistent, of low energy, or appreciably less frequently produced. Nuclei suitable for these purposes are given in Table 10.

2.5

Inelastic Scattering

7 Rays

from nuclei, give some of their kinetic The neutrons gives up the The available energy intrinsic energy by emission of one or more 7-ray photons. will be distributed among the photons according to the energy levels available to the nucleus in the readjustment process, that to the nuclear levels between that to which the nucleus excited by neutron impact and the ground state. The Nuclear collecting and publishing data on inelastic scattering Cross Section Committee Both cross sections and photon energy 7-ray production as they become available.12 distributions are required for shielding calculations, so the compilation of that com mittee In the absence of such data, is, of course, conservative to assume needed. single photon carries that all neutron energy given to the struck nucleus and that off (see also Art. 3.22). Fast neutrons, on scattering inelastically

a

is

is

it

is

is

is,

energy to the struck nucleus in the form of intrinsic nuclear energy. The nucleus at once proceed with altered course and lower energy.

it


evident below 5 Mev.


7-78

2.6

PROTECTION

[SEC.

7

X Rays

Characteristic X rays are produced following photoelectric interaction of y rays or other processes in which the atomic electrons are disturbed. These X rays are, in general, soft and of negligible consequence in reactor shield design. Continuous X rays, or bremsstrahlung, are produced when a high-energy electron is deflected in the coulomb field of a nucleus.13 This is the common process by which X rays are produced in commercial machines. The only case in which this is of significance in reactors is that of Li7 coolant. The reaction Li'(n,7)Lia, for which the thermal-neutron cross section is 0.033 barn, gives Li8, which decays with a half-life of 0.88 sec, yielding a /3 ray of maximum energy 13.1 Mev. This /S ray is sufficiently energetic to produce appreciable X rays by the bremsstrahlung process. 3

ATTENUATION PRINCIPLES

NEUTRON

Neutron shielding is not at present so well understood as is that of 7 rays, so that In general, best estimates of shield thickness safety factors are called for. should be increased by 10 per cent to ensure adequate attenuation in those cases in which a full-thickness shield cannot be tested experimentally. The method to be described of calculating neutron shielding differs from that of the preceding article on y rays, in that the build-up factor is not taken as a separate factor. Instead, the scattered neutrons are taken into account by using cross sections This is tantamount to somewhat lower than the measured total cross sections. using build-up factors which are exponential in mathematical form. This approach ignores such questions as the order in which materials may appear in a laminated shield, but no reliable theory is available to cope with these problems with the brevity The recommended safety factor (10 per cent in thickness) should required here. suffice for this shortcoming, provided the recommendations regarding configuration are adhered

to. 3.1

Neutron Sources in a Reactor*

The dominant source of neutrons in a reactor is the fission process itself. On U*36 gives 2.5 neutrons on the average, distributed in energy from near 0 to about 17 Mev, approximately according to the empirical formula of Watt : fission,

N(E)

=

/—

e~*

sinh

y/2E

(10)

= probability (normalized to one) that a virgin fission neutron will have energy E Mev, per unit Mev energy interval E = neutron energy, Mev This function, plus integrals above and below a given energy, are given in Table 15. There are several things to note about this spectrum of neutrons: The most probable Above this energy is about % Mev; the average energy per neutron is about 2 Mev. energy, the neutrons become increasingly less frequently produced, and at energies Most reactors contain a large as high as 15 Mev there are extremely few neutrons. quantity of low-atomic-weight material which serves as a moderator to reduce the Some reactors also include a reflector energy of the neutrons within the reactor. which, likewise, usually reduces the average energy of the neutrons. The result in general, a large number of low-energy neutrons entering the shield is that there and a small number of high-energy neutrons. The low-energy neutrons are usually large 7-ray source in this captured in the first layers of the shield, giving rise to The high-energy neutrons, on the other hand, are more penetrating, and region. the bulk of the biological shield required to stop tins component. In addit ion to the prompt fission neutrons just described, there are delayed neutrons These delayed neutrons have which are emitted by the fission products themselves. half-lives ranging from a few seconds to many minutes. The delayed neutrons can

N(E)

See also Art.

4

is

a

is,

where

*


greater

on Neutron Sources.


Sec. 7-3]

7-79

characterized as belonging to a set of individual radioactive emitters, each having The delayed neutron emitters give characteristic energy spectrum and half-life. neutron spectra which are peaked approximately at the energies shown in Table 16. The delayed neutrons are appreciably lower in energy than the prompt fission neutrons, and it is evident from the recent work of investigators at the Atomic Energy Research Establishment at Harwell, England, that there are no very high neutron energies among the delayed neutrons. be

a

3.2

Neutron Attenuation Processes

is

1

it

it

is

is

is,

3.21 Elastic Scattering. In this process the neutrons are changed in direction and reduced somewhat in energy. For heavy nuclei, the reduction in energy is insig nificant; for hydrogen, the reduction in energy is one-half of the initial energy, on the average; for high-energy neutrons of several Mev or- greater, the elastic scattering is dominantly in the forward direction; that is to say, the change in direction occasioned by elastic scattering on the average, not very great. For this reason, the elastic not so effective in attenuation as might be otherwise thought. scattering process Elastic scattering in hydrogen an exception to this because of the large energy degradation introduces. 3.22 Inelastic Scattering. For neutrons of energies greater than about Mcv possible for the struck nucleus to be raised to an excited state by transfer of some of the kinetic energy of the neutron to the intrinsic nuclear energy of the struck

y

y

it

is

is,

F,

is,

is

7

is

is

is

y

is

y

y

is

is

a

a

a

is

y

is

y

is

is

y

7

y

is

y

it

is


is

particle. This process nonexistent for hydrogen but relatively quite probable for the larger nuclei. It very effective in reducing the energy of the neutron, but gives rise to inelastic scattering rays which are produced when the excited nucleus returns to the ground state. The emission of the inelastic scattering rays occurs These inelastic scattering essentially immediately following the neutron collision. rays constitute an additional hazard which must be taken into account in the design of the reactor shield. 3.23 All neutrons which do not undergo particle reactions such Capture {n,y). as (n,a) or (n,p) with nuclei are eventually captured. The cross section for this process is, however, significant only for low neutron energies. There some capture in the resonant region, that at about 100 kv or less, and there in general, very The most pronounced strong capture for thermal neutrons in most materials. exceptions are He, Be, C, O, Mg, and Bi. Immediately following neutron capture, rays are given off with total energy equal to the binding energy of the extra neutron in the new nucleus. For hydrogen, this energy 2.23 Mev; for most other nuclei of the order of Mev. These new rays constitute an additional source of radiation against which shielding must be provided. In some nuclei only a single then a very hard or penetrating capture ray given off, and this ray. In other nuclei many rays are given off, and since the total energy just equal to the binding energy of the neutron, they are of lower energy per photon. It almost never necessary to add material to shield to ensure neutron capture. Almost all elements have sufficient thermal-neutron capture cross section to ensure that neutrons which are sufficiently reduced in energy will be captured before traveling very far. Exceptions to this include the materials of interest for moderators or reflectors in reactor cores, such as beryllium or graphite. These are seldom used in biological shields in any case, although as reflectors they often constitute the first shield layer. 3.24 Particle Reactions. Examples of these are the (n,a) reactions in B10 and Li* and the (n,p) reaction in N14. When a charged particle such as proton or an a no accompanying particle released, there either ray or only one of low energy. rays, and For the cases of particle reactions in lithium and nitrogen, there are no one 0.493-Mev ray which for the case of B10, there produced in 93 per cent of the particle reactions induced by thermal neutrons. This fact legislates in favor of adding B'°, N", or Li6 to a biological shield. B10 and Li* arc particularly desirable because of the fact that they exhibit large cross sections for these particle reactions. Tabic 10 gives They can thus be thought of as strong capture 7-ray suppressors. to be noted that there also a cross section of pertinent data on these nuclei. It


7-80

PROTECTION

[Sec.

7

.08 barn for pure capture (n,y) in N11, and in this case 10.8 Mev of capture y rays are given off. 3.25 Effect of Neutron Energy. Because of the fact that there are essentially no capture processes occurring at high energy, the attenuation of the most penetrating neutrons in a biological shield is accomplished largely by one inelastic followed by many elastic collisions. The total cross section of most materials decreases with increasing neutron energy, this effect being stronger in some elements than others, notable examples being hydrogen and lead, which have significantly greater cross sections at 2 than at 8 Mev. Because of this general trend in cross sections, the neutrons travel farthest on their first flight, on the average. Following each collision they are of lower energy and consequently travel shorter and shorter distances between collisions as they are slowed down within the shield. This effect is especially pro nounced in shields which are highly hydrogenous. In dry shields, such as ordinary

Table 11.

Dominant Energy in Hydrogenous 'H,0'

10 20 30 40 50 60 70 80 90 100 120 140 160 180 200

lHf0

— thickness

EDom"\ Mev

aw,

£Dom11',

Mev

2. 1 3.3 4. 2 4.9 5.6

4'5

6.2 6.8 7.3 7.8 8.2 9.0 9.9

6.3 6 6

8^0

10.6 11.3 12.0

cm-1

Mev

50 100 200 300 400 500 600 700 800 900 1.000 1.200 1.400 1.600 1.800

1.8 2.7 4.2 5.3 6.2 7.0 7.7 8.4 9. 1 9.7 10.2 11.2 12.2 13. 1 14.0

of water at N. T. P. in shield, cm

— density of hydrogen atoms, cm-1, X I0~", in shield, average — 6.7 for HiO so that, e.g., ih,o ™ '00 cm corresponds to NHt ~ 670 cm 1 / ™ shield thickness, cm /•j'..m 1 ■»dominant energy, Mev, calculated on basis of first collision in hydrogen ^Dora'1* ~ dominant energy, Mev, based on calculations for water attenuation by Aronson, Certaine, and Goldstein, NDA-15C-60. 1954, reflecting effect of variations in oxygen cross section as well

Nfi

the high-energy neutrons do not predominate so strongly but the trend is The result of the trend to lower cross sections at higher qualitatively the same. neutron energies is that those neutrons which escape the shield are dominantly those which were born at relatively high energies. For water shields, for example, in com mon thicknesses, the neutrons which penetrate the shield were born* in the region of In concrete shields the dominant energy is about 8 Mev.f This most impor 8 Mev. tant energy, or dominant energy, as we shall call determined by the interplay of the two factors of increasing penetrability with increasing neutron energy on the one The hand and the decreasing population in the fission spectrum on the other hand. dominant energy in hydrogenous shield given as a function of shield thickness in Table 11. It will be noted that below about Mev there little or no inelastic scattering. small energy reduction Furthermore, elastic scattering on heavy nuclei gives only As a consequence, a neutron shield made entirely of heavy elements per collision. The neutrons would be reduced to energies of the order would be unsatisfactory. of Mev by inelastic scattering and subsequently would suffer many elastic scatterit,

1

a

is

1

a

is

is

concrete,

is

is

* This does not imply that the spectrum of effluent neutrwna at the shield outer surface peaked at this energy. These high-energy neutrons are moderated in the outer shield layers and escape mostly at much lower energies. about 2.5 Mev. For concrete shields less than 100 cm thick the dominant energy

t


cm

Shields with U,a' Fission Spectrum

Sec. 7-3]


7-81

In doing so they would be reduced only slowly in energy and diffuse through shield. Another reason for abnormally high penetration of these shields is that in their neutron cross section with cor many heavy elements exhibit resonances Neutrons in those energy regions find responding regions of low total cross section. A particular example of this effect is noticed in the it easy to penetrate the shield. case of iron, which has a minimum cross section of only about 0.5 barn for neutrons of 26-kev energy. A shield of pure iron would be ineffective against neutrons for this reason. This effect is so serious that it is not permissible to have long pieces of a single element penetrating through the shield in the direction of outward travel. It is to be noted, however, that alloys consisting of several elements seldom exhibit minima in their total cross sections because the minima for the individual elements seldom occur in the same energy region. Nevertheless, it is essential that each biological shield contain a large amount of moderating material, hydrogen being pre ferred, to ensure that the neutrons do not diffuse at intermediate energies through the shield. The criterion for ensuring that sufficient moderator is present is that one sixth of the mean squared slowing-down distance shall be less than the square of the mean free path for high-energy neutrons; i.e., ings. the

W

(11)

Effective Removal Cross Sections

3.3

Of the foregoing attenuation processes only two are significant for high-energy these being elastic and inelastic scattering. It is possible to predict the performance of a biological shield, subject to the limitations of the previous paragraph, in terms of a linear combination of these two cross sections. In general, the entire inelastic scattering cross section and about one-half of the elastic scattering cross section are considered effective. This sum is a fair approximation to the effective removal cross section of the shielding material. It has been demonstrated that the attenuation of fission neutrons through most shields can be expressed by a simple exponential, using this effective removal cross section. Of course, the dominant energy, at which the elastic and inelastic scattering cross sections are taken for determination of the effective removal cross section, will vary from shield to shield depending on the composition and the thickness. For hydrogenous shields the dominant energy can be seen from Table 11 to vary strongly with shield thickness, this being a consequence of the strong variation of hydrogen cross section with energy. A good approximation of the hydrogen cross section for the region from 1 to 12 Mev is neutrons,

10.97

E

+ 1.66

,

,10, (12)

barns

if

3

is

if

is

is

is a

is,

where E is the neutron energy in million electron volts. Table 11 was computed using this cross section and Watt's fission spectrum, obtaining the maximum in the product of free-flight penetration times population in the fission source. More exact calculations by Aronson, Certaine, and Goldstein14 show less hardening of the neutron spectrum in a water shield, as is seen by their values reported in the table. For thick C>100 cm of H20 or equivalent) hydrogenous shields many removal cross sections have been measured. These are given in Table 12. It is to be noted that these cross sections are strictly applicable only when the extraneous material It is acceptable, however, to use them for other precedes the hydrogenous material. cases as well, provided a pure heavy component, such as a slab of iron or lead, does not constitute the outermost layer. In the latter case, the slab cannot be counted on for appreciable attenuation (a relaxation length of about 35 cm is generally used). For nonhydrogenous shields, such as most of the concretes, it is necessary to deter mine the dominant energy in the same way, that by the maximum in the product of long and difficult task, penetration and population in the fission spectrum. This not a very strong function of the dominant energy, about and since the attenuation the thickness Mev generally used 100 cm or more, Mev less.

8


< X2

7-82

EADIATION AND RADIOLOGICAL PROTECTION Table 12.

Cross section,

Material

barns/atom

Nickel

1.31 0.97 1.07 3. 49 0.81 1.2 2 04 1.29 1.98 1.01 1. 89

Lead Tungsten

3.5 2.5

Beryllium

Copper Iron

Cross section, barns /molecule

CiFit CjF.Cl CHj B.C

26.3

6.6 2.8 4.3 2.8

0.99

ORNL-1843, Aug. 31, 1955.

and C. L. Storrs,

3.4


7

Effective Removal Cross Sections for Fission Neutrons* Material

« G. T. Chapman

[Sec.

Dose Rate from Point Isotropic Fission Source

The process of calculating the attenuation of a point isotropic fission source in an infinite medium is as follows: 3.41 For hydrogenous shields (other than Hydrogenous Nonaqueous Shields. aqueous shields) of thickness / cm, the biological dose D at a distance R cm from unit source is

D(R,t)

=

e"s''

ivR*

mrem/hr

(13)

r = pHW0.111, cm = density of hydrogen, g/cms, in hydrogenous shield multiplied by the If there are several layers of thickness of hydrogenous shield, cm. different hydrogen concentration, then pH
ph
D(K,t)

=

fr~

4W£S

^

e

^

mrem/hr

(14)

mrem/hr, in a pure water shield multiplied by 4wr* (obtained from curve 6 of Fig. 5)* Sri = macroscopic removal cross section, cm-1, for all elements in the shield other than those in water, multiplied by the shield thick ness, cm (use Table 12 or Fig. 6) PHjoinio = water density, g/cm3, multiplied by thickness of water portion of shield, cm

where 4irr,.Dn,o(r) = dose,

=

Pii,o
* For thick thickneaa.

water

shields

g/cm'

beyond

the range of Fig. 5 the 7-ray dose predominates

and

governs

Fia.

5. Dose as a

AO • SO 60 TO BO 90 r, THICKNESS ATNTP(cml OFWATER

function of distance from

a point

100 MO 120

isotropic fission source.

10

£r,

Ir,

TOTAL CROSS- SECTION VALUE

ATS Mev

(TAKEN FROM LA-1655) REMOVAL CROSS-SECTION VALUE DERIVED MEASUREMENTS BEHIND COMPOUND OF THE ELEMENT

FROM

tr.

fl' M -[J

30

VALUE FROM REMOVAL CROSS-SECTION MEASUREMENTS BEHIND ELEMENT

DENSITY OF MATERIAL

'2 Fio.

6.

weight.

Neutron

shielding

5


a WHYDROGEN OFDENSITY O.HIg/cm'(SAMEASTHATINH.O ATNTP){ARONSON,N0A-ISC-39». j | | | I
10

20

A, ATOMIC

50

ability per unit weight of 7-83

100

200

500

1000

WEIGHT a material

as a function

of atomic


7-84

PROTECTION

[Sec.

7

For nonhydrogenous shields, such as the con Nonhydrogenous Shields. 3.43 cretes (hydrogen atomic per cent less than ~50), the dose is calculated by several First the sum of the removal cross sections for the various materials in the steps. shield is determined as a function of energy: tr(E) -f{&i{E) + f&*(E) + /323(E) where 2i(i?) = macroscopic hydrogen cross section,

cm-1

Tii(E) = macroscopic total cross section of oxygen plus carbon, cm-1 /j = 0.5, E < 4 Mev (i.e., no inelastic scattering) /, = 0.75, E > 4 Mev 2i(£) = macroscopic total cross section of all other elements in the shield, cm-1 /j = 0.75 (i.e., all of the inelastic scattering and half of the elastic scattering cross sections are effective) If the elastic and inelastic cross sections 2,/ and 2,„ are known, however, it is better to compute the removal cross section of the shield as follows:

ME)

/,Zi(£) + y2X,,(E) + 2i.(E)

=

The maximum of the function


Q(E)

=

N(E)

e-2rW<

is then computed where t is a first estimate of the shield thickness. The value of E for which Q(E) is a maximum [dQ(E)/dE = 0] is the dominant energy E<, Mev. For

Table Neutron energy-, Mev

Biological Effect of Neutrons*

13.

to give Flux. neutron8/(cm1)(sec), laboratory tolerance ™ 7.5 mrem/hr (assuming

2.5 X 10 8 (thermal)

40 hr/week)

(Mrem/hr) / [(neutrons)/(cm") (sec)] 3.75 X I0-" X 10 • 4.4 X 10 ->

2.000 1.550 1.700 830 250 90 60 60 55 50 50

io-« 4 X 10 > 2 X 10-1 0. 1

0.5 1.0 2.5

5.0 7.5 10.0

4.8 9.0

X X 8.3 X 1.25 X 1.25 X 1.36 X 1.5 X 1.5 X 3

* From data of W. S. Snyder. Oak Ridge National Laboratory, submitted trons of National Committee on Radiological Protection, 1956.

10'

10 ' 10-'

10'

10-' 10-' 10-i IO-<

to Subcommittee

most concretes computation of the maximum is difficult, and a value of 8 be used for E<,. The biological dose is then given by DB(R,t)

=

mrem/hr

on Neu

Mev can

(15)

strength, neutrons/sec (regardless of energy — for conservatism — in the fission source) b(Ea) = biological effect of unit flux of neutrons of energy En, (mrem/hr) /[neu trons/ (cm!) (sec)] (given in Table 13) A better estimate of the biological dose can be had by integrating over all source energies, thus:

where

DB{R,t)

- ~ //_„ N(E)

e-Wb(B)

dE

(16)

Sec. 7-3]


7-85

Because the integrand is not analytic, the integration must be done by hand, but since for energies above about 5 Mev both b(E) and 2,(2?) are nearly constant, at least this part of the integration can be done easily, making use of the integrated fission spectrum in Table 15. As is evident from Table 16, the delayed Shielding of Delayed Neutrons. 3.44 neutrons are of energies below the threshold for inelastic scattering for many elements. As a consequence it is highly desirable to shield delayed neutrons with hydrogenous materials, for which the cross sections are large and the energy degradation per col lision high. However, the cross section of hydrogen below 1 Mev does not increase

Table 14.

Attenuation by Water of 2.0*- and 0.9t-Mev Neutrons, Unit Point Source (Neutrons/Sec), Detector a Distance R Cm away 10) (RBE

-

R, cm


10 20 30 40 50 60

Dose, mrem/hr, 0.9-Mev source

2. 7 X 10-« 1.3 X 10-'

4.8 X I0«

Dose, mrem/hr, 2.0-Mev source

3.7 X 1.5 X 8. 1 X 4. 5 X 3. 5 X 2.2 X

10* 10-' I0"« 10-' 10-'°

10-"

J. Certaine, and H. Goldstein, NDA 15C-60 (NYO-6269), 1954. Rockwell HI (ed.), "Reactor Shield Design Manual," TID-7004. 1956; see also K. Shure and August, 1955. P. A. Roys, Report WAPD-T-226. • R. Aronson,

tT.

(fractionally) with decreasing energy as much as it does for energies above that energy, with the result that for low energies the build-up factors in hydrogenous shields are quite high. Attenuation data for 2.0- and 0.9-Mev sources in water are given in Table 14. These can be corrected safely for other shielding materials on the basis of Of course, the delayed neutrons have lower energies, so hydrogen content alone. that it is conservative to use the values in the table for estimating doses. 3.6

Heating in Shields by Neutrons

In the first place, in Neutrons cause heating in shields by each of four processes. collisions they cause nuclei to recoil, and these recoiling nuclei deposit their kinetic energy as heat in the shield. In inelastic scattering, the nuclei which have been excited give off y rays as well as depositing some kinetic energy, and both of these components heat the shields. Of these, in general, the kinetic energy can be ignored. Neutrons are captured in shields, and the capture y rays cause heating in the shield. The fourth process by which neutrons cause heating in shields is by In charged particle reactions, such as (n,a) and (n,p) as described previously. general, the important aspect of heating in shields is the thermal stresses which are Therefore, any calculational method which tends to overemphasize induced thereby. the average gradient in the heating distribution will be a conservative calculation. The exact calculation of heating in a shield is a very complicated problem and will, in general, require solution by a high-speed computing machine, perhaps utilizing the Monte Carlo method. Some approximations by hand computation, however, may be desirable. Estimates of the neutron heating will be possible by the method to be outlined in the following paragraphs. Heating by elastic scattering can be estimated to be accomplished on the first elastic scattering collision. This will be conservative if the heat is removed from the outside (far from the source). For hydrogen, on the average, one-half of the energy is given to the recoiling proton in an elastic collision. However, the neutron will suffer subsequently another collision at some distance not far removed from the first and give up again one-half of its remaining energy on the average. Thus the heat elastic

7-86


[SeC. 7

deposition due to elastic collisions in hydrogen is at least as far from the core (or source) as would be calculated on the basis that the total neutron energy is released on the first elastic collision. The calculation for this deposition then is a simple exponential, using the total hydrogen cross section. If the heat removal is from the inside instead, it is conservative to assume that the This distance can energy is deposited at some distance beyond the first collision. be taken to be the "displacement distance" t/\ where t is the Fermi age to thermal energy and \ is the relaxation length of the uncollided flux.* W. S. Snyder gives curves of physical and biological dose as a function of distance into a 30-cm-thick slab of tissue (see Figs. 5 to 13 of Sec. 7-2). Since the physical dose is primarily attributable to recoil protons, it is permissible to use the dose curves to predict heating in hydrogenous media. Of course, at neutron energies appreciably below about 1 Mev, the recoil protons do not override in importance those from the N14(n,p)C14 reaction, so that the biological curves do not properly predict heating Thus, the heating at a depth x in a hydrogenous medium is given except in tissue. in terms of the physical dose in tissue as follows:

H(£,i) ^ 10-»^5 I)P(E,x') A'h

(watts)(g)/(neutrons)(cm')(scc)

(17)

where A'h = hydrogen density, atoms/cm3 X 10~", for material in which heating is to be calculated Arn' = hydrogen density in tissue X 10"" Generated for wjivans (University of Florida) on 2015-09-23 02:52 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

= 5.98

x' = (Nu/Nb.')x, cm Dp(E,x') = physical dose, rads/(neutron)(cms), from Snyder's curves (Figs. 5 to 13 of Sec. 7-2). Two other interesting facts, from a heating-in-shields point of view, are to be seen in Snyder's curves. The energy deposited by recoils of heavier nuclei such as oxygen, carbon, and nitrogen accounts for less than 10 per cent of that from all recoils. The heating from capture y rays is given also, and it is seen to be dominant at the greater depths, owing, of course, to the greater penetration of y rays in tissue. For oxygen and carbon in the absence of hydrogen many collisions are required to reduce the neutron energy, so that age calculations, multigroup calculations, or Monte Carlo calculations are required. For all other nuclei, the fast neutron collisions Since probably result about equally in inelastic scattering and elastic scattering. in the elastic scattering only a small fraction of the energy is given to the recoiling nucleus, this will represent a negligible fraction of the total heating due to neutrons. The inelastic scattering y rays will dominate strongly. In conclusion, it may be said that if the shield contains either hydrogen or a sig nificant quantity of heavy element, then these two components will do the large The ratio of inelastic scatterings in the heavy elements part of energy absorption. and elastic scattering in hydrogen will depend, of course, on the relative cross sections for these two processes in the shield material. In case the shield is largely made up of other elements such as oxygen and carbon, then the energy will be transferred by The calculation of energy deposition elastic collisions in either of these light elements. in this case is most properly carried out by the methods described in neutron modera A rough approximation of the tion in the reactor physics section of this handbook. heat deposition can, of course, be had even in this case, using exponential attenuation with the effective removal cross section. As an example of the calculation of the heat deposition in a shield, the following situation will be studied: A point source of fission neutrons will be presumed to be The problem will be to calculate the heat surrounded by a shield of barytes concrete. distribution due to the incidence of the neutrons on the barytes. The composition of barytes concrete is given in Table 22 in Art. 6 on Shield Mate rials. The source will be assumed to be at the center of a spherical hole of radius o in the concrete. Examination of the concrete materials reveals that energy may be •See. for example, Ref. 15. pt. 2. pp. lOOff.

Sec. 7-3]


7-87

to be deposited largely by two processes: clastic scattering in hydrogen and inelastic scattering in Ba, Fe, Ca, and Si. Other processes, such as elastic scattering in oxygen, may contribute to the attenuation, largely because of neutron deflection, but will not reduce neutron energy by a large fraction. Therefore, the ratio of elastic scattering in hydrogen to inelastic scatterings in heavy elements will be the ratio of the two macroscopic cross sections. The neutron heating (excluding secondary 7 heating) is given by expected

H"(R>

" Aji L" NW

Eoxp[-(R-

o)2,l dE

where S = source strength, neutrons/sec R ~ distance from source to measuring point, cm a = radius of hole containing source, cm N(E) = distribution function for neutron energv in source,

2r(£)

= 1

(18)

normalized so that

= removal cross section of barytes concrete for neutrons of energy

E

Mev,

cm-1

-ME) =

+f,?i(E) +h?3(E)

macroscopic hydrogen total cross section as it appears in barytes concrete, cm-1 2?(/7) = macroscopic total cross section of oxygen as it appears in barytes concrete for neutrons of energy E Mev, cm-1 fi(E) - effectiveness factor for oxvgen to obtain removal from total cross sections: h » V% for E < 4 Mev; /2 = % for E > 4 Mev — macroscopic total cross section of inelastic scatterers, Ba, Fc, Ca, Si, Xi(E) as they appear in the concrete for neutrons of energy E Mev, cm-1 ft(E) — effectiveness factor to obtain removal from total cross section for inelastic scatterers: fi » % for E > 1 Mev;/i » H for E < 1 Mev While the integral above appears to be difficult to evaluate, actually relatively few energy intervals will be needed, since the largest contribution to the heating will be for the energy range of about }^ to 5 Mev. The sources of inelastic scattering can be calculated by an integral much like that of Eq. (18) except that in place of 2i in the numerator, 2;„, the inelastic scattering cross section, is used. In the absence of exact data, Yi 2a can be used. The spectrum of inelastic scattering 7 rays is known only for a few isotopes, so some assumption must be made. The simplest reasonable assumption is that the ys have nearly all the neutron energy and that it comes off in a single photon. For high neutron energies (E > 6 Mev), the inelastic scattering y rays may be expected to come off in several photons of lower energy if the capture ys of that element do so. This is not a very reliable rule, however, as the emitting isotopes arc different in the two cases. The inelastic scattering 7 rays will, of course, travel isotropically from their points of production before losing energy to the medium, and the calculation of this effect is one largely of geometry. The total result of this travel will be to alleviate the Again, if cooling is from the outside, it is conservative gradient in heat production. to ignore this further travel in considerations of heat stress and to assume that the y energy is deposited at the point of scatter. The neutrons which are slowed down in the concrete are captured and give rise to capture y rays. Because neutron capture takes place dominantly at or near thermal energies, the neutrons for this case are calculated according to the age diffusion picture, and these calculational methods are described in the reactor physics section of this handbook. Similarly, the deposition of heat due to particle reaction in the shield is again an effect proportional to thermal neutron flux and can be calculated according to standard neutron diffusion techniques. Very often it is necessary to calculate the heating due to capture 7 rays produced within a slab of material, such as a steel thermal shield or pressure shell. Enlund" has treated this problem, using exponential distribution of the neutrons being cap For the 7-ray attenuation he used an expression derived by H. Goldstein" tured.

2i(£)


}N(E) dE

Mcv/(cm»)(sec)

7-88

RADIATION

AND RADIOLOGICAL

PROTECTION

[SEC.

7

2.2

l\a~

4.7

2.0

1.8

0=2.6 1.6

1.4

— o

1.2

3 1.0

0.8

0.6

f /

a ■ 1.2 c = 1.0

a = 0.8

0.4

0.2

0.2

Fig.

0.4

0.6

0.8

1.0

M* 7. Energy- absorption from capture

1.2

18

y rays: the function F(jiz, a).

H(R)

*r)

on the basis of conservation of energy. Thus the heating at R cm due to a point source of unit strength (one photon per second) of photons of energy E Mev in an infinite medium is = En* I 1 +

4WT2

Mev/(cm')(sec)

(19)

1.

and n* are described in Art. for The problem Enlund has solved

where

/i

an infinite plane source of neutrons emitting /o neutrons/(cm2)(sec) into an infinite plane slab with source on one side, giving a flux within the slab described by is


£

a current

(x) =

KD


Sec. 7-3]

7-89

where Io = neutron current, cm-1 sec-1 = —D d^/dz at z = 0

K = \/32«2,, cm-1 2„ = macroscopic absorption cross section, cm-1 S, = macroscopic transport cross section, cm-1 D = diffusion coefficient, cm z = coordinate of distance into slab from source, cm Kite-*' 2„*(z) = density of absorptions at z, cm-3 sec-1 N(E) = fraction of neutron captures which give a photon of energy E Mev ME) = energy absorption coefficient, cm-1, for y rays p(E) = linear attenuation coefficient, cm-1 a = K/fi The heating at z due to y rays of energy E is H(z,E). Total heating is obtained by summing over-all capture y rays:

-

H(z,E)

= 8.0

X lO-l*N(E)Ehi*(E)

+ a

-

— k(jtz,a)

J

watts/ (20)

where

FGiz,«)


F(pz,«) In 7 FG«z,a)

e-aiu

+ Et[„z(«

l^t-A'tX-^)]

e~i"\ei"[-Ei(-vz)] -f

In

juz

[wi-EH-na)] I

1)]

+ In

+ In (27)]

0.5772 = e""*"

-

+ EH

-Ml - «)]

for a >

for a =

+ In

1

^-^1 1 — acJ

for a <

F(jiz,a) are given in Fig. 7 and values of K(nz,a) are given in Fig. 8. functions —Ei(—x) and ~El(x) see Table 15 of Sec. 1-2.

Values of the

08

0.6

Fig.

8. Energy absorption

3.6

1

1

For

(.0

from capture y rays: the function K(jiz,a).

Secondary

7

Rays — Biological Effects

As has already been mentioned, reactor shields become sources of 7 rays owing to capture and inelastic scattering of neutrons. These sources of 7 rays constitute a biological hazard which must be taken account of in calculating shield effectiveness. The first requirement is to determine the distribution of capture and inelastic scatter ing 7 rays in the medium. The distribution of inelastic scattering 7 rays has already been described in the discussion on heating in the shield. The distribution of capture 7 rays, on the other hand, will vary somewhat from that for the inelastic scattering

7-90

RADIATION

AND RADIOLOGICAL

PROTECTION

[Sec.

7

because of the age displacement between the first collision of the neutrons and their In general, the rate of removal of neutrons from actual capture after slowing down. the penetrating beam is equal simply to the derivative of the neutron flux at any point in the shield. That is to say, it is equal to the macroscopic removal cross section multiplied by the neutron flux as calculated by the methods described in the foregoing paragraphs; i.e.,

R

_ _ dm a w dz X

(21)

where R = rate of removal of fast neutrons / = current of fast neutrons X = relaxation length, cm, or inverse of macroscopic removal cross section, average These removed neutrons can be presumed to slow down according to a Gaussian distribution. Convolution* of this distribution with an assumed exponential attenua tion of fast neutrons gives the total arrivals at thermal energies as a function of Ignoring diffusion at thermal energies, the rate of arrivals at thermal energy position. is equal to the rate of absorption, i.e., 7:„MtK(z) ■ The result is


2..«»*»(*) =

=

where E(m) =

t

—2— /fm

-^L

Vrt

«"*«

~ e^'ll

Jf" Vf_ *

*—• dx

(22)

Vir

+ *(m)]

(22a)

e~'' dx

= Fermi age to thermal of dominant

m = («/4r)

t_

- (V^/X)

energy, cm*

The term E(m) is given as the "error integral" in Table 17 of Sec. 1-2. For m = 1, E(m) = 0.84, so for thick shields it is sufficient to use the value 2 for the quantity in the brackets. The next step in dealing with secondary y rays is to determine the relative attenua tion of the shield for neutrons and y rays. If the shield has a longer relaxation length for 7 rays than for neutrons, then the y rays which are produced in the first couple of mean free paths of the shield will constitute the important source of y rays in so far One can therefore take the source of y rays as as secondary y rays are concerned. those produced within two mean free paths of the inner surface. Presume that these are produced as predicted above, or approximately at a distance equal to t/X from the inner surface, and attenuate them through the shield by the standard methods described in the 7-ray section. If, on the other hand, the neutrons penetrate the shield more easily than the 7 rays, then because of the greater biological importance of the neutrons no account need be taken of the secondary 7 rays produced by the neutrons in the shield, the neutrons themselves constituting the only significant source of radiation. This does not say that the 7 rays produced in the core and the reflector may not be important. In general, the biological shield thickness is deter mined by the most penetrating component of radiation. 4

NEUTRON SOURCES

The fission products emit neutrons which are classed as "prompt" if the time between fission and emission is not measurable or "delayed" if it is measurable. These fission neutrons are discussed below, along with the production of neutrons from artificial sources. • Integration of the product over all space.


Sec. 7-3]

7-91

Prompt Fission Neutrons

4.1

The neutrons emitted at the time of fission of U2" from its fission fragments are distributed in energy from 0.075 Mev»« (or lower) to about 17 Mev. 19-21 The dis tribution of these "prompt" neutrons is quite closely described by the approximate formula of Watt,*0

N(E) dE where

N(E) dE

Values of \N(E) dE

E N(E)

\/2E e~*

= number of neutrons of energy

emitted — neutron energy, Mev dE are given in Table 15.

Table 15. The U236 Fission is the fraction of neutrons of energy E, Mev

0. 1


= 0.484 sinh

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1. 1 1.2 1.3 1.4 1.6 1.8 2.0 2.5

N(E), Mev-'

10.0 11.0 12.0 13.0 14.0 15.0

4.384 1.994 8.953 X 3.997 1.771

4.5 5.0 5.5

6.0 6.5 7.0

E

4-

dE

per fission

A simpler approximate

jl

N(E> dE

0 014 0.377 0.663 0 987 0 1321 0 1669 0 2027 0 2385 0 2729 0 3082 0 3425 0 3759 0 4074 0 4383 0 4983 0 5631 0 6028 0 7073 0 7876 0 8476 0 8914 0 9192 0 9463 0 9624 0 9739 0 9821 0 9876 0 9916 0 9943 0 9960 0 9973 0 9982 0 9988 0 9994 0 9998 0 9999 1 0 1 0

0.3446

7.5 8.0 8.5 9.0 9.5

3.0 3.5 4.0

to

(23)

expression

Spectrum of Prompt Neutrons* E to E + dE from thermal-neutron nation

0. 2023 0. 2676 0.3068 0.3300 0.3450 0.3528 0.3557 0.3542 0.3513 0.3368 0.3271 0.3179 0.3073 0.2841 0.2604 0.2371 0. 1839 0. 1384 0. 1026 0.07472 0.05381 0.03847 0.02729 0.01916 0.01336 9.316 X 6.436 4.423 3.034 2.076 1.418

E

dE

I0~«

9.53 X 10' 10»

fg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 2 1 4 2

neutron

that is

of UM,1

N(g) dE

9860 9623 9337 9013 8679 8331 7973 7615 7271 6918 6575 6241 5926 5617 5017 4369 3972 2927 2124 1524 1086 0808 0537 0376 0261 0179 0124 0084 0057 0040 0027 0018 211 X 501 X 473 103 875 X 148

IO-» 10-'

IO-i

•Calculated from the formula of B. E. Watt, PhVs. Rm„ 87: 1037 (1952). From J. E. Evans, "Fast Neutron Spectra from the Water Boiler," LA- 1395, AECD 3073. Los Alamos Scientific Labora tory. March, 1951. good to

15

per cent from

E

= 4 to 12 Mev is

N{E) dE although

1.8*-°

"» dE

(24)

has also been measured21 and is fitted fairly well by there appears to be a somewhat greater abundance of higher

The fission spectrum of Pu*"

Eq. (23),

-

7-92


The statistical energy neutrons. difference, however. 4.2

accuracy

is not

adequate

[SEC.

to certify

7

a genuine

Delayed Neutrons

Some of the fission products decay radioactivcly by neutron emission. These are of lower energy than the prompt neutrons and are much less numerous2* (Table 16).

"delayed" neutrons

Table 16.

Half-life,

Delayed Neutrons*"

Mean energy, kev

Absolute yields of delayed neutrons 10* total neutrons emitted

U'"

V>-

55.07

± ± ± 2.26 ± 0.611 ± 0.244 ±


21. 13 5. 56

0.49 0.33 0. 14

0.08 0.074 0.020

250 ± 20 460 ■ 10 405 ± 20 450 ± 20

2. II 14. 1 12.6 25 4

2.24 7.76 6.54 7.24

7.40 2.68 Total 64.3

1.34 0 87

26.0

per

Pu»»

0.73 6.32 4.48 6.91 1.81

0.94 21.2

* Half-lives and yields from LA-21 18, G. R. Keepin, T. F. Wimett, R. K. Ziegler, "Delayed Neutrons from Fissionable Isotopes of Uranium, Plutonium, and Thorium," p. 32 (February, 1957). using r = 2.46, 2.54, and 2.88 for U,J\ U9", and Pua" respectively, and taking mean values for the six halflives. Mean energies from R. Batchelor and H. R. McK. Hyder, Jour. Nuc. En., 3: 7 (1956).

4.3

Artificial Sources

4.31 It is possible to produce neutrons from a-particle bombard (a,n) Sources. ment of certain nuclei. The a particles can come from radioactive nuclei, so that a small pill which includes an a emitter (e.g., radium) and a target material (e.g.,

Table 17. a emitter

Target

Po

Li

Ra Pu«"

8b>"

La»»

Target nucleus

lie DtO I)*0 Be

Ysfl

Be

Na"

D.O DtO Be

Mn"

5 2. 3 6 2 1.7 7 1.5

Be

Table 18.

Ga"

Yield, neutrons/(sec) (curie of as)

Be B Al Be B

* See also Wilmot N. Hess, UORL-3839.

7-ray emitter

Alpha Neutron Sources*

X X X X X X X

I0< I0« 10' 10* 10' 10' 10'

"Neutrons from (cr.n) Sources," July, 1957.

Photoneutron

Sources

Neutron energy. kev

30 135 130 730 160 210 260 960

Half-life

60 days 14.35 hr 40 hr 40 hr 105 days

2.59hr 15 hr 15 hr


Sec. 7-3]

7-93

6

Data on some useful boron) can be made which acts as a convenient neutron source. sources are given in Table 17. It is also possible to produce neutrons by the inter 4.32 Photoneutron Sources. action of 7 rays on nuclei. Certain nuclei have sufficiently low thresholds in 7-ray energy for this process to enable the construction of convenient small photoneutron sources. The yield is small, but the neutrons are monoenergetic for each 7-ray Data on these are given in Table 18. energy.

GEOMETRY

In the previous articles attenuation from point sources of 7 rays or neutrons has described. In practice, radioactive sources such as cesium-137, cobalt-60, or polonium-beryllium neutron sources can be thought of as point sources; however, reactor shielding problems deal with arrays of sources distributed throughout a volume of absorbing material. In order to determine the biological effects of arrays of sources, it is necessary to integrate over all the sources. The net resultant biological dose due to an array of sources is simply the sum of the effects of the individual sources. This section describes the processes of integration over arrays of sources. Trans formations for surface sources in homogeneous shields with unspecified attenuation functions are first discussed, after which those with partially specified attenuation functions are described. Methods for determining the leakage flux from volumedistributed sources are then outlined, and methods of estimating build-up factors Finally, formulas to determine the flux in for dose from distributions are given. various source-shield geometries are listed. 6.1

Transformations

for Surface Sources in Homogeneous Shields with Unspecified Attenuation Functions

The response of a detector to a nearby point source can be described in terms of an attenuation kernel, which in general is a function of the type of source, the type of detector (or receptor), the type of material interposed between the two, and the separation distance between source and detector. The source can for the moment he any source of unit strength. It is presumed that the detector records biological dose and has no directional dependence. 7'/ie medium is presumed to be uniform throughout with no preferred direction for attenuation. In this situation the dose due to a point source of strength
DP,(R)

SG(R)

(25)

is

a

is

it

If

1

is,

In this equation the function (7(7?) is the attenuation kernel. It in fact, the subject of Arts. and 3. 6.11 the attenuation kernel G(fi) known for all Point to Plane Source. values of R, surface distribution of sources in a possible to obtain the dose from medium by integration: homogeneous (26)

where

s

surface

dA R

= source strength per unit area = element of area on source plane = distance from area element to detector

When G(R) known either from measurement with a point source or from calculations and 3), the following equation useful: (such as were described in Arts. 1

is

is

(26a)

is

it

a

is

is

z

the dose at distance from an infinite plane source. where Dp,(z, «) If the information which known the dose from 6.12 Plane to Point Source. convenient to differentiate Eq. (26a) to large plane distribution of sources, then is


been

7-94

RADIATION

obtain the kernel

AND RADIOLOGICAL

PROTECTION

[SEC.

7

:

-5^ 2t.s

=

«)1J<-». [--£/>,.(*■ Lz dz

(27)*

5.13 Point to Plane Disk Source. In case the plane isotropic source of surface strength s is limited to a disk of radius a (Fig. 9), then the dose at a point on the axis a distance z from the disk is given by = 2-ks

DP\(z,a)

f y/t'+o G(R)RdR

(28)

5.14 Plane Disk to Point Source. Unless the form of the attenuation kernel is However, the attenuation known, the foregoing integral cannot be evaluated. kernel can be obtained if the dose from the disk source is known everywhere on the disk axis." Thus,

G(R)

itzS^'^

+

^L

(29)'


Equation (29) is used to obtain the point-to-point kernel from data from a finite disk source (such as that at the Oak Ridge National Laboratory Lid Tank Shielding

DETECTOR

VP**1 SOURCE

F10. 9. Geometry for disk source and homogeneous

Facility) in the following way:

For

(1)

shield.

a given value of R, for which the kernel

is

+ a', y/R* + 2a1, . . . , etc.; (2) from the experimental data, find the derivative of the dose at the abscissas 7?, y/R* + a*, y/R1 + 2a', . . . , i.e., for v = 0, 1, 2, etc.; (3) add these together to form the indicated sum, divide by 2rsR, and the negative of the result is the dose to be expected at a distance R from a point source of unit strength. For large enough source plates the foregoing sum converges rapidly and only one term [i.e., Kq. (27)], or at most a few terms, is required for accurate evaluation. For very small source plates and large distances the source can be considered to be a Equation (25) then applies. point source of strength As, A being the source area. The derivative of the dose function can be characterized by a relaxation length, or "e-folding length," X, which is defined for this dose function as required, calculate

\/R*

Hz)

=

DP, (z,a)

If X(z) is only slowly varying in z, then an approximate used which obviates the summation: L \4x\jsX * The quantities in the brackets

(30)

(d/dz)Dpl(z,a)

ra's/

are to be evaluated at z = R.

form for Eq. (29) can be

J*

(31)«

Sec. 7-3]


This should be used only

if

(1) X is nearly a constant in the region z

a» < 4Xz.

1 is not satisfied, Eq. (29) must will give a good estimate of G(R). Finite Disk to Infinite Plane Source.

If condition Eq. (27)

7-95

be used.

If condition

2 is

R and

(2)

not satisfied,

It is possible to determine the dose to from an infinite plane of sources in terms of the dose at all axial points to sources spread evenly on a finite disk:*

6.15

be expected due

A>i(z.«)

-

Vz'

Dpl(

£

+ *»',

o)

(32)

w-0 {Note: In this equation it is assumed that surface source strengths for disk and infinite plane sources are the same. If they are not, a true equation can be had by dividing each dose by the corresponding source strength.) The use of this equation is similar to that of Eq. (29). The summation can be avoided by the use of the following

approximate inequality

:

D„,(z,a) a


where

—

— (a' VX

2

+

1 )

/

and X is defined by Eq. (30). For the approximate equality in Eq. (33) to be valid, 1. X(z) must be nearly constant in the region of use. 2. a* < 4Xz + 4X». If condition 1 is not satisfied, Eq. (32) must be used. If condition 2 is not satisfied, the dose on the disk axis will be nearly equal to that at corresponding distances from an infinite plane.

6.16 Plane to Spherical Shell Source. If an isotropic source of a given strength per unit area is spread on a spherical surface which is wholly within an attenuating medium, it is possible to predict the dose as a function of distance from this surface in terms of the dose from'an infinite plane isotropic source of the same strength per unit area. The material inside the sphere must be the same in attenuation as that out-side the source, and the source itself must be thin enough so that it does not affect the attenuation. This transformation can be used for the case of spherical or nearspherical reactors in which the spherical shell source will be an element of the volume source. The dose at a distance R from the center of the sphere whose radius is r,(R > r0) is

D,(R,r0)

=

5

[Dpl(R

-

r„,

»)

- Dpi(R

+ r„,

=o)]

(34)

The quantities in the brackets

are the doses to be expected at distances of R — r0 + r0 from infinite plane sources of the same surface source strength. If the attenuation length for the plane source data is sufficiently small in comparison with the diameter of the sphere, then the second term in the brackets can be ignored and

and

R

D.(R,r„)

« £ Dpl(R K

-

r„,

»)

(35)

6.17 Plane to Cylindrical Shell Source. For conditions similar to the previous if the source is spread on an infinitely long cylinder of radius r0, the transformation, dose at a distance r from the cylinder axis is i>,(r,r„)

^

Dpl(R

-

* Hurwitz transformation; for derivation see Ref. 15, part I.

r„, «,)

(36)

7-96 6.2


Transformations

for Surface Sources in Homogeneous Specified Attenuation Functions

Shields

[SEC. 7 with Partially

It is possible to extend the transformation geometry somewhat further if some The form of the assumption is made about the form of the attenuation kernel. attenuation function is unspecified for a distance z from the detector to the nearest source element, but for distances R > z to other source elements the increased attenua tion is assumed to be a pure exponential with a relaxation length X, which can be taken from data as described by Kq. (30). Thus G(R) = G(z)e-<«-')A

(37)


If the unit strength per unit area isotropic source is 5.21 Quadric Sources. assumed to be spread uniformly on a quadric surface tangent to the XY plane (Fig.

Fio.

10.

Geometry for quadric surface source and homogeneous

shield.

10), the equation describing the source surface near the origin is

where x,, yit z\ = coordinates of curved source surface a = radius of curvature in XZ plane 6 = radius of curvature in YZ plane The dose at the detector is then given by D,{z,a,b)

«

2*sG(z)

1

(i+iY(-Y ia*(-+-Y(-+-Y \hJ \\z \b/ Xa/

\Xz

rv,+

"

+

\Xz

Xa/

\Xz

+

(39)

\Xz For a,b,

D,(z,a,b)

For

Xa/

VXz

X6/

.

S> X,

a derivation

= 2rtO(z)

(±+±y(i \Xz VXz Xa/

+

of this expression see Welton and Blizard."

±y X6/

(39a)


Sec. 7-3] Special Cases.

For the special

case of a plane surface source,

7-97 a,b — > °o

Bp, (2,00) = 2x2\C(2) This is directly comparable to Eq. (27). and for r X,

»

For

(39//

a spherical surface source,

D(z,r) at 2ttXG(2)

a = b = r,

(39c)

+ r

2

and

The same result can be found from combination of Eqs. (27) and (34).


5.3

Leakage Flux from Volume-distributed

Sources

For a self-attenuating medium containing sources, the leakage can usually be satis factorily calculated using first-flight methods. First flight will not give satisfactory answers when there is little attenuation per collision in the medium (as with diffusing thermal neutrons) or when the sources are concentrated largely in regions far from However, for most neutron shielding calculations it is the source-shield boundary. desired to determine the leakage of the penetrating component, and for this purpose cither first-flight or exponential attenuation with removal cross sections is adequate. In the foregoing paragraphs the use of surface distributions of sources has been discussed. These, with the attenuation kernels obtainable from Arts. 1 and 3, enable the calculation of shield performance. It remains to determine surface dis tributions which are equivalent to the volume sources such as reactor cores. At the surface of a volume containing 6.31 Uniformly Distributed Sources. uniformly distributed sources the leakage flux as a function of angle to the surface normal is 1(B)

. th 4»

COB g

(40)

8 = angle to the surface normal 1(8) = leakage rate (of photons or neutrons) per unit area of source volume sur face, per unit solid angle about a direction at an angle 8 to the normal, cm"1 sec-1 p = source strength, particles per unit volume per unit time, cm-' sec-1 Xc ~ relaxation length in the source-containing material (core) for the radiation = (2r)-1, when 2, = macroscopic removal cross section for the core, cm"1, when fast neutrons are considered = (m)~'i where m = total linear attenuation coefficient for the core, cm-1, when 7 rays are considered. It is required that \e be small compared with the source dimensions and the local radius of curvature of the source-shield interface. The total leakage per unit area

where

of surface is

l

=

/;=o2

w aQ («>

This

expression is useful when

it is necessary

to calculate the total heating, for

example, of escaping y rays, or the total escape of neutrons for purposes of obtaining a capture -y-ray source. For shields it is necessary to calculate the leakage of the most important components With thick shields especially, those neutrons or photons that penetrate of radiation. Con are generally those that leave the surface in directions close to the normal. sequently, the equivalent isotropic surface source is chosen to match the leakage flux in the normal direction [8 = 0, Eq. (40)] s,„

S pX„

(42)

7-98


[SEC. 7

where s«, is that isotropic surface source, particles per square centimeter per second, which, if placed at the interface between the volume source and shield, gives the sameIt is here assumed that this source gives the same radia dose at the shield exterior. tion per unit solid angle about the normal direction. In case the dimensions of the volume source are not large compared with X0, a correction may be made on the basis of a simple integration over the interior points For very thin source regions, of the source volume.

s„ « ph

(42a)

where h is the thickness of the source region in centimeters, and it is small compared

with

X,.

Corrections for curved surfaces, such as spherical source volumes, are in general small. The expression for the dose from a sphere of volume source density p becomes"

R

where

T"

~RK

=

DSy(R,r0)

^f

-

Dpl(R

r„,

»)

(43)

= distance from center of sphere to receptor, cm

r0 =

radius of sphere, cm

= source strength for plane data, Dpi{R — r0,

is

is

a

is,

a

R

r0
and

r0

«)«-toA.

« Xe,

(44)

-

~

U

In

(f^)

]

Rrc

j

at 2>rpSG(fi)

(**

DBr(R,r0)

If

-r*.

same case the dose can be determined also in terms of the kernel

[

For this

dp'(r

^r.^r

Dsv(R,r0)

= %*Tt*pQ(fi)

Dsv(R,ro)

directly: («> (46)

is

S

= fiirro'p. The last equation follows from the fact that the total source In case the sources are not distributed Nonuniformly Distributed Sources. 6.32 In this case, required. uniformly over the volume, an integration

»-/.

vol

v(R,) G(R„R) dV

(47) R

+

a

a

the attenuation for distance point source Ri away, the measured in the shield and Ri in the source-laden volume (e.g.,

»

is

is

Xe, knowledge of the variation attenuating source volume, i.e., r0 near the boundary adequate to make proper adjustments to the given by Thus, the source density

if

R

is

where G(Ri,R) being distance reactor core). For a strongly in source density foregoing results.

p(x)

= po

+ Pix

(48)

constant source strength term, cm-' sec-1 Pi = first derivative of source term with respect to distance from surface, cm-4 sec-1 x = normal distance from surface, measured inward, cm then the leakage per unit surface area, per unit solid angle about a direction at an to the normal, angle po =

P»Xc cos

4r

piK*

cos2

—^

e

-

+

W

0

is

where

6


is

is

is

it

s

«>), em'/sec seen that the equivalent to be compared with Eq. (35), from which surface source very nearly p\c. The correction from r0/R to (r0 — K)/R takes curved away from the receptor. into account that the surface situation encountered in calculating y For small spherical cores, that r0
This

(49)


Sec 7-3].

7-99

The total leakage for thia case is

L

=

-

fir/2

/

4

l(e) da

+

(50)

6

The equivalent isotropic surface source is s«,

6.4

-

PoK + PiXc1

Approximations for 7-ray Build-up

(61)

Factors

In calculations of doses from source arrays, which usually involve integrations, it is convenient to approximate the build-up factors by an analytic function. The The effective removal cross section concept simplest of these is an exponential. for fast neutrons implies an exponential build-up factor. It has been demonstrated''27 that build-up factors for y rays can be well approximated by a sum of two exponential functions, as follows: B(V) = Ae-"<>" + (1 A)e-"i't (52)


-

where B(iii) = build-up factor, either for dose or energy absorption, for a given attenuating medium, such as water, iron, lead, or concrete, for a given The use of this factor, which is energy per photon in the source. dimensionless, is described in Art. 1 A, ai, at = constants which make Eq. (52) give the same build-up factors as are given in the tables of Art. 1 It = total linear attenuation coefficient, cm-1, from Table 1 t = thickness of attenuating material, cm The advantage of using this mathematical form for the build-up factor is that it entails no extra complication in the calculation of dose from an array of sources. The parameters for the dose build-up factors in water, concrete, iron, and lead are given in Table 19. 6.6

Formulas for Flux in Various Source-Shield

Geometries

If the algebraic form of the attenuation function is known, it may be possible to The simplest attenua perform the integration at once over the source distribution. tion function (or kernel) is that with exponential attenuation and unitary build-up factor. For this,

°w)-S

(53)

If removal cross sections for fast neutrons are used, then it is, of course, just the macroscopic removal cross section and the build-up factor is taken to be unitary. This equation gives the current (of photons or neutrons) at a distance li from a source of unit strength with an interposed shield of thickness ( and attenuation coefficient #1. If the approximations to -y-ray build-up factors8'27 are to be used, then the attenua tion kernel would be Ae-
If

just

a linear build-up factor is to be used as is sometimes done for y rays of energy below the minimum in total cross section in light elements (Z
m'i) Better than the foregoing,

because

'S

(55)

it gives the correct values for small id, is

a


7-100 Table 19.

Values of A,

— oi,

[SEC.

7

and «j for Various Media and a Point Isotropic Source* Value of constant

Energy, Mev

Worst error in fit,

— O]

A

at

%

In Water Medium 0.5 1 2 4 6 B 10

0.5


1 2 4 6 6 10

23 11

0. 136 0. 104 0.076 0.055 0.0495 0.045 0.042

6.4 4.5 3.55 3.05

2.7

0.028 0.092 0. 1165 0. 124 0. 128 0. 13

In Ordinary Concrete

Medium

12.5

0.111 0.088

0.01 0.029

0.068

0.058

0.059 0 0585 0.057

0.079 0.083 0.0855 0.0835

9.9 6.3 3.9 3. 1

2.7 2.6

0.05

In Lead Medium 0.5 1 2 3 4 5. 1097 6 S 10

0.032

1.65 2.45 2.60 2. 15 1.65 1. 20

0.296 0. 178 0. 103 0.077 0.064 0.059 0 059 0.067

0.045 0 071 0.097 0. 123 0. 152 0. 175 0 204 0. 214

0.96 0.67 0.50

0.08

- 1.5 -3.9

+ 4.9 + 5.0 + 3.8 -3. 1

-3.9 + 5.2

-3.3

I n Iron Medium 0.5 1 2 4 6 8 10 * aouBCE:

Kef.

10

8.0 5.5 3.75

2.9 2 35 2.0

0.0948 0.0895 0.0788 0.075 0.0825 0.0883 0.095

0.012

0.04 0 07 0 082 0 075 0 0546 0.01 16

6 8 8.0

5.9 6.25 4.95

2.8 1.85

9. pp. 416 423.

combination of the unitary and linear build-up factors,

OiM

= (1

+

ff*

inn'

Somewhat more general than Eq. (56) is

(56)

47rA2

where

It

k is a constant to be chosen for best fit to the actual kernel. has been shown*-" that for heating calculations with y rays ~ Mo M k £* Mo

* Derivation given in Ref. 17. vol. 1, chap. 2.7.

(57a)


Sec. 7-3]

7-101

This formula gives results between % and % of the correct build-up factors for E ~ 2 Mev in elements for which Z

...

(R)

#o(mO =

Enbd)

for id <

id

= (id)n~l

f" — dx

J tit

2"™

(see Figs,

10

for id <

F(0,id) 5.61

=

L

m<

«""' sec

Point Source.

0' d0'

(fty

for id >

(see Figs,

10

_rgr1_» + »(» + i)_.(» m/

lla-c)

10

+ i)(» + (mO3

2)

+

lla-c) .

. .1

(see Figs, \2a-d)

forM(>

J

10

~

The flux beyond a slab shield due to a point source is given by

0(R)

=

SG(R,i)

(58)

-oij -ii

',t(-3

Wff 90 10

V

j-W)

„

pu« UPO'ff ■» P1

-

°*

:r0

fl

-o

01

\L

•» jrf

-#

,,.£.

(/"

^ -*- L*«

i«

—

"oasJ £

4

r

noixvkivh any ivoidoioicivh NOixoaxoud

?

5

*

?

?

9

-

s

50

•
ro

l

7

to

0

1

3

20

t

7

'j-

s

'j _/>-(/*)

»

t


Z0l~L

v»

<-0'

^\

/*)>-*

•-(

^j- — 'P

'P-j^ft'-W)'

/*-

oi

= e *>■» "01


Sec. 7-3]

NUCLEAR RADIATION SHIELDING 7-103


7-104

1

"1

[Sec.

7

7*

.

T""~" s

\

>

k

/ n\ /C

V

13. Geometry

6.62 Line Source. At line source is given by

P,

and

P,

for line source and slab shield.

beyond a slab shield (Fig. 13), the flux due to

a

*

A = ■=- [F(6,,,U)

- Ftfunt)]

(59)

At P,,

*

- -A- if

+ F(e2lld)]

(59a)

AtP,,

4> =

4irZ

(eltMi)

(596)

F(8UIU) along the axis of

14)

disk

I

t

z

Disk Source. For a point at distance (Fig. 6.63 source and beyond a shield of thickness (i.e., < z),

a

-P-

4x2

(60i

Infinite Plane Source. given by

The flux beyond a slab shield due to an infinite plane

lid

1

»

for

»

1

S-—

ut

E*(jd)

^

-

2

=

2

/

The outbound particle current density

for /it

61

is

Z

6.64 source

is

(62)

is

5.66 Thick Slab Volume Source. If the source distributed throughout a thick slab volume the flux beyond the slab shield given by is


Fio.


Sec. 7-3]

14. Geometry

for disk source and slab shield.

SHIELD

Fio.

15. Geometry

and the outbound

for cylindrical volume source and slab shield at side.

current density is

2n,

(04)

fit

a

is

a

Cylindrical Volume Source with Shield at Side. To obtain flux from sources 6.66 slab distributed uniformly in a cylinder of attenuating material (nc, cm-1), with necessary to obtain shield exterior and parallel to the cylinder axis (Fig. 15), it


Fia.

7-105


7-106


50

I

1

0

Fig.

16.

1

4

2

self-absorption distance tions. At Px (Fig. 15), =

6

1

8

1

1

1

10

12

<4

1

1

(6

exponent, y.ete, of a cylinder us a function of its radius,

Self-absorption

> 10.

*

1

[SEC. 7

lc

from Figs. 16 to

tt^tV;

4(2 + /«)

tf'(92'

18

a, for 2 'a

and to use the F(d,itl) attenuation

& + "A)

- F($u

yi +

func

(65)

Mctc)}

at P2, i{z + Note:

nAJm

tc)

To obtain

pA<, use Fig. 16 if z/a > 10. and then obtain m from r"ig. 18.

If

z/a <

10,

use Fig.

17 to

obtain

5.67 For this geometry upper Cylindrical Volume Source with Shield at End. and lower limits, based on truncated cones, are given for the flux. The upper limit is for the upper truncation surface equal to the actual cylinder base. The lower limit is for the truncated cone with lower base equal to the actual cylinder base

(Fig. 19).

The upper limit is

Sec. 7-3] =


JL EtW 2ft

- Et(fA

+ nch) +

£2|(m*

+

Vl

Hch)

Vl

+

7-107

+ («/*)']

(a/«)«

E,\nl V~l 'l +(a/z)']\ + (a/*)'

(66)

|

where nJc

is

jtf.W

+

found from Figs.

*0 21

- Ei

nJt.)

+

(^qhT'l

and 22 using a calculated value of

(67) }

| pa

^1

=

\(jd

*

+

The lower limit is the same expression with z replaced by (z + h). Nole: for pch > 3, the second and third E functions in the above expression can be neglected with an error of about 5 per cent or less. 6.68 Spherical Volume Source. For the geometry in Fig. 20,

^ca.

3.8

• 8


3.6

jrIO, 17. Self-absorption

tion of

Mi for 2/a <

!©■

exponent divided by self-absorption

20

22

parameter,

24

26

iictc/m, as a func


7-108


0<

[SEC.

7

23456789 I/O

Fio. 18. Self-absorption t/a < 10.

parameter,

m,

for a cylinder as a function

of its radius, o, for

f!

Fio. 19. Geometry for cylindrical source and slab shield at end.

volume

Flo. 20. Geometry for spherical source and slab shield.

volume


Sec. 7-3] 5.0

I

1

-I

,

1

1

1

1

7-109 1

,


4.5

Fig.

21. Self-absorption by n, for z/a > 1.

exponent, fide, of a sphere as a function of its radius, o, multiplied

6

6.1

SHIELD MATERIALS

Criteria for Selection of Shield Materials

Choice of shield materials varies drastically with application, as does the relative The moat, important commonly used criteria are importance of the several criteria. radiation attenuation, ease of heat removal, resistance to radiation damage, expense, and structural strength. 6.11 Attenuation. The attenuation of fast neutrons is described in 1. Neutrons. Art. 3. The removal cross section is the best criterion, subject to the limitations due to diffusion of intermediate neutrons as discussed there. The lightest shields are usually hydrogenous, whereas the thinnest shields will contain a large volume fraction of iron, copper, or some other material in which the atoms are closely spaced. 2. Gamma Rays. The attenuation of y rays is described in Art. 1. In general, the high-atomic-number elements are the best -y-ray shields, except that hydrogen itself is good on a weight basis. However, the build-up of scattered y rays in hydrogen is high and no really dense forms exist, so that this element is not often considered for 7-ray shields. 3. Capture y-ray Suppression. Gamma rays produced within the shield by neutrons are often serious sources of radiation. Capture 7 rays may be suppressed by the elements boron, lithium, and nitrogen, as shown in Art. 2 (Table 10). Of these, the first two excel because of their large capture cross sections. 6.12 Heat Removal. It is often necessary to remove heat from the inner layers of a shield. For this reason the shield in those regions must have good heat con ductivity or must be capable of being circulated to a heat exchanger. If the exchanger

7-110


[SEC.

7

is not shielded, the circulated material must not become radioactive. Fixed shields may be steel or copper, cooled by water, helium, or nitrogen, for example. The water will become activated, however, by fast neutrons, so that the heat exchanger must be shielded. Shields which are circulated may be water, molten lead or bismuth, or an alloy of the two.

0 75

0 70

0.65

060 0.55

0.50


0 45

0 40

0 35

0.30

0.25

0.20

0(5 0 10 0

2

4

6

8

10 +

/)

12

ficio Ratio of self-absorption distance to radius, tc/a, for distance to its center, z + a, multiplied by nc for z/a < 1. Fio.

22.

(4

a sphere

(6

as

18

a

function of

It is, of course, essential that the radiation Radiation-damage Resistance. 6.13 which the shield attenuates does not have a significantly deleterious effect on the shield itself, impairing either its structural integrity or its ability to attenuate. This problem is most acute in the innermost layers in which the radiation is greatest. In general, the metals are most resistant to damage, although they are subject to some rearrangement within the crystal lattices which affect their strength. Con cretes seem to hold up well, although if heated they will lose water of crystallisation, becoming somewhat weaker and less effective in neutron attenuation. Hydrogenous


Sec. 7-3]

7-111

materials, such as the plastics and wood, are particularly tender, often undergoing Simple molecules such as II20 or NH3 seem to be exceptions in that gross changes. recombination can be effected and the decomposition products can be removed with no impairment of the attenuating property of the residual material. Several review See Sees. 10-4 and 10-5. articles on radiation damage have been published.*132 In many shielding problems expense is the paramount considera Expense. 6.14 In this case a careful balance must first be made between the intrinsic cost tion. of the shield and the effect of shield size and configuration on other aspects of the installation, such as building size, support structure, etc. In the choice of the shield material itself, transportation costs should be carefully considered. Local material such as natural minerals (e.g., barytes in cast Tennessee) and waste material (e.g., steel scrap) should be evaluated as aggregates to be used with portland cement. Earth itself provides a fair shield. Water is an excellent shield if large thicknesses can be tolerated, and the thickness can be strongly reduced by adding a layer of lead or steel for 7-ray attenuation.


6.2

Shielding Properties of Specific Materials

The macroscopic removal cross sections for several hydrogen-containing materials used for neutron attenuation are listed in Table 20. The attenuation coefficients for some of the heavy elements, particularly effective as 7-ray shields, are given in Table 21. Table 22 illustrates the method of computing both the neutron and the For this example the effectiveness of each 7-ray shielding properties of a mixture. element in two different concretes, portland and barytes, is computed. Table 20.

Macroscopic Neutron Removal Cross Sections for Materials Containing Hydrogen (<•)

Compound

Chemical formula

Density, g/cm1

NHi LiBH. LiH H,0

0.771 (78%) 0.686 0 820 1.00

Water

em'/gt

8. 18 7.58 6.21

6.69

0. 0. 0. 0.

144 165 152 100

Zr, cm-1

Ill

0. 0. 113 0. 125 0. 100

(f>)

Material

Assumed chemical

Fuel oB

CH1.1

Natural rubber *

Sh

Density, g/cni*

em'/gt

formula

CsHu (C.H«)n

0.89 0.71-0 73 0.92

6 4 6. 7

6.5

is the atomic density of hydrogen,

10" atoms/cm1. the macroscopic removal cross section divided by the compound purposes of rough comparison, hydrogen is assigned I barn.

t Xr/p is

density

0. 106 0. 124 0. 106

Sr. cm-1

0 095 0.093 0.098

(see Art. 3).

For

Tables 23 to 27 give data on cost, composition, and property of high-density con crete taken from an article by H. S. Davis." Material cost in Table 25 includes costs for raw material, transportation, process Cost can be adjusted for local prices using the mix data. Costs ing, and handling. Limonite, combined with barite, magnetite, increase with density and water content. or steel aggregate, raises fixed water content to about 5 per cent. If the amounts of

7-112

RADIATION AND RADIOLOGICAL PROTECTION Table 21.

Attenuation

[SEC. 7

Coefficients for 7-ray Shield Materials »i/p, em'/g*

Element

K/em'

7.78

Iron Tungsten Mercury. Lead Bismuth. Uranium.

19.3 13.5 11.3

9.8 18 7

* Mass attenuation coefficient,

2

0.5

Mev

Mev

Mev

0.0840 0.0598 0.0422 0. 131 0.0655 0.0432 0. 147 0.0692 0.0451 0. 132 0.0703 0.0456 0. 156 0.0714 0.0461 0. 185

0.0779

3

6

0.5

Mev

Mev

Mev

0.0359 0.0400 0.0411 0 0413

0.0305 0.0426 0.0441 0.0445 0.0417 0.0449 0 0483 0.0435 0.0471

i.e., total linear attenuation coefficient

Table


Element

Atomic weight,

A

pi, density in concrete, g/cm'

2

3

6

Mev

Mev

Mev

0.279 0.772 0.557 0.466 0.409 0.814

0.237 0.823 0.598 0.503 0.440

0.653 0.465 0.329 2.528 1.263 0.835 1.993 0.939 0.611 1.718 0.794 0.515 1.53 0.700 0.455 3 46 1.458 0.904

0 882

divided by density (see Art. I).

22. Composition and Properties of Concretes (Illustrating method of computation of attenuation coefficients Neutron attenuation*

Mev

for Shielding for mixtures)

7-ray attenuation for 6 Mevf

f/f,

Sr/P. em'/g

em'/g

Ordinary (Portland) Concrete

0

Si

Al

Fe Ca C Na

K

H Mg

16.0

28.06

1.103 0.2815

55.85 40.08

0.0330 0.0183 0.7712

26.97

12.01

0.0761

23.00 39.10

0.0116 0.0079

1.0

0.0250

24.32

0.0426

Total

0.041 0.0295 0.0301 0.0200

0.024 0.050 0.033 0.0245 0.602 0.032

2.37

0.0452

0.0038 0.0004 0.0002

0.0255 0.0279 0.0266 0.0305 0.0302 0.0246 0.0255 0.0289

0.0150

0.0449

0.0014

0.0267

0.0083

0.0010 0.0004 0.0185

0.0942

0.0282 0.0079

0.0009 0.0006 0.0233

0.0019 0.0003 0.0002 0.001 1 0.0011 0.0655

Barytes Concrete Ba O S

Fe Ca Si

Al H Mg Na

Mn Total

137.36 16.0

32.07 55.85 40.08 28.06 26.97 1.0 24.32

23.0 54.93

1.470 1.090

0.348 0.307 0.159 0.061

0.020 0.015 0.013 0.005 0.003 3.49

0.0539 0.0278

0.0124 0.041 0.0275

0.0182

0.0200 0.024 0.0295

0.0061 0.0038

0.0305

0.0302

0.0094 0.0048

0.0018

0.0279

0.0017

0.0006 0.0090 0.0004 0.0002 0.0001

0.0266

0.0005

0.0449 0.0267 0.0255

0.0007 0.0003 0.0001 0.0001

0.0301

0.602 0.032 0.033 0.0202

0.0447

0.0096

0.0945

0.0367

0.0255 0.0287

0.0294

0.0100

0. 1093

* -Sr/p is the macroscopic removal cross section divided by the compound density (see Art. 3). t m/p is the mass attenuation coefficient, i.e., total linear attenuation coefficient divided by (see Art. 1).

density

Sec. 7-3]

nuclear radiation shielding Table 23.

Concrete Aggregates Per cent by

Specific

gravity* Heavy

aggregate

Source

Punching!), sheared bars

Chilled


2FeiOi.3H>0

FeiOi.HiO FeiO«, etc.

Hydrous iron! >92% BaSO. Nev. >90% BaSOi Tenn., Mo., Fo.P, FejP, FeP Mont.

Barite

Dollars/ton

weight

Composition Fine

Coarse pieces

Limonite-goethitet. ■. Mich., Utah Nev. Mont. Barite Tenn.

7-113

sand

3.75 3. 45 4 62 4.30 4.20 4.28 6.30

Iron

3.80 3.70 4.68 4.34 4.24 4.31

58 55 64 60 1-10

6.28

SAE

7.50

standard

FOB source

9

Processed

at jobt

70

0 0 0

7 11-20 9 7 18 19 80

99

0

120

130

98

0

120

130

7.78

Fixed water

II

1

2-5

25-40 40-50 20- SO 18 22-30 30-40 90

* Material water saturated, with its surface dry. t Processing charges include grinding and other operations to make the ore suitable for use in concrete. Cost of grout sand is in italics. Freight charge of SlO/ton is included. %This ore occurs as a mixture of the two minerals. 1 This ore is primarily magnetite, with some hematite (FeiOa) and Umonite. source: H. 8. Davis, Nucleonics, Ref. 33.

Table 24.

Grouts

Sand/cement ratio Type*

Weight

S-270 S-250

M-170 B-155

L-146

Water/cement weight ratio

Grout sand

No. 110 steel shot No. 110 steel shot Magnetite Barite Limonite

3.75 3.30 2. 13 1.49 1.28

Volume

1.58 1.39 1.04 1. 10 1.09

0.35 0.35 0.55

0.54 0.55

Wet weight,

lb/ft'

270 250 170 155 146

* Grout sands processed from heavy ores should have a fineness modulus of 1.0 to 1.5; the heavier the sand, the finer it must be to stay in suspension. Sands having 20 to 40 per cent finer than No. 200 screen may be used if tensile strength at high temperature is not important. Each of the above grouts is pumpable; however, the metallic ones should not be pumped into large maKses of aggregates. Grout S-250 may be pumped more easily than grout S-270. source: H. S. Davis, Nucleonics, Ref. 33.

magnetite and limonite are varied, density can he varied from 175 to 250 lb /ft', with corresponding fixed water content from 20 to 30 lb. Using iron aggregate, Natural-aggregate concretes cost 1 to l.SfVlb for materials. densities from 240 to 425 can be obtained at 2 to 6^/lb. This concrete can be justified only where space is at a premium. The greatest potential cost savings are in simplifying form work, reducing labor, Estimated costs at left show and using natural heavy aggregates rather than steel. that usually form cost is over half the total cost. In the first three cases, temporary plywood forms were used, while permanent steel forms were used in the others. On one job using elaborate steel forms cost breakdown was: 16 per cent, form fabrication; 20 per cent, materials other than aggregate; 30 per cent, labor at the site; 12 per cent, equipment; 15 per cent, steel aggregate; 7 per cent, limonite and Limonite and magnetite are over 50 per cent of the volume magnetite aggregate. of the concrete.

AB C

D

#1

10 shot 63; punchings 324 Magnetite 44; punchings 270 107; 161 Ferrophosphorus Ferrophosphorus 92; 171 42; magnetite punchings Magnetite Limonite' 28; limonite punchings 60 Limonite shot No. 330 65; 1320; 49 + barite 70 35: 70 Ferrophosphorus 42; magnetite Magnetite punchings Magnetite 37; 180 86; 110 Magnetite Barite 29; 168 Barite 86; 105 Hydrous iron ore 82; 100 Limonite + magnetite 28 29; limonite Limonite 62; 76 Local sand 50; 61

aggregate: coarse, lb /ft16

-f 1

EF +

H

IJ

1

122

160 40 100 + 50 122 + 67

67 -r-

K

L

Flast. I.A. Flast. Plast.

I.A.

I.A. I.A.

A. I.A. Plast.

A. A.

Admix ture*

Shielding

12 10.9 12.2 13.5 12.8 10.9 9.7 11.5 10.5 11.6 12 12.5 15.4 11.5

5.9 11.3

Mix water. lb/ft»

2.5

Min

~9.5 3.6 4.1 13.0 12.3 3.6 4.7 4.8 5.7 2.9 2.9 9.2 10.9 17. 4.7

5.9 11.8 15.0 12.7 12.0 21.9 20.9 12.8 12.6 11.9 13.5 10.5 11.6 17.5 20.0 27.8 11.5

Max

Water content, lb /ft"

1

0 a is

+

+

+

+

+

1

is

2.

3.

ia

0PQ

L-146

B-155

M-170 M- 70

M-170 L-146 Mortar

S-270 M-170

type

Grout

Puddled Prepacked Conv. Conv. Pre. Pre. Conv. Conv. Pre. Pre. Conv. Pre. Conv. Conv. Pre. Conv. Conv.

Place ment

Tested after number of months indicated in parentheses. Where only one material used for both fine and coarse aggregate. given, Intrusion added where shown. I.A. (Intrusion Aid, 1.5 per cent by weight of the Portland cement, special grout admixture designed and used by Aid) Plaat. (Plastiment) concrete densifier manufactured Prepakt Concrete Co., Cleveland, Ohio. Corp., Passaic, N.J. by Sika Chemical Maximum Difference water content water weight when concrete wet. Minimum water content amount left after drying to constant weight at B5°C. between maximum water content and amount of mix water added water content water of crystallization held by aggregate. The difference between minimum and water of crystallization water retained by the hardened cement paste. This concrete made using magnesium-oxychloride cement, made of 12.2 lb/ft3 of MgO powder and 19.8 lb of MgCh solution. - Utah limonite used except for mix P, which uaea Michigan limonite. H. source: Davis, Nucleonics, Ref. 33.

16.8 20.6 MO« 24. 19.8 22.2 34.0 23.7 19.8 17.5 24.3 19.3 19.3 24.9 22.7 31.3 31.3

Heavy fine, lb/ft3;

for

7

N

3.000 (3) 3.000 (3) 4.700 (3) 3,870 (4) 5.000 (3) 6.000 (3) 7.000 (3) 5.350 (4) 4,800 (3) 5.000 (3) 6.500 (3) 3.500 (3) 6,000 (3) 6.500 (3) 5.000 (3) 5.800 (1) 8.800(1)

Cement, lb /ft1

Concretes

I.

M

25.40 18.40 15.40 12.05 11.90 11.20 11.30 7.75 5.75 2.55 2.20 3.30 3.05 10 2.75 10 0.50

Compressive strength, psi"

High-density

■+-

G

410 346 300 300 300 263 262 262 262 244 232 227 222 219 215 185 154

Weight, lb/ft*

Material cost, dollars/ ft'

26.

I. I.

-4

type

Con crete

Table


5 3

1

is

is

0 * e

ia

a is

is

is

it

is

is

*

*

is

S.


Sec. 7-3]

Table 26.

Concrete*

Conv. (15) Q (16) P (16) 0(17) M (15) L(I5) K' (17)

F(I7)

Physical Properties of Concretes

Com Rupture Klast. Utt. bond Shrink pressive Lb/ft> Bags/yd* strength, modulus, modulus strength, age, % psit paif (X 10 «)t psit

153 154 185 210 226 230 224 273

~5 0 9.0 9.0 6 55 5 1 5 0 7.0 6.55

4.960 8.870 5.865 4.640 6.130 3.340 5.780 3.180

7-115

4 6 5 9 4.4 5 0 4.3 3 7 4 3 5.4

630 1.005 700 516 445 630 684 406

Expan sion, in./(in.) (°F)

0.041 0 012 0 021 0 023 0 029 0 029 0 018 0 013

900 970 1.3%

1.423 1.266

5.5

5 6 10 10 5 7 5.9

Specific Conduc heat, tivity, Btu/ BtuAhr) (lb)CF) (ft)(°F)

Diffusivity, ft'/hr

0 23

1.50

0.042

0 20 0 157 0 146 0 21 0.18

1.20 0 884 0 867 1,68

0 0 0 0 0

2 75

030 025 026 034 056

* Concretecomposition is similar to that given in table of concretes,except for K', which has46-lb limonite and 0.6-lb fixed Titer per cubic foot. Conv. is Grand Coulee concrete. Placement method is given in Table 25. The limonite usedwas fromMichigan; the barite was from Nevada; specific gravity of magnetite was 4.4. t Strength values are Formoist cured specimens,28 days old. source: H. S. Davis, Nucleonics, Ref. 33.

Concrete Shield Costs


Table 27.

Cost,

Ordinary structural. Ordinary Ilijrb density Higb density High density source: H. S. Davis, Nucleonics,

dollars/yd1

Type

Concrete

Q

K

0 F Ref.

or N

Total cost/lb, cents

Forms

Concrete

45 60 100 300 600

25 40 120 250 500

1.7

2.4 3.7 ».5 15.5

33.

REFERENCES 1.

Fano, U.: Gamma-ray Attenuation, Nucleonics,

d.

June, 1953. Way, K., et al. "Nuclear Data,"' Natl. Bur. Standards Circ. 499 and Supplements

11(1): 8 (August, 1953); 11(2): 55 (September, 1953). 2. Goldstein, H., and J. E. Wilkins: "Calculations of the Penetration of Gamma Hays," NYO-3075, June 30, 1954 (available from the Office of Technical Services, Depart ment of Commerce, Washington). 3. "Reactor Handbook," Vol. I, chap. 2.3, Technical Information Service, U.S. Atomic Energy Commission, 1955. 4. Grodstein, Gladys White: "X Ray Attenuation Coefficients from 10 Kev to 100 Mev," Natl. Bur. Standards Circ. 583, April, 1957. 5. Davisson, C. M„ and R. D. Evans: Revs. Mod. Phys., 24: 79 (1952). 6. Nelms, A.: Graphs of the Compton Energy — Angle Relationship and the Klcin-Nishina Formula from 10 Kev to 500 Mev, Natl. Bur. Standards Circ. 542. 7. Gamble, R. L.: "Prompt Fission Gamma Rays from Uranium 235," Dissertation, University of Texas, Austin, Tex., June, 1955. 8a. Hollander, J. M., I. Perlman, and G. T. Seaborg: Table of Isotopes, Revs. Mod. Phys., 26: 469 (1953). 6. Clark, F. H.: "Decay of Fission Product Gammas," NDA-27-39, Dec. 30, 1954. Product Decay Gamma Energy Spectrum," APEX-134, c. Moteff, John: "Fission

and 2. e. Supplements 9.

and Nuclear Data Compilations in NSA. Rockwell, T. (ed.): "Reactor Shielding Design Manual." TID-7004, March,

1956.

1

7-116


[SEC. 7

"Chart of the Nuclides," 1950, free from Dept. 2-119, General Electric Company, Schenectady, N.Y. 11. Mittelman, P. S., and Robert A. Liedtke: Nucleonics, 13(5): 50 (1955), revised and brought up to date as of April, 1956. 12. Hughes, D. J., and J. A. Harvey: "Neutron Cross Sections," BNL-235, July 1, 1955. 13. Heitler, W. : "Quantum Theory of Radiation," 3d ed., Oxford University Press, New 10.

York,

1954.

15.

R., J. Certaine, and H. Goldstein: "Penetration of Neutrons from Point Isotropic Monoenergetic Sources in Water," NDA-15C-60, 1954. Blizard, E. P.: "Introduction to Shield Design," pt. 2, pp. lOOff., ORNL-CF-51-10-70,

16.

Enlund, H. L. F.: " Energy Absorption of Capture Gammas," p.

14.

Aronson,

November,

1951.

4,

ORNL-CF-52-«-99

(suppl.),


Oct. 16, 1952. 17. United States Atomic Energy Commission: "The Reactor Handbook," vol. 1, p. 701, AECD-3646, Government Printing Office, Washington, D.C., June, 1953. 18. Bonner, T. W., R. A. Ferrell. and M. C. Rinehart: Phys. Rev., 87: 1032 (1952). 19. Hill, D. L.: Phys. Rev., 87: 1034 (1952). 20. Watt, B. E.: Phys. Rev., 87: 1037 (1952). 21. Nerenson, N.: Phya. Rev., 85: 600 (1952). 22. Nerenson, N., et ol.: "The Fission Neutron Spectrum of Plutonium," LA-1078, Los Alamos Scientific Laboratory, Mar. 8, 1950. 23a. Keepin, G. R.: "Delayed Neutrons— A Review as of October 1955," LA-1970, Los Alamos Scientific Laboratory, March, 1956. 6. Keepin, G. R., T. F. Wimelt, R. K. Ziegler: "Delayed Neutrons from Fissionable Isotopes of Uranium, Plutonium, and Thorium," LA-2118, Los Alamos Scientific

Laboratory, February, 1957. Blizard, E. P.: "Transformation from Disc to Point Source Geometry," ORNL-CF52-3-219, Mar. 27, 1952. 25. Welton, T. A., and E. P. Blizard: The Shielding of Mobile Reactors, II, Nuclear Set. Technol., 2(2): (August, 1952). (Classified.) 26. Blizard, E. P.: "Reactor Leakage for Shielding Calculations," ORNL-CF-53-7-170, Nov. 27, 1954. 27. Tavlor, J. J.: "Application of Gamma-ray Buildup Data to Shield Design," WAPDRM-217, 1954. NDA. 28. Goldstein, H.: Private communication, 29. Kirn, F. 8., R. J. Kennedy, and H. O. Wykoff: "Oblique Attenuation of GammaConcrete, and Lead," NBSrays from Cobalt-60 and Cesium-137 in Polyethylene, 24.

2125, Dec. 23, 1952. 30. Auslender, S. A. (Pratt and Whitney Aircraft Corp.), and C. D. Zerby (Oak Ridge National Laboratory) : Private communication. 31. Bopp, C. D., and O. Sisman: Radiation Stability of Plastics and Elastomers, Nucle onics, 13(7): 28-33 (1955). 32. Faris, F. E., et of.: Report NAA-SR-241, 091 pp., Nov. 2, 1953 (classified). 33. Davis, H. S.: Nucleonics, 13(6): 02 (1955).

7-4

MECHANICAL HANDLING OF RADIOACTIVE

MATERIALS

BY Raymond C. Goertz, Kenneth R. Ferguson, and William B. Doe 1

GENERAL SOLUTIONS

TO HANDLING PROBLEMS

Introduction


Special facilities and equipment, as well as certain precautions, are required for the safe handling of radioactive material. It is necessary to protect the personnel both from the bodily intake of this toxic material and from exposure to the penetrating radiation. To guard against bodily intake, the general practice is to carry out fuel processing or laboratory experiments within enclosures designed to isolate the radioactivity from the worker and the building. The enclosures range from the size of a large room down to a few cubic feet. The walls of the enclosure tend to keep the material within its boundaries, but it is customary to maintain a lower pressure on the inside than on the outside to ensure an inflow of air through all apertures. The degree of containment required is directly related to the state and toxicity of the material being handled. Radioactive gases must be trapped, or, if the quantity is small or the half-life short enough, they can be dispersed by discharge through sufficiently high stacks. Radioactive material is usually easier to contain in the liquid than in the solid state. The surface of some solids may oxidize and form very fine particles which are easily dispersed in air. Some of the more toxic materials are pyrophoric, and accidental ignition of these materials may result in the generation of radioactive smoke. Alpha emitters are highly toxic, and containment must be very good. Alpha radiation, however, is easily stopped, and this type of material can be handled in thin-walled enclosures equipped with rubber gloves. Most y emitters have much lower toxicity, so that the containment requirement is less stringent than for a emitters, but thick shielding walls are required to attenuate the penetrating radiation. The shielding may be placed around the enclosure or be an integral part of it. Viewing and mechanical handling within these enclosures must.be provided by special equipment. Large shielding windows and periscopes generally are used for viewing. Specific-purpose remotely controlled equipment and general-purpose manipulators are used to perforin the operations. When large quantities of a and y emitters are to be handled together, the remote operations must be performed in facilities that incorporate both the required shielding and a high degree of containment. Under these conditions the design of these facili ties should emphasize a minimum need for human access. Hence, completely remote handling, viewing, and maintenance of all equipment within the enclosure arc needed. For example, several proposed power reactors using highly toxic materials will require completely remote operation of certain parts of the plant. 1.1

Methods of Achieving Radiological Protection

1.11 Handling in Canals. Water provides a convenient shielding and transparent medium for carrying out submerged operations on jacketed specimens. Operations are usually carried out with simple long-handled tongs from a platform just above the surface of the water. The water conveniently provides thermal cooling for the

7-117


7-118


[Sec. 7

fuel elements after discharge. Disassembly of fuel element structures can be per formed by machines designed to operate under water. Small amounts of dissolved radioactive material can be removed by continuous recirculation of the water through a mixed-bed ion-exchange unit. However, operations which would produce large quantities of particulate radioactive material cannot be performed conveniently. Handling "7-free" Highly Toxic Materials. 1.12 Experiments involving a emitters can be performed in open-faced ventilated hoods if extensive precautions are followed to prevent spread of the material. Generally, however, this type of facility is used only when the material is in solution and the quantities are reasonably small. Dry experiments are usually carried out in nearly sealed enclosures under negative pressure to ensure more complete containment. In some cases, boxes or facilities equipped with gloves are completely sealed and filled with an inert gas to prevent ignition of pyrophoric materials. •>>•* The box pressure is always kept below room pressure to ensure that the flow is inward in the case of a leak. These facilities vary in size from small "glove boxes" to long process lines' with numerous gloves. Gram quantities of a-emitting materials may contain appreciable amounts of impuri ties which emit penetrating radiation. In some cases limited working time will be necessary to avoid overexposure of the hands. Equipment may be set up in a box and used for a number of similar experiments. The box will eventually have to be cleaned sufficiently for reuse or discarded. Equip ment may be removed by means of a plastic pouch transfer system which maintains the seal of the box enclosure and leaves the equipment in a sealed bag. Engineers at the Atomic Energy Research Establishment, Harwell, England, have developed a technique of removing, cleaning, and setting up new equipment in boxes by sealing one side of the box to a service room which is entered only by personnel wearing "frog suits." These suits are completely sealed, and air is supplied through long hoses. A series of three showers is used to clean the suits before they are removed from the worker. Handling Small Quantities of -, -emitting Materials (to about 0.01 Curies). 1.13 Toxicity of these materials is generally lower than that of a emitters, and operations can be performed quite safely in open-faced ventilated hoods or ventilated glove boxes. It may be necessary to provide local shielding within the enclosure, depending upon the working distance and time. When shielding is required, manipulation can usually be performed by using straight- or bent-handled tongs and tools; viewing may require mirrors or small lead-glass windows inserted into the local shielding. Handling Intermediate Quantities of -/-emitting Materials (to about 10 1.14 Experiments involving these materials, but with little or no amount of or Curies). emitters, are usually carried out in ventilated hoods, boxes, or open-top junior caves (Figs. 1 to 4). The y shielding can be placed around the enclosure or incorporated as a permanent part of the facility. General-purpose manipulation is visually pro vided by ball-joint manipulators, castle manipulators, or master-slave manipulators, and viewing is provided by moderately large shielding windows. Access to the enclosure for setting up experiments and transferring of equipment or samples is provided by access doors and ports. 1.16 Handling Large Quantities of 7-emitting Materials (to about 106 Curies). Laboratory operations involving high-intensity y radiation, with only small amounts of highly toxic materials, are usually performed in ventilated enclosures adequately shielded on all sides. This type of facility is usually called a "cave" (Figs. 5, 6), and the bulk shielding generally is ordinary concrete, dense aggregate concrete,4 or steel. Equipment and personnel access is through large doors and, in special cases, through removable roof sections. Radioactive samples are introduced either by placing the shipping carrier within the facility or by transferring the samples through the wall from specially designed transfer carriers. Most caves are equipped with masterslave manipulators for handling samples and small tools and for operating some of the The handling of objects weighing more than 20 lb requires experimental apparatus. a crane or heavy-duty rectilinear manipulator. General viewing required for these operations is provided by large shielding windows. Utilities are usually supplied * Superscript numbers

refer to References



Sec. 7-4]

handling of radioactive materials

7-119

and controlled at the face of the cave and introduced into the cave, as required, through shielding plugs in the front wall. Personnel access into the facility is required for making equipment changes and Frequent remote cleaning is desirable to prevent the accumulation and repairs. Those operations which spread of radioactivity during the course of an experiment. produce quantities of particulate matter, such as machining or abrasive sawing, are sometimes completely or partially hooded to facilitate the cleaning operations. 1.16 Handling of 0- and 7-emitting Highly Toxic Materials. When these mate rials are handled, careful containment, along with shielding, is necessary. Experi ments involving 0- or a moderate amount of 7-emitting materials can be carried out within essentially sealed boxes which have shielding placed around them. Leadglass windows are generally used for viewing, and ball-joint, castle, or master-slave Although this type of facility provides manipulators are used for general handling. adequate protection during the experiment, the problem remains of cleaning the facility and equipment for new experiments. Design studies are in progress on general-purpose cave faciUties for handling large The general design approach quantities of materials emitting both a and 7 radiation. at Argonne National Laboratory is based on a completely sealed cave with suitable y and neutron shielding. The cave will be equipped with mechanical and elec tronically controlled manipulators for performing all the general experimental opera tions. Emphasis is placed on providing equipment, manipulation, and crane facili ties not only for performing the experiments but also for performing all the routine setup and maintenance of equipment without the need for personnel to enter the shielded enclosure. For special servicing, provision will be made for partial remote cleaning followed by personnel entry in sealed suits with forced air supply. Engineers at the Atomic Energy Research Establishment, Harwell, England, are considering using "frog suits" for personnel access into the high-level caves for routine The plan is to construct several caves in a row setup of experiments and servicing. with an a-sealed front wall and with shielding doors opening into an a-sealed room at the rear. Sufficient decontamination of equipment will be performed remotely so that the workers in suits can enter the cave proper to decontaminate it further and to set up new experiments. Personnel egress from this area is through three suc cessive showers before the suits are removed. Ventilation of Enclosures. The atmosphere within all the above enclosures 1.17 must be conditioned to meet the operational requirements. This may be accom plished by passing a suitable amount of properly conditioned air through the system Also, in systems that are more tightly once and exhausting it outside the building. sealed, conditioning can be accomplished by recirculating the air or gas through a suit In either case the discharge to the outside atmosphere able purification system.*'' must always pass through an adequate air-cleaning system to reduce the radioactivity Useful general concentration to within the accepted limits for the surrounding area. information on air cleaning has been collected.* Fundamental studies and extensive engineering tests have been carried out to improve the techniques and media used in air cleaning. Air cleaning for many enclosures is accomplished by an exhaust prefilter on the Tightly sealed enclosures filter on the outside. inside and a "high-efficiency" designed for handling highly toxic materials may require high-efficiency inlet as well as exhaust filters to contain contamination in the case of accidental loss of negative A noninflammable medium, such as glass fiber, should be used for all pressure. filters if there is any possibility of ignition or chemical reaction with an organic filter medium. Some commercial types of air filters have been tahulated.9 A number of methods have been developed for the safe replacement of dirty filters. Generally, a covering is placed over the contaminated side before it is removed. A recent technique utilizes sliding seals to achieve a fairly high degree of containment, and in addition, it permits the filter to be removed into a shielded carrier.10 The pattern of air flow within t he enclosure and the rate of exhaust must be designed to meet the operational requirements of the facility. Heated air and light gases • See Ref. 1, p. 33.

7-120


[SEC.

7

near the ceiling, and heavy vapors and particulate matter can best be exhausted In addition, the rate of exhaust must be sufficient to ensure a from near the floor. minimum air flow of about 150 lin ft/min through all cracks.


1.2

Manipulation

within Enclosures

Manipulation is best performed directly with the hands through gloved ports when heavy shielding is not required. It is often necessary to use tongs or other tools to give the hands additional radiological protection or to perform the operation better. If the enclosure has thick shielding, all the manipulations must be performed remotely. Engineering solutions to the remote handling problems can emphasize either (1) special equipment to perform specific operations or (2) general-purpose The early manipulators for performing a wide variety of handling and manipulations. solutions, in both laboratory and process plants, relied principally on specific-purpose Recent shielded enclosures are nearly always equipped with generalequipment. purpose manipulators in addition to specific-purpose remote-control equipment. General-purpose manipulators" generally have seven or more independent motions. The objective is to grasp an object, translate, and orient it in space. Three inde pendent motions (rectilinear or polar) are required for translation within the working volume, three independent motions are required for angular orientation, and a mini mum of one motion for grasping. Additional motions of the arm have been incor porated in some manipulators essentially to change the shape of the arm and /or add Three different types of these manipulators are widely used. gripping fingers. 1.21 Master-Slave Type of Manipulator. Master-slave manipulators have two similar arms, one for performing the work within the enclosure and the other for control outside the enclosure (Figs. 7, 8, 9). 11 The seven or more independent motions of the master arm and handle drive the similar motions of the slave arm and tongs by efficient mechanical linkages or force-reflecting positional servomechanisms. The motions of the slave arm, wrist joint, and tongs follow the motions of the similar parts of the master arm and handle in a simple proportional manner. Counter Forces weights are provided to balance the manipulator in nearly all positions. reacting on the slave arm are reflected back to the master arm by the mechanical or servo linkages. This type of manipulator is easy to operate and gives an adequate sense of force reflection back to the operator's hand. The mechanically connected manipulators The are economical to construct and are in wide use in junior and high-level caves. servo-manipulator is in its early stages of development, and a few working models have been built.13 It appears that after further development this type of manipula tor will help solve many of the general-purpose handling problems in high-level caves and process facilities. For example, it can be constructed in a variety of load capaci ties with multiplication of the forces from master to slave for the high-load capacity units. And, of course, the manipulator has the advantage of being able to operate over a large working volume when mounted on a suitable carriage. " 1.22 The "rectilinear electric manipu Rectilinear Electric Type of Manipulator. lator has a bridge mounted on rails in the caves, a carriage that provides horizontal motion in two directions, and a rigid vertical boom that provides vertical motion (Fig. 10). The lower end of this vertical boom is equipped with a wrist joint having at least three independent degrees of freedom and gripping tongs which are usually The seven or more motions are each driven by an electric motor interchangeable. which is controlled by a switch or a more elaborate speed control located in a portable console. The switches or controls may be combined so that only a few control levers are needed.

This type of manipulator lias found fairly extensive application for general handling especially whore loads are quite high. Various units have been designed to handle ,4_" The maximum load capacities from one pound to several hundred pounds. two principal limitations of this type of manipulator are that it does not have a sense of force reflection and lacks directional correspondence between the control levers and

Sec. 7-4]


7-121

Considerable care must be exercised the manipulator response for some motions. in controlling the manipulator to avoid damage to apparatus. The ball-joint manipulator passes Ball-joint and Castle-type Manipulators. 1.23 The manipulator through a shielding ball mounted in the shielding wall (Fig. 11). is usually in the form of a shaft with tongs on one end and a handle on the other. The shaft passes through the ball and is free to slide along its own axis. Thus, the manip The manipulator often has exchange ulator can cover a conical working volume. Its operation is able tongs and sometimes is equipped with articulated wrist joints. moderately easy, but it is limited in volume coverage, dexterity, and useful size. The castle-type manipulator (Fig. 12) is like the ball-joint manipulator except the These shielding ball is replaced with two shielding cylinders, one within the other. These cylinders may be mounted in low-friction bearings to allow ease of movement. manipulators are usually used in shielded boxes or sometimes in junior caves.


1.3

Viewing within Enclosures

The viewing necessary for performing mechanical handling within enclosures can be achieved with a number of systems such as shielding windows, periscopes, mirrors, Maximum visual information may demand television, and underwater operations.17 (1) an unimpaired view of the work, (2) a wide angle of view, (3) a means for scanning the working volume, (4) depth perception, and (5) a means for viewing from various No single viewing method has been devised which completely vantage points. Often two or more of these systems are used together satisfies all these requirements. to obtain the desired results. The choice of the viewing system depends upon the type of remote operation to be For example, many phases of chemicalperformed and the visual detail required. process operations require general viewing, i.e., only a broad field of view at a con Usually windows, periscopes, and some siderable distance without magnification. times television are used for viewing such operations. Shielded enclosures for research usually require viewing systems of the highest performance practicable. General viewing for these facilities is by large shielding windows, and magnified viewing is provided by periscopes. A shielding window is essentially a transparent section 1.31 Shielding Windows. It usually consists of a suitable container for a dense liquid and/or glass of shielding. (Fig. 13); however, if neutron radiation is present, it may be supplemented with The visual information obtainable with windows is only water or plastic.18-*0 slightly less than that for normal vision. Windows can provide a useful field of view approaching 120° and stereo vision by several operators at a minimum optical dis tance to the work. Chromatic aberration causes a significant loss of resolution when the work is closer to the inside of the window than several times its thickness. This loss can be elimi nated by the use of monochromatic lighting, which can, in practice, be provided by Ix)w-pressure mercury-vapor lamps are nearly as effective, sodhim-vapor lamps. provided that the blue light emitted is absorbed either by the window or by an auxiliary filter. Viewing contrast is not significantly impaired by most windows provided such factors as haze in liquid windows, incomplete polishing on glass surfaces, and surface reflections are suitably controlled. For all types of mechanical handling with general-purpose manipulators, shielding windows should be large enough to allow the principal operating area to be viewed directly. In high-level applications, they are usually 3 to 5 ft wide by 2 to 3 ft high on the operator's side. The inner side may have larger dimensions to obtain a wider The optical distance to the work can be reduced by using high-density field of vision. These materials have a high index of refraction, which transparent materials. further decreases the apparent thickness. Transparent Window Materials. It is usually desirable that the transparent 1.32 window material have approximately the same shielding effectiveness per unit of Hence, development has been conthickness as the surrounding biological shield.

7-122


[SEC.

7

centrated on materials with an effectiveness comparable to those of ordinary con crete, dense aggregate concrete, and steel. Table 1 gives the physical properties of commonly used transparent shielding materials. Optical-grade zinc bromide solution21 suitable for use in shielding windows has the following significant properties: freedom from optical imperfections such as occur in glass, high light transmittance, and relatively low cost. Hydroxylamine hydrochloride added as a stabilizer in a concentration of 0.01 per cent is usually sufficient for many years of service providing the window is not over For thicker windows, the required stabilizer concentration can be cal 3 ft thick. culated from an estimate of the radiation exposure that the window will receive during a given period of operation." The stabilizer content should be chemically analyzed at suitable periods to ensure that it is not depleted, resulting in the release of bromine, which will attack the tank liner. Table

Physical Properties of Transparent Shielding Materials

1.

Type

Density

Index of refraction

Internal luminous transmittance per inch,


Tungsten light

Zinc bromide solution. Cominerciul lime glass. Water-white lime glass Protected silicate glass Protected lead glass. . . Dense lead glass

2. 52 2.52 2.52 2.68 3.27 6. 20

1.56

I.S2 1. 52 1.53 1.59 1.98

Sodium

%

light

98.5

99.0

94.5 99.0 96.5 97.5

94.2 99.0 96.7

92.0

98. 1

93.0

Another precaut ion is to purchase zinc bromide solution in batches of adequate size to fill an integral number of windows to prevent subsequent hazing which might occur when different batches are mixed. Commercial lime glass (ordinary plate glass) is available in thicknesses up to 1 in. Because of radiat ion darkening, it can be used in laminated shielding windows up to only 12 in. thick. Water-white lime glass may be used in place of commercial lime glass, as in the case of very thick all-glass windows, in order to gain increased light transmission. Protected silicate glass is available as l-in.-thick plate and as cast glass in thick nesses up to 10 in. Both the plate and cast glass have been made in maximum sheet sizes between 5 and 6 ft on a side. The glass is used as tank cover plates for zincbromide-filled windows or in glass sections of windows where protection from radia tion darkening is required. Protected lead glass is used in windows where the extra density is desirable and where a radiation-protected glass is required. The glass is available in castings up to 10 in. thick. The surface of this glass must be kept dry to prevent haze formation. This Dense lead glass is available in castings up to 6 in. thick and 4 ft on a side. glass is generally used as a complete window in a steel wall or as a section of a com The surface of this glass must be kept dry to prevent haze formation. posite window. This glass is easily darkened by radiation but clears more rapidly than other glasses. Up to 8 in. of the glass can be used for a period of 100 hr when exposed to 1-Mev y However, thicknesses radiation of an intensity to give 6 mr/hr on the shielded side. up to 14 in. can be used for only a few hours but will clear for reuse in a few hundred hours. The limiting factor in using thick 1.33 Radiation Damage to Window Materials. Properly engi shielding windows is the darkening of the transparent materials." neered windows can receive a total incident exposure up to 10* or possibly 10* r with out prohibitive darkening of the glass providing the exposure is over a period of at least 1,000 hr. The 1, 000-hr period is significant because a clearing process in glass.


Sec. 7-4]

7-123

is

5

is

is

is

a

is

it

a

it

a

a

is

is


it

y

ft

a

A

a

}■£

having a time dependence, takes place simultaneously with the darkening. Higher intensity exposures over shorter periods will cause greater darkening for the same total exposure. However, at incident radiation intensities above 10' r/hr y heating of the glass may be sufficient to increase the rate of clearing. Gas release in zinc bromide solution commences at incident radiation intensities of 5 X 10* r/hr. Viewing, however, is not impaired until the intensity is about 5 X 10* or greater. 1.34 A periscope is basically a system of lenses for transferring a real Periscopes. optical image from one end to the other (Fig. 14). In effect, it optically transfers the observer's eye to the far end of the periscope. Usually periscopes are designed to have a magnification in the range of to 10 power. Shielded enclosures for research usually are equipped with periscopes for magnified vision, as a supplement to large shielding windows. They also are used to transfer images from other optical devices such as microscope. Scanning-type periscopes often are used for viewing objects at some distance beyond the radiation shield. typical application would be in process facility where the work may be 50 or more from the operator. In some instances, these applications require binocular periscopes to prevent fatigue during prolonged observa tion or to provide depth perception.21 1.36 Radiation Damage to Optical Materials. The lenses in periscopes and other optical instruments may be darkened by radiation having an order of magnitude of 10 curies with an energy range between 0.5 and 3.0 Mev. Special radiation-pro tected optical glasses have been developed for use at high intensities."-2' In the case of high radiation intensities, may be necessary to provide local shielding around Some use has been made of plastic lenses, which are more resistant to the lenses. radiation than the commercial optical glasses.28 Some lens cements fail completely at a total incident exposure of still satis X 107 r. Canada balsam, however, factory after a similar exposure. 1.36 The optical resolution achieved with industrial-wired television Television. about one-sixth that of periscopes for a given field size. chains For this reason, television useful only when the observing distance too long for periscopes or windows or for temporary viewing when other means are not available. Stereo television systems have been demonstrated and were found to improve depth perception but to impair resolution somewhat. *• 1.37 Mirror Systems. In low-level facilities, mirrors may be used for viewing over the top of shielding wall. The level of radioactivity must be limited to about 10 curies because of the scattered radiation. In high-level enclosures, the optical path for mirror system must follow through a labyrinth-type passage in the shield. Hence, for this type of mirror system, difficult to obtain wide angle of view and the optical distance to the work will always be greater than the effective optical distance through a window with equivalent Because of these limitations, shielding. impractical to use mirror systems in high-level enclosures for general-purpose manipulation. 1.38 Underwater Viewing. When the nature of the material and equipment allows submerged operation, underwater handling provides a simple means for remote the usual method of viewing when large quantities of radioactive It viewing. material, such as fresh fuel elements from a reactor, must be handled or stored for cooling. Ripples on the surface of the water can be controlled by the use of glassbottomed float box. Water usually must be specially treated to give the clarity required for viewing through 15 or 20 ft. Underwater periscopes, for detailed viewing, are commercially Any radioactive contamination of the water by the material being handled available. must be removed by filters and ion-exchange beds. 1.4

Specific-purpose

Remote -control

Equipment

Nearly all types of processes and experiments performed on nonradioactive mate This requires nearly all types of rials are also performed on radioactive materials. In process and testing equipment to be operated under remote-control conditions.

7-124


[SEC. 7

some cases standard commercial equipment can be used with a minimum of modifica tion, and in other cases completely new designs are required to achieve adequate remote operation. In addition, there may be no commercial equipment which meets the requirements of the specific experiment or process. It is impossible to describe the many pieces of equipment which have been designed. A catalogue briefly describ A revised ing some of the more useful pieces of equipment has been prepared.30 and expanded catalogue is being prepared under the auspices of the United States Atomic Energy Commission. 1.5

Radiation

Damage

Radiation damage of materials used in carrying out experiments on moderate quantities of radioactive materials is generally limited to the darkening of viewing materials described in the article Viewing within Enclosures. Radiation damage on other materials is covered in Sees. 10-4 and 10-5.


1.6

Fuel Processing and Fabrication

Fuel elements must be processed to extract useful materials, such as plutonium, All processing, at the present and to reconstitute the fuel to its original condition. time, is performed by a variety of chemical operations in large shielded rooms, or "canyons." Present reactors are fueled with materials which can be handled manu ally. However, proposed reactor systems will recycle the fuel without complete decontamination. A canyon equipped with 1.61 Chemical Processing with Mechanical Servicing. a special overhead bridge crane (Fig. 15) provides a moderately simple means of performing a limited number of mechanical operations on the various chemical vessels located within the canyon.31 The vessels are connected to pipes cast in the canyon wall by means of simple piped connections with special clamps for attachment. These connections can be made by using an impact tool suspended by the crane. In this manner vessels can have simple operations performed on them or be replaced. A chemical-processing plant 1.62 Chemical Processing with Manual Servicing. has been designed and operated with manual servicing of the equipment.32-34 The chemical vessels are sufficiently decontaminated by a series of remote cleaning opera tions. The problem of mechanical handling under these conditions is very small indeed, but extensive decontamination is required before major repairs or alterations can be carried out. 1.63 Remote Processing and Refabrication. Certain "short-cut" fuel-purifica tion methods, such as pyrometallurgy, are under development as an economical means of processing fuel for heterogeneous power reactors. Since the fuel remains radio active after this type of processing, the refabrication must also be done remotely. An economical solution to a completely remote operating plant must place emphasis However, the equipment on the reliability of the remotely controlled equipment. will be highly contaminated with toxic materials and cannot be economically cleaned Also, because of the complexity of the operations to he for direct maintenance. This maintenance performed, it appears that remote maintenance will be required. can be simplified by designing the equipment so that it can be easily taken apart into subassemblies. In this way the repair of a specific machine would be accomplished by replacing a subassembly which could then be later repaired or discarded. Cranes and manipulators would be provided to replace or service all the equipment, including themselves, within the enclosure. 1.7

Reactor Servicing

Present reactor systems are designed to require very little remote repair or servicing for the fuel and control elements which are removed for processing. When servicing is required on highly radioactive or contaminated parts of the reactor, special tools and shielding must usually be provided to perform the specific operations.5* except

Sec. 7-4]


7-125

For some types of proposed power reactors, remote servicing of a major portion of the plant appears to be the most economical approach. This will require the exten sive use of general-purpose remote handling devices, such as manipulators, special cranes, and robots, as well as special viewing systems. The plant layout and com ponent parts of the reactor system must be designed so that they can be serviced remotely with a minimum of plant downtime. 1.71 Reactor Charging and Discharging Methods. The basic design of each heterogeneous reactor includes provision for removing and replacing fuel elements.

The particular method and auxiliary handling equipment may be quite simple or complex." Methods of charging and discharging heterogeneous reactors which have been designed

include (1) removing the fuel into a shielded carrier placed on top (2) pushing new fuel in one vertical face of the reactor, causing spent fuel to be discharged from the opposite face into a canal (Fig. 16) ; (3) in a light watercooled reactor, flooding it so that there is sufficient shielding for personnel to transfer the fuel to a discharge chute with simple long tongs (Fig. 17); (4) providing a large shielding room, above the reactor, equipped with special charging and discharging cranes (Fig. 18) ; and (5) performing reactor discharge and storage within a large tank of liquid metal by a special mechanism (Fig. 19).

of the reactor;

2

TYPICAL EQUIPMENT FOR HANDLING RADIOACTIVE MATERIALS 2.1

Glove Boxes

The glove boxes illustrated Glove Box with Interconnecting Vacuum Lock. These par representative in size and shape of units commonly used. ticular units arc made of welded stainless steel, but boxes also are constructed from wood, aluminum, and plastic. The illustrated boxes are gaslight, and materials


2.11

in Fig.

1 are

Fia.

1.

Glove boxes with interconnecting

vacuum lock.

are admitted by means of plastic pouches attached to the box or vacuum lock. Metal covers over the pouch openings and glove ports are used to reduce leakage through the rubber gloves and to add to the safety of the facility when the gloves or ports are not in use. These boxes are 24 in. deep, but many boxes range in depth up to

in. The deeper boxes require tongs or other means for reaching the extremities. In some eases boxes are 48 in. deep and various heights with gloves on both sides.

30


7-126

STEEL CLAMP RING

O-RINGIDO NOT DISTURB)

REMOVE

GLOVEBOX

(INSIDE)

[SEC. 7

GLOVE-INSTALLED READY FOR USE

GLOVEBOX

_Jj

p CLAMP RING (STEEL)

-OLD O-RING 8 GLOVE (REMOVE AS ACTIVE

(INSIDE)

WASTE)

OLD GLOVE

GLOVEBOX-


(INSIDE)

i/

GLOVE

REPLACEMENT 2ND STEP

NEW GLOVE (SLIP OVER OLD

8 CLAMP AS SHOWN)

Fig.

GLOVE BOX JNSIDE)

—

2.

GLOVE

REPLACEMENT 3RD STEP

Glove-replacement procedure.

GLOVEBOX (INSIDE)

-Q-RING(NEOPRENE) ^ (BAG TUBULAR

£_(ORIGINAL)

(SEE ENLARGED SECTION

ABOVE)

A

-

-I

ST STEP (TUCK BAG END IN AND INSERT OBJECT INTO POUCH FORMED)

TRANSFER INTO GLOVEBOX

/

DO NOT BREAK

THIS

.

GLOVEBOX(INSIDE)

CUT-AXIS -

SEALtIm*

GLOVEBOX(INSIDE)~£aJl TRANSFER INTO GLOVEBOX- 3RD STEP (DRAW POUCH INTO GLOVEBOX REMOVE OBJECT)

X ALL HEAT SEALING DONE

TRANSFER INTO GLOVEBOX - 2 ND STEP (HEAT SEAL BAGENDAS SHOWN)

AND

DIELECTRICALLV

Fig.

- TRANSFER OUT . GLOVEBOX (INSERT OBJECT INTO BAG.TRIPLThEAT SEAL" (SEE ENLARGED SECTION a CUT OFF AS SHOWN)

3. Pouch-transfer procedure.

Sec. 7-4]


7-127


The boxes are always kept under negative pressure in the range of 0.25 to 2.0 in. of water to ensure that any leakage will be inward. A rubber-glove-replacement procedure used 2.12 Glove-replacement Procedure. A gastight seal between is illustrated in Fig. 2. at Argonne National Laboratory To minimize the spread of the glove and the port is made with a metal clamp ring. contamination resulting from repeated glove changes, before proceeding to change a glove, care is taken to remove all dust from the inside of the glove port by wiping and Three steps for changing the glove sometimes by the use of a small vacuum cleaner. are illustrated in B, C, and D. The metal clamp ring is then removed, and the bead of the glove folded over the end of the port leaving the tight-fitting O ring as the seal as shown in B. Next, the rib of the new glove is slipped over the old glove and into the groove adjacent to the box, after which the metal clamp ring is fastened in

Fio.

4.

Junior cave.

place as shown in C. Finally, the old glove and O-ring are pulled into the box and a new O ring is put into the outer groove as shown in D. Other systems are similar

but may use only two grooves instead of three for the glove mounting. 2.13 Pouch -transfer Procedure. Objects may be transferred into or out of a glove box without breaking the seal, as illustrated in Fig. 3. The system illustrated here uses a dielectric heat

located outside of the box. To transfer an object end of the pouch as shown in A and four thick heat sealed as shown in B. The completely enclosed object is then transferred into the box where the tucked-in enclosure is cut open, thus exposing the object to the inside of the box without disturbing the four-thickness seal as shown in C. Removal of objects from the box is done simply by inserting the object into the end of the pouch, separating it from the interior of the box by making three parallel dielectric heat seals, and cutting through the center seal as shown in D. New pouches may be exchanged by a method similar to the method used for rubber gloves (Fig. 2). sealer

into the system, it is tucked into the nesses of vinyl film are dielectrically

7-128

RADIATION AND RADIOLOGICAL PROTECTION 2.2

[SEC.

7

Caves

2.21 Junior Cave. The junior cave, illustrated in Fig. 4, has a working volume It is designed primarily for low-level chemical 30 in. deep, 60 in. wide, and 40 in. high. research and can be equipped with two model 7 master-slave manipulators through

ports A or ball-joint manipulators

through ports B.

The front shielding wall can


be moved from side to side about 10 in. or straight away from the enclosure, leaving a plastic contamination barrier which can be equipped with rubber gloves at ports Materials can be admitted or removed by moving the front shielding wall to C. one side and lifting one of the sliding panels G or by passing material through the

0

Fig.

5.

Typical high-level

12

4 3 SCALE, FEET

cave, elevation

5

section.

transfer chamber F. The enclosure has cross-flow ventilation with the inlet at D and the exhaust at E. 2.22 An elevation sectional view of Typical High-level Cave, Elevation Section. a typical high-level cave is illustrated in Fig. 5. The cave has a dense concrete Recent front wall, steel sliding rear doors, and a dense aggregate fill for the roof. cave designs use steel forming for the concrete which is left in place as an integral The interior is painted The steel forming is fabricated by welding. part of the cave. and further protected with stripable material, such as industrial adhesive tape, to facilitate decontamination. A large shielding window A is mounted in the front wall with lights B above and below it. Numerous access holes C are placed around the window and at other strategic places in the wall. Two master-slave manipulators A crane E is used for lifting heavy D are used for general manipulation and handling. objects. The cave is continually exhausted at F through prefilters G and H.


Sec. 7-4]

7-129

High-level Cave, Plan View.

A plan view of a typical high-level is illustrated in Fig. 6. This illustration shows the concrete Access holes B are located plug doors A at the roar of each cell, partially removed. The steel partitions C separating the cells are often movable around the windows. in a vertical or horizontal direction to accommodate a large experimental setup or for transfer of materials and equipment. Ventilation ducts D are located on either side of each cell. 2.23 Typical rave with three

cells


BC

D

01

2 3 4 5 SCALE, FEET

Flo.

Fig.

7.

6.

Typical high-level

Mechanical master-slave

cave, plan view.

manipulator. ANL Model

7.

7-130

RADIATION

AND RADIOLOGICAL 2.3

PROTECTION

[SEC. 7

Manipulators


Mechanical Master-Slave Manipulator, 2.31 ANL Model 7. The mechanical master-slave manipulator, ANL model 7 (Fig. 7), is useful in junior or intermediatelevel caves. It has a load capacity of about 10 lb in any direction. The master and slave arms are coupled together with a 4-in. support tube B, cables, and tapes. The arms are kept parallel to each other with a connecting link C. The conical working volume covered by the slave wrist joint has an included angle of 70 to 90" and a height of 27 in. The wrist joints have angular movements as follows: twist ± 180°, elevation up 30° and down 120°, azimuth ± 170°. The manipulator enters

Fig.

8.

Mechanical master-slave

manipulator, ANL Model

8.

the enclosure through a hole in the roof. The slave arm may be completely lxxited (A) for contamination and chemical-corrosion control. The tongs clamp over the boot and are remotely removable with a special jig. The manipulator is supported by bearings attached to the top of the enclosure. With the tongs removed, the manipulator can bo removed vertically from the enclosure and its boot. 2.32 Master-Slave Manipulator, ANL Model 8. The master-slave manipulator ANL model 8 (Fig. 8) is designed for use in high-level caves. The usual length of the arms from horizontal support to the wrist joint is 90 in., fully extended, and the telescopic range is 36 in. It has a load capacity of about 20 lb in any direction. The wrist joints have angular movements as follows: twist ± 180°, elevation up 30° and down 120°, azimuth ± 170°. The 8-in. horizontal tube connecting the master and

Sec. 7-4]


7-131

The slave slave arms passes through a hole or slot in the front wall of the enclosure. and master arms are kept in synchronism by mechanical linkages within the tube. An electric indexing mechanism adjusts the angular relation between the master arm and slave arm, causing the slave arm to be closer or farther from the master arm. This allows the slave arm to cover a larger working volume or to be indexed to the


The slave arm of the horizontal position for manipulator installation or removal. The manipulator may be booted for contamination and chemical-corrosion control. tongs are clamped over the boot and are remotely removable with a special jig. 2.33 Master-Slave Servomanipulator, ANL Model 2. The master-slave servomanipulator ANL model 2 (Fig. 9) uses special electrical force-reflecting positional servomechanisms to couple each of its seven motions from master to slave. There

Fig.

9.

Master-slave servomanipulator,

ANL

Model

2.

is a correspondence of motion of the slave and master arms and a reflection of the load forces from the slave back to the master. The manipulator has a load capacity of The wrist-joint angular movements are twist ± 180 , eleva 10 lb in any direction. Both the master and slave units of tion up 30° and down 120°, azimuth + 345°. the manipulator are preferably mounted on traversing mechanisms with rigid vertical This allows the manipulator to cover a large working volume. In fact, booms. it is possible to design the system so that the tongs can reach any point within an enclosure. Rectilinear Electric Manipulator. The rectilinear electric manipulator (Fig. 2.34 10) produced by General Mills, Inc., is the most widely used of this particular type The mechanical arm is mounted on a crane bridge to provide hori of manipulator. The vertical motion is provided by a rigid boom zontal movement in the cave. telescoping through the carriage. The tongs are replaceable, or tools may be sub stituted for the tongs. The manipulator is electrically driven and controlled at a

7-132

RADIATION

AND RADIOLOGICAL

PROTECTION

[Sec. 7


The rectilinear motions are console equipped with two pistol-grip control handles. controlled with one of the handles, and the angular motions are controlled with the Meters are provided to indicate the gripping force and wrist torque. An other. alternate force-indicating system can be provided by using a pulsed-audible signal which has the frequency of pulses proportional to the force. It has a load capacity

0. Rectilinear electric

manipulator. General Mills, Inc., Model E.

of 150 lb in any direction, with the exception of a vertical lifting force of 750 lb. Also, it has a wrist torque of 30 ft-lb. 2.35 Ball-joint Manipulator. The ball-joint manipulator (Fig. 11) is representa tive of a variety of similar units used in shielded boxes and junior caves. The shaft A of the manipulator slides through a dense ball B placed in the front shielding wall. This allows the tongs C of the manipulator to cover a conical working volume by


Sec. 7-4]


Flu.

11.

7-133

Ball-joint manipulator. B

0

Fio.

12.

1

2

3

4

5

SCALE, INCHES

Castle-typo manipulator.

movement of the ball and by sliding the shaft along its axis through the ball. The handle has a lever D for actuating the squeeze motion of the tongs, and in this par ticular manipulator, the handle may be indexed along the shaft A and locked at a convenient position by a small lever E. Some ball-joint manipulators have wrist joints for controlling the orientation of the tongs with respect to the shaft. Also, they can be designed so that a boot can be placed over the inner portion.

7-134


[SEC. 7

The castle-type manipulator illustrated in Fig. Castle-type Manipulator. is similar to the ball-joint manipulator except that the ball is replaced by two cylinders A and B, one within the other, to provide the conical motion of the manip This arrangement has the advantage that the cylinders may be mounted on ulator. low-friction bearings and thus provides easy movement of the manipulator. 2.36

12

2.4

Viewing


The composite window Shielding Window for High -intensity y Radiation. 2.41 (Fig. 13) illustrates the use of several transparent materials to achieve an optimum It is essentially a modified zinc bromide window A, used for its high trans design. parency and low cost, shielded from radiation intensity greater than about 10* r/hr

Fin.

13.

SCALE, FEET Shielding window for high-intensity y radiation.

by the protected silicate or lead cast glass section B, and with the dense lead glass section C added to reduce the over-all thickness. The reduced size on the operator's side conserves window materials but does not Design features include an internal step D appreciably reduce the field of view. to allow for thermal expansion of the liquid, a 2-in.-thick protected — silicate cast — glass tank cover plate E and a similar laminated commercial lime glass cover plate F, a commercial lime glass cover plate O used to prevent tarnishing of the lead glass, a


Sec. 7-4]

SECOND

TELESCOPE

PARALLEL REGION OF


RAYS

FIRST

TELESCOPE

Fig.

14. Bush-

optical system for

-OBJECT

a periscope.

7-135

REGION

PARALLEL

OF

REGION OF

RAYS

,

PARALLEL RAYS

7-136


[SfiC.

7


spout H for filling and stirring the zinc bromide solution, lead crack-shielding /, and cast-iron pads to facilitate insertion and removal of the window J. 2.42 Basic Optical Systems for a Periscope. Figure 14 illustrates the basic optical system for a simple form of a periscope (A). It is essentially one telescope The optical system B shows the connected so that it looks into a second telescope. This process of same two telescopes moved apart by adding two extender lenses. The lens system can be bent lengthening can be continued by adding more lenses. These with mirrors for convenience of viewing or to achieve adequate shielding.

SCALE, FEET

Fig.

15.

Canyon crane (Hanford).

periscopes may be used with a real object as shown or may be used with a standard optical instrument by placing the entrance pupil of the periscope so that it coincides with the exit pupil of the standard instrument . The system may be converted to a scanning-type periscope by incorporating scanning prisms or mirrors in the optical beam near the entrance pupil. 2.5

Canyon Crane

Simple operations can be performed in a shielded canyon as illustrated in Fig. 15. The chemical vessels A or other apparatus may be connected by pipes B which have a hook C directly above their center of gravity. These pipes have special clamps on either end which can be tightened with impact tools carried by the crane. The


Sec. 7-4]

7-137

crane is of the bridge type with several hooks and cables to perform the variety of lifting jobs; for instance, high-capacity hooks and carriages are required for lifting A periscope the shielding slabs, and lighter carriages can be used for carrying tools. D is used for viewing from the shielded crane cab E. The motions of the crane are controlled by variable velocity drives. 2.6

Fuel Handling


2.61 Charging and Discharging Horizontally through a Reactor. Figure 16 illus trates the charging and discharging of reactors which have their fuel in horizontal The process of loading or unloading can be accomplished by pushing the fuel tubes. elements in one end, causing the spent fuel elements to be discharged at the opposite face and fall into a canal. The charging can be accomplished manually or with a

Fig.

16.

Charging and discharging

horizontally through a reactor (Hanford).

An elevator at the charging face is needed for personnel or charging machine. An elevator at the rear face is needed for servicing, such machines while charging. as making pipe connections before and after discharge. 2.62 Manual Discharging of Fuel into a Canal below the Reactor. Figure 17 After schematically illustrates the discharge system used for the MTR reactor. the reactor is shut down, the top plug containing the shim-rod drives and magnets and regulating rod drives are lifted from the reactor top by the overhead building crane and placed in a "drydoek" on the first floor where they can be serviced (A). The upper assembly grid B is removed from the reactor and stored in the canal. This grid is somewhat radioactive, but it can be handled safely at a distance with the overhead crane. A working platform C is then placed over the reactor. Since the operators are working straight above the reactor core and are looking through a column of about 18 ft of water, only simple tools are needed to remove the fuel ele ments from the reactor lattice. The elements are then lowered through the discharge



7-138

f

SCALE, FEET

Fio.

Fig.

18.

17.

Special

Munual discharging

charging

cranes in a shielded

room

7

-j

of furl into a canal below the reactor

and discharging

[SEC.

(MTR).

above the reactor.

chute G into the tilting mechanism E. The tilting mechanism is disconnected from the discharge chute at F and lowered to the horizontal position for removing the fuel elements to the canal for storage. 2.63 Special Charging and Discharging Cranes in a Shielded Room above the Reactor. Figure 18 is a schematic illustration of a loading and unloading system


Sec. 7-4]

7-139

The which uses one or two special cranes in a large shielded room above the reactor. cranes are driven by electric servomechanisms controlled behind the observation window. The discharge canal extends into the shielded room so that the cranes can One discharge the fuel elements directly into a holder located within the canal. crane can be used for loading and the other for unloading. Figure 19 illustrates 2.64 Liquid Metal Submerged Charging and Discharging. a

system of loading and unloading a reactor submerged in liquid metal.


ROTATING

0

Fig.

19.

12

After the

PLUGS

3 4 5

SCALE,

FEET

6

Liquid metal submerged charging and discharging.

cover of the reactor is removed, a mechanism is lowered to hold down the fuel sub assemblies around the one to be removed. The gripper is then lowered and is attached. Alignment of the hold-down mechanism and gripper is accomplished by rotating plugs on the top of the tank. After the gripper and hold-down mechanism are lifted, the rotating plugs are moved to a suitable position, and a transfer arm takes the fuel element from the gripper and transfers it to a storage rack. The process can be This system is repeated for all elements, and the reverse process used for charging. designed for the EBR-II reactor.

REFERENCES 1.

2. 3.

Kelman, L. R., W. D. Wilkinson, A. B. Shuck, and R. C. Goertz: "The Safe Handling of Radioactive-pyrophoric Materials," ANL-5509, Argonne National Laboratory, Decem ber, 1955. Edited version in Nucleonics, 14(3): 61-65, 14(4): 65-71, 14(5): 77-82. Hall, G. R. (ed.): "Symposium on the Handling of Radioactive and Toxic Substances," AERC C/R 958, Ministry of Supply, HarweU, Berkshire, England, March, 1952. Shuck, A. B., and R. M. Mayfield: "The Process and Protective Enclosures Designed for the Fuel Fabrication Facility No. 350," ANL-5499, Argonne National Laboratory,

January,

1956.


7-140 4.

5.

6.

8.

9. 10.

7

Davis, Harold S., Frederick L. Browne, and Harry C. Witter: Properties of Highdensity Concrete Made with Iron Aggregate, J. Am. Concrete rut., 87(7): 705-726 (March, 1956). Garden, N. B.: "Successful Multicurie Operations in Equipment Designed with Mobility Complete Control," pp. 338-346, TID-5280, Department of Commerce. Office of Technical Services, September, 1955. Friedlander, S. K., L. Silverman, P. Drinker, and M. \V. First: Particulate Removal in "Handbook on Air Cleaning," Government Printing Office, Washington, D.C..

I

September,

7.

[Sec.

1952.

"Development of a High Temperature High Efficiency Air Filter," NYO-4527, A. D. Little Co., Inc., Aug. 18, 1953. Blasewitz, A. G., R. V. Carlisle, B. F. Judson, M. F. Katzer, E. F. Kurti, W. C. Schmidt, and B. Weidenham: "Filtration of Radioactive Aerosols by Glass Fibers," HW-20847, Hanford Works, Wash., Apr. 16, 1951. Disposable Air Filters, Nucleonics, 13(3): 92-94 (March, 1955). Blomgren, R. A., and N. J. G. Bohlin: "Proposed Methods for Remote Transfer of Alpha and Gamma Active Materials into and out of High Level Caves," pp. 61-72, TID-5280 (Suppl. 1), Department of Commerce, Office of Technic al Services, January, 1956.

11. 12.

Goertz, R. C: Fundamentals of General-purpose Remote Manipulators, Nucleonics, 10(11): 36-42 (November, 1952). Burnett, J. R., R. C. Goertz, and W. M. Thompson: "Mechanical Arms Incorporating a Sense of Feel for Conducting Experiments with Radioactive Materials," A/Conf. 8/P/69, USA, International Conference on the Peaceful Uses of Atomic Energy, June 23, 1955.


13. 14.

Thompson, W. M., and R. C. Goertz: "Master-Slave Servo-manipulator," pp. 1-70, TID-5280, Department of Commerce, Office of Technical Services, September. 1955. Gitzendanner, L. G., W. A. Hartman, G. H. Sittner, and H. M. Steven: Nuclear Engineering, pt. 3, General Purpose Overhead Manipulator, Chem. Eng. Progr. Sympo

sium Ser., 50(13): 18-26 (1954). Board, Jr., Samuel S.: "One Hundred Pound — One Ton Hook General Purpose Manip ulator," pp. G-81-103, Conference on Nuclear Engineering, University of California. Sept. 9-11, 1953, California Book Company, Berkeley, Calif., 1953. 16. James, T. R.: "Some Recent Manipulator Progress," pp. G-71-79, Conference on Nuclear Engineering, University of California, Sept. 9-11, 1953, California Book Company, Berkeley, Calif., 1953. 17. Monk, G. S., K. R. Ferguson, and D. F. Uecker: Remote Viewing in "The Reactor Handbook," vol. 2, chap. 7.2, AECD-3646, Government Printing Office, Washington.

15.

D.C.,

1955.

K. R.: "Windows for Remote Viewing," pp. 73-93, TID-5280 (suppl. 1), Department of Commerce, Office of Technical Services, January, 1956. 19. Ferguson, K. R.: Design and Construction of Shielding Windows, Nucleonics, 10(11): 46-51 (November, 1952). Edited 20. Pedersen, L. T.: "Reactor Shielding Windows," pp. 81-83, September, 1955. version in Nucleonics, 13(10): 78 (October, 1955). 21. Doe, W. B.: "Zinc Bromide for Use in Shielding Windows," ANL-4879, Argonne (Specifications as amended Feb. 15, 1954.) National Laboratory, September, 1952. 22. Doe, W. B.: "Stabilizer Consumption in Zinc Bromide Windows," pp. 78-80, TID5280, Department of Commerce, Office of Technical Services, January, 1956. 23. Ferguson, K. R., and W. B. Doe: "The Performance of Shielding Windows at High Reprints of the Radiation Intensity," pp. 74-78, Brookhaven National Laboratory. "official-use-only" and unclassified papers from classified report BNL-302, May, 1954. 24. Holeman, J. M.: Binocular Periscope Viewers, Nucleonics, 12(1 1) : 64 (November, 19541. 25. Kreidl, N. J.: "Irradiation Damage to Glass," USAEC Report NYO-3777, Bausch and Lomb Optical Co., March, 1953. Damage to Glass," USAEC Report 26. Kreidl, N. J., and J. R. Hensler: "Irradiation NYO-3780, Bausch and Lomb Optical Co., November, 1954. 27. Ferguson, K. R. : "The Performance of Radiation Protected Microscope Objectives." pp. 95-105, TID-5280 (suppl. 1), Department of Commerce, Office of Technical Services, January, 1956. 28. Monk, G. S., and W. H. McCorkle: "Optical Instrumentation," pp. 179-194, Nuclear Engineering Series, div. 4, vol. 8, McGraw-Hill Book Company, Inc., New York, 1954. 29. Johnston, H. R., C. A. Hermanson, and H. L. Hull: Stereo Television in Remote Con trol, Elec. Eng.. 69(12): 1058-1062 (December, 1950). 18.

Ferguson,

Sec. 7-4] 30. 31.

Laboratory Equipment, "Chemical Processing 5276, Government Printing Office, Washington, Harty, W. M.: Nuclear Engineering, pt. 3, The tion and Maintenance of Separation Facilities, 50(13):

32.

33.

34. 35.

36.

handling of radioactive materials and Equipment,"

D.C.,

7-141 pp. 47-302,

TID-

1955.

Design Philosophy of Remote Opera Chem. Eng. Progr. Symposium Ser.,

115-121 (1954).

Chemical Processing of Reactor Fuel Elements at the Idaho Chemical Processing Plant, "Chemical Processing and Equipment," pp. 3-44, TID-5276, Government Printing Office, Washington, D.C., 1955. Kennedy, K. K., and D. G. Reid: Radiochemical Plant Design Philosophy for Direct Maintenance, Am. Inst. Chem. Engrs. Preprint 312, Nuclear Engineering and Science Congress, Cleveland, Ohio, Dec. 12-16, 1955. Jackson, H. K„ and G. S. Sadowski: Design and Operation for Direct-maintenance Fuel Separation, Nucleonics, 13(8): 22-25 (August, 1955). Campbell, K. K.: "Maintenance Work in the Field of Nuclear Energy," pp. G-9-18, University of California, Sept. 9-11, 1953, Cali Conference on Nuclear Engineering, fornia Book Cbmpany, Berkeley, Calif., 1953. McLain, Stuart, R. C. Goertz, and A. H. Barnes: Methods of Replacing Fuel in Hetero geneous Reactors, Am. Inst. Chem. Engrs. Preprint 356, Nuclear Engineering and Science Congress, Cleveland, Ohio, Dec. 12-16, 1955.

BIBLIOGRAPHY 1.


2. 3.

First Information Meeting on Hot Laboratories and Equipment, Argonne National Laboratory, ANL-4670, May, 1951. (Classified.) Second Information Meeting on Hot Laboratories and Equipment, Oak Ridge National Laboratory, ORNL-CF-52-10-230, October, 1952. (Classified.) Third Information Meeting on Hot Laboratories, Brookhaven National Laboratory, " BNL-302, May, 1954. Reprints of for-official-use-only " and unclassi (Classified.) Laboratory, papers, fied Brookhaven National 12-54-200. Edited version, Nucleonics, 12(11):

4.

5. 6.

7. 8. 9.

10. 11.

12.

35-100 (November,

1954).

Annual Symposium on Hot Laboratories and Equipment, Department of Commerce, Office of Technical Services, TID-5280, September, 1955; TID-5280 (suppl. 1), January, 1956. Hall, G. R. (ed.): "Symposium on the Handling of Radioactive and Toxic Substances," AERC C/R 958, Ministry of Supply, Harwell, Berkshire, England, March, 1952. Laboratory Design for Handling Radioactive Materials, research-correlation conference sponsored by American Institute of Architects and USAEC, NP-3875, May, 1952. (Copies available from Building Research Advisory Board of National Research Council.) Nucleonics, 10(11): 34-55 (November, 1952). University of California, Berkeley, Calif., Cali Conference on Nuclear Engineering, fornia Book Company, Berkeley, Calif., September, 1953. Yakovlev, G. N., E. P. Dergunov, I. A. Reformatsky, and V. B. Dedov: "A Hot Analytical Laboratory," International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/P/672, U.S.S.R., June 30, 1955. Pravdjuk, N. F.: "Metal-Research Hot Laboratory," International Conference on the Peaceful Uses of Atomic Energy, A/Conf. 8/P/673, U.S.S.R., July 5, 1955. Kelman, L. R., W. D. Wilkinson, A. B. Shuck, and R. C. Goertz: "The Safe Handling of Radioactive-Pyrophoric Materials," Argonne National Laboratory, ANL-5509, December, 1955. Edited version in Nucleonics, 14(4): 61-65, 14(4): 65-71 (pt. 3, subsequent issue). Nuclear Engineering and Science Congress, Cleveland, Ohio, Dec. 12-16, 1955. (Pub lished by the American Institute of Chemical Engineers.) The following papers given at this conference pertain to mechanical handling or related subjects: Fourth

Glen, H. M.: "Structural Aspects of the Multicurie Fission Products Pilot Plant," Preprint 16. Goeller, H. E., and E. J. Frederick: "Design of Radiation Analytical Facilities at Oak Ridge National Laboratory," Preprint 17. Kershaw, M. G.: " Ventilating Radiological Laboratories," Preprint 156. Hill, A. J., Jr.: "The 'Containment' Philosophy at the Savannah River Laboratory," Preprint 159. Dykes, F. W., R. D. Fletcher, E. H. Turk, J. E. Rein, and R. C. Shank: "Idaho Chemi cal Processing Plant Remote Analytical Facility," Preprint 309.

7-142

RADIATION

AND RADIOLOGICAL

PROTECTION

[SEC. 7

Kennedy, K. K., and D. G. Reid: "Radiochemical Plant Design Philosophy for Direct Maintenance," Preprint 312. McLain, Stuart, R. C. Goertz, and A. H. Barnes: "Methods of Replacing Fuel in Reactors," Preprint 350. Heterogeneous 13. 14. 15.

Vol. 1, No. 1 to date. Handling and Control in "The Reactor Handbook," Vol. 2, sec. Government Printing Office, Washington, D.C., May, 1955. "Chemical Processing and Equipment," TID-5276, Government Washington, D.C., 1955.

Nucleonics,

7,

AECD-3646,

Printing

Office,

Sources of Illustrations


Except where otherwise indicated, most of the illustrations in Sec. 7-4.

Argonne National

Laboratory

is the source of

SECTION

8

CONTROL OF REACTORS BY SIDNEY KRASIK, Ph.D., Westinghouse Electric Corporation M. A. SCHULTZ, B.S.E.E., Project Manager, Westinghouse

Testing

Reactor,

Westinghouse Electric Corporation

JOSEPH M. HARRER, B.S.E.E., M.S.E.E., Head, Nuclear Reactor Control Section, Argonne National Laboratory


C.

ROGERS McCULLOUGH,

Ph.D., Chairman, Advisory Committee on Reactor Safeguards, and Deputy Director, Division of Civilian Application for Hazard Evaluation, U.S. Atomic Energy Commission, formerly Assistant Director, Development Department, Monsanto Chemical Company

JOHN C. MOISE, Ph.D., Assistant Principal Engineer, Liquid Rocket Plant, AerojetGeneral Corporation, formerly Assistant Project Engineer, Fox Project, Pratt and Whitney Aircraft

CONTENTS PHYSICS OF CONTROL BY SIDNEY KRASIK

8-1

1 Reactor Kinetics - Control-rod Effectiveness

in

Thermal

Reactors

8-3

PAGE

8-2

1 2 3 4 5 6 7

THE CONTROL SYSTEM BY M. A. 8CHULTZ

1 Reactor Startup and ation 2 Reactor Operation 3 Reactor Shutdown References

Low-power Oper

Introduction Nature of the Problem Controls Administrative Control Containment Location Future Reactor Types References

8-21

PAGE 8-76 8-77 8-81 8-82 8-84 8-85 8-87

8-88

8-32 8- 42 8-411

CONTROL INSTRUMENTS AND DRIVES BY JOSEPH M. HARRER

1 Neutron-sensitive Detectors 2 Control Drives References

REACTOR SAFEGUARDS

BY C. ROGERS McCULLOUGH

8-12 8-19

Reference*

8-2

8-4

8-49 8-
8-1

8-5

REACTOR-SYSTEM DYNAMIC SIMULATION BY JOHN C. MOISE

1 Mathematical Description of System. . . Reactor-system Setup 2 Computer of

Equations

References

8-90 8-4}7

8-102

8-1

PHYSICS OF CONTROL BY Sidney Krasik 1

REACTOR KINETICS Delayed Neutrons

1.1

The kinetic behavior of a reactor is strongly affected by the delayed neutrons pro duced in fission. These neutrons are emitted by certain radioactive fission-product The characteristic periods of the delayednuclides following 0 decay of the nuclide. neutron emissions are, in fact, the period of the 0 decay. For example, the decay scheme of the mass 137 fission product is given as follows:tpl n


0

HI"7

►

„Xe>"

22s

»

MXe"6

inst.

The Xe1'7 decays by neutron emission to Xe13'. The apparent half-life of the neutron emission is about 22 sec, which is the half-life of the I117 decay.* While perhaps as many as ten delayed-neutron emitters have been found, some five or six of these are of importance in reactor kinetics. The others, because of their yield or period, or both, do not strongly affect the kinetic behavior of a reactor. The measurement of the delayed-neutron yields from fission and their associated For the thermal-neutron periods has been the subject of intensive investigation. fission of uranium-235, the measurements of Hughes et al.' have in the past been most widely accepted. These are given in Table 1 together with data on the energies of the delayed neutrons. An average mean life for delayed neutrons can be defined as

i where ft is the fractional yield of the tth group per neutron emitted in fission and n is the mean life of the ith group. Using the values of Table 1 the value obtained for f is 12.3 sec ; the corresponding half-life is 8.55 sec. In a similar way a mean neutron energy for the delayed neutrons may be defined:

E

From Table t Superscript

1

the value obtained for

numbers

refer to References

I*** =

E

-k

I-

is 515 kev.


8-2

Table

1.

Delayed-neutron

Periods and Yields from Thermal Fission of Dabba,

(Data of Hughes,

rW

half-life,

Group

sec

55.6 ±0.2

i 2 3 4 5 6

22.0 ±0.2 4.51 ± 0. 1 1. 52 ± 0.05 0.43 ± 0.05 0.05 ± 0.02

Ti mean

life,

X,

sec"'

80 2

0.0125 0.0315 0. 154 0.456 1.61 14.0

31.7 6.51 2. 19

0.62 0.07

IOO(0./0) percentage of delayed neutrons

decay constant,

sec

Calm, and Hall>)

0.

fractional yield

3.4 + 0.9 22.0 ±2.3

0 00026 0.00166 0 00213 0.00241 0 00084 0.00025

28. 2 ± 1.7

31.9 ± 1.7 11.2 ± 3 3

Totals

I.I

100.0

Table 2.

0.00755

Si

fiiTi

product mean life X fractional yield

2.06 X 5. 27 X 1. 39 X 5.27 X 5.2 X 1.7 X 9.30 X

10 '

\0'

10 ■

10'

I0 « 10-' 10 «

Delayed-neutron Periods and Yields from Fission by High-energy Neutrons (Data of Keepin, Wimett, and Zeigler4)


Fissile isotope

Group

100 fii/fi percentage of delayed neutrons

Tl/J' half-life.

54 51 21.84 6.00 2.23 0.496 0. 179

± ± ± ± ± ±

0.94

52.38 21.58 5.00 1.93 0.490 0. 172

± ± ± ± ± ±

1.29

55. II 20.74 5.30 2.29 0.546 0 221

±1.

53.75 22.29 5. 19 2.09 0 549 0.216

±0

53 56 22. 14 5. 14 2 08 0.511 0. 172

± ± ± ± +

56 03 20.75 5.74 2 16 0.571 0.211

± ± ± ± ± ±

0 54 0. 17

0.06 0.029 0.017

U«> total

U"'

rjm

U"'

Pu«»

total

total

± ± ± ± ±

± ± ± + ±

0.39 0. 19

0.07 0.023 0.009

0 0. 0. 0. 0

0 1 0 0 0

5 6 7 a 3

100 ± 0 ± 0 ± 2 ± 1 ± 1 7.5 ± 0

1 2 0 2 3 5

Pu"" total Th"»

± ± ± ± ±

1.3 13.7 16.2 38.8 22.5

100 86 86 19 18 108 042

100 ± 0. ± 0. ± 1 ± 1 ± 0 3.5 ± 0

3 4 8 0 9 5

0 0070 ± 0.0004 0.00024 0.00176 0.00136 0.00207 0.00065 0.00022

2.8 27.3 19.2 35 0 12.8 2.9

0.66 0.24 0.08 0.042 0.019

Th"» total

100 ± 0 3 ± 0 4 ± 5. 3 ± 2 0 ± 1. t ± 0 6

6 0063 + 0.0003 0.00022 0.00238 0 00162 0.00315 0.00119 0.00024

100

0 0088 ± 0.0006 0.00169 0.00744 0.00769 0.02212 0.00853 0.00213

3.4 + 0.2 15.0 ± 0.5 15.5 44.6 17.2 4.3

± ± ± ± 100

8-3

0.0412 5 5 I 4 7

1.21

0.95

± 0.0005 0.00054 0.00564 0.00667 0.01599 0.00927 0.00309

27.4 ± 0 22.7 ± 3 31.7 ± 1 7 3+1 2.3 ± 0 3.8 28.0 21.6 32.8 10.3

0. 19 0.077 ± 0.033

0.0165

± 0.0017 0.00060 0.00192 0.00159 0.00222 0.00051 0.00016

95 0 36 0 12 0 08 0.049 0.017

0.38 0.42

0.00063 0.00351 0.00310 0.00672 0.00211 0.00043

8.6 ± 0 3

Pu"» total Pu140

Absolute yield, neutrons /fission

3.8 ± 0 3 21.3 18.8 40.7 12.8 2.6

2. 1 1.5 1.3 0 b

0.0496

U"'

± 0.0020

mean energy of neutrons, kev

250 560 430 620 420

± ± ± + +

60 60 60 60 60

8-4

CONTROL

OF REACTORS

[Sec.

8

/

0i

The total fractional yield of delayed neutrons per neutron emitted in fission /3.

For the thermal fission of U"6 Hughes obtained the value

0.00755

1.

is designated as as noted in Table

i

a

More recently the matter of delayed-neutron yields has been reinvestigated by Keepin and collaborators at Los Alamos.419 Their most recent publications* give delayed-neutron yields for fission of U"', U"8, U"», Pu2", PuS4°, and Th»2 by highfission energy distribution) and for fission of U"*, Pu*", energy neutrons (essentially and UM* by thermal energy neutrons. Table 3. Delayed-neutron Periods and Yields from Fission by Thermal Neutrons (Data of Keepin, Wimett, and Zeigler5)

Ji

I.I

±

0.0005 0.00021 0.00182 0.00129 00199 00052 0.00027

1.6

±0.3 ±0.4 ±4.0 2.0

2.4 1.4

0

0.242 0.047

8.6 29.9 25.2 27.8 5.1

+ 0.0003 0.00057 0.00197 0.00166 0.00184 0.00034 00022

2.9

3 4 ± ± ±

± ± ±

0.21

±0.20 +±

0 0 2

U»» total

0.54 0.38

0.0061

3.5

±4.8 ±3.3 100

55 00 20.57 5.00 13 615 277

0.0158

±0.9

0 0

0.213 0.045

3.5 29.8 21.1 32.6 8.6

±

4.2

4. 4 ± ±

■'

0.40

±0.24 ± t

0 2

6 5 4 3 2 i

±1.67

total 6 5 4 3 2 1

U'»

54 28 23 04 5.60 13 618 0.257

0.00052 0.00346 0.00310 0.00624 0.00182 0.00066

0.9 0.8

100

Pu»»

Pu'"

3.3 21 19.6 39.5

0.0066

100

±

2.34

0.23 0.09 0.083 0.025

Absolute yield* neutrons /fission

0.3 0.9 2.2

5 9 ± ± ± ± ± +

30 610 0.230

0.71

100 0, percentage of delayed neutrons

11

6.22

±1.28 ± ± ± ± ±

55.72 22 72

±

6 5 4 3 2 1

half-life, sec

U"» total

0.0003

0,

is

a

0

is

is

is

0.

i

In another report' Keepin, Wimett, and Zeigler have included from literature sur Divid veys best estimates of », the average number of neutrons emitted per fission. These results are ing the total number of delayed neutrons per fission by gives given in Table 4. the In examining delayed-neutron data, there are two areas of concern. One total delayed-neutron yield per neutron emitted in fission. The other the half-lives assigned to the various groups. For U2" the value of deduced by Keepin et al 0.0064 compared with 0.00755 given by Hughes et al, difference of about 15 per cent. Even taking into account the difficulties of the experiment, this difference perplexing. Presumably the recent, more elaborate experiments have yielded better data. The half-lives and fractional yields of the delayed-neutron emitters given by al are determined by fit to the neutron emission data after irradiation. Keepin These experimenters report that six periods are required to fit their data. The errors assigned to the periods are deduced from the fit. Recently Perlow and Stehney7 have made chemical separations of the delayed-neutron emitters. They report that the second group period of 22 sec assigned to I1*7 results from the mixture of I137, with half-life of 24.2 0.2 sec, and Br88, with a half-life of 15.5 0.4 sec. They report the U"5 fission yield of Br88 3.5 1.5 times the yield of Br87, the precursor of group

1.

±

is

±

±

a

a

el


rjm

Tl/J1

Group

0 2

Fissile isotope

Sec. 8-1]

8-5

PHYSICS OF CONTROL

This means the yield of Br87 would be about 13 per cent of the total yield of delayed neutron emitters in fission of Um. Keepin el al have reported6 that, despite a search, their method of data analysis It should be noted that, because of the method of does not resolve a 15.5-sec period. data analysis they use, introduction of the additional period could change the yields At this time it does not appear that conclusive assigned to the first three periods. and definite values have been assigned for yields and periods of delayed neutron emitters. Table 4.

Absolute Yield of Delayed Neutrons per Fission and Deduced Values of 0 (Data of Keepin, Wimett and Zeigler)

Fission by thermal neutrons

Fission by fast neutrons Absolute yield, Fissile isotope neutrons/fission


[■:--, 0 I'm 0 0 0«M Pu«> 0 Pu>» 0 Th"" 0

Absolute yield, neutrons/fUsion

Fissile isotope

0070+ 0 0004 2.63 ± 0 08 0. 00268 ± 0 00017 0165+ 0.0005 2 56 ± 0.06 0.00645 ± 0 0003.1 04I2± 0.0017 2.62± 0.130.0157 ± 0 00IJ 0063± 0. 0003 3.01 ± 0.09 0.00209± (I mtdIf; 0088 ± 0 0006 3.42 ± 0.20 0.00257+ 0.00030 04% ±0.0020 ~2.3 ~0.022

1.2

U*» Pu»»

V

0

0 0066+ 0 0003 2.55 ±0.06 0.00259+ 0 00018 0.0158 ± 0.0005 2.47 ±0.05 0.00640 ± 0.00032 0.0061 i 0.0003 2.91 ± 0.06 0.00210 ' 0 000IS

Reactor Kinetics — One-group

Diffusion Theory

The simplest reactor kinetic problem to treat is that of a bare reactor analyzed by The conservation equation, i.e., rate of increase of one-group diffusion theory. neut ron density equals rate of production of neutrons per unit volume less leakage and absorption per unit volume, is formulated as Eq. (1).

DV*

-

Sa
+

k(l

-

0)2a
+

Y Xid

=

-

/I0

(i)

neutron flux = (j>{x,y,z,i), neutrons/(cm,)(sec) neutron velocity, cm/sec D = diffusion constant, cm 20 = macroscopic absorption constant, cm-1 0 = fraction of delayed neutrons per neutron emitted in fission \i = decay constant of the ith group of delayed neutrons, sec-1 Ct = density of delayed -neutron emitters of ith group, number /cm1 k — multiplication constant of the reactor material (k = tj/) with 77the number of neutrons emitted per neutron absorbed in fuel and /the thermal utiliza tion, e the fast fission factor and p the resonance escape probability are k is sometimes written fc„ assumed unity in a one-group theory, The companion equation is that for the density of dclayed-neutron emitters. This

where

<* =

v —

is

Eq.

(2).

^ dt

= /s.-fcz,,*

- x.c-

(2)

In this equation 0; is the fractional yield of the ith delayed group in fission. The The solution of this problem has been treated by a number of authors.8-14 usual treatment is to describe the solution as a product of a dominant space mode and The time dependence is separated by assuming various exponential time factors. the flux
where T is to be determined.

=
Assuming that

(\

has the same time dependence as the

8-6

OF REACTORS

CONTROL

[SEC.

8

flux, Eqs. (1) and (2) may be combined to yield Eq. (3).

(1+B,L,)_fc[(1_,)

^_^_].0

+

(3)

+

i

A.

is

=

is

L

is

where B1 the buckling of the mode. In addition, the diffusion length

BV

+

W

space

0

is

To obtain this equation use made of the fact that
introduced

and the mean lifetime of neutrons t>2.

1

constant of the reactor material multiplied by

+ B*L* In addition

*«//

~

and the expression

?

ffi k

frequently called the excess

——

(4)

+

y

-A_

+

-

obtained:

1

is

—

1

~

The term fc,//

B*L*

the following equation 1

After algebraic manipulation

an effective neutron

A° 1

A

A is

-

is

1

1

is

the multiplication

+ B»L«

the nonleakage probability for the particular mode. lifetime defined:

is

n

1,

+

is

is

T

is

is

T

is

is

called the reactivity and designated by the symbol p. the famous inhour equation. It relates the reactivity of the reactor Equation (4) to the parameters of the delayed-neutron emitters and the neutron lifetime of the reactor. whore of order n Equation (4) considered as an algebraic equation in the total number of delayed-neutron groups. For six delayed groups, for example, seventh order in and has seven solutions. The time-dependent solution for the neutron flux therefore of the form

it

£

l

n+

4>(x,y,z,t) =
A^n

is

where the jth solution of Eq. (4) and the A, are amplitude constants which are fixed by the initial conditions. The solution of a reactor kinetics problem involves solution of Eq. (4) for the various and Eqs. (1) and (2) for the various A,- with the given initial conditions.

Tj


k.ff

'_

k

the k,ff for the reactor:

to define

,

is

The next step

physics of control

Sec. 8-1]

Their values

Equation (4) has n + 1 poles for negative T. by rewriting Eq. (4) in terms of p. " =

(a/d

i +

8-7

(I + X ft i

are most clearly shown

(5)

+1x,r)

t

is,

of the equation occur for values of T for which (1 + \iT) vanishes; that = -Ti. = -A. The n The roots of are all pole occurs at real; n roots lie between the poles of the equation and are negative numbers — t;+i

T,

<

-n.

T

T

1

-l/Xi

4-

-

<

T

n poles

T

is

is

is

Id

I

n+

c

= c.o

V

1

>

I

and for times

— koSao
j-i

Bi«en

and

V

l

n+

Ajt*/T>

The Bji may

a

be written in terms of the Aj by use of Eq. (2), and matrix equation written by use of Eq. (1). While this constitutes solution of the problem few groups of delayed an exceedingly laborious method to carry out for even neutrons. A much more elegant procedure for numerical solution of the kinetics problem the Laplace transform method. This has been discussed at length by Goertzel." For practical purposes, however, the only reasonable approach to a detailed solution of the reactor kinetics problem, number of delayed-neutron groups are involved, In particular, analogue computers of the differential by automatic computation. dis inalyzer type are highly adaptable to this kind of problem. This approach Its particular value not only that cussed by Schultz.18 permits solution of the adaptable to problems in which the transient condition of the reactor but that reactor load must be coupled to the reactor. Essentially all engineering work in carried out by analogue computation techniques. reactor kinetics the exact behavior required detailed solution of the reactor kinetics equations needed. Eventually the reactor will be on a of the neutron flux during its transient stable period T, and the time behavior will be simply a

Aj

is

Determination of T,

is

is

is

if

is

it

is

it

is

is

a

if

is

a

it is

for

A


I

iB

A

A

a

a

is

T

it

p

is

is

The remaining root infinite for zero, positive algebraically the largest root. It for positive p, and negative for negative p. Since the exponentials involving the other roots damp out in time prior to that involving this root, this value of which describes the kinetic behavior of the reactor after the initial transient condi tions have disappeared. This value of often referred to as the stable reactor period or the e-folding time. It will be designated as T, in the following. The roots of Eq. (5) may be determined by trial for specified value of p. In principle the amplitude constants Aj may be determined from Eqs. (1) and (2) by specifying the initial conditions, for example the values of the flux, the delayedneutron emitter concentrations together with sufficient number of derivatives at the initial time U. problem in this form seldom arises in reactor kinetics. problem of considerably greater frequency one in which the reactivity zero for all time For time 2 the reactivity either a constant or a specified < t0. function of time. The initial delayed emitter concentrations Co for this case are given in terms of the initial flux ipn(x,y,z) and the original k0 and 2o0 as

relatively simple for many cases of importance.

8-8

OF REACTORS

CONTROL 1.3

[Sec.

8

The Stable Reactor Period

The inhour equation may be solved approximately for the stable reactor period These illustrate the physical effect of the delayed in a number of important cases. neutrons on the kinetics of the reactor. No Delayed Neutrons.

1.31

r.- — Since fc,// is essentially unity for p small, the reactor period is approximately A /p. '/'., a Large Positive 1.32 Number. For this case all the Delayed Neutrons. \iT, 1 by assumption. Furthermore A/7', may be neglected compared with

»

unity:

r. p

(A + pr)

f

is the mean lifetime of delayed neutrons as denned in Art. 1.1. case indicates that for fairly long positive periods the principal effect of the From Table 1, delayed neutrons is to increase the effective lifetime of the neutrons. 0f is about 9.3 X 10_». This is 100 to 1,000 times larger than the value of A for highly thermal, water-moderated reactors. Thus the reactor period for a given reactivity is increased by several orders of magnitude by the presence of the delayed neutrons in the case of long periods. For this case a useful approximate form of the inhour equation is

where

This

p

T, a Small Positive 1.33 Delayed Neutrons. and the inhour equation becomes A

T.

or

-

In this

Number.

case

X.T,

+P A

,

is

is

p

0 is

p

0

is

0,

is

p

is

0.

p

if

/3,

This is the other important effect of delayed neutrons. For very short periods, when the reactivity exceeds the expression for T, becomes exactly the same as there were no delayed neutrons with the exception that the effective value of measured not by its excess over zero, as in Art. 1.31, but by its excess over The significance of this the reactor critical on prompt neutrons only and simply that with > = the period The condition referred to as governed by this condition. = prompt criticality, whereas sometimes designated as delayed crilicality. 1.4

Units of Reactivity

is

is

f)

p

0

is

a

is is

a

p 0

it

p

2

is

is

Several kinds of units have been used to measure reactivity. As used in the preced ing equations, periods and lifetimes are measured in seconds and reactivity is given as a dimensionless fraction. expressed in per cent; for Frequently this fraction referred to as example, a reactivity of 0.02 fractionally per cent reactivity. — occur. By measuring In the preceding article the combinations p//3 and in units of somewhat greater generality to certain equations. possible to give The number called the dollar. sometimes used as a unit of reactivity and Thus a reactivity called one dollar of reactivity. One onenumerically equal to hundredth of dollar called a cent of reactivity. /3


=

T.

or

physics of control

Sec. 8-1]

8-9

The other common unit of reactivity is the inhour, which is short for inverse hour. is the unit of reactivity which produces a reactor period of 1 hr or 3,600 sec. The value of p corresponding to one inhour may be found by substituting T, = 3,600 in Eq. (5). The reactivity corresponding to any other period T, may be measured in inhours by dividing p in fractional units by the value of p for T, = 3,600. The utility of the inhour unit is most easily noted for large values of 7',. For long periods, p expressed in inhours is

It

...

1_

3600_ =

T.(hr)

7-.(sec)

is,

For long periods, measurement of reactivity in inhours does not depend on any physical properties of the fissionable material of the reactor. This measurement simply makes use of the fact that the product pT, is a constant for long reactor periods. While the inhour or dollar are convenient units for reactor operation, if the measured properties of the reactor are to be related to calculated properties, the value of the This is particularly true in the use of a critical inhour or dollar must be determined. experiment to establish the design of a proposed reactor. The data of Hughes et al. given in Table 1 and that of Keepin and Wimett given in Table 2, both for U"', lead to significantly different values of the inhour and the dollar. Using the data of Table 1, 1 inhour in a reactor using U"6 as the fissionable material is a reactivity of about 2.58 X 10~6 fractionally; 1 per cent in reactivity thus corresponds to 387 inhours, and 1 dollar of reactivity is 292 inhours and is, of course, 0.755 per cent. Using the data of Tables 2 and 4, 1 inhour in a reactor using U"6 as the fissionable material is a reactivity of about 2.31 X 10~6 fractionally; 1 per cent in reactivity thus of course, corresponds to 433 inhours, and 1 dollar of reactivity is 277 inhours and 0.64 per cent. The data of Keepin and Wimett are being increasingly accepted as the better data. Negative Stable Periods

1.5

p

is

p

is

Equations (4) and (5) give the reactor period as a function of reactivity for both As noted previously, the stable period T, positive and negative reactivities. positive for positive and negative for negative. The interesting approximations for T, were all derived for T, positive. For very large negative periods the approxima Thus for a small reactivity variation around zero, tion of Art. 1.32 applicable.

a

a

given absolute the resultant reactor periods are the same absolute magnitude for magnitude of p. For shorter periods, i.e., those approaching the longest delayed-neutron period, With positive the symmetry between positive and negative periods disappears. limit fixed by the increases, T, becomes smaller, finally approaching periods as that of the generation time A. For negative periods the limiting stable period longest period delayed-neutron group. For large negative reactivities the reactor ultimately decays with a period determined by the longest delayed group; this mean life of approximately 80 sec.

is a

is

p

It

is of importance

to know the magnitude of the initial

transient when

a

The Prompt Jump

1.6

reac

is

a

a

is

made in reactor. When sudden reactivity change made, there tivity change due to the change in reproduction of is a very sharp change in flux level. This prompt neutrons before the delayed-neutron emitter concentration has changed. is

solved, assuming that the reactor the magnitude of this change Eq. (1) At time = suddenly changed flux level 0just critical with completed, assuming no change in the Cs. To make from zero. The calculation after the necessary to assume < The ratio of the flux the calculation, /S.

p

~

-

p) o)

o

= AC1 (0

>0

is

change to the initial flux

±_

prompt

is

is

is

p

t

a

initially

0,

is

To calculate

was

it


PU l) =

8-10

CONTROL

OF REACTORS

The flux increment above 0o is reached in time as Tp =

-A

(1

(3

-p) -p)

(1

—

[SEC. 8 exp t/TT), where

The prompt jump or prompt drop may be made the basis of experimental reactivity measurements. 1.7

Reactor Kinetics — Energy-dependent

Effects

In Art. 1.2 the kinetic behavior of a bare reactor was studied by use of a one-group diffusion theory model. The concepts and results obtained by this model may be extended to a few-group diffusion theory model. In most cases of interest, the errors introduced are less than 25 per cent. By the use of correction factors these errors may be reduced to the order of 5 per cent or less. The theory as extended remains a bare reactor theory. Little development has been done on the kinetics of multiregion reactors, and the bare reactor theory is generally assumed to be applicable. In the extension of the theory to the few-group model certain obvious parameter modifications must be made. First, the multiplication constant is written in the four-factor form : k = rrftp

1 1

or

1

+ B'L* 1

1

+

BW

+ B'L1

depending upon whether a two-group model or an age slowing-down model is used. In these expressions is the age of fission energy neutrons to thermal energy. The for k,fl and the effective neutron lifetime are modified by using this expression nonleakage probability instead of the expression 1 + B'L* used in the one-group treatment. The errors introduced in the use of the one-group results are in the relation of reactivity to the stable period. The corrections are made by modifying the fas and While these corrections modify the kinetic behavior introducing "effective" fts. of the reactor, their chief importance is in the detailed analysis of the properties of a reactor such as a critical experiment. 1.71 Effects Due to the Energy of the Delayed Neutrons. The delayed neutrons are emitted with an energy spectrum considerably lower than that of the fission neutrons. For this reason, in a finite reactor the delayed neutrons are less likely to leak from the reactor than are fission neutrons. This can be expressed by assigning a greater importance or worth to the delayed neutrons than to the fission neutrons. The increased importance of delayed neutrons has been pointed out by deHoffman," Hurwitz,10 and Goertzel." This importance when evaluated amounts to increasing the fraction 8 of delayed neutrons by an appropriate factor. The effective value of which will be denoted by 8*, obtained by finding the effective value ft* of each Hurwitz12 defines Si* /Si in this way: "[The delayed group and summing these. essentially the probability that a delayed neutron of the ith kind will produce ratio] a fission divided by the probability that a prompt neutron will produce a fission." For simple reactor this probability given by the ratio of the nonleakage proba bilities of the respective types of neutrons. For a bare reactor described by two-group theory, + +

_

1

Bi

1

is

is

is

f).

rj

a


where p = resonance escape probability t = fast fission factor The nonleakage probability is written as

BW BW

Sec. 8-1]

physics of control

8-11

tJ

is the age of fission neutrons to the energy of the thermal group while t,* For a bare reactor described by age the age of the ith group of delayed neutrons. theory,

where is

ft!

■

ft

where Pi is the nonleakage probability of the ith group of delayed neutrons and P0 the This is the correction discussed previ nonleakage probability of fission neutrons. ously. 0i" is the fractional yield of the ith group in U"6 fission, while ft" is the yield of the same group in U"8 fission. vw is the number of neutrons per U"8 fission; k" the corresponding number for U"6 fission. & is the ratio of U"8 fast fissions to UJSi thermal fissions, while e*8 is the fast-fission factor arising from the presence of U"8. The magnitude of this correction can be estimated roughly by taking Then

/Si" = ft™ = Pi

x« =

and

►»« '„ 0

h>

«

Thus the existence of fast fissions enhances

.—

«"

r\' '„ 0

0 by approximately

the fast-fission factor t.

Reactivity Coefficients

1.8

_

dT

fop (1

dT

d

dk,t!

is

is,

A number of reactivity coefficients can be defined which arc associated with a Of these the temperature coefficient of reactivity is one of the most impor reactor. tant with respect to the stability of power reactors. The temperature coefficient may be defined as the rate of change of reactivity per unit temperature change. In general, it is a function of temperature. Ordinarily reactors are designed to have negative temperature coefficients of These customarily lie in the range 10~6 to reactivity to ensure adequate stability. 10~4, reactivity per degree Fahrenheit. The temperature coefficient of reactivity is made up of a number of effects. Some of these effects respond to a power change of the reactor more quickly than others. For this reason a reactor may have a prompt temperature coefficient of one value and a delayed or isothermal temperature coefficient which is substantially different. These differences are associated with equilibration of the temperature rise in the reactor. The calculation of the temperature coefficient of a reactor in principle, simple. For a two-group formulation it

+BV)(l

+ B*L*)

is

In evaluating this derivative every term which depends upon temperature must be In general the result a number of terms both positive and taken into account. negative whose values are substantial compared with the net coefficient. This scheme for evaluating the coefficient frequently not used because of the difficulty of An alternate procedure evaluating the various terms with sufficient precision. to is

is


Clearly the amount by which 0 is increased will depend on the nature of the moderator In a small water-moderated reactor, the increase and the buckling of the reactor. in 0 can amount to 25 per cent. A more complex situation arises if fast fission 1.72 Effects Due to Fast Fission. This would be the case with a reactor having a sub must be taken into account. stantial U"8 content. While the delayed neutrons, in general, are too low in energy to cause fast fission in USM, the fast fissions do modify the effective value of 0. Hellens18 has suggested an expression of the form

8-12

CONTROL

OF REACTORS

[Sec. 8

calculate the reactor at various temperatures near the temperature for which the coefficient is desired. The reactor parameters are evaluated at these temperatures, for example 25°F above and below the operating temperature, and the k effective calculated for the three cases. From the change in k effective and the temperature interval, the value of the coefficient is deduced. The temperature coefficient not only is a function of the reactor temperature but also depends upon the control-rod configuration, state of depletion or poisoning of the reactor, etc. For this reason, it is seldom possible to associate a unique value of coefficient with a reactor. Another coefficient of reactivity of importance is the pressure coefficient of reac tivity. The pressure coefficient of reactivity of a reactor is the change in reactivity In pressurized water reactors the tempera per unit pressure change of the reactor. ture coefficient of reactivity is defined to be the coefficient at constant pressure and The interaction between the pressure coefficient is taken at constant temperature. these coefficients in a reactor plant results from the finite time required for the pressurizer to reestablish the initial pressure following a temperature rise. For this reason the effective temperature coefficient of reactivity upon initiation of a tempera In pressurized water ture transient is not the isothermal temperature coefficient. reactors the pressure coefficient of reactivity is of the order of 10-6 in reactivity per It is generally positive. pound per square inch pressure rise.


2

CONTROL-ROD EFFECTIVENESS IN THERMAL REACTORS 2.1

Introduction

The direct measure of the worth or effectiveness of a control rod or system of con trol rods is the change produced in the k effective of the reactor upon movement of the rod or system of rods over the maximum available range of mechanical motion. If the Sk/k change produced by the rods is substantial compared with 1, there may be some ambiguity in the interpretation to be placed upon the k effective when it is markedly different from 1. Nevertheless the suggested measure of control-rod worth is unambiguous and provides a numerical scale for comparison of control systems. The theory of control-rod worths is one of the more poorly developed phases of reactor analysis today primarily because of the absence of a body of analyzable experimental information with which to compare the various theoretical results. In applying control-rod theory the uncertainty in the estimates of control-rod worth While the results may frequently be considerably can amount to 25 to 50 per cent. better, extreme caution should be exercised in application of theoretical results to design problems. By The treatment outlined here is a theory of rods which are thermally black. this it is meant that the probability of capture of a thermal neutron which penetrates The theory of weakly absorbing rods is simple and the rod surface is near unity. The theory of gray rods has been may be handled by homogenizing the absorber. little developed, principally because of the lack of interest. The first is deter Generally the control-rod problem is made up of two problems. mination of the worth of a single control rod of a given material and geometrical The second problem is the configuration located in a specified region of the reactor. determination of the worth of an array of control rods distributed in the reactor. 2.2

Control-rod

Boundary Conditions

It is well known that the neutron flux does not vanish on the surface of a perfect For a plane absorber but extrapolates to zero some distance inside the absorber. surface, transport theory gives for this extrapolation distance19 0.71X,

where

X( is

the transport mean free path for neutrons of the energy considered in the

physics of control

Sec. 8-1]

8-13

material bounding the absorber. This result is derived for monoenergetic neutrons and applies only for weak neutron absorption in the medium around the absorber; for neutron absorption and multiplication Davison20 has published a table of corrections. For a curved absorbing surface the extrapolation distance increases above the value for a plane.51 For cylinders of 1-cm radius the value is about 0.94X, and increases These results hold, as in the plane case, for weakly to about 1 .3A< for very small radii. absorbing media. The extrapolation distance is used by setting the logarithmic derivative of the flux taken normal to the bounding surface equal to the reciprocal of the extrapolation distance. For a plane boundary, for example, 1

d*\ dx boundary

0.7 IX*

and for a cylindrical or spherical boundary

If**


dr boundary

- JL **X(

where a is a constant ranging from 0.71 to about 1.3 depending upon the radius of the rod. Corrections for absorption and multiplication may be applied to these constants. This type of boundary condition permits treatment of the case in which the controlFor such cases an appropriate extrapola rod material is not completely absorbing. tion distance may be derived for the interface between the reactor material and rod material by use of either transport theory or an approximation to it. This extrapola tion distance may then be used in a calculation of the rod worth instead of the perfect absorber extrapolation distance given above. This same procedure may be utilized to treat the epithermal absorption of controlrod materials which are thermally black. The black boundary condition may be applied to the thermal-neutron group, and the epithermal boundary condition to a higher energy group. This may be important in certain cases, since materials such as hafnium, silver-cadmium alloy, or boron which have significant epithermal capture can have appreciably greater worth in a thermal reactor than a thermal absorber such This enhanced worth can range from 10 to 30 per cent depending on the as cadmium. material and the ratio of epithermal to thermal flux in the reactor. 2.3

Effective Radius of a Cylindrical Control Rod

The actual

extrapolation distance can be used to calculate the surface inside or beyond the control rod at which the neutron flux vanishes. For sufficiently thick control rods this surface lies entirely within the boundary of the actual rod and has a shape similar to that of the control rod. It is sometimes more convenient to use the surface at which the neutron flux vanishes rather than the actual control-rod boundary. This can be done, of course, only if the extrapolated surface lies within the control-rod It is particularly convenient to use this extrapolated boundary for control boundary. rods which are right circular cylinders. The radius of the extrapolated boundary is referred to as the "effective" rod radius and is designated r«//. The concept of "effective" radius is also used in a related sense in control rod theory. For control-rod shapes which are not circular cylinders, such as flat slabs, crossshaped slabs, Y-shaped slabs, etc., it is frequently convenient to express the worth of the rod in terms of an equivalent circular cylinder. This is generally expressed by giving the "effective" radius of the equivalent circular cylinder. 2.4

The

located

Worth of

a

Single Central Circular Cylinder Control Rod

effectiveness of a single central control rod in the form of a circular cylinder in a circularly cylindrical reactor has been calculated by Wigner et al."

8-14

CONTBOL

OF REACTORS

[Sec. 8

The result of the calculation is Ak = 7.45

Ml

[0.116

(1

+

f)

+ In

+

f

In

L-^V

where Ak is the change in multiplication constant of the reactor, Ro is the outer boundary of the equivalent bare reactor, r,u the control-rod "effective" radius, r is the two-group age, Ls the thermal diffusion length, and M* is the migration area

r +L*. This calculation

has been carried out on a two-group basis using diffusion theory. Within this framework there are approximations in the analysis which are valid for a reactor sufficiently large so that it can be treated as a bare reactor and such that k is not much greater than unity. The axial leakage is assumed small compared with the radial leakage. Furthermore, the control-rod size cannot be a large fraction of the reactor radius. In general the approximations appear quite reasonable if the Ak/k worth of the rod is of the order of or less than Y*,{k — \)/k for the reactor. To get some idea of the control effectiveness associated with fast-neutron absorption, it can be assumed that the control rod is black to fast neutrons as well as thermal neutrons and that the "effective" radius is the same for both energy groups. For this case an equivalent one-group analysis may be made which gives for the rod worth


Ak = 7.45

Ml Ro*

fo.116 + In

\

-^V'

2.4r„„/

While this result is a gross overestimate of the control-rod worth, it does provide an upper limit to the effect of epithermal neutron absorption. 2.6

Worth of a Single Off-center Circular Cylinder Control Rod

The analysis of a single off-center control rod is of importance not only as an evalua tion of a single rod in such a location but as a step in one type of control-rod inter action analysis. The transfer of the central, circular control rod to an off-center position is accom The problem plished analytically by use of the addition theorem for Bcssel functions. While has been solved by Nordheim and Scalletar" in 1945 and by Adler" in 1947. the two treatments differ slightly in detail, one using the "effective" rod radius and the other the actual radius, they are essentially identical. Both neglect the angular variation of the neutron flux over the control-rod boundary and, in effect, treat the rod as a line "sink" for thermal neutrons. The result is given in the form of a critical equation which must be solved using one of the parameters as a variable. The critical equation given by Adler is approxi mately ao

XB,K,(Brr„)

+ F„(Brr„)

=

£

m=0

In this equation B,

[e«Jm*(BJ)

J'ffff ]

is the critical radial buckling of the reactor with the control rod

*■

- *-* -

(5)'

R<, is the outer B^1 is the material buckling and H is the reactor height. radius of the equivalent bare reactor, r0 is the geometrical radius of the control rod, d is the distance from the center of the reactor to the center of the control rod, X is the control-rod extrapolation length, and the Y0, Ym, J„, etc., refer to the appro priate Bessel functions. The expression tm is a factor which is 1 for m = 0 and 2 for all to > 1. This critical equation may be used to determine the control worth of a single off-center control rod in the following manner. The geometry is fixed; i.e., ffo,

where

physics of control

Sec. 8-1]

8-15

H, and d are chosen; a reactor composition is chosen; and B.v* is calculated. one-group treatment, for example, Bm* =

k

-

In

a

1

M*

where A/5 is the migration area. The extrapolation distance X is estimated from the thermal diffusion length, and the various values are substituted in the critical equa tion. In general this equation will not be satisfied, and a new trial must be made. From a series of trials the value of Bu' which satisfies the equation is determined. Next, with the control rod removed (assuming that all the reactor constants remain the same for simplicity), the critical buckling is

The

excess buckling

with the rod removed is then Afl» = BM*

and the worth of the rod is Sk

m


k

/

\l

- Ba*

ABM/' \ BMW*)

+

Instead of varying B.w2, it would be equally feasible to vary the rod radius r<>or one of the other geometrical parameters to achieve criticality. The flux-squared weighting rule is a result derived from perturbation theory and permits an estimate to be made of the relative effectiveness of a neutron absorber located in various positions in the reactor. The rule states that the relative effective ness of an absorber in one position in the reactor to its effectiveness in another position is proportional to the ratio of the squares of the thermal-neutron fluxes at the two The rule is an approximation and depends on the assumption that the positions. flux is not perturbed significantly by movement of the absorber. It is, however, Adler has quite useful and may be applied to the problem of the off-center rod. computed off-center control-rod worth for a given reactor" and compared this with the results obtained by the flux-squared weighting rule. The weighting rule is in error by 25 per cent or less for the example calculated. For rough approximations the flux-squared weighting rule is apparently adequate and considerably simpler to calculate than the critical equation given above. 2.6

Control-rod

Interaction — Line-sink Method

Two schemes have been used to calculate control-rod worths when a number of rods are in the reactor. One of these, the so-called "poison-cell method," will be discussed later. The other may be called a line-sink method. In this scheme, the flux in the immediate neighborhood of a control rod is calculated, neglecting the The over-all flux distribution is calculated taking angular variation about the rod. into account all the rods. In effect, this amounts to inserting a line sink for neutrons into the reactor at the axis of each rod and calculating the gross flux distribution on this basis. This scheme is most readily calculated when the rods are placed in the reactor in angularly symmetrical fashion at a fixed radius. Adler has given the result14 for a single ring of jV rods located symmetrically at a fixed radius in a cylindrical reactor. An approximate form of the critical equation is Y„(Brr») + Br\Yt(B.ro) Jo(Brrtt) + B,x/t(/?rro)

=

N

S

•mJ*mH{BJ)

^|^

N

- y

Y0(Brr«)

where the syral>ols have the same significance as for the single off-center rod case. N is the total number of rods in the ring; r„i is the distance from the axis of the /th to the axis of the nth rod. The denominator of the expression on the left side of the

8-16

[SEC. 8

OF REACTORS

CONTROL

equation is essentially unity, so that the case of a single off-center rod may be con sidered a special case of this expression with N = 1. A critical equation for two rings of rods and a central rod has been given by Cam

bodian."

The multiple control rod critical equation may be handled in the same manner as the single off-center case, i.e., by variation of Bj#'. For practical purposes it is generally simpler By computing the excess buckling to fix Br and vary either m or d to achieve criticality. In general, with the rods removed a Sk/k worth of the rod group may be obtained as before. it will be necessary to calculate a family of curves for various values of Br to obtain a sufficient number of cases to permit interpolation.

It The interaction of control rods has been studied by Wheeler" and by Adler." has been noted that, for two control rods close together, their total worth is less than the sum of their individual worths. This is referred to as control-rod "shadowing." There are circumstances, however, in which the insertion of one control rod may enhance the worth of another rod. This is the case with two rods which are near the center of the reactor and spaced an optimum distance.


2.7

Equivalent Effective Radius for Rods of Special Shape

2.71 The worth of a control rod of shape other than a right Wheeler's Results. circular cylinder is frequently expressed by giving the effective radius of the circular rod having the equivalent worth. Expressions for the effective radius of noncircular rods were first given by Wheeler." Two results of particular interest taken from Wheeler are given below. In both cases the reactor material is presumed to fit snugly around the control-rod surface. The moderator in which the rod is embedded is presumed to be a good scatterer and only weakly absorbing. 1. Flat Ribbon. The width W of the ribbon is presumed to be comparable to or The thickness of the ribbon larger than X„, the neutron mean free path in the reactor. is assumed to be small compared with A„. The effective radius of the equivalent cylindrical rod is

r.ti

-

W — exp

-

- 0.114?/

(y

y/ZW

y =

where

+ 0.048!/')-'

r\«

Cross-shaped Rod. The rod is presumed to be made of thin ribbon as before. The span of the cross-shaped rod is W; i.e., each of the four legs is W /2 in width. The effective radius of the equivalent cylindrical rod is l!

2

y/2

exp

- (z

0.114*

+ 0.048zs)-'

2.72 Hurwitz-Roe Results. More recently Large Diffusion Length Limit. Hurwitz and Roe have given results for control rods of various shapes." They have given values of the effective rod radius in both the large diffusion length limit, i.e.,

Ttij/L

«

1, the

small diffusion length limit

rr///L

»

1,

and for intermediate

(L designates the thermal diffusion length in the reactor material.)

cases.

Their results

are based on one-group diffusion theory assuming a constant source strength.

For the large diffusion length limit Hurwitz and Hoe give their results directly A number of eases of interest, arc listed below. terms of the effective radius. 1. Flat Ribbon. Using, as before, W as the width of the ribbon

r.n

=

W

T

in

physics of control

Sec. 8-1] 2. Cross-shaped

Rod.

With W

as the span of the rod,

r.it 3.

Ellipse.

With

8-17

w 2

7.

\/2

a and 6 as the semimajor and semiminor axes, respectively, a + b

4. Rectangular Bar. If R, is the short side of the rectangle and J?„ the long side, r,i! is given in terms of ft*, by Table 5.

Table

5.

Effective Radius for Rectangular Bars

R.IR.

r.n/R. 0.25


0 0 0432 0.2219 0. 3968

0.2732 0 3939 0.4038

0 5085 0 6418 0. 8030 1.000

0. 0 0. 0.

4401 4820 5314 5902

2.73 Hurwitz-Roe Results. Intermediate and Small Diffusion Length Limit. The results obtained by Hurwitz and Roe for other than the large diffusion length limit are best presented by introducing their absorption-area concept. The absorp tion area C is defined as the ratio of the number of neutrons absorbed by the control rod per unit length per unit time to the number of neutrons thermalized per unit time per unit volume in the reactor material outside the rod. For a circular cylindri cal control rod, C and rtll are related by the equation

L

=

&rr,,,\

—

—■-

Ao(r,//A)

where KB and K\ are Besscl functions which vanish at infinity. For two cases of interest, namely a ribbon and a cross, they have calculated by variational methods the correction to be applied to this simple concept. This is tabulated below as a

of the ratio of the diffusion length to a characteristic rod dimension. I. Flat Ribbon. The correction factor is to be applied to the absorption area when it is calculated using r,fI = W/4. function

Table 6.

Absorption-area

L/W 0 0. 1 0 2

0.3 0.4

0 5

0.6 0. 7

2. Cross-shaped

Rod.

Correction Factor for Flat Ribbons

Correction factor for absorption area

1.27 1. 17 1.12 1.09 1.07 1.06 1.05 1.04

Correction factor for

L/W

absorption

0.8 0.9

area

1.03 1.03 1.02 1.02 1.01 1.00 1.00

1.0 1.2 1.5

2.0 3.0

The correction factor is in this case to be applied to the

absorption area when it is calculated using r,ts

—

W/2 y/2.

8-18

OF REACTORS

CONTROL Table 7.

Absorption-area

L/W

Correction Factor for Cross-shaped Rods

Correction factor for absorption area

0 0. 1

0.2 0.3 0.4

O.J 0.6 0.7

2.8

[SEC. 8

L/W

1.70 1.55 1.40 1.25 1. 18 1. 15 1.12 1. 10

Correction factor for absorption

0.8 0.9

1.08 1.07 1.06 1.05 1.04 1.02 1.01

1.0 1.2 1.5

2.0 3.0

Control-rod

area

Interaction — Cell Method

The results given in the preceding article for special shapes of control rods could the reactor assuming that the actual rods were replaced by the equivalent cylindrical rods. In fact, the correction factors for the absorption area could be applied to correct the effective radius of the equivalent cylinder. In most cases, however, the resulting problem would not be particularly tractable unless the rods happened to be positioned symmetrically on concentric circles. The cell method, to be described next, is less limited in this respect. Furthermore, the method is particularly adaptable to use of the results of the area absorption calculation. In essence, the cell technique is to associate with each control rod in the reactor a cell of reactor material. The absorption of neutrons by the control rod in the cell is represented by introducing a fictitious thermal-neutron absorption cross section into the cell. This cross section is of such magnitude that the thermal-neutron absorption by it is just equivalent to that which would be absorbed by the rod. It is of some importance to choose the cell area properly, since the choice of cell size will determine if the change in leakage out of the reactor with insertion of the control rod is properly given by this technique. The choice of cell size is relatively simple if a large number of control rods are uniformly distributed throughout the reactor. Even with two rings of control rods near the center of the reactor, the choice of cell size is geometrically determined by the inner ring if all rods are of the same size. The most difficult choice is that of a single central rod. Even in such a case the cell may be chosen to give the proper change in reactivity as determined by direct calculation of a central cylindrical rod. More recently Stewart and Smith88 have applied this technique, adapting Hurwiti and Roe's absorption area for this purpose. If C is the absorption area of the control rod as defined above, A the cross-sectional area of the reactor cell, 2, the macroscopic thermal-neutron absorption cross section in the reactor, and 2P the fictitious control poison equivalent to the rod, then


be applied by calculating

C

A

_ Xp

s„ + 2.

Once C is determined in a given reactor material and the cell size chosen, 2P may be calculated. The reactivity behavior of the reactor with various control rods inserted may then be calculated in general by a two-dimensional solut ion of the reactor. If the cells comprise the entire cross section of the reactor, the situation with all rods inserted is simple to calculate. For a bare reactor in which the entire cross section of the reactor is occupied by cells, the reactivity worth of the entire group of rods is then just Sk

k

_ ~

2, 2„ + 2.

_ =

C A

Sec. 8-1]

PHYSICS OF CONTROL

8-19

Other special configurations with rings of rods down may be calculated onedimensionally by arranging the poisoned cells to form a poisoned annular region. Table 8.

Absorption Areas for Control Rods LtjW

Ribbon rod

Cross rod

C/1WL

C/4WL

0. 1

0.2 0.5 1 2 2.5 5 10

0.6295 0.6973 0.8802 1. 1478

1.061 1.4073 1.909 2 786

1.8328 5.021 8. 2440

4. 5291 I

Table 8 gives values of the absorption area for a ribbon and cross rod as calculated by Hurwitz and Roe. The absorption areas are given as the ratio of C/2WL for the ribbon and C/4WL for the cross-shaped rod, i.e., as the ratio of absorption area to rod perimeter times diffusion length. The absorption areas are calculated using the effective radius given above, namely, W/A and W/2 two cases. These may be corrected using Tables 6 and 7.

respectively, for the


REFERENCES 1. 2. 3.

4.

5. 6.

7.

8. 9. 10. 11. 12. 13.

14. 15. 16. 17.

18.

Coryell, C. D., and N. Sugarman: "Radiochemical Studies: the Fission Products," book 3, McGraw-Hill Book Company, Inc., New York, 1951. Snell, A. H., et al.: Studies of Delayed Neutrons, Phys. Rec, 72 : 541-549 (1947). Hughes, D. J., J. Dabbs, A. Cahn, and D. Hall: Delayed Neutrons from Fission of U">, Phys. Rev., 73: 111-124 (1948). Keepin, G. R., and T. F. Wimett: "Delayed Neutrons," International Conference on Peaceful Uses of Atomic Energy, P/831, 1955. Keepin, G. R.: "Progress in Nuclear Energy," Scries I, Physics and Mathematics, Vol. I, "Delayed Neutrons," Pergamon Press, Ltd., 1956. Keepin, G. R., T. F. Wimett, and R. K. Zeigler: Delayed Neutrons from Fissionable Isotopes of Uranium, Plutonium, and Thorium, Phys. Rev., 107: 1044 (1957). Keepin, G. R., T. F. Wimett, and R. K. Zeigler: "Delayed Neutrons from Fissionable Isotopes of Uranium, Plutonium, and Thorium," LA-2118, July, 1957. Also Jour. Nuc. Energy, 6, Nos. 1/2 (1957). Perlow, G. J., and A. F. Stehney: Delayed Neutrons from 15.5-sec Br", Phys. Rev., 107: 776 (1957). Nordheim, L. W.: Pile Kinetics (Abstract), Phys. Ren., 70: 115 (1946). Fuchs, K.: Proc. Phys. Soc, A62 : 791 (1949). Hurwitz, H.: Derivation and Integration of the Pile-Kinetic Equations, Nucleonics, 6:61-67 (July, 1949). Soodak, H.: "The Science and Engineering of Nuclear Power," vol. 2, chap 8, AddisonWesley Publishing Company, Cambridge, Mass.. 1949. Soodak, H., and E. C. Campbell: "Elementary Pile Theory," John Wiley & Sons, Inc., New York, 1950. Glasstone, S., and M. Edlund: "The Elements of Nuclear Reactor Theory," chap. 10, Time Behavior of a Bare Thermal Reactor, D. Van Nostrand Company, Inc., Prince ton, N.J., 1952. Ehrlich, R., and H. Hurwitz: Multigroup Methods for Neutron Diffusion Problems, Nucleonics, IS: 23-30 (February, 1954). deHoffman, F., et al.: " ltelotive Effectiveness of Delayed Neutrons in the Water Boiler," LA-471, Dec. 13, 1945. Hellens, R. L. : Private communication. United States Atomic Energy Commission: "The Reactor Handbook." vol. I, Physics, chap. 1.6, Reactor Dynamics, by G. Goertzel, AECD-3645, Government Printing Office, Washington. D.C., February, 1955. Schultz, M. A.: "Control of Nuclear Reactors and Power Plants," chap. 11, McGrawHill Book Company, Inc., New York, 1955.

8-20

CONTROL

OF REACTORS

[Sec.

8

Placzek, G., and W. Seidel: Phys. Rev., 72: 550 (1947). Davison, B.: CRT-358, June, 1940. 21. Davison, B., and S. Kushneriuk: MT-214, March, 1946. 22. Davison, B.: Proc. Phys. Soc. London, 64 : 881 (1951). 23. Wigner, E. P., A. M. Weinberg, and R. R. Williamson: CP-1461 (classified). " Efficiency of Control Rods as a Function of Their Position in a Cylindri 24. Adler, F. T.: cal Pile in the One-group Picture," MT-222, February, 1947. 25. Nordheim, L. W., and R. Scalletar: "Theory of Pile Control Rods," MDDC-42. " Principles of Nuclear Power," chap. 22, N-2292 26. Wheeler, J. A.: (classified). Hurwitz, Jr., H., and G. M. Roe: "Absorption of Neutrons by Black Control Rods," 27. KAPL-1336, March, 1955. 28. Stewart, J. C, and J. H. Smith: "Interaction of Black Control Rods," KAPL-1471, January, 1956. 29. Garabedian, H. L.: "Control Rod Theory for a Cylindrical Reactor," WAPD-18, August, 1950. 19.


20.

8-2

THE CONTROL SYSTEM BY

M. A. Schultz REACTOR


1

STARTUP AND LOW-POWER OPERATION

A shutdown reactor in a steady state may be considered one whose Introduction. The absolute reactivity is negative or whose multiplication factor is less than unity. value of the reactivity at shutdown is a function of fuel loading, neutron absorption To start a reac and reflection, and environmental conditions such as temperature. tor implies that this reactivity state must be changed and generally some form of Manipulations of fuel, reflec mechanical motion in or about the reactor is needed. Ncutrontor, and absorbing or reflecting types of control rods may be performed. Refer absorptive-type rods are presently most commonly used to change reactivity. Hydraulically ence 1 gives specific designs of mechanical control-rod mechanisms. operated mechanisms have also been described, f1' See also Art. 2 of Sec. 8-3. 1.1

Control-rod

Nomenclature and Combinations

Control rods may be operated individually or connected together electrically, mechanically, or by other means in banks or gangs so that more than one rod moves at a time. Faster Slow-moving rods or rod banks are generally called shim-banks. moving individual rods which do not contain much reactivity and are used for auto matic control functions are called regulator rods. Regulating rods or other fastmoving rods should be limited in reactivity worth to an amount smaller than that Rods whose principal necessary to take the reactor from critical to prompt critical. purpose is to shut down a reactor quickly are called safety rods or scramming rods. The design of safety rods is predicated on the principle that the ever-ready safetyrod bank should always possess more reactivity than any rod or rod combination in the reactor which can be moved at any given time. Rods are frequently designed as combination shim and safety rods. Control rods may be converted to have a safety function depending upon their position in the reactor and the reactor life. For example, a shim-rod bank could be completely withdrawn from the reactor as the depletion of fuel ensued. rod bank then could be converted to have the function of a safety bank. 1.2

Subcritical

This shim-

Operation

Subcritical Multiplication. In a reactor containing a source, the number of which exist in this multiplying medium at the end of a sufficiently long interval of time is n = Sl*(l +*+**■•• fc"-1) (1) 1.21

neutrons

where n = S = I* = k = m =

total number of neutrons in reactor source strength, neutrons/sec mean neutron lifetime, sec multiplication factor of the medium

number of neutron generations that have elapsed source

t Superscript

numbers

refer to References


8-21

since

insertion of the

8-22 In

CONTROL

OF REACTORS

8

[Sec.

closed form, for long intervals of time, this expression becomes n

SI*

"^

-* 1

1

(2i

This ratio

if

is known as the subcritical multiplication factor. Kinetically, is suddenly inserted into a subcritical multiplying medium containing neutron population, the reactor responds according to the equation

a source

no initial

N(/)

(3>

where the effects of delayed neutrons arc ignored. A plot of this equation for various multiplication factors is given in Fig. 1. 1.22 Subcritical Period. The period of a reactor is defined as 1

(\/n){dn/dt)

A subcritical period may be obtained from this formula for slowly increasing reac tivity as in reactor startup by performing the suggested differentiation on Eq. (2). The assumptions involved are that tran sient dist urbances due to the insertion of the source and the reactivity changes have died out and the rates of change of reactivity with time are slow compared with delayed-neutron emissions. The re sulting expression is Subcritical period = 0

2

4

6

8

10 12 14 16 18 20 22 24

TIME IN NEUTRON

1.3

—

\

(5)

dk/dl

Rule of Power Increase

LIFETIMES

1.31 Power Levels Caused by Re activity Changes. In a reactor startup, the reactor is initially subcritical and its multiplication factor is brought up to unity by any designed method. The elapsed time needed for the reactor to reach criticality from a fixed subcritical situation at the beginning of the startup operation depends on the rate of change of reactivity. This change of reactivity is rarely linear with time because of the nonlinearities imposed by such factors as control-rod effectiveness. Reactivities that are linear functions of time may be used in calculating the approxi mate behavior of reactor level with time. If the reactivity is assumed to be of the form ik = a + yt, with a being negative for the initial subcritical condition, the neutron level vs. time can be calculated from the subcritical through the supercritical ranges by the following expression* exp

x(Sk

-

xH ~2y

x +

x^'y

dx

(til

t

t

x

=

to

yt

is

a

k)

factor that brings n(0) to value of (source level)/(l — at = transformation factor which used to evaluate integral from x = oo for each value of desired in the expression ik = a + fraction of delayed neutrons, approximately 0.0075 for U"' constant of delayed neutrons, approximately average decay for U»6 I* = mean lifetime between succeeding neutron generations, sec

where a

0

-•

0

»(0

0)

Subcritical multiplication.

[

1.

!o~

Flo.

x


Period

0.1 /sec

THE CONTROL

Sec. 8-2]

8-23

SYSTEM

This expression is for lumped delayed emitters but gives a good approximation to multiple delayed emitter case. In Eq. (6) the as* term in the exponential is always negative. The x term is negative when Sk is below prompt critical, zero at prompt critical, and positive above prompt critical. The level response below critical is not particularly sensitive to I*. Figure 2 shows this equation plotted for an example in which the

Sk

=

-0.333 +

7

X

10-<

I* = 10_4sec, 0 = 0.0075, and X = 0.1 /sec. In this example a neutron multiplication of approximately 100 has occurred at criticality. Neutron multiplications between 30 and 1,000 are generally encountered up to criticality (see Fig. 3). The effect of different linear reactivity change rates y upon a reactor of I* = 10-4 sec and a = —0.13 is shown in Fig. 3. These curves are plotted as a function of the reactivity remaining in the reactor rather than against time. The subcritical mul tiplication curve is furnished as a reference curve. As the reactivity rate is increased, the critical point comes at lower neutron levels.


108 e

1

1

n

cm

LJ

.o'l

I I i 480 490 460 470 TIME AFTER REACTIVITY CHANGE STARTS, SECONDS

Fig.

2.

Relative neutron level vs. time for input in Sk, I* = 10-4 see.

ramp-function

6

5

.

i

.

i

4 3 PERCENT

,

i

.

i

.

i

10

2 REACTIVITY

Relative neutron levels vs. reactivity in a reactor for various rampfunction reactivity changes, I* = 10-4 sec. Fio.

3.

remaining

An approximate solution of the linear reactivity change problem giving the neutron level in a reactor as a function of time for the condition in which the reactor is initially critical is Eq. (7).4 ' It is assumed that the reactor is operating at a steady state before the application of the reactivity disturbance and that the disturbance starting at t = 0 is of the form Sk = yt, then

where the symbols have the same meaning as in Eq. (6). for the short time intervals during which the contribution can be considered to be (0/2*)no.

This equation

is accurate of the delayed emitters

Figure 4 shows Eq. (7) plotted to give examples of the effects of changing reactivity for a reactor having /* = 10~4 sec and 0 = 0.0075. Equations (6) and (7) hold only when reactivity is a linear function of time. Control rods are normally rates

8-24

CONTROL

OF REACTORS

[Sec. 8

withdrawn at a constant speed corresponding to a given rate of reactivity change at the position of maximum effectiveness. Nonlinearities in control-rod effectiveness, as a function of position and time, therefore tend to stretch out a startup curve in the time scale. The worth of a control rod in most reactors as a function of its position resembles a curve of the form / sin1 x dx. If the neutron levels vs. time are ■

0006

STEP 6k f 6k CHANG) AT 0O6/SEC

/ L

6k( XANGE ATOC JI2/SEC

200

100

Fig.

300

400

TIME -MILLISECONDS

TIME

Relative neutron levels vs. time for reactivity change various using rates 4.

formula.

Power level vs. time for a reactor being started up on either linear change of reactivity or linear rod motion. The curves are for full-length travel of the control rod in the indicated time. 5.

125

o

2 5

I

1

8

AT 000045/SEC

2

6k CHANGE

8 a IS

I S

0

o

°

3

H

CHANGE AT 0.0016/SEC

i 25

6k

0.90

0.92

/

20 100

J/

8 10

40

10 0.94 096 0.98 MULTIPLICATION FACTOR k

r

80

120

160

200

240

TIME IN SECONDS

Flo. 6. Reactor period vs. multiplication fac tor for various ramp-function reactivity

Fio. 7. Reactor period vs. time for various ramp-function reactivity change rates.

change rates. as a linear function of time as against linear rod motion will form, curves similar to Fig. such an effectiveness result. The rod worth in the linear reactivity case taken to be the maximum rod worth in the linear rod motion curve. The shape of the curves is, in general, similar, but the time scale longer for linear rod motion. Harrer1 suggests that, for tentative design considerations, the speed of reactivity factor which includes an allowance for change should be 10~*/F in S/c/sec, where a

compared for reactivity

is

5

with time for rods having

is a

F

is


approximate

Fio.

Sec. 8-2]

THE CONTROL

SYSTEM

8-25


the rod effectiveness at a given operating point as compared with average rod effec tiveness. He suggests 4 as a reasonable number for F. The period of a reactor during startup may be obtained 1.32 Startup Periods. by differentiating kinetic neutron level equations such as Eq. (6) or by differentiating any experimental level vs. time curve. A special case is the reactor period which Figure 6 indicates the periods results from constant rates of reactivity change. which result as the multiplication factor of a reactor becomes closer to unity at con The reactor for which these curves were plotted stant rates of change of reactivity. had an I* of 2 X 10-* sec. At high reactivity change rates a short period exists at quite low multiplication factors, whereas if the rate of change of reactivity is small, the multiplication factor must be very nearly unity before a short period occurs. Figure 7 is a plot of the same information as a function of time, and it can be seen that a more gradual operating approach to a given period is available as reactivity is inserted into the reactor at a slower rate. Of importance is the period at criticality at given reactivity rates of change. Figure 8 indicates the period at which criticality is reached as a function of the reactivity change rates in per cent for a reactor having an I* of approximately 10-* sec. 1.4

Initial Startup

As the 1.41 Radioactive Sources. power level of a reactor must be under continuous observation, neutrons for the instruments to measure must be available at all times even when a reactor is shut down. Cosmic rays or spontaneous fis sion generally create a few neutrons per minute in a reactor, but this number is too small to be satisfactorily detected. 001 • The basic startup problem is getting the nl . I I . 1 I ■ i i initial neutron level up high enough that 6 6 10 12 14 16 16 20 0 2 4 detecting instruments can measure this CRITICAL PERIOD-SECONDS level with good statistics. To accomplish Fio. 8. Period on which a reactor would go this purpose, artificial radioactive sources through criticality by inserting reactivity of neutrons are strategically inserted into at given linear rates. These sources emit neutrons a reactor. either by (y,n) reactions or by (a,n) reactions. The (a,n) sources, such as Ra-Be and Po-Be, are generally used because of their higher efficiency. A typical source consists of an intimate mixture of radium and beryllium or polonium and beryllium which has been highly compressed into a small pellet. The most common (y,n) source is Sb-Be, which is easy to manufacture and is presently less expensive than Ra-Be A disadvantage of this (t,ti) source is that its emitted neutrons and Po-Be sources. have less energy than those of the (or,n) emitters. Consequently in some reactors greater attenuation of the lower energy neutrons would result in the path between the source and the detecting instrument. Physically, sources are quite small, and commercial sources of neutron radiation are available as small capsules approximately y± in. in diameter by a few inches long. Also available are thin sandwich-type plates about 1 in. in diameter. Source efficiency depends upon both cross sections and geometrical constructions. Efficiencies Source strengths are measured in curies of the radioactive elements. are such that an Ra-Be source yields approximately 1.6 X 107 neutrons/(sec) (curie), a Po-Be source, 3 X 10" neutrons/(sec) (curie). The efficiency of the photo-neutron source, Sb'"-Be, is such as to yield approximately 3 X 10' neutrons/(sec) (curie)."'7


8-26

CONTROL

OF REACTORS

[Sec.

8

For safe reactor startup, sources between 10' and 10' neutrons/sec are presently desirable. Source strengths diminish with time if the radioactive element has a short halflife. Po"°, which is commonly used in Po-Be sources, has a half-life of 138 days. If the source must be installed early in the construction of a reactor, its strength may be considerably depleted at the end of reactor life or even at its initial startup. The source half-life must be considered in the design of neutron-detection instruments. The absolute magnitude of the source strength need not be known. The number of counts per second or current produced in a neutron-detecting instrument located at a given position with respect to the reactor is what is used in a startup. These counts or current should be a direct indication of the multiplication in the reactor. Consequently, the location of the source or sources in the reactor must be chosen with care. Sources in the geometrical center of a reactor generally produce satisfactory indications of multiplication. Source lo cations outside the reactor core or on the -INSTRUMENT TOO edge of a reactor core such that they are CLOSE TO SOURCE seen directly or partially by the neutron detector should not be used. The larger t he source rating in curies, the more counts per second will be indi cated by the detecting instruments. The only limit to the size of a radioactive source is the practical one of how large a source can be safely handled in the fabricating and installation technique of placing the source in the reactor. In some reactor designs, small factors in source strength may be gained by splitting the source into a number of discrete pieces, each piece being more easily handled than the over all source itself. 1.42 Initial Startup. Initial startups 01 0.2 0.3 0.4 0 5 0.6 07 08 09 10 differ from subsequent reactor startups KILOGRAMS OF URANIUM in that the reactor is cold, clean, and un Flo. 9. Plot of reciprocal counting rnte vs. known. Considerable care is generally amount of uranium solution for a small taken in initial startups, and they are ac water-boiler reactor, showing the effect of instrument placement. complished with very low rates of change of reactivity. The objective usually is to get the reactor to criticality and not go beyond this point to appreciable power levels. Because of the slowness of the startups, the counting rates on the detecting instruments follow the subcritical multiplication equation; i.e., Counting rate =

1

-

h

(8)

A is a constant of the instrument system geometry and k is the multiplication factor. Reciprocal counting rates are generally recorded as a function of some Figure 9 shows the type of plot obtained parameter directly related to reactivity. As k approaches unity very slowly, for a small water boiler on three instruments. the counting rate approaches infinity. Consequently, the reciprocal counting rate The shape of the curve will approaches zero as the reactor approaches critical. If the instrument is depend on the position and type of detecting instrument. located very close to the reactor, it will see the source for a long period of time and the The condition of the instrument being shape of the curve will be concave downward. too far from the reactor is a safe one, whereas if the instrument were too close the extrapolation to criticality might be dangerously misleading. If the source is removable from the reactor during the startup, then another tech where

Sec. 8-2]

THE CONTROL

The reactor is brought nique can be used. quickly removed from the reactor. If the rate will stay constant. If the reactor is rate will slowly drop off, and if the reactor


1.5

SYSTEM

8-27

close to critieality, and then the source is reactor is exactly at critical, the counting subcritical a Blight amount, the counting is supercritical, the level will rise.

Subsequent Startups

Once the reactor has been operated at any appreciable power level, the following items may influence the number of neutrons reaching detecting instruments. 1.61 A large y flux exists within most reactors Photoneutron Induced Sources.8'9 after initial operation. Use may be made of this y radiation to create new neutrons by (y,n) reactions. In order to produce neutrons by this method, the y energy must be at least equal to the binding energy of the neutron to the nucleus in question. This energy is about 8 Mev for most nuclei except for some of the light ones. In particular, deuterium and beryllium have '0T, threshold energies of 2.21 and 1.62 Mev, respectively, and are presently used in the design of some reactors. The number of neutrons produced from such induced sources depends upon the quantity of material involved, the y in tensity, and the time after shutdown. ,.. AFTER 100HOURS OPERATION ^ AT FIXED PO\VER LEV EL The relative levels of a natural water1 moderated plant as a function of time AFT iR 10 HOURSOPERATION after shutdown and previous operating ^Tats. \ME FIXE DPOWE * LEVEL time arc given in Fig. 10. The concen tration of heavy water in normal water S^-AFT AT& by weight is 0.016 per cent . Water-mod would contain normally erated plants --AFT :roiho UR0PER ATI0N many thousands of pounds of water. ATS/1MEFIXE.DPOWE R LEVEL 1.52 Fission-product Delayed Neu trons. After any reactor operation, de layed neutrons are emitted from the fis sion products. These delayed neutrons are given off at various times (sec Table 5 of Sec. 1-1). Ultimately, they die down in accordance with the longest lived of 0 10 20 30 40 50 60 the delayed emitters, i.e., on an approxi TIME AFTER SHUTDOWN-HOURS In a shutdown after mate 80-sec period. Flo. 10. Relative core flux after shutdown a power-level operation, the initial reduc production on Dj caused by photoneutron tion in power level is a function of how for various power-level operation times. much negative reactivity is injected into the reactor in a given amount of time. After a few minutes, this effect, disappears and the 80-sec-period decay predominates. After approximately 20 min, no usable neutrons are available from this source. The temperature coefficient influences multiplication in a 1.53 Temperature. startup operation by directly changing the subcritical multiplication factor. Some power reactors may have to be preheated before a reactor startup operation. If the temperature coefficient of the reactor is negative, the reactivity will be reduced and the detecting instruments will count less neutrons after this preheating. The multiplication factor of a reactor is directly affected Fuel Depletion. 1.54 by the amount of fuel loading. As the fuel is burned out, the multiplication factor An approximate expression for the percentage reactivity change resulting decreases. from fuel depletion is Sk = Sm/2, where 6m is the percentage of the fissionable mate rial depiction. In thermal reactors, the multiplication factor Fission-product Poisoning. 1.66 is drastically affected by the creation of fission products which absorb neutrons. Because of their large thermal-neutron absorption cross section, Xe136 and Sm1" are

W.

8-28

CONTROL

OF REACTORS

[Sec.

8

the principal poison contributors. Xe13* is principally formed from the decay of the direct fission product Te136. Te13' consists of over 5 per cent of the direct fission products and decays by 0 emission in the following manner: 1

Te1"

min >

I'«

6.7 hr

9.2 hr

2.1

X

10«

year

> Cs13'

» Xe13'

> Ba136

Ba136 is stable. Xe13' has an absorption cross section for thermal neutrons of approxi mately 2.9 X 10' barns, and it decays to Cs with a half-life of approximately 9.2 hr. Sm1" iB the stable end product of the chain

1.7 hr

Nd»«

>

47 hr

Pm»»

> Sm1"

(stable)

This reaction occurs in approximately 1.5 per cent of the fissions, and Sm149 has a somewhat lower cross section for thermal neutrons of approximately 5.3 X 104 barns. Xe13' effects usually predominate over Sm119 effects during power operation and behave in accordance with the following equations: dX ~U

dl


where

X

y,

0 X/ 1

X,

2/ yi

= = = = = = = = =

= (7^2/ =

7,2/*

-

+ X,/

- X,X

(9)

- X,7

(10)

number of atoms of Xe135 present per cm3 at any time t fractional yield of xenon as a direct fission product microscopic thermal-neutron absorption cross section of Xem thermal-neutron flux decay constant of I135 number of atoms of I136 present per cm3 at any time t decay constant of Xe13' macroscopic fission cross section of fuel in reactor fractional yield of I135 from direct fission process

0

10

20

30

TIME AFTER SHUTDOWN

Fio. 11. Relative peak xenon poisoning a thermal reactor.

reactivity

as a

40

IN HOURS

function of time after shutdown for

Values for the decay constants and yields for thermal reactors are X, = 2.9 X 10"', X 10-', y, = 0.067, yt = 0.004, a, = 2.9 X 10"18. From these equations, two effects occur which concern reactor control. The first effect is the equilibrium poisoning during operation, and the second effect is peak poisoning which occurs after shutdown of a reactor. Equilibrium xenon poisoning reaches a steady plateau in approximately 40 hr. If a thermal reactor is shut down quickly from a high-power-level operating con dition, the thermal-neutron level is reduced to a relatively low level, and consequently, X, = 2.1

Sec. 8-2]

THE CONTROL

8-29

SYSTEM

tbe burn-up term due to thermal-neutron absorption of Xem no longer exists in Eq. (9). The Xem concentration builds up to a maximum from the I1" which has been previously formed. Ultimately a peak occurs, and the radioactive decay of the Xe1" to CslJ* concentration drops off. The time involved in this process is shown in Fig. 11, with a peak in Xe concentration appearing at approximately 11 hr The magnitude of this peak depends upon the initial steady power after shutdown. In some reactors insufficient loading may level of the reactor and its specific design. make it impossible to completely "override" this peak in poisoning during the startup Under conditions where an insufficient amount of reactivity is present, operation. unless the reactor is started up quickly after a shutdown, it will be impossible to start the reactor until the Xe decays down. For example, in the Materials Testing Reactor sufficient loading is provided to give 20 min worth of xenon override time dur ing the first half to two-thirds of the fuel cycle. The magnitude of the equilibrium and peak xenon poisoning has been computed410 when poisoning is defined as the ratio of the thermal neutrons absorbed by the poison to the thermal neutrons absorbed in the fissionable material. Table 1 gives these values as a function of flux level. Table


Long-time operating flux, cmVsec

10" 10"

X

Equilibrium poison, % reactivity

Maximum poison after shutdown, % reactivity

0.54 2.8 4.5 4.6

10»

2

1

10"

1.6

Startups

0.55 3.0 20.0

38.0

to Power

Level

In power reactor operations, 7 to 10 decades of neutron flux may be involved in a startup from source level to power level. Criticality usually occurs during the first two decades, and the remaining level rise is Periods for accomplished on a fixed period. practical startup operations range from 10 to 60 sec with periods less than 10 sec presently considered unsafe. Instruments are required to monitor both level and period. Four basic types of control systems with varia tions have been mentioned, f

1.61 Manual Control. Figure 12 indi cates a typical level vs. time curve for a startup operation. In manual control, re actor control-rod motion is such as to move the control rods at a given rate until the re actor reaches a desirable period and then rod motion is stopped or changed manually to maintain this period. The operator watches two sets of meters, one indicating period and the other power level. The reactor is main tained at the fixed period until the specified TIME Fig. 12. Typical reactor startup to power level is reached. The reactor then is returned to exact criticality, and this power power level. level maintained. 1.62 Period Permissive Control. Because of the hazards involved in manual reactor startup, some form of override or interlock protection is usually provided.

t Sec

Ref. 4, p. 239.

8-30

CONTROL

OF REACTORS

[Sec. 8


In a period permissive type of startup operation, control rods may be moved freely If shorter periods provided the reactor period is not shorter than a fixed amount. are achieved, control-rod motion is automatically stopped by the interlocks or they Figure 13 are moved in the opposite direction either manually or automatically. The reactor is illustrates a permissive type of operation around a 20-sec period. gradually brought from long periods toward a 20-sec period by control-rod movements. For periods longer than 25 sec, the control rods may be moved either in or out by an As the period shortens to 20 sec, interlocks prevent the rods from being operator. Manually initiated inward withdrawn any further and rod motion is stopped. In the event that a 15-sec period is reached, inter motion, however, is permitted. locks and controls take over and automatically cause control-rod motion inward. 1.63 Automatic Period Control. Figure 14 shows a block diagram of an automatic This system is similar to automatic level control control system for period control. The principal difference is that instead of a level demand signal (see Fig. 16). placed into the control-loop comparator, a period demand signal is used. For exam ple, a period demand of 20 sec might be set into the comparator as a fixed voltage. The reactor, through the neutron detector and period circuits, would create a matching

TIME

Flo.

13.

Permissive-type startup control.

Period controls direction of rod motion.

voltage, and any error signal would be amplified to initiate control-rod motion to reduce the period error signal to zero. 1.64 Combination Period and Level Startup Control. Figure 15 indicates a combination period and level automatic control loop. Both types of signals feed information into the comparator. A typical startup program is as follows: If, initially, the reactor is at a very low level, no significant output is available from the level detector. Arbitrary numbers may be assigned to the voltages one might For example, a variation of level from 0 to 100 per cent expect into the comparator. A variation full power may be translated into 0 to 100 volts into the comparator. in period or period demand from infinity to 10 sec might correspond to a voltage variation from 0 to 10 volts. The level detector would be expected to give out In startup sensible information only during the top two decades of power operation. from some arbitrary low level, if the demand signal is 10 volts, it would correspond to either 10 per cent of full power or a 10-sec period. The control rods would be manip ulated in an attempt to match this error signal. Even though the rods are moved at their maximum permitted rate for some time, the reactor would not be able to attain a 10-sec period. Consequently, a large error signal is inserted into the amplifier As soon as the period detector and rods are moved at the maximum permit ted speeds. gives out a 10-volt signal, the error is reduced to zero and rod motion is stopped or

THE CONTROL

Sec. 8-2]

SYSTEM

8-31

The period of the reactor is then leveled off at 10 sec and is main possibly reversed. tained by the servo loop. As soon as the reactor approaches the power operating range, some information begins to come from the level detector. This information plus the output from the period detector must now match up against the demanded This situation automatically calls for the output of the period detector 10 volts. to be reduced, and the reactor must now operate on a longer period as the level becomes still higher. Now the servo system gets less and less information from the period Hence, the reactor comes up channel and more information from the level channel. through the startup range on a 10-sec PERIO0 LEVEL period, and this period slowly levels off DEMAND DEMAND into infinity as the reactor approaches the I 0-100% 00-IO SEC 0-IOV 0-100 V desired power level. 1.7

IDEMAND

Startup Accidents

The startup accident

10V


is that condition in which positive reactivity is continually inserted into a reactor at a given rate and nothing is done to stop it. It is presumed that safety stops or emergency means are

not available until the reactor power level In the reaches a given overpower point. case of a reactor being started up by with drawing its control rods, some situation is hypothesized whereby the rods are ex tracted from the reactor until they are all the way out. In this way, the reactor is brought from a subcritical condition

CONTROL RODS

COMPARATORH AMPLIFIER

!REACTOR

ACTUATOR

RODS

(a)

DESIRED POWER LEVEL

NEUTRON DETECTOR

REACTOR

PERIOD CIRCUITS

ACTUATOR

PERIOD SIGNAL ERROR SIGNAlIJ

AMPLIFIER

Fia. 14. Automatic period control.

control

system

Fia.

15. Combination period and level con trol: (a) block diagram of control syistem,

for

(fj) approach

to power level.

through critical and through prompt critical. The parameter which is of importance is the period attained by the reactor in that this period is a measure of the time available to do something about the accident. It has been shown that, for rates of change of $k consistent with reasonable startup times, prompt critical may be reached before the instruments indicate properly the resulting neutron flux level. An upper period bound can be derived from the reactor kinetic equations for uniformly increas ing reactivity which states that Period <

Bk

-

0

(ID

for the region above prompt critical or alternately Period nod > „

/— —L

\27 In

(ni/n0) 0

(12)

8-32 where

CONTROL

OF REACTORS

[SEC. 8

= reactivity 0 = fraction of delayed neutrons 1 = prompt neutron lifetime 7 = rate of increase of k

6k

tii = neutron level at which safety devices operate

A

no = startup neutron level before rods are moved lower bound to the period for subcritical operation

can also be derived.

This

subcritical bound is that Subcritical

2

7

(13)

REACTOR OPERATION 2.1


period >

Manual Operation

Operation of a reactor means maintaining or changing its power level once startup procedures have brought the reactor to a critical state. To maintain a given level requires that control rods or other means of changing reactivity be manipulated to keep the multiplication factor of the reactor equal to unity in the face of slow changes such as depletion or poisoning. In some reactors, continuous precise level control is not required and consequently manual control may be used. Changing the power level also requires rod motions or equivalent reactivity changes. When power level is changed, two conditions must ultimately be satisfied by any control system: The power-output level of the reactor must be the demanded power level within a given error, and the multiplication factor must ultimately be 1. The method of changing power level in a simple reactor possessing a zero temperature coefficient is as follows: Assuming that the reactor is initially critical at a low power level and it is desired to increase this level, the first step is to extract a control rod a small amount and change the multiplication factor k from 1 to a value slightly greater than 1. The neutron level then starts to rise in an exponential manner. As this power level rises and approaches the desired ultimate level, the control rod must then be inserted back to where k = 1. In a manual control system with no anticipation provided by the operator, the control rod will oscillate about the k = 1 position but ultimately it will settle down at the original position from which it started. This system is one in which the power level is independent of rod position, and to change the power level a control rod is moved temporarily in or out of the reactor and then returned back to its original position. When the reactor possesses a finite temperature coefficient, the required rod motions For example, when the may be different, depending upon the coolant program. reactor coolant is supplied to the reactor at a constant temperature, the average At any power temperature of the reactor depends directly upon its power output. level, in order to keep the reactor critical, control-rod motion is required to balance In this case, the the reactivity variation caused by the temperature coefficient. ultimate position of the control rods changes progressively with the power level. In general, changing conditions of coolant programing, temperature, poisons, and depletion make it difficult to predict where the control rods will be located as a func However, as long as the required motions arc slow and precise tion of power level. output is not needed over long periods of time, manual operation backed up by auto matic safety devices may be used for reactor control. 2.2

Automatic Control

An automatic control system consists of a control loop around the regulator rod or a shim-rod group and can be operated either as a proportional regulating system A proportional regulating system is one or as a discontinuous regulating system. in which the position of the control rod or rods is changed in proportion to and in phase opposition with any error created either by a power-demand change or by an A discontinuous regulating system is one in which no internal system transient.

THE CONTROL


Sec. 8-2]

8-33

SYSTEM

control is exercised unless an error which is some fixed percentage away from a demanded set of conditions is created in the control loop. When sufficient deviation occurs from the demanded conditions, control-rod position is changed, usually at a fixed velocity. Present-day discontinuous regulating systems can be designed to Better accuracy can be obtained hold reactor power level to within 0.5 per cent. The discontinuous system offers the advantage with a proportional type of control. that it is less sensitive to noise which may originate in the detecting elements. In high-flux reactors, noise is usually not a serious problem and for many purposes the increased accuracy obtainable warrants the use of the proportional system. Figure 16 shows a block diagram of the type of control system used to control a reactor in the power-level range. Here the reactor multiplication is changed by movement of the control rods. The output of the reactor, in terms of power output or neutron level, is measured by a neutron detector, generally an ionization chamber. This reactor output is then compared with the desired power demanded in the com parator, and any error between the actual output and the demanded output is noted This amplifier finally controls an actuator and amplified in the error-signal amplifier. which moves the rods the proper amount and in the proper direction to eliminate COMPARATOR ACTUATOR the error. The comparator, error-signal AMPLIHtH amplifier, and actuator may be of any suitable hydraulic, type. Pneumatic, electrical, and mechanical devices have CONTROL REACTOR A brief description follows RODS all been used. of the major components in a loop from a Flo. 16. Block diagram of the essential ele control-system point of view. The trans ments of a reactor automatic control loop. fer function approach will be used. In using this technique the reactor and all loop components are assumed to be "black box" circuit elements having specified amplitude and phase shift output responses to a small-amplitude variable-frequency sinusoidal input signal. The ratio of the output to input is defined as the transfer function. 2.21 Reactor as a Control Element— No Temperature Coefficient. The reactor transfer function has been derived by several authors, f'11-1' The form of the transfer function is given in Eq. (14):

[1+ H !'()

+

J

(14)

where the reactor transfer function = KrGu{s), with Gr(s) being the frequencydependent portion of the transfer function and Kb a constant factor,14 in(s) is the output for a given small sinusoidal change in reactivity input, Sk(s). no = operating power level of the reactor I* = mean neutron lifetime 0i = fraction of delayed neutrons in the t'th group = decay constant of ith group 8 = ju, Laplace transform operator For a U"5 reactor of I* = 1.25 X 10~3 sec and five principal groups of delayed neutrons, the reactor transfer function has been given as"

\i

Sn(s)

+ 0.456) Q + 0.154) (a + 0.0315) l*s(s + 14.4)0 + 5.91)0 + 1.41)0 + 0.32)0 + 0.08)

= n0Q + 14) Q + 1.61)f>

«*(»)

Reference 4 indicates the transfer function for six groups of delayed neutrons as

a

U'"

reactor with an I* = 10-4

Snjs) = n0Q + 14)Q + 1.61)Q + 0.456) Q + 0.151)(a + 0.0315)(s + 0.0124) Sk(s) U*(s + 77)0 + 13.38)0 + 1.43) (s + 0.336)0 + 0.0805)0 + 0.0147)

t See

Ref. 4, chap. 3.

(15) sec and

(16)

8-34

CONTROL

OF REACTORS

[Sec.

8

Figures 17 and 18 indicate the transfer function gain and phase shift for different for reactors containing zero temperature coefficient. The gain is normalized to The principal control features to bo equal 0 db at 1 cps for I* = 10_< sec in Fig. 17. noted from Eq. (14) are the nonlinearity of the transfer function; i.e., the gain of the reactor as a circuit element depends upon no, the level at which the reactor is operating;, and the fact that at zero frequency the gain is infinite.


Z*s

FREQUENCY-CYCLES PER SECOND

Fio.

17.

Amplitude of Um reactor transfer functions.

001

01

1.0

FREQUENCY-CYCLES PER SECOND

Fio.

18. Phase

shift of U"s reactor transfer functions.

It will be noted also, from a transfer function point of view, that the break point of highest frequency in the reactor transfer function is determined in Eqs. (14) to (16) by the largest root, which in turn depends upon the value of fi/l*. It is the value of this root which specifies the time behavior which distinguishes, control-wise, a thermal reactor from an intermediate or fast reactor. In a fast reactor, the value of I* is so small that the last break in the transfer function occurs at a very high frequency, Difficult component problems arise in control systems operate usually above 100 cps. ing in this frequency range. From a physical point of view, control out to this last

Sec. 8-2]

THE CONTROL

SYSTEM

8-35

break represents control on prompt neutrons. This procedure is generally unneces sary, and slower control systems are more desirable. 2.22 Reactor as a Control Element — with Negative Temperature Coefficient. "■" Negative temperature coefficient in an elementary reactor can be considered as an internal feedback loop around the reactor transfer function. This feedback loop consists of many branches which provide components of both negative and positive feedback. The combined effects generally provide over-all negative feedbacks around the reactor. This complex feedback loop in simplified form can be repre sented in two parts: first, a part lumping the transfer of heat from the reactor fuel to the moderator and, second, a change in reactivity caused by the change in tempera ture and density of the moderator and fuel. Figure 19a indicates in block form these

KrGr(S) (S)

COMBINED REACTOR AND

LOCAL TEMPERATURE COEFFICIENT EFFECT TRANSFER FUNCTION

KtcGtc'S) LOCAL

TEMPERATURE


COEFFICIENT EFFECT

Fig.

19.

(a) Combination of reactor

local temperature

functions.

coefficient:

(b) function with feedback transfer function of (a) individual transfer functions, (6) combination transfer transfer

parallel transfer functions, with the temperature coefficient effect having the transfer function notation KtcGtc(s). These two parallel transfer functions can be combined, as indicated in Fig. 19b, into a new over-all combined reactor and local temperature coefficient effect KrtcGrtc(s). This combined reactor transfer function with tempera ture coefficient is given by

v

n

/ \ =

KrtcGrtc(s)

,

,

KrGrM „ , - i. „— rr

■■

(17)

where KrGr(s) is the transfer function of the plain reactor as given in Figs. 17 and 18. An elementary form of Ktc(*tc(s) can be used with fair accuracy by considering the two-part process to consist of a lumped time lag in transferring the heat from the reactor to the moderator and a gain term related to the j>ower level and the value of the negative temperature coefficient. This type of approximation is quite accurate for homogeneous-type reactors and has been treated in detail by Weinberg and Ergen.17 For some heterogeneous reactors, the double heat-transfer characteristics between fuel and moderator must be taken into account in that both fuel- and moderatortemperature changes contribute to the over-all temperature coefficient. This situa tion has been treated in detail by Lipkin.18 The homogeneous case in servo notation assumes that only one simple time lag exists between the neutron power output and the ultimate reactivity change caused by the temperature coefficient. The transfer function of this lag is given by

KtcGtcW where

Ktc

- ts—^r +

1

= constant multiplied by temperature coefficient = over-all lumped time constant of thermal lag s = jw, Laplace transform operator

t

(18)

8-36

OF REACTORS

CONTROL

[Sec. 8


A combined reactor temperature coefficient transfer function is presented as an example in Figs. 20 and 21 for several values of Ktc and t = 0.159 sec. The ampli tude curves of Fig. 20 indicate that at very low frequencies the over-all combined transfer function is determined solely by the temperature coefficient effect feedback.

0.1

1.0

FREQUENCY, CYCLES PER SECOND

Fig.

20.

Amplitude response of combination

transfer functions

0001

for several temperature

001

reactor coefficient

01

and temperature gains.

coefficient

feedback

coefficient

feedback

I

FREQUENCY, CYCLES PER SECOND

Fio. 21. Phase-shift response of combination reactor transfer functions for several temperature coefficient

and temperature gains.

The gain at zero frequency is no longer infinite, but a finite value equal to 1 /Ktc- At very high frequencies, the combined transfer function gain takes on the shape of the plain reactor transfer function which, in turn, depends principally on I*. A gen eralization can be made that if the break frequency caused by the time delay t is

Sec. 8-2]

THE CONTROL

8-37

SYSTEM

low compared with the highest natural break frequency of the reactor set by fl/l*, the high-frequency response of KrtcGrtc(s) is not affected by the temperature coefficient. Figure 21 indicates that at low frequencies the phase shift approaches zero for reactors with negative temperature coefficients instead of 90° as in the case of the At high frequencies again the phase shift of the combined elementary reactors alone. transfer function approaches the phase shift of the plain reactor. Some phase lead may result from the combination of reactor and temperature coefficient. The position and magnitude of this lead depend upon r. This simple representation of the combination of the two transfer functions is quite stable. A reactor with this type of representation for the local temperature coefficient effect cannot oscillate, as this form of transfer function as seen in Figs. 20 and 21 can never encircle the —1,0 point on a Nyquist plot for any value of the openloop gain. Harrer and DeShong19 have indicated the transfer function for the CP-3 reactor, and from their work it is apparent that the elementary transfer function form for the


COMBINED REACTOR AND TEMPERATURE COEFFICIENTl TRANSFER FUNCTION

POISONING FEEDBACK

K„G,(S)

Fig.

(b)

(o)

(a) Elementary block diagram representation of poisoning feedback. 22.

of

poisoning

feedback,

negative temperature coefficient feedback of Eq. (18) represents

(6)

transfer-function

a good approximation

for some reactors. The behavior of circulating

fuel reactors has been formulated as a function of Reference 20 temperature coefficient under conditions of constant inlet temperature. indicates that circulating fuel reactors are also very stable and that they may be made as stable as desired by making the parameter Sktt/l* sufficiently small. In this expression Sk is the reactivity,
.v,,,

8a (s) =

His)

(Tx — VzXo) (a

(a V,

-\

+ X,)(s +

n' 7z —

do

.. +

+ X,)

\i\

J

£10.

where the symbols have the same meaning as in F,qs. (9) and (10) and where Xo is the through the steady-state poisoning level created by the steady-state flux level

o

t

Sec Ref. 4. p. 56.

8-38

CONTROL

OP REACTORS

[Sec. 8

relationship

Xo

=

(y* +

7j)
(20)

the constants X; = 2.9 X 10"6, X* = 2.1 X 10~6, yi = 0.056, y, = 0.003, X IO-18, Figs. 23 and 24 can be plotted giving the amplitude and phase shift of the poisoning transfer function for various flux levels. The level curves are normalized in gain about the

70

60 50


40

v

30

a

z

s

^-n= IO"

20 10

s -10

X

=I015

/%*I0"

\

-20 4>o«io"'

-(Ho'lO*

-30

-40

ioFia.

10'' 23.

10"'

IO"5 FREQUENCY,

10"*

CYCLES PER SEC

IO"3

io-'

Amplitude of transfer function of xenon feedback factor SX(a)/S(s).

o

= 3 X 10" the over-all loop of Fig. 22 stable. On the other hand, for values above small negative temperature coefficients are needed to prevent very long period oscillations. 2.24 The function of the comparator in an automatic control loop Comparator. First, it provides that the error signal is essentially the subtraction is twofold. between the output of the neutron-detector signal and a power-demand signal. The preferred Second, it is used to compensate for the nonlinearity of the reactor. form of the comparator output signal should be error /level. Figure 25 shows several forms of elementary comparator circuits. Figure 25a is the simplest form. Here the signal from the neutron detector is V„, which is proportional to the neutron level If a reference voltage Fo is arbitrarily denned as Vo = Eb/K, then of the reactor. V, KVn — Ei = (EbV„/V») — Eb » Eb(n — n0)/n0, where V, is the output error signal, n is the actual reactor operating level, and no is the demanded steady-state level. This circuit, when connected to an output level signal from the reactor, approximately cancels out the reactor gain dependence upon level. Actually, a signal inversely proportional to n rather than no is more desirable, but the circuitry is

-


Fio.

(C)

25. Elementary forms of comparator

circuits.

8-40

CONTROL

OF REACTORS

[Sec.

8

usually more complex. The difference between the actual level n and the demanded steady-state level no is usually quite small, as the control system acts to make the two quantities the same. Comparators having perfect cancellation over an infinite range cannot be con structed practically, but good cancellation can take place over a limited range of about 100 to I. Figure 256 shows an acceptable comparator circuit in which the demand signal is linearly proportional to the displacement D of the potentiometer P when no current is taken from V„ To ensure that no current is taken from Y„ computing techniques may be used and an operational amplifier connected to V, as indicated in Fig. 25c. Here R' and R" have the same resistance as the original R of Fig. 256 and the bias voltage of the amplifier becomes the equivalent of the battery Ei,. For some applications, it is undesirable to use vacuum tubes in a comparator circuit, and consequently, magnetic amplifiers have been used. Figure 25d indicates one such circuit which operates on the principle that if sufficient feedback is used in an amplifier circuit, the gain of the amplifier depends inversely on the amount of feedback In Fig. 25d the load current that is used. equals [A(n — n0)]/(l + AB), where A If .-IB is now made large is the gain of the amplifier and B is the feedback factor. = (n — no)/B. It now becomes necessary only to make B compared with 1, proportional to n or no in order for the comparator output to have the proper form. Again, in practical circuits it is usually easier to obtain no than n.

II

II

Table Generated for wjivans (University of Florida) on 2015-09-23 02:52 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

(n,/no

peak power ratio

2 — I + 30
, radians

0.3 0.75 1.5

3.0

7.5 15.0 30.0

i» = 10-" sec

f* = I0"« sec

2.00 1.50 1.25 1.00 0.75

2.00 1.50 1. 25 1 00 0 85 0.75

0.50 0.25

0.65

I» = 10 '

2.00 1.50 1.25 1.00 1.00 1.00 1.00

The actuating mechanisms for control-rod 2.25 Other Control-loop Components. The transfer functions for these drives are covered in Sec. 8-3 of this handbook. actuators as well as the accompanying amplifiers are well known and can be obtained from several servo texts. 14,2Si" The transfer functions of the neutron detectors are usually treated as pure gain functions in that the frequency response of most ioniza tion chambers does not begin to fall off until above 200 cps. Many types of components and mechanisms 2.26 Automatic Control Philosophy. A reactor can be used in an automatic control loop of the type indicated in Fig. 16. control loop may be examined by four criteria: Absolute stability — no exponentially increasing oscillations Relative stability — the number of oscillations before a transient has essentially died out 3. Maximum power-level excursion — the peak power reached by the reactor for a fixed shape input disturbance 4. The restoring time of the system — the absolute time for a given transient to die out 1.

2.

The stability of a control loop can be examined hy the well-known Nyquist criterion." The transient response and the restoring time of a system may be obtained by direct analysis or simulation techniques. The peak power attained by the reactor can be limited by the automatic control system or by the negative temperature coefficient.

THE CONTROL

Sec. 8-2]

8-41

SYSTEM

and DeShong have shown that the peak power level attained by a reactor having no temperature coefficient when a Sk step reactivity change is inserted into a proportional-type control loop similar to Fig. 16 is a function of the natural frequency Table 2 indicates these of the system and the magnitude of the Sk step input."' t peak power levels. In Table 2, the natural frequency of the control system a is selected as is the mag The peak power n, referenced to the nitude of the Sk step inserted into the reactor. original operating power level no is then ACTUATORl DEMAND obtained from the table for reactors hav AMPLIFIER MOTOR ing given I* values. TACHOMETER When the reactor has a strong nega FEEDBACK tive temperature coefficient, this coeffi transients, and cient limits the peaks of all REACTOR] [CONTROL ROOH for Sk step inputs into a reactor control system the peak power that the reactor (a) can attain during the initial front edge of the transient because of the temperature DEMAND Harrer

coefficient

limiting np


no

AMPLIFIER

is

_

Sk*t*p &

— Sktup

Equation (21) is valid only when the time constant of the negative temperature co efficient is short compared with the time constant of the delayed-neutron emitters. It is presumed that the automatic control system restores the reactor power level to its normal demanded value after the initial peak in power has been reached. To improve the performance of a reac tor automatic control loop, compensating Figure 26 in devices are generally used. dicates the common methods used to ob tain more loop gain or better transient response.

NEUTRON IDETECTOR

(21)

ACTUATOR

POSITION FEEDBACK REACTOR! :tor| i control

rqdH

(b) DEMAND,

êtV^kH^pêrHactuatcr|

NEUTRON [DETECTOR] REACTOR] [CONTROL ROOh (c)

Fio.

Types of compensation networks commonly used in reactor automatic con trol loops: (a) tachometer feedback, (b) position feedback, (c) derivative network. 26.

The above methods of reactor control must be modified in the case of a boiling reactor in that, for these reactors, moderator pressure and density become as important The basic method of analysis and control, however, as the temperature coefficient. remains the same."" 2.27 Procedure for the Selection of Automatic-control-system Constants. Pres ent-day reactor control systems have safety problems and automatic control systems must be extremely reliable. The following procedure has been suggested as a guide for the design of an automatic control system of a reactor: t 1. Determine the temperature coefficient in magnitude and time constant or the ranges that the temperature coefficient might have during the reactor lifetime. 2. Determine the integrated or peak energy permitted from the reactor as a function of time. 3. Decide the form of the control-system mechanisms, and derive the transfer functions of all the loop components. 4. Determine the adequacy of existing components of the type chosen as a function of the natural frequency of the control loop. 5. Consider the performance of the reactor control system by simulation techniques or others, and evaluate this performance by means of the four criteria given under Art. 2.26, Automatic Control Philosophy, for the type of desired response. 6. Choose the slowest, safest system that fits the above conditions. t

See Ref. 13, p. 941.

J See

Ref. 4, chap. 4.

8-42

CONTROL

OF REACTORS

REACTOR

3

[Sec.

8

SHUTDOWN

Normal Reactor Shutdown

3.1

Control For a reactor operating routinely, shutdown is an elementary process. rods are inserted into the reactor at any normal rate, and the reactor neutron level falls off to its shutdown value. The initial dropoff in neutron level depends upon the amount and speed of the reactivity inserted. After a few minutes, however, the neutron-level dropoff becomes solely a function of the delayed-neutron emitters. Figure 27 indicates the dropoff in neutron level as a function of step reactivity inserted

>k=-O.OC)4


10"'

2

-~6k»-0.0IS

>

U

ik ;-0.02 5

icr

icr-

40

20

60

80

100

120

140

TIME IN SECONDS

Fia. I* =

27. Relative neutron 1
level

vs. time

for negative

step-function reactivity

changes.

for a reactor of /* = 10-4 sec. It can be seen that, regardless of the amount of initial reactivity involved, the dropoff ultimately proceeds on a minus 80-sec period. The initial dropoff slope depends on I*. For a step input in reactivity this slope equals Sk/l*. 3.2

Beta-Gamma

Power

A reactor may be shut down neutron-wise but still continue to give out considerable power in the form of heat caused by delayed-fission-product emission of (3 and > energy. Provisions must be made to handle this energy, particularly in large power reactors. Some reactors may not be able to dissipate even a small fraction of normal power output without some form of designed-in cooling. The power to be antici

the control system

Sec. 8-2] pated as a function of

time"'"

8-43

is

O07P„ (••«

(22)

where Po — long-time power level before shutdown Pi = P-y power I sec after shutdown in same units as Pa

3.3

Methods of Reactor Shutdown


Shutdowns, other than normal shutdown, caused by something wrong in the plant Whenever possible, it is desirable to limit or ward may require special rod motions. off accidents by moans other than scramming. Slower means of protection are Some of these slower types of protection are listed below in order of preferred. increasing effectiveness: 1. Sound or light up an alarm. This permits the operator to determine what has actually happened and then to perform a correction if he deems necessary. 2. Stop all rod motions. In many reactors, the negative temperature coefficient is sufficiently strong to handle certain classes of accidents, provided additional com plications are not introduced by rod motions. 3. Permit rod motion in the negative-reactivity direction only. Regardless of whether the automatic control system or operator wants to perform a given function, once this safety condition is switched in, only negative reactivity can be obtained. 4. Automatically cause rods to move into the reactor at their normal rate. This condition is sometimes called "reverse." 5. Cause rods to move inward at a faster rate than normal. This condition is sometimes called "cutback." It is not a scramming operation, as the rods are driven inward either by the normal control motor with different excitation or by a separate motor with different gearing or clutching. Rod-insertion rates of between 2 and 10 times normal excitation rates can usually restrain most accidents. This is the condition of inserting the rods as quickly as possible. 6. Scram. A given reactor system can use some or all of these devices in two ways. The first method is a parallel operation whereby a given accident or class of accidents can be assigned a given type of correction. For example, if an electrical interlock connection is broken, an alarm might be shown. If a pump fails, a slow inward rod motion might be called for, etc. The second method is a series operation in which the weakest type of correction in the system is always used first and then the corrective action progresses to more drastic types of protection if the severity of the accident continues to increase. For example, protection devices of the following type might he applied to the condition of excessive power input from a reactor: 1.

2. 3. 4.

105 1 10 120 130

per cent demanded power per cent demanded power per cent demanded power per cent demanded power

level — alarm level — rod motion frozen level- -fast cutback level — scram

3.4

Scram

As mentioned previously, scramming consists of inserting negative reactivity

into

the reactor as quickly as possible. Control rods are apt to be relatively heavy pieces of neutron-absorbing material. Present-day techniques restrict the motions of these rods as a function of time. Inserting rods into a reactor in less than 10"3 sec is probably impossible. Inserting control rods completely into a reactor in 10~* sec probably requires explosives. Inserting rods into a reactor in 10_1 sec requires a Inserting rods into a reactor in 1 sec is a relatively complicated mechanical design. simple matter, usually not needing more than a gravity drop.

8-44

CONTROL

OF REACTORS

[Sec.

8

There are many possible reasons for scramming a reactor, some of which are listed below: 1.

Excessive neutron level

2. Short period 3. Excessive core temperature 4. Excessive outlet coolant temperature

Excessive inlet coolant temperature Low system pressure High differential pressure across reactor Pump failure Electrical-system power failure 10. Control-rod coolant-system failure 11. Radioactivity level in secondary steam system too high 12. Radioactivity level in building too high 5.

6. 7. 8. 9.

A control or safety rod that is used in scramming is 3.41 Scramming Circuits. usually connected to its drive mechanism by some form of electromagnet. The electromagnet coupling may be either direct or by means of a latch. The physical process of scramming is generally one of breaking the current in the electrical circuit feeding the magnet. Three basic means of interrupting this current have been circuits, employed in reactor scramming circuits. These are contact-relay-type magnetic amplifier relay circuits, and electronic circuits. MAGNET Generated for wjivans (University of Florida) on 2015-09-23 02:52 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

HOLDING

J

SUPPLY

RELAY POWER SUPPLY

L

MANUAL SCRAMS ROD POWER

RELAY

RELEASE MAGNETS

5 ■ AUX. • RELAY

SCRAM SIGNAL N0.I

- AUX.

-RELAY

SCRAM SIGNAL NO. 2

Fio.

28.

3

AUX. RELAY

SCRAM SIGNAL NO. 3

Relay-type series-feed scramming signal circuit.

All the auxiliary Figure 28 is an illustration of the contact relay type of circuit. The power relay is relay contacts are normally closed and are connected in series. normally operated with current through its coil, and a power failure of any sort This type of Front contacts are used throughout the circuit. releases the magnet. circuit is inherently slow because two mechanical relays are always in series. An example of a magnetic amplifier scramming protection channel is indicated in Fig. 29. An ionization chamber is the primary detector in this system. The current from this detector is amplified directly in a conventional magnetic amplifier relay.

THE CONTROL

Sec. 8-2]

magnetic

current Amplifier IONIZATION CHAMBER DETECTOR

j

8-45

SYSTEM

XMAGNETICf TYPE £

jr itrv xamplifier,
7— 'CONVENTIONAL' RELAY

NORMAL CONDITION ZERO CURRENT


Fig.

29. Magnetic-amplifier scramming

SCRAMMING SIGNAL

protection

channel.

"SIGMA"

AMPLIFIER

NO. I

SLECTRONICj MAGNET

AMPLIFIER ROD

SCRAMMING SIGNAL NO.Z

RELEASE MAGNET

"SIGMA"

AMPLIFIER

7

SCRAMMING SIGNAL NO. 3

/

"SIGMA"

AMPLIFIER MAGNET

AMPLIFIER POWER SUPPLY

Flo.

30. Block diagram

of electronic

scramming

system.

The normal condition for the auxiliary relay is to supply minimum rod current. The auxiliary relay feeds a scramming bus through the coil of a conventional electro magnetic relay. This relay is used to indicate which channel has caused the scram. The auxiliary relay is isolated from the scramming bus by means of a rectifier. The scramming bus feeds the power magnetic amplifier relay which is biased such that This load current feeds the rod-release magnet. its load current is normally on.

8-46

CONTROL

OF REACTORS

[Sec.

8

A channel of this sort may be Mechanical scrams are provided in the output circuit. constructed from conventional components operating at 400 cps to have an over-all time delay from detector signal to rod motion of less than 60 msec. To When the utmost in scramming speed is desired, electronic circuits are used. take advantage of the high speeds obtainable from these circuits, special magnets In the use of electronic circuits, with quick release times must also be provided. Direct-coupled tube circuits problems concerned with safe failure are magnified. are generally used. Figure 30 shows a simplified block diagram of the basic type of electronic circuit. The terminology is that which originated at the Oak Ridge National Laboratories, and a so-called sigma or summation bus is set up which is fed from a group of summation amplifiers.30'*1 This summing bus feeds a magnetholding amplifier of a vacuum-tube type. Because of the large current requirements of the magnet, more than one magnet is usually not connected to a magnet amplifier. The current to the rod-release magnet is a rectified alternating current, and the Manual output tubes of the magnet amplifier are fed from a separate transformer. scrams and other low-power interlocks may be inserted by controlling the primary If this primary voltage fails, the system scrams. Cir voltage to this transformer. cuits of this sort have been constructed with an over-all time delay from the detector to rod motion of less than 40 msec.


3.6

Last-ditch

Emergency Shutdown

Because of the destructive capabilities of a reactor plant and the resulting cleanup problems, most power reactors have last-ditch backup safety devices which operate in the event that all the above-mentioned types of protection fail. The insertion of chemicals with high-absorption cross sections and the use of boron shot in control-rod For reactors of the water-boiler type, it is possible to dump passages are feasible. out the fuel into an auxiliary tank having a safe geometry. For reactors which have separate moderator and coolant, the moderator may also be dumped. The Oak Ridge graphite-moderated reactor employs boron shot. The Brookhaven reactor contains a means of injecting liquid trichlorobenzene into tubes in the reactor. The Canadian NRX reactor can have its moderator removed. The speed of control required for a last-ditch emergency action is usually slow. Presumably some damage has already been done. This device would be used either to prevent further damage or to permit repair work on the reactor. These devices should not be capable of easy and quick usage because of the difficulty entailed in the operation of cleaning the poison out of the system. Operation times from 1 min to 1 hr are generally employed.

REFERENCES 1. 2.

Harrer, J. M.: Control Rod Mechanisms, Nucleonics, 13(6): 48 (June, 1955). Rice, C. M.: Hydraulic Control-drive Mechanisms, Nucleonics, 13(11): 116 (November, 1955).

3.

Wallach, S.: "Solutions of the Pile Kinetic Equations When Reactivity Ib a Linear Function of the Time," WAPD-13, Westinghouse Atomic Power Division, Pittsburgh. Pa., 1951

4. 6. 6. 7. 8. 9. 10.

Schultz, M. A.: "Control of Nuclear Reactors and Power Plants," p. 40, McGrawHill Book Company, Inc., New York, 1955. Lundby, Arne: Kinetic Behavior of a Thermal Heavy Water Reactor, Proc. 1953 Con), on Nuclear Eng., D-39, University of California Press, Berkeley, Calif., 1953. Anderson, Herbert L. : " Neutrons from Alpha Emitters," Report 3, National Research Council Nuclear Science Series, Washington, December, 1949. Wattenberg, A.: "Photo-neutron Sources," Report 6, National Research Council Nuclear Science Series, Washington, July, 1949. Bernstein, S., F. L. Talbott, J. K. Leslie, and C. P. Stanford: "Yield of Photoneutronfrom U"« Fission Products in Be," AECD-1833, Feb. 20, 1948. Ergen, William K.: "Hard Gamma Emitters among Fission Fragments," ANP-59. May 3, 1951. Gladstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952.

the control system

Sec. 8-2]

8-47

11. Franz, Joseph P.: "Pile Transfer Functions," AECD-3260, Oct. 26, 1951. 12. Harrer, J. M., R. E. Boyar, and Darwin Krucoff : Transfer Function of Argonne CP-2 Reactor, Nucleonics, 10(8): 32-36 (August, 1952). 13. United States Atomic Energy Commission: "The Reactor Handbook," vol. 2, Engineer ing, AECD-3646, Government Printing Office, Washington, D.C., May, 1955. 14. Brown, G. S., and D. P. Campbell: "Principles of Servomechanisms," John Wiley & Sons, Inc., New York, 1951. of Reactor Kinetics on Temperature," BNL-173, 15. Chernick, J.: "The Dependence Brookhaven National Laboratory, Upton, Long Island, N.Y., Dec. 20, 1951. 16.

Robinson, L. B.: Concept of Stability for Nuclear Reactors,

516-518

J.

Appl. Phys., 28(4):

(1954).

Weinberg, A. M., and W. K. Ergen: Some Aspects of Nonlinear Reactor Kinetics, Proc. Kjeller Conf. on Heavy Water Reactors, JENER, 1953. 18. Lipkin, H. J.: A Study of the Non-linear Kinetics of the Chatillon Reactor, J. Nuclear 17.

Energy,

1 : 203 (1955).

Harrer, J. M., and J. A. DeShong, Jr.: Discontinuous Servo for Control of Power Reactors, Nucleonics, 12(1): 44 (January, 1954). 20. Fleck, Jr., J. A.: "Temperature-dependent Kinetics of Circulating Fuol Reactors," Convention Record of the Nuclear Engineering and Science Congress, Cleveland, Ohio, 19.

December,

22.


23. 24. 25. 26.

J.

1955.

N., M. A. Schultz, and T. E. Fairey: Inherent Reactor Stability, Proc. 1965 Conf. on Nuclear Ena., University of California Press, Berkeley, Calif. Chestnut, H., and R. W. Mayer: "Servomechanisms and Regulating System Design," John Wiley & Sons, Inc., New York, 1951. Truxal, John G.: "Automatic Feedback Control System Synthesis," McGraw-Hill Book Company, Inc., New York, 1955. Nyquist, H.: Regeneration Theory, Bell System Tech. J., 11: 126 (January, 1932). DeShong, J. A.: "Performance of Servo Motors for Reactor Regulating Rods," AECD-3391, Argonne National Laboratory, Naval Reactor Division, Lemont, 111., Feb. 5, 1951. Macphee, J.: How to Control a Boiling Reactor, Nucleonics, 13(12): 42 (December,

21. Grace,

1955).

27. Untermyer, S.: "Reactor Controls and Instrumentation," Convention Record of the Nuclear Engineering and Science Congress, Cleveland, Ohio, December, 1955. 28. Way, K., and E. P. Wigner: Rate of Decay of Fission Products, Phys. Rev., 53 : 1318 (1948).

29. Untermyer, S., and J. T. Weills: "Heat Generation in Irradiated Uranium," ANL4790, Argonne National Laboratory, Lemont, 111., Feb. 25, 1952. 30. Cole, T. E. : Design of a Control System for a Low-cost Research Reactor, Nucleonics, 11(2): 32 (1953). 31. Trimmer, J. D., and W. H. Jordan: Instrumentation Control of Reactors, Nucleonics, 9(4): 60 (1951).

BIBLIOGRAPHY Bo wen, J. H.

: Automatic Control Characteristics of Thermal Neutron Reactors, Proc. Inst. Elec. Enors. (London), 100(1): 102 (1953). Breazeale, W. M.: The "Swimming Pool," a Low-cost Research Reactor, Nucleonics, 10(11): 6 (November, 1952). Brown, G. S., and D. P. Campbell: "Principles of Servomechanisms," John Wiley & Sons, Inc., New York, 1951. Cochran, D., and C. A. Hansen, Jr.: Instrumentation for a Nuclear Reactor, Nucleonics, (August, 1949). Cole, T. E.: Design of a Control System for a Low-cost Research Reactor, Nucleonics, 11(2): 32 (1953). Colmer, F. C. W., and D. G. Littler: Gleep: Design, Construction, and Use, Nucleonics, 8(1): 3 (1951). Commonwealth Edison et al.: "Reports to the United States Atomic Energy Commission on Nuclear Power Reactor Technology," Government Printing Office, Washington, D.C.. May, 1953. Dahl, A., and G. Randers: Heavy Water Reactor at Kjeller, Norway, Nucleonics, 9(5): 5 (November, 1951). Geyger, W. A.: "Magnetic Amplifier Circuits," McGraw-Hill Book Company, Inc., New

York,

1954.

Glosstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952.

8-48

CONTROL

OF REACTORS

[Sec. 8

C: "The Science and Engineering of Nuclear Power," vol. 2, Addison-Wesley Publishing Company, Cambridge, Mass., 1947. Goss, Clinton G.: Reactor Control Instruments, Proc. 1953 Conf. on Nuclear Eng., Univer sity of California Press, Berkeley, Calif., 195.3. Grace, J. N. : "Synthesis of Control Systems for Nuclear Power Plants," Convention Record of the IRE 1954 National Convention, pt. 9, Medical and Nuclear Electronics, Institute of Radio Engineers, New York, 1954. Hamilton, W. H., el al.: "Power and Temperature Control of Pressurized Water-Cooled Reactors," Convention Record of the Nuclear Engineering and Science Congress, Cleve Goodman,

land, Ohio, December,

Harrer, Harrcr, Harrer,

1955.

J. M.: Control Rod Mechanisms, Nucleonics, 13(6): 48 (June, 1955). J. M.: Controlling a Power-Producing Nuclear Reactor, Nucleonics, 6(3): 58 (1950). J. M., R. E. Boyer, and D. Krucoff: Transfer Function of Argonne CP-2 Reac tor.

10(8): 32-36 (August, 1952). J. A. DeShong, Jr.: Discontinuous Servo for Control of Power Reactor.-?. Nucleonics, 11(1): 44 (1954). Hurwitz, H., Jr.: Derivation and Integration of the Pile-Kinetic Equations, Nucleonirs. 6(1): 61 (1949). Hurwitz, H., Jr.: Safeguard Considerations for Nuclear Power Plants, Proc. 1953 Conf. on Nuclear Eng., University of California Press, Berkeley, Calif., 1953. Isbin, H. S., and J. W. Gorman: Applications of Pile-Kinetic Equations, Nucleonics, Nucleonics,

Harrer,

J.

M., and

10(11): 68 (1952).


Jaques, T. A., H. A. Ballinger, and F. Wade: Neutron Detectors for Reactor Instrumenta tion, Proc. Inst. Elec. Engrs. (London), 100(1): 110 (1953). Jordan, W. H., H. B. Frey, and G. Kelley : An Instrument for Measuring the Logarithms of Neutron Level and the Period of a Pile, Oak Ridge Natl. Lab. Rept. 1 10, November, 194S. Lansing, N. F., comp.: "The Role of Engineering in Nuelear Energy Development."

AEC-TID-5031.

Lauer, H., R. Lesnick, and L. E. Matson: " Servomechanisms Fundamentals," McGrawHill Book Company, Inc., New York, 1947. Los Alamos Scientific Laboratory: An Enriched Homogeneous Reactor, Ret. .Sci. Instr.. 22: 489 (1951). Lundby, Arne: Kinetic Behavior of a Thermal Heavy Water Reactor, Proc. 1953 Conf. on Nuclear Eng., University of California Press, Berkeley, Calif., 1953. MacColl, L. A.: "Fundamental Theory of Servomechanisms," D. Van Nostrand Company. Inc., Princeton, N.J., 1945. Moore, R. V. : The Control of a Thermal Neutron Reactor, Proc. Inst. Elec. Engrs. (.London) , 100(1): 90 (1953).

Murray, R. L.: "Introduction to Nuclear Engineering," Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954. Schultz, M. A.: "Automatic Control of Power Reactors," AECD-3163, 1950. Schultz, M. A.: "Control of Nuclear Reactors and Power Plants," McGraw-Hill Book Company, Inc., New York, 1955. Schultz, M. A.: Temperature Programs and Control Systems for a Nuclear Power Plant. Proc. 1953 Conf. on Nuclear Eng., University of California Press, Berkeley, Calif., 1953. Schultz, M. A., and J. C. Connor: Reactor Power Calibration, Nucleonics, 12(2) : (February. 1954).

Siddall, E.: "The Control of Heavy-water-moderated Massive Fuel Reactors," Convention Record of the Nuclear Engineering and Science Congress, Cleveland, Ohio, December, 1955.

H., and E. C. Campbell: "Elementary Pile Theory," John Wiley & Sons, Inc.. New York, 1950. Book Company. Stephenson, R.: "Introduction to Nuclear Engineering," McGraw-Hill Inc., New York, 1954. Trimmer, J. D., and W. H. Jordon: Instrumentation and Control of Reactors, Nucleonu-s, Soodak,

9(4): 60 (1951).

Truxal, John G.: "Automatic Feedback Control System Synthesis," McGraw-Hill Book Company, Inc., New York, 1955. Van Rennes, A. B., J. C. Simons, Jr., and T. S. Gray: "Dynamics and Control of Nuclear Power Plants," Convention Record of the Nuclear Engineering and Science Congress, Cleveland, Ohio, December, 1955. Wade, E. J.: "Instruments Used with Experimental Reactors," Convention Record of " Medical and Nuclear Electronics," Institute the IRE 1954 National Convention, pt. 9, of Radio Engineers, New York.

8-3

CONTROL INSTRUMENTS AND DRIVES BY Joseph. M. Harrer NEUTRON-SENSITIVE DETECTORS

Neutron-sensitive ion chambers (neutron detectors) are used to measure neutron flux. They are located in the reactor shield, as far from the reactor core as is con sistent with sensitivity so that local flux perturbations are minimized. The average flux at the chamber is then proportional to the average neutron density in the core. Average neutron density is an almost instantaneous measure of reactor power. To calibrate chamber current in reactor kilowatt output, a thermal power measurement is used. Control of reactor neutron density is always from a neutron flux measurement. For low-power research reactors this may be the only power measurement. For power reactors, less emphasis is placed on the flux measurement, thermal power being In both cases flux measurements are used as limiting the important parameter. parameters for maximum reactor power. For all subcooled reactors, power output over time intervals of minutes is controlled at a desired value through reactivity control based upon flux measurements. For boiling reactors, control is from steam pressure, but neutron flux is measured and displayed as a guide to operation and for peak power limiting.

1.1

Source Considerations

is

is

is

is

A

is,

Reactivity is independent of the neutron density. A source of neutrons is essential therefore, an important consideration for design of reactor to reactivity control and source of neutrons instrumentation. obtained from the photoneutron effect in suitable materials and should be one which activated by reactor flux to maintain An antimony-beryllium (y,n) source containing about 30 gr of antimony strength. will produce about 10' neutrons/sec. Radium-beryllium (cr,n) sources are equally effective but lose strength in time. placed next to the core, the attenuation of the flux out to the Unless the chamber the next consideration. chamber location Exact flux calculations can be done (Sec. 7-3), but an estimate of the attenuation can be made by allowing a factor of 10 for each foot of water or concrete interposed. The flux at the instrument location can be estimated from Eq. (1).

I*

source strength, neutrons/sec core volume, cm'

attenuation factor (about 10" for a typical power reactor) neutron velocity, cm/sec (about X 106 for thermal reactor) multiplication mean effective neutron lifetime 2

M

= = = = = =

neutrons/(em*)(sec)

4

where

A V S

Instrument nv =

v


1

8-49

(1)

8-50 M

is obtained from Eq. (2) (given in asymptotic form).

-

k is

where

(ft

1

= 1

M

k

The multiplication

[SEC. 8

OF REACTORS

CONTROL

(2)

the effective number of neutrons produced per neutron absorbed. Instrument

1.2

Chains

1

is

1.

In Display instruments associated with chambers are summarized in Fig. order to give an adequate flux reading throughout the normal range from source based strength to full power, several instrument chains must be used. Figure The flux at the chamber for full on a nominal chamber sensitivity of 10~14 amp/nv. power (unity) was taken as 10* n/(cm2)(sec). RELATIVE LEVEL

T 9

T 8

IO-1

I0"2 I0"3

PERIOD RANGE

±

10-" 10'

~r

LEVEL

SOURCE

,n-7 IU

J

COUNTER RANGE 1 J

lO"8-

I0"9*n.v AT DETECTOR ION

CHAMBER

I09 NEUTRONS/CM2 SEC AT NORMAL SENSITIVITY IO~l4AMP/rw.

POWER

LEVEL.

INSTRUMENT CHAINS 6. LOG METER 7. SAFETY TRIP NO. SAFETY TRIP N0.2 9. SAFETY TRIP N0.3

CHAINS 3,4 AND INCORPORATE COMPENSATED USE UNCOMPENSATED ION CHAMBERS.

Fig.

1.

6

ION

CHAMBERS,

CHAINS

5,7,8 AND

9

5.

8.

I

2 I

I.

COUNTER NO. PROPORTIONAL 2. COUNTER NO FISSION 3. ELECTROMETER 4. MICRO-MICRO-AMMETER GALVANOMETER

Reactor neutron flux instrumentation.

(ft

is

a

severalThe ranges are named with reference to the normal experience with In the source range source multiplication alone hundred-kilowatt reactor output. critical or above critical observed < 1). In the period range the reactor

(k is

7

is

is

7

a

y

is

a

is

> normally observed in the flux measure period and a continuous rise or high enough to raise the temperature ment. In the power range the reactor power of the coolant. After reactor has operated at power and then has been shut down, the radiation at the instrument will be relatively high because of induced activity in reactor struc Counters and chambers to be used under this condition are designed tural material. radiation. Proportional counting chambers, to discriminate against the signal from liFa-fillcd, are most sensitive and are used in the lowest signal range 10~9 to 10~B of The full power. Fission counting chambers are used from 10~7 to 10~4 of full power. about one-tenth as sensitive as a BFj counting chamber fission counting chamber useful in discrimination but has very high neutron-induced pulses and, therefore, In most applications both counter types are used. radiation. against 1)


T T

T

——

NORM AL

7

10

I I

'

POWER RANGE

10

°

POWER

Sec. 8-3J

CONTROL

INSTRUMENTS AND DRIVES

a-51


Compensated ion current chambers are used from 10~8 to 10+1 of full power to In the upper three decades provide some discrimination against the y background. 10-' to 10+1 of full power either compensated or uncompensated current chambers can be used. The range of a compensated chamber is about 100 times greater in the The chief reason for using presence of y radiation than an uncompensated chamber. uncompensated chambers at high levels is to gain sensitivity, compensation being unimportant in this range; also, the chamber construction is simpler, the cost is lower, and the chamber requires only one high-voltage lead. Details of the nine instrument chains shown in Fig. 1 are discussed later. Nine chains are a maximum complement for reactor operation.

Fio.

2.

Typical layout of instrument holes

The ion current chamber flux at full power Location of Neutron Detectors. 1010 or less to avoid radiation damage to insulators and rapid boron Chambers are placed in holes, through the reactor shield, tangential to depletion. the reflector circumference of the reactor. A typical arrangement is shown in Fig. 2. As chambers are moved backward from the inner end of the hole, more shield material In Fig. 2 Thus optimum chamber location is found experimentally. is introduced. a nominal 6-in. hole is shown with a step up to 8 in. This allows insertion of plugs Moisture drainage should be pro behind the chambers to maintain shield integrity. vided for the holes to prevent nitrogen fixation as nitric acid by neutron radiation.*'1 Silica gel packs can be inserted, or a dry helium atmosphere kept on the hole. Flux reductions at the chamber, larger than can be obtained by pulling it back, 1.21

should be


refer to References


OF REACTORS

CONTROL

8-52

[Sec.

8

Graphite or concrete plugs with sections of cadmium or are sometimes necessary. boral (boron-filled aluminum) serve this purpose by preventing streaming. The plugs behind chambers should be 2 to 3 ft long with grooves to allow passage Tapped holes in the plug face provide for attachment of a handling rod. of cables. Adjacent plugs in the same hole are oriented so that wire way grooves do not line up and provide a straight-line radiation path. Detector Cables. Coaxial cables are used to connect the current chamber 1.22 or pulse chamber to an associated amplifier located outside the biological shield. A For the voltage supply to the minimum of two cables for each chamber is required. high-voltage chamber plate, the cable need only meet the voltage breakdown require The shield braid should provide ment. The pickup or signal cable is more critical. Two shield braids are frequently used to minimize close to 100 per cent coverage.

OUTER

JACKET POLY. .250" 00 SHIELD BRAID

OUTER

INNER

BRAID

SHIELD

SEMI

SOLID

DIELECTRIC .146" O.D. CONDUCTOR

22CW

DIELECTRIC SPACER


-

Fia. DETECTOR

-

-PULSE

3.

RG71U cable. - DISPLAYS -

AMPLIFICATION-

AURAL INDICATOR B10

PULSE

CHAMBER ELECTRONIC SCALER

ELECTRO MECHANICAL REGISTER

AMPl JFIER PROPORTIONAL

PULSE

PRE AMR CLIPPER

CHAMBER

LINEAR DISCRIM AMP.

INATOR

LFISSION

PULSE

CHAMBER

COUNTING UNIVIBRATOR

RATE

METER

AVERAGING COUNTING CIRCUIT RATE

._J LOG. COUNTING UNIVIBRATOR

RATE METER

AVE. LOG. COUNTING CIRC. CIRC. RATE

Fig. 4. Counting chains for low-level measurement of reactor neutron flux. Cable capacity per unit length should be low, pickup from external power sources. 10 to 20 /ifif /ft. A solid dielectric keeps the center conductor from movement relative to the shield, reducing microphonic pickup from capacity variation. Polyethylene is a good signal cable dielectric. RG71U shown in Fig. 3 is a commonly accepted standard. A cable designed for very low current (10~" amp) application has a conducting coating between the dielectric and shield braid to prevent electrostaticcharge buildup from cable movement. Dry helium can be applied to the cables through specially designed connectors to keep the cable dry. This same gas can be allowed to pass through boron-coated chambers and serves to maintain a known chamber atmosphere. A leakage resistance upward of 1017 ohms is advisable. As little brass and copper as possible should be employed in the chamber construction and in the connectors to reduce residual activity and facilitate repair. The outer shield of the cables which is connected to the outer can of the chamber should be insulated from ground except at one point. This single point of grounding is normally at the amplifier.

Sec. 8-3]

CONTROL


8-53

1.23 Neutron Counting Chains. During reactor startup a sensitive and positive After the reactor has been operated at a significant radiation detector is needed. A useful power, the unit must detect neutrons and discriminate against y radiation. detector is a B10F3-gas-filled counting chamber. The output of a typical unit con sists of pulses proportional to the ionizing ability of the ionizing particle. NeutronPulses produced a radiation gives high specific ionization compared with y radiation. The amplifier used should faithfully are amplified to between 10 and 100 volts. After being amplified the pulse is "clipped"; i.e., the reproduce the pulse shape. pulse has a long tail which is cut off by the discriminator circuit, allowing higher counting rates and rejecting pulses below a selected magnitude to facilitate the 7-radiation discrimination. The arrangement for cascading elements of counting chains is shown in Fig. 4. The dotted lines indicate alternate methods of counting or alternate displays which can be used singly or in combination. The most useful display is the electromechani cal register plus a log counting rate meter.


1.3

Detectors

BFj Pulse Chamber. A typical pulse chamber containing BF3 gas is shown 1.31 in Fig. 5, and its characteristic curve is given in Fig. 6. This chamber is similar to a current ion chamber except for the wide electrode spacing which keeps chamber capacity low and improves pulse resolu METAL TO GLASS SEAL The J^-in .-diameter center elec tion. trode keeps the field strength relatively GLASS GUARD OR SHIELD RING low so that no gas amplification is ob The normal voltage supply is tained. 1,500 to 2,000 volts, electronically regu lated, with about 0.005 volt ripple. This chamber has a collection time of 0.2 X 10-* sec with 1 atm of BFj. Using B'°Fj gas filling improves sensitivity. Maximum counting rate is 150,000 counts per second. The n,a reaction in B10 pro COLLECTOR , ELECTR0C€-*DIA. duces 4 X 10~4 volt pulses in a normal circuit. The y radiation produces pulses of 10-' volt for similar conditions. The factor of 400 is available for discrimina , /OUTER The counter efficiency is about 10 tion. V ELECTROOE-S'DIA per cent, or 10 counts for every 100 neu trons entering the sensitive region. The chief disadvantage of this chamber is that the pulse height observed is a function of how far from the collection electrode ion Fia. 5. B'°Fa pulse chamber. ization takes place.* 1.32 B10F3 Proportional Pulse Chamber. A typical B10Fj counting chamber is shown in Fig. 7. The principal difference between this and the pulse chamber of Fig. 5 is the size of the center electrode, in this case about 0.47-mm wire. The advan tage is a resultant high field strength at relatively low voltages which accelerates the electrons from a collisions producing secondary ionization or gas amplification. A typical curve for this type of counter is given in Fig. 8, which also shows the regions of interest for various types of counter operation. Gas amplification of 10* to 10* is Values of amplification above this approach the typical for proportional counters.3 Geiger-Muller counter operation. The proportional counter is preferred over a simple pulse chamber, because pulse height is nearly independent of the distance from the collector at which ionization takes place. This produces shorter pulses, shorter resolving time, and higher count ing rates and requires lower external amplification.

8-54

CONTROL

OF REACTORS

500

Fig.

C.

1000

B'"Fj

[Sec. 8

2000

1500

COUNTER VOLTAGE pulse-chamber characteristics.


-GUARD ELECTRODE

I DIA

-COLLECTING

Fio.

7.

B'°F3 proportional-counter tube.

ZERO GAS AMPLIFICATION

0.6

,

0.8 RELATIVE

Fio.

8.

OUTER ELECTRODE

WIRE

GAS

1.0

AMPLIFICATION

1.2

COUNTER VOLTAGE

Counter pulse height vs. supply voltage.

1.4

Sec. 8-3]

CONTROL


8-55

DISCRIMINATOR BIAS SETTING 9. Effects

pulse height at output of discriminator.

of nonuniform

\

s

TT CO-AXIAL LENGTH

CROSS

SECTION

CABLE

TO SUIT

DIAMETER. COATING 10.

'/2"

0.014" APART,

NEUTRON ELEMENT. STRIPS ARE WIDE AND ROLL TO

OF TYPICAL

FISSION CHAMBER

Fio.

- CONNECTOR HOUSING

r— TUBE FOR FILLING ^GAS

r— NEUTRON SENSITIVE ELEMENT CONTAINER

A

Fig.

-90% U-Z35 Fission counter.

direct func and active volume.* Neutron counter efficiency being tion of active volume, the sensitivity of counters to neutrons in the presence of radia tion several small-diameter counters are higher (discrimination improved) one in The optimum counter design grouped together and operated in parallel. which the plate spacing very close to the a range for its particular gas pressure. Pulse chambers are also made with B10 coating on both the center and outer electrodes and filled with argon or helium gas. These counters are less sensitive than discrimination. B10Fj-gas-filled chambers by a factor of 10 or more but arc useful for 1.33 fission pulse chamber makes use of the ioniza Fission Pulse Chambers. These tion resulting from fission fragments. shown in Fig. 10. typical chauiber chambers have a counting efficiency of about 90 per cent, but because of the a back-

if

See Ref. See Ref.

4, 3,

«

is

t

A

A

7

a

is

is

is

is

y

a

to gas pressure

t


In Fig. 9 a curve is given to show the effect of nonuniform pulse height on the A figure of merit for any counting chain is the ratio of maxi entire counting chain. mum to minimum discriminator bias setting for no change in counting rate. Point A shows where the external circuit noise itself triggers the circuit, and point B where A pulse chamber where the the pulse height discrimination bias setting is too high. pulse height variation is large (as for the chamber of Fig. 5) would give the dashedline break at B'. Counter and circuit must be designed together for best performance and rated or tested together to determine their usefulness. The effects to be considered in pulse are space charge, electron attachment, and the like.4 The counting chambers It is proportional efficiency for a proportional counter is between 10 and 20 per cent.

p. 53. p. 218.

8-56

CONTROL

OF REACTORS

[Sec.

8

1.4

Pulse Amplifiers and Counting Units

8

pp. Sec Ref. and 9. Sec Ref. 5. p. 704. 2,

*

is

is

is

A

a

s

is

a

is

f

is

is

is

is

is

ft

is

is

is

a

it

it

is

ft

is

is a

is

is

4

The amplifiers and counting units shown in Fig. and used with pulse chambers for reactor operation are available commercially in many forms and have been described in pertinent literature The following notes are intended as a guide to application showing important factors to be considered. 1.41 The pulse preamplifier functions to reproduce the pulse Pulse Preamplifier. It called a linear preamplifier5 and commonly referred to com shape faithfully. combined The AIA amplifier mercially as an AIA or an AIB preamplifier. used for The AIB has no gain and amplifier and impedance-matching unit. Pulse distortion increases as the cable length between impedance matching only. of cable can be used, making About 10 to 12 increased. chamber and amplifier can be repaired, possible to locate the preamplifier outside the reactor shield where remote point. but not at matched to a low-impedance coaxial cable, which in The preamplifier output terminated in its characteristic impedance. In this way the preamplifier turn connected to the linear amplifier at the control desk where the at the shield face long but may be In practice this cable usually about 200 set up. display unit run over any distance necessary. Signal and Noise Considerations. Amplification ahead of the clipper or 1.42 differentiating circuit produces a better signal-to-noise ratio for the entire chain. limited by reduced.* The gain of the preamplifier Noise due to the input resistor pulse pile-up due to the pulse tail.' The pulse amplifier normally used to amplify the 1.43 Linear Pulse Amplifier. to Its function called a linear amplifier, pulses to between 10 and 100 volts provide a pulse proportional to the input pulse for use in the discriminator circuit of the type generally classed as widewhich follows in the chain. This amplifier somewhat determined by the The output pulse shape frequency bandpass unit. wide bandpass sec. rise time of 10 passband because the typical pulse has If high sensi amplifier readily passes circuit noise originating at the input stage. lowered to reduce the effect tivity desired, the upper frequency of the passband of the noise component. t


7

in

is,

I:

ground the useful sensitivity is lower by a factor of 10 than B10Fj-filled chambers. This effect is demonstrated in Fig. 11. The background a is the chief noise con tribution and limits the over-all range in which the counter can be used. FLUX 10nv The chief advantage of fission pulse cham NOTE I bers is the relatively good neutron counting in a y background as high as 1000 r/hr. They can be subjected to flux well above the counting region with only a small initial change in sensitivity. 1.34 Radiation Damage to Pulse Cham bers. Pulse chambers containing P»10Fj or B10 coating tested prior to and after being subjected to high neutron or y radiation show 1.6 2.0 2.4 1.2 marked changes in characteristic curves. 0.4 0.8 BIAS SETTING DISCRIMINATOR Experience shows that this effect is notice able in neutron flux of 10* and in y of 10' NOTE MULTIPLY ORDINATE BY 128 TO OBTAIN ACTUAL COUNTING RATE r/hr. The neutron flux varies a factor of 10* Fio. 11. Discriminator output for fission or more in power reactor applications and in chamber. normal installations may go well above 10s. It therefore, necessary to shield the cham ber from both and neutrons after the startup period either by moving shield around the chamber or by drawing the chamber out of the field.

Sec. 8-3]

CONTROL


8-57

The discriminator is usually included as part of the linear-amplifier circuit. It consist of a simple resistance-capacitance network in which time constants can The time constant is made longer to readily be changed with a selector switch. increase sensitivity and shorter to obtain higher discrimination. The setting in a particular application is adjusted to suit the radiation conditions encountered. For this reason absolute counting rates are not considered a useful reactor-control param eter. For reactor control the change in counting rate is the important quantity may

observed.

The ideal discriminator develops a standard output pulse whenever a sufficiently input pulse is received. It has a discrete and stable threshold, accepts short pulses, has a high input impedance, and does not easily overload.* 1.44 An aural display is considered necessary in connection Counter Displays. with reactor control. The operator should watch the continuous-current instruments but listen to the counters. Although a speaker can be used to make a popping noise at low counting rates, a scaler which clicks as it indexes serves both as aural and as visual indication. A mechanical scaler, which saturates at about 50 counts per second, can be employed up to 104 or 10s counts per second by using an electronic high


scaler,

t

One type of electronic scaler in common use has eight binary or "scale of two" units to include 256 counts in each scaler cycle.' A switch is provided so that the mechanical scaler can be connected into whichever binary circuit output is counting at a suitable rate. Decade scalers are also used in a similar way having three or more decades, each decade displayed with 10 neon lights, and a switch for changing the mechanical scaler connection between the outputs of each decade. 1.45 A useful display is the counting rate, which quantity Counting Rate Meter. is frequently recorded as an aid to operation. The pulse output of the discriminator is first applied to a pulse-shaping circuit or "univibrator" unit which puts out a uniformly shaped pulse, t The pulses are then applied to a condenser which is bled through a resistor and meter. The voltage of the condenser is proportional to the counting rate. A switch is used to change the bleeder resistance value and select The necessity for range changing is eliminated by feeding ranges of counting rate. the current from a fixed resistor-capacitor combination into a thermionic diode whose This voltage is then displayed output voltage is the logarithm of the input current. and calibrated as counting rate over four decades.* 1.5

Neutron-sensitive

Current Chains

The important instruments used in reactor operation are ion current chambers. The useful range of pulse chambers is usually exceeded as soon as the reactor is The various chains of units used with neutron-sensitive critical or above critical. ion current chambers is shown in Fig. 12. The current from a chamber fed directly to meter movement relay with a 20 X 10"'Current measurement in the amp full scale forms a reliable trip circuit element. power range (Fig. 1) with a galvanometer and shunt is useful, since the power can be The galvanometer being a supplied by a battery making the unit self-contained. low-impedance device, the chamber impedance (described later) is not a consideration. An eighWlecade range can be covered with an electronic amplifier either with or without feedback, the output being displayed on a recorder. The use of a feedback amplifier improves the speed of response by nullifying some of the input time constant A time con resulting from cable capacitance across the amplifier input resistance. stant of 10 sec is reduced to 0.001 sec with an amplifier forward gain of 1,000 in a feedback circuit with unity gain. A typical circuit for this type of amplifier is shown in

Fig. 13. For chamber current

as low as 10~16 amp a vibrating-reed electrometer is used An electrometer is not useful below 10-13 amp with an indicator or recorder display. • See Ref. 6, p. 203. See Ref. 8. p. 213. See Ref. 6, p. 87.

t

J

8-58

CONTROL

OF REACTORS

[Sec.

8

in reactor applications. Statistical variations in current from the ion chamber are The upper range is set by the maximum chamber output of large below nu = 10. The elec about 10~* amp. Standard electrometer units are commercially available.' trometer does not cover the entire range of 10*. Two instrument chains are necessary (Fig. 1, chains 3 and 4). Normal practice is to use a micromicroammeter,10 which is a chopper-type amplifier, in the range 10"* to 10-4 amp for full-scale deflection. Both the electrometer and micromicroammeter require range change switching. A convenient instrument is a logarithmic amplifier such as shown in Fig. 14 plus a recorder or meter to show about eight decades of flux change. Another circuit for a logarithmic amplifier is shown in Ref. 9.* A feedback logarithmic amplifier is used to some advantage, especially in experimental work where a short input time constant is important.11 The slope of the log of flux curve vs. time is denned as inverse reactor period. A differentiator attached to the output of a logarithmic amplifier supplies current RELAY (TRIP CIRCUIT) 20 x I0~6 omp SHUNT ATTENUATOR SWITCH Ronqe 300-1 D.C.

GALVANOMETER (CURRENT) 7 sec. period

RELAY, METER,

.

OR RECORDER


AMPLIFIER

(CURRENT) RELAY, METER,

FEEDBACK

OR RECORDER

AMPLIFIER

(CURRENT) METER OR RECORDER

REED ELECTROMETER

VIBRATING

(CURRENT) METER OR RECORDER (LOG CURRENT)

LOGARITHMIC AMPLIFIER

LOGARITHMIC

DIFFERENTIATOR

AMPLIFIER

RELAY, METER, OR RECORDER (PERIOD)

Fig.

12.

Current channels.

inversely proportional to reactor period. This in turn is a measure of reactivity, the The period is displayed on indicating basic control parameter for a nuclear reactor. and recording instruments with a zero at the center representing an infinite period A relay (zero reactivity) and calibrated to +1- or + 0.1-sec period at the extremes. operated from this output signal is used as an overpower trip or reactor shutdown signal. The trip setting is normally at 5 sec but may vary to much lower values in a particular application, such as with a boiling-water-reactor application where the normal rapid-signal variation is between 2 per cent and 5 per cent of the total A typical period circuit is shown on Fig. 15, current representing power operation. and another is shown in Ref. 9.f 1.61 The normal range for current ion chambers Neutron Ion Current Chambers. is 10~14 to 10~4 amp. The 10~4-amp figure corresponds to 6.25 X 104 electrons/sec. Therefore, at the lower figure of 10~14 amp the over-all signal-to-noise ratio for the entire measuring chain is low and becomes the limiting factor. The upper value of to materials on a normal chamber of 10~4 is set by considering radiation damage * See Ref. 9, AT46A M1C-22B. t See Ref. 9, AT-44A.


8-59


8-60


»

8-61

8-62 COLLECTOR

CONTROL

OF REACTORS

[Sec.

ELECTRODE INSULATOR

HIGH VOLTAGE

AMPLIFIER

ELECTRODE

ROUNDED GUARD RING

" Fig.

16.

HIGH

VOLTAGE

GROUNDED

SHIELD

SUPPLY

Guarding for chamber insulators.

-H.V. ELECTRODE

COLLECTOR ELECTRODE CO-AXIAL


8

CONNECTORS

FEED THROUGH GLASS INSULATOR SEALED TO BRASS MOUNTING RING -SOLDER

L_. L,

-LAVITE INSULATOR

HOLE IN PLATE

± rl

—

TL±" 4

IONIZATION CHAMBER B,0F3 GAS I ATM

Fig.

17.

Neutron-sensitive ion current chamber.

1,000 cm8, with a normal 10~" amp/mi sensitivity. A current as high as 5 X 10~14 can be obtained by using a larger chamber but is not considered practical for most installations. Chambers 2 to 3 ft long can be placed close to the reactor core parallel to the core axis in special cases to obtain somewhat higher than normal current signals. The advantage realized from this arrangement is questionable, since the flux at the chamber will vary along the length.

Sec. 8-3]

CONTROL

INSTRUMENTS AND DRIVES B^Fj IONIZATION

8-63

CHAMBER o

I

--NV =2.4> I09 UJ

a u

0.

—1

J—

A

2

UJ or or

L

3

o

K

—I


1

Fig.

/

^NV

=2 4x 10s

CHAMBER VOLTAGE 18.

Collector current vs. applied voltage, ion chamber.

The principal problem encountered in trying to obtain higher current from small chambers is the high voltage which must be applied in order to saturate the cham ber. A measure of good chamber design is the inverse of the slope of the plateau curve AE/Al called the dynamic cham ber impedance in a radiation field. The chamber impedance should be 100 times as high as the measuring circuit input im pedance to maintain 1 per cent accuracy. Characteristic curves of chambers are given later in terms of this dynamic im pedance, which is both a function of volt age applied to electrodes and an inverse function of current. 1.52 Chamber Design. Current ion chambers generally used employ the (n,a) reaction in B10 for ionization. The B10 can be in the form of a B10F3 gas filling or coated on the plates. Where low ion cur rent 10-10 to 10-14 amp is to be measured, the chamber insulators should be guarde Figure 16 shows both the guarding meth and the normal connections for current measurement. Ri is the measuring Jpircuit input resistance; Rt is the collector leakage resistance to ground which aWunts Ri; Ri is the high-voltage plate-lA-collector resistance. Where the insulator is guarded, as shown, current between the

B'°F, IONIZATION

500

1000 CHAMBER

Flo.

19.

CHAMBER

1500

2000

2500

VOLTAGE

Dynamic impedance, ion chamber.


*H4

20. Fission

chamber.

FISSION CHAMBER

CHAMBER V0(-TS I. 31. Current vs. voltage, fission chamber.

1000

CONTROL

Sec. 8-3]


8-65

plate and ground does not reach the collector, the collector itself being The dynamic impedance (operating in radiation) of the ground potential. chamber is not shown. It would shunt Ri. Guarding is not necessary in the current range above 10~* amp. The handling of ion chambers, i.e., replacement, repair, and inspection, is simplified if the materials of construction have low intensity or short-lived neutron-induced Aluminum is recommended, with iron and brass as second choices. radioactivity. Cable connectors are usually copper and should be located on the chamber housing for easy replacement. high-voltage

close to

tO«l

I 100

0

I

200

I

300

I

I

400

500

I

I

600

700

CHAMBER VOLTS

Fig.

22.

Dynamic impedance, fission chamber.

Insulator materials suitable for use in chamber construction are quartz, lavite, and oolystyrene. Teflon insulation in cable connectors should be avoided. 1.63

Testing Neutron Ion Current Chambers.

Chambers of the various types are

A few typical chambers of simple construction are shown ivailable commercially.* lere along with their characteristic curves. The important design . considerations ire a dynamic impedance suitable for the circuit application, leakage resistance above I017 ohms, and a B10 coating which does not flake off.

In Fig.

chamber with parallel plates is shown. The plates could be The feature of construction is the simplicity sensitivity. electrodes and plates, the high-voltage electrode serving as a mounting The collector plates are held in or both high-voltage plates and collector plates. therefore, not accurate ilace on lavite insulators. The chamber iB not guarded and • See Kef. MIC-3A. 17 a B10Fa

B10 to increase

is,

oated with f mounting

9,


FISSION CHAMBER

8-66

CONTROL

OF REACTORS

[Sec.

8

below about 10-10 amp. The plateau curve, collector current vs. applied voltage, is shown in Fig. 18, and the counterpart of this curve, chamber dynamic impedance, The curves of Fig. 19 are the measured characteristic important is shown in Fig. 19. The sensitivity is 2.8 X 10~u amp/nv. to circuit design.

-SIDE RETAINING PLATE l%x&4*li POLYSTYRENE

UPPER PLATE RETAINER | %>x POLYSTYRENE

V

i^xjg GROOVES

/iX/ÂND^2x^GROOVES - PICTURE FRAME GUARD RINGS

A PLATE

pUPPER SUPPORT

tHGH VOLTAGE COLLECTOR -HGH VOLTAGE GROUND

GLASS TO METAL SEAL O'RING

2^x5x^2

HELIARCWELD


COAXIAL CONNECTOR

B10 COATING i PLATE RETAINER | %x 2 x% POLYSTYRENE ^x^ GROOVES

WELD

FILLED WITH I ATM ARGON.ALL ALUM CONSTRUCTION. PLATE SPACING 5^"SENSITIVITY 1.6AMR/NV. COATING 0.5 MILLIGRAM CM^S10

Fig. 23. Simple compensated ion chamber. COMPENSATED

70

IONIZATION CHAMBER

£ o.

i1

'

1 NV=3.<5xl09

<

60

5

1 5

40

30 N EUTRO MSENS

tive m

If

20

s NV=3

/

10

0

100

200

300

400

500

CHAMBER

6 x10s

600

700

800

900

VOLTAGE

Fig. 24. Current vs. voltage, compensated ion chamber.

A typical fission chamber, an ion current chamber in which fissionable material is coated on the plates, is shown in Fig. 20. The filling is nitrogen at 1 atm; the plate The coating is 0.2 mg/cm2 of uranium. The sensitivity is 1.1 X 10_M amp/ni'. plateau characteristic is shown in Fig. 21, and the dynamic resistance in Fig. 22. A simple compensated ion chamber is shown in Fig. 23. The plates are rectangular and mounted in grooves in the polystyrene side support. Between each pair of plates is a picture-frame plate which forms a guard ring for the insulator. The

Sec. 8-3]


CONTROL

8-67

wiring and coating plan is shown in the figure. The sensitivity is 1.6 X 10_u arap/nu. The filling is 1 atm of argon gas, and the coating is 0.5 rag/rm1 of B10. Figure 24 shows the plateau, and Fig. 25 the dynamic impedance. 1.64 Testing a Neutron Ion Current Chamber. The chambers to be used on a nuclear reactor must be tested with a neutron source prior to installation. The

chambers shown in Figs. 17 and 20 present no internal polarity problem, although a proper polarity is usually marked. As long as the polarity is proper for the external measuring circuit, neutrons will produce an up-scale read ing. Thus any source of ionizing radiation is suitable for checking the circuit operation. For a compensated chamber an error in con necting the electrode will be found only by testing with a strong neutron source. A re actor test source is preferable, but a radium beryllium neutron source may be used if the neutrons are thermalized with water or par affin around the chamber. When the chamber is being used at currents of 10~* amp or higher and the measuring circuit input resistance is better than 10s ohms, it is advisable to check the dynamic impedance. This is particularly true if absolute accuracy of measurement is desired. The basic meas uring accuracy depends upon a high ratio of dynamic impedance to circuit input imped A circuit used to measure dynamic im ance. pedance is shown in Fig. 26. Ei is the mean high voltage applied. i?2 is a bucking volt-

COMPENSATED CHAMBER


IONIZATION

HIGH VOLTAGE PLATE

/GROUNO

SHIELD

COLLECTOR

200 300 CHAMBER VOLTAGE

100

PLATE

400

Flo. 25. Dynamic impedance, pensated ion chamber.

com

Flo.

26.

Circuit for measuring

dynamic

imped

ance.

with adjustable Rt, is used to make /i = h with E> =0. Ea is The square-wave voltage generator usually with a peak to peak of 30-volt change. meter Aft reads the Al which results from E3. This meter is usually an oscilloscope with a calibrated deflection. The average scope deflection is observed as A/. This method eliminates any error due to drift in neutron source intensity. Curves of Figs. 19, 22, and 25 were obtained in this way. age which, in series

a

2

CONTROL

DRIVES

Ether rotary or linear motion is required from a mechanism to move control rods in a nuclear reactor. In most designs linear movement is required. Water-cooled reactors operate at temperatures up to 625°F and pressures as high as 2,000 psi. reactors are pressurized with helium gas to slightly above atmos Liquid-metal-cooled phere. In either case the mechanisms are frequently kept in the external atmosphere and either linear or rotary motion is transmitted through seals to drive the rods.

8-68

CONTROL

OF REACTORS

[Sec.

Completely sealed drives are available in which all components operate at atmosphere, but they are costly and cannot be readily serviced.

reactor

Drive Specifications

10~4 Sfc/sec and 1. Control speed should be positive or negative at a nominal adjustable in a 4:1 continuous range to allow final selection of the control speed after the reactor is built and the control rods are calibrated. 2. Safety release or k reduction should start in 10 to 50 msec after a shutdown signal is received. of 3. Speed of k reduction should be the equivalent of 0.9 of the acceleration gravity acting through the travel distance. A spring may be used to obtain rapid Deceleration acceleration at about 5 g, in which case the velocity should be limited. at the most effective rod position should take place in 10 per cent or less of the total rod travel. with the mechanism should be to + 2 X 10-6 6k. This may be 4. Positioning called position repeatibility and fixes the corresponding gear backlash to about 10 per cent of this value. Motors should be braked to stop the drive in about 0.2 sec after power is removed. When placed at a position, the rod should hold or remain fixed and not drift under the effect of forces acting on the rod, not supplied by the motor. 5. Position indication should be continuous, and the position-measuring element should be attached directly to the control-rod extension used to move the control rod proper. The accuracy of position measurement should correspond to repeati bility given above. Limit switches should show at least the all-out and all-in position Knowing where the rod is under all and if possible several intermediate positions. circumstances is so important that more than one method of indication is advisable.

2.2

Drive-design

Considerations

However, Design principles are best discussed by examining typical mechanisms. general design considerations should be noted. 1. The mechanism itself is connected to the control rod proper by an extension, usually a rod 10 to 20 ft long. Alignment of the rod in place, assembly, and repair Invariably joints are required in this rod, and must be considered in the design. where possible welded joints are preferred. Assembly joints should be threaded and A section of the rod exten Securely bolted flanged joints are advisable. pinned. sion or the control rod itself should be designed to rest on a mechanical stop so that the rod remains in place in the core without the mechanism attached. 2. Overtravel or extra movement should be provided for about 5 per cent of the The stroke should then be set with limit switches specified rod travel in the core. in the rod drive motor circuit or with adjustable mechanical stops, whichever the mechanism design. electrical, pneumatic, or hydraulic motors are 3. Direct- and alternating-current easiest to apply, suitable for mechanism driving power. Electrical direct current used to stop the having a simple means of speed control. Dynamic braking motor, which operated with the field always energized. Two-phase servo motors again having a high armature torque-to-inertia ratio are useful, and speed control simple. This class usually needs no braking method, the gear ratios being selected eliminated. Three-phase a-c motors are needed for applications so that motor coast hp. Solenoid brakes are required for stopping this class requiring more than low. motors because the armature torque-to-inertia ratio is

is

fit

some important

is

of

is

is

is


2.1

8

2.3

Seals

Shaft Seals. The seals used on the drive shafts 2.31 are commonly referred to as rod plunger packings or shaft Commercially available floating metal-ring seals which down principle are suitable and give satisfactory service

of a control-rod mechanism packings.1,is work on the pressure break when used according to the

Sec. 8-3]

CONTROL


8-69

manufacturer's specifications. The leakage rate of these seals is 0.2 to 1 gpm at 1,000 psig. In application, the sealed shaft may either rotate or move linearly. A linear Necessary seal allows locating the entire drive mechanism outside the reactor shield. maintenance can then be performed readily. Friction in this type of seal is about 100 times greater than for rotary motion. This advantage of the rotary seal must be balanced against the difficulty of performing maintenance of the moving parts which convert rotary motion from the sealed shaft to linear motion for the rod. Zero net leakage is obtained by placing a shaft packing outside the rotary seal and pumping the fluid which leaks through the labyrinth back to the reactor system. 2.32 Bellows Seals. Where the reactor pressure is 150 lb or less, commercially available bellows may be used to seal the drive shaft. In most cases the manu facturer recommends holding the compressibility to 30 per cent of the extended length for reliable long-term performance. Where rod movement is short, 10 to 20 in., the use of bellows is feasible. Improved designs are being tried which allow up to 70 per cent compressibility with reasonable service-life expectancy.


2.4

Typical Mechanisms

The functions demanded of control-rod drives make careful mechanical design mandatory. No element or combination of elements should be relied upon unless a reasonable operational test is given to it. On the other hand simplicity is of utmost The drives shown in the fol importance. lowing figures are typical of designs which GEAR MOTOR have been used successfully. Designers frequently differ on exactly how a unit should be manufactured, and the MAGNETIC strength and sizes of each part must be tai CLUTCH lored to a particular application. The drives are therefore not a catalogue of methods but were chosen to demonstrate good design principles. 2.41 Ball-nut Drive. Figure 27 is a drive in which the mechanical advantage was obtained in a ball-nut jack. Rotary motor motion is first converted to linear motion, then back to rotary motion. In the appli cation, swinging rods were raised through a 55° rotation to a horizontal position above the core. As the screw is turned, the ball nut moves upward and compresses the spring to the position shown. The magnetic clutch provides 40 in.-lb torque at 9 watts d-c DASH POT The motor is direct current, % hp, power. geared 850: 1. Fully compressed, the spring supplies a 1,000-lb force which drives the 90 per cent efficient ball nut backward when the clutch is released. The clutch releases in 0.01 sec. This quick return motion was completed in from 0.25 to 0.45 sec for 55° Fio. 27. Control-rod mechanism, ballrotation on a set of six mechanisms tested. nut drive. Desirable features are (1) the continuous application of a reliable return force, (2) continuous position indication directly from the rod drive shaft for both slow control and fast rod-return movement, (3) no reset time required, the clutch can be reenergized and the rod raised immediately because the mechanism has no latch to make up. In this application the reactor rod seal was a simple rod packing, the reactor being pressurized at a few inches above atmosphere with helium.

8-70

CONTROL

OF REACTORS

[Sec. 8

In Fig. 28, the principal mechanism component is Rack-and-pinion Drive. 2.42 The force for quick return action is supplied by the drive. the rack-and-pinion The air cylinder serves as a velocity limiter 150-psig air from an accumulator tank. The force and as a dashpot to stop the rod at the end of the quick return stroke. of deceleration is normally large because control rods weigh upward of 100 lb, and the decelerating force is controlled in this way. A J|-hp two-phase a-c gear head motor was used in the application. The mag The 150-psi air supply through an accumula netic clutch developed 500-in.-lb torque. tor tank drove the rod through a quick return stroke in 0.2 to 0.3 sec. Position is measured by gearing a synchro pair directly to the rod drive shaft, and a set of cams AIR SUPPLY-


AIR CYLINDER FOR FAST RETI

-POSITION v TRANSMITTER

CONTROL

ROD

FlO. 28. Control-rod mechanism,

rai;k-and-pinion drive.

operates the limit switches. This method of installing limit switches eliminates wiring along the rod mechanism. The advantages of this drive are the same as those for the drive shown in Fig. 27; gravity is acting directly on the portion of the mechanism having the lowest mechani cal advantage. The rod, therefore, falls freely even if the air supply is lost. Figure 2914 is similar in many respects to the drive of Fig. 28 except that a rotary seal is used. Advantage is taken of the low friction of a rotary seal, and a 75-Ib A clock spring is applied to the rotary drive shaft for the fast rod return stroke. The details of helipot is geared directly to the drive shaft for position indication. the seal are also shown on the figure. The seal is 6 in. long; the clearance between the shaft and floating rings is 0.0003 in. The rated leakage is 10 lb/hr at 1,200 psig. The advantages are similar to those for the drive of Fig. 28. Since the mechanism runs in water, the bearings and mechanical parts are water-lubricated." Details of materials used for various parts are shown.

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CONTROL

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14,

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2.43 Piston Drive. The drive of Fig. 30 is typical of the combination mechanical and pneumatic or hydraulic piston drive. Basically, a hydraulic or pneumatic piston drive does not meet the standstill requirements (Art. 2.1, Item 4) for a controlrod drive. Some method of controlling positions is necessary. In this drive a mov able stop supplies this requirement. The pneumatic piston and cylinder could be replaced with hydraulic units. In the application the stop is attached to a pair of nuts moved by a lead screw. The rod proper is driven by the 50-psi air supply applied to the lower portion of the drive cylinder. This forces the rod against the movable stop. The drive motor is J4 hp alternating current with a gear A solenoid brake is needed to stop head. PRESSURE Fâ the motor when power is removed for ac EQUALIZER^ curate positioning of the stop. A 1 50-psi air supply is available to the upper por tion of the cylinder to drive the rod in the A three-way sole quick return stroke. noid valve either vents or connects this cylinder to the supply of air. The lowpressure side is vented through a relief valve set at 60 psi. The quick return stroke was completed in about 0.2 sec; the solenoid valve operated in 0.05 sec. The rod position is indicated by a syn chro transmitter attached to the lead screw and nut drive. A limit switch travels with the stop to show that the rod is in contact with the stop. A second -HIGH PRESSURE limit switch actuated by the drive rod FOR FAST RETURN itself shows that the rod is fully down. An electrical interlock prevents applica tion of 50-psi air and applies the 1 50-psi /-LOW PRESSURE air if the "in-contact" limit switch is opened. This higher pressure air returns the rod to the safe position of greatest k reduction. Reactor steam pressure was applied to a balancing cylinder above the drive so that the forces due to reactor pressure were balanced. mechanism, pneumatic Flo. 30. Control-rod The chief advantage of this drive is drive and return. the large force which can be developed If the air pressure fails, the drive acts as a backup inser for the rod return stroke. tion method. 2.44 Figure 31 shows a drive designed to operate from under Inverted Drive. No pressure balancing is used, and the motor must drive the rod neath the reactor. against the pressure force of the reactor. The drive rod which passes through a linear The neutron-absorber section is pushed up seal is attached to the control rod proper. The drive rod is attached to the nut with a roller carriage. The out of the core. rollers are forced into the slot by a guide attached to the frame when the nut moves to As the latch enters the slot, a spring is compressed and a the lower position as shown. solenoid latch retains the rollers in the slot. The rod is then attached to the nut. In the application, a J-^-hp three-phase a-c motor is used with a 30:1 ratio gear head. The slip clutch is used to prevent damage to the mechanism should the drive limit switches fail. An oil-filled dashpot 5 in. long is used, the total rod travel being The rod passes through a labyrinth-type seal backed up with a commercial 48 in. Water leakage is drawn off between the seal and packing and returned rod packing. gpm at 600 psig. The seal to the reactor water system. The leakage rate is between 0.0004 and 0.0010 in. ring clearance is


8-72

Sec. 8-3]

CONTROL

8-73


For the quick return stroke the solenoid is deenergized, opening in about 0.010 sec. The roller latch is forced out of the slot, and the rod is forced down by the reactor The spring develops about 1,000 lb force and assists pressure, gravity, and a spring. At atmospheric pressure the 48-in. quick return travel in accelerating the rod. requires 0.5 sec; at 600 psig, 0.25 sec. The speed during the quick return stroke is limited by the piston in the thimble above the drive to 16 fps. The seal housing is attached to the thimble with a spool piece, which makes assembly The control rod proper is inserted from above the reactor, of the mechanism possible. and the drive rod is attached to it at the bottom of the thimble. The seal is then placed over the rod and bolted in place on the thimble flange. CONTROL THIMBLE

CONTROL ROD EXTENSION

DIAPHRAGM STELUTE*6 2 MICR-INCH FINISH

GUIDE BUSHING

STELLITE*6

FLOATING RING STELLITE*3 2MICROHNCH FINISH

SEAL DISCHARGE

-2" H20

LAPPED SURFACE

LANTERN RING STELLITE»6

■SEAL HOUSING STAINLESS STEEL

"0"RING SEAL

SIRVENE GARTER SPRING SEAL WITH STAINLESS

STEEL CHROME PLATED SEAL SHAFT


STAINLESS

MEEHANITE

ROD

STEEL CASE

HOUSING

1040 HEAT TREATED SCREW SHAFT

ROLLER CARRIAGE PUSH SOLENOID Jie STROKE 4.3LBS. THRUST® 115VOLTS60 C r. AC 3AMP. FOR MAX.

SAE 52100 ROLLERS ROLLER RESET GUIDE

STROKE

BRONZE NUT

EXTERNAL OIL SCRAM

-MAGNETIC BRAKE -GEAR MOTOR J$H.P-I7I IN.LB. TORQUE

3^-60

220V-

ANGLE DRIVE

CY. AC

SYNCHRO GENERATOR

SLIP CLI

- DRIVE

Fio.

31. Control-rod mechanism,

SPRING

SHAFT

drive designed to operate underneath

the reactor.

In Fig. 32* a completely sealed drive is shown. The 2.46 Canned Motor Drive. drive has the desirable advantage of zero leakage of primary coolant to atmosphere without the necessity of leakage collection imposed by a sealed drive.

In operation the unit is basically a lead screw and nut drive in which the lead screw moves through a stationary nut. When the motor stator coils are energized, The the rotor segment arms move outward and the nut below engages the screw. motor stator field is then rotated electrically as in a three-phase motor, and the nut When the motor stator field is deener rotates, driving the screw in either direction. gized, the rotor segments are driven inward by springs and the lead screw is free to fall under the force both of the spring and of gravity. A magnetic extension of the drive screw passes through the position indicator coils, giving an indication of the rod position. This drive is especially useful in pressurized power-producing reactors. The chief disadvantage is cost. * Submitted by Westinghouse Atomic Power Division, Pittsburgh, Pa.

8-74

CONTROL

OF REACTORS

[Sec. 8

•*,W*,-i-.w.

SPLINE

MAGNETIC SECTION

SCRAM

SCREW EXTENSION

SPRING

SUPPORT HOUSING

TERMINAL

PRESSURE

BOX

WALL


COOUNG WATER INLET

AIR SPACE COOLING

PRESSURE

WALL

WATER JACKET

SEAL WELD STATOR

SUPPORT TUBE

LOWER HOUS1N8

THERMAL SLEEVE

STRENGTH WELD

SCRAM SHAFT

PRESSURE

VESSEL HEAD

... Fio.

32.

Control-rod mechanism,

completely

sealed drive.

Sec. 8-3]

CONTROL

8-75


Other drive types are under development but have not yet been proved. One popular idea is to utilize the cooling water for the reactor in a hydraulic piston applica The chief difficulty here is drift of the piston under the effect of forces on the tion. rod. This type would be well suited to safety-rod operation in which the rod was held ready for insertion by a fixed mechanical stop. Another type is generally known as a magnetic jack in which solenoid coils outside a pressuretight enclosure move magnetic grippers which are immersed in the reactor coolant. Finite steps are taken each time the rod is moved in the order of 0.1 in. When perfected, this type of com pletely sealed rod could be very useful in power reactor application. At present the chief difficulty lies in developing forces large enough to handle rods weighing several hundred pounds plus the force to compress a rapid return spring.

ACKNOWLEDGMENTS


The author received valuable assistance from J. A. DeShong in setting up the counter and counting circuit sections and from W. C. Lipinski in setting up the current chamber and circuit data. The resistance curves were taken from an unpublished report by W. C. Lipinski. The control-rod drive designs were the result of contributions as follows: Figures 27 and 28 —J. J. Dickson (Argonne National Laboratory) Figure 29— A. L. Bock (Oak Ridge National Laboratory) Figure 30 — H. V. Lictenberger (Argonne National Laboratory) Figure 31 — C. F. Bullinger (Argonne National Laboratory) Figure 32 — N. B. Dewees (Westinghouse Electric Corporation)

REFERENCES 1. Priraak, W., and L. H. Fuchs: Nucleonics, 13(3): 38 (1955). 2. Gillespie, A. B. : "Signal Noise and Resolution in Nuclear Counter Amplifiers," p. 12, Pergamon Press, Ltd., 1953. 3. Kroff, Serge: "Electron and Nuclear Counters," p. 51, D. Van Nostrand Company,

Inc., Princeton, N.J.,

1946.

4. Rossi, B. B.. and H. H. Staub: "Ionization

Hill Book Company, Inc., New York,

Chambers

and Counters," p. 90, McGraw-

1949.

6. Jordan, W. H., and P. R. Bell: Rev. Sci. Instr., 18: 705 (1947). 6. Elmore, W. C, and M. Sands: "Electronics," p. 125, McGraw-Hill 7. 8. 9. 10. 11.

Book Company, Inc., New York, 1949. Higgenbotham, W. N., J. Gallagher, and M. Sands: Rev. Sci. Inatr., 18: 709 (1947). Bousgnet, A. S.: Nucleonics, 4(2): 67 (1949). AEC Radiation Instrument Catalog 3, MIC-2c. MIC-15A and B. MIC-22A, MIC-22B, Technical Information Service, Oak Ridge, Tenn., 1952. "Williams, A. J., Jr., R. E. Tarpley, and W. R. Clark: D-C Amplifier for Zero and Gain, AIEE Trans., vol. 67, 1948. DeShong, J. A., Jr.: Logarithmic Amplifier with Fast Response, Electronics, (March,

1954). 12. Marks,

L. S. (ed.): "Mechanical Engineers' Handbook," McGraw-Hill Book Company, Inc., New York, 1951.

5th ed., sec.

13. Smaardyk, A.: "Summary of Shaft Seal Tests," ANL-5220, April, 1954. 14. Livingston, R. S., and H. L. Boch: Power Reactor Package, Nucleonics, (1955).

15. Smaardyk, A.: Water Lubricated Bearing Studies,

ASM E Paper

8,

p. 1000,

13(5):

27

55, SA42, June, 1955.

8-4

REACTOR SAFEGUARDS BY C. Rogers McCullough


1

INTRODUCTION

As soon as World War II was over, an urge developed to build reactors for research and the production of power. It was realized that nuclear reactors had certain peculiar characteristics which needed to be carefully considered in order to protect There was a fear the machines themselves, and especially the public. employees, that the new groups entering the field were not sufficiently experienced to be aware of all the pitfalls. It was thought that the construction and operation of a reactor without adequate care might result in a disastrous accident with large numbers of It was believed this persons injured and killed and widespread property damage. would set back the whole field to a very serious extent. Accordingly, in 1948 the Atomic Energy Commission established the Reactor Safe guard Committee, consisting of some of the leading experts in the field and headed by This committee was Dr. Edward Teller, to review all existing and new reactors. merged with the Industrial Committee on Reactor Location Problems on Apr. 27, On September 1953, and renamed the Advisory Committee on Reactor Safeguards. 2, 1957, the Atomic Energy Act was amended again and "Section 29, Advisory Com mittee on Reactor Safeguards,"12* was added, establishing this committee on a The essential provisions were that this committee would consist of a statutory basis. maximum of fifteen members appointed by the Commission for terms of 4 years each. The responsibilities given were quite broad: "The Committee shall review safety studies and facility license applications referred to it and shall make reports thereon, shall advise the Commission with regard to the hazards of proposed or existing reactor facilities and the adequacy of proposed reactor safety standards, and shall perform One member shall be designated such other duties as the Commission may request. The Committee must review all applications by the Committee as its Chairman." for testing and power reactors privately owned and issue reports which are made part of the public record. This latest action emphasizes the great stress placed on protecting the health and Under this act reactors must safety of the public by the Atomic Energy Act of 1954.' be regulated by the AEC to protect the workers in the reactor plant and the public. " Before a license The act requires licensing for any "facility which includes reactors. (or a construction permit) is granted, the reactor proposal is reviewed on the basis of detailed and complete technical information submitted by the applicant for a license to determine that the construction and operation of the reactor will not result in an unreasonable hazard to the employees or the public. The AEC has a Reactor Hazard Evaluation Staff whose duty it is to receive the technical information, study it, obtain corrections or additional data as required, and On routine matters and make an evaluation as to the amount of hazard involved. where standards have been established, this staff can make definite recommendations. In the case of major or controversial proposals and those required by law the data and evaluations presented to the Reactor Hazard Evaluation Staff are referred to the * Superscript numbers

refer to References


8-76

Sec. 8-4]

REACTOR

SAFEGUARDS

8-77

Advisory Committee on Reactor Safeguards for their evaluation and recommendations. Several groups are at work attempting to write standards and codes for reactor construction and operation to assure protection of the public. The industry is too new and reactor types too diverse to expect to get a code within the next few years, and each reactor will have to be judged largely on its own merits. A reactor is defined as any machine or device in which a nuclear chain reaction can take place (except a nuclear weapon). Thus, critical assemblies are included but not exponential experiments, although, since these can become critical by error, it is customary to notify the Reactor Hazard Evaluation Staff and the Advisory Committee on Reactor Safeguards of them as a matter of courtesy. The radiological problems of day-to-day operations are covered by regulations from other parts of the AEC and are not the responsibility of the Reactor Hazard Evaluation Staff. The criticality problems of fuel-element fabrication and transportation and of fuelThe element chemical-processing plants arc much less serious than those of reactors. responsibility for evaluating these hazards has been included in the duties of the Reactor Hazard Evaluation Staff. 2

NATURE OF THE PROBLEM


2.1

Fission Products

The fission process results in the accumulation of poisons (the fission products) which arc more toxic than any other materials of industrial significance. This toxicity is due principally to radioactivity. The manner in which the radioactivity damages the body varies with each particular element but that is discussed in Sec. 7-2. Table 1

Table

1.

Comparison of Toxic Substances in Air (Concentration in nig/m3)

Substance

Chemical

poisons:

Radioactive poisons:' Pu"» Sr'° +

Tolerance*

Y"

2.9< 0. I6< 1.5 X 10-' 1650 X I0"« 32 X 10-' 1.4 X 10-'

* Tolerance tor chemical poisons is defined as the maximum tolerable level for 8-hr /day exposure. the case of radioactive poisons tolerance is the maximum level which can be tolerated 8 hr/day.

In

compares the tolerance doses of several radioisotopes with those for a few chemical poisons. This table is presented mainly to impress the reader with the magnitude of the problem. It should be noted that the quantities of fission products differ greatly from the amounts of common chemical poisons encountered. Suppose we compare the amount of poison in a 1,000-M\v (of heat) reactor with a storage tank Assume that the reactor contains fission products in containing 250 tons of chlorine. fuel elements at equilibrium for 1,000-Mw operation for an average period of 1 year (Table 2). The strontium 90 alone in the 1,000-Mw reactor is more toxic than 250 tons of chlorine by factors as follows: Tolerance, 1.35 X

10»

8-78

CONTROL

OF REACTORS

[Sec.

One must remember that there are other fission products in addition also must be considered.

8

to SrM which

The problem, very obviously, is not only to shield the operating reactor so that the neutrons and y rays do not leak and hurt employees and the public but also to be quite sure that no appreciable fraction of the fission products ever escapes. This escape must be guarded against both when the reactor is operating and when it is shut down. Table 2 Sr»in

Toxicity, mg/mJ: Volume

of air contaminated,

Lethal


2.2

Chlorine in

reactor

storage tank

1.4 X I0<

2.3 X 10"

1.3 X IO-> 1.3 X IO-«

2.9

1.08 X 10" 1.08 X 10"

0.8 X 10" 0.8 X 10'

m»:

290

Nuclear Runaway

A reactor at steady state has its reactivity at a value of 1. The number of neutrons produced per unit time is exactly equal to the sum of the number of neutrons needed to carry on the chain reaction, absorbed in fertile material, lost by escape, or lost by parasitic capture. Suppose reactivity should be increased by removal of fertile material, reduction in escape, or reduction of parasitic capture. Figure 1 shows the rate of power rise for various amounts of excess reactivity instantly introduced in a graphite-moderated thermal reactor. These curves are only approximate, as there are various modifying effects in as complex a machine as an actual reactor. The curve does show that even in the case of a sluggish reactor such as a graphite-moderated one, once "prompt critical" (the delayed-neutron fraction = 0.75 per cent excess reactivity) is exceeded, the power rises very rapidly. Depending upon the kind of reactor being considered, a variety of things can happen. If the fuel elements are water cooled, the water may start to boil or be ejected from the channel. If the reactor is not immediately shut down, this will probably result in melting or vaporizing one or more fuel elements. If the reactor is of the type that has the same water acting as moderator and coolant, ejection of the water or even its boiling will result, in a decrease in reactivity which may shut off the chain reaction. Because of the very large increases in power which are possible, a nuclear runaway might result in melting or vaporizing fuel elements. Sudden heating of fuel elements could result in serious pressure effects owing to the sudden increase in volume due to steam, or molten or vaporized metal. These sudden pressures could result in rupture of the reactorcontaining vessel. Sudden changes in temperature of fuel plates may result in breaking of the cladding, Although rupture of the cladding warping or breaking of welds or brazed joints. in itself probably will release only relatively small amounts of fission products, if the "meat" of the fuel element is more corrodible than the cladding, its rapid dissolution could lead to serious consequences. For example, if all the fission products from the fuel elements are released, this may be a major hazard in itself. If these corrosion products block cooling channels, then a chain of events could ensue, resulting in a major accident. Warping of elements may result in important changes of reactivity, but the design should be such as to minimize this. Elements broken away from their Of more importance, probably, is moorings may also result in reactivity changes. the possible touching of plates or restriction of coolant flow, resulting in serious over heating or melting.

Sec. 8-4]

REACTOR

8-79

SAFEGUARDS

In passing, it is worth noting that the introduction of stresses repeatedly in a reactor, whether by mechanical shock or temperature changes, is not good practice. Thus, too sensitive a scram circuit may be bad rather than good. A reactor structure including fuel elements, spacers or grids, control rods, and container may fail by could be very serious. Fatigue fatigue on repeated stressing, and the consequences failures probably will only begin to show up some years from now, although there is already the suspicion that fuel cladding may be subject to this effect. If fuel elements containing appreciable amounts of metallic aluminum, zirconium, or uranium vaporize, they will most certainly react vigorously with water on contact. It is a moot question 1000

800

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400 300

k

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• FRACTIONAL INCREASE IN REACTIVITY.

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TIME-SECONDS

Fio.

1.

Effect on power level of

a reactor

of sudden increase in reactivity.

whether or not they will react with water if only molten. This is a highly contro versial point, requiring additional study. Reaction is thermodynamically possible and can be demonstrated under certain conditions. Whether or not these conditions are likely to exist in a reactor is speculative. However, if a major amount of metal reacts with water suddenly, the energy release from chemical reaction can be much more than from the nuclear reaction — a queer and amusing situation. One method of minimizing or avoiding the metal-water reaction is to use "already reacted" materials in the fuel element such as uranium oxide and metals such as stainless steel, which have little or no thermodynamic potential for reaction with water.

Another approach is to use liquid-metal coolants, such as sodium, sodium-potassium alloys, or lead, or organic coolants, such as biphenyl, terphenyls, etc., in connection with aluminum, zirconium, and uranium fuel elements.

8-80

CONTROL

OF REACTORS

[Sec.

8

The important feature in avoiding nuclear runaway is to be sure that the intro duction of reactivity is slow enough so that control elements can always counteract It is not enough to the introduced reactivity with an appreciable safety factor. depend upon one channel of control to reduce reactivity; several should be employed, and the rate of control should always be appreciably more rapid than the maximum Best of all is to have a reactor design possible rate of introduction of reactivity. which inherently compensates itself without instruments, control circuits, and control rods. Negative temperature coefficients (water moderated) and void compensation (boiling reactors) are examples of how this might be done. Aqueous homogeneous reactors have both a strong negative temperature coefficient and a void coefficient. It should be emphasized, however, that the analysis of these effects and their time constants is complex. An analysis of the interrelation of all the coefficients is posi tively required, as short-time instabilities might appear from too strong negative coefficients, which might otherwise seem desirable. Neutron-generation time — the average time from neutron fission to fission — is obviously of great importance. The shorter this time, the more care must be taken to avoid sudden increases in reactivity and the quicker and more reliable the counter acting mechanism. Fast reactors arc an example of devices with extremely short generation times and require inherent compensating features.


2.3

Loss of Cooling

When the reactivity is reduced below the value of 1, by a control rod, by introduc However, tion of voids, or by whatever means, the chain reaction dies very rapidly. heat production does not cease so rapidly. First, the delayed neutrons, if present, cause fissions at a rapidly decreasing rate over a period of a second or so. Then, when fission has ceased for all practical purposes, heat is still generated by the radioactivity of the fission products. Since these emit both 0 and y rays, which for all practical purposes are absorbed and turned into heat within the core, t his heat must be removed. For power reactors, this delayed heat, if not removed, is sufficient to melt down the core.

The delayed heat is given by the following equation for reactors which have operating long enough to have the fission products at an equilibrium value: Pd.Uyd = (0.07)Pn„rm.l[<]'0-2 where

been

t > 1 8«!

PDorm.i = normal operating power level of the reactor PdeLyod = delayed heat power level of the reactor in the same

units as the normal power level t = time, sec 0.07 = experimentally determined coefficient This is an empirical equation sufficiently accurate for practical purposes. For the interval for 1 sec after shutdown, the delayed heat production is about 7 per cent of the normal power level. Reactors with the highest heat flux per unit area of fuel element obviously are the most vulnerable to disaster if coolant is lost. Thus, the advantages of liquid-metal or water cooling must be weighed against possible reactor damage or public disaster if cooling cannot be maintained. Gas-cooled reactors with low specific heat transfer rates and large thermal capacity might cool themselves sufficiently to avoid a major melt-down, even if all coolant were lost. Obviously, each reactor design should be considered individually and great care taken to ensure the integrity of the cooling system, to have emergency power supplies for circulating pumps or emergency cooling means. Some of the devices used are cooling sprays if water is lost, flywheels on pumps, automatically starting diesel generators, and overhead water tanks. One very good method is to have the whole reactor system in a tank or excavation which would always be full of coolant, even if the primary coolant system should rupture. Natural convection or boiling then takes care of the afterheat.

Sec. 8-4]

REACTOR

SAFEGUARDS

8-81

CONTROLS

3

3.1

Operating

five main ways of controlling reactors during operation, by (1) control poison, and (5) neutron leakage especially heterogeneous ones, use control rods. The homogeneous reactor is especially susceptible to control by coefficient, but also some of the water-moderated ones and the air-cooled temperature Brookhaven reactor can be so controlled at least within limits. The use of voids is reactor and for the boiling reactor. The depended on partially for the homogeneous use of soluble poison has been proposed but has not been used practically as far as is There are

rods, (2) temperature coefficient, (3) voids, (4) soluble control (for fast reactors). To date most reactors,

known.

all other things being equal, one would rather depend phenomena than on any mechanism. Although the record to date has been excellent, one must expect that sometime a chain of events will lead to a failure of a control mechanism. From a safety standpoint,

on natural

3.2

Safety

would like to have a device which will positively and promptly shut off the reaction when safe limits of temperature, neutron flux, or rate of rise of neutron flux are exceeded and to have this an inherent tamperproof part of the reactor. Work is going forward on such devices, called "reactor fuses." It appears that satisfactory devices for critical experiments and low-power reactors can be made. For highpower reactors, this is considerably more difficult and a practical device has not yet One


chain

been developed.

The main reliance for positive shutdown of reactors today is "poison" rods, which may be combined with the removal of some fissionable material. These rods must have sufficient reactivity effect so that they positively cannot be overridden by loss of coolant, decay of xenon, reasonable errors in loading or unloading, or any other reasonable reactivity change. One would like to specify that their reactivity value should be at least twice the calculated reactivity change that might conceivably take place, unless the reactor has other equivalent ways of neutralizing the reactivity change. There should be at least three scram rods, and each of these should be independently operated. Preferably, there should be more so that the failure of several rods still does not endanger the shutdown of the reactor. Obviously, all scram rods, their There should mechanisms, and control circuits should be of the fail-safe character. he at least two different types of actuation. Gravity Scram rods should act rapidly. actuation is generally used, but spring acceleration has definite advantages. The discussion here is based on the concept of safety "rods." If three rods were used and their total value was only twice the possible introduced reactivity, the factor of safety is 2. If one rod failed, it would be reduced to 1.7 and if two failed, to 1.3; obviously other means of accomplishing the same objective would be satisfactory. Other systems are more complex to explain. One cardinal principle should be observed: One should know the poshion of all control and safety rods at all times. This feature of the reactor is too important for estimates or guesswork. Safety rods should be so designed that they cannot be accidentally removed from the reactor by a reasonable loading or unloading mistake, normal maintenance, accidental starting of a coolant pump, or other conceivable mishap. Each system should be carefully analyzed for such possibilities. It is highly desirable that an additional method of reducing reactivity be available. poison scram rods are used, a heavy-water-moderated reactor ran be additionally protected by dumping the heavy water. If either light or heavy water is used as a moderator, the introduction of boron or other high-cross-section poison is a possi bility. Boron shot can also be introduced into appropriate channels.

If

8-82

CONTROL 3.3

OF REACTORS

[Sec. 8

Interlocks

It is common practice in reactor control systems to have numerous interlocks or cross channels of information. Such interlocks are important from a safety point of view. During steady operation, information from any part of the reactor system indicating abnormal operation should be fed either to correcting devices or to the For example, if a cooling-channel operator's attention at the control panel or both. outlet indicates an abnormally low temperature, this information should be compared with the inlet temperature and coolant flow rate of the channel. If these are normal, then either the heat transfer has decreased, because of dirt on the fuel-element surface


perhaps, which might result in undesirably high temperatures within the fuel element, or the power output of the channel has decreased, which might be owing to local "poisoning" or shadowing, etc. The foregoing is to illustrate the ramifications of such information. Interlocks are also desirable to prevent the operator from inad vertently operating any of the reactor controls in an incorrect manner. In startup it is usual to have an interlocked system so that certain definite steps must be taken in the proper sequence. The interlocks also make sure that various important parts of the reactor control system are in place and functioning properly. For example, the source is in place, the ionization chambers are functioning properly, there is current on the amplifying system, etc. As yet, there are no definite requirements for interlocks, but because of their importance some minimum set of standards for them should be devised and made a part of the requirements for reactor safety. 4

ADMINISTRATIVE

CONTROL

One would like to think that reactors could be so designed that they would start up, run, and shut down all automatically; they would be so designed that automatic controls would prevent them from operating incorrectly, and therefore, there would be no hazard. In the first place, we cannot build such machines, and in the second place, it is doubted that they would have fewer hazards than those at least partly dependent on human control. Maintenance, of necessity, must be by men, and the assurance that the maintenance is done properly is distinctly an administrative control problem. One cannot depend on human control alone, since, as Fig. 1 shows, under certain conditions of trouble things happen so fast that human reactions are nowhere nearly fast enough. In these cases, one must rely on automatic, mechanical, or electronic shutdown devices. There must be a reasonable balance between auto matic safeties and highly skilled operators. 4.1

Operation

After construction there must be a thorough and careful checkout and testing of every vital part of the reactor. This should be from a carefully prepared and written Although not specifically required in the license regulations, this checkout cheek list. should be a part of the record of the licensee for his own protection. Such a check list is much too long to print here, but it should include among many other items the following: a record of the fuel loading and its position to go critical; the amount of excess reactivity, cold and at operating temperature and power; a complete calibration of all control, shim, and safety rods separately and in all combinations; the speed of operation of these rods; a test of their functioning; the presence of and strength of the neutron source; calibration of the power level and period meters; calibration of other vital control points; etc. The initial startup should be in the presence of the immedi ate supervisor and at least one other qualified person in addition to the reactor opera tor and should be by a definite standard written procedure. This should also be the case during any loading of the reactor and after any important maintenance. Once the reactor is in steady-state operation, there is little for the operator to do. Actually it becomes quite a boring job, and therefore, one should select the proper A high school graduate is probably type of person for such a routine position. adequate after training on the reactor itself. There should be at least two qualified


Sec. 8-4]

REACTOR

SAFEGUARDS

8-83

They should be in contact with each other either by operators on duty at all times. The operator both being in the control room or by an intercom system at all times. in charge of the reactor should never leave his post without proper and formal relief. Any communication with his assistant should not require the use of his hands. (For example, a telephone system is bad, but a microphone-loudspeaker system is excellent.) The principal operator should not work over 8 hr in any one day. In addition, he should be relieved at frequent intervals, perhaps even every 2 hr. A permanent record of the operation should be kept, including comments on unusual events. It goes without saying that the operator and his assistant should be thoroughly trained on the particular reactor they are to operate. They should understand the machine well enough to know what the essential control elements are and how they Manual last-ditch measures should be thoroughly understood and prac operate. ticed if only in simulation. The procedure for shutdown should be in writing and rigidly adhered to and should cover all foreseeable cases. The immediate supervision should be notified promptly if unexpected shutdown occurs. If the reactor is to be unloaded, then the assistant Unload operator and the supervisor should be present and a record of progress made. ing, shifting, or replacing fuel elements should be done only according to a written procedure which has been reviewed and approved by the Reactor Hazard Evaluation Staff or the Advisory Committee on Reactor Safeguards. There should be an operator on duty in the control room or at an auxiliary control panel, and essential monitoring and control elements should be maintained in opera tion, as far as possible, throughout any unloading and loading operation. After reloading, the startup should be by an approved written procedure and in the presence of the operator, assistant operator, and supervisor.* 4.2

Supervision

In

discussing this, we include shift supervisors, area supervisors, and senior super The shift supervisors at a minimum should be educated or trained to understand the possible kinds of deviations from standard To illustrate, if the operator discovers that a conditions and their probable causes. channel is running hotter than normal but not hot enough to require a power cutback, his official responsibilities would be discharged having made a note of the fact. He would be expected to be on the alert for a further temperature rise. The shift super visor would be expected to understand that this higher temperature was due to a low water flow, a partial channel stoppage, a peaking of flux, etc. Somewhere in the supervisory staff above the level of shift supervisor should be at least one who understands the principles of the nuclear reactor and its associated To continue with the He should be a fully qualified nuclear engineer. equipment. illustration of a hot channel, this nuclear engineer should understand the possible reasons for a low water flow, a partial channel stoppage, or a peaking of flux and understand the interrelation of the information of the hot channel and the other data of the reactor operation. By his full understanding of nuclear engineering, this man or men can interpret the data and forestall the development of serious failures. The nuclear engineer would review any changes to be made in the reactor and decide if they were reasonable and safe and if they were significant enough to be referred to the Reactor Hazard Evaluation Staff and the Advisory Committee on Reactor Safeguards. The nuclear engineer would also inspect and approve all maintenance on the reactor system.

visory personnel by whatever title.

4.3

Maintenance

Faulty maintenance of

The replacement original condition

a reactor is perhaps the most likely cause of an accident. or repair of a part of the system must put it back in its identical, unless one is certain that the change is an unimportant one or an

* These are suggestions rather For very small power reactors. two persons need be present.

procedures and are especially applicable to largo than mandatory reactors with small accumulations of fission products, perhaps only

8-84

CONTROL

OF REACTORS

[Sec. 8

The point to be made is that these reactor systems are so new and be made without careful consideration and It must be certain after repair that motors do run in the correct direction, study. relays are still "fail-safe," and the meter is reading on the correct scale. This can be done only by thoroughly competent workers, supervision, and a careful As men and thorough checkout of the repaired part and its associated equipment. tioned before, supervision of maintenance is another area requiring the services of a nuclear engineer. improvement.

little understood that changes cannot

6

CONTAINMENT

6.1

Homogeneous

Reactors

is

is

is a

a

is

a

it

it,

Aqueous homogeneous reactors for research purposes usually operate at close to atmospheric pressure. The first line of defense against the escape of fission products is the primary container, the tanks and pipes containing the solution of fissionable The homogeneous reactor has a gratifyingly large negative temperature material. coefficient and, in addition, a still larger (about five times) void coefficient owing to the formation of H2 and 02 and steam. Nevertheless, if there is a possibility of the sudden addition of an appreciable amount of reactivity, there is the possibility that The amount of reac the container may be ruptured and fission products released. tivity introduced as a "step function" necessary to disrupt the vessel obviously However, on the basis of present varies with the vessel design and the power level. knowledge, a step increase of about 8 per cent is estimated to be the maximum which might be introduced without rupture. It should be remembered that research reactors by their very philosophy are subject to varying conditions; therefore, there is a greater possibility of accidental introduction of excess reactivity than is the case for reactors operated in a routine manner. Because of the excellent inherent safety of a low-power homogeneous reactor, a secondary container would probably not be required if a reasonable exclusion area If one desires the reactor in the midst of a university campus or in the is available. heart of a city, a tight secondary container is likely to be required. An aqueous homogeneous reactor for the generation of power will have much larger barring continuous processing. amounts of fission products stored in Also, will operate under pressure. In case of leakage, the escaping solution will flash into steam carrying fission products. Certainly, secondary container would be required having the necessary volume-strength characteristics to contain all the steam which might result from major rupture of the system. This the case unless the power were quite low and suitable isolation were available. Aqueous homogeneous reactors have the advantage of excellent stability as already mentioned. There are no problems of a possible metal-water reaction. However, there problem of possibly all kinds of corrosion and possible change of metal characteristics by neutron bombardment. Until more knowledge obtained, one a chance of rupture of the pressure vessel. must assume that there So few data are available on homogeneous reactor systems other than aqueous that no discussion worth while. of them is


After one has done his best to design a reactor that is reliable and trained a corps of competent and careful operators and supervisors, one considers what might happen to the reactor. As of today there is no reactor of any appreciable power level, perhaps 1 Mw of heat or more, which cannot have a credible major accident. The con sequences of such an accident are possibly so great that one must either locate it in an uninhabited, remote location (the greater the power, the more remote the location) or surround the reactor with a container which will not permit fission products to It is doubtful if there is any place in the continental United States remote escape. enough that release to the atmosphere of the equilibrium fission products from a reactor of several hundred megawatts of heat power or more would not constitute a One doubts that such a potential hazard can be justified great potential hazard. by economic reasons alone.1

reactor safeguards

Sec. 8-4]

8-85

should be emphasized again that each reactor facility must be considered on a basis and all features of the design, operation, supervision, containment, and It may be possible to come up with a design of a power location considered together. homogeneous reactor not requiring a secondary container.

It

case


5.2

Heterogeneous

Reactors

Heterogeneous research reactors operating at a few megawatts of heat power will usually not be pressure systems. The lower power will be offset by the greater possibility of misoperation simply because of the experimental use of the machine. If the fuel elements are made of aluminum or zirconium and these elements should Also, there is the possibility of reaction with melt, fission products will be released. In the case of uranium, contact of the metal with hot water (as by the failure water. Therefore, of the cladding) will result in rapid reaction and release of fission products. if the reactor must be placed in a populated area, some means of avoiding the escape Means are provided (limitations on excess reac of fission products must be found. tivity available, operating procedures, and automatic and manual controls) so that Again one can reduce there is reasonable assurance that fuel elements will not melt. The relation of these the containment requirements by greater exclusion area. factors will be considered in the article treating location. Power reactors of the heterogeneous type are considered less liable to misoperation than research reactors, but this is offset by the greater storage of fission products and probably closer approach to maximum safe operating levels. These operating limitations may be upper temperature limits set by corrosion, surface boiling, crystal phase transition, etc. But power reactors using water as a coolant or moderator will inevitably be under pressure, probably relatively high pressure. One is then faced with the possible rupture of the pressure system. If there were a major failure of the pressure system, water would flash into steam, the fuel elements might lose their cooling and melt, It appears, therefore, a reasonable precaution to provide releasing fission products. a container with the proper volume and strength characteristics to contain the steam and fission products of a major primary system failure. If these reactors contain metallic uranium, aluminum, or zirconium fuel elements, there is still the question of a possible metal-water reaction. Power reactors cooled with sodium or other liquid metals or hydrocarbons not requiring pressure pose different problems. The lack of pressure is certainly an If sodium is used as the coolant, it becomes highly radioactive. There advantage. However, there is also the possibility of the violent reaction of sodium with water. is not the possibility of a violent chemical reaction between metallic fuel elements containing fission products and sodium or hydrocarbon coolant. With proper design, nonaqueous coolants appear to have certain outstanding advantages over aqueous coolants from a hazard point of view. It should not be forgotten that, with a perfectly tight steel container, there is still a If one must assume in the maxi possibility of serious hazard to the surroundings. mum credible accident that the container becomes completely filled with fission products, then it will become an intense source of y rays. The public hazard can be Unless minimized by using a concrete building or concrete lining of the container. the concrete is required for other purposes, it is usually cheaper to have the protection of distance. A relatively short distance will suffice. The layer of air reducing the average fission-product 7-ray intensity to one-half is about 86.5 meters, and the inverse square law reduces intensity still more rapidly.*

LOCATION

6

The discussion in this article emphasizes the concern with the health and safety of It should be clear from the other articles that if the reactor is of the the public. * This

m

-

0.81

is

baaed

on

X 10-' cm"'.

assumed

energy

of

7

rays of

1.0

Mev

and

an

absorption coefficient

8-86

CONTROL

OF REACTORS

[SEC. 8

proper design and is operated and supervised adequately, it will have a low probability of a serious accident. In addition, if it is properly contained and this containment can be guaranteed, the reactor can be located almost anywhere. These necessary precautions may well turn out to be excessively expensive. Moreover, it is not at all clear that one can design, operate, and supervise so there will be no accident, and one cannot "guarantee" containment in an absolute sense. It is entirely reasonable and proper, therefore, to look at the location as a part of the whole picture to help make the entire facility as free of hazard as possible and, at the same time, economical. The location also has a clear bearing on what one might be able to do to alleviate injury and loss of life if some most unusual and unforeseen set of circumstances should cause a major accident. In picking a suitable location, one must consider the character of the surrounding area: population density, nearness to large cities, and nearness to vital defense installa tions, hospitals, recreation areas, etc. One must consider meteorology, hydrology, seismology, and geology. Meteorology

There has been a very considerable amount of study of the possible path and dilution of radioactive material in connection both with atomic-bomb tests and with postulated reactor accidents. As was mentioned in the article on containment, if the equilibrium fission products from a reactor operating at several hundred mega watts of heat power were released, the consequences could be very serious quite a few miles away as a result both of whole body y radiation and, even more importantly, of the possible inhalation and retention of the more toxic fission products. The possibility of a rainout that could cause serious contamination extends to even greater distances. The proper estimation of these possibilities requires the services of an experienced meteorologist and a study of the meteorology of the particular site. This study results in a sort of probability. these probabilities deal with the Unfortunately, weather only. One needs to have a better appreciation of the probability of mecha nism failure, the actual form and size in which the fission products might be released, and the possible leakage from a container in order to arrive at some idea of the probable over-all hazard. It is doubted if these probabilities can ever be made precisely quanti tative, but certainly more study and experiment will make them more meaningful than they are today. At one time the Reactor Safeguard Committee derived a formula for the exclusion

9, a

8,

is

7,

it,

area around a reactor which was r = 0.01 V*> where r was the radius of the exclusion area in miles and P was the operating power of the reactor in kilowatts of heat. Thus, a reactor producing 360,000 kw of heat would require an exclusion radius of 6 miles. This formula was based on a number of assumptions regarding weather conditions, amount of fission products released, and upon an external whole body radiation dose of 300 r. A period for warning affected populations was also in the thinking. Inges tion was not considered. It was never intended that this formula would be uni versally and rigidly applied. It was useful in one specific case and formed background for other cases. Also, it should be noted that in case of a reactor accident, as one Reactor Safeguard member put "the formula guaranteed no safety outside the exclusion area but guaranteed trouble within it." Since this time, the U.S. Weather Bureau has studied the problem of dissipation of fission products in the atmosphere. The results of their study are contained in book, "Meteorology and Atomic Energy," July, 1955, especially Chapters and 10. The problem much too complex for any single formula. 6.2

Hydrology

is

a

a

Just as reactor accident may pollute the air with fission products, the accident Since may also contaminate the ground. large nuclear power reactor very river or a lake, one must examine the consequences of major pollution probably on a


6.1

Sec. 8-4]

REACTOR

SAFEGUARDS

8-87

of this water. If the river passes by or through a large eity or, even worse, if it forms the water supply of a city, then contamination past the allowable limit is not tolerable. Theoretically, the water might be purified, but there are no facilities set up to do this. One cannot permit large amounts of fission products to enter the ground water Not infrequently, one is not which may be the supply for people some miles away. entirely clear where the ground water does actually go, so unknown persons might be

damaged by the contaminated ground water. Experience with a reactor accident has shown that very large quantities of con Since these cannot be discharged directly, large taminated water are accumulated. The water can then be held until radioactive decay holding basins are required. will bring it within tolerance or bled off a little at a time, taking advantage of dilution, or pumped to a definite disposal area. Finally, the site must be adequately protected from flood and the design so arranged that flood will cause no major radioactive hazard.


6.3

Seismology

One thinks again and hard before locating a reactor in a region prone to earth The earth tremors have been known to do queer things. Warping of the quakes. structure so as to increase reactivity; preventing control, shim, or scram rods from entering; breaking coolant supply lines; breeching the container are all events which If the area has any could contribute to a major disaster by a reactor accident. A "faults," then the reactor building or container design must be made accordingly. P-wave meter to detect the approach of any earthquake is a necessity so the reactor can be shut down before the arrival of the damaging earth shock. 6.4

Geology

One should know thoroughly the rock and soil structure of the site chosen for a nuclear reactor. This is important for the foundation, the probable path of liquid effluent, and the determination of the probability of earthquake. 7

FUTURE REACTOR

TYPES

The possible hazards of present reactor designs are causing a great deal of extra effort to minimize the possibility of a serious reactor accident. There is a great need for new designs which do not have the hazard drawbacks of the present ones. In passing, one can also remark that the generation of electric power from nuclear energy by the heating of water and the operation of a steam turbine seems a rather Perhaps one can at least alleviate the hazard problem in a clumsy way to proceed. design which also makes an advance in power generation. 7.1

Fission -pro duct Storage

The outstanding hazard problem is the storage of large amounts of these highly poisonous fission products in a reactor. Because of their presence all the possible malfunctions and their consequences must be examined. If one could materially reduce the amount of fission products in the reactor at any time, the problem immedi reactors have the possibility of ately becomes less difficult. Circulating-fuel-type In fact, this is being continuous treatment of the fuel to remove the fission products. The aqueous homogeneous power reactor is one already given serious consideration. At Brookhaven National in the pilot-plant stage at Oak Ridge National Laboratory. Laboratory, work is in progress on a circulating liquid-metal fuel reactor which also has the possibility of continuous removal of fission products. It is to be hoped that other designs will be forthcoming. The fission products do not cease to become toxic by their removal from the reactor, but they now can be kept in a place which does not have the dynamic properties of a nuclear reactor with its extremely small time constants.

8-88

CONTROL


7.2

OF REACTORS

[Sec.

8

Temperature Coefficients

Present thinking in the United States, at least, is to the effect that any power reactor must have a negative over-all temperature coefficient. The over-all tem perature coefficient is made up of a number of temperature coefficients: the fuel, coolant, moderator, lattice, and blanket coefficients where these parts are present. There are other less important temperature effects such as reflector, control rods, Each of these coefficients must be examined, and their value and interrelation etc. determined. A strong negative temperature coefficient is greatly to be desired, but it has its drawbacks as well. Such a coefficient makes the reactor sensitive to cooling below its operating temperature, and one must guard against the so-called "cold-water accident" (for water-moderated reactors) in which moderator significantly below the operating temperature is suddenly introduced into the core. Likewise if the reactor must be cooled for shutdown and reloading, there is considerable excess reactivity to manage in starting up and bringing the reactor to power again. In heterogeneous reactors there is a relatively slow coupling between fuel plates and the moderator. If the fuel plates themselves have insufficient negative tempera ture coefficient in a water-moderated machine, it is possible to melt the fuel plates before any significant rise in water temperature can take place. What is greatly to be desired is a reactor design in which in the region from room temperature to operating temperature there is a small positive temperature coefficient changing to negative at the operating temperature. In addition, the fuel elements themselves should have these temperature characteristics or have an effective rapid coupling with the other elements of the reactor. The Doppler effect in natural uranium fuel elements is a good negative effect, and this is being studied. However, it may be that this effect will come into significant play only after such high temperatures are reached as to damage the reactor seriously. In summary, one hopes that the inventive genius in the field of reactor design will result in a reactor design with no fission products stored in the machine and which has such stable characteristics that it cannot have a nuclear runaway.

REFERENCES 1.

Membership of Advisory Committee on Reactor Safeguards at time of going to Manson Benedict Harvey Brooks Willard P. Conner, Jr. R. L. Doan C. Rogers McCullough, Chairman

K.

2. 3. 4. 5. 0.

7.

press:

R. Osborn

D. A. Rogers R. C. Stratton Abel Wolman Public Law 85-256, 85th Congress, H. R. Atomic Energy Act of 1954, Public Law Chap. 1073.

7383, 703,

September 2, 1957. 83rd Congress, 2nd Session. H. R. 9757.

Conference of Governmental and Industrial Hygienists in Atlantic City, N.J., April, 1951. Patty, Frank H. (ed.): "Industrial Hygiene and Toxicology," Interscience Publishers, Inc., New York, 1949. Maximum Permissible Amounts of Radioisotopes in the Human Body and Maximum Permissible Concentrations in Air and Water, Natl. Bur. Standards Handbook 52, Mar. 20, 1953. " Theoretical Possibilities and Consequences of Major Accidents in Large Nuclear Power Plants," WASH-740. Values adopted at meeting of the American

8-5

REACTOR-SYSTEM

DYNAMIC SIMULATION BY

John C. Moise Analysis of reactor dynamics is essential to the evaluation of reactor stability and specification and design of reactor control systems. This analysis involves solu tion of both the reactor kinetic equations and the auxiliary equations required to describe the dynamics of all system components which influence the nuclear behavior of the reactor. Solution of an adequate system of equations can be done most Where great accuracy of solution is effectively with the aid of machine methods. required or an extreme range of variables must be covered, the digital computer may Investigation of startup provide the best method of obtaining problem solution. dynamics where flux values range over many decades is an example of the type of problem best adapted to digital solution. However, digital computers are in some cases slow and inflexible compared with analogue computers for investigating the effect of system parameters and designing control systems. The reactor-system equations can best be solved by using commercially available analogue computing equipment rather than by attempting to construct a specialThe advantages of using commercially available equipment of purpose simulator. the electronic analogue computer type are the use of reliable, proved equipment which provides accurate results and permits more effective use of computing time; the elimination of electronic development and construction programs; and the adaptability of such equipment to the solution of a wide variety of problems. Commercial equip ment is available which performs the linear operations of integration, addition, and multiplication by a constant as well as the nonlinear operations of multiplication of division, and function generation. A general discussion of electronic variables, analogue computers and their operation may be found in such references as Korn and Korn.1'* Specific problems relating to reactor simulation will be discussed below.


the

Nomenclature

A, A,

—
m

c. <-„

= —

f

-

U

fuel-element surface area, ft* surface area, ft* specific heat of reactor coolant, Btu/(lb)(°F) specific heat of fuel elements, Btu/(lb)(°F) specific heat of secondary heat-exchange fluid (assuming no change of state),

- heat-exchanger Btu/(lb)(°F)

specific heat of moderator, Btu/(lb)(°F) precursor concentration for ith delay group, Btuf/sec fraction of fission energy not absorbed in fuel change in t'th variable effecting reactivity

* Superscript numbers refer to References at end of subsection. The expression of precursor concentration in energy-release-rate

,

units rather than the conventional nuclei per cubic centimeter is consistent with the subsequent treatment of the neutron balance equation in nuclei per cubic centimeter as a power relationship. C%is proportional to the precursor concentration and consequently to the energy-release rate due to neutrons resulting from the decay of the ith delay compared generation time group. Since the precursor decay rate is C»X. and the corresponding with the generation time r for prompt neutrons, the steady-state energy production due to delayed neutrons will be Ciktr.

t

8-89

8-90 fc„

M/

P R S I

Te

Tf T, Ta,in

Tu Tm 7',,,,,,,

T,

U/

U, V,

Vm v

W, W„ Wm

OF REACTORS

= source strength, Btu/sec = time, sec = temperature of primary reactor coolant, °F = fuel-element temperature, °F = temperature of secondary heat-exchange fluid, °F = inlet temperature of secondary heat-exchange fluid, °F = coolant temperature in heat exchanger of simplified model assuming com

plete mixing, °F

= moderator temperature, °F = moderator inlet temperature, °F = absolute neutron temperature, °F

= fuel-element heat transmittance, Btu/(ft,)(sec)(°F) = heat-exchanger heat transmittance, Btu/(ft!)(sec)(°F)

= coolant volume in region under consideration, = moderator volume, ft' = neutron velocity, cm/sec

Pc

•=

p„ X<

r

ft*

= flow of primary reactor coolant, lb /sec = flow of secondary heat-exchange fluid, lb/see = flow of liquid moderator, lb /sec

Pi

0

[SEC. 8

= excess neutron multiplication = fuel-element weight, lb = reactor power, Btu/sec = control-rod reactivity contribution

= reactivity coefficient associated with = total delayed-neutron fraction = delay fraction for ith delay group

a<


CONTROL

ith variable

coolant density, lb/ft3

= moderator density, lb /ft* = decay constant for ith delay group, sec-1 = effective neutron lifetime, sec

Unprimed variables are physical variables. Primed variables are computer variables. The subscript zero denotes steady-state value of the variable. In reactivity rela tion it indicates the design-point steady-state value. The subscript n indicates that the variable applies to the particular region used in the analysis of fuel elements and heat exchanger. The subscript d denotes design-point value. 1

MATHEMATICAL DESCRIPTION OF SYSTEM

For the purposes of dynamic analysis, the reactor system can be considerably simplified and assumed to be made up of the following subsections: the "core," consisting of the fuel-bearing region and the coolant required to remove the energy absorbed in this region; the moderator and its associated coolant (in many reactors the moderator and core are integral and have a common heat-removal system); the reflector and its coolant; and the external system, consisting of the heat exchangers, pumps, heat engines, etc., which utilizes or rejects the reactor heat. The technique of analysis will be presented by discussing the general equations relating to a particular subsection of the reactor system and then applying these equations to a specific hypothetical reactor. 1.1

Specification of Illustrative Problem

Computer setup will be discussed relative to the specific reactor which employs liq uid-cooled solid fuel elements, a liquid moderator, and a liquid-to-gas heat exchanger. Under design conditions, the primary coolant enters the core at 650°F, removes 20 Mw of heat from the fuel elements in a single pass, and leaves the fueled section at 850°F. The primary coolant transfers this energy to a flowing gas in a counterflow heat

reactor-system dynamic simulation

Sec. 8-5]

8-91

The liquid moderator circulates independently of the primary coolant exchanger. and at a rate sufficient to remove the heat generated in this region with a temperature rise of 95°F. The following specifications describe the example reactor.

p

= 20 Mw = 18,960 Btu/sec = 0.05 = 0.5 msec

T

«/ at am

Mtcf wci

U,A,

<■»

w« V.A.

Cm PmVmC„

-

-2 -5

X 10~s Ak/k per °F X 10-* Ak/k per °F +1X10-' Ak/k per °F

= = =

= = = =

12 Btu/°F 316 lb/sec = design coolant flow 2Wco» 0.3 Btu/(lb)(°F)

total coolant heat capacitance; in fuel-element region = 9 Btu/°F; in heat exchanger = 120 Btu/°F = 380 lb /sec = design gas flow

= = = = =

0.25

Btu/(lb)(°F)

1.65HV-8 10 lb/scc = design moderator flow 1

f

Btu/(lb)(°F) Btu/°F

2000

Nuclear System

5

a

is

if

is

a

1.21. Physical Equations. Equations (1) through (.3) represent the nuclear charac teristics of the reactor by one-group neutron model, assuming that no coupling exists between space- and time-dependent phenomena and neglecting the slow dynamic effects due to xenon-concentration changes. The representation of neutron multi plication also approximate but adequate even for reactivity excursions considerably in excess of prompt critical. While xenon poisoning changes, discussed in such references as Glasstone and Edlund,2 are too slow to be considered in the usual reactor dynamic analysis, they can be conveniently treated on an analogue computer condensed time scale used. Subject to the above limitations the time-dependent nuclear behavior of a reactor can be described by the following set of equations:

T

(1)

a»/»

R

• •

+

aifi

1—1

i-1 •

=

T

+

k„

T

+

at

(2) (3)

t

at

is

a

a

is

is

it

P,

is

Equation (1) a frequently used approximate representation of the transient behavior of reactor neutron flux. In Eq. (1) the reactor power, assumed to be propor tional to the effective neutron flux. Although the assumption of a fixed ratio between flux and power common, an approximation, since for a given reactor at given time the variation of fission cross section (and hence the fission rate for given flux) with neutron temperature and the fission-product generated afterheat both prevent exact proportionality. If these effects are to be taken into account as a first approxi

/

1

a

6

if

1

y/T\ and the afterheat the cross section can be assumed to vary as /v or constant at approximately per cent of the preceding steady-state power for 10 sec, then decays from this value in complex manner. These effects should be taken into account large variations in either power or neutron temperature (approxi mately the moderator temperature in thermal reactors) are to be considered. Since moderator temperature varies slowly relative to fuel temperature and coolant temmation, ia


1.2

8-92

CONTROL

[Sec. 8

OF REACTORS

perature in most reactors, the fission cross-sectional effects will not appreciably influence the faster reactor transients. In Eq. (1) k,x represents the excess neutron multiplication, t is a properly chosen average neutron lifetime, and S represents the neutron source necessary for initiation of the reaction. The/ terms in Eq. (2) represent thermodynamic variables associated with the fuel, moderator, coolant, and core structure, while the corresponding as are the reactivity coefficients associated with the particular variable. It should be noted that the reactivity coefficients will vary with temperature and that this varia tion should be taken into account by means of function generation if the nonlinearity is severe and large temperature variations are being considered. The R term repre sents control-rod reactivity contribution, 0,- represents the delayed neutron fraction, X,- the decay constant, and C,- the precursor concentration for a particular delayedneutron group. The pertinent characteristics of the five definitely established delayed-neutron groups are listed in Table 1. If it is desired to minimize computing components the five groups may be represented adequately for most dynamics problems by the two equivalent groups indicated in the last two rows of Table 1.

Table

1.

of Delayed

Properties

Decay constant, sec-1, Xi

Neutrons

from Fission of Uranium

Mean life (time constant), sec, l/Xt 0.62

1.61

0.456 Generated for wjivans (University of Florida) on 2015-09-23 02:53 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Resulting

Fraction, 0i

0.00084 0.0024 0.0021 0.0017 0.00026

2. 19

6.50 31.7 80.2

0.151 0.0315 0.0124

236

Equivalent Two Groups 0.758

1.32 10.51

0.095

0.00326 0.00404

In circulating fluid-fuel reactors where an appreciable portion of the precursor decay may take place in the external system, the delayed-neutron fraction will be smaller than indicated in Table 1. While accurate representation of the delayedneutron dynamics for these reactors is quite difficult, a good approximation may be obtained by altering the effective (3< values to correspond to the steady-state dis tribution which may be calculated from the decay rates and the system volume ratio. Applying Eqs. (1) through (3) to the specific reactor discussed above, it is assumed that the thermodynamic variables influencing reactivity are the fuel-element tem perature, the coolant temperature, and the moderator temperature; that two equiva lent groups provide adequate delayed-neutron representation; and that power is proportional to neutron flux. It is also assumed that reactivity can be influenced by a maximum of 5 per cent by a control rod. Using the numerical values listed for the example reactor Eqs. (1) through (3) become dP

2,0OOk„P

dl

-0.00002(77

^ at

-

T,o)

- 15P + V XiC, + -S

- 0.00005(7', -

+ 0.095C, = 8.08P

dC%

dl

7V„)

+

+ 0.00001 (Tm

(4)

- T„o)

0.758C, = 6.52P

+ R

(5) (6)

1.22 In order to solve this set of relations on an analogue Computer Equations. computer, the equations must be transformed to corresponding computer relationships. This scaling process is conveniently accomplished by letting the maximum expected


Sec. 8-5]

8-93

value of the variable being considered be equivalent to the limiting voltage of the computer, which will be referred to as one "machine unit." The use of machine units rather than computer voltages in scaling is convenient when multiplication and division are involved. In general for any variable F, F' = aF + b, where a is the voltage range in machine units divided by the maximum range of the variable and 6 is the computer bias For example, if the maxi required to permit maximum computer range utilization. mum range of P is expected to be 40 Mw (37,900 Btu/sec), a would be equal to On the other hand, for 2.64 X 10""' machine unit/(Btu)(sec) and 6 would be zero. maximum sensitivity, temperatures, which appear as differences only, are scaled so that zero computer voltage corresponds to the temperature midrange; i.e., if 500°F < T/ < 1000°F, then a = 0.004 machine unit/°F and b = -3 machine units. Assuming the expected range of variables to be 0 < <

-0.05

P k„

-

< 40 Mw < 0.05

0 < C, < 3,210,000

0 <


500 < 500 <

c,

<

Btu/sec

< 326,000 Btu/sec < 1,000°F < 1,000°F

T, T.

£

300 < Tm

-0.025

31,900 Btu/sec

R

c no

~~

= 3,210,000 0.095 the precursor Btu/sec expressing concentration in energy units. See (i.e., 37,900

X

footnote in nomenclature article)

500°F

<

0.025

^

the computer equations corresponding to Eqs. (4), (5), and (6) will be 0.0667

k„'

dP' at

0.10(T/

-

10.52

+

P'

= 6.67A„'P'

7'/„')

^ dt

+ 0.553C,' + 0.447C7 + S'

- 0.25(7*/ -

+ CV =

P'

7V) 1.32

+ 0.02(7V

^ dt

+

(Y

=

-

Tm0') + 0.5'R

(7) (8)

P'

The block diagram of the computer setup for these equations is shown as a part of the general computer block diagram in Fig. 1. The delayed neutrons are simulated by passive networks to save computing amplifiers. 1.3

Heat-generation

System

Fission heat will be generated either in a solid-fucl-element system or within a fluid in the case of fluid-fuel reactors. In either case partial differential equations are needed to represent accurately the space and time dependence of reactor coolant For accurate representation of reactors with compressible fluids as temperature. coolants, internal energy and enthalpy must be considered as variables in place of temperature and continuity and state equations must be employed to supplement In either a gas-cooled or boiling reactor, the resulting the energy relationship. equations are too complex for computer solution as they stand. Simplification and at least partial linearization of the equations should be undertaken. Linearization of the equations results in a set of relations which represent the system dynamics for small perturbations and can be readily solved on an analogue computer. This type of approach is discussed in the heat-exchanger analogue article. The fact that the fuel-element and coolant heat-balance equations are partial differential equations means that only an approximate solution can be obtained in any Spatial effects are approximated by dividing the fuel element and coolant case. system into a number of axial regions with a single fuel element and coolant tern-


8-94 CONTROL

NON-LINEAR ANALOG

Fio.

1.

OF REACTORS

SET UP FOR EXAMPLE

[Sec.

POWER

REACTOR

Nonlinear analogue setup for example reactor.

8


Sec. 8-5]

perature for each region. below:

The equations representing this approximation

M"Cl^r at PcV^Cc^ at

8-95

-

=

0

-flP»

-W.fi.lTn.

- V,An,{T., -

- Tn-ie]

are given

Tnc)

+ V,An,{Tnl

(9)

-

TV

(10)

Equations fuel

tor

(9) and (10) imply that the fraction of fission energy not absorbed in the If the primary coolant removes modera is not removed by the primary coolant. heat, as it does in many reactors, this energy will have to be added to the right-

/


hand side of Eq. (10) for the regions where energy absorption takes place. The larger the number of regions, the better the approximation to the correct spatial temperature distribution. If variable flow and variable heat-transfer coefficients are considered, the equations are nonlinear with respect to multiplication of temperatures by these variables. To illustrate the above points for the example reactor a six-region representation of the fuel element and coolant system with variable flow and heat-transfer coefficients will be used. It is assumed that the heat generation is uniform throughout the reactor core and that the fuel-element heat-transfer coefficient varies with the eight-tenths power of the coolant flow. If the heat generation is nonuniform, the coefficient of the power term will vary from region to region and will be determined by the per Using the listed data, the equations become centage power generated in each region.

2d-I^ at

1.5^5 at

If

- 0.333ire»-«(r./ - TV - -0.30W,[Tnc - IV,).] + = 0.158P

0.333W,.°

(11)

"(TV

- TV

(12)

the expected maximum range of Wc is 350 lb /sec, the scaled equations become 0.0554 0.0413

~T^ at

dT

= 0.663P'

-

(W.')a»(TRf'

'

--^ at

The computer setup of

=

-2.90WV[TV

- TV)

- IWl

+

these equations is illustrated

(RY)"(7V

(13)

- TV)

(14)

in the appropriate section of the

general block diagram. 1.4

Piping

Accurate representation of temperature transient of the fluid flowing in piping is difficult, since in most cases a pure dead time must be simulated. This dead time can be mocked up with magnetic tape or approximated by various electrical networks. In the case of a fixed flow, the electrical network gives a reasonable representation if rates of change of temperature are slow relative to the dead time. Magnetic-tape systems are expensive and still in the development stage with respect to this applica tion. An approximation to dead time in a variable-flow system can be made by treating the pipe as a number of single lag regions represented by equations of the following type:

PcVncC,^ at

= -WcCclTn,

-

TV-!).]

(15)

In the example reactor the fuel-element system is considered to be so closely coupled to the heat exchanger that dead-time simulation is unnecessary. 1.5

Heat Exchanger

Treatment of a heat exchanger is similar to that of the fuel-element and coolant Again with systems involving a compressible fluid linearization should be system.

8-96

OF REACTORS

CONTROL

[Sec.

8

seriously considered. If the time constants involved in compressibility effects are short relative to other dynamic effects in the exchanger, the internal compressibility can be neglected and the exchanger treated in the same manner as one involving incompressible fluids. This will often be the case in a liquid-to-gas exchanger, since gases have much lower volumetric specific heats and therefore much shorter transit times than liquids. An approximation which has been used for representation of a steam generator simulates this element by a single first-order lag with mean reactor coolant tempera ture and power demand as forcing functions.3

gî at


where

= a(Br

-

d.)

- r„

(16)

g = heat $, = mean

capacity of steam generator steam temperature a = over-all supply side heat-transfer coefficient Pi, = power demanded from steam generator 6, = reactor coolant temperature The steam generator forcing on the demand side, of course, depends on steam If is quite complex. pressure (temperature) and steam flow, but this dependence the load is loosely coupled to the reactor, representing these effects by a term such as Pi, is probably justified in small perturbation stability analysis. To illustrate a typical multiregion heat-exchanger analogue setup, consider that the example reactor rejects heat to a counterflow liquid-to-gas heat exchanger. Because of the high liquid side heat-transfer coefficient, the heat-exchanger meal will be at approximately the liquid temperature and its heat capacitance may b; The over-all heat-transfer coefficient will be primarily lumped with that of the liquid. a function of gas side flow, since the gas side resistance is controlling. An eighttenths power variation in heat-transfer coefficient with gas flow will be assumed. Subject to these assumptions, a six-region approximation of the exchanger is repre sented by the following equations: W,C0[T„i, PeVncCc

at

=

— T(n+l),]

I ;iAn,,[Tnr

= — WcCc[Tnc — jT(i._l)c] —

I

— 7,(»+l),] — Tf^+l),] „A nj[7\>c

(17) (18)

It should be noted that the heat transfer in a particular region is expressed in terms of the inlet temperature of one of the fluids and the outlet temperature of the other fluid. This representation gives satisfactory static and dynamic results with a minimum of computing equipment. Correct steady-state terminal temperatures and reasonable transients are given by this six-region model for variations of more than two to one in the flow of either fluid. Some experimentation is warranted in determining the best analogue for a given heat exchanger. Inserting values for the example reactor gives = 0.273WV '[THC

0.25WAT„e- r(„+1)J

20

^ at

0.30We[7'„c

-

r(,+l„]

- r(„_„e] - 0.273W,<"{T„ -

(19 (20)

Using the additional scaling relations,

T, < 1000°F < 500 lb /sec

500°F <

W,

the scaled equations are

0.506

-

r,.+»,'] 3.16WY[7V AT ' = _2.66W7[7V

-

=

- 7Wn.'] - (HV)°'[7V - 7*(.+i,/]

(WV)"[7V

(21) (22)


Sec. 8-5]

Computer setup of these equations is illustrated general block diagram.

8-97

in the appropriate section of the

Moderator and Reflector System

1.6

The moderator and reflector systems can be treated in the same manner as the fuel-element system if fluid-cooled solids are involved. If fluid moderators or reflec tors are used, the treatment is modified slightly to apply to a volume-heated fluid. If the primary coolant serves as the moderator or the moderator coolant, the energy must, of course, be added to the primary coolant system. Since the power density is much lower in the moderator and reflector systems than in the fuel system, the moderator and reflector respond slowly compared with the primary system. This indicates that a simplified treatment is often adequate as far as reactor dynamics is concerned. For the purpose of illustration consider that the liquid moderator in the example reactor can be represented by a single well-mixed volume-heated region. The heatbalance equation will then be dT

=/F PmF„r„^f at 2000

^= at

-

= 0.05P

- Wm(Tm -

Wmrm(Tm


Use of the additional scaling relation W„ < 160

The computer setup of

at

= 1.52/"

(23)

r„.,i„)

(24)

lb/sec gives

125

-

Tm.
- 7V,„')

WV(7V

(25)

these equations is illustrated in the appropriate section of the

general block diagram. 1.7

In

Mechanical

Equipment Associated with Reactor

designing a satisfactory over-all reactor system

control it is often necessary

to

take into account the dynamic behavior of such mechanical equipment as pumps, Analysis of these devices can be carried through fans, turbines, and condensers. manner. The resulting equations will often involve nonin a straightforward linearities such as Y = f(X) or Y = f(X,Z) as well as multiplication and division.

The

decision of whether to solve the nonlinear system on the computer or to linearize the expense of more thorough analysis

the equations must be made by comparing with the degree of accuracy required.

COMPUTER SETUP OF REACTOR-SYSTEM EQUATIONS

2

2.1

Basic Linear Operations

Computer representation of the basic linear operations represents a high-gain d-c amplifier. Addition:

AX R/ Ra

_=

+

BY

=

A

/f,

D

Rb

is as follows: The symbol >

-DZ _ B D

Addition

Integration:

EX +FY

G^"e

/AW// ///AW

— 1\

i

> Integration

8-98

OF IlEACTORS

CONTROL

[Sec. 8

Lagged integration:*

HX+JY=-(Z+L%) RfC li Rf Rh


Rj

=

L

—

" ti

=

X *

J

«AWRj

I

WWW

I

Lagged integration

Note that an operational amplifier always inverts (changes sign). If a positive value is needed, an additional unity gain amplifier must be used. The relations determining resistance and capacitance values always permit free choice of one component. Condensers are usually chosen to be even-valued and of such value that all resistances will fall somewhere between 50,000 ohms and 10 megohms. Low values of input resistance may load the previous amplifier, while excessively high values may introduce noise problems. If potentiometers are used in In the the system, careful selection of components will minimize loading errors. illustrative block diagram the linear operations have been combined with symbolically represented nonlinear operations. The coefficients of the scaled equations for the example reactor are such as to require no computing gains greater than 10. Gains as high as 50 are feasible with most commercial equipment, but high gains are to be avoided because of the accentua tion of computer noise and stability problems. If equations scaled on the basis of the expected range of the variables result in a wide range of coefficients and con sequently high gains, the physical analysis should be carefully examined because a small coefficient in the scaled equation implies a small influence on the physical problem. In some cases different scale factors for a particular variable in different equations will result in more reasonable coefficients. To facilitate problem checkout, the computer used should permit easy readout of the voltage representing every variable. Before dynamic analysis is attempted, the computer voltages should be checked against calculated sets of steady-state voltages at a number of operating points. If "real time" (computer time scale same as physical time scale) simulation is used, dynamic responses can be conveniently X-Y plotters recorded on any of a number of commercially available oscillographs. If a con for use in phase plane analysis or other applications are also available. densed time scale repetitive computer is used, responses may be displayed and photo graphed on an oscilloscope. Once the reactor analogue is set up and checked out, the effect of any design parameter on system stability can be rapidly evaluated and a mocked-up control system can be designed which will provide specifications for reactor control components. Certain remarks are pertinent to the specific computer setups representing nuclear reactor equations. The multiplication in the neutron balance equation (8) must be carried out rapidly if short reactor periods are to be considered and use of an elec Drift, present in many electronic multipliers, is not a tronic multiplier is warranted. serious problem in this application, since the reactor system feedback will stabilize the system. The flow and heat-transfer-coeffieient multiplication by temperature is best accom plished with scrvomultipliers whose response is adequate for this purpose, since multiplication of many temperatures by the particular variable requires only a single Generation of the heat-transfer coefficient as a unit with ganged potentiometers. function of flow may be accomplished by using a separate servo or electronic function generator. Often adequate representation is obtained by assuming that the heattransfer coefficient is proportional to flow and using a single servo to take care of both variables. The results obtained from reactor simulators should be used in the light of what * Lagged integration refers to a timo constant

or first-order response as opposed to pure integration.

Sec. 8-5]

8-99

REACTOR-SYSTEM DYNAMIC SIMULATION

they represent — the best available information on a difficult problem. A serious attempt should be made to evaluate the effect of approximations used in obtaining the computer equations. Some of the numbers supplied to the control designer by physicists and design engineers are far from accurate, and many simplifications in the system must be made to come up with a set of equations which will fit on the available The best hedge against erroneous data is to vary parameters computing equipment. over a wide range in stability analysis and control-system design. Testing Control Components on a Simulator

2.2

The dynamic performance of the reactor system coupled to physical control com ponents can be evaluated through the use of suitable transducers to couple the control components to a real-time analogue. This is of particular value in evaluating nonlinearities such as backlash, nonviscous friction, speed-limiting overrides, and power saturation in such devices as control-rod drives. Simplification and Linearization


2.3

To set up the system of equations chosen to represent the example reactor 40 opera tional amplifiers, two electronic multipliers, and four servo multipliers are required. This relatively large amount of equipment was needed adequately to represent the Reactor stability information heat-transfer process in the core and heat exchanger. can be obtained from a simpler model involving linearization and using a single wellmixed region fuel clement and coolant system and a single well-mixed region heat exchanger. However, if relatively accurate control-design information is to be obtained, the heat-transfer systems should be split up into as many elements as is To illustrate this simplified approach linearization will be discussed and feasible. then applied to a simplified model of the example reactor. Linearization is based on approximating nonlinear functions by the linear terms of their Taylor series expansion about an operating point in question. For example, if a particular variable X was known to depend on two other variables Y and Z a small perturbation of that variable AX would be represented by AX

It

=

^

follows that linearization of a product

AX

= Z0

AY +

M

X

YZ

AY

=

AZ 0

gives

+ Y„AZ

Similar operations may

be carried out on other nonlinear functions occurring in reactor dynamic analysis. Linearization of the scaled equations for the example reactor modified to take into account the single-region model is discussed and listed below. It should be rioted that since scaled equations were linearized, scaled steady-state values must be used in investigating a particular operating point. For example, a fV of 0.5 would correspond to design-point operation at 20 Mw. When a linearized analysis is used, the dynamic characteristics should be investigated for small perturbations about an adequate number of operating points. The design-point values were used in obtaining the numerical values for the computer block diagram of the linear model. Note that 11 operational amplifiers are the only active components in this analogue of the simplified and linearized system. 0.0667

dt

Ak„' 10.5

=

+ AP' = 6.67/Y Afc„' + 0.553AC + 0.447AC

-O.IOAT/

±^f± AC,' dt

=

- 0.25A7V

+ AP'

+ 0.02A7V + 0.50AR' 1.32

dt

+

AC

=

AP'

(26) (27) (28)

8-100 0.0554

rt&T/

(UV)0-8 rco')°"

dt

+ at/ T '

6(Hrc0')08

-

[^r^r, (r'°' 3.036

2.66HV +

rf

6(HV)"

r<°')

A2V

,

=

_' , _

+" AT/,' =

««

!

(^£„')0

"

[ ■'

-0.527[7V +

-

WV

IW)]

(J*/.'

- ?V)AWV

8

(29)

at/I

ait.' + ear..')"

(30)

2.6GHVT^(WVF»[2-66Tre0'Are'

+ 6(TF,.')« AT,'

-

- A7W

AT./

- -M

AP'

[0.527MV + (WV)°'] {a527^»'Ar»--' +

[Sec.

2.8261F,

- 2.826(rc„' -

- 2.66(7W - 7V)AHY

Ar"'

0 663

A7Y + T

=

dATV

0-2478

2.826UV + +

OF REACTORS

CONTROL

+ +

^

+

- 4-8(7'"-'

~

^°')

AWV]

^0U4W

TV)] ATT.'J ^ W1 AP-

_2j*_ (IW

(325

(33,

2.31 Discussion of Results Obtained from Simplified Linear Equations. Figure '& presents typical response transients obtained from the simplified model of the example


ALL CAPACITANCE VALUES

JO

Fio.

IN MICROFARADS AMPLIFIER □ -INDEPENDENT INPUT ALL RESISTANCE VALUES IN MEGOHMS

[>- OPERATIONAL

2. Simplified

linear analogue setup for example reactor.

reactor whose set up block diagram is shown in Figure 2. Certain of the results are worthy of comment: 1. The heat exchanger provides a large amount of damping for the reactor output temperature, which means that rapid power excursions may be tolerated without

Sec. 8-5]

.


affecting the external system.

seriously

8-101

However, the power excursion rate may be

limited by transient temperature considerations in the fuel elements.


2. Control-rod reactivity changes and reactor-coolant inlet-temperature changes produce relatively rapid changes in reactor power level. Moderator temperature

0

12

TIME, SEC

Fia.

3.

3

12

3

TIME, SEC

4

5

I

2

TIME, SEC

3

0

12

TIME, SEC

3

Typical response curves from simplified analogue of example reactor.

responses (not illustrated) were much more sluggish, indicating that moderator tem perature might be used as a shim control but would be unsatisfactory for dynamic control. 3. The step response of Ta to 7'0,„ due to the single-region heat-exchanger model is obviously unrealistic and represents one of the reasons for employing a multiregion model. 4. The stable nature of this reactor is one of the primary facts which can be deter mined from the simplified linear analogue. The tolerable range of reactivity coeffi

8-102

CONTROL

OF REACTORS

cients which will still maintain adequate stability can also be the aid of this model. 5. The responses obtained from the simplified linear reactor viding preliminary design data for the reactor and control such as those mentioned in item 3 above do not preclude quantitative information.

.

[SEC. 8

easily investigated with model are useful in pro system. Discrepancies obtaining useful semi

REFERENCES 1. 2. 3.

Korn, G. A., and T. M. Korn: "Electronic Analog Computers," 2d ed., McGraw-Hill Book Company, Inc., New York, 1956. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952. Collier, D. M., L. A. Meeks, and J. P. Palmer: "The HRE Simulator," ORNL-1572, 1954. (Unclassified.)

BIBLIOGRAPHY 1. 2.

Bell, P. R., and H. A. Straus: Electronic Pile Simulator, Rev. Sci. Inst. 21(8): (August, This article discusses a special-purpose reactor simulator. 1950). Collier, D. M., L. A. Meeks, and J. P. Palmer: "The HRE Simulator," ORNL-1572, 1954.

3. 4.


5.

6. 7.

Fischbeck, Kenneth H. : "Nuclear Reactor Simulators," 1954 Convention Record at the IRE, pt. 9, Medical and Nuclear Electronics. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952. Howard, R. C: "Analysis of the Nonlinear Kinetic Behavior of a Nuclear Power Reactor," M.S. Thesis, Massachusetts Institute of Technology, June, 1955. This thesis contains a good discussion of the effects of the variation of fission cross section with neutron temperature and the effect of variable reactivity coefficients and presents a heat-exchanger representation based on the use of dead time units. Korn, G. A., and T. M. Korn: "Electronic Analog Computers," McGraw-Hill Book Company, Inc., New York, 1952. Moise, J. C: Application of Analog Computing Techniques to Reactor Dynamic Analysis, Chem. Eng. Symposium Ser. 12, Nuclear Engineering, pt. 2, p. 96, Ann Arbor, 1954.

8. 9.

O'Meara, F. E.: Reactor Simulators, J. Appl. Phys., September, Pagels, Walter: A Portable Kinetic Simulator, AIEE Trans., vol.

1953. 70,

Paper 51- 262, 1951.

SECTION

9

FLUID AND HEAT FLOW BY

WAYNE H. JENS,

B.S., M.S., Ph.D., Assistant Technical Director, Atomic Power Development Associates, formerly Senior Engineer, Nuclear Development Cor poration of America.

CHARLES F. BONILLA, Ch-E., Ph.D.,

Professor of Chemical Engineering, Columbia

University.

ROBERT DAANE, B.S., M.S.,


Research Engineer, Bcloit Iron Works, formerly Senior Mechanical Engineer, Nuclear Development Corporation of America, and Associate Mechanical Engineer, Argonne National Laboratory.

CONTENTS PHYSICAL PROPERTIES OF HEAT-TRANSPORT MEDIUMS

9-1

PAOE

9-2

9-2 9-22

FLUID FLOW IN REACTOR SYSTEMS BY CHARLES

F. BONILLA

1 Viscosity 2 Material and Mechanical Balances 3 Streamline Flow 4 Turbulent Flow in Ducts 5 Flow of Slurries (Bingham Bodies) f> Flow of Vapor-Liquid Mixtures 7 Flow-distribution Control 8 Losses in External Flow 9 Mechanical Pumps and Blowers References

HEAT REMOVAL FROM NUCLEAR REACTORS

BY CHARLES

BY WAYNE H. JENS Tables References

9-3

9-25 9-27 9-29

F. BONILLA

PAOE 1 The Spatial Distribution of Heat Gener ation in Reactors 9-51 2 Heat Conduction 9-52 3 Convection 9-57 4 Condensing Vapors 9 -65 5 Boiling Liquids 9-Ofi 6 Steady-state Thermal Design of Reactors 9-84 7 Special Steady- and Unsteady-state Cal 9-93 culation and Test Methods 9-100 References

9-4

0-31

9-40 9-40 9-44 9-46 9-48 9-49

THERMAL STRESS AND DISTORTION BY ROBERT

1 Thermal Stress and

ticity Theory

2 Thermal-stress Fracture References

9-1

DAANE

Distortion by Elas 9-1 10 9-114 9—116

PHYSICAL PROPERTIES OF HEAT-TRANSPORT MEDIUMS

9-1

BY Wayne H. Jens Table

Physical Properties of Air at Atmospheric Pressure

1.

Dynamic viscosity


Temperature

Thermal conductivity *,

Specific heat Cp,

Prandtl number

°K

°R

lb/(hr)(H)*

Btu/(ft)(hr)(°F)»

Btu/(lb)(°F)t

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1300 1600 1700 1800 1900 2000

180 360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700 2880 3060 3240 3420 3600

0.0167 0.0321 0.0447 0.0553 0.0645

0.00535 0.01044

0.246

0.01518

0.240 0.242 0.246

0.0729 0.0806 0.0876

0.0942 0. 1003 0. 1063 0. 1117

Npr

0.251

0.257 0.262 0.268 0.273 0.277 0.282 0.286 0.290 0.294 0.298 0.303 0.308 0.313

0.1172 0.1223 0.1270 0.1320 0. 1366 0.1410 0.1455

JlCp/tf

0.770 0.739 0.708 0.689 0.680 0.680 0.684

0.241

0.01943 0.02340 0.02700 0.03030 0.03340 0.03630 0.03900

-

0.689

0.696 0.702

0.320

* J. Hilaenrath and Y. S. Touloukian. The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oi, Hj, NO, Hi. CO, COi, HrO, He, and A, Trans. ASMS, 56: 967-985 (August, 1954). t J- F. Masi, Survey of Experimental Determinations of Heat Capacity of Ten Technically Important Gases. Trant. ASME, 56: 1067-1073 (October, 1954).

Table 2. Temperature, »F 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200

Enthalpy of Air at Low Pressures*

Enthalpy,

Btu/lb 23.74 35.71 47.67

59.64 71.61

83.57 95.53 107.50 119.48 131.46 143.47 155.50 167.56 179.66 191.81 204.01 216.26 228.58 240.98 253.45 265.99 278.61 291.30

* Reprinted with permission, New York, 1948.

from

J.

Temperature, °F

Enthalpy,

1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300 2350

304.08 316.94 329.88 342.90 356.00 369.17

Btu/lb

382.42 395.74 409.13 422.59 436.12 449.71 463.37 477.09 490.88 504.71 518.61 532.55 546.54 560.59 574.69 588.82 603 00

H. Keenan and

9-2

J.

Temperature, °F

Enthalpy,

2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950 3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500

617.22 631.48 645.78 660.12 674. 49 688.90 703.35 717.83 732.33 746.88 761.45 776.05 790.68 805.34 820.03 834.70 849.48 864. 24 879.02 893.83 908.66 923.52 938 40

Btu/lb

Kaye, 'Gas Tables," John Wiley & Sons, Ino.,

heat-transport

Sec. 9-1]

9-3

mediums

Specific Heat of Air at High Pressure* [c„ Btu/(lb)(°F)]

Table 3.

Pressure,

atm

Temperature, °C '

-

0.240 0.239 0.238 0.238 0.237

100 0 50

-100 -150 *

J.

York,

10

H. Perry (ed.)f "Chemical

0.311

0.259 0.246 0.242 0.239

20

40

70

100

0.505 0.258 0.257

0.370 0.279

0.247

0.251

0.240

0.245

0.846 0.332 0.277 0.250

0.412 0.298 0.258

Engineers' Handbook,"

McGraw-Hill Book Company, Inc., New

1950.

Table 4. Absolute

Physical Properties of Argon

Dynamic

temperature

Thermal conductivity

viscoBity

Specific heat

Prandtl number,


*,

T. °K

T, °R

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

180 360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700

lb/(hr)(ft)*

Btu/(hr)(it)(°F)*

0 0201 0. 0393 0.0555 0.0693 0.0817 0.0931 0. 1033 0 1130 0.1266 0.1313 0.1400 0.1480 0.1560 0.1640 0. 1710

0.00372 0.00718 0.01020 0.01285 0.01520 0.01740

Btu/(lb)(°F)t

N,r

- Cpll/k*

0.692 0.676 0.670 0.664 0.658 0.655 0.654 0.653 0.654 0.654 0.656 0.657 0.659

0. 128

0.125 0.124 0. 123

0.122 0. 122

(0.122) (0.122) (0.122) (0.122) (0.122) (0.122) (0.122) (0.122) (0.122)

0.661 0.663

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air. Oi, Ni, NO. Hi, CO, CO., HjO, He, and A, 7Vanj. ASMS, 66: 967 (August, 1954). t Values in parentheses added by author.

Table 6.

Viscosity of Argon under Pressure* |m, lb/(hr)((t>] Pressure,

atm

Temperature, "C 1

20 50 100 150 200 300

0.0582 0.0655 0.0712 0.0772 0.0873

20

0.0550 0.0593 0.0664

0.0722 0.0782 0.0880

40

60

0.0567

0 0585

0.0610

0.0630

0.0673 0.0743

0.0693

0.0797 0 0890

0.0748 0.0814 0.0906

80

100

0 0654

0.0717 0.0770

0.0832 0.0922

(0.0745) (0.0800)

(0.0860) (0.0949)

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oi, Nj, NO, Hj, CO, COi, HjO, He, and A, Trans. ASME, 56: 967-985 (August, 1954).

FLUID AND HEAT FLOW

[Sec. 9

Effect of Pressure on Thermal Conductivity of Nitrogen and Argon at 127°F*

Table 6.

Pressure, atm



of nitrogen,

of argon,

Btu/(hr)(ft)(°F)

1 30 90 150 215

Btu/(hr)(ft)(°F)

0.0162 0.0175 0.0185 0.0205 0.0226

0.0114 0.0119 0.0136 0.0156 0.0168

* J. M. I^cnior, W. A. Junk, and E. W. Comings, Measurement and Correlation of Thermal Con ductivity of Gases at High Pressure, Chem. Eng. Progr., 49 (10): 539 (October, 1953).

Table 7.

Physical Properties of Carbon Dioxide at Atmospheric Pressure

Absolute temperature

Dynamic


viscosity

Specific heat


k,

T, °K

T, °R

200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700

360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700 2880 3060

lb/(hr)(ft)»

0.0245 0.0362 0.0467 0.0561 0.0648 0.0727 0.0803 0.0873 0.0941 0. 1005 0. 1065 0. 1125 0. 1183 0. 1240 0. 1295 0. 1345

Btu/(lb)(°F)t

Btu/(ft)(hr)(°F)»

0.00554 0.00959 0.01425 0.01940 0.02430 0.02850 0.03240 0.03600 0.03930 0.04230 0.04520 0.04770

,

Prandtl number

0. 189

0.204 0.222 0.242 0.256 0.269 0.279

0.288 0.294 0.300 0. 306 0.310 0.313 0.317

(0.839) 0 770 (0.727) 0.702 0.685 0.687 0.692 0.698 0 705

0.712 0.721 0.729

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity and Prandtl Number for Air, Ot, Ni, NO, H;, GO, COi, H.O. He. and A, Trans. ASMB, 56: 967-985 (August, 1954). t J. F. Masi. Survey of Experimental Determinations of Heat Capacity of Ten Technically Important

Gases, Trans. ASME, 56: 1067-1073 (October, X Values in parentheses added by author.

Table 8.

1954).

Viscosity of Carbon Dioxide under Pressure* [M, lb/(hr)(ft)] Pressure,

atm

°c

50 100 150 200 250 300

1

20

0 0380 0.0442 0.0494 0.0542 0.0592 0.0637

0.0406 0.0452 0 0504 0.0558 0.0603 0 0647

40

0 0 0 0 0 0

0437 0476 0521 0577 0620 0633

60

80

0.0494 0.0517 0.0571 0.0608 0.0642 0.0681

0 0584 0.0616 0.0630 0.0652 0 0676 0 0702

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oi, Ni, NO, Hj, CO, COt, HiO, He, and A, Trans. ASME, 66: 967-985 (August, 1954).

heat-transport

Sec. 9-1]

Table 9. Absolute

Physical Properties of Helium*

Dynamic viscosity

temperature


Specific heat

*,

T, °K

T. °R

100 200 300 400 500 600 700 800 900 1000 1100

180 360 540 720 900 1080 1260 1440 1620 1800 1980

9-5

mediums

lb/(hr)(ft)

Btu/(hr)(ft)("F)

Btu/(lb)(°F)t

0.0231 0.0366 0.0481 0.0585 0.0678 0.0767 0.0852 0 0930 0.1025 0.1080 0.1150

0.0422 0.0673 0.0868 0. 1031 0.1175 0.1300

( 1. 237) ( 1. 237)

(1.237)

( 1. 237)

(1.237)

( 1. 237)

Prandtl number Npr = c,it/k

0.677 0.674 0.686 0 700 0.715 0.729

(1.237) (1 237)

(1.237) (1.237) (1.237)

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal, Conductivity, and Prandtl Number for Air, Oi. Ni, NO, Hi, CO, COt, H»0, He, and A, Trans. ASME, 56: 967-985 (August. 1954). f Values in parentheses added by author.


Table 10. Absolute temperature

Dynamic viscosity

Physical Properties of Hydrogen Thermal conductivity

Specific heat

*,

°K

T, °R

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

180 360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700

T.

lh/(hr)(ft)»

Btu/(hr)(ft)(°F)*

Btu/(Ib)(°F)t

0.0102 0.0165 0.0217 0.0263 0.0306 0.0346 0.0383 0.0420 0.0453 0.0487

0.0385 0.0740 0.1050 0.1323 0.1 580 0.1828 0.2080

2.68 3.24 3.42 3.46

0.0518

3.47

3.48 3.50 3.51

Prandtl number NP, cr„/k*-X

-

(0.718) (0.721) (0.707) (0.687) (0.671) (0.659) (0.661)

3.52 3.58 3.61 3 68 3 72

3.78 3.83

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oi, Nj, NO, Hj, CO, COi. H»0, He, and A, Trans, ASME, 56: 967-985 (August, 1954).

t J. F. Masi, Survey of Experimental Determinations of Heat Capacity of Ten Technically Important Gases, Trans. ASME, 86: 1067 1073 (October, 1954). X Values in parentheses added by author.

9-6

FLUID AND HEAT FLOW Table 11.

[SEC.

9

Viscosity of Hydrogen under Pressure* 1m,lb/(hr)(ft)] Pressure,

atm

Temperature,

°C 1

-

-150

0.0119 0.0149 0.0203 0.0250 0.0294 0.0337

0.01165 0.0148 0.0202 0.0250 0.0294 0.0337

100 0 100 200 300

40

20

0.0120 0.0150 0.0204 0.0251 0.0295 0.0337

60

80

100

0.0122

0.0123 0 0152 0 0206 0.0252 0.0298 0.0338

0.0129 0 0153 0 0207 0 0254 0.0298 0.0338

O.OISI 0.0205 0.0252 0.0297 0.0338

* J. Hilsenrath and Y. S. Tuuloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air, O.. Nt, NO, Hi, CO, CO., H.O, He, and A, Tram. ASME, 66: 967-985 (August, 1954).

Table 12.

Physical Properties of Nitrogen at Atmospheric Pressure


Abe >lute tempc rature

r. "K

T, °R

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

180 360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700

Dynamio


viscosity

lb/(hrj(ft)«

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Specific heat

*,

Btu/(ft)(hr)(°F)* Btu/(lb)(°F)t

0166 0314 0433 0533 0622 0705 0783 0858 0922 0990 1055 1114 1173 1232 1290

0.00546 0.01052 0.01510 0.01928 0.02300 0.02650 0 02960 0.03240 0. 03500 0.03740 0.03960 0.04150

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

256 250 249 250 252 257 262 268 274 279 284 288 291 294 297

Prandtl number

N,r

-

yc,lk*

0.780 0.747 0.713 0.691 0 684 0.686 0.695 0.708 0 723 0.740 0.757 0.774

* J. Hilsenrath and Y. S. Touloukiun, The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oi. N«, NO, H,, CO. COt, H.O, He, and A, Tran,. ASME, 56: 967-985 (August, 1954). t J, F. Masi, Survey of Experimental Determinations of Heat Capacity of Ten Technically Important Gases. 7Von«.

ASME.

56:

1067-1073 (October,

1954).

heat-transport

Sec. 9-1]

d-7

mediums

Viscosity of Nitrogen under Pressure*

Table 13.

fj,, lb/(hr)(ft)] Pressure,

atm

°c

20 50 100 200 400 600 800 1000 1200

1

20

40

60

80

100

0.0425 0.0455 0.0535 0.0599 0.0761 0.0907 0. 1040 0. 1160 0. 1270

0.0431 0.0460 0.0512 0.0603 0.0765 0.0910 0. 1040 0. 1160 0. 1280

0.0446 0.0470 0.0515 0.0606 0.0767 0.0910 0. 1040 0. 1160 0. 1280

0.0460 0.0478 0.0523 0.0610 0.0770 0.0920 0. 1050 0. 1170 0. 1290

0.0476 0.0487 0.0533 0.0618 0.0775 0 0920 0. 1050 0. 1170 0. 1290

0.0490 0.0499 0.0540 0.0625 0.0780 0.0921 0. 1050 0. 1170 0. 1290

* J. Hilsenrath and Y. S. Touloukian, The Viscosity, Thermal Conductivity, and Prandtl Number for Air. Oi, N», NO, Hi, CO, COj, HiO, He, and A, Tram. ASME, 86: 967-985 (August, 1954).


Table 14.

Specific Volume of Steam* (», ft'/lb) Pressure,

psia

Temperature,

°F

32 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

15

1.000

2.000

3.000

4.000

5.000

6.000

26. 29 29.91 33.97 37.99 41.99 45.98 49.97 53.95 57.93 61.91 65.89 69.86 73.84 77.92 81.79

0.5140 0.6084 0.6878 0. 7604 0.8294 0.8962 0.9615 1.0257 1.0893 1. 1524 1.2146

0.2489 0.3074 0.3532 0.3935 0.4311 0. 4668 0.5021 0.5352 0.5693 0.6011

0.0984 0. 1760 0.2159 0.2476 0.2757 0.3018 0.3277 0.3505 0.3752 0.3966

0.0287 0. 1052 0. 1462 0. 1743 0. 1979 0. 2192 0. 2409 0. 2581 0.2786 0. 2943

0.0268 0.0593 0. 1036 0. 1303 0. 1513 0. 1696 0. 1890 0. 2027 0. 2207 0.2329

0.0256 0.0393 0.0758 0. 1021 0. 1223 0. 1392 0. 1545 0. 1688 0. 1825 0. 1954

* Reprinted, with permission, from J. H. Keenan and F. G. Keyes," " Thermodynamic Propertied Steam," John Wiley & Sons, Inc., New York, 1936.

of

9-8


[SEC.

9

Dynamic Viscosity of Steam* [m, lb/(hr)(ft)l Pressure,

psia

Temperature,

°F 0


32 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

026 028 032 036 040 044 049 055 059 065 068 070 073 075 078 081 085

1.000

2.000

3.000

4,000

5.000

6.000

0.070 0.072 0.075 0.077 0.080 0 082 0.085 0.087 0.092 0.094 0.097

0.086 0.087 0.088 0.090 0.092 0.094 0.097 0.099 0. 102 0. 104

0. 0. 0. 0 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0

0.252 0. 167 0. 148 0. 140 0. 138 0. 135 0. 135 0.137 0.139

106 104 104 104 104 106 109 110

Ill

133 118 116 114 114 116 118 121 123

186 140 130 126 126 126 126 127 128

• A. W. Lemmon, Jr., D. J. Daniels. D. E. Sparrow, C. J. Geankoplis, J. J. Ward, and J. W. Cleg?. "Empirical Evaluation of the Properties of Steam at Elevated Temperatures and Pressures," BM1-858, Battelle Memorial Institute, Aug. 25. 1953.

Table 16.

Thermal Conductivity of Steam* [*, Btu/(hr)(ft)(°F)] Pressure,

psia

Temperature, °F 0

32 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

0.011 0.012 0.014 0.016 0.018 0.020 0.023 0 026 0.029 0.032 0.035 0.037 0.040 0.043 0.046

1.000

0.048

0.030 0.031 0.032 0.034 0.036 0.039 0.042 0.044 0.047 0.049

0.051

0.052

2.000

0 034 0.035 0.036 0.038 0.041 0.043

0.046 0.049 0.051

0.054

3.000

4.000

5,000

6.000

0.041 0.040 0.041 0.043 0.045 0.048 0.051 0.053 0.055

0.050 0.044 0.044 0.046 0.048 0.050 0.052 0.055 0.057

0.071 0.050 0 047 0.048

0.096 0.056

0.050 0.052

0 . 051 0 051 0 053 0 054

0.057

0.057 0.059

0.059

0.061

0.055

* A. W. Lemmon, Jr., D. J. Daniels, D. E. Sparrow, C. J. GeankopUs, J. J. Ward, and J. W. Cleg*. "Empirical Evaluation of the Properties of Steam at Elevated Temperatures and Pressures," BMI-65S. Battelle Memorial Institute, Aug. 25. 1953.

heat-transport

Sec. 9-1] Table 17.

mediums

Constant-pressure Specific Heat of Steam* kr . Btu/(lb)(°F)] Pressure,

°F

32 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

15

1.000

2.000

psia

3.000

4.000

5.000

6.000

1.20

2.03 1.06

3.60

0.82 0.72 0.68 0.65 0.64 0.63 0.63

0.95 0.80 0.72 0.70 0.68 0.66 0.65

2.80 1.77 1.07

0.42 0.43 0.44 0.45 0.46 0 47

0.48

0 88

0.68

0 48

0.49 0.50 0.50 0.51 0 52

0.53 0.54 0.55 0.56

0.61

1.20 0.81

0.58 0.56 0.56 0.57 0.57 0.57 0.57 0.57

0.70 0.63 0.60 0.59 0.58 0.58 0.59 0.59

0.83 0.72 0.66 0 63

0.62 0.62 0.61 0.61

* Reprinted, with permission, from J. II. Keenan and F. G. Keyea, Steam," John Wiley A Sons, Inc., New York, 1936.


9-9

Table

18 .

1.28

0.89 0.78 0.71

0.69 0.68 0.67

Thermodynamic Properties of

Prandtl Number crti/k for Steam Pressure,

psia

Temperature. op

0

32 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

1.000

2.000

4.000

5.000

6.000

3. 10 2 18 83 60 52 43 29 27 23

5.40

9.40 3.58

7.30

0.99 1 00 1.01 1.02 1.02 1.03 1.02 1.01 1.00 1.00

0.98 0 97

0.95 0.94 0.92 0.93 0.94

Table 19.

2 05 1.58 1.43 1.31 1.24 1.18 1.15 1.13 1.10 1.09 1.08

3 04 2 01 71 49 34 29 22

IS 16 13

2.84 2. 16 1.78 1.61 1.51 1.45 1.39 1.36

2 63 2. 10 1.81 1.70 1.55 1.47 1.41

5.10

3.10 2.94 2.32 1.77 1.63 1.58 1.53

Critical Properties of Light Water*

705. 40 °F 374. 2 °C Critical pressure 3.206 psia 218.2 atm 19. 9 lb /ft" Critical density 0.320 c/ml " International Critical Tables of the Numerical Data of Physics, Chemistry, and Tech West. McGraw-Hill Book Company, Inc., New York, 1933.

Critical temperature

* C. J. nology,"

3.000

9-10


Temper ature,


°F

32 40 SO 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 212 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700

[Sec.

9

Specific Volume of Liquid Light Water* (r, ft' /lb) Pressure,

psia compressed

liquid

Saturated

liquid

1.000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01602 01602 01603 01604 01606 01608 01610 01613 01617 01620 01625 01629 01634 01639 01645 01651 01657 01663 01670 01672 01677 01684 01692 01700 01709 01717 01726 01735 01745 01755 01765 01776 01787 01799 01811 01823 01836 01850 01864 01878 01894 01910 01926 01940 0196 0198 0200 0202 0204 0207 0209 0212 0215 0218 0221 0224 0228 0232 0236 0241 0247 0253 0260 0268 0278 0290 0305 0328 0369

0.01597 0.01597 0.01598 0.01599 0.01601 0.01603 0.01605 0.01608 0.01612 0.01615 0.01620 0.01624 0.01629 0.01634 0.01640 0.01647 0.01652 0.01658 0.01665 0.01667 0.01672 0.01679 0.01687 0.01695 0.01703 0 0171 1 0.01720 0.01728 0.01738 0.01748 0.01758 0.01768 0.01779 0.01791 0.01803 0.01815 0.01828 0.01842 0.01856 0.01870 0.01897 0.01903 0.01919 0.01933 0.0195 0.0197 0.0199 0.0201 0.0203 0.0206 0.0209 0.0212 0 0215

2.000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01591 01591 01592 01594 01596 01598 01600 01603 01607 01610 01615 01619 01624 01629 01635 01641 01646 01652 01659 01661 01666 01672 01680 01688 01696 01704 01713 01721 01731 01741 01750 01765 01772 01783 01793 01804 01817 01830 01844 01858 01873 01888 01923 01916 0193 0195 0197 0199 0201 0204 0206 0209 0212 0215 0218 0221 0225 0229 0233 0239 0245 0252

3.000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01586 01586 01587 01588 01589 01591 01594 01598 01602 01605 01610 01614 01620 01624 01630 01635 01640 01647 01654 01655 01660 01667 01674 01682 01690 01698 01706 01715 01725 01735 01745 01755 01765 01775 01787 01796 01808 01820 01832 01847 01861 01876 01890 01902 0191 0193 0195 0197 0199 0202 0204 0207 0209 0213 0215 0217 0220 0225 0228 0233 0237 0242 0248 0254 0263 0275 0288 0311 0359

4.000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01580 01580 01581 01583 01586 01588 01590 01593 01597 01600 01605 01609 01614 01619 01626 01630 01636 01642 01649 01650 01656 01662 01670 01677 01686 01690 01699 01707 01717 01725 01735 01746 01755 01764 01775 01787 01798 01812 01824 01837 01850 01866 01880 01890 0190 0192 0194 0196 0198 0201 0203 0205 0207 0209 0211 0214 0217 0220 0223 0226 0231 0233 0240 0243 0251 0255 0265 0275 0287

6.000

5.000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01575 01575 01576 01578 01583 01585 01587 01589 01592 01596 01600 01604 01609 01614 01620 01626 01631 01539 01642 01646 01649 01654 01659 01669 01680 01681 01694 01701 0171 1 01719 01728 01741 01751 01761 01771 01780 01790 01802 01814 01825 01830 01850 01866 01876 0189 0191 0193 0195 0197 0199 0201 0203 0204 0206 0208 0210 0213 0216 0219 0223 0228 0230 0236 0239 0245 0249 0255 0260 0268

* Reprinted, with permission, from J. H. Kcenan and F. G. Keyes, "Thermodynamic Steam," John Wiley & Sons, Inc., New York, 1936.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

01570 01570 01571 OIS73 01575 01579 01 580 01582 01588 01590 01595 01599 01604 01609 01615 01621 01627 01633 01639 01642 01645 01650 01655 01665 01676 01677 01688 01695 01705 01715 01724 01734 01744 01755 01765 01771 01780 01792 01805 01815 01820 01840 01856 01865 0189 0190 0192 0193 0195 0197 0199 0200 0201 0203 0205 0207 0210 0213 0216 0220 0224 0228 0232 0236 0240 0244 0248

0.0252 0.0256

Properties of

J*

Sec. 9-1]

HEAT-TRANSPORT

Table 21.

Constant-pressure

9-11

MEDIUMS

Specific Heat

cp

of Liquid Light Water*

[Conversion factor: From Btu/(lb)(*F) to cal/(g)(°C)

multiply by I]

1

c,, Btu/(lb)(°F)


Temper ature. °F

32 40 SO 60 70 80 90 100 110 120 130 140 ISO 160 170 180 190 7.00 210 212 220 230 240 2S0 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690

Pressure,

psia, compressed

liquid

Saturated liquid

1.0083 1. 0048 1.0004 0.9990 0. 9980 0.9975 0.9974 0.9976 0.9976 0.9977 0.9981 0.9988 0.9996 1. 0004 1.0012 1.0022 1.0034 1. 0047 1.0061 1. 0064 1. 0079 1. 0097 1.01 19 1.0142 1.0165 1 0191 1.0222 1.0255 1.0289 1.0322 1.0354 1.0404 1.0455 1 0511 1.0564 1.0614 1.0669 1.0728 1.0794 1.0861 1.0941 1. 1023 1. 111-4 1. 1212 1. 1319 1 1431 1. 1545 1 1698 1. 1861 1.21 1.23 1.25 1.28 1.31 1.34 1.37 1.41 1.46 1.51 1.58 1.65 1.71 1.88 2 09 2.34 2.84

3.5

5 5

1.000

2.000

3.000

1.0032 1.0014 0.9991

1. 0004 0.9986 0.9963 0.9939 0.9920

0.9979 0.9959 0.9933 0 9907 0.9888 0.9878 0.9871 0.9864 0.9860

0.9968 0.9949 0.9943

0.9938 0.9932 0.9932 0.9934 0.9937 0.9940 0.9950 0.9959 0.9969 0.9980 0.9984 1. 0008 1.0021 1.0024 1.0039 1.0057 1.0075 1. 0095 1.0117 1.0140 1.0163 1.0194 1.0232 1. 0270 1.0307 1.0353 1.0399 1.0445 1. 0496 1.0553 10611 1. 0668 1.074 1.080 1.087 1.095 1.105 1.114 1.124 1.136 1.149 1.163 1. 176 1.19 1.21 1.23

0.9912 0.9904 0.9897 0.9895 0.9895 0.9896 0.9897 0.9905 0.9913 0.9921 0.9931 0.9945 0.9958 0.9971 0.9974 0.9988 1.0005 1.0023 1.0041 1.0061 1.0082 1.0102 1.0131 1.0166 1. 0200 1.0235 1.0278 1.0322 1.0365 1.0411 1.0461 1.0510 1.0560 1.062 1.068 1.075 1.082 1.091 1. 100 1. 109 1.120 1. 131 1.143 1. 154 1.171 1.188 1.205 1.225 1.252 1.278 1.304 1.341 1.382 1.448 1.52 1.62

0.9858 0.9856

0.9854 0.9861 0.9868 0.9876 0.9885 0.9896 0.9907 0.9919 0.9921 0.9934 0.9951 0.9967 0.9984 1.0003 1.0023 1.0042 1.0069 1.0101 1.0132 1.0164 1.0205 1.0246 1.0287 1.0331 1.0377 1.0424 1.0471 1.053 1.059 1.065 1.072 1.079 1.087 1.095 1.104 1. 114 1.125 1. 135 1.131 1.166 1.182 1. 199 1.218 1.238 1.257 1.283 1.310 1.346 1.385 1.441 1.509 1.597 1.701 1.817 2. 14 2.56

4.000

5.000

6.000

0.9948

0.992 0.991 0.987 0.984 0.982 0.981 0.980

0.989 0 988 0 984 0.981 0.979 0.978 0.977 0.975

0.9926 0.9899 0.9872

0.9848 0.9840 0.9832 0.9824 0.9819 0.9817 0.9814 0.9811 0.9817 0.9823 0.9829 0.9836 0.9847 0.9857 0.9868 0.9870 0.9881 0.9896 0.9910 0.9925 0.9943 0.9961 0.9979 1.0005 1.0037 1. 0068

1010

1.014 1.017 1.021 1.025 1.029 1.034 1.038 1.043 1.048 1.053 1.059 1.066 1.074 1.081 1.090 1.099 1.109 1.118 1.132 1.146 1.160 1. 175 1. 192 1.209 1.227 1.246 1.265 1.290 1.317 1.352 1.392 1.442 1.505 1.576

0.978 0.978 0.978 0.977

0.977 0.978 0.978 0.978 0.979 0.980 0.981

0.982 0.982 0.983 0.985 0.986

0.988 0.989 0.991 0.993

0.995 0.998

1.001 1.004 1.008 1 Oil 1.014 1.018 1.021 1.026 1.030 1.035 1.040 1.045 1.051 1.057 1.065 1.071 1.079 1.087 1.095 1.102 1.115 1. 127 1. 139 1.152 1.168 1. 182 1. 198 1.216 1.230 1.25 1.27 1.29 1.32 1.36 1.40 1.43

0.974 0.974 0.973 0.973

0.974 0.974 0.974

0.975 0.975 0 976

0.977 0.977

0.978 0.980 0.981 0.983

0.984 0.986 0.988 0.989

0.992 0.995

0.998

1.002 1.005 1.008 1.012 1.014 1.019 1.022 1.027 1.032 1.037 1.043 1.048 1.056 1.061 1.068 1.075 1.082 1.088 1.100 1.110 1. 121 1.132 1.147 1.158 1.172 1. 189 1.200 1.21 1.23 1.25 1.27 1.31 1.34 1.36

* E. J. Well man and W. L. Sibbitt, A Survey of Some of the Thermodynamic and Physical Properties of Water, Combustion, April, 1955. pp. 51-56.

9-12

FLUID AND HEAT PLOW

[Sec. 9

Table 22. Thermal Conductivity k of Liquid Light Water* [Conversion Factor: From Btu/(hr)(ft)(°F) to cal/(sec)(cm)(°C) multiply by 0.00413379] k,

Temper


ature, •F

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 212 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690

Btu/(hr)(ft)(°F)

Pressure,

psia, compressed

liquid

Saturated liquid

0.3185 0.3245 0.3319 0.3397 0.3472 0.3532 0.3589 0.3641 0.3689 0.3733 0.3772 0.3810 0.3836 0.3861 0.3884 0.3905 0.3923 0.3935 0.3943 0.3944 0.3950 0.3957 0.3961 0.3964 0.3964 0.3962 0.3959 0.3955 0.3952 0.3948 0.3944 0.3933 0.3921

0.3906 0.3891

0.3876 0.3857 0.3834 0.3809 0.3783 0.3753 0.3724 0.3693 0.3670 0.3640

0.3608 0.3575 0.3535 0.3494

0.3446 0.3397 0.3348 0.3298 0.3246 0.3189 0.3130

0.3064 0.2997 0.2919 0.2839 0.2753 0.2662 0.2565 0.2454 0.2335 0.2198

0.2056

0. 1854

1.000

2.000

3.000

4.000

5.000

6.000

0.3198 0.3260 0.3337 0.3414 0.3487 0.3537 0.3608 0.3659 0.3707 0.3751 0.3790 0.3828 0.3855 0.3880 0.3903

0.3211 0.3275

0.3227 0.3294 0.3377 0.3454 0.3530 0.3590 0.3651 0.3703 0.3760 0.3792 0.3831 0.3868

0.3246 0.3314

0.3266

0.3286 0.3358 0.3447 0.3528 0.3604 0.3665 0.3723 0.3777 0.3822 0.3862

0.3924 0.3942 0.3957 0.3970

0.3972 0.3977 0.3983

0.3988 0.3992 0.3992 0.3991 0.3987 0.3984 0.3981 0.3975 0.3969 0.3958 0.3947 0.3933 0.3919 0.3904 0.3885 0.3863 0.3840 0.3817 0.3787 0.3757

0.3728 0.3697 0.3664 0.3630 0.3595 0.3553

0.3510 0.3461

0.3410 0.3356

0.3356 0.3433 0.3508 0.3570 0.3629 0.3680 0.3727 0.3771 0.3810 0.3847 0.3875 0.3902 0.3925 0.3945 0.3964 0.3980 0.3995 0.3998 0.4003 0.4010 0.4016 0.4021 0.4021 0.4020 0.4018 0.4015 0.4013 0. 4006 0.3998 0.3988 0.3977 0.3965 0.3951

0.3936 0.3919 0.3900 0.3880 0.3860 0.3833 0.3806 0.3776 0.3745 0.3713 0.3678 0.3642 0.3603 0.3562 0.3519 0.3475

0.3424 0.3371

0.3318 0.3256 0.3190 0.3118 0.3045

0.2962 0.2875 0.2778

0.3897 0.3924 0.3947 0.3968 0.3986 0.4003 0.4017 0.4020 0.4028 0.4037 0.4043 0.4048 0.4048 0. 4048 0.4048 0.4047 0.4045

0.4038 0.4029 0.4019 0.4009 0.3997 0.3984 0.3970 0.3954 0.3937 0.3920 0.3903 0.3877 0.3850 0.3821 0.3791 0.3761 0.3727 0.3689 0.3650 0.3611 0.3574 0.3535

0.3486

0.3437 0.3386 0.3328 0.3265 0.3198 0.3130 0.3056 0 2979 0 2892 0. 2800 0 2700 0 2588 0 2469 0 2319 0.2161

0.3398 0.3476 0.3551 0.3616 0.3676 0.3727 0.3773 0.3815 0.3853 0.3891 0.3920 0.3948 0.3971 0.3992 0.4010 0.4027 0.4044 0.4047 0.4055 0.4064 0.4071 0. 4077 0. 4077 0. 4078 0.4079 0.4078 0.4076 0.4069 0.4060 0. 4050 0. 4040 0.4031 0. 4020 0.4006 0.3991 0.3975 0.3960 0.3944 0.3920 0.3894 0.3865 0.3836 0.3809 0.3775 0.3738 0.3698 0.3658 0.3626 0.3591 0.3545 0.3499 0.3452 0.3398 0.3339 0.3277 0.3215 0.3146 0.3074 0.2997 0.2916 0.2831 0.2736 0.2634 0.2516 0.2395 0.2242

0.3335

0.3422 0.3500

0.3576 0.3640 0. 3700 0.3751 0.3797 0.3839 0.3877 0.3914 0.3944 0.3973 0.3996 0.4017 0.4034 0.4051 0. 4067 0.4070 0.4079 0.4091 0.4099

0.4106 0.4106 0.4107

0.4109 0.4110 0.4109 0.4102

0.4094 0.4084 0.4075 0.4069 0.4059

0.4046 0.4031 0.4017 0. 4000 0.3983 0.3960 0.3936 0.3907 0.3880 0.3857 0.3823 0.3784 0.3744 0.3705 0.3677 0.3645

0.3602 0.3559 0.3515

0.3465 0.3410 0.3354 0.3298 0.3232 0.3167 0.3100 0.3030 0.2953 0. 2869 0. 2779

0.2679 0.2577 0.2463

0.3900

0.3938 0.3968 0.3997

0.4020 0.4041 0.4059 0 4074

0.4088 0.4091

0.4103 0.4118 0.4127 0 4135

0.4135 0.4136 0.4140 0.4141 0.4141 0 4137 0.4131

0.4122 0.4114 0.4108 0 4101

0.4089 0.4077 0 4063 0 4043 0.4021

0.3999 0.3976 0.3947 0.3922 0 3904

0.3872 0.3832 0.3791 0 3750 0 3725

0.3697 0 0 0 0 0 0 0

3657 3617 3577 3530 3478 3429 3380

0.3318 0.3257 0.3199 0.3137 0 3068

0.2993 0.2914 0.2830 0.2744 0.2661

* E. J. Wcllman and W. L. Sibbitt, A Survey of Some of the Thermodynamic and Physical Properties of Water, Combustion, April, 1955. pp. 51-56.

heat-transport

Sec. 9-1]

9-13

mediums

Table 23. Dynamic Viscosity m of Liquid Light Water* (Conversion factor: From lb/(hr)(ft) to poises multiply by 0.00413379] M. lb/(hr)(ft)

Temper ature,


°F

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 212 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690

Pressure,

psia, compressed

liquid

Saturated liquid 1.000

2.000

3.000

4.000

5.000

6.000

4.340 3.742 3. 169 2.731 2.372 2.084 1.847 1.650 1.487 1.353 1.236 1. 137 1.050 0.970

4.309 3.721 3. 154 2.722 2.369 2.084 1.848 1.654 1.492 1.360 1.244 1. 145 1.059 0.979

4.279 3.699 3. 140 2.714 2.365 2.083 1.850 1.658 1.498 1.366 1.252 1.154 1.068

4.248 3.677

4.218 3.656

4. 187 3.643 3.097 2.688 2.354 2.082 1.854 1.671 1.514 1.387 1.275 1. 179 1.095 1.015

4.157 3 613 3.082 2.679 2.350 2.081 1 855 1.675 1.519 1.394 1.283 1. 187 1. 103 1.024

0.839

0.849

0.902 0.786

0.912

0.738

0.796 0.748

0.660

0.670

0.595

0.604 0.578

0.702 0.687 0.624 0.568 0.542

0 515 0.494

0.472 0.452 0.436 0.420 0.405 0.391

0.378 0.366 0.355

0.346 0.336 0.327 0.318 0.310

0.302 0.294 0.287 0.280 0.274 0.267

0.262 0.256 0.251

0.246

0.240 0.235 0.230 0.225 0.221 0.217 0.214 0.210 0.205 0.200 0.195 0. 190 0.183 0.177 0. 169 0. 161 0. 148

0.711 0.697 0.633

0.551

0.524 0.502 0.480

0.460 0.442 0.426 0.411

0.396 0.384 0.372 0.361 0.351 0.340 0.330 0.321

0.312 0.304

0.296 0.289 0.282 0.276 0.270 0.263 0.257 0.251

0.246 0.241

0.988 0.921

0.858 0.806 0.757 0.721

0.706 0.680 0.643 0.614 0.587

0.560 0.533 0.511

0.488 0.468 0.450 0.433 0.418 0.404 0.390 0.378

0.367 0.356 0.345 0.335

0.326

0.317 0.309 0.301 0 293

0.286 0.280

0 273 0 267

0.260

0.255 0 249

0.244

0 239 0.235 0.231

0.226 0.222 0.217

0.212 0.207

0.202

3.126

2.705 2.361 2.083 1.851 1.663 1.503 1.373 1.259 1. 162 1.077 0.997 0.930 0 868

0.815 0.767 0.731

0.716

0 689

0.652 0.623

0.596 0.569 0.541 0.519

0.496 0.476 0.459 0.441

0.426 0.411

0.396 0.383

0.372 0.361 0.350 0.340 0.331

0.322 0.313 0.305

0.298

0.290 0.284 0.277 0 270

0.264

0.258 0.253 0 248 0.243 0.239

0.234

0 229 0.225 0.221 0 217

0.212

0 208

0.202 0. 0. 0. 0. 0.

196 189 182 174 166

Ill

3. 2.696 2.357 2.082 1.853 1.667 1.509 1.380 1.267 1. 170 1.086 1.006

0.939 0.877 0.825 0.777 0.740 0.726 0.699 0.661

0.632 0.605

0 578

0.550 0.528 0.505 0.484 0.467 0.449 0.433

0.418 0.403

0.390 0.378 0.366 0.355 0.345 0.336 0.327 0.318 0.310 0.302 0.294 0.287 0.280

0 274 0 267

0.262 0.256 0.251

0.247 0.243

0.238 0 234

0.230

0 226

0.222 0.218 0.214 0 0 0. 0.

209 203 197 192

0.186 0.180

0. 172

0.949 0.887 0.835 0.787 0.750 0.735 0.709 0.671 0.641

0.614 0.587 0.559

0.536

0.513 0 492 0.475 0.457 0.440 0.425

0.407 0.396 0.384 0.372 0.360 0.350

0.340

0.331 0 322

0.314 0.306 0.298 0.291

0.284 0.278

0 271 0.266 0 260 0 255 0 250 0.247 0.243

0.238 0.234 0.230

0.227 0.224 0.220

0.216 0.211

0.206 0.202

0. 198 0. 194 0. 188

0.958 0.896 0.845 0.796 0.760 0.745 0.718 0.680 0 650

0.624 0.596 0.567 0 545 0.521

0.500 0.484 0.464 0.448 0.432 0.416

0.401 0.389 0 377

0.365 0.355 0.346 0.336 0.327 0.319 0.310

0.302

0 295

0.288

0 281

0.274 0.269 0.264 0.259 0.254 0.250

0.246

0 242

0.239

0.235 0.233 0.230 0 226

0.222 0.218 0.215

0.212

0 210 0 208 0.203

* E. J. Wellman and W. L. Sibbitt, A Survey of Some of tho Thermodynamic and Physical Properties of Water, Combustion, April, !955, pp. 51-56.

9-14


[SEC. 9

Prandtl Number cpp/k for Liquid Light Water* c»m/*

Temper

Pressure,

ature.

°F


1.000

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 212 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690

psia, compressed

liquid

Saturated liquid

13.74 11.59 9.55 8.03

6.82 5.89 5.13

4.52 4.02 3.61

3.27 2.98 2.74

2.51 2.33 2. 15 2.01 1.88 1.79 1.75 1.69 1.59 1.52 1.45 1.39 1.33 1.27 1.22 1. 18 1. 14 1. 10 1.07 1.04 1.02 0.993 0.973 0.956 0.941

0.927 0.913 0.903

0.894 0.886 0.876 0.871

0.867 0.863 0.865 0.870

0.88 0.89 0.90 0.90 0.93 0.95 0.97

1.00 1.04 1.09 1. 14 1.20 1.25 1.39 1.56 1.77 2 19 2.75 4.40

13.52 11.43

9.44 7.95 6.76 5.86 5.09 4.49 4.00 3.60

2.000

13.33 11.28

9.32 7.86 6.69

5.78 5.05 4.46

3.98

2.73 2.51 2.33 2. 16

3.59 3.25 2.97 2.73 2.51 2.33 2.16

1.89 1.80 1.76 1.69 1.60 1.53 1.46 1.40 1.33 1 28 1.23 1. 18 1. 14

1.89 1.80 1.76 1.70 1.60 1.53 1.47 1.40 1.34 1.28 1.23 1. 19 1.15

3.26 2.97

2.02

111 1.07 1.04 1.02

0.997 0.977 0.958

0.939 0.924 0.909 0.897 0.886 0.878 0.871

0.866 0.863 0.861

0.862

2.02

III

1.08 1.05 1.02

0.996 0.975

0.954 0.934 0.918 0.903 0.890

0.877 0.869

0.861 0 855 0.852 0.849 0.847

0.86

0.844 0.848

0.88

0.859

0.861 0 87

0 852 0.870 0.886 0.905 0.924 0.953 0.985 1.015 1.09 1.18

3.000

13.14 11. 12 9. 19 7.76 6.61 5.73 5.01 4.43 3.94

3.57 3.24

2.96 2.72

2.51 2.33 2.16 2.02 1.90 1.80 1.77 1.70 1.61 1.54 1.47 1.41 1.34 1.29 1.24 1.19 1. 15

111

1.08 1.05 1.02 0.993

0.972 0.952 0.930 0.914 0.898 0.884 0.872 0.862 0.854 0.845 0.840

0.836 0.833 0.830

0.832 0.834 0.841

0.848 0.859

0.871 0 882 0.903 0.925 0 954 1.003 1.034 1.089 1.158 1.241 1.337 1.61 1.97

4.000

12.93 10.95

9.06 7.66 6.54

5.67 4.96 4.39 3.93 3.55 3.23 2.95 2.72 2.50 2.33 2.16 2.03 1.90 1.81 1.77 1.70 1.61 1.54 1.47 1.41 1.34 1.29 1.24 1. 19 1. 16 1. 12 1.08 1.05 1.02 0.994 0.971 0.949 0.926

0.908 0.892 0.877

0.864 0.854 0.845 0.835

0.829 0.824 0.821

0.816 0.818 0.819 0.822 0.826 0.837 0.849 0.860

0.874 0.889

0.908 0.934

0.964

0.997 1.034 1.086 1. 147

5.000

12.72 10.80

8.93 7.56

6.46 5.61 4.91

4.36 3.90 3.53 3.21

2.94 2.71 2.50 2.32 2.16 2.03 1.90 1.81 1.77 1.71 1.61 1.54 1.48 1.41 1.35 1.30 1.24 1.19 1.16 1.12 1.09 1.05 1.01

0.994 0.968 0.946 0.922

0.905 0.889 0.874 0.861

0.850 0.841

0.829 0.822 0.816 0.812 0.806 0.805

0.804 0.806 0.811

0.820 0.828 0.837 0.849 0.859 0.879 0.899 0.916 0.941

0.970 1.006 1.039

6.000

12.51 10.63

8.80 7.47

6.38

5.55 4.87 4 32

3.87 3.52

3.20 2.93 2.71 2.49 2.32 2. It 2.03 1.91 1.82 1.78 1.71 1.62 1.55 1.48 1.42 1.35 1.30 1.24 1.20 1 16 1.12 1.09 1.06 1.02 0.991 0 965

0.942 0.917 0.902 0.887

0.872 0.858 0.846 0.836 0.822 0.813

0.807 0.802 0.796 0.794 0.792 0.793 0.796 0.803 0.809 0.817 0.827 0.835 0.849 0.869 0.884 0.901

0.933 0.962 0.990

* E. J. Wcllman and W. L. Sibhitt, A Survey of Some of the Thermodynamic and Physical Properties of Water, Combustion. April, 1955. pp. 51-56.

Sec. 9-1]

HEAT-TRANSPORT MEDIUMS Table 25.

Critical Properties of Heavy Water*

Critical temperature

700 . 7°F 371. 5°C 3212 psia 218.6 atm 22. 7 lb/ft» 0.363 g/ml

Critical pressure Critical density

* C. J. West, "International Critical Tables of the Numerical Data of Physics, Technology," McGraw-Hill Book Company, Inc., New York, 1933.

Table 26.


*

I.

and

°C

°F

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 371.5

32 50 68 86 104 122 140 158 176 194 212 230 248 266 284 302 320 338 356 374 392 410 428 446 464 700.7

Pressure

Atm (absolute)

0.00480 0.01025 0.0200 0.0368 0.0647 0. 1100 0. 1797 0.2843 0.4363 0.6520 0.9503 1.353 1.888 2.584 3.474 4.596 5.993 7.707 9.788 12.29 15.26 18.77 22.87 27.63 33. 17 218.6

Psia

0.0705 0. 1506 0.2939 0.5408 0.9508 1.616 2.641 4.178 6.412 9. 582 13.96 19.88 27.75 37.97 51.05 67.54 88.07 113.3 143.8 180.6 224.3 275.8 336. 1 406.0 487. 5 3.212

Kirshenbaum, " Physical Properties and Analysis of Heavy Water," McGraw-Hill 1951.

ChemiBtry,

Vapor Pressure of Liquid Heavy Water*

Temperature

Inc., New York,

9-15

Book Company,

9-16

FLUID AND HEAT FLOW

[Sec. 9

Latent Heat of Vaporization of Heavy Water*

Table 27.

Aff

Temperature

°C

°F

3.82 10 25 40 60 80 100 120 140 160 180 200 220

* I. Kirshenbaum, "Physical Inc., New York, 1951.

38.9 SO 77 104 140 176 212 248 284 320 356 392 428

Cal/g

Btu/lb

554.66 549.47 541.53 531.50 519.86 507.98 495.65 482.76 469.08 454.70 439.28 422.70 404.63

998.39 989.05 974.76 956 69 935.75 914.36 892. 16 868. 98 844.35 818.47 790.69 760 86 728.33

Properties and Analysis of Heavy Water," McGraw-Hill Book Com

pany,

Table 28.

Absolute Density

p

of Liquid Heavy Water at Saturation*


Temperature p.

°F

°C

50 68 86 104 122 140 158 176 194 212 230 248 266 284 302 320 338 356 374 392 410 428 446 464 482

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

fj/cm'

lb /iff

1.1059 1.1054 1.1032 1. 0999 1.0957 1. 0906 1.0847 1.0782 1.0708 1.0630 1.0547 1.0459 1. 0366 1.0268 1.0167 1.0058 0.9950

69. 172 69.010 68.872 68.667 68. 404 68.086 67.718 67.312 66.850 66.363 65.845 65.295 64.715 64. 103 63.473 62.792 62.118 61.344 60.557 59.746 58.871 57 935 56 999 55.999 55.001

0.9826 0.970 0.957 0.943 0.928 0.913 0.897 0.881

fmo

pH,o

1.1080 1. 1079 1. 1080 1. 1085 1.1090 1 1092 1 1093 1. 1095 1. 1093 1. 1092 1 1090 1. 1086 1. 1084 1. 1084 1. 1084 1. 1082 1.1080 1. 1073 1 1070 1. 106 1. 106 1.104 1. 104 1.102 1.102

* J. It. Hciks, M. K. Barnott. L. V. Jones, and E. Orl>an, "The Physical Properties of Heavy Water from Room Temperature to 250°C," MLM-934. Jan. 12. 1954.

heat-transport

Sec. 9-1]

9-17

mediums

Dynamic Viscosity n of Liquid Heavy Water at Saturation*

Table 29.

f

Temperature

"DiO

°F

°C

Centipoise

Lb/(hr)(ft)

50 68 86 113 MO 167 194 212 257 302 347 392 437 482

10 20 30 45 60 75 90 100 125 150 175 200 225 250

1.684 1.260

4.074 3.0480

0.713

1.725 1.335 1. 076

0.969

2.344

0.552 0.445

0.365 0.323 0.252 0.208

0.883 0.782 0.610 0.485 0.423 0.365

0. 175 0. 151

0.135

0 327 0 300

0. 124

"H*)

1 1 1 1 1 1 1 1 1 1 1 1 1 1

25 23 21 19 18 17 15 14 13 12 ■ 11

II

10 10

* J. B. Heiks, M. K. Harnett, L. V. Jones, and E. Orban, "The Physical Properties of Heavy Water from Room Temperature to 250°C." MLM-934, Jan. 12. 1954.


Table 30.

Specific Heat

cp

of Liquid Heavy Water at

1

Atm*

Temperature

* I. Kirahenbaum, "Physical pany, Inc., New York, 1951.

Table

°C

°F

10 15 20 25 30 35 40 45 50

50 59 68 77 86 95 104 113 122

Btu/(lb)(°F)

1.009 1.008 1.006 1.005 1.005 1.004 1.004 1.004 1.004

Properties and Analysis of Heavy Water," McGraw-Hill Book Com

Thermal Conductivity k of Liquid Heavy Water at

31.

Temperature

1

Atm*

k *r>iO

*HjO °F

°C

50 68 86 104 122 140

10 20 30 40 50 60

Cal/(cm)(sec)(°C)

0.00134 0.00138 0 00142 0 00145 0.00148 0.00151

Btu/(hr)(ft)(°F) 0.324 0.334 0.344 0.350 0.358 0.365

0.968 0.968 0.968 0.967 0.967 0.968

* J. R. Heiks, M. K. Barnntt, L. V. Jones, and E. Orban. "The Physical Properties of Heavy Water from Room Temperature to 250°C," MLM-934, Jan. 12, 1954.

FLUID AND HEAT FLOW

9-18

[Sec.

Physical Properties of Molten Sodium Hydroxide

Table 32.

Molecular weight: 40.01 Melting point: 605°F. 318.4°C Latent heat of fusion: 72 Btu/lb, Boiling point: 2534°F, I390°C Density, lb /ft"

Temperature,

650 700 800 900 1000 1100

Viscosity, lb/(hr)(ft)»

117.7 116. 9 115 2 113.5

40 cal/g

Thermal conductivity,

Specific

Btu/(lb)(ft)(°F)

10. 1 8. 2 5.9

0 57 0. 58 0 59

4.7 3.8

0.64

heat,

Btu/(lb)(°F)t

0. 555 0. 540 0. 509 0 475 0.442 0.410

1.00

3. 1

* Kurt Arndt and Georg Ploetz, Leitfahigheit und Zahigheit von geschmolzenem Kaliumhydroxyd, Z. physik. Chcm., 121 (5-6): 439-455 (1926). t V. R. Terashkewich and Vrashmcrshu, J. Cen. Chem. USSR, 7 (1937).

Table 33.

a Generated for wjivans (University of Florida) on 2015-09-23 02:53 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

"3

Vapor

Density

pressure

3

.j

Natrium- und

Physical Properties of Bismuth*

Atomic number: 83 Atomic weight: 209 Melting point: 520°F, 271°C Latent heat of fusion: 21.6 Btu/lh, 12 cal/g Boiling point: 269 l°C, 1477°F Latent heat of vaporization: 368 Btu/lb. 204.3 cal/g

2

c

9

Specific


Viscosity

heat

Mm of Hg

°F

°F

Ll> /ft1

°F

Btu/(lb) (°F)

°F

Lb/(ft)(hr)

°F

1 10 100 200 400

1683 1953 2295 2417 2552

572 752 1112 1476 1764

626 619 603 587 574

520 752 1112 1472 1832

0. 0340 0.0354 0.0376 0.0397 0.0419

579 844 1112

4.02 3. 10 2.41

572 752 932 1112 1292

* R. N. Lyon, "Liquid-Metals Washington, D.C., June, 1952.

Btu/(hr)(ft)(°F)

9.90 8.95 8.95 8.95 8.95

Handbook," NAVEXOS P-733 (rev.), Government Printing Office.

Table 34. Physical Properties of Gallium* Atomic number: 31 Atomic weight: 69.72 Melting point: 85.9°F, 29.75°C Latent heat of fusion: 34.5 Btu/lb, 19.16 cal/g Boiling point: 360I°F, 1983°C Latent heat of vaporization: 1825 Btu/lb, 1014 cal/g Vapor

Density

pressure

Specific

Mm of Hg

op

°F

Lb /ft'

°F

1 10 100 200 400

2399 2727 3139 3285 3443

90 574 1112 1483 2012

380 368 357 350 340

54 to 392


Btu/(lb)(°F)

Handbook,

0 082


Viscosity

heat

°F

Lb/(ft)(hr)

127 574 756 932 1483

NAVEXOS

4. 58 2.49 2. 13 1.96 1. 58 P-733

(rev.).

°F

85.9

Government

Btu/(hr)(ft)m

17-22

Printinn Office,

HEAT-TRANSPORT MEDIUMS

Sec. 9-1]

9-19

Table 35. Physical Properties of Lead* Atomic number: 82 Atomic weight: 207.21 Melting point: 62I°F. 327.4°C Latent heat of fusion: 10.6 Btu/lb, 5.89 cal/g Boiling point: 3I59°F, I737°C Latent heat of vaporization: 369 lit u .It . 205 cal/g Vapor

Density

pressure

Specific

Mm of Hg

°F

°F

Lb/ft1

°F

1 10 100 200 400

1809 2133 2583 2746 2932

752 932 1112 1472 1832

656 649 641 627 612

621 752 932

Btu/(lb)(°F)

0 039 0.037 0.037

* R. N. Lyon, "Liquid- Metals Handbook." D.C., June. 1952.

Washington.


Table 36.


Viscosity

heat

op

826 853 1024 1298 1551

NAVEXOS

-F

Lb/(ft)(hr)

5. 12 4.98 4 11 3. 27 2. 86

P-733

(rev.).

Btu/(hr)(ft)(°F)

626 752 932 1112 1292

9. 40

9.2 9.0 8.7 8.7

Government Printing Office.

Physical Properties of Lead-Bismuth Eutectic (44.5 % Pb 56.5 % Bi) (Weight Per Cent)* Atomic weight: 208 Melting point: 257°F, I25°C Boiling point: 3038°F, I670°C

Density

Specific

°F

Lb /ft'

°F

392 752 1112 1472 1832

653 636 619 602 584

291

Btu/(lb)(°F)

to

0.035

676

• R. N. Lyon, "Liquid-Metals Washington. D.C., June, 1952.


Viscosity

heat

°F

630 842 932 1022 1112

°F

Lb/(ft)(hr)

4. 11 3.34 3. 12 2 98 2.83

Btu/fhrXftJCF)

320 392 464 572 608

5.30 5 60 5 80 6 30 6 50

Handbook," NAVEXOS P-733 (rev.), Government Printing Office,

Table 37. Physical Properties of Lithium* Atomic number: 3 Atomic weight: 6.94 Melting point: 354°F, I 79°C Latent heat of fusion: 284 Btu/lb. 158 cal/g Boiling point: 2403°F, 13I7°C Latent heat of vaporization: 8420 Btu/lb, 4680 cal/g Vapor

Density

pressure

Specific

Mm of Hg

»F

°F

Lb/ft>

°F

1 10 100 200 400

1373 1634 1983 2113 2257

392 752 1112 1472 1832

317

392 572 752 932

30 29 28 27

6 6 5 5

Btu/(lb)(°F)

1.40 1.23 1 09 1 00

Thermal conductivity!

Viscosity

heat

°F

362 380 421 483 545

Lb/(ft)(hr)

1.43 1.39 1.31 1 19 1 10

°F

Btu/(hr)(ft)(°F)

420 662 838 1002

26.5 24.2 21.5 17.6

* R. N. Lyon, "Liquid-Metals Handbook," NAVEXOS P-733 (rev.). Government Printing Office. Washington. D.C.. June. 1952. t H. A. Webber, D. Goldstein, and R. C. Fellinger, Determination of Thermal Conductivity of Molten Lithium, Tram. ASMS, 77 (2) (February, 1955). ASME Paper 53-A-79, pp. 97-102.

a-20

FLUID AND HEAT FLOW of Magnesium*

Physical Properties

Table 38.

[Sec. 9

Atomic number: I 2 Atomic weight: 24.32 Melting point: I204°F, 651°C Latent heat of fusion: 148 Btu/lb, 82.2 cal/g Boiling point: 20I7°F. 1103°C Latent heat of vaporization: 2406 Btu/lb. 1337 oal/g Vapor

Density

pressure

Mm of Hg

°F

°F

1 10 100 200 400

1150 1296 1668 1773 1893

1205 1252 1292 1328 1382


Handbook." See also

Lb /ft'

°F

Btu/(lb)(°F)

98.3

1204 1341 1701 1881 2048

0.317

96.8 95 9

94.3 91.8

TID

NAVEXOS

P-733

Vapor

Density

Mm of Hg

°F

1 10 100 200 400

259 363 503 555 613

°F

-4 68 212 392 572

0.321 0.332 0.337

0.342 (rev.), Government Printing

5277".

Atomio number: 80 Atomic weight: 200.61 Melting point: -37.97°F. -38.87°C Latent heat of fusion: 5.0, 2.8 cal/g Boiling point: 675°F, 357°C Latent heat of vaporisation: 125 Btu/lb,

pressure

heat

Office.

Physical Properties of Mercury*

Table 39.


Specific

Specific

°F

Btu/(lb)(°F)

852 846 834 819 804

32 212 392 572 842

0.03334 0.03279 0.03245 0.03234 0.03256


Visoosity

heat

Lb /ft"

69.7 cal/g

°F

Lb/(ft)(hr)

°F

Btu/(hr)(ft)(">F)

-4

4.48

32 68 212 392

4.74 5.59

3.75 2.93

32 140 248 320 428

4.07 2.44

6.31

6.75 7.33

* R. N. Lyon. "Liquid-Metals Handbook." NAVEXOS P-733 (rev.). Government Printing Office. Washington, D.C., June, 1952. See also TID 5277".

Physical Properties of Potassium*

Table 40.

Atomic number: 19 Atomio weight: 39.096 Melting point: I47°F, 63.7°C Latent heat of fusion: 26.3 Btu/lb, 14.6 cal/g Boiling point: 1400°F. 760°C Latent heat of vaporization: 893 Btu/lb, 496 cal/g Vapor

Density

pressure

Mm of Hg

°F

°F

1 10 100 200 400

648 829 1078 1175 1285

212 482 752 1022 1292

Specific

Lb /ft'

51. 48 46 44

1 8 6 3

42.2

• R. N. Lyon. "Liquid-Metals Washington, D.C., June, 1952.

»F

167 392 752 1112 1472

Btu/(lb)(°F) 0 0 0 0 0

Handbook,"

1956 1887 1826 1825 1884


Viscosity

heat

°F

Lb/(ft)(hr)

F°

Btu/(hr)(ft)(°F)

157 333 482 752 1292

1.25 0 801 0.624 0.462 0.329

392 572 752 932 1112

26.0 24.5 23.2 21.7 20.5

NAVEXOS

P-733

(rev.),

Government Printing

Office,

heat-transport

Sec. 9-1]

mediums

9-21

Table 41. Physical Properties of Sodium* Atomic number: 11 Atomic weight: 22.997 Melting point: 208°F, 97.9°C Latent heat of fusion: 49 Btu/lb, 27.2 oal/g Boiling point: I621°F, 883°C Latent heat of vaporization: 1810 Btu/lb, 1005 cal/g Vapor

Density

pressure

Mm

Hg

of

1 10 100 200 400

Specific


Viscosity

heat

°F

°F

Lb /ft'

°F

Btu/(lb)(°F)

op

Lb/(ft)(hr)

op

Btu/(hr)(ft)(°F)

824 1018 I28S 1386 1499

212 482 752 1022 1292

57.9 55.7

212 392 752 1112 1472

0.3305 0.3199 0.3055 0. 2998 0.3030

212 392 572 752 932

1.66 1.22

212 392 572 752 932

49.7

53 2 50.9

48.7

* R. N. Lyon, "Liquid-Metals Washington. D.C., June, 1952.

Table 42.

Handbook," See also

TID

NAVEXOS

0.922 0.651

0.440

P-733

5277".

47.1

43.8 41.2 38.6

(rev.), Government Printing Office,

Physical Properties of Sodium-Potassium Alloy (Eutectic Mixture, 22% Na-78% K) (Weight Per Cent)* Atomic weight: 33.9 Melting point: 12°F. 11°C Boiling point: I442°F, 784°C


-

Vapor

Density

pressure

Mm

of

°F

op

Lb /ft'

°F

671 856 1117 1218 1330

212 482 752 1022 1292

52.8 50.7 48.3

32 392 752 1112 1472

Hg

1 10 100 200 400

Specific

46.1

43.8

* R. N. Lyon, "Liquid-Metals Washington, D.C., June. 1952.

Table

43.

Btu/(lb)(°F)

°F

Lb/(ft)(hr)

°F

Btu/(hr)(ft)(°F)

0.238

219 334 482 752 1292

1.13

212

14.1

0.869 0.674 0.496

752

15.4

0.217 0.210 0.209 0.213

Handbook." See also


Viscosity

heat

TID

NAVEXOS

0.353

P-733

(rev.), Government Printing Offioe.

5277".

Physical Properties of Sodium -Potassium (Weight Per Cent)*

Alloy (66 % Na-34 %

K)

Atomic weight: 28.1 Melting point: 66.2°F, I9°C Boiling point: 151 8°K, 825°C

Vapor

Density

pressure

Mm

of

Hg

1 10 100 200 400

Specific


Viscosity

heat

°F

°F

Lb /ft'

°F

Btu/(lb)(°F)

°F

Lb/(ft)(hr)

op

Btu/(hr)(ft)(°F)

720 914 MHO 1285 1400

212 482 752 1022 1292

55.3 53.0

212 572 932 1112 1472

0.269 0.255 0.249

219 334 482 752 1292

1.32 0.997 0 765 0.557 0.353

212 392 572 752 932

14.9 15.3 15.7 16.0 16.3

50.8 48.6 46.3

» R. N. Lyon. "Liquid-Metals Washington, D.C., June, 1952.

0.248 0.253

Handbook,"

NAVEXOS

P-733

(rev.), Government Printing Office.

FLUID AND HEAT FLOW

9-22

[Sec.

9

Table 44. Physical Properties of Tin* Atomic number: 50 Atomic weight: 118.70 Melting point: 449°F, 231.9°C Latent heat of fusion: 26.1 Btu/lb, 14.5 cal/g Boiling point: 41 18°F, 2270°C Latent heat of vaporization: 1031 Btu/lb, 573 cal/g Vapor

Density

pressure

Specific

Mm of Hg

op

°F

Lb /ft"

°F

1 10 100 200 400

2718 3097 3574 3745 3931

768 973 1065 1198 1298

427.0 422.0 420.0 417.0 415.0

482

* R. N. Lyon, "Liquid-Metals D.C., June, 1952.

Btu/(lb)(°F)

°F

0.058

464 572 752 932 1112

0.0758

2012

Handbook,


Viscosity

heat

NAVEXOS

°F

Lb/(ft)(hr)

464 558 783 928

4.62 4.04 3.34 2.86 2.54

P-733

Btu/(hr)(ft)(°F)

(rev.),

19.4 19.6 19. 1 18.9

Government Printing Office.

Washington,

Physical Properties of Diphenyl*

Molecular weight: 154.2 Melting point: 157°F, 69.4°C Latent heat of fusion: 177.8 Btu/lb, 98.8 cal/K Boiling point: 49I.5°F, 255. 3°C Latent heat of vaporization: 136.5 Btu/lb, 75.8 cal/g Critical temperature: 980°F. 526.7°C Critical pressure: 607 psi, 41,3 atm Temperature,

"F

Vapor pressure, psia

Specific heat

c,

Latent heat,

Btu/lb

Density of liquid,

Density of vapor,

lb /ft'

lb /ft'


Viscosity. lb/(hr)(ft)

Btu/(lb)(ft)(°F)t I

0.701 4.20 16.30

47.0

110. 1 221.8 399.7 607.0

0.42 0.50 0.56 0.61

0.65 0.67 0.69 0.69

0.70

175.0 154.0 148.0 135.0 117.0 103.0

91.0 67.5

0.0

60.85 58.08 55.24 52.32 49.04 45.38 40. 89 34.58 19.60

0.00108 0.0117 0.0616

2.42 1.35 0.90

0.236 0.750

0.51

0.62

1.730 3.355 6.850 19.60

0.27

0.38 0.34 0.29

0.078 0.076 , 0.073 0.070 0.067 064 . 060 057 6154 to.

0.060

0 0 0

200 300 400 500 600 700 800 900 980

J.

A

t *

Kermit Anderson, "Engineering Properties of Diphenyl," ANL-5121, Aug. 11. 1953. R. Woolf and W. L. Sibbitt, Thermal Conductivity for Dowtherm (eutectic mixture 2^-5 P" cent diphenyl and 73.5 per cent diphcnyloxide), Ind. Eng. Chem., 46: 1947 (1954).

2.

Arndt, Kurt, Natrium- und Ellenwood, F. Wide Ranges

<.

REFERENCES 1.

and Georg Ploetz: Leitfahigkeit und Zahigkeit von geschmolsenen.1 Kaliumhydroxyd, Z. physik. Chem., 121 (5-6): 439-455 (1926). O., N. Kulik, and N. R. Gay: The Specific Heats of Certain Gases over of Pressures and Temperatures, Cornell Univ. Eng. Expt. Sta. Bull. 30,

October, 1942. 3. Heiks, J. R., M. K. Barnett, L. V. Jones, and E. Orban: "The Physical Properties of Heavy Water from Room Temperature to 250°C," MLM-934, Jan. 12, 1954. 4. Hilsenrath, J., and Y. 8. Touloukian: The Viscosity, Thermal Conductivity, and Prandtl Number for Air, Oj, N,, NO, H,, CO, COj, HsO, He, and A, Tram. ASME, 56: 967-985 (August, 1954). Sons, Inc., New York, 194S. 5. Keenan, J. H., and J. Kaye: "Gas Tables," John Wiley 6. Keenan, J. H., and F. G. Keyes: "Thermodynamic Properties of Steam," John Wiley Sons, Inc., New York, 1936.

A

&


Table 46.

Sec. 9-1] 7.

8. 9. 10.

11.

12.

13. 14. 15. 16. 17.


18. 19.

heat-transport

mediums

9-23

Keyes, F. G.: A Summary of Viscosity and Thermal Conduction Data for He, A, Hi, ASME, 73: 589-590 (1951). Kirshenbaum, I.: "Physical Properties and Analysis of Heavy Water," 1st ed., McGraw-Hill Book Company, Inc., New York, 1951. Lee, J. F.: Thermodynamic Properties of Liquid Sodium, Nucleonics, 12 (4): 74-77 (April, 1954). Lemmon, A. W., Jr., D. J. Daniels, D. E. Sparrow, C. J. Geankoplis, J. J. Ward, and J. W. Clegg: "Empirical Evaluation of the Properties of Steam at Elevated Tempera tures and Pressures," BMI-858, Battelle Memorial Institute, Aug. 25, 1953. Lenoir, J. M., W. A. Junk, and E. W. Comings: Measurement and Correlation of Thermal Conductivity of Gases at High Pressure, Chem. Eng. Progr., 49 (10): 539 (October, 1953). Lyon, R. N.: "Liquid-Metals Handbook," NAVEXOS P-733 (rev.), Government Printing Office, Washington, D.C., June, 1952. Also C. B. Jackson (ed.): "LiquidSupplement," TID 5277, Government Printing Metals Handbook, Sodium-NaK Office, Washington, D.C., July, 1955. Masi, J. F.: Survey of Experimental Determinations of Heat Capacity of Ten Tech nically Important Gases, Trans. ASME, 66: 1067-1073 (October, 1954). Perry, J. H. (ed.): "Chemical Engineers' Handbook," McGraw-Hill Book Company, Inc., New York, 1950. U.S. Atomic Energy Commission: "Reactor Handbook/' vol. 2, Engineering, McGrawHill Book Company, Inc., New York, 1955. Terashkewich, V. R, and Vrashmershu: J. Cen. Chem. USSR, 7 (1937). Webber, H. A., D. Goldstein, and R. C. Fellinger: Determination of Thermal Con ductivity of Molten Lithium, Trans. ASME, 77 (2) (February, 1955). ASME Paper 53-A-79. Wellman, E. J., and W. L. Sibbitt: A Survey of Some of the Thermodynamic and Physical Properties of Water, Combustion, April, 1955, pp. 51-56. West, C. J.: "International Critical Tables of the Numerical Data of Physics, Chemis try, and Technology," McGraw-Hill Book Company, Inc., New York, 1933. Os, N», CO, COa, HjO, and Air, Trans.

9-2

FLUID FLOW IN REACTOR SYSTEMS BY Charles

F. Bonilla

PRINCIPAL NOMENCLATURE A

= wall area

C

—

volume fraction of solids in a suspension [Eqs. (11) to (13)] heat, when cp substantially = c„ coefficient of stream-area contraction orifice discharge coefficient specific heat at constant pressure specific heat at constant volume pipe diameter total energy per unit mass pump efficiency wall roughness energy lost as friction

e — specific

Cc = Cd = cp =

c, =


D

= = E = e = F = F = force

E

fF

m

0

Fanning friction factor

= mass velocity, w/S g = acceleration due to gravity gc = dimensionless ratio F / ma for units employed h enthalpy per unit mass t = internal energy per unit mass K = number of pipe average-velocity heads lost in a fitting m = mass; hydraulic radius L = length L, = length of straight pipe having same friction loss as a fitting n = rotational speed of pump or blower, revolutions P = pressure P = pump shaft power

"

Q — rate of flow of volume [except heat absorbed from surroundings and (58)] q = rate of heat flow R = pipe radius R' = gas constant S = cross-sectional area of coolant stream T = absolute temperature t

-

thermomctric temperature

u = velocity u = average velocity in a channel u, = superficial velocity in a porous body, based on total cross section u+ = dimensionless local velocity V — specific volume = l/p =» work done by surroundings w — rate of flow of mass x = fraction of vapor in vapor-liquid

W

mixture a-24

in Eqs. (IT)

Sec. 9-2]

fluid flow in reactor

systems

9-25

y+ = dimensionless wall distance z = elevation above horizontal datum plane a = ratio between actual kinetic energy of a stream and that at u j9 = volumetric coefficient of thermal expansion S = &' = e —

=

ri 8 = ix = v =

p = =

t

All

thickness of a boundary layer thickness of layer equivalent to wall slippage eddy diffusivity [except Eq. (21)] coefficient of rigidity time viscosity kinematic viscosity density shear stress nuclear reactors are cooled by one or more fluids in motion.

In

order to main

tain proper cooling and safe operation at high powers it is necessary that the coolant distribute itself appropriately throughout the core, shields, and blankets under all

It is also desirable that the coolant circuit be designed to achieve optimum conditions. over-all economy. Accordingly, fluid-flow principles and design methods are very pertinent in reactor design. VISCOSITY

Viscosity it defines the capacity of a fluid to transmit shear stress through the rela

tive motion of individual molecules or atoms. Under shear stress t a velocity gradient du/dy will be obtained normal to the plane of the shear. These quantities are related by rffc = n(du/dy)

(1)

If m is constant over a wide range of t, the fluid is designated Newtonian. Thia includes all gases; most liquids, in particular all proposed reactor coolants except slurries and other two-phase mixtures; and solid particles. Equation (1) is valid only in the absence of turbulence, namely, for streamline (i.e., laminar or viscous) flow Art. 3). The cgs unit of viscosity is the poise (grams per centimeter per second), which requires t in Eq. (1) in dynes per square centimeter, du/dy in sec-1, and gc, as in all (see

absolute systems, to equal unity. Viscosity in poises must be multiplied by 0.0672 to convert it to pounds per foot-second, by 242 for pounds per foot-hour, and by 0.00209 for slugs per foot-second. For other conversions see Table 17 of Sec. 1-3. 1.1

Viscosity of Gases

Nitrogen:1 Air:1 Argon :■•' Carbon dioxide:1

it

-

it

=

it it it

= 15.24r^/(71 = 17.Q0T^-/(T

in micropoises and

T

formulas are suitable.

is

pheric pressure the following 273.1. degrees centigrade

/«'

The viscosity of gases is independent of pressure at normal pressures (such as up to 10 per cent of the critical pressure) while above the critical temperature and is also independent of shear stress. Values of viscosity are tabulated in Sec. 9-1. For higher temperatures and atmos in

= 14.057'"V(7' 1

14.607'?V(r + 9.07^/(7" +

109.1) 109.6) 135.4)

-

+

+

4- 219.4) 126.8) = 4.237'5V(7,|)-826 0.409) = 24.407T?V(7' 1137) = 25.13r?V(71 + 1142)

=

6.422'5V(r

+

it M n

Oxygen:1 Helium:1 HjO (steam):' D2<) vapor:* Hydrogen:1

it

+

+


1

71.7)

(2) (3) (4) (5) (6) (7) (8) (9) (10)

9-26

FLUID AND HEAT FLOW

[SEC. 9

The effect of pressure on n will generally be negligible at practical reactor pressures. However, correction can be applied if desired.*-' The viscosity of a mixture of gases of similar molecular weight, such as oxygen and nitrogen, may be accurately obtained by interpolation by mole fraction between the viscosities of the pure components at the same temperature and pressure as the mix ture. With other gases the viscosity of the mixture may range up to 25 per cent or more higher than the interpolated value and may roughly be estimated graphically' or calculated by available formulas.4 Viscosity of Liquids

1.2

The viscosities of liquids are much higher than those of gases, as shown in Sec. 9-1, on account of their greater density and, thus, molecular interactions. Among the methods of interpolating or extrapolating to other temperatures, the following are most used: (1) A plot of y. versus t yields a smooth curve with a maximum deviation from a straight line of the order of 1 per cent per 20°C interval over useful ranges. Thus linear interpolation over short intervals is permissible. (2) A plot of fluidity 1/m versus t is closely linear over fairly long temperature intervals and thus suitable for reasonably long interpolations or extrapolations. is (3) A plot of log p versus straight over temperature ranges having a constant energy of activation E and is useful for interpolation or extrapolation with two or more known values of m and T. If viscosity is known at only one temperature, it can be extrapolated by a line parallel to the log n versus plot for a similar compound. The effect on liquid viscosities of pressures that are appropriate for nuclear reactors is generally negligible at the (low) temperatures employed.

l/T


l/T

1.3

Flow Properties of Suspensions

The properties of suspensions are of interest in nuclear engineering in view of the possibility of the use of finely divided fuel and/or fertile elements, alloys, and com Dilute suspensions of solid particles tend to be pounds suspended in a liquid. Newtonian, following the Einstein equation: v./nm =

1

+ 2.5C

(11)

up to a volume fraction C of solid of the order of 0.05. m- and m™ arc the viscosities of the suspension and medium, respectively. The coefficient 2.5 holds for any reason ably equiaxed shape and is constant or increases only slightly in going to the smallest sizes or the most irregular shapes. A straight line through the origin on a plot of du/dy versus t indicates Newtonian behavior [Eq. (1)], the slope of the line being the fluidity l/p,. Suspensions with equiaxed particles that interfere more with each other the higher the shear rate show a downward concavity in Fig. 1 and are dilatant, or inverted, plastics. Long particles that line up more closely parallel with each other the higher the shear show the opposite curvature and are pseudoplastic; the shear diagram usually becomes straight at high shear. If the lining-up requires an appreciable amount of time or work, the substance is thixotropic. In Fig. 1 the line D represents a rapid increase in shear and G the infinitely slow or equilibrium increase or decrease. If the rate of shear is held constant after a rapid increase, the shear stress will gradually decrease along a line such as E, reaching F and finally G. Solids and some concentrated suspensions require a definite yield shear stress r, before any motion occurs. These materials are not therefore lluids and are designated plastic substances. Many plastic suspensions approach an idealized straight -line behavior, that of the "Bingham body." The ratio of the shear stress to the velocity gradient at any point on the shear curve of a substance is the effective viscosity n" at that point. For instance, the slope of * Superscript numbers

refer to References


FLUID FLOW IN REACTOK

Sec. 9-2]

9-27

SYSTEMS

The reciprocal of the local slope line A is \/n" for point B. is designated the rigidity »j at that point. Several equations are available for predicting the effective at practical shear stresses. For volume concentrations up to reasonable agreement with equiaxed particles is obtained by Einstein's derivation, as in Vand's equation:

at any point, such as B, viscosity of suspensions the order of 25 per cent, retaining more terms in

= 1 + 2.5C + 7.17(7' + 16.20'


Orr has found substantially Newtonian behavior in suspensions and sizes of particles with 10 to 20 per cent solids by volume.

Shear stress

Fig.

1.

Types of shear diagrams in streamline

(12)

of different types

In streamline flow

r

flow.

(From C. F. Bonilla, Re/. 1.)

they roughly follow Eq. (13) up to the fraction of solids by volume, (?„, to which the particles settle in infinite time.

„"./m-

= [(1

- C)/CU->"

(13)

been proposed for high concentrations, but it is not to rely on such relations without experimental confirmation at the identical conditions.

Many other equations have

safe in general

2

MATERIAL AND MECHANICAL BALANCES 2.1

Material or Mass Balances

Applying the law of conservation of mass to a flow process yields a material balance, The steady-state flow w in mass per unit time is the same "continuity equation." through any surface S, completely cutting the stream. Standard methods of com putation are or

S w = Qp = upS = OS =

V o

(jiÂS)

-

Jo

updS

(14)

9-28

FLUID AND HKAT FLOW

[SEC.

9

The bars designate averages over S. The summation is commonly employed to evaluate w in a test when velocity and temperature vary irregularly over a large duct. The integral is useful for radially symmetrical flow in a pipe. Pressure of Force Balances

2.2

Fluid pressure forces may be very large in high-pressure or large reactors. If z is the elevation above a datum surface at which the pressure is constant, the difference in pressure between elevations z* and zt in a static fluid (no friction or acceleration) is

Pi

—

-

Pi

I

= pg(zi

pg dz/gc

- zt)/gc

(15)

This expression is useful in computing free-convection driving pressures, nonisothermal manometer lead corrections, etc. Integrating completely around a closed circuit gives AP*

= ^

(\7« [ "Padz

gc

+

f" pidz)/ J'i

-

f" (p„ -

9-

p„) dz =

Jzi

gc

S5* gc

f" (A - t.) dz

Jzi

j5o

(16)

if

2.3

Total Energy Balances

a

is

The law of conservation of energy particularly useful in cases in which the velocity, expansion, and heat flow arc all high, as in many optimized nuclear-reactor stream in steady flow along a differential length designs. Following unit mass of of its path,

x+d(p)+i±+?*m.dk ' g-c

v>

1g*

+

dQ+dW

(I?)

h

c

a

h

i

is a

is

is

heat gained All terms must be in the same units of energy per unit of mass. dQ from the outside, and dW work done on the fluid from the outside in addition to the "flow work" d(P/p). a factor to obtain the average kinetic energy from the are available for many fluids, and generalized average velocity u. Tables of and and cp are reasonably plots of versus reduced temperature and pressure for gases, constant over limited A7', except near the critical point, and Ai — c. AT" and Ah = cpAT. Equation (17) may be integrated between any two positions by merely replacing each differential by the finite increment. Mechanical

2.4

Energy Balances

+

(dP)/p

adm/2gc

=

dW

- dF

(18)

the energy lost as friction. For instance, in stationary These energy balances simplify in many applications. fluid the last three terms are zero, and in horizontal flow dz = 0. If constant, the p

is

dF

dz/gc

+

g

p

it

a

is is

F

= ma to a steady stream yields the mechanical energy Applying Newton's law more convenient than the total balance, or Bernoulli's principle.18 This relation energy balance when thermal effects are small and necessary in evaluating the fric However, the relation tion in a test, predicting coolant pressure drop in reactor, etc. can be integrated. It may be written: between and P must be known before

is


is

a

is

where a and 6 represent the two sides and the average value of the coefficient of thermal expansion over the temperature range, based on po. With stationary fluid the buoyant pressure APb = 0. If the fluid moving, APb equals the pressure spent in losses, plus acceleration not in steady state.

Sec. 9-2]

FLUID FLOW IN REACTOR

SYSTEMS

9-29

integral of the second term is AP/p, and if in addition the stream cross section is If p varies only moderately, it is usually sufficiently accurate to constant, d(u2) =0. W = 0 unless there is a pump or fluid motor in the system replace it by its average p. between the limits of integration.

For constant or moderately changing p or for isothermal gas flow Eq. (18) can be For flow through a nozzle or orifice dF may be neglected and integrated analytically. Eq. (18) integrated by obtaining the relation between P and p from Eq. (17) or the adiabatic expansion relation. For the general case in a reactor of heat generation and a considerably expanding fluid, it is convenient to proceed in succession over short sections of the channel such that p does not vary much. When the conditions at one end of a section are known, Eqs. (17) and (18) are solved simultaneously for P and p at the other end. This is continued over the whole channel. If stated conditions at the second end of the channel are not matched, the process is repeated with a different flow or different assumed first end conditions. Some 10 to 20 increments usually yield an accuracy comparable to that of the friction calculation. The method is evidently also applicable to adiabatic flow on setting AQ = 0 and to isentropic flow on setting AF = 0. 2.5

Momentum

Balances


Applying Newton's law over an interval of time AO yields the equality between impulse and momentum change, ¥ A8 = m Au, where F is the net or resultant force and Au is the vectorial change in velocity. Dividing by AS for a steady flow process Tgc = Aw(m/Afl)

= Au(dm/dB)

= w Au

(19)

In certain problems requiring the calculation of a force or pressure due to a change in direction or magnitude of the velocity of a stream, the momentum balance is more convenient than an energy balance. Equations (18) and (19) both hold for any steady flow process. Some cases exhibit an apparent discrepancy between the two methods, but this is the result of an incorrect application. 3

3.1

STREAMLINE FLOW Isothermal Streamline Flow

3.11 Isothermal Newtonian Streamline Flow in Ducts of Regular and Constant Cross Section. Streamline flow in symmetrical channels, such that the relation between the shear stress and the distance y from the wall is known, may be analyzed The principal results are given under streamline flow in Table 1. directly by Eq. (1). To illustrate the use of the table, for the average velocity in circular pipe upL APgc

or

AP

=

1 _R

8

^

(20)

The relations for mean momentum and mean kinetic energy of the streams per unit time are used in unsteady-state or start-up and coast-down calculations. In short channels with a smooth inlet nozzle, extra pressure drop is required near the inlet to convert the uniform velocity so obtained to the parabolic velocity distribu tion characteristic of laminar flow. This is given for pipes and wide flat channels in Table

2.

For channels of irregular cross 3.12 Streamline Flow in Irregular Channels. section (i.e. triangular) such that the spatial distribution of t is not known, numerical methods of computation are available.1 Certain cases are already in the literature.5 For a narrow annulus with an eccentricity, or distance between the two axes, of t, an approximate flow equation is obtained by taking the annular width as (Ri — Ri + e cos 0), substituting into the relation for flow in a parallel wall channel, and inte

9-30

FLUID AND HEAT FLOW Table

Flow at Constant Density in Long Horizontal

1.

/F

-

radius R

Streamline flow: velocity,

APge

flat channel. spacing b

Concentric annulus, radii ■- Ri, Rt

pipe.

-

R'

... .

R'

Local velocity,

-

r>

4

In

I Ri* — r* H 4 V

In

)

Ri/Ri

Ri/K,

(») mft* u

2

.

1.333 2

...

(3)

(2)

1.5180

1.5077

1.207 1.568

1.204 1.552

- v'

12

In — 1

yb

Rx)

«,/«.

Maximum velocity,

Square channel. side = b

b' 8 V,

8

(12) 1.5019

~I2

~ 1.543

2

b' 28 6 See Ref. 1

(1) 1.500

2.07

1.200 1.543

1.36 2.22

1.035 1.018 1 010 1.229 1.055 1.022

1.078 1.038 1.02

Turbulent flow:


Umax i Maximum velocity, .. u

0.01 0.005 0.0025 0.01 0.005 0.0025 0.01 0.005 0.0025

1.286 1.202 1. 143 1.055 1.030 1.017 1.135 1.073 1.040

~I035 ~l 018 ~l 010

kinetic energy per unit length Momentum per unit time kinetic energy per unit length at u Momentum per unit time at u Kinetic energy per unit time Momentum per unit length Kinetic energy per unit time at u Momentum per unit length at u ♦Source: C. F. Bonilla, "Nuclear Engineering," Ref. I.

Table

2.

Excess Streamline Pressure Drop in Short Tubes and Wide Slots Due to Uniform Inlet Velocity*

(To be added to pressure drop by Table 1 to obtain total pressure drop from start of tube or slot. Does not include pressure drop across the nozzle.) Excess velocity pressures

Pipe diameters

or slot widths downstream

AP.,/(p u'/le.)

of nozzle outlet

0 Ro 0.000125 Re 0 00025 Re 0.0005 Re 0.0010 Re 0.0015 Re 0.0020 Re 0.0025 Re 0.0050 Re

0.010 Re 0.015 Re 0.030 Re

0 040 Re > 0.060 Re Total AP in long channel (parabolie velocity distribution, first line, Table 1): Each 1. 0 Re

♦Source: C. F. Bonilla, "Nuclear

Engineering," Ref. I.

Pipe

Slot

0 0. 14

0.104

0.20 0.288

0.396

0 0. 148

0.208

0.522

0.290 0.339 0.370

0.570

0.39

0 464

0.74 0.99 1.14 1.32 1.37 1.41

64.00

9

Channels* Wide

Circular

Average

[SEC.

0.448 0.522 0.560 0.601 0 601 0.601

48.00


Sec. 9-2]

If Ri/R\

grating around the semicircle.

systems

9-31

< 1.5, the formula obtained

-

u = (gc/l2ri(dP/dL)l(R,

fl,)> + H«*J

(21)

is correct to 1 per cent or better. For a flat wide channel of nonparallcl walls and constant cross section, integrating across the channel from 6i to bt gives

ii

-

(gem»HdP/dL)(bl* +

(22)

62»)

Wall roughness generally has negligible effect in streamline flow. It has been shown' that the minimum height e of roughness necessary to disturb the flow is (4ZWu/>)w in a pipe or (9b^/2up)''i in a wide channel with parallel walls a distance apart. Two-dimensional plane flow problems, such as coolant distribution through narrow channels between thermal shields (neglecting thermal buoyancy and inertia effects), can be attacked by several methods. For streamline flow the same general methods of solution are available as for heat Analytical solution by complex variable' or numerical solution by conduction. actual and fictitious sources and sinks' is rigorous for the assumed conditions. For many cases an adequate and quick solution is obtainable by trial-and-error drawing of the "flow net." A system of curved streamlines between source and sink A second system of curved estimated to divide up the flow equally is first drawn. isobars is then sketched in, of which the source is the first, so as to be perpendicular to the assumed streamlines at every intersection and so spaced as to form curvilinear and When these conditions of perpendicularity squares with the streamlines. the squareness can be maintained and the last isobar fits the sink or is parallel to For flow between parallel planes the total pressure correct flow net has been obtained. then1 equal to 12spQ/pgJ>', where the total volumetric flow, the spacing drop the number of squares in scries and parallel, respectively. of the planes, and and For a porous slab the quantity 12/i/gcb1 replaced by (dP/dL)/iril, where dP/dL the pressure gradient caused by average or "superficial" velocity u. Electrical analogues are also readily applicable to streamline flow problems. continuous conductor or semiconductor may be used having the same shape as the flow field, or a set of discrete "nodes" covering the field (usually in square lattice) Flow conveniently and connected with adjacent nodes by appropriate resistors.1 measured as electrical current, and pressure difference as voltage difference. it,

6

Q is

is

a

A

is

is

p

s

is

3.2

If the viscosity

Nonisothermal

Streamline Flow

For gases + 0.46fc.

V

is used.

= 0.25<«

+

V

I'

tt,

is

is

it

is

U,

a

a

is

higher than that at the average temperature at the wall of pipe /i„ the stream u, as in cooling a liquid or heating gas, the pressure gradient —dP/dL will be higher than that computed for isothermal flow at the average stream tempera difficult, but empiri ture and vice versa. Theoretical calculation of the correction found satisfactory4 to divide the isothermal pressure drop by (w/Va)0-28 when cally cooling the stream and by (juAn.)0'" when heating, or by l.l(/i»/j*«)0M for cither case, though with less accuracy. to employ the isothermal formula with the viscosity at a tem Another method and the wall temperature <„. For oils intermediate between perature

of

0.75fe

= 0.58C + 0.42/4 has been recommended,' and for liquid metals

= 0.54t„

4

('


b

TURBULENT FLOW IN DUCTS 4.1

All

fundamental

Dimensionless

laws are dimensionally

Ratios

consistent.

also be, whether obtained theoretically or empirically.

Thus all derived laws must For maximum compactness

9-32

FLUID AND HEAT FLOW

[Sec. 9

and generality, the variables involved in any phenomenon are best grouped into the This is particularly true of turbu simplest and fewest possible dimensionless ratios. The most used dimensionless lent flow, the knowledge of which is mainly empirical. ratios in fluid and heat flow are listed in Table 3. Table

Common Dimensionless

3.

Symbol

Name

Reynolds No

Definitions

D.up/ii

Re

&PrV'D./lLpu' U'rttDSe/Ln*

fr

Ka

Diameters Relative roughness Maoh Not

L/D

Nu

St

h/cG

Gr

Fo

(D'pW /m'W AO DQc/k = Re Pr ke/cpL'. a8/L'

in

St

d

Gi Pe

j factor

-

e/D u/c 2APgt/pu* hD/k, hL/k

M

Pr

-

Pr

Re' 2/F

Nu/(Ro Pr)

*

* Source: C. F. Bonilla, "Nuclear Engineering," Ref. I. t The Weisbach friction factor, also widely used, is 4/p. t The symbol c here represents velocity of sound rather than specific heat. II The symbol h here represents the heat-transfer coefficient rather than enthalpy.

Turbulent

4.2

Flow Friction

Many turbulent pressure-drop results have been accumulated for widely different fluids and conditions, including wetting and nonwetting liquid metals. For any one With rougher pipe, results plotted as fw versus Re invariably lie in a narrow band. pipes /f is higher and levels off sooner, as shown in Fig. 2. The roughness e in the parameter in Fig. 2 is the diameter of sand particles glued to cover uniformly the inner wall of the pipe. Commercial pipe must be tested for pressure drop to evaluate the equivalent e. Recommended values10 of equivalent e are as follows: drawn tubing 0.000,005 ft, commercial steel pipe 0.00015 ft, galvanized iron 0.0005 ft, cast iron 0.00085 ft, and riveted pipe 0.003 to 0.03 ft. Figure 2 is based10 on the Colebrook relation for the transition and turbulent regions :

1/VA=-4iog,0(^ e = 0,

J5=) *<>VfF

this agrees with Prandtl's original equation for hydraulically smooth 1/

For full

+

Vh

= 4 log.o (Re

turbulence, setting Re =

h

-

« in Eq.

(228

V/7)

-

-4 log

(25)

The following relations are

= 0.00140 0.125(Re)-°" = 0.046(Re)-°M

+

fr

(24)

(23) :

Most nuclear-reactor channels will have smooth walls. equivalent to but more convenient than Eq. (24): Whole turbulent range: 5.000 < Re < 200,000:

0.4

l)"

Sotting pipes:

'ft


Ratios*

(26)

(27)


Sec. 9-2]

9-33

SYSTEMS

The walls are usually rougher in other industrial applications. The curve so labeled in Fig. 2 is typical of commercial operation. By whatever method fr is obtained, friction pressure gradient is then computed, aa in Table 3, by

dPr dh

_

n

dF dL

_

2/fmV gcD

_

2/t
gj)p

_

2frw*

(28)

gJ)pS*

In substantially equiaxed cross sections and in rectangular cross sections or con annuli in which the turbulence in the main stream would extend almost


centric

0002

5 6 8 I04

Fig.

2.

2

2 3 4 5 6 8 I06 3 A 5 6 8 I05 Reynolds number Re ■DaQ^/fi

Fluid friction in pipes.

■45 6 8 I0T

(From C. F. Bonilla, Re/. 1.)

equally over the cross section, the pressure gradient for a given fluid and average velocity will be substantially the same as for a circle having the same ratio of (cross sect ion) /(wetted perimeter), or hydraulic radius m. Since a circle has m

=

ttD*/4*D

=

D/4

the equivalent diameter D. for any section to use Fig. 2 is 4m. Dc equals the side of a square section, the difference in outer and inner diameters for an annulus, and twice the spacing b for a wide duct with parallel walls. For a concentric annulus, however, if the inner diameter is less than some two-thirds of the outer diameter, the pressure drop starts to exceed that given by the usual hydraulic radius." When De = 2Rt is used, agreement with Fig. 2 is obtained. bundles, Fig. 2 is some 40 per cent low.

ft2»

R2 In

- RS

(«./«,)

For flow parallel to uniform tube and rod

4.21 Nonisothermal Turbulent Flow. Isothermal flow and small temperature gradients are minimized in nuclear power plants, as they represent unproductive

FLUID AND HEAT FLOW

9-34

[Sec.

9

weight, space, cost, and pumping power. Thus highly nonisothermal flow is impor tant for gases in pipes. Figure 2 holds12 if p, and p are used at V = (U + W/2 or13p at I' = OAh + 0.6t„ and p at h- For liquids' fi and p may be taken at tt and the com In high-speed flow11 the pipe walls puted pressure drop divided by 1.020n/j*„)011*. is the will heat up by an additional amount y'v?/2gtCpJ because of frictional heat. mechanical equivalent of heat, and y', the "recovery factor," approximately equals

J

Pr».

4.22 Turbulent Flow of Compressible Fluids. In isothermal or substantially isothermal flow of gases, Eqs. (18) and (28) can be combined to eliminate dF and integrated analytically. Setting dW and dz = 0,

Pl%

_ iV

=

252?

Lf \nEl + Vl™t)/ V

g<

P2

T).

(29)

Either unknown

pressure is usually obtainable on the second trial by substituting for pi/ps the first trial value of Pi/Pi. Equation (29) is also applicable in spite of large temperature variation provided the friction term considerably exceeds the expansion term. In this case the average of /f?7 over L is employed. Normally p. and therefore Re do not vary with P and only with about the three-fourth power of T. By Eq. (27) will thus vary only as the one-seventh power of T, approximately, and may be taken, for simplicity, as constant at an average value /. then is averaged over L, and thus is the midplane tempera ture for a nuclear reactor with symmetrical heat generation and a coolant of constant specific heat. Multiplying through by V before integrating yields, for any fluid,

/


f

Pl _

P2 = 91

\ (F,

_

Vl)a +

W* + *>]

(30)

The quantity 2g(zt — z\)/(Vt + Vi)gc should be added outside the brackets if appre ciable. This relation is usually sufficiently accurate if 2 > (Vi/Vi) > 0.5 and is par ticularly useful in an integration by finite increments of channel length. Special methods are available for adiabatic flow.5 Equations (29) and (30) apply to stream line flow as well as turbulent if does not vary too much to permit the use of /. The 4.23 Velocity Distribution and Eddy Diffusivity in Turbulent Streams. velocity distribution that is obtained in turbulent flow in pipes and flat channels has been correlated in various ways1 in terms of the dimensionless local velocity u+ and distance from the wall y+, where

/

p.

>T»Je/

r„ is (D/4)(dP/dL) in a pipe and {b/2)(dP/dL) velocity distribution is given by 0 < y+ < 5 5 < y+ < 30 30
u+

in a flat channel.

= y+

3.05 + 5.0 In y+ u+ = 5.5 + 2.5 In y+ u+

(31)

Von Karman's

(32) (331

(34)

Equation (34) is normally used all the way to the axis or midplane. However, there it yields a definite velocity gradient instead of zero. Accordingly Eq. (34) is at times15 only used halfway to the axis, from which point u+ is continued constant to the axis. One application of Eqs. (32) to (34) is in the calculation of mean and maximum stream velocity in turbulent flow, also mean momentum and kinetic energy. Values are listed in Table 1. Another important application is in the prediction of heat- and mass-transfer coefficients, temperature distributions, etc., in channels of the shapes to which they apply and for conditions not so far investigated experimentally. For this purpose the


Sec. 9-2] eddy diffusivity by means of

systems

9-35

of shear force tr is computed for each region from Eqs. (32) to (34)

p

du/dy of heat tit or the eddy diffusivity of

tr is then taken as equal to the eddy diffusivity* mass e.« and employed in the equations

4 A

- (k +

£ dy

pcTt„)

NM = (Dv +

iv)

^ dy

(36) (37)


When these are integrated over the radius or thickness of the channel, the desired local and average temperature or concentration relations are obtained. 4.24 Wall Boundary Layers. At the inlet of channels, leading edges of strips, etc., "boundary layers" of slowed-down fluid begin to form. They progressively thicken, finally for long channels becoming the steady velocity distribution of Eqs. If the thickness S of the boundary layer is less than the available space, (32) to (34). boundary-layer relations rather than long-channel relations will apply in the calcula These are as follows: tion of wall shear forces. Laminar boundary layer (for Lupin < about 400,000) :

!-*(fr

W, (f.W

0.332

-0.064^

Turbulent boundary layer (for Lup/n

i

L

=

(39) (40)

> about 400,000):

= 0.377

MlW...

SL'feY*

0.0295

(^)~H \ ji /

(41)

>JLS(^VH

(42)

fe)"M

(43)

0.0369^'

u is the relative velocity of the wall and the approaching stream. At high velocity or with simultaneous heat transfer, p and p. may be evaluated" at 0.751* + 0.25k. 4.25 Initiation of Turbulence in Channels. From visual observation of threads of dye solution in pipes17 turbulence seems to occur first at the axis and at Re = 900, though with a smooth trumpet-shaped inlet much higher flow may remain streamline. An over-all phenomenon such as friction pressure drop (Fig. 2) generally first shows in In the long pipes observable departure from streamline flow at about Re = 2,100. "critical region," approximately 2,100 < Re < 3,500 if flow rate is given or 70,000 < Ka < 270,000 if pressure drop is given, flow is irregular, specially "in short pipes. It is generally desirable to avoid this region in nuclear reactors and heat exchangers, since pulsing may develop in which flow alternates between streamline and turbulent, the velocity of the stream controlling the period of the pulses. Such flow irregu larities would cause undesirable temperature and thermal-stress cycling. In other cross sections, such as annuli and square or flat ducts, turbulence initiates less abruptly and a pulsing is less likely to be encountered. * This in inaccurate

for Prandtl numbers much smaller than unity, namely, for liquid metals.

FLUID AND HEAT FLOW

9-36

[Sec. 9

Pressure Losses in Fittings

4.3

Pressure and energy losses in fittings cannot be reliably predicted, since fittings are not highly reproducible, differ widely among different brands, and affect one another unless widely separated. However, approximate rules have been developed to esti mate the losses for rough design purposes. In turbulent flow, the irregularities in most fittings are analogous to large values of e in Fig. 2, and the friction loss is thus generally proportional to fl*. A convenient measure is the number K of pipe average-velocity heads or pressures lost in the fitting. Another convenient unit is the length of straight pipe expressed in pipe internal diame ters L./D having the same friction loss as the fitting. These quantities are related by K «■ 4/f(L,/Z)). An average value of /p = 0.0055 may be used to compute L,/D approximately from K instead of the actual /p in the pipe. The friction computed from Table 4 is added to that for the same actual total distance as if it were straight pipe. Screwed and flanged fittings give similar K values. The computed drop in pressure is in addition to any drop or rise by Bernoulli's principle. When the discharge coefficient Co is provided, the pressure drop can be calculated Losses at an expansion may be estimated by by Eq. (60).


K„,

= 1

- 2y,Ai/At

+

(A,/Aty(2y,

-

1)

where 71 and yi are the momentum ratios from Table 1, based on and downstream pipes, respectively. At a contraction, Kc.„ =

(1

/Ce)

-

2

+ Cc(2y,

-

/f for

the upstream

1)

where Ce is the coefficient of stream-area contraction.* In streamline flow these calculated values may average 19 per cent high, and in turbulent flow 11 per cent K„p and Kcm are used in Eq. (60) with the smaller area. high.18 When many fittings are near one another, as frequently occurs in reactor designs, the pressure drop may be up to several times that calculated by Table 4, which is for isolated fittings. 4.4

Optimum Size of Pipes

To compute the optimum pipe or channel to conduct a given stream it is necessary to reduce all the factors involved to a common unit, preferably net cost per unit of Ideally, total time, and determine the size at which the total cost is a minimum. every cost item should include all advantages and disadvantages, such as space, weight, availability, probability of unexpected troubles, etc., in terms of money as well as the actual money cost. The total annual fixed charge per unit length for pipe and fittings, completely installed, is generally approximately proportional to £)', or, say, equal to XD* The total annual fixed charge on the fluid in the pipes is generally dollars/(year)(ft). negligible, relatively. However, for DjO, an expensive liquid metal, or a fuel solu tion, it may be important, say equal to YD* dollars/(year)(ft). The total annual net cost of the required pumping energy is inversely proportional to a high power of D, say equal to ZD~" dollars/(year)(ft). The exponent n =

5

+ d(log/)/d(Iog Re)

from Fig. 2, ranging from 4 for streamline flow to 5 for full turbulence (likely in many nuclear-power-reactor applications). The optimum diameter from the standpoint of these costs is obtained by adding • Ce minimum cross section of the jet downstream of the orifice nr slot.

of the orifice or slot, divided by the cross section


Sec. 9-2]

Table 4.

systems

9-37

Friction in Typical Pipe Fittings, etc. * Convergence

L./D

A't

Fittin«

Cd

ratio, Dt/Dx or

Ct + Co

61/61

Turbulent Flow t


One average- velocity head

Pipe fittings: Elbow, 45° standard Elbow, 45° long radiua Elbow, 90° standard Elbow, 90° long radius Elbow, 90° square Return bend, 180° close Return bend, 180° medium Tee, 0° through Tee, 90° side outlet Tee, 90° side inlet Coupling, union, 0° through Valves: Valve, open, 0° gate Valve, open, 0° globe Valve, open, 90° angle Valve, open, 0° swing check Pipe bends: 0. 5 Bend. 90°; r/D Bend, 90°; r/D 1 Bend, 90°; r/D 4 Bend, 90°; r/D 10 Coil, each turn. 360°; r/D 1 Coil, each turn, 360°; r/D Coil, each turn, 360°; r/D 10 Contraction and expansion: Sudden contraction, Di/Dt — 1. 33 Sudden contraction, Di/Dt = 2 Sudden contraction, Di/Dj — 4 Sudden expansion, Di/Di — 1. 33 Sudden expansion, Dj/Di — 2 Sudden expansion, Di/Di ■■4 Gradual conical expansion in pipe Gradual conical convergence in pipe Gradual conical converging inlet Orifices and nozzles:^ Small square-edged orifice Rounded nozzle Excess in tube below nozzle Short tube downstream of plate (L < 3D) Long tube downstream of plate (L > 3D) . Short tube upstream of plate (L < 3D). - . Long tube upstream of plate (L > 3D). .

-

-4 -

45 (pipe)

0.35 0.2 0.82 0.53 1.3

2.0 1.2 0 4 1.3 1.5 0 04

0.16 6 3 2 0 8

0.35 0. 16 0. 16

2.2 0.55 0.3 0.19 0.33 0.42 0 19

0.56 0.92 0.2 -0.5 0.06-0. 15 0. 13-0.4 0 05

0.05 0.02 0 05

0.5 0.06 0 8

15 10 35 25 60 90 55 20 60 70 2 7 270 135 90 35 16 7 7 100 25 14 9 15 19 9 25 42

35-0.94 0 60 0 98 fl 0 0 0

60 82 53 72

Streamline Flow Conical

or straight slot convergence

\

(135°)

0

0.4 0.8 1.0

Sharp-edged orifice or slot (90°)

0

1

0.4 0.8

I

1.0

0 746 0 749 0.789 1.0 0.611 0.631

0.722 1.0

* Source: C. F. Bonilla, "Nuclear Engineering." Ref. I. smaller cross section. t Where there is a change in cross section, K applies to the velocity in the " Applied Fluid Mechanics," % For additional data see particularly M. P. O'Brien and G. H. Hickox. p. 350. McGraw-Hill Book Company, Inc., New York, 1937. losses, which can be calculated as for an expansion. 1 K does not include downstream

FLUID AND HEAT FLOW

9-38

them and differentiating with respect to D.

Thus

[Sec. 9

is obtained

(turbulent)

(streamline)

Since X, Y, and Z normally vary somewhat with D, after a preliminary calculation of Dopt they should be recomputed from the data for the problem and a second approxi mation of D0pi computed. D may be changed somewhat (roughly ±25 per cent) from Dovt without making a large change in total annual cost, since at D„pt the rate of change of fixed costs with increasing D is exactly equal and opposite to the rate of change of operating costs with D. Flow in Complex Circuits


4.6

Complex circuits and networks in steady-state flow in reactors can be solved by principles analogous to Kirchhoff's laws for electrical circuits. Thus, the net flow to any location is zero, and the pressure change when integrated around any closed path is zero. With streamline flow at known temperatures, the simultaneous equations are all linear in mass-flow rate w, and enough can be written to solve any network directly. In the more common case of turbulent flow, the net-flow equations are linear in w but Thus in the general case simultaneous solution the pressure-drop equations are not. is not possible, though trial and error may be satisfactory. Alternatively, and particularly where the direction of all flows is evident, the con cept of summation of series resistances and parallel conductances is useful for com From Eq. (20) in streamline flow, the puting total effective resistance of a network. The equivalent resistance of series resistance AP/w of a pipe is (Y2&/-rgc){iL/F*). pipes is

(S).-S2(b> rg,

The equivalent resistance of parallel pipes is

(401

For turbulent flow a "turbulent resistance" AP/w' is convenient. pipes in series

(^)

From Eq.

(28) for

-i2y(/^)

(4;r,

For pipes in parallel

Similar procedures can be used with other channels. 4.6

Liquid Flow with Thermal Buoyancy

Letting subscripts 1 and 2 refer to the inlet and the outlet, respectively, Eqs. and (28) yield for a vertical channel with symmetrical heat generation

p

_p _

pifftes

- zi)

ffc

m

2/ric' pS'gc D,

_

p,gB,q(zt 2wcgc

- zi)

(1<»)

Sec. 9-2] This relation

FLUID FLOW IN REACTOR ia

9-39

SYSTEMS

useful for computing any one variable for a channel provided all

others are known.

.

For parallel channels, the left-hand side is constant for all the tubes; therefore the right-hand side also is. The ratio of the buoyancy and the friction terms on the right side is

p'Pgjh

-

4/^

_ Gr

(&U)

Ka

is

is

is

is

it in

is

t2

1,

»

1,

«

is

t\,

where the Grashoff number Gr includes the temperature rise tt — not the usual If Gr/Ka wall-to-fluid At. minor and Eq. (20) or (28) yields the buoyancy flow in any particular channel regardless of variations in heat generation among the tubes. If Gr/Ka friction minor and the flow adjusts itself in each channel so that the outlet temperature the same throughout, provided flow upward. Downward flow should be avoided, at least unless Gr/Ka always «1. The reason for this that instability may develop under certain conditions, so that a slight decrease flow increases the buoyancy opposing the flow, thus decreasing the flow further until stops and reverses. During the reversal burnout may occur. Differentiating Eq. (49) with respect to w and equating d{P\ — P2)/dw to zero, the following critical flow wc obtained for downflow

_

\mi^Bi^V\s 8/f

(si)

J

J

L

L

8/Vc

L

rpggggÂg-iM L

w< =

4.7

Flow

Unsteady-state

a

1

a

is

is

is

a

is

The calculation of unsteady-state fluid flow of importance in evaluating tempera tures and stresses in rapid start-up, load change, or scram of nuclear reactor. Initial maintenance of the flow by the momentum of the stream and by any flywheel effects, and subsequently by thermal buoyancy, also important or necessary in the event of sudden power or pump failure in order to prevent fuel-element burnout by the delayed heat generation. This characteristic the "flow coast-down" curve. The calculation can be carried out by equating the pressure drops around the flow loop that oppose the flow to those which maintain the flow. Friction pressure drop during velocity changes given adequately by Eq. (28). The momentum of stream per unit length may be shown by Table to equal w/gc, as at uniform velocity. Thus the total forward pressure generated at any instant by acceleration or deceleration in —(dw/gcde)X(L/S). loop Calling APp the pressure generated by the pump in the direction of flow, APb that of thermal buoyancy in the direction of flow [the last term of Eq. (49)], and ERr the total of energy losses in the loop in the form Z(APp/w2), the time required to change from wi to u)2 is

is

VL,

\S)

J

gc

J»t w'XRr

- APb - APp

-

Rei /"Rei

Re, Me,

u

fT

LZVp

AS

J /f

L.it

2 L-11

0,

is

which may be integrated graphically. If most of the flow resistance localized in one size of pipe (thus at the same Re) and APp = Eq. (52) can be generalized by replacing to by (jiS/D,) Re:

jnP dlie

3

— -

Re* Rc!

where E = p*PgDetAzq/2CSii'Lt. Equation (53) from Rei = given in Fig. smooth pipes and values of E. is


is

Higher flow rates are stable, and lower flow rates unstable. One evident cause of cutting back the flow in a reactor under low load or during shutdown to instability maintain the coolant temperatures constant and minimize thermal stresses. At low enough load we would be reached.1

E/Bie £/Re

for to lower Re values for several

has been evaluated graphically 107

(53)

9-40

FLUID AND HEAT FLOW

o

IT

1

h) hi

h)

o o T 7

-

r

o

o

o

i m o C =

o

c

II h)

h]

II h)

II <0

i hi I4

ii h]

1

1

1

[Sec. 9 16

2

o

o

O

II hi

II hi

h)

14

V

1

-1

-

— No thermal '|

'

—2

-3

\

-

L-^-

10

-55^

|

8

.Steady state. (6 -=oo )

~

12

>:

,

^

6*

1

4

'■^ ^^L _

2 0

-A 1

2

3

4

5

6

log Re

Fia. 3. Generalized solution for steady- and unsteady-state natural convection flow in a pipe loop with thermal buoyancy. (From C. F. Bonilla, Ref. 1.)


6

FLOW OF SLURRIES (BINGHAM BODIES)

Slurries may be useful in nuclear power on account of poor solubility of a fuel or fertile element in an otherwise desirable fluid medium, provided settling out does not occur or can be controlled. They may also offer processing advantages over a solution or a stationary solid. Direct experimental determination of the flow-characteristic curve (Fig. 1) of a non-Newtonian fluid can in general be carried out only in a pipe and with streamline flow. A complete curve is obtainable from a series of tests at different flows. For each flow test, the volumetric flow Q, pressure gradient dP/dL, and radius R are known. The wall shear stress t„ is (R/2)(dP/dL). The wall velocity gradient can be computed from the slope of the curve of Q versus t„ by the Rabinowitsch-Mooncy equation:19

(du/dy).

= [3Q +

T„(dQ/drMrR>

(54)

Each pair of values of t„ and {du/dy)w is a point, on Fig. 1. Most slurries when dilute are Newtonian (see Art. 1.3) and when concentrated are For the former case Table 1 applies. For the latter, in Bingham bodies (Fig. 1). streamline flow:1 Pipes: ■

Flat ducts:

— <£[' -fc ifcfl+3 +

W = T*Qc

DdP 4dL b

dP

2

dh

(55)

(56)

In the turbulent flow of suspensions Fig. 2 holds, using for p. the effective viscosity at infinite shear (i.e., the rigidity for Bingham bodies). For Newtonian suspensions p of the slurry and n of the pure medium give the best agreement with Fig. 2. 6

FLOW OF VAPOR-LIQUID MIXTURES 6.1

Friction Pressure Gradient

Knowledge of the friction pressure gradient for vapor-liquid mixtures is useful in designing wet vapor-flow lines but is particularly valuable in predicting the effect of


systems

9-41

intentional or unintentional boiling on flow distribution among parallel channels in a reactor. In general, high gas- or vapor-flow rates tend to carry a coexistent liquid phase along as slugs in a small pipe or as a film on the wall of a large pipe, whereas high liquid rates tend to carry the gas or vapor as bubbles. When both rates are low, fluids in a horizontal pipe tend to stratify. The friction pressure gradient near atmospheric pressure in such varied flows correlates with an average deviation of some 25 per cent80 against the friction pressure To find the gradients of the two phases flowing separately, each at its own flow rate. two-phase friction pressure drop APip, Re is first calculated for each fluid separately. For Re under 1,000 flow is assumed streamline and over 2,000 turbulent. AP is then computed from Fig. 2 for each fluid alone, and Table 5 entered with the ratio APl/aPq at the proper column. The corresponding APjp/Aft is read off or interpolated and multiplied by APl to give AP2p. The fraction by volume of liquid and of gas also That for the liquid Rl is listed in the table; that for the gas correlate reasonably well. is 1 — Rl- The columns headed Pc are for liquid and vapor at the critical pressure. For intermediate pressures log (AP2p/APl) may be interpolated linearly with P. The column headed "down " is for downward flow of boiling water in a J-jj-in. annulus, and the columns headed "up" for upward flow in a 1-in. tube. An alternative method of predicting the friction pressure drop employs Fig. 2 and Specific volume and fluidity are considers the two-phase mixture as a single fluid. A straight-line plot of l//i versus fraction assumed additive on a mass-flow rate basis. vapor x does not change rapidly with pressure above the boiling point. The former method is more accurate, but the latter more convenient for repeated calculations.1 In wall or "local" boiling of subcooled liquid coolant the pressure gradient exceeds that for nonboiling at the same heat and coolant flows. However, the expansion and therefore pressure gradient are less than for "bulk boiling" at the saturation tempera In a %-in.-ID pipe the local to nonboiling pressure gradient ratio for water at ture. 30 to 85 psig" is cosh [(4.6 X lO^'q/A + 1.2)L/Lt] = cosh (aL/L,) (57) where q/A is in Btu per hour per square foot, L is the distance downstream from the incidence of local boiling, and L< is the total length that would be required in local boiling before the liquid reaches the boiling point. Integrating over the local boiling length AjPis, the total pressure drop at constant q/A is obtained as the nonboiling The average density of localpressure gradient multiplied by (Lt/a) (sinh aL/Lt). boiling water has also been studied for evaluation as a moderator." Local-boiling results cannot as yet be extrapolated safely beyond test conditions. 6.2

Pressure

Drop in a Bulk-boiling Tube

= Sm + AQn/n =

M Qc

+ fc,+

20„

yj = ^ yc

h'n

+ xji"» +

V

(V„

+

is

are the enthalpy and specific volume of the saturated are the increases on complete evaporation. It desirable that

where h' and

and V"

^

En

+

Article 6.1 yielded the pressure gradient at any point in a boiling tube. The gradient must be integrated along the tube to obtain the total APf ■ Since many quantities vary irregularly a step-wise calculation is usually best. The "equivalentfluid" method of Art. 6.1 will be outlined. The Rl values of Table 5 generally However, the equal indicate a higher vapor than liquid velocity, or "slip flow." velocity or "fog flow" assumption is simpler, and results agree well with experimental data. Going downstream over an increment of length from m to n with dW = 0, Eq. (17) may be written

Sec. 9-2]

x.V".Y

(58)

liquid and h" G'a A(F8)/2yc

FLUID AND HEAT FLOW

9-42 Table 6.

Friction Pressure Gradient Correlation for Flow of Liquid-Gas Mixtures*' t log

, log

APl APo

- 1.0

1.0 1.5

1.790 1.389 1.050 0.771 0.567


2.0 2.5 3.0 3.5 4.0

P.

Down

- 1.5 0.0 0.5

Volume fraction liquid Hl Liquid viscous, gas

-4.0 -3.5 -3.0 -2.5 -2.0

-0.5

(APV/\PL)

Liquid and gas turbulent

Up

2 1 1 1 1 0

042 727 473 267 088 925

[Sec. 9

4 3 3 2 2 I 1 0 0 0 0 0 0 0 0 0 0

000 505 012 526 052 598 180 817 527 317 181 099 0529 0279 0146 0076 0040

turbulent

Pa

4 3 3 2 2 2 1 1 1 1 0 0 0 0 0 0 0

214 762 340 930 534 172 832 519 246 027 831 651 486 366 262 169 091

4 3 3 2 2 1 1 1 1 0 0 0 0 0 0 0 0

Liquid turbu

Liquid

lent, gas viscous

158 705 249 800 364 978 635 328 083 879 696 534 403 31 1 227 159 091

4 3 3 2 2 1 1 1 1 0 0 0 0 0 0 0 0

098 624 167 732 323 954 616 321 083 879 723 580 440 338 236 159 091

and gas viscous

4 3 3 2 2 1 1 1 0 0 0 0 0 0 0 0 0

042 556 084 630 087 774 404 086 833 663 537 440 352 281 219 159 091

Up

0 0 0 0 0 0 0 0 0 0 0

010 019 034 06 11 17 30 45 60 81 91

P.

0 0 0 0 0 0 0 0 0 0 0 0 0 0

03 05 08 12 17 23 30 37 45 53 63 72 81 "0

P.

0.005 0 010 0 019 0 036 0 067 0. 12 0.21 0 34 0 50 0 66 0.79 0 88 0 93 0.96 0 98 0 99 0.995

* R. W. Lockhart and R. C. Martinelli, Chem. Eng. Progr.. 45: 39 (1949) ; R. C. Martinelli and D. B. Nelson, Tram. ASMS. 70: 695 (1948); R. P. Stein, el at., Chem. Eng. Progr. Sympotium Ser..S0: II " Heat Transmission," 3d ed., McGraw-Hill Book Company, Inc., New York, ( 1954); W. H. McAdams, 1954. t Values are for pressures near atmospheric and horizontal flow unless marked otherwise.

Enthalpy h. Btu/lb

Flo.

4.

Thermodynamic properties of wet steam.

(From C. F. Bonilla, Ref. I.)


Sec. 9-2]

SYSTEMS

9-43

Thus smaller length be small compared with the total energy per unit mass E. One increment may be adequate for the increments are preferable near the outlet. nonboiling portion. A chart of h versus V for the two-phase region (Fig. 4) is drawn by joining the points for each saturated phase at a given P with straight lines. These isobars arc then subdivided into equal fractions of their lengths, and curves of constant x drawn. The following procedure is used: 1. Eu is evaluated as (Em + AQm/*). 2. Vn is estimated (somewhat > V„). 3. h„ is calculated by Eq. (58) and x„ read from the chart.

' and " are defined as is computed as D«G[(1//*)' + t„/n(l/ti)"}, where and of and above xm. x„ £„/„ is the average 5. fr is read from Fig. 2. 6. Pn is calculated by Eq. 29. 7. The chart is consulted to see if the estimated Vn and computed h„ and P„ agree. If not, a new Vn is estimated and steps 2 through 8 repeated. 8. The next increment is started. It is evident that fr will not change rapidly, so that Re and fr will not ordinarily The three curved lines in Fig. 4 are need to be determined for every increment. Integrated solu condition curves for different flows through a given heated tube.1 tions not requiring the finite increment calculations are available for the special case of APr P and Q linear in tube length.11 If Table 5 is employed to yield friction pressure gradient, the procedure is the same, except that use of this table replaces steps 4, 5, and 6. 4. Re„/„


«

6.3

Maximum

Velocities of Compressible Fluids

A compressible fluid in a pipe can nowhere exceed the mass velocity G, at which all the energy obtained from a differential pressure drop is used up in the corresponding increase in kinetic energy of the stream. At a point at which Gc is reached, the expan sion is isentropic, as it will take place so rapidly that heat transfer and friction may be On further expansion the velocity would rise even more, which is not neglected. possible in a pipe of constant cross section; thus Gc is reached, if at all, at the outlet of the pipe. The maximum mass velocity Gc for a boiling fluid can be shown1 to be given by the

relation

(59)

which

Table 6 is the same as for sonic velocity in a homogeneous compressible fluid. is a skeleton table of Gc and E (kinetic energy plus enthalpy) as a function of exit quality and pressure, as computed by finite increments from steam tables. If in a downstream step-wise calculation at a given Pi, G, and hi with unknown P„, as per Art. 6.2, Gc for the local values of P and x is reached before the pipe outlet, too low a Pi was employed.* The calculation should be repeated starting from the downstream end at the necessary P„ to yield Gc = the desired G. If there is a constriction in the pipe upstream of the outlet, the local maximum velocity may be reached there as well as or instead of in the outlet. Complete equilibrium between the two phases is usually not attained, particularly when x is small and the pipe size large. For instance, experience indicates that 221 lb/(ft1)(sec)(3.7 fps) of boiling water at atmospheric pressure with a trace of steam can be readily exceeded in a pipe. Preliminary results show that in Y\- to 1-in. pipes2* the ratio of observed to theoretical Gc for fog flow is roughly 2/(x + 1), though rising much higher for x under 2 per cent. For a Jjj-in. annulus the ratio is similar at low x but falls rapidly to about 1.1 at x = 0.1. Nevertheless, reasonable agreement between calculated and observed values of Pi is obtained by the above theoretical methods. * The subscripts i

and o indicate inlet and outlet

respectively.

FLUID AND HEAT FLOW

9-44

[Sec. 9

Maximum Mass Velocity Gc in Feet per Second and Energy E in Btu Table 6. per Pound above 32°F for Fog Flow of Saturated Steam-Water Mixtures Pressure,

Quality, x, vapor fraction

14.7

a.

53.8 61.4

0.995

0.75 0.50 0.25

73.7

98.9

0. 10 0.005

For all practical

136.9 221.0

psia

50

B

G,

1186.6 938.2 684.7 431.4 279.8 184.9

177.0 201. 2 240.7 317.5 430.7 609. 1

200

E

B

1214.4 976. 1 733.2 490.3 345.4 254.7

680.9 761 3 918. 2 1151.0 1522.6 2153.3

1242. 1 1021.8 799. 1 575.3 442.6 359.7

purposes, data for H20 are valid also for D20 pressure-drop calcula 10 per cent higher density.

tions, except for the

7


7.1

FLOW-DISTRIBUTION

CONTROL

Protection with Orifice against Burnout

A heated tube with a fixed outlet pressure and liquid coolant in its boiling range has a Pi versus to* characteristic as in Fig. 5. If the full inlet header pressure, say However, if there is an Pn, is available at the tube inlet, a flow of wr would result. initial fixed orifice or nozzle, a flow-regulating valve, a length of pipe, a temporary accidental obstruction, or merely pressure loss at the inlet of the pipe, P.- will be less than Pr. In this case, P; to the heated tube when computed from the inlet header will be given by a substantially straight line of negative slope, such as PrF , which yields a flow of wm . Operation at R or M will be stable, since if w should drop momentarily for external or statistical reasons, a net accelerating AP becomes available, and if it should rise, a net decelerating AP is produced, restricting the change in flow and returning the flow to the original value when the cause of the flow change disappears. A positive value of However, S and S' are metastable (unstable) operating points. d(X&P)/dw at the intersection indicates stability, and a negative value instability. Permanent inlet resistances have a constant or almost constant ratio AP/tc', which may be called their turbulent resistance R: ±P = R w*

1

2!W>itVSo1! (orifices. nozzles)

=

=

OrPiD.S* (pipes. etc.)

_JL_ 2i7rp,S»

(60)

(Table 4)

where S is the smaller area involved. The slope of any one of the inlet pressure lines, such as PrF, is — Si? summed up for all the inlet resistances that are in series. Figure 5 shows possible design situations: 1. If wr is rated flow and no inlet resistance is employed (P< = Pr), the tangent from Pr to the curve shows that an added inlet obstruction of resistance RG/wr' is possible before start of boiling. 2. If an inlet, pressure Pq is employed, a nozzle of resistance (Pq — Pr) /itrs is needed to cut the normal flow to wr. Drawing the new tangent, an obstruction of up

RV/wr*

can now be tolerated before instability appears. cases 1 and 2, if boiling starts, flow will drop to much less than tc« and will not recover to wr if the obstruction disappears unless the power is decreased sufficiently to stop the boiling. To permit recovery to wr without necessity of power decrease.

io

3.

In

Sec. 9-2]

9-45

FLUID FLOW IN REACTOR SYSTEMS

Pj

the tangent from R to H is drawn, yielding as the necessary P,- and (Pj as the nozzle resistance. 4. If it is desired to be able to obtain stable flow at all possible flow rates by setting a valve at the channel inlet, as in an experimental setup, the inlet pressure must

Pr)/W

exceed Pg, the intercept given by the highest tangent to the curve.


In the design of protective nozzles, the accidental restriction Ri to be protected against should first be decided on, from the estimated probability, extent, and duration of the possible accidental restrictions; the fixed and pumping charges of the protective nozzles; required reliability and other characteristics of operation; hazards and

Fio.

drop in a tube with heat generation, and its protection with an inlet series Outlet header constant at P„. (From C. F. Bonilla, Ref. 1.)

5. Pressure

nozzle.

The lowest desired flow in the tube, say expenses of a burnout; etc. The corresponding inlet pressure is Pm. The necessary selected. pressure may be shown to be

must also be inlet manifold

wm,

(61)

In actual practice, the inlet header pressure is commonly selected merely by adding an arbitrary reasonable excess above some minimum pressure, for instance 100 per cent of (P« — P„) above PR or 30 per cent of (Pm„, — /'„) above /'m„ have been used, the inlet nozzle and piping dissipating the excess AP above Ph. Upward flow in parallel tubes with boiling is inherently stable. However, in this case inlet nozzles are likely to decrease the stable range and would then be unde Protective nozzles and other resistances may also be located downstream sirable.1 of the heated channels, which considerably alters the flow curve after boiling has This arrangement must be calculated in detail but in general has the started.

FLUID AND HEAT FLOW

9-46

[Sec. 9

advantage that boiling will not start until a lower flow is reached, due to the higher static pressure in the heated section. Once boiling has started, however, the pressure drop will rise faster. This arrangement may thus be preferable over a limited range. The use of a common nozzle for protection of several channels in parallel is impractical, since the necessary excess pressure for the same degree of protection increases rapidly with the number of channels.1 7.2

Use of Nozzles for Flow-distribution

Control

Optimum design in a reactor for power generation will normally require the coolant to issue at the same temperature from all channels or at otherwise related temperatures throughout the reactor. Thus the flow through the main lower power channels must be decreased by using smaller channels, separate pumps and inlet headers, or series nozzles. Whether or not protective nozzles are used in the high power channels, noz zles are usually the most practical flow control for the lower power channels.

LOSSES IN EXTERNAL FLOW

8

Flow Past Isolated Objects

8.1


The force F developed when a flowing fluid envelopes an object is usually correlated u is the relative by the drag coefficient Cj, the dimensionless ratio 2Fgc/pAu*.

-rtt f

1

'4

—Lh

Ia

—

-Cn

—

4

a? she d silica

y

v\

|

i

>s

:: := t

t- t i_

—

II

1

10"2

Fig.

6.

:

10"'

1

2

: t-

s 10'

J—

r

4 6 io

WT 102

_XJ

CO

Sd h e res

i i i

D da

Su

1 103

10'

10s

10'

Drag coefficients for steady motion in a fluid. Solid lines, solid particles; broken and drops. The parameters are as follows: (From C. F. Bonilla, Re}. 1.)

lines, bubbles

li

= Hp — lia

bu ~

velocity of approach, p the fluid density, and A an arbitrarily defined area or (dimen For any given sion)2 of the object, generally the area projected in the direction of u. geometry, depends only on a Reynolds number up to a considerable fraction

d

of sonic velocity. Figure 6 gives the relationship for shapes of interest in computing settling velocities of slurry particles and liquid droplets, pertinent to slurry fuels and breeding blankets, liquid-liquid fission-product extraction columns, etc. Corrections are necessary because of molecular motion for particles under about 0.003 mm in diameter, because of the wall or other particles if they are not far away,57 and because of acceleration.18 In addition, suspended particles may flocculate by collision and adhesion and then settle out more rapidly.

fluid flow

Sec. 9-2]

in keactor systems

9-47

For long fuel elements, the drag due to the flowing fluid is more readily obtained by a force balance, the total drag equaling the friction pressure drop by Arts. 3.11 through 4.22 multiplied by the total cross section of the stream. Flow through Tube Banks

8.2

Flow parallel to the tubes in heat-exchanger shells can be handled by Arts. 3.12, For flow perpendicular to tube banks the average velocity of the 4.2, and 4.21. stream um»« in passing between the tubes at their closest approach is employed. Each of five or more rows of tubes has a loss K of approximately 0.72 velocity heads or pres sures for staggered tubes and 0.32 for tubes in line. Within 25 per cent accuracy, for 10 or more rows of staggered tubes on an equilateral triangular grid spaced up to D/2 between the tubes, the pressure drop per row is AP

= 1.5p»-«iZra„1-V-Vfl.o-20e

(62)

where D, is the open space between tubes. If baffles are employed in the shell, two additional velocity heads each. Actual pressure drops with baffles may be much lower at a result of leakage around or through them, but for the same heat leakage will require a larger flow and may actually thus increase the


8.3

Flow through Beds of Particles

Fixed packed beds of the normal 35 to 45 below within ±25 per cent, on the average: DyV>P

n

U,flL

< 40

voids follow the expressions

per cent

=

Dp1

gcAP

1,7004/

=

\

geAP

n

L

may be lost at given flow as a transfer, baffle pressure drop.

76Af

fi

(63) (64)

)

is the bed depth, Dp is the particle diameter, u, is average velocity disregarding

packing, and A/ is the wall leakage factor, varying linearly from 0.81 in streamline and 0.7 in turbulent flow at Dp/Dâ = 0.1 to 1 at Dp//)bed = 0. Five- to sixty-mesh-per-inch wire screens, as well as beds of spheres, fall" within some 20 per cent of the following correlation: Usp/lta iiPl>ca>/pN0ii.'

0.5 10

2.5 2 5

where a is the fractional volume of voids in wire diameter) or bed thickness L and 0 is and thickness L of the screen or bed. 8.4

If

a particulate

Fluidization

10

0.85

25

0.5

125 0. 25

the full screen thickness (usually twice the the total surface area per unit face area S

of Particulate

Solids

bed is not confined, at a fluid-flow rate such that the pressure

drop

across it (Art. 8.3) roughly equals its net weight per unit area but is below the trans port velocity for the separate particles (Art. 8.1), the bed will become "fluidized." Good fluidization exhibits fast and thorough mixing of the bed and temperature uni formity throughout the bed and fluid and offers possibilities for nuclear fuel process ing and reactor operation. Best fluidization is obtained with uniform particles fluid ized by a liquid of low viscosity, and in the fluidized bed the particles occupy some 5 per cent of the total volume. Accurate predictions are difficult,30 and tests are required on the specific system for quantitative design purposes.

9-48

FLUID AND HEAT FLOW


9

[Sec. 9

MECHANICAL PUMPS AND BLOWERS

Centrifugal and axial-flow pumps and blowers are suitable for most fluid moving in nuclear power installations. They are available in a wide range of sizes, can stand reasonably high temperatures and pressures, and can be sealed well in case leakage is expensive or dangerous. Erosive fluids would damage them but would rule out other These pumps are available with an inert-gas seal on the shaft or equipment also. without emergent shaft thanks to a built-in motor or to a thin scaled nonmagnetic diaphragm through which magnetic drive occurs. Variables in substantially constant-density operation of a pump or blower of a given geometrical design include rotational speed n in revolutions, a dimension such as the rotor diameter D, fluid pressure rise AP, density p, viscosity ji, and volumetric flow rate Q. In common practice, the characteristics of a pump are plotted as constant-speed curves of AP (or of pressure head APgc/pg), shaft horsepower P, and efficiency E versus Q for the desired fluid. If, instead, the appropriate dimensionless ratios are employed, operation for any fluid and conditions can be predicted, the size of a geometrically similar pump selected for a different Q, etc. The dimensionless ratios may be picked from the following: the rotational Reynolds number D*np/n, the "capacity coeffi cient" Cq = Q/nD', the "head coefficient" CH = APgr/pn'D2, the "specific speed" n. = nQîp/APgc)*'*, and APgcD* / pQ1, a von Karman number. When pump data are so handled, it is found that unless Dhip/n is unusually small, it has no significant effect on the other ratios. Viscous friction losses are evidently relatively small. Thus, over its practical range, the characteristics of a centrifugal pump or blower can be expressed as any one of the other four above ratios plotted against any other one of them. Cavitation occurs with a given liquid when the absolute pressure at any location Noise, harmful ham in the pump falls to the vapor pressure at the same location. If the absolute static pressure at the inlet is mering, and fall off in efficiency ensue. Pi and the absolute vapor pressure of the inlet fluid P„ the "cavitation ratio" (Pi — Pr)/AP below which cavitation develops has been found to be roughly constant for a given pump shape. For centrifugal pumps handling water it is recommended Centrifugal pump and blower that Pi — Pv be at least some 7 or 8 psi (15 to 20 ft). tip speeds are usually under 200 fps, and inlet opening velocities commonly below 10 fps for water and 100 fps for air. The shaft power P can be expressed dimensionlessly as the "power coefficient" C'p = Pgc/pn*D*, which also plots as a function of any one of the four groups above, provided viscous and cavitation effects are absent. Efficiency (APgcQ/P) is also a dimensionless ratio dependent on any one of the others. The maximum efficiency of a pump is obtained at some one point on the generalized efficiency curve, and thus at one value of each of the four other ratios above. The specific speed is particularly convenient in that 1) is absent and the other quantities in it arc usually specified in a given pumping job (except for any possible Table

7.

Dimensionless

Performance Coefficients for Commercial Single-stage Centrifugal Pumps and Fans Pumps

Coefficient

Co t, .

At E,

At Q

-

3.5 -5.8 0.01-0.4 0.03-0. 3 0.4 -0.9

4 .4

-6.5

0

0 0 0 0 06-0.8

At E,

"

3.5-11. 0.2- 1.3 0 1 0.3 0.4 0.9

JAt'6 "HI 3 6-12 0 0 o 0 5-10

Sec. 9-2]


systems

9-49

alternate values of n). Thus, for any given pumping operation the specific speed can be computed and a pump or blower of the desired type having the desired specific speed at its maximum efficiency selected from manufacturers' data or scaled up or down from data on a given shape. For commercial single-stage centrifugal pumps and fans of various designs the dimcnsionless coefficients usually fall in the ranges indicated in Table 7. For single-stage axial flow pumps n. at Emn, usually ranges from 0.3 to 0.8. Actu ally, the optimum operating point is usually not exactly at maximum efficiency. Minimum pump size, investment, and total pumping cost are obtained by operation slightly beyond maximum efficiency (higher Q); high emergency reserve pumping capacity is obtained by normal operation below maximum efficiency flower Q). The combined characteristic AP versus Q curve of dissimilar pumps in series can be obtained by plotting the sum of the separate AP values at the same Q versus the value of Q. Similarly, for pumps in parallel 2Q at each AP is plotted versus AP.

REFERENCES Bonilla, C. F. (ed.): "Nuclear Engineering," McGraw-Hill Book Company, Inc., New York, 1957. 2. Bonilla, C. F., S. J. Wang, and H. Weincr: Trans. ASME, 1955. 3. Bonilla, C. F., R. D. Brooks, and P. L. Walker: "General Discussion of Heat Transfer," American Society of Mechanical Engineers, New York, 1953. 4. Bromley, L. A., and C. R. Wilke: Ind. Eng. Chem., 43: 1641 (1951). Handbook," 3d ed., McGraw-Hill Book 5. Perry, J. H. (ed.): "Chemical Engineers' Company, Inc., New York, 1950. in Fluid Dynamics," Oxford University Press, 6. Goldstein, S.: "Modern Developments New York, 1938. Book Company, Inc., New York, 7. Streeter, V. L.: "Fluid Dynamics," McGraw-Hill


1.

1948.

8. 9. 10.

11. 12. 13. 14.

15. 16.

17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Jakob, M.: "Heat Transfer," vol. 1, John Wiley & Sons, Inc., New York, 1949. Deissler, R. G.: NACA TN-2410, 1951. Moody, L. F.: Trans. ASME, 66: 671 (1944). Rothfus, R. R., et al.: Ind. Eng. Chem., 31: 618 (1939). Deissler, R. G.: NACA TN-2138, 1950. Deissler. R. G.: ASME Paper 53-S-41, 1953. Keenan, J. H., and E. P. Neumann: J. Appl. Mechanic*. 68: A91-100 (June. 1946). Poppendiek, H. F., and L. D. Palmer: ORNL-1395. Rubesin, M. W., and H. A. Johnson: Trans. ASME, 71: 383 (1949). Rothfus, R. R., and R. S. Prengle: Intl. Eng. Chem., 44: 1686 (1952). Kays, W. M.: Tram. ASME, 7S : 1067-1074 (1950). Alves, G. E., D. F. Boucher, and R. L. Pigford: Chem. Eng. Progr.. 48: 385 (1952). Lockhart, R. W., and R. C. Martinelli: Chem. Eng. Progr., 46: 39 (1949). Martinelli, R. C, and D. B. Nelson: Trans. ASME, 70: 695 (1948). Stein, R. P., et al.: Chem. Eng. Progr. Symposium Ser., 50: 11 (1954). McAdams, W. H.: "Heat Transmission," 3d ed., McGraw-Hill Book Company, Inc., New York, 1954. Reynolds, J.: ANL-5178, 1954. Jens, W. H., and P. A. Lottes: ANL-4627, 1951. 1953. Isbin, H. S.: Personal communication, Lapple, C. E., et al.: "Fluid and Particle Mechanics," University of Delaware Press, Newark, Del., 1951. Hughes, R. R.. and E. R. Gilliland: Chem. Eng. Progr., 48: 497 (1952). Coppage, J. E.: Ph.D. thesis in mechanical engineering, Stanford University, Stanford, Calif., 1952. Weintraub, M., and M. Leva: Ind. Eng. Chem., 45: 76-7 (1953).

9-3

HEAT REMOVAL

FROM NUCLEAR REACTORS BY

Charles F. Bonilla

PRINCIPAL NOMENCLATURE A C c, c. D D'

surface area heat; also concentration electrical capacity — specific heat at constant pressure
D.

_


e = specific

-

.

..

. equivalent diameter =

, 4

cross section :——— wetted perimeter

diffusivity electrical potential F' - mass-transfer coefficient = Fanning friction factor mass velocity 0 = gravitational acceleration 9 0' = ratio between units of force and of mass X acceleration H = rate of heat generation per unit volume. Hi = rate at inner face h = local film heat-transfer coefficient electrical current K o = zero-order Bessel functions = radiation intensity; also mechanical equivalent of heat k — thermal conductivity L length; also dimension in general diffusion length L M molecular weight

Dv

E

= =

h-

Io,

J

hi o,

J

-

m

N P P

mass =

number of atoms per unit volume

= pressure = pump shaft

power per cent gas by volume; partial pressure quantity of heat rate of heat flow q q/A •m heat flux. (q/A)c = critical heat flux R - fraction by volume; also thickness; also resistance in general fraction by volume T — radius T, = electrical resistivity S = cross-sectional area < = surface tension T = absolute temperature 9-50 V Q

=

; Ry =

vapor

heat removal from nuclear reactors

Sec. 9-3] t =

u = =

V

w x = x' = y = z, Z =

thermometric temperature. I, t/, /„ = bulk, film, wall temperatures, respectively ul, u, = liquid, slip velocity, respectively velocity, volume; specific volume mass rate of flow fraction by weight; also vapor quality; also distance along a channel fraction by volume distance transverse to a flow, or away from a channel wall height above a horizontal datum

a = thermal diffusivity f)

r

i

= = = = = = =

9-51

= — cp

volumetric coefficient of thermal expansion mass flow rate per unit of width

eddy diffusivity time X latent heat of vaporization molecular viscosity, -y-ray attenuation coefficient It kinematic viscosity v — density. p pf, pL, pv = film, liquid, vapor densities, respectively a = microscopic cross section ; also standard deviation = wetting angle

(see

Table

3

Nu, Pe, Pr, Ra, Re, Sc, Sh, St = dimensionless ratios

of Sec. 9-2 or context)

a

is

is

Removal of the generated heat an absolute requirement for continuous operation of a nuclear reactor, since otherwise the temperature will rise indefinitely until failure of the fuel elements or container and dissemination of the fuel and the fission prod ucts occur. In any commercially valuable power reactor the rate of heat generation per unit volume and of heat removal per unit surface area will be high in order to optimize the fixed charges, i.e., minimize the total cost. Temperatures will tend to be high to favor efficient conversion to electrical power. Thus, thermal design of fuel elements and coolant systems number of research reactors very exacting. For For power the cost of the coolant system has been about 18 per cent of the total cost. reactors some 25 per cent may be closer to the optimum allocation of funds.*'1 1

THE SPATIAL DISTRIBUTION OF HEAT GENERATION IN REACTORS

The distribution of the total energy generated Table The prompt energy includes 168

is

5 ± ± 5

y

is y

y

0

it

is

is

is

is

is

a

is,

is

1.

it

it

/3

8

1

±

1.

shown in by an average fission Mev of fission-fragment kinetic energy, expended within 0.001 in. in metals, 0.5 Mev of neutron kinetic energy, and 4.6 Mev of rays. Radioactive decay of fission products and of unstable nuclei produced by reactions of stable nuclei with neutrons yields about Mev of short-range rays and 12.5 Mev of rays. The prompt fission-fragment and energy appears in the fuel. The energy dissipated roughly according to the mass through which passing, so that in a small reactor most of appears in the shield. How ever, in a large reactor may distribute itself roughly equally among the shield, fuel, and moderator. Typical distributions are given in Table If the heat in the reflector and/or thermal shield not gainfully recovered, the percentage of the useful power in the fuel of course, greater. The prompt energy more significant in power surges, and higher fraction This difference usually dissipated in the fuel. neglected, however. Detailed calculations of the fissioning (generally, thermal) neutron flux nv through out the reactor are described in Sec. 6-2. If this information available, the heating density H/ per unit volume of the fuel due to fission obtained for any location by is


Gr, Gz,

j,

(f>

8

* Superscript

numbers

refer to References


FLUID AND HEAT FLOW

9-52 H,

= Q'nv

£
[Sec. 9

X WQ'nvp

= 6.02

£

^

(1)

Q' = a; = N/ = p = X/ = Mi =

heat generated within the fuel per fission, any heat unit fission cross section of fuel, cm8 number of atoms of fuel per cubic centimeter of fuel element, cm-3 density of fuel element, g/cms weight fraction of fuel in fuel element atomic weight of fissionable nuclide or nuclides (i.e., Um and Pu"9), g A simpler method of computing the distribution of y heat rate per unit volume H is frequently employed when the fuel is uniformly distributed, as in a homogeneous The fraction of the total reactor heat reactor or in a heterogeneous uniform lattice. qt appearing in the moderator and in the fuel elements is first determined or estimated Then q, is accordingly divided, and the over-all average values H (e.g., by Table 1). per unit volume of fuel and of moderator are computed by dividing by the correspond ing volumes. The value of H in any location is then obtained by multiplying the corresponding H by the appropriate ratio of neutron flux to over-all average neutron flux (see Sec. 6-2). where

Table

1.

Approximate Distribution of Steady-state Heat Dissipation in Representative Reactor Types Approx


core size, ft

Large

thermal

Fuel

Reflector

Moderator

element

Thermal shield

reactor

(natural U): Total Mev per fiasion. . . % of total % of heat generated in fuel and moderator. Small thermal reactor

..

(U«"): Total Mev per fission. . . % of heat generated in fuel and moderator. . . Specific reactors: Fast breeder (EBR-I) . . . Thermal water-cooled. water- and berylliummoderated (MTR)

Intermediate, sodiumcooled, beryllium-

1 3

175 90

12 6

93

7

168 86

7 4

96

4

72% (core)

94.3% (including water)

4

2 1

S 3 Unrecovered

2 1

17 9 Unrecovered

28%. (blankets)

3.9% (Be)

1.6%

Unrecovered 0. 16% (steel)

(graphite)

95 % (including 4 % Be)

1 %. (steel) (biological

moderated (SIR) Thermal, graphitemoderated, air-cooled,

natural L'(BNL)

Sodium-cooled,

U(SRE)

Sodium-cooled,

U(SGR)

25

92%

6%

>1%

0. 1 %)

<1S

enriched 6

91.5%

7%

1.5%

7

92.3%

7%

0.7%

3

91%

5.5%

3. 5%

enriched

Thermal, diphenylmoderated and -cooled

(OMRK)

2

2.1

HEAT CONDUCTION Heat-conduction

Relations

Fourier's law gives the time rate of heat conduction dQ/d0 or A as q = dQ/de = -kA dt/dx

q

through a plane area (2)


Sec. 9-3]

where t is temperature and distance x is measured normally of the heat flow. Equation (2) also serves as the defining conductivity k. Usually k is a known function of /, and for many cases known or experimentally measured functions of x. Thus, steady-state heat conduction between isothermal surfaces

9-53

to A and in the direction equation for the thermal

A and q are theoretically to integrate Eq. (2) for at x and Xi, it is usually

rewritten

Since k may usually be assumed constant or at most a linear function of (, Eq. (3) If q is not constant equals k(t — U), where k is the arithmetic mean of k at I and I,. along the heat-flow path under consideration, for steady-state heat flow it can be

obtained by integration of

H:


9

=

?° +

jo

"A dl

(4)

then substituted into Eq. (3). Results of this ?o is any heat entering at x — 0. double integration are given in Arts. 2.3 to 2.5 for the one-dimensional geometries and the simple distributions of H. Including motion of the conduction field (i.e., a flowing fluid), unsteady-state heat conduction, and two- and three-dimensional shapes, the appropriate differential equation is — - = a V«i H 06

where

The

c is

cp

I

\

u,

Bx

+ u, — + u, — dy

j

heat, it is the velocity, and the thermal diffusivity Vs is defined in Art. 4.4 of Sec. 3-2.

the specific

Iâplacian

(5)

dz/

S.2

a is k/cp.

Thermal Conductivity

Gases and Liquids. Thermal conductivity for most reactor fluids is tabu The NBS-NACA Thermal Tables' also provide the thermal con heat, enthalpy, and many other properties of gases over fairly wide For interpolation of k or extrapolation to higher temperature and pressure ranges. temperatures, the Sutherland equation [Eqs. 2 to 10 of Sec. 9-2] is satisfactory; 2.21

lated in Sec. 9-1. ductivity, specific

■\/~T

/k plots reasonably straight against

l/T.

For monatomic

gases k can also be

+ 0.016 \/M), and for the common linear molecules except Ht as m(1-45cp — 0.13c,). The thermal conductivity for a mixture of gases can be higher or lower than the molar interpolation from the pure components but can be predicted4 within several per cent. The thermal conductivity of gases increases exponentially with the ratio P/T. For coolants and conditions so far considered, however, the effect is not significant. CO2 shows the greatest increase, 33-i per cent at 10 atm and 20O"F. The thermal conductivity of nonmetallic liquids is not readily predicted or extrapo lated theoretically, and empirical values are desirable (see Sec. 9-1). 2.22 Metals. The thermal conductivity of solid materials used in reactors is given in Sec. 10-2. Among solid pure metals k varies twentyfold, and empirical values should be used when available. It usually decreases slightly as temperature rises, then on melting falls abruptly to one-third the solid value. Cold-worked metal has k somewhat lower than annealed, and cast metal even lower. Impurities in solid metal, including fission products, decrease k considerably. Alloys generally have a conductivity of 50 to 100 per cent of the linearly interpolated value. Practically all the heat conducted in metals is transported by the free electrons. The Lorenz relation between thermal and electrical conduction computed3 as

juc»(2.4

FLUID AND HEAT FLOW

9-54

[Sec.

9

watts/(cm)(°C)

k

is useful for predicting k if the electrical resistivity r, is known or can be measured. In Eq. (6) T is in degrees Kelvin and r, in ohm-centimeters. The constant is 23 ± 1

for most solid and liquid metals and ranges from two-thirds to twice this for alloys.4 2.23 The thermal conductivity km of two-phase mixtures Two-phase Mixtures. in which the discontinuous-phase particles seldom touch each other or have their conductivity kd lower than kc of the continuous phase can be approximately predicted" by Maxwell's formula 2xj(ke 2ke + kd kd) km —

-

2kc

+

-

+ Xd(kc

kd

—

^

kd)

volume fraction of discontinuous phase. beds of solid particles of conductivity k, permeated with a stagnant fluid of lower conductivity kf, km can be conservatively estimated7 within some 20 per cent If the fluid is flowing through the bed, the effective conductivity normal by Table 2. where

Xd is the

For packed

\/

to the flow is given8 by km = 3.7(/ifp/x/)(G a/£)-4, where G is the over-all or super ficial mass velocity, a is the surface area per particle, and £ is the average viscosity.


Table 2.

Thermal-conductivity

Ratio km/kf for Packed Beds Filled with Stagnant Fluid

a

Fractional void volume kmIki

o 0.2

1 3 10 30 100 300 1.000

1 3 10 30 100 300 1.000

I 2.4 5.4 10.7 22 42 83

0.3

0.4

1 2. 1 4 3

1 1.9 3. 4 5. 4 8.3

7.5 13 20 32

II

15

0.5

0.6

1

0.8

1 1.5

17

2.8 4.0 5. 5 7.0

8.5

2.2 3.0 3.7

1 1.2 1.5 1.7 1.9

4. 5 5.3

2.0 2.2

1

j | | |

Steady-state Conduction in an Infinite Slab

2.3

In nuclear reactors, the surface temperature of the fuel elements, moderator, and shielding is controlled by the coolant. Thus, internal temperatures are computed For Eqs. (8) and (9) one surface is at x = 0, with respect to surface temperature. and the cooled surface at xi is at temperature d. Constant Heat Flow. For constant heat flow q = constant and H = 0, as 2.31 that in fuel-element cladding generates relatively little heat:

k

kA

A

For ?o = 0 and constant heating density Uniform Heating Density. 2.32 = constant ), as in homogeneous fuel elements, slabs of moderator, cladding, shields, etc., with appreciable heat generation but thin enough so H is uniform: (H

,

_

«,

- xjL^ H 2k

u-U=X-^2ic

= X-^

2kA

If there are simultaneously qa entering at x = 0 and heating density the total t — /i or to —
(9.

H from 0 to x..


Sec. 9-3]

9-55

Heating. In a bare reflector or shield of inner surface the reactor midplane, for cooling at the inner surface:

2.33 Thermal-neutron at xt and outer surface at

x% from

1 A '

=

H 'L

coah

sinh

5

-

[(is/i)

.

(x2/L)]

sinh l(x,/L)

I

k

- (x/L)] - I

M*'/L)

-

(x2/L)]f

where H, is the inner surface (maximum) heating density and diffusion length. Similarly, for cooling at the outer surface:

L is

the average neutron

--ir[
is available instead of

Hi, the latter

by

_

- (*,/£)] (xt/L)] sinh [(*,/£) - (xt/L)] [(x,/I)

-

cosh

[(x,/L)

can be eliminated from the above expressions 1

(13)

2.34 Gamma-ray Heating. As in a reflector, blanket, or thermal shield, the thickness Ar of separately cooled thermal shields and other special -y-ray absorbers is small compared with r. In this case Hy can be represented approximately by Hy =• be'*', where n is a linear energy attenuation coefficient. Hi and Hi are calcu lated as described in Sec. 7-3, and

ri r2, the

r3, the

Ml Hi

H

~

=

n(xi

inner surface h

For cooling at

In

_

u

_

=

g'(r'

—

-

(14)

i2)

r>)

(15)

outer surface:

^^)_^^

= (16)
k~H

For cooling at

— r2

(16)

fc/i2

is

2.36 Actual Reflectors and Shields. These usually have heat generation by several simultaneous methods, and the heat removed to various extents and at different temperatures through the two faces. Assuming that the coolant temperature known on both sides at the desired level, the distribution of the heat generated between the two sides assumed, the two surface temperatures are computed from estimated heat-transfer coefficients, and the intermediate plane located at which the the same when computed from either side with the known distribution temperature of H. Then the assumed distribution on both sides of the heat dissipation must be checked, and the calculation repeated with the new distribution agreement poor. The dissipation on each side can also be calculated as the temperatures on both sides were equal, then adding to the cold side and subtracting from the hot side the heat that would be conducted through the plate as a result of the given surface temperatures alone. In best design, however, all coolant streams are usually so apportioned that their about the same, and the simpler case of equal wall temperatures temperature rise In this case, even with ratio H%/Hs as high as 10, the hottest plane adequate. located within 10 per cent of the midplane, at least one-third of the heat leaves is

is is

a

is

if

if

is

is

is


H Hi

(12)

FLUID AND HEAT FLOW

9-56

[Sec.

through the outer surface, and the maximum temperature rise is within several of the value it would be with uniform H3i:

9

per

cent

8k

Thermal shields and other slabs with high heat generation are generally subdivided into thinner parallel layers with intermediate coolant streams as necessary to keep („„, or the thermal stress below their permissible maximum values. The layers are made thicker the farther away they are from the axis of the reactor. Steady-state Conduction in an Infinite Cylinder

2.4

2rlcL

r*

- 0.367

log k~L

r-i

In

5

—

=

|

tl\

-

(18)

n

_ fffi* /,

_l r'J

r'4

4.

+

Fuel

In

Element.

\

,

,

a

is

the rate of heat flow per unit length. where q/L 2.42 Fission Heating in Homogeneous Cylindrical elements, with Io(r/L) neutron flux distribution:

such

(19)

-

r2)

-

= — r,2 = 0.0795

t,

= — (r,2 4£

to

_

t,

I

:

a

Solid Cylinder. This holds for the case of 2.43 Constant Heating Density in homogeneous cylindrical fuel elements of radius less than about half of the neutron diffusion length

4f

In

(20)

kL

a

q.

L

h

f0

are both the frequent case that the axial temperature and surface temperature of fuel elements can be directly computed from specified, the total required length the power 2.44 With external cooling Constant Heating Density in Hollow Cylinder.

_

(r,/r,)>(l

2

r,i[l

+

at rs: In r,/r,)]

(21)

/

peratures yields the maximum temperature

rm

r,'

at

-

r,»

(22;

In (r,/r,)

it

3

2

or by m. obtained from Eq. (21) by replacing subscript Though cooling at the outer surface Homogeneous Cylindrical Reactor. might be useful for removing entirely impractical for power production,

2.46 alone

is

tm

\2

tm

3.

2

and With internal cooling at rs Eq. (21) holds on interchanging the subscripts With cooling to the same temperature at both rt and r3, equating the internal tem

is


|/,

is,

Cylindrical fuel elements are useful because they are easy to fabricate and replace and because tubes are desirable as fuel-element jackets and coolant channels. The cylindrical shape is also desirable for reactors in which through-flow cooling channels are necessary, as is the usual case. 2.41 Constant Heat Flow Through a Hollow Cylinder. Here q = constant and H = 0, as in cylindrical fuel-element cladding. The temperature drop in either direc tion of heat flow between radii r« and r3 by Eq. (3),

Sec. 9-3]


9-57

fission-product heat in small reactors in the event of coolant-system failure, particu £ must be an appropriate average larly on complete loss of the coolant in the core. for radial conduction through the core. Based on a Jo radial distribution of H:

where r'i = 2A04Srl/R„ R, being the extrapolated radius to zero neutron flux. Steady-state Conduction in a Sphere

2.6

Although spherical fuel elements are not normally practical in power reactors for mechanical reasons, their resonance escape probability can be higher, and suitable This case also approaches that of fuel suspensions. designs may evolve. 2.51 Constant Heat Flow Through a Hollow Sphere. This, as for fission heat passing through cladding on a spherical fuel element, is equal to

- U - X (1-1) ^, r*f

tl

(24)

4dE


2.62 Constant Heating Density in a Solid Sphere. As in homogeneous spherical fuel elements of radius less than about three-fourths of the neutron diffusion length, this equals

t-tl-*L6£ (r,« -

r„

r«)

- i, -

—r r,« 6Jt

(25)

For an externally cooled spherical shell

-U

U

2.63

=

—

(2

^

+ r,»

- 3^

(26)

Fission Heating in a Homogeneous Spherical Fuel Element. neutron flux distribution, the result is

r/L)/(r/L)

t

=

Hif'1

2.64 Homogeneous Spherical distribution, the result is 0

, =

ri*

4-

Reactor.

nr L3! 3

3.1

(1

-1-

With

" 5!/?t»

+

r«4

4- • •

(sin

7m?

A

(27)

vr/Rt)/{rr/Re)

—

With (sinh

1

neutron flux

(

CONVECTION

Heat-transfer Coefficients

Heat generated in nuclear reactors is usually removed by fluids flowing past solid In general, the heat gained or given up by a fluid determines or is determined walls. by its average, bulk, or "mixing-cup" temperature I. The temperature <„of the solid This tempera wall must exceed < in order to transfer heat to the fluid, or vice versa. ture difference is designated At. The ratio between the heat velocity q/A and At at any point is known as the local film heat-transfer coefficient h. Thus:

4 A

- \k(t. - D| - h At

(29)

FLUID AND HEAT FLOW

9-58

[Sec.

9

A principal phase of thermal design is the prediction of the magnitude of h and thus At for the desired coolant-flow conditions. For pure streamline flow this can be done mathematically by solving simultaneously the appropriate heat and fluid-flow rela For turbulent flow, certain tions, particularly Eq. (1) of Sec. 9-2 and Eq. (5), etc. cases have been solved semitheoretically by also using Eqs. (31) through (36) of Sec. 9-2. The bulk of available heat-transfer-coefneient information, however, consists of generalized dimensionless correlations of actual test values of h for standard simple shapes and flow directions in terms of the significant dimensions, physical properties, and flows. According to the Buckingham "pi theorem" the number of dimensionless ratios needed is the least number of necessary variables (including h) minus the number of independent physical dimensions. The principal dimensionless ratios in heat transfer are listed in Table 3 of Sec. 9-2. In most cases the variation of a physical property throughout At has a negligible The property is then generally taken at the arithmetic mean to moderate effect on h. or "film" temperature

3.2

Fuel elements are generally thin compared with their length, so that heat leaves the Thus the distribu fuel element at or very near the section at which it was generated. tion of superficial g/A for Eq. (29) is closely obtainable from Art. 1. The information also yields the local average coolant temperature I (see Art. 6.1). 3.21 Turbulent Flow. Figure 1 shows graphically the principal cases for metals. Table 3 gives in algebraic form the pertinent correlations for nonmetals. From Table 3, case b, it is seen that there is an appreciable decrease in h for a short distance downstream from the entrance to a channel. Since this "thermal entry length" is short, it is neglected in most experimental studies and correlations, which

NM

Post short strip or p.nsfins

/«l/MrffKcM//0,/s

Re'PGmo,/^,

JKP

No mo!to bondsof staggeredtubes

j'-WcGmaJ[Cp/*)*n

ELF

Parollelto o plane

/-(/i/cGJlc^/*),,*'3

Re'DGmoiZ/v R**LG*/pa

GH

Normalto 0 simplecylinder

/.(/>/rGKr>/A),z/s

ABC

Turbulentflowtns.detubes Flowtnroughbedsof conye*solids

/H/>/cC){Cfx,/A)tn

ST

Fig. 1. Mean heat-transfer C. F. Bonilla, Re/. 98.)

/•(Ayfc6")(c/*AV,/S

coefficients for nonmetallic

Ret,

Re'DC.V, Re'DG/M,

fluids in forced convection.

{Prom


Sec. 9-3]

Turbulent-flow Heat Transfer for Nonmetallic Fluids

Table 3. Desig

Limitations, etc.

Case

nation

Flow in long

a

Nu

-

Nut

-

2.000 < Re < 300.000

Local h in short channels: down stream distance Local h in short channels

e

60.000 < Re

Flow in coils Flow in wide con centric annuli Flow in wide con centric annuli Liquid in tubes with p' % of gas by volume

d e

I 9

K

Nu Nu

-

■

Gases at high u

Case a holds with A<

i

Flow parallel to

Axial spacing /diameter

tube bundles Flow parallel to tube bundles

Table Desig nation

4.

< 1.3 Axial spacing /diameter = 1.46

m

Flow Flow Flow Flow Flow Local Flow

/

Flow in a flat channel

g

Flow in a flat channel

h

Flow in a concentric

long tube long tube long tube a short tube

in a flat channel

annulus

i

Flow in a concentric

i

annulus Flow in a concentric annulus Flow along an unbaffled bundle

*

thus

in a long tube

in a long tube in a in a in a A in

-

0.87

Nu

-

(1 + 6. \0 >

Nun

-

0. 236f />«(*/*)..<]•-"

1.

9 11

VOi/DtNu"

" logi.

10

12 p') Nu»

13

14

-iNu

-

2gJe,

Nu" (using D.)

15

Nu

-

1. 4 Nu» (using D.)

16

«'

PrV4

9

Turbulent-flow Heat Transfer for Molten Metals

Cose

a a1 b b' e d

Nu

surface

Upflow with 25 < O < 100 lb/(ft')(sec)

9

Nu»

[r^(^)W']-

Nu» (using D.)

surface

0.41. + 0.6i

-

Pr^

-

Heat transfer at inner

-

Re"

0.023

ence

- 1. 5Nu« at L/D. - 1 - I.I Nu'at L/D. - 5 - (1 + 3.5Z>/Bcoil) Nu»

Nut NUi

Heat transfer at outer

Gases in long tubes at high A<

k


2.000 < Re < 60.000

L

Refer

Relation

channels b

9-59

Limitations, etc.

Constant q/A Constant q/A Constant Constant 50 < Pe < 20.000 Constant (»

-

One wall q/A 0. other q/A constant Both walls with q/A constant Each wall with differ ent constant q/A D./D, < 1.4

Nu Nu Nu Nu Nu

-

Nut Nu Nu

-

5.8 + 0.02

17

10. 5 + 0.036

(using

Pe"

D.)

(using D.)

17

Relation e. /, or g, as applicable

Constant q/A at Di

Nui

D — tube or rod outer

NuD

-

h.

"'

Pe"

17

-

give, in effect, the "long tube"

-

17 21 17 21 18 19

Nut

diam

ence

7 + 0.025 Pe" Nu« 6 . 3 + 0 .016 Pe° ••«Pr° •• 4. 8 + 0.025 Pe" 4.8 + 0.015 Pe"' Pr° > 0 625 Pe" 1.31(Pe D /£,)»

> 1. 4

D,/Di

Refer

Relation

0.

75(Di/Di)" Nu"

(5.25 + 0.0l75Pe)(O./Di)"«

ll.6(PeD)"(D./t)'

>

20 20

Since the downstream h is lower and also the

For I is higher, long-tube h values are correct for calculating fuel-element hot spots. a more accurate profile of the temperature along a fuel element, however, local h

values

are used.

Figure

2 shows the principal data for liquid metals, recalculated on a consistent The large discrepancies seem to be due mainly to errors in temperature readings or to the presence of entrained gas from incorrectly designed gas-pressurizing

basis."

FLUID AND HEAT FLOW

9-60

[Sec.

s

9

'try*

R a. Id CD

2 3

?'

CO to

£^

ss

/


■

i

/ / /

/ / /

/

i

/

/

<

i i i

i

i

—1

1 1 1

PECLET NUMBER Pe

Fio.

2.

Solid lines A to F, theoretical U, unwctted; If, wetted. (From

Heating of liquid metals in turbulent pipe flow.

relations; dashed lines G to T, experimental C. F. BonUla, Ref. 98.)

results.

A. Martinelli (Pr = 10-J) B. Martinelli (Pr = 10~3) C. Lyon (Table 4a) D. Deissler E. Kennison F. Colburn (Pr = 0.027) G. Isakoff and Drew (xld H. Isakoff and Drew (x/d /. Untermeyer (W) J. Johnson et al. (1951)

K. Trefethan (W and U) L. Doody and Younger (W) If. Doody and Younger (U) A'. English and Barrett (U) 0. Johnson et al. (1950) P. Lubarsky (U) Q. Untermeyer (U) R. Werner, King, and Tidbnll (W) S. Keen T. Lubarsky and Kaufman (Table

= 58) = 138)

(U) (U)

4c)

Gas impairs heat transfer more if the liquid metal does not liquid expansion tanks.*7 wet the walls, as has been shown for flow of mercury through a tube bank." The principal relations that have been obtained are listed in Table 4. The constant term represents the strong conduction contribution, and the Pe term the turbulent convection contribution. The lowest h computed among all appropriate relations, further decreased by a moderate safety factor, should be used for conservative designs. 3.22 Laminar Flow. In laminar flow h is smaller than in turbulent flow under conditions that are otherwise the same, since radial heat convection by eddies is absent. However, laminar flow may be entirely adequate for full-power operation when Al is low in any case, such as with low heat velocities, liquid-metal coolanis, and/or small channels. In addition, laminar flow may be employed at low circulation rates or at low-power operation or obtained unintentionally during coast-down and during steady-state natural circulation following pump failure. The principal laminar-flow relations for "short" channels are listed in Table 5. In laminar flow the effect of the thermal entry length is greater than in turbulent flow (i.e., see case /). However, even in laminar flow the long-channel relations are suit able for hot-spot calculations, since they are almost correct, simpler, and somewhat


Sec. 9-3]

9-61

conservative. Niu.if (cases d and e) employs Ham = q/[A(&l)A3i\, where Mam is the arithmetic mean of the inlet and local At. Tables 6 and 7 give long-channel values of local h for the principal channel shapes. The Aim., values are useful in computing from the usual Atn„wm>an [At in Eq. (29)] the maximum temperature variation in the stream, for thermal shock estimating, etc. The At,,, values yield the volumetric average temperature of the coolant, useful for calculating the mass of coolant in the tube, its buoyancy, etc.

Table

6.

Li nutations

Case

Local A in a short tube — parabolic flow

Local A in a short flat duct — parabolic flow

Local A in a tube — plug flow

Heat Transfer in "Short" Tubes

Laminar-flow

Reference

Relations

Constant tte',Gz cOD'/kL > 12

Nu

-

I.ISGiM

23

Constant (»; Gz' cGD.'/kL > 70

Nu

-

l.3(Gi')M

23

Nu Nu

-

1.3 + 3.7 ±

24

-

-

-

Constant (»: Gz ™ I, 10 Gz

Average A in a tube — Constant^; actual flow

Re > 10 Nu.

9

..76G.*(ei)°"


2.25(1 + 0.01 Gr!4) log Re Average A in a flat duct — actual flow Local A in a tube — parabolic flow

Constant ir; Re > 10 D = D, Constant q/A

15 % above relation

d at same

Gz Nu 4.36 5 6 8 10 14 19

Gz 0-10 34.4 73.7 191 378 632 2000

Because of viscosity change with temperature, h for a liquid being heated or a gas The same being cooled will fall between the parabolic and uniform velocity cases. method applies, in either cooling or heating, for estimating h for slurries that have Bingham body or pscudoplastic properties.

3.3

Forced Convection Normal to Cylinders, Tube Banks, Screens, and Beds

The principal correlating equations for these cases are given in Table 8. In all these cases the local h or Nu varies considerably around the perimeter of the body. The values correlated herein are all experimental surface-average results. Theoreti cally, average coefficients are slightly higher at constant q/A than at constant t„. Most experimental results and most nuclear applications involve small, solid elements and are close to constant t„. Table 8 thus is reasonably accurate or else slightly con servative under all practical conditions and may be safely employed for design pur poses regardless of the precise thermal conditions. 3.4

Free

Free Convection

convection is the circulation by gravity of a fluid having density variations due

to thermal expansion caused by heat transfer. If the circulation is inside an open channel that is at least several diameters long or a closed loop, it is internal convection, or natural circulation. In this case the rate of circulation can be predicted approxi-

Table 6.

Local Nusselt Number and Temperature Differentials Flow in Long Channels

in Streamline

(Axial conduction neglected)

on flow mean tem perature"

Velocity

heating

conditions

q/A

q/A f»

u

q/A

Flat channel (infinite width)

q/A


Nu baaed

Constant

Channel shape

-

4.36' 8 3 66 5.80

2 aides

5.384 8.240

1.636 2.013

A'apacr men Alflow

0.727 1.000 1.000

1.346 1 288

0.942 0.824

1. 500 1 500

1.000

1.571 1.571 2 00'

1 000" 1.000"

Uniform: Heated Heated

Parabolic:

1.

Uniform:

Heated Heated Heated Uniform^

1 side 2 sides

6 12

2 sides

7.60

1 aide 2 sides

4.94" 9. 88< 6<

Parabolic

3.68'

q/A q/A

Uniform Uniform Uniform

422>

q/A q/A q/A q/A

Parabolic Parabolic Parabolic Parabolic

4. I6«

q/A

-

Heated

u

q/A q/A

Triangular:' 60° X 60° X 60° 45° X 45° X 90° 30° X 60° X 90° Rectangular: 2 o/6 3 a/b 5 alb 10 alb

Parabolic Uniform Parabolic Uniform Parabolic : Heated 1 aide

A/ max A'Uow menu

I.00O

3.00' 4. 10'

4.78'

5 55' 6. 77-

Note: The length in Nu is Dt for all channels. "R. H. Norris and D. D. Streid, Trans. ASMS.

62: 525 (1940). b For gases with large M take k at 1.271 — 0.27ft*. Similarly, for liquid metals multiply 4.36 by See Ref. 25. (m»/m«)9 Ue To include axial conduction divide by { I + [I + («*/&*)*] ! /2. d "Liquid Metals Handbook" (2d ed., R. N. Lyon, editor; 3d ed., C. B. Jackson, editor), Govern D.C., 1956. ment Printing Office, Washington, • Based on mean wall temperature f Conservative, since circulating currents are neglected.

'[(«») m«

- t)HU -

Table 7.

t).

Local Nusselt Number for Streamline Flow in Long Concentric Cylindrical Annuli Heated at Constant q/A on One Side* (Nu based on £>,; D, =* Di Dt)

-

"Parabolic" £>insuUted

54 60 7.389 2.718 1.649 1 284

It

0. 7788 0.6065 0.3679 0. 1353 01

»L. M. Trcfethen, NP-1788. t Flat duct.

Uniform velocity. Nu

32.96 9.71 6.748 6. 168 6.040 6 6 036 6 132 6 442 7. 144 8

Nu

34 78 10. 18 6. 758 5.898 5.606 5 384 5. 222 5. 104 4. 962 4 854 4.364

1950.

t Round pipe.

9-62

velocity (/i constant)

Mm Aiflow mcaD

1 1 1. 16 1.23 1. 29 1.31 1.35 1.37 1.40 1 46 1. 55 1.64

Sec. 9-3]


9-63

mately by analysis (see Art. 6.21) and h obtained by the usual forced-convection correlations (Tables 3 through 7). The product Gr X Pr = L3p2g0 At c/nk, which applies to laminar free convection, is As in forced convection, the physical proper abbreviated Ra, the Rayleigh number. ties should be evaluated at 1/ for either heating or cooling. The thermal and velocity boundary layers are rather thin (2.7 Z/Nu for case e, Table 9), so that free-convection h values are generally the same for the unsteady-state heating or cooling of a fluid and for the steady-state process with another sink or source of heat, respectively. Table 8.

Surface Average Nusselt Number in Forced Convection Normal to Tubes, Screens, and Beds ( 'MP

Nonmetals normal to a circular or streamline section cylinder daws normal to a cylinder Metals normal to a cylinder

Refer ence

Limitations

Nu = (0.35 + 0.47Rc° ")Pr» « 1000< Re < 50.000 Nu = 0.26 Re" «Pr» ■« High Af; Re" = Mpi/ii/ > 100 Nu = 0.46(Re") + 0.00128Re' "Turbulent" Nu = 0.80 Pe>'2 0.1 < Re < 1000


Nonmetals through a staggeredbank of tubes: 5th and later banks

500 < Re < 75.000

5th and later banks 1st. 2d, 3d, and 4th banks

75.000< Re

Nonmetals through tubes in line: 1st and 2d banks 3d and 4th banka 5th and later banks Mercury ( 100°F) through a staggered bank of >£-in. tubes He in. apart (all but 1st,3d, and last banks)

10<< Rem., < 10' nonwetting nonwetting

NaK through a staggeredtube bank

Nu = 0.36(Re) max"'" Pr" Nu = 0.38(Re)m.*° ••PrM 60. 75.94. and 97 % of r, above 60 % of e, above 70% of e, above 80% of e, above K

- 11.6 Re"."

i - 3.45 Re» " Nu = 4 + 0.23 Pe„ Nu

-

34

l,17(Pen,»,)« >Pr:

Any liquid on the shell side of a heat D = tube OD, iy. = fl. between NuD =0.19(D",)»«ReD»'Pr tubes, in. exchangerwith segmentalbaffles pOmaa ■< 20,000 flowing 200 past norma) pins Nip = 1.0 Rer>4PrW Nonmetal < Re„ c parallel strips or small where p = perimeter Heat transfer between packed bed 1.96 ReD» "PrW ReD<350 NuD and nonmetal Sowing through D = diameter of sphere of same Re„>350 Nu„- 1.06 Re°" Pr*4 surface

-

-

Heat transfer between cylindrical wall of packed bed of diameter Dw and nonmetal flowing through

Nu„

■0.813 e"V>- Re„

26 31 32,37 33. 34

35 33 36

If

there are no other walls near the heat-transfer surface or surfaces, "external" or free circulation is obtained. Table 9 and Fig. 3 give available empirical correlations. These have generally been determined for uniform <«, or for cases intermediate between uniform tw and uniform q/A. The former case gives somewhat lower h values. Thus Table 9 is reasonably accurate for cases of constant and slightly conservative for constant q/A. More detailed summaries are available" of

"local"

natural convection under various conditions, including with heat generation within the fluid.

When conditions are such that free convection and forced convection exist simul If taneously, the values of h expected by each separate mode should be calculated. one is considerably larger, it controls the heat transfer. Approximate rules for inter mediate cases are given in Table 10; for rough estimates they can be extended to other geometries.


9-64

FLUID AND HEAT FLOW

[SEC. 9

LOG (Gr X Pr)

Fia.

3.

liquids. Table

Heat transfer by free gravity convection (From C. F. Bonilla, Ref. 98.) 9.

between solid surfaces and nonmetallic

Local Free- or Gravity-convection Heat Transfer without Fluid-phase Heat Generation Limitations

( 'as.-

Nonmetals at a vertical wall or large tube (L = height)

I0 »
= 1.12Ra» >'

10"< Ra < 10" I0« < Ra < 3 X I0'«

■0.555 RaW . 0.73 RaM

3 X I0>°< Ra < 10"

• 0.13 Ra-4

Any fluid at a vertical wall

Ra < 10"

Nonmetals at a vertical wire (/. °=diameter, Z = height)

10"

Nonmetals with heat flow between enclosed parallel flat vertical walla{I. = spar ing, H = height, 3 < H/L <42)

Relation

<

Ra|<3X

. 0.667[PrT.r/(Pr

I0

-,. ,51c, »i L

[.+ 4.47 (luf)*"]

Ra < I0> 10" < Ra < I0>

Nu ■

10' < Ra < 10'

Nu • 0.072

0.20H.M(!)«

Ra';'(i)H

Any fluid at a horizontal Ra < 5.4O07.ft cylinder or bankof cylinders {L = cylinder diameter; add twice the mean free /) > 0.01 ft path if appreciable)

J_

Nonmetals itetween concen tric horiiontal or vertical cylinders {/, = Dt — /Ji)

Ra < 10" 5 X 10" < Ra < 10' 10' < Ra < 10"

I Nu Nu 0.060 Ra» •» Nu = 0.26 Ra°'°

Nonmetals at a horizontal Btrip facing upward (/. «=

10' < Ra < 10"

Nu

10" < Ra

Nu

width)

+ 0.952)]H

Nu' No ■ 0.53

-

5 5

»:

*"*

(VPr+ 0.952/)H

- 0.71 Ra^ - 0.162 RaH

42 5 5 5

Sec. 9-3]

heat removal from nuclear reactors Table 10.

Desig nation

» e 4

l) k

Combined Free and Forced Convention

Cue

Rcfer-

Relation

Limitations

NonmetalB flowing normal to zontal cylinder (turbulent) Nonmetals flowing normal to tontal cylinder (turbulent) Nonmetals flowing normal to zontal cylinder (turbulent) Nonmetals flowing normal to lontal cylinder (turbulent)

a

9-05

H

Nu

- the larger Nu - 0.9 X the larger Nu

a hori-

3 < NuIo«rd/Nu(«e < 2 > Nui„rc,.,l.'Nuir„o > Counterflow

a hori

Cross(low

Nu = I.I X the larger Nu

a hori-

Parallel flow

Nu

a hori

M

Nil

Nu -

1.2 X the larger Nu

43 43 43 43

■•< 8.25 Nonmetals flowing upward in a short, Re/Ra° 8.25 < Re/Ha" < 15 heated, vertical tube (turbulent) Rc/Ra° <> 15

Nufroe Nu > 0.75 Nult»™d Nu =- Nu,„rced

44 44 44

Nonmetab flowing downward in a short, heated, vertical tube (turbu

Nu > Nufr,.

44

Re/RaW

> 18

lent)

i \

i

Nonmetals flowing in vertical pipes at constant qIA (laminar) 1

45

500 < Gi < 2500; 10' < Ra

CONDENSING

5

< 10' Adown — 2ûp ±

46

VAPORS

The theoretical formulas assume that the (At) / across the condensate film is uniform. At substantially all reasonable condensation rates' the outer condensate layer tem perature may be assumed equal to the saturation temperature of the vapor, though "temperature jump" occurs at the vapor-condensate interface at high g/A and low pressure." Streamline film-wise condensation on vertical surfaces:

the latent heat of vaporization and r the condensate mass flow rate per unit L down. The local h at L is three-fourths of h down to L. For a plane 4>off the vertical, multiply h by cos d>. Turbulent film-wise condensation on vertical surfaces (actual for nonmetals): where

width

X is

at distance

•—"WOT Streamline film-wise condensation on n horizontal tubes in a vertical bank: S

- 0.724 (*gL)H - 0.76 \ ViinD At)

Mr

m

/

for the nth tube = h,[n-* — (n — 1)'*]. Valid for a single tube with n = 1. Theory for metals:" Use Eq. (30) for 0 < 4r/M < 104 + . Actual h may be much g/A in Btu per hour per square foot for Hg vapor up to lower at high At;" that is,

A„

15 psia = 400,000

+ lOOOAi for At in degrees

Fahrenheit and = 22,000A<

- 36,000

h

for Na vapor whenever these At values > At by Eq. (30) or (32). 10-fold. " so High downward vapor velocity can increase Ripples start in vertical film-wise condensation when (s'p/M4f/) = surface tension). ing appreciably

< 0.3 Re0-8, increas

(s

h

It

Drop-wise condensation on vertical surfaces occurs on smooth surfaces when the may be maintained by vapor-phase wetting angle appreciably less than 180°. With adding an unwetted nonvolatile compound that will adhere to the surface. is


4

Theory

FLUID AND HEAT FLOW

9-66

[Sec. 9

nonmetals h is usually increased 5 to 10 times over Eq. (30). With metals h may be lower than in film condensation; i.e., for Hg vapor at 15 psia.47 4-

A

- 500,000

Btu/(hr)(ft!) at

At = 300°F

Presence of noncondensible gas even in traces" will greatly decrease A if it is not swept away effectively parallel to the surface, and somewhat even if it is. Small diameters decrease h on horizontal tubes below Eq. (32), as the pendant drops cover appreciable fractions of the condensing surface. Yet smaller diameters increase h above Eq. (32) because surface tension keeps the condensate film thin by squeezing it over to preferred spots, from which it drops off. 6


5.1

BOILING LIQUIDS

Boiling in Nuclear Reactors

Originally it was generally assumed that saturated boiling could not be permitted in nuclear reactors. " This assumption was due primarily to fear of the flow instability obtainable in parallel coolant channels, which might immediately decrease the flow rate until boiling dry and burnout occurred in any channel in which boiling might commence. Other adverse possibilities were burnout due to film boiling even without flow instability (see Fig. 4) and burnout due to scale formation, as occurs in fuel-fired boilers. It was also anticipated that jumps in reactivity and thus power level might take place, and possibly nuclear runaway, if the bubbles collapsed as a result of a pressure or cold-liquid surge. Scramming might also occur with increased turbine load owing to reactor-pressure falloff and coolant flashing just when more power is needed. Thermal fatigue of the fuel elements might also occur as a result of local oscillation of coolant density and thus of cooling power and surface temperature. Local thermal-neutron flux and therefore heating density fluctuations would also occur because of the boiling, particularly with H20 on account of its short neutronWhile subcooled boiling is diffusion length, and might also cause thermal fatigue. much less unstable than saturated boiling, in that the increase in pressure drop and in volume are much smaller, some fluctuation would occur, and it was considered too close to saturated boiling for safety. However, tests with the Borax reactors and with EBWR have shown that boiling reactors can operate smoothly and protect themselves against reasonable sudden Borax 1 was not damaged until a reactivity of nearly 4 per cent, reactivity jumps. In the corresponding to a period of 0.0026 sec, was suddenly applied to the cold core. hot or boiling conditions the protection would be better, as further bubble formation would occur sooner. EBWR in normal operation bypasses straight to the condenser, through an automatically controlled valve, some 5 per cent of the steam generated. This corrects for any sudden reactivity changes, control rods taking care of slower load When the reactor was running at 600 psi and full load, for instance, a sudden changes. closing of the turbine inlet valve was accomplished without the reactor pressure rising even to 630 psi, the scram setting. It is evident that boiling reactors, with their con siderable saving in operating pressure, will have importance in commercial power. In addition, the study of boiling is necessary in predicting the course of power surges and in design to prevent accidents. Boiling can occur at a heated surface in two ways according to the temperature of the liquid, three according to the temperature of the surface, and three according If the liquid is at the boiling to the method of circulation of the liquid and vapor. " boiling is said to occur. If the liquid is below point, "saturated," "bulk," or "net "local," or "surface" boiling is obtained. If the the boiling point, "subcooled," temperature of the surface is not much above the boiling point, "nucleate" boiling occurs; if it is well above, "film" or "stable-film" boiling; and if it is intermediate, " boiling. If the or "metastablc-film "partial-film," "mixed," "unstable-film," circulation is by natural convection in a large vessel, "pool" boiling is occurring: if it

Sec. 9-3]

heat removal prom nuclear reactors

9-67

is in a restricted pipe or loop, "thermal-siphon" or "thermal-circulation" boiling. If a pump maintains the flow regardless of buoyancy, "forced-convection" boiling is The 18 possible combinations of these characteristics have almost all been present. studied, and almost all have actual or potential interest in nuclear power. All these, plus homogeneous boiling, are listed in Table 11. Table 11.

Types of Boiling

Method of fluid circulation post heated surface

Free or external natural convection (in a large volume or pool of liquid)

Liquid temperature

Subcooled

Saturated* * Confined or internal natural convection (by " thermal siphon in a closed loop or conduit)

Subcooled

Saturated* Forced convection (past open surfaces or inside of conduits)

Subcooled


Saturated* No heated surface — homogeneous

heat generation

and boiling. . . . Saturated*

* Actually slightly superheated at the lower pressures. (M. Jakob, Sons, Inc., New York, 1949.) t These cases have not been studied experimentally so far.

Method of vaporization

Nucleate Mixed Film Nucleate Mixed Film Nucleate Mixed t Film Nucleate Mixed t Film Nucleate Mixed Film Nucleate Mixedf Film Nucleate

Type

1 2 3 4 5 6 7 8 9 10 11 12 13

H

15 16 17 18 19

"Heat Transfer," John Wiley

&

The stages of saturated pool boiling, shown in Fig. 4, will be discussed qualitatively to illustrate certain general characteristics. In most equipment low heat velocities can be dissipated without boiling, usually by natural convection of the liquid to a cooled wall or to the free surface, vaporization When natural convection cannot occurring there because of the lower boiling point. maintain the heating surface below the boiling point (point A), bubbles begin to form Natural convec at favored nuclei in the surface, which constitutes nucleate boiling. tion of the liquid is still the principal cause of circulation at low boiling rates (i.e., zone A-A'). If the liquid has not yet heated up to its boiling point or is kept below it by simul taneous cooling, type 1 boiling occurs, or even type 2 or 3 at high enough At. If and when the liquid rises to its boiling point, type 4 boiling commences. Beyond A' the bubbles increase the natural convection circulation, so that beyond A' in strong nucleate boiling the whole mass of liquid is usually circulating rapidly at a very uniform temperature, except for a small amount of superheat near the heating surface. As l„ and thus At are increased, the number of nuclei for which At is sufficient superheat to initiate ebullition goes up rapidly, as well as the agitation by the bubbles formed. The line A'DC usually has a slope from 3.4 to 3.6 and is fairly straight. Once the heating surface and immediately adjacent liquid film have reached the necessary superheats, vapor formation starts immediately, several milliseconds being adequate with a large power burst for expulsion of a small volume of liquid.61 At some point B a maximum is reached in the rate at which the vapor bubbles can leave the boiling surface and the liquid can yet reach it by simultaneous counterflow. When liquid cannot reach the surface at the same rate at which it is boiling away, depletion of liquid at the surface occurs immediately, the surface temperature rises further (at least locally), and the increasing molecular velocity of the vapor finally " film " that holds the liquid away from the surface. This vapor binding forms a vapor

FLUID AND HEAT FLOW

9-G8

[Sec. 9

or "film boiling," which causes the "spheroidal state" of water drops on a hot stove, irregularly but increasingly replaces nucleate boiling over the region BEG of film boiling (type 5). Beyond G, film boiling (type 6), and radiation to an increasing amount, arc the methods of heat transfer. If the heated surface is maintained at a definite temperature <«, by a condensing vapor or a rapidly circulating fluid stream and the boiling liquid temperature I is fixed by pressure control, Al is constant and the corresponding q/A by Fig. 4 will be obtained. No over-all instability will be observed at any operating condition, except the local instantaneous fluctuations due to bubbling, or to alternate nucleate and film boiling in Most design is in the region ADC, to obtain the smallest necessary Al. region BEGF. Film boiling, however, is useful if it is desired to "waste temperature," as in cooling It may also be a high-temperature loop by boiling water at atmospheric pressure. employed at low q/A for more accurate control. 10'

\ / 10s

/

X

(ffi

//II; / / /

6I03

i

yi

i

NUCLEATE 1

J~ NATURAL

I04

£ ^BOILING 10

Fio. 4. Saturated Bonilla, Ref. 98.)

STABLE FILM BOILING

UNSTABLE FILM BOILING

G

1

1

/III /ill /III h—

BURNOUT

i

/II /II /

'(Si

1 1

1

i BOILtt* G (SLOPE

-3

1 to 7±)

| 1

1

1 1

CONVECTION (SLOPE =1.25 to 1.33) '

'

1

i

! COMMENCES 1

1

1

At,

Atc

pool boiling

1

!

1 at3 to3

At2 10*

10*

(t.-T)»At,°F at a submerged horizontal plate or tube.

\

(From

C. F.

\

a

G,

it

is

is

A

if

a

is\

4

is

4,

a}t

(q/A)i in Fig. However, if q/A to the boiling surface is set as in nuclearreactor fuel element, there are evidently three passible values of At. E shown as follows to be unstable. If an instantaneous increase occurs in heat velocity to the boiling surface, its temperature will rise. But by\ the curve this means that q/A removed by boiling will drop. Thus an even larger net q/A remains to heat the surface further. This rate of heating of the surface proportional to A(q/A) in Fig. and accelerates to maximum at then decelerates a»id again yields steady operation at the surface or the heat -generating element hiehind have not melted or other wise "burned out" in the process. By the same' analysis, an instantaneous decrease in q/A carries down to All at D. similar consideration shows that D and F are stable operating points. If operation at high \film-boiling temperatures to be avoided, the "critical" heat velocity (q/A)e must;, of course, not be reached. Available information on the 19 types of boiling summarized in Arts. 5.2 to 5.5. Equations are in consistent units when dimensioifless. Other quantities are in units of feet, pounds, hours, degrees Fahrenheit, and Btu, particularly primed, otherwise stated. F if


t

/

y/h™.

CONSTANT h (SLOPE =1)

/

Sec. 9-3]

heat removal from nuclear reactors 6.2

Free Natural-convection

9-69

Boiling

6.21 Subcooled Liquid.* Subcooled free- or gravity-convection boiling could be significant in nuclear reactors cooled by internal natural convection or slow forced convection. The following can However, very little information on it is available. be used as a guide in design work, but tests under actual conditions are called for in most cases. 1. General. This type of boiling has been studied over a wide range of At by The quenching the desired shapes in cold water and following their temperature. results are probably also reasonably valid for steady-state operation. A general correlation has been proposed'2 for the quenching of steel rods of radius R\


(33)

However, other experience" shows very where s is the surface tension of the liquid. Considerable noise irregular effects of small changes in subcooling and gas content. and vibration usually develop in subcooled boiling. The average life of a bubble is of the order of 0.001 sec and correlates with bubble size (q/A)c and the physical properties.'4 2. Effect of Diameter on (q/A) e. Steady data for wires are also available. Platinum wire of 0.0039-in. diameter burns out' at a (q/A)c of 1,200,000 Btu/(hr)(fts), 55 per cent of that for the 2-in. cylinder,! and 0.0079-in. nickel wire" at about 1,320,000. Values of (q/A)c for other diameters in water at 14.7 psia and 144°F subcooling could be approximated by a log-log interpolation of (q/A)c versus D. A tight horizontal coil with turns as little as D/2 apart still agrees with straight wire." 3. Effect of Liquid, Subcooling, and Pressure on (q/A)c. For other liquids, subcoolings, and/or pressures, (q/A)c may be obtained within some 20 per cent by the

correlation"

(34)

Al,«i is f«tf — I; at t,a, pressure of the liquid plus any dissolved gas equals that on the system." (q/A)t.,at is obtained from Fig. 7 and seems to vary little with diameter. (q/A)e,^, by Eq. (34) is for small wires; thus it should be conservative for most fuel elements (according to 2, above). Values of (q/A)c,,Hb for type 1 boiling could also be approximated from reliable correlations for type 11 boiling by substituting zero

velocity.

4. Effect of Subcooling on At,. Available data6 show t, for a wire constant as sub is changed all the way to zero. Thus At, for any subcooling can be computed if it is known at one subcooling or for saturated boiling (i.e., Fig. 6). If (q/A), and (At), are known or can be estimated, At 5. Type 1 Boiling on Wires. can be approximated for lower values of q/A because q/A falls off about as (Ai,,,)*-1. For 0.0079- and 0.01-in. wires in water at, 0 psig" q/A = 0.54(A/.ol)» 1 Btu/(hr) (ft8) (°F) for At > 30°F. No large effect of diameter on this curve would be expected, the same as in type 4 boiling, since the circulation is similar to that in liquid-phase natural convection, in which h is independent of L or almost so (Table 9, items d, k, and p). Dissolved gas is of interest because of the dissociation 6. Effect of Dissolved Gas. of HjO and DjO, etc., in reactors. Increasing the dissolved-gas content decreases t,at at a given pressure. Since At,at is primarily a function of q/A for each liquid, t,at, At and tu, decrease about equally." Accordingly, a given q/A can be dissipated with a

cooling

* Table 11, types 1, 2, and 3. This lower (rj/A)e must in part be due to the small heat capacity per unit surface for wires and thus high susceptibility at constant heat-generation rates to overheating from local instantaneous fluctuations in heat-removal rate caused by bubbles. It must also in part be due to nonuniformities in wires, which cause hot spots, and to the (usual) increase of electrical resistance with temperature, which causes an "f

Direct-current heated wires thus yield conservative overheated spot to generate yet more heat. compared with nuclear heating and larger shapes, and a-c oven more so.

of iq/A),

values

FLUID AND HEAT FLOW

9-70

[Sec.

9

On the other hand, vapor phase would considerably lower t„ in the presence of gas. start to form at a lower q/A than with degassed water, which would be undesirable for smooth operation because of the change in moderation or neutron economy, though it might be desirable as a safety feature. Gas also decreases (q/A)e significantly," as is shown in Fig. 5. Saturated nucleate pool-boiling data 6.22 Saturated Liquid, Nucleate Boiling.* for useful reactor coolants are shown in Fig. 6. D20 may be assumed to follow the HsO curves. 6000


4000

200 20

30

Fiq.

40

60 80_ 100 at -- t.-t, *f

5. Effect of dissolved gas on natural convection pressure from wires. X = burnout (nickel wire).

200

300

subcooled boiling water at atmospheric (From C. F. BonUla, Ref. 98.)

1. Effect of Shape. Cylinders and plates and even 10 vertical banks of tubes" Addition of shallow fins decreases yield the same coefficients at the same P and q/A. At for a given heat dissipation, as would be predicted with the fin area and efficiency. Wires" give up to about one-fourth higher (q/A), and h, because of the greater At low q/A, h at wires also falls less because of accessibility to surrounding liquid. more effective natural convection and because of surface tension squeezing the vapor to fewer locations before release. •* 2. Effect of Pressure. Increase of pressure and temperature always decreases AtM A correlation" for any one liquid is or increases h in nucleate boiling.

X

♦Table 11, type 4.

L uX

VoAp/

J

(35:

Sec. 9-3]


9-71

Other surfaces K = 0.013 for water on clean platinum and 0.0060 on clean brass. X is the Uncertainty is about ± 30 per cent. and liquids require other values of K. latent heat of vaporization, and s is the surface tension in force per unit length. Increase of pressure narrows down the range in which boiling can occur by bringing together the natural-convection limit below and the threshold of film boiling above." Moderate increases in P at normal pressures" roughly increase h proportionally to P*M to Po t. The highest h values are near the critical pressure and are about ISPC


Btu/(hr)(ft!)(°F) for Pc in pounds per square inch absolute.

100

Fio. C.

F.

6. Heat transfer to liquids in pool boiling at plates and tubes under pressure. Bonilla, Ref. 98.)

3. Effect of Agitation.

bubbles or particularly

(From

At low q/A additional agitation by forced convection or by A liquid velocity u vapor jets may increase h several fold.69 0.18 1 H

q/A

1

/

approximately.

M

At high q/A there is no appreciable increase in h, nor in (q/A)e or (At)c. Vapor flow past the top of a vertical plate, tube, or deep bundle may even decrease the local h. Long operation can cause h to decrease as much as 4. Effect of Surface Roughness. Cleaning or aeration restores it. Coarse 70 per cent without apparent soiling. scratches increase h more than fine ones, and separated deep scratches even more so; 0.25 in. apart is best for water at atmospheric pressure.61 5. General Correlations. A rough general correlation of type 4 boiling is Lukomskii's data, he suggests a straight Averaging many data, hemany suggests a straight plot18 of (q/A)/(q/A)e versus A
9-72

FLUID AND HEAT FLOW

[Sec.

9

is

is

is

if

is

A

is

V

V

is

\j/

6. Liquid Mixtures. Both miscible and immiscible mixtures require a A/ up to several times that interpolated or estimated, respectively, from the pure components. 7. Effect of Metal Surface. There is a definite effect of the metal, due probably to wetting angle, centers for nucleation, and thermal diffusivity. For instance, in boiling ethanol, freshly polished Cu, Au, and Cr and aged Cr show successive 20 per cent increases in At at a given q/A.st 8. Low Boiling Rates. If At is so small that (Ra) < 1010, liquid natural convection predominates and Table 9, item b, or equivalent applies for vertical surfaces and Table 9, item o, for horizontal surfaces.6 9. Bubble Volume and Frequency.*** The theoretical bubble volume V on release as a function of surface tension for liquid-phase wetting angles up to 140° in degrees If a good average. unknown, 50° O.OO0OO48(^2s<7c/j7 Ap)'s. Above the boiling in rapid boiling grows to an ultimate value some 50 per cent larger (several surface times larger in slow boiling) because of the liquid superheat. Bubble frequency per second from a boiling nucleus at moderate pressures roughly 64K~° ■*" for breakoff in cubic millimeters. bubble stable in superheated liquid its diameter = As/Ap, where Ap the increase in vapor tension due to the superheat. Larger bubbles grow, and smaller ones condense.

500

yy*

/

400 <

100

DIRTY SURFACE-^

80

^"""^ CLEAN SURFACE

4G

Vi

^

200

60

\\ \\

300

(-

m 100

0.01

1

1 1 ! 1

1

1

1

1 i i i i

i

i

1

I

1

1

—

1 I I 1 1 i

20

0.001

1.0

0.1 REDUCED

PRESSURE P„

Fio.

7. Correlation of Atc and (
liquids

is

it

is

7

is

<„

is

is 6

is

is

is

a

+

6,

is

it

is

Vapor volume holdup causes important in predicting the change in moderation and in selecting the volume of expansion tanks required. General experience that up to 50 per cent of the total volume between closely spaced tubes or flat vertical plates can safely be, and will be, vapor at high heat-transfer rates when there is negligible restriction of the recirculating liquid. The relative or "slip" velocity u, of the vapor bubbles with respect to the liquid Sec. 9-2. For low q/A boiling, a conservatively high value can be estimated by Fig. For of per cent vapor volume could be estimated by neglecting the liquid velocity. higher q/A a liquid velocity ul up from the heat source can be assumed, the vapor volume Vv at vapor velocity it, ul computed, and then check made as to whether the liquid velocity head &i*/'2ge equals the buoyant head Vripi — pv)l Ahpl, where This method rough, partly the horizontally projected area of the heat source. h observed than given by Fig. of Sec. 9-2 (i.e., Fig. 81, vapor because higher u, For reliable design, momentum uncertain. neglected, and stream cross section tests should be carried out for the specific conditions of interest. of importance in Although high required for rapid initiation of boiling, providing safety for boiling reactors that no additional time delay has been found to occur prior to steam formation." 10. Maximum The critical (q/A)r at which film boiling Boiling Ralr, (q/A)e. initiates for liquids normally boiling above room tempera obtainable from Fig.

A


a.


Sec. 9-3]

9-73

ture." (q/A)e is increased slightly by stirring." The following correlations also yield accuracy of about ±20 per cent but require physical properties: Addoms" for I',

< 0.7 + :

(i).-«-(5): Rohsenow," using

feet and hours:

©.-—(:-;-')•• Q-)

Kutateladze:*7

(37)

- Pv)\*

= 0.16X[sMpk!(px.

(38)

is

h

is

h

h

6,

±

if

4

is

4)

O,

is

a

in

1

h

1

is 1

5

a

it

a

4)

it

6

a

a

If

Table Table

type S. 11. type B.

1 .1

*

f


is

1

a

7) is

Soiled surfaces average 25 per cent higher (q/ A)c, Though these relations include no effect of the metal, other data on wires show over 2-to-l variation in (q/A)c under identical conditions with different metals.6 The temperature difference Ate at or causing the termination of saturated 11. Aie. It increases with increased slightly by stirring." pool nucleate boiling (see Fig. viscosity,41 going, for instance, from 58 to 220°F on changing from pure water to pure glycerin boiling at atmospheric pressure from platinum wire. It also varies with the atm Pt black gave 55°; shiny Pt, 58°; Fe, 59°; Ag, 72°; Ni, 83°; metal; for water at Cu, 87°; and Pb, 104°F (this series Of these the same as for hydrogen overvoltage). metals, only Pt did not melt on going into film boiling at (q/A)c. Additives, 12. Surface-active for boiling water at moderate pressures can be for mercury increased some 20 per cent by adding soluble wetting agents." also Effect of the increase in (q/A)c. improved by Na or Mg, but more important additives on (q/A), for HsO has not been reported. Soluble salts generally decrease h. As per Fig. values comparable to water 13. Liquid Metals. liquid metals give when they wet the boiling surface. However, Hg and Cd tend not to wet stainless steel even with traces of Na and Mg, respectively, as dissolved wetting agents." Pure Hg on stainless steel and on plain steel tends to shift on use and deaeration from to boiling type q/A > 10,000 Btu/(hr)(ft2). On wetted steel or other surfaces tends to superheat and bump because of the high surface tension and bubble-formation superheat of liquid metals.7' Saturated Liquid, Mixed Boiling.* 5.23 On log scales the region BE (Fig. mirror image of BD. BEG may thus be approximated by drawing mirror roughly to tangent ially intersect FO (see image of DB from (q/A)c and Atct then curving Lukomskii plot (see paragraph under Art. Art. 5.24 discussion). Alternatively, and Al/Atc = 5.22) can be extended on linear coordinates by a curve from q/qc = Strong stirring can raise the curve roughly to (0.9,1.1), then straight to (0.2,1.8). 100 per cent." 5.24 Saturated Liquid Film Boiling.f Saturated natural-convection "stablefilm " pool boiling at atmospheric pressure yields values of only some 15 to 80 Btu/ The blanketing vapor per cent of those in nucleate boiling. (hr) (ft') (°F), about film falls by the same way that condensate film rises because of its buoyancy no appreciable effect gravity in condensation, and similar relations apply. There of surface metal or roughness except on the radiation contribution. Fig. ranges from about 10 to The minimum q/A (at 1. Minimum q/A. 30 per cent of (q/A)c, regardless of pressure or liquid, averaging about 20 per cent of (q/A)c." At pressures at least up to V, = J.J, (q/A)m-,a on tubes" roughly directly proportional to P. a fuel element has gone over into film boiling (and not burned out), q/A must This takes place by jump decrease to (q/A)mio to leave the film-boiling region. along the curve GEBCD and down on the nucleate-boiling branch to the existing q/A.

9-74

FLUID AND HEAT FLOW

[SEC. 9

2. h without Radiation. Bromley has shown," for the boiling film on horizontal cylinders 0.19 to 0.47 in. in OD at atmospheric pressure, that hcc„ for conduction is given within ± 15 per cent by hcon = 0.62

[fcv'MPi L Dfii

- Pv)g*y J

At

- Pr)gXl»

_

[>>-(>>.

L

Duvq/A

J

(39)

where subscripts V and L represent vapor (at its average temperature) and liquid, The coefficient 0.62 is halfway between 0.724, theoretically derivable respectively. for no resistance to vapor flow by the liquid [see Eq. (32)) and 0.512 if the liquid is stationary. By the same method, for a vertical wall,


= 0.80

pv*K»t Lhy L

~ At

">HM

J

= 0.74fc,

r^fo L

~

Pv)9Xf J Lnvq/A

(40)

where 0.80 is halfway between 0.943 [Eq. (30)] and 0.666 for stationary liquid. 3. Effect of Superheat. The sensible heat of the vapor stream is usually significant. Correction can be made" by multiplying X by (1 + 0.4Atep/X)2. 4. Effect of Diameter. On small wires A is 30 to 100 per cent higher than predicted.19 To fit wires as well as larger cylinders, Banchero et al.'8 replace 0.62/)"^ in Eq. (39) by aJD + a,. From boiling 02 and N2, a, = 0.00375 ft?* and a, = 1.64 fr>» for D from 0.025 to 0.75 in. 5. Effect of Radiation. Radiation transfers additional heat but in so doing increases the film thickness, decreasing If the The approximate effect is to add %h,. hco„. other quantities are given, the wall temperature T„ can be most accurately computed*7 by trial and error from q'

/._„.

S=l°-53H

- Pv)g\V*\K

[pv(Pl

DTv

J

/

(7'„

-

Tl)*

(q/A)H

,

+

1.73

X 1Q-'(7V

l/«.

+

l/,t

-

-

7V) 1

1411

•» is the wall emissivity,
6.3

Confined Natural-convection

Boiling*

Thermal-siphon boiling has characteristics intermediate between the local-naturalconvection and the forced-convection types. It differs from local natural convection in that the rest of the loop affects the circulating flow and h. It differs from forced It occurs convection in that the flow is directly dependent on the heat transferred. in water-tube boilers, evaporators, and free-boiling nuclear reactors with vertical Boiling bet ween vertical parallel plates and in tube bundles in closed vessels tubes. also falls in this type, particularly if the circulation is reinforced by a vertical mixedflow tube or chimney. There are practically no data on subcooled or on film natural-convection boiling in tubes (types 7, 9, and 12). However, subcooled natural-convection boiling usually cannot be very effective by itself because of the small vapor volume, and film boiling probably also cannot promote strong circulation because of the film location of much of the vapor. Thus the relations for types 1, 3, and 6 boiling should apply, respectively, to types 7, 9, and 12. Natural-convection nucleate boiling in tubes (type 10) has been studied frequently, n but again no useful general correlations are available to give h or w directly from given dimensions and conditions. Accordingly, one of the following three methods must be used to obtain rough design information: 1. Rough interpolation between values estimated for free natural convection and for slow forced convection by the methods of this chapter. ♦Table 11, typos 7 to 12.

Sec. 9-3]


9-75

2. Use of actual operating data for type 10 boiling in commercial boilers, etc. boilers" generally employ 2.5- to Large, 1,200- to 1,400-psi natural-circulation 3-in.-ID tubes and yield inlet water velocities of 1.5 to 2 fps. Outlet qualities are As with commercial type 16 3 to 10 per cent, rising to 15 to 20 per cent at 2,400 psi. boiling, their average q/A and other operating conditions are too low for optimum boiling nuclear-reactor design. The highest q/A is about 250,000 Btu/(hr)(ft») of internal projected area but is seldom reached. The actual local maximum is lower However, these conditions are useful because of conduction around the tube wall. If the circulating flow is regulated to a desired value, as conservative lower limits. Some direct data on h the methods for type 16 boiling may be used to estimate h. are also available'8 for unregulated flow.

90 V50kw 'liter NÔ

80

A \

, 70

IK 60

\


<

£50

H^°

o

o

.

>40

30

-STEA * SLIP

u

RELA1 IVE TO WA TER

> 20

IS^H

200

60

Jt — A DTUAL

STEAM VELOC ITY

Q.

50 "•

w

^

\^ ^V ^^

10

100

V \

70

40

\

30.

\

w ^

w

\\

20 CRITI CAL 1'EMPER ATURE I

I

300 400 500 600 700 SATURATION TEMPERATURE t,„i,*F

10

800

Flo.

8. Typical exit-steam velocity and exit-steam "slip" velocity at 50 kw/liter, and over-all average steam volume, in natural convection boiling of saturated water between vertical plates 5 ft high and }^ in. apart. (From C. F. BoniUa, Ref. 98.)

Some experimental results are also available for possible boiling reactor geometries.

Untermyer" reports a definite slip velocity u., decreasing with pressure but inde u, and Rv for upflow between parallel plates J^ in. apart pendent of liquid velocity, are given in Fig. 8. Bailey70 also found u, independent of ut; for water at room tem

perature and pressure, u, equaled 1/Rl fps. Cook,71 for ^£-in. spacing at 600 psi, found instead that uv/ul remains roughly constant at 2.5 ± 0.5. Rv was substan tially proportional to boiling length at constant q/A, which facilitates prediction of buoyancy pressure APb from an assumed outlet Rv. However, with J^-in. spacing Fx>ttes" reports uv/Hl about 1.5 at 250 psia, 2 at 100 psia, and 8 at 25 psia. 3. Estimating the flow rate by a trial-and-error calculation. A flow rate w is first assumed. The over-all average "voids" or fractional volume Rv of vapor held up is next estimated. For the lower pressures Rv = 1 — Rl, from the upflow column of Tabic 5, Sec. 9-2. For higher pressures Rv is obtainable by interpolation between the low pressure and the Pc columns of Table 5, Sec. 9-2, or from the slip u., or from the relative velocity of vapor and liquid (see Art. 6.2 of Sec. 9-2). The buoyant pressure

9-76

FLUID AND HEAT PLOW APb

= (pL — Pv) — &v Az =

(pl

- Pv) —

[Sec. 9

J

Rvdz

(42)

is computed. Then the sum of the frictional and acceleration pressure drops APfa around the loop is evaluated as per Art. 6.2 of Sec. 9-2. If LID, is low, friction is negligible, and APfa approximately equals 1 to 2 times the velocity pressure of the stream leaving the boiling channel. If APb < APfa, the trial to or G was too high, and vice versa.* After circulation has been determined, Eq. (62) can be used to predict h, or the h by Table 3, item o, and Eq. (35) added." Between closely spaced parallel cylinders or vertical plates, Rv increases approxi mately as the square root of q/S, the heat dissipation per unit horizontal cross section, t Vertical plates 5 ft high and in. apart in a tank, as an illustration, can safely reach over 90 per cent voids at the top and an average Rv of about 0.8 and can dissipate a q/S of about 20 million Btu/(hr)(ft*) [see also Eq. (64)1. Figure 8 gives data for a Burnout occurs when Rv reaches 1 at the top, but it is not readily ^-in. spacing." For conservative design predictable; tests at the desired conditions are necessary. (EBWR, etc.) an exit quality of 20 per cent is not generally exceeded. 6.4

Forced-convection

Boiling


When a liquid is pumped through a pipe and heated at constant q/A, the flow-mean temperature I of the subcooled liquid has a substantially constant axial gradient Wall temperature i„ rises at almost the same rate: dl/dx = (q/A)rD/wC.

l„ rises until it is somewhat above the boiling point before type 11 boiling, commonly In this phenomenon small flattened bubbles designated "local boiling," initiates. form at nuclei on the wall, grow very rapidly to a small thickness, then are condensed The total vapor volume is by the subcooled liquid flowing past and collapse noisily. The very small, so that no important loss in moderation (reactivity) would occur. pressure gradient increases somewhat [Eq. (57), Sec. 9-2). After the necessary length of channel in local boiling, the liquid has reached its boiling point and type 16 forcedconvection "bulk" boiling starts. If q/A is high enough, type 18 film boiling occurs and the tube generally burns out. Forced convection sweeps the bubbles away strongly, so the necessary q/A to initiate type 18 boiling exceeds that for type 6 for the same liquid and pressure. Though not so far reported, short L/D may exert some effect, since the radial distribution of subcooling (or of vapor content) will differ from low to high L/D. Strong mechanical vibrations or "singing" of small tubes with internal cooling water flow operating at high pressure and subcooling near burnout have been reported. For such reasons it is necessary to conduct tests under the exact desired operating con ditions to obtain reliable design data. Dissolved gas has some effect; unless otherwise stated, t,ai refers to gas-free liquid. 5.41 Subcooled Liquid, Nucleate Boiling. | Forced-convection subcooled boiling has been studied under many conditions, but no fundamental correlation has as yet been proved. Results are usually given in dimensional equations in terms of liquid subcooling A/,„t = tap — I, and average linear velocity of the liquid before boiling * Employing u. from Fit*. 8 and setting the buoyant head up to any level equal to two liquid-phaae. heads at that level yield a close check on the observed Rv in Fig. 8. Eliminating ul: tGj, — ulplO — Rv) and Gv m («t + u,)pyRy for upward flow.

velocity

Since Oy = qy/\S and Oi = Gtoul — Qy. Rv at any elevation heat qy for trial values of (rtoul{ Table 11. type 13.

z can be computed

from the vaporising


Sec. 9-3]

9-77

(' is in degrees Fahrenheit, q'/A in Btu per hour per square foot, and u'l in feet per There are a number of relations available; the most applicable should be second. used in each case, with an appropriate safety factor. McAdams et al.78 give, for water in an annulus 0.09 to 0.26 in. 1. q/A versus At. thick heated at its ID of 0.25 in., clean 304 stainless steel 3.75 to 11.5 in. long, 20 to

150°F subcooling, 30 to 90 psia,

1

to 36 fps upward flow.

At'.*, = t'a

- t\

,.©;-

ai = 1.97 for degassed water and 1.54 when saturated with air. At,*, increased slightly with pressure. (q'/A), is the total heat velocity. Clark and Rohsenow" subtract the computed nonboiling (q/A)^, (by Table 3, 5, For water heated in a tube at 0 to or 6) from (q/A), to obtain the boiling (q/A)t. where

2,000 psig, 0 to 30 fps upward flow and 0 to 300°F subcooling,

«- -«,^r^>C^vrw' *g(PL - Pv)J |ii

(44)

L

CL

within ±50 per cent, where a2 = 0.006 in a 0.18-in.-ID nickel tube and 0.014 in a 0.54-in.-ID 304 stainless tube. Given (q/A),, I, and I,*, at a given location in a tube, can be found by trial-and-error

"

(l)

(-)' T-W-H \°^^^T

(t,

- U).

+ hL(U

Lottes80 conclude, for water at /•" = 85 to 2,500 psia and

0

Dissolved gas and nature of the surface have little effect. Other approximate simple relations for water: With 0.226-in.-ID 347 stainless tube, to 900 cm' Ns per liter, 3,800,000, and

(A<).»6

to 236°F:81 At'. a,

for 1,000 <

P'

=

123

ul

- l)

(45)

= 3 to 40 fps,

«-"/»"

(46)

5

At'.*, = 1.9

0£\

Jens and

or other solution of the equation

to 40 fps, (q'/A) to

- 35 log P'

(47)

< 2,500 psia, also with reasonable accuracy down to P' 10 to 30 fps, P' = 1,000 to 2,000 psia:

= 15 psia.

With 0.18-in.-ID L-nickel tube,

At'.*,

= 28

-

0.012P'

(48)

of

2. Incidence Local Boiling. Forced-convection subcooled boiling commences at From Table 32 and subcooling (A/,„4)0 equal to that for nonboiling heat transfer. Eq. (43), in units of degrees Fahrenheit, Btu, feet, and hours,

... For q/A low and

.

43.5D.o-V'*

4

a

AV'"

,JQ,

I

is

if

G

high, Eq. (49) will be negative, indicating that nonboiling heating This passes directly into saturated boiling without the appearance of local boiling. can occur at moderate pressures but has not been observed at high pressures." Dissolved gas decreases by up to 20 per cent the q/A at which local boiling initiates At.*, < 36°F, and at 500 psi at 2,000 psi A<.„, = 100°F.81 In any study of reactor cooling in which subcooled or saturated boiling anticipated, complete plots of the nonboiling and boiling regimes should be made for all anticipated The feasible regime giving the lowest tw for any particular value of will conditions. be the one obtained. if


U.

FLUID AND HEAT FLOW

9-78

[SEC. 9 annulus

= (400'000

+

(a)

Mc Adams78 gives, for water in an 3. Maximum Heat Velocity (q/A)e* 0.26 in. thick, 0.25 in. ID, and 3.75 in. long (L/D. = 7.2), At'.y, = 20-100°F, to 12 fps upward flow, and 30 to 90 psia,

u'l

=

1

(50)

4,800A/'.«4)(?7'i,)W

u'l

=

o

P

Gunther8' gives, for water heated on one 3ide of a J.g-in.-square channel at = 14 to 160 psia, and Al'.ut 20 to 280°F, to 40 fps, = 7000(u't)»

At'.*

(51)

-

1

3

P

5

10, to 30 fps, L/D Buchberg81 gives, for water in a 0.226-in.-ID tube at u'l = = 250 to 3,000 psia, and A<',„4 = to 160°F, for G' in pounds per hour per square

foot,

(£)

= 520(G')W(A<'.„6)» •»»

(52)

under similar conditions for ID = 0.143 by replacing 520 by 530 and 0.20 by 0.28. Jens and Lottes80 give, for Buchberg's data,

McGill and Sibbitt's

in. and

results80

— 21 are represented

©.

■

10,03

(&)"

(st'^-"

(53)

"

H (£)"

- B§d

" " •» FT*

"

CO.

+

1

is

is

0.16, 0.28, 0.50, and 0.73, respec where a, 0.817, 0.626, 0.445, and 0.250 and m tively, at 500, 1,000, 2,000, and 3,000 psia. For published data on water in round, square, and annular ducts from 14 to to 54 fps, At'.* from 65 to 380°F, and (q'/A). from 360,000 to 3,000 psia, u'l from 1,130,000 Btu/(hr)(ft»), Bcrnath obtained"

3

a

±

is

is

is

The equivalent diameter D', of the stream and the heated diameter £)'*(= heated The second in pounds per square inch absolute. perimeter /tt) are in feet and P' This method correlates the data bracket the over-all Atc, and thus the first hc. 15 per cent on the average, and ±30 per cent at worst. mentioned previously within Thus design These equations do not agree particularly well among themselves. with q/A at all locations and times less than some two-thirds of the lowest pertinent correlation and no flow instability seems necessary for safety. For any desired higher q/A, burnout and stability tests at the identical conditions and dimensions are called for. For flow normal to tubes, (q'/A), about double Eq. (50) has been reached at com For upward flow of atmospheric-pressure water subcooled 50 to parable conditions. within an average 125°F past K6-in. wires, at to 10 fps, (q'/A)c = 27,500Ai,„t,(u)^ of 10 per cent.84

is

a

G,

is

It evident from the previous relations that a heated tube with uniform q/A and given on increasing q/A will burn out at the outlet, where At,* has its lowest value. Although the formulas do not correlate burnout at varying conditions very usually obtainaccurately, tests in a single tube show high stability; safe operation is

In general, and certainly at the higher velocities, no effect of direction of flow on (q/A). observed. also the same for substantially steady state or for rapidly changing conditions. These equa (q/ A), tions are for degassed water; presumably dissolved gas decreases (q/A). somewhat in subcooled boilinc* is

*


L/D


Sec. 9-3]

9-79

Zj

qt

1.33 |

w

wC

/

\

(gMh/pjgXK

;

+ 4,800(A<,.t)i,e 1

400,000

is

is

h,

is

If

U

q'u. =

1.95 |

w

I

V

qt

(g/AWpsSV'

(

7,000(A<^t),_ we

* pi

_

15,600p,,

1,

I"

Am

a

is,

a

is

is

<

In the second case [(g/A)2 (q/A)e], the two simultaneous equations! q/A = (q/A)e and d(q/ A) /dA = d(q/A)c/dA must be solvedj to find the burnout conditions This analytically. generally impractical, and in any case the full temperature profiles for a number of operating conditions are usually desirable for stress and other Thus best. purposes. graphical solution The vapor quality x in the bubble boundary layer 4. Density and Vapor Volume. at (q/A)e has been estimated" to be near 0.8. However, this corresponds to a very small volume of vapor in any reasonable channel. The decrease in mass of water within ratio of about 2: per unit of heated surface80 q/A

"]'■'

q is

of

is

is

is

is

t

?•«*,«— ti. w in Eqs. (55) and (56) in lb/sec. for the liquid phase only. (M*ub)i-e the total time rat*; has been used for the local heat velocity dqfdA, in which Similarly, d(q/A)/dA more properly d^q/dA1. heat Rain by the stream up to the given position. For the simple case of a bare reactor with cosine distribution of q/A along the channel and sub stantially constant U
For brevity, q/A

Units

are the same as for Eq. (55).

AWOAi

B

1

v to w

/

7200u>ci_c

(58)

V

(Afw»)t-«

— = tan c(wpS)' coa B + sin B

= tan

1

X

1

2

X


is

is

is

it

If

a

is

<«,

able indefinitely at constant q and G within several per cent of (q/A)r. However, a small increase in q/A or decrease in G also decreases At,j, and will thus cause burnout to approach faster than otherwise. Furthermore, an increase in q/A will decrease G more or less, depending on orificing and other system characteristics. Thus a careful calculation of coolant pressure and temperature throughout the reactor under all possible conditions is desirable in order to estimate the safety of the operation and the power at which boiling and /or burnout will occur. If bulk boiling occurs, calculation by finite increments is desirable (Art. 6.2 of Sec. 9-2). A con venient procedure is to plot for each desired channel the q/A versus position, then the P, I, and t,ai from given inlet or outlet conditions. Next t» is computed by the possible methods of heat transfer, the lowest fixing the method and (q/A)e by that method. If (q/A)t comparable to (q/A)„ or larger, as in fast or intermediate reactors with neutron-thermalizing reflector or thermal reactors having higher neutron losses in the core than in the reflector, on burnout the (q/A) and (q/A)c curves will intersect at the outlet. <_q/A)t falls to a small fraction of (q/A)*,, as in thermal reactors with low neutron losses in the core, the two curves will meet by tangency at some earlier position Ac. In some cases In possible to predict directly the burnout power of a reactor. in known proportion the first ease, (q/A)i, which from the neutron-flux distribution to total power q,, If given, any one of Eqs. equated to (q/ A)e at the outlet. (50) to (53) can be directly solved for w, P2 [Eq. (53) requires trial and error] or (At„j,)i = (A<,„»)i_e — (qt/wC) given or desired instead of (q/A)t and qt. substituted for At.ub and the unknown quantity obtained directly or by trial and error, as required. Equations (50) and (51) can also bo solved directly for q,, yielding, respectively,*

9-80

FLUID AND HEAT FLOW

[Sec. 9

in units of pounds, feet, seconds, and degrees Fahrenheit, (q/A)^, being at incipient local boiling. Visual observation of subcooled boiling near atmospheric pressure" seems to show a significant and fluctuating volume of vapor. However, quantitative methods yield volumes low enough and fluctuations rapid enough so as probably to cause no difficulty in reactors if the subcooling is at least 20°F at high pressures'0 or somewhat more at low pressures for q/A of the usual magnitude desired in nuclear power reactors. For instance, water at 100 psia, 80°F subcooling, and 1 fps showed less than 0.0001 in. of Also" water at 23.5 psia subcooled 155°F, flowing at vapor at q'/A = 400,000." 10 fps with q'/A = 2,330,000 [60 per cent of (q'/A)c\, had only 4 per cent of the surface covered with bubbles, which had maximum radii of 0.01 in. and lifetimes of 0.0002 sec. Water at 2,000 psia81 in a )£-in.-ID tube showed less than 1 per cent Ap with local boiling if At,„t > 35°F. A decrease of 3 per cent was observed at At,*t = 4°F. Ap and its fluctuations increased the lower the pressure; at 100 psia the standard deviation of p was about 10 per cent. 6.42 Subcooled Liquid, Mixed Boiling.* This case has been studied'4 for water at 16 to 60 psia, subcooled 50 to 100°F, flowing at 1 to 5 fps along the outside of a Ji-in. tube. At'M = 600°F and agreed well with %


A

= 880,000(A<'..,)-»-«

(59)

6.43 Subcooled Liquid, Film Boiling, f Forced-convection subcooled film boiling in tubes has not yet been studied, except for the conditions required for its initiation, summarized earlier (Art. 5.42, paragraph 2). However, nonaqueous liquids sub cooled 20 to 80°F have been flowed upward at 3 to 13 fps" past film-boiling horizontal The subcooling raised the values of h almost to those for %- to %-in.-OD cylinders. nucleate boiling. The results are approximately correlated by the dimensionless relation (see Art. 5.24, paragraphs 2 and 5) :

0.5*

SlniriPi.'-''

=

—

* Average deviation ™ ±1; maximum deviation =

100

3*

5*

7*

9*

200

300

500

800

± 2.

where u is the velocity past the cylinder, m0 and D0 are for the duct creating the turbulence in the stream, and X is the enthalpy decrease from ly to saturated liquid. For ethanol, benzene, and hexane hco„ was roughly 10u' Btu/(hr)(°F)(ft') at Al.*i = 0 The total h = h„n + 7sAr. and varied linearly with A(Iui to 30u' at Atlui, = 80°F. 6.44 Saturated Liquid, Nucleate Boiling. J Forced-convection saturated nucleate boiling in channels yields rather low values of At, which are fairly independent of dissolved gas content" and of velocity and channel dimensions. Because of the considerable pressure drop and its high negative slope vs. flow rate at constant q, type 10 boiling in parallel channels in nuclear reactors may be dangerously unstable unless the channels are roughly equiaxed in cross section and heavily orificed indi vidually at the inlet (Art. 7.1, Sec. 9-2) or have individual pumps or flow is upward § and the channels are large so that the buoyant pressure of the vapor is significant compared with the friction. The negative pressure coefficient is even important in burnout tests on individual tubes. In a system in which the flow drops appreciably if AP across the heated tube » Table 11, type 14. 11. type 15. X Table 11. type 10. § Carter87 predicts that boiling downflow is more stable than upflow, but his tests show that in parallel downflow channels q cannot safely exceed about one-half that for a single channel.

t Table


Sec. 9-3]

9-81

rises (as with a low-pressure centrifugal pump and no inlet orifice), the values of (q/A)c observed may be considerably lower than for constant flow and may be mis leading. Nonuniformity of q/A or of velocity distribution, as in a square duct87, also lowers (q/A)c. 1. q/A versus At. In general, h near the inlet of a tube containing boiling water is not very different from that in type 4 pool boiling if the velocity is low or from nonboiling flow if the velocity is high, and correlations for these cases may be accepted as conservative. It is generally observed at low pressures that h approximately doubles" as the quality (and thus velocity) increases downstream to about 50 per cent. There after wetting is poorer and h starts to drop, particularly after 80 per cent vaporization. Turbulence promoters are not advantageous. For light petroleum oils at 3 to 30 fps in an annulus of 0.226-in. heated ID and 0.012-in. width and At'.*, 50 to 250°F,88

(fQ

-

= 0.000097(4/'.,,,)'

(60)

Pr«

is

h

(^)

*

= 0.0086(Re„)»-8

(61)

"

(°2)

*]

—

)'

y

0.0005

(-'

+

Nvu =

[4.3

G,

6

is

s

where Re„ employs the log mean of the average inlet and exit velocities, neglecting surface tension, and all physical properties are for the liquid. vapor slip, For water entering 0.465-in.-ID 347 stainless-steel tubes at 1.5 to fps and boiling at 45 to 200 psia and q/A = 50,000 to 250,000 Btu/(hr)(ft'), the local Nul, in terms of is" the quality x and total flow

(62) held within 10 per cent on the average and was independent of L/D, Above x = 0.5, to 0.4. decreased toward the value for steam at 1 = about 0.7. Above x = 0.6, temperatures fluctuated widely. In Nut and Ret, ki, and ixl are used. 2. Maximum Heat Velocity (q/A)e. The highest values of (q/A)c are obtained8* near or somewhat below reduced pressure of one-third, as in saturated pool boiling h

a

Equation from 1 =

0

(Fig. 7). (q/A)c compared with nonboiling (q/A)\g by Table

3,

item a, calculated for the same fluid and wall temperatures given for JP-3 (gasoline) and JP-4 (80 per cent gasoline, 20 per cent kerosene) hydrocarbon jet fuels88 of critical pressure Pc approxi mately = 550 psia, in tubes at uL 15 to to 80 fps, P' = 30 to 500 psia, I)e = 0.6 in. and G' in pounds per hour per square foot by

- 1,
=

A

,(g/,A)'

(q/A) sb

3

-

0.

is

n.wB

(63)

a

In soiled tubes burnout may ensue'0 below (q/A)c for clean tubes. In square or other angular sections at constant q/A, (q/A), may be only fraction of that in tubes. For q'/A > 100,000 burnout with water in general occurs when x reaches about 0.7" to in q/A.

0.8. 7*

Burnout x seems to increase

slowly with pressure

and with decrease

Re»

> 20,000,

if

Deissler has correlated all the available data for (q/A)e for water in tubes in satu rated and slightly subcooled boiling, within +25 per cent on the average, by the ratios Ret < 20,000, Gy/GL = 0.006; and Gv/Gl = (q/A)c/\GL and GLD/UV = Re*.

If


Nu

«

5

0

For a number of liquids heated near 15 psia at constant q/A in slow vertical upflow, per cent quality and leaving at under in pipes entering at per cent, the average correlated within several per cent by"

9-82

FLUID

AND HEAT FLOW

[Sec. 9

Gv = 1.25(Itet)-°-»

Gi

Gl

is for liquid in the pipe and Gv for vapor generation at the wall (based on wall area). A rough general correlation" of effect of velocity on the safe q/A for water [a lower limit of (q/A)v for saturated boiling in tubes], for quality x from 0.05 to 0.6, is utx2 = 0.8

(64)

This yields the following value of safe production rate of saturated vapor G'v and heat of vaporization q'v/S per second per square foot of boiling channel cross section S: P', psia

14.7

a-,

47.8

Q'r/S

46.400

In general,

L/D to

500

45.0

1.000

40.6

37 0

30,700

40.000

these values of burnout

the burnout

L/D.

100

24.000

quality do not hold below

1,500 34 0 18.900

L/D

be considered to decrease

xt can (conservatively)

2.000 31.1 14.400

At lower proportionally

= 25.

3. Commercial Boiler

Some commercial high-pressure boilers operate Operation. The main purpose of the pumps is to maintain the desired distribution of the water among the tubes and thus decrease the danger of burnout by boiling dry or scaling; make up for low buoyancy at low heads or high pressures; It permit smaller, lighter, and thinner-walled tubes; and obtain higher exit quality. is also possible to force the inlet water along any desired path, such as first past the region of highest q/A, rather than being limited to substantially vertical upward flow. Each tube usually has a flow-distribution orifice (Art. 7.2 of Sec. 9-2) with AP, approximately equal to the friction and acceleration APfa in the highest flow tubes and greater in the others. Recirculation is used commercially above about 1,500 psi with 1- to l^-in.-ID The maximum local q/Ai ranges tubes and 2.5 to 5 fps inlet liquid water velocities.* At the exit, quality is usually 10 to 30 per cent up to about 200,000 Btu/(hr)(fts). Because (95 or 100 per cent in the Sulzer and Benson boilersf), but q/A is much lower. of the presence of some scale-forming salts in the usual boiler water and the limited q/A values obtainable in fuel-fired furnaces, these operating conditions may be con sidered conservative lower limits for nuclear reactors. Free bubbles tend to accumulate in the center of the stream 4. Location of Bubbles. This decreases Rv and the because of shear stress distribution and Bernoulli forces. Whirling of the coolant as it passes through the effect of the bubbles on moderation. tubes has the same effect, owing to centrifugal acceleration, and should considerably increase boiling reactor stability and permissible heating density. Vapor volume or the total coolant mass in boiling coolant channels can be computed subject to any desired assumptions. For instance, for a slip velocity u, and heat pickup distribution q(x), the liquid velocity ul can be computed as a function of i, the distance from start of boiling, by trial and error from


with forced convection."

Plul

X

b

1

Pl(UL +

U.)

P/.ul-

(65)

The total vapor volume in the channel is then the graphical integral

rx JO

I

p,X

rx Jo

Uh + u.

* It is found permissible to operate for an hour or mure at about 75 per cent of these normal velocities (half the pumps shut down). t Superheater tubes are used at 2,500 psi and 1050°F.

(66) inlet I

heat removal from nuclear reactors Sec. 9-3] If u, =0 and q/A and pressure are substantially constant, the correct

9-83

mean specific volume V over the boiling length is the log mean of the liquid and outlet specific For a bare reactor with cosine distribution of heating and saturated inlet volumes. For this case and weighting the density by the flux coolant V is the geometric mean. distribution, as would be appropriate in computing coolant irradiation, the geometric mean is again correct, in conjunction with the mean neutron flux and the total resi dence time. In a well-wetted tube, obtainable by Na or K additions or 5. Liquid Metals. strong turbulence, h for mercury is reported" to be substantially the same function of dP/dx whether the mercury is below the boiling point, boiling, or superheated vapor. It could thus be approximated from relations for liquid metals or gases in tubes; also" by h = 6100(AfVAz)o ««(£>)w Btu/(hr)(ft*)(°F) for D in., x ft, and P psi. Poorly wetted tubes fluctuate unpredictably in temperature between wetted and unwetted (hotter) operation. 5.45 Saturated Liquid, Film Boiling.* Forced-convection saturated film boiling, as for type 15 boiling, has been studied for upward flow of nonaqueous liquids past horizontal cylinders of to %-in. diameter.'4 It was found that at low liquid velocity [u/(gD)M < 1], the same relation for hro„ [Eq. (39)) as for natural convection is valid, and h = A„„ + %hr. For u/(gD)tt > 1,


fc„.

- 2.7

yWpi Ml

+ 0.4 At c/X)

(67)

and h = hecn + J<&hT. The local Ae<>»and q/A are considerably larger along the leading edge than farther around. In natural-convection nucleate boiling at a given q/A, burnout of the heating surface generally occurs when (q/A)c is reached and film boiling starts. Only highmelting surfaces like platinum and graphite, which can stand the temperature neces sary to radiate a large part of the (q/A)c, generally avoid burnout. In forced con vection, however, the vapor film is thinner and can conduct more heat, particularly at For instance, high pressures. Thus burnout in film boiling is more readily avoided. for hydrocarbon jet fuels" above 25 fps and 300 psia, stable-film boiling can be 200 to some 200 to obtained above (q/A)c in 347 stainless tubes. At (q/A)c, A< jumps some jumps 300°F but then stabilizes there. For further increases in the film-boiling region, wall tem thewall burnout until log q/A increases about three times as fast as log A
Homogeneous Boilingf

Saturated homogeneous boiling would be expected in a homogeneous reactor or in a heterogeneous reactor using a liquid-fuel solution if the fuel solution reached the boiling point plus the necessary small superheat to nucleate bubbles. Actually, because gaseous fission and decomposition products increase the vapor pressure and because fission fragment recoil forms nuclei, boiling would no doubt occur sooner, probahly with negligible superheat. It does not seem likely that local subcooled boiling with bubble collapse could be obtained in a homogeneous fuel solution or even in aqueous suspensions of fuel-element compounds. Boiling of an aqueous fuel solution may eventually become a desirable or feasible normal operating condition in power reactors. In any case, however, it offers a valuable safety feature for homogeneous reactors — a negative coefficient of sub•

t

Tabic

11, type 18. Table 11, type 19.


9-84

FLUID AND HEAT FLOW

[SEC. 9

stantially infinity. Accordingly it has been studied primarily from the standpoint of speed of response to a sudden heat pulse. 6.61 Available theories predict higher superheat Superheat to Initiate Boiling. than is actually observed*6 before bubbling of pure liquid occurs in a container. For water at 100, 400, and 1,000 psia the maximum superheat required was, respectively, Theory 85, 35, and 10°F, the average superheat required being 25 per cent lower. In another study" 40°F yields 275, 175, and 80°F, respectively, for pure liquid. superheat was sufficient at 15 psia with an exponential power rise period of 0.017 sec, corresponding to a time delay of only 0.005 sec. The higher the heating density //, the higher the 6.62 Steady-state Boiling. necessary Buperheat, since more new nuclei need to develop into vapor bubbles per unit of time, and the lower the density of the vapor liquid mixture. Circulation rate and mixture density along each assumed vertical streamline can be roughly estimated by trial and error from a reasonable slip velocity and a method similar to that outlined In pool homogeneous boiling the friction is negligible and the under type 10 boiling. The "slip" velocity buoyancy pressure APb would all go to acceleration of the fluids. in feet per second for water up to moderate pressures'0 roughly equals Rv, the fraction of vapor by volume. 6.63 Rate of Bubble Growth." Bubble growth is steady for any one bubble at Rate of collapse of about 0.02 to 0.09 fps radially, which checks available theory. bubbles of substable size has also been studied. 6.64 Rate of Density Change. The decrease in density with time due to an increase in the heating density H follows available theory" for bubble initiation and A sudden jump from low power to about 4 to 30 kw /liter for water boiling growth. at atmospheric pressure requires a further superheat of about 1°C before the volume At 135 psia some 2°C is required starts to increase, or about 0.3°C if gas is present. either way, gas having little effect. The volume then increases at about 50 per cent/sec." This delay of 0.1 to 0.3 sec decreases in higher power pulses but does not disappear. 6

STEADY-STATE THERMAL DESIGN OF REACTORS 6.1

Temperature

Rise of the Coolant

Although power reactors will have high q/A and thus high transverse temperature Thus gradients in the coolant streams, the coolant channels will generally be thin. for heat balances it will usually be adequate to consider the coolant temperature If the axial heat flow in the solid uniform at I at any given cross section of a stream. metal and coolant due to the coolant temperature rise is computed, it will generally Usually also the heat transferred be found negligible compared with the total power. from one channel to another in a direction normal to coolant flow is found to be Thus all the heat q generated per unit time in the "cell" of fuel, coolant, negligible. and moderator nearest to each coolant channel, from coolant inlet at X\ to position- x, is assumed to have gone into the coolant stream by position x. The total heat dq/dx transferred to the coolant per unit length in any channel can be obtained by integrating Eq. (1) over the appropriate cross section of fuel cooled by the channel and multiplying by a factor to include the additional heating by y The absorption, moderation, and radioactive disintegration after neutron capture. total heat

q

transferred from

ii to x is then

the integral

/ Jxi

(dq/dx) dx.

It is usually more convenient to work with the total desired reactor heat rate qt and the ratios H /Hi and H /Hi of average and local H to the maximum, Hi. The total heat rate qi for a given channel is

«-£©.

••

where Nc is the total number of channels and (H ///)„ is the local to average ratio for the position of the channel in the plane of the reactor normal to the coolant flow. The


Sec. 9-3]

local dq /dx at position x along the channel, for x = 0 at the midplane so that Xi =

The total

9-85 — x2,

q at x for the channel is thus

^ (») (' («)

2x».

&

=

SL

[' (») *

(70)

The only practically important orientation for coolant channels is normal flow between For homogeneous fuel loading, parallel inlet and outlet planes, or through a "slab." For "roof-topped" the cosine distribution obtained can be integrated analytically. or other irregular distributions

a numerical or graphical

integration

is necessary for

Eq.

(70). If the coolant undergoes a significant change in density and velocity while traversing the reactor, such as in boiling at low pressures, the method of Art. 6.2 of Sec. 9-2 is used to obtain the relation between I and x from that between q and x.

If the pressure is high and the coolant expands significantly, as with supercritical or subcooled water at high temperatures, but velocities remain low, Eq. (17) is used in the form


h' =

h\

+

i

(71)

w

If the pressure is substantially constant, I is obtainable directly from the calculated If the pressure decreases enthalpy h' and a table of h' versus t at the given pressure. with x, P is calculated by Eq. (28) and I obtained from the table at the corresponding values of k and P. For small expansions and moderate temperature rises the specific heat c as well as This is usually the case with liquid metals, velocity remain substantially constant. with subcooled water under about 500°F, etc. In this case I is directly obtainable as a function of q: I

= ?, + .'L = ?, + wc

For the cosine distribution

(1,-1)

-1

(72)

qj.

with reflector,

H

x'i cos x'

H

sin x'i

(73)

and Eqs. (70) and (72) yield I where x' =

- I, +*LZi! (l+$^ \ sin x i/

tx/2X,.

2

Without reflector, A', 6.2

Fuel-element

= h

+12.

2tvc

(l \

+

$^-\

sin x 2/

(74)

= x«.

Temperature

Control of Fuel-element The normal or 6.21 Temperatures by the Coolant. local average wall temperature tw at any position x is obtained by adding to 2 the Al from the wall to the fluid at that position. If q/A and the coolant velocity are uniform over the heated perimeter P' at each position along a given channel or stream, as is usually substantially true, t„ will also be uniform and will be given by

^J + l^J + ^f^ A 2xJiP -■a h

(75)

9-86

FLUID AND HEAT FLOW

[SeC. 9

The normal maximum temperature to of the fuel element at x, which is at its axis or midplane, is obtained by adding to Eq. (75) the temperature drop At, across the fuel element from Eqs. (9), (20), etc. Expressing /0 — /* as equal to q/A times the ratio At,/ (q/A),, from the above equations

A plot of tx or to against x for a symmetrical bare reactor will always show a maxi mum ix between x = 0 and x = x2 and a maximum U between x = 0 and x at (!,)„,. A reactor with reflector may show the maximum tx either at x«, or before Xj. For a cosine distribution by reflector, substituting for I by Eq. (74) and for H '11, by Eq. (73)

fc.i.

+

iu.l f1 + *L^\+-ri + ^.]s-4i 2X,Plh (q/A), J x'jj x'2/ \ 2

sin

\wc

The position xm„ of maximum temperature is obtained by setting dto/dx' yields

ir - <- Eq.

which (78)

(77) and simplifying

yield

x maT)

2ioc(
1

+

/_^^Vfi+_^]M» V2X.P7 (?M)iJ

(79)

J

_J_l1 sin x',

—

l\)

x

L^

4-

; ?l)

;

2wc((o,m«, —

\

1

1

= 0,

iil/(sin— —sin— —

+

i

—

tan"

(78) for x' in Eq.

2

x'm,„ from 9L =

is

it

is

a

I,

t,

l,

it

I,

a

a

is

h.

t

I,,

is

if

is

if

is

is

is

I,

if

U

For l„ instead of the same equations (77), (78), and (79) may be employed by deleting the term At,/ (q/A),. It evident from Eq. (79) that and w have been fixed and the maximum satis set by the uranium phase change, strength, or other reasons, the factory to.mmi maximum permissible power gz, of the channel fixed. the maximum Similarly, set by corrosion, incidence of boiling, or other factors, the permissible permissible t„ gi, fixed.* and maximum tr or r<>are fixed, the necessary u> can be Again, <7l However, w appears twice in Eq. (79) and also affects It must thus computed. be found by repeated trials. It seen that all the temperatures in nuclear reactor depend directly on the coolant inlet temperature and flow rate and the power. In nuclear power plant the reactor coolant inlet temperature may, of course, be kept constant by suitable instrumentation and controls, or might be allowed to drop as low as possible or to swing between limits, with changes in the power generated, main-condenser-coolant In the latter case the temperature would have to be temperature, and other variables. traced from the condenser-coolant inlet to the outlet, through the condenser, and through any intermediate boilers and heat exchangers before could be estimated. 6.22 Hot -channel Factors. By Sees. 6.1 and 6.21 theoretical or "average" values of (..max,
a

ia

*

Maximum *o Rencrally limiting in liquid-mctal-cooled reactors, and maximum /* in water-cooled reactors. As in computing coolant distribution among the channels of reactor to obtain constant tw or t» instead of h.

t


Substituting

(77;

sin


Sec. 9-3]

9-87

the axis of a reactor will all have identical values of lo.im> by Eq. (79) if all other factors In actual fact, however, there arc bound in the equation arc (substantially) equal. to be appreciable differences among them in qL, in w and thus in
Table 12.

Typical Hot-channel Coolant temperature rise,

Uncertainty in

i

Fuel channel concentration Eccentricity and diameter

Fuel-element

thermal

-

i,

Film

temperature difference,

1 20 1. 10 1.40 1.02 1.04

1 20 1.20 1.25 1.02 1.03

1.20 1.20 1. 10

1.95

1.90 1.20

1.60

conductivity. *

2. 28

-

Maximum possible coolant temperature =■ti + l.95(i — fi)ca!e Maximum possible surface hot spot °= ti -f- I.95(t — (i), ,1. 28(Ai/)Ca)c Maximum possible metal hot spot = + l.95(t (Ode + 28U//)c«lc •S. McLain, ANL-5424, 1955.

1 05 ! 05

F,

1 76

- F.

+

2.

-f2.

-

ii

Fuel

temperature difference,

A//

1.95- F,

1.76(AMc»ic

h

U

is

is

a

is a

The hot-channel factor for a given source of inaccuracy multiplier applied to the normal design value of temperature differential to yield the maximum possible value of that differential. Such factors must be determined by careful analysis of manu facturing and inspection procedures and tolerances, the reliability of the physical properties and design relations used, methods of reactor control* and fuel-element routing in the reactor, tests on models, critical assemblies or "mock-up" reactors, etc. LeTourneau and Grimble" outline details of estimating hot-channel factors from dimensional tolerance data, etc. Some can also be measured from model tests.'8 Table 12 lists possible values of these factors for the three significant temperature differentials in computing the hottest spot. These or similar factors apply at all posi tions in the reactor, but the values of most importance are usually those at the posi tions of maximum and to, found in Art. 6.21. It seen that the maximum possible about twice as much above the coolant inlet temperature as the to in this case normal design hot spot.f To design with hot-channel factors, in Eqs. (74) through (79) w should be replaced by te/Ft, by h/F/, and At. by F, At.. Equations (78) and (79) become, respectively,

In ot***in

and

L

h

sin x'2 sin a:'",,.**

by the ratio

changed. As before, to,mn% can be * For instance, the neutron- flux density or fuel solution level control.

(81)

^

+

F,

-

p

/Fr, and Fn F/, and F, are equal, by tw by setting Ate — 0.

xmax

is

decreased

2wc(U.m**

m

[F,^T F.At.1 («/A),J

if

qL

taD

1

Evidently,

is

Ql =

m"

Zi)

X

2X.

not

replaced

varies much more with poison-control

rods than with reflector

later power reactors, considerably smaller values are perforce being employed in order to reasonable designs. These are justifiable by smaller tolerances, greater experience and optimism, a qualitative or subconscious application of the statistical method mentioned herein.

F

■f


Factors

9-88

FLUID AND HEAT FLOW

[Sec.

9

It is seen in Table 12 that all the factors are assumed to apply simultaneously at the hottest spot. It is true that some of them are interrelated, such as neutron flux, average fuel concentration, and warping of the fuel element as a result of high burnup. However, this method of design is bound to be overconservative if each hot-channel factor is the maximum possible effect of the corresponding uncertainty. A sounder procedure, based on statistics, would be to determine or estimate the fractional standard deviation a' for each independent uncertainty,* combine them to find the total a of the limiting temperature, then design by the normal equations (without hot-channel factors) to an agreed-on number of as below the limiting tem perature. "•los The probability of a hot spot exceeding the limiting temperature is 16 per cent if the highest design temperature is a lower, 2.3 per cent if 2a lower, 0.13 per cent if 3a lower, etc. 6.3

Heat Generation in Flowing Liquids and Slurries

Homogeneous reactors employing solutions of fuel plus moderator, such as the generate fission heat internally when the solution is in the enlarged vessel that constitutes the core. Reactors employing fuel slurries instead of solutions or dissolved or suspended fuel with separate moderator, such as the externally cooled LMFR, are similar cases. In these reactors there is usually no separate coolant in the core. Thus all the heat generated appears as temperature rise of the fuel fluid. This does not mean, however, that the outlet temperature h is uniform even across any one channel. Apart from the radial variation of H, small for small channels and long neutrondiffusion lengths, and vice versa, there is a variation due to the nonuniformity of The consequent tem coolant velocity u and therefore residence time in the core. perature nonuniformity is continually being decreased by radial molecular heat con duction in laminar flow and much more effectively by eddy conduction (Art. 4.23 of Sec. 9-2) in turbulent flow. However, radial temperature differentials may be of importance because of their effect on average density of the stream, because of thermal Accordingly, their evalua shock of surfaces on which the stream may impinge, etc. tion is necessary in optimizing the design of such a reactor. Volume heating of fuel solutions also occurs outside the reactor core because of the Heating of both coolant and fuel solu radioactive disintegration of fission products. tions also takes place within the core as a result of 7 absorption, fast-neutron slowing These other sources of heat within the core can usually down, and neutron reactions. (conservatively) be neglected (since they are much smaller than the fission-product However, they can plus 0 energy), being included with the fission heat generation. be important in evaluating behavior of the reactor in the event of coolant flow failure. If it is necessary to consider the axial or radial variation in H, the calculation must In the case of uniform H, general generally be made by finite increments (Art. 7.2). mathematical solutions have been worked out for long tubes."100 This case gives the highest temperature differentials; thus it is somewhat conservative for design purposes. From Eq. (5), the differential equation for a wide flat duct is


HRE,

(821

For streamline flow where x is measured along the length and y along the thickness. Equation (82) has been integrated t/i = 0 and u is obtained from Table 1, Sec. 9-2. for the conservative case of long ducts to give the transverse temperature distribution for the boundary conditions that both walls (at y = 0 and y = L) are at U, and that the fraction of the heat generated that remains in the stream is F' (F' = O for iso Substituting the position of the midpbuie thermal walls and 1 for adiabatic walls). gives the axial temperature
fractional deviation from the mean due to each uncertainty.


Sec. 9-3]

—

'axial

U„

HL'/l

5F'\

9-89 17F'\

HL>(2

The integrations have also been carried out for turbulent flow, taking «» as equal to tr in Eq. (36), Sec. 9-2, and using a generalized velocity distribution similar to that of Results for the principal conditions for flat plates and for Eqs. (32) to (34), Sec. 9-2. tubes are given in Table 13. Table 13.

Values of

k(t

-

*„)

for Liquid with Uniform Heating Density Value of t in (t

Wall conditions

Velocity distribution

-

Wide flat duct (L — wall spacing)

'axial*

'flow mean*

<«)

Round pipe

(L

'space mean 'nai.lt

■■diameter)

fepscc mean

/flow meant


Streamline

Adiabatic One well isothermal and one adiabatic t . One wall isothermal and one adiabaticf. .

Parabolic Uniform Parabolic Uniform

-32

Parabolic

2

20/7

3

Uniform

2

3

3

8 8

12

10 12

-46.

ac

16 16

- 12 15

7

00

□0

24 32

-32

-64

103 143 400 2,200

216 296 920 4,200 785 5,750 2,860 40,000 13,000 300,000

00

32 32 96

so

Turbulent

Adiabatic Adiabatic Adiabatic Adiabatic Adiabatic Adiabatic Adiabatic Adiabatic Adiabatic

Adiabatic

Ro

Pr

5,000 10,000 100,000 1,000,000 10,000 100.000 10.000 100,000 10,000 100,000

0.01 0.01 0.01 0.01

67 94 230 10.000

100 182 364 1.700 290 2.000 820 12,900 3,330 85,000

* H. F. Poppendiek and L. D. Palmer, ORNL-1701. t H. F. Poppendiek and L. D. Palmer, ORNL-1395.

I 1. 1954.

J fvi-at

1954. 1952; Chem. Eng. Progr. Symposium

Ser., No.

the isothermal wall, and fatal at the adiabatic wall.

Experiments are lacking on laminar free convection in vertical fuel channels with uniform H, but theoretical analyses are available. They give conservatively high values of the maximum (axis-to-wall) temperature difference AtV across a horizontal section of the liquid fuel. For fuel sealed within channels of high height-breadth ratio Z/D, with an external coolant flowing upward which rises by Atc in height Z:101

Mcp> Zk„

-9—

k

M,

q

A

(flat channel)

1

10

I0»

I01

I0<

4

4

4.08

4.55

7.7

13.5

4

4

4.02

4.4

7.85

25.5

10*

9-90

FLUID AND HEAT FLOW 6.4

Steady-state

Thermal

[Sec. 9

Circulation

Because of its simplicity

and dependability, thermal circulation or gravity convec tion of coolant or liquid fuel in a loop is frequently ' considered for the primary method of heat re moval from a nuclear reactor. When forced cir culation of the coolant is the normal method, COOLER thermal circulation is still of importance; by (HEAT locating the heat exchanger higher than the EXCHANGER) reactor, partial flow is maintained in the event of pump failure. The pertinent driving heads in a thermalcirculation loop are shown in Fig. 9. Usually there is little, if any, temperature change along the exterior piping, and it is convenient to ex press the buoyancy pressure AP« over the loop in terms of the (average) temperature differ ence between the hot and cold legs and the effective driving head Z,. Zt evidently equals the minimum head Zo plus the effective values of w If Za exceeds some five times Zk Zk and of Zc. and Zt, one-dimensional flow and

' 3-e ,

t_

i

\

\

Zo

'

HEATER (REACTOR)

Lh


1

Z.

PRIMARY COOL ANT

Flo.

9.

heads. 98.)

Thermal circulation driving (From C. F. Bonilla, Re/.

For streamline flow in

/

- Zo + Zk+Zc

may be assumed without sensible error. For convenience, the following explicit tions for w and h — U are given:

solu

a pipe loop,

t

f}p*gZ.D*q\*

For streamline flow in other

,

7

_

/128

pL,q

cross sections, employing the ratio

V*

uiiL/AP

gc

from Table

1, Sec. 9-2,

(upL fo*gZ,Sqy* \AlJ gc pL.c /

=

■ *

,

'

_ (AP

gc

\ upL

mL^_\m pp'gZ.ScJ

(85)

For turbulent flow,

\

J

ZLjje

(86)

\0p*gZeD.S*c'J

or, employing Eq. (27), Sec. 9-2, for fF,

_ fffptgZ,D.ltSliq\'""

\

The

0.092L,cM»J

/

,

_

,

_ /

\'w

0.092L.M1' a7" \pp*gZ.D.l-'Sl-tc1*/

(87)

case of smaller Zo is covered elsewhere. •• 6.5

Comparison of Coolants

Many factors are involved in the selection of coolants, particularly primary coolants. These factors include cost, nuclear properties, hazards, and physical properties over the desired ranges of conditions. In designing a nuclear power plant, any coolant should properly be considered only

Sec. 9-3]


9-91

optimized over-all design. If only a few factors are rigidly fixed, such as output it is impossible to solve conveniently for the optimum values of all the other factors and thence the minimum cost. However, considerable guidance can be obtained by computing important derived quantities, pumping power in particular for For a rough comparison of coolants, the ratio of their pumping power reactors. powers for any desired fixed conditions is helpful. For a more thorough analysis, one or more of the conditions can be optimized for each coolant before computing the pumping power. Employing Eq. (28), Sec. 9-2, for AP and Eq. (27), Sec. 9-2, for/F, the pumping power in turbulent flow for substantially nonexpanding fluid is in an

and fuel,

gcD.'-'S"

p

For streamline flow, using (upX/AP

ge)

p'

L

from Table

J

K

1, Sec. 9-2,

P-^'^J^' 6.51

«

Coolant Temperature

Rise

If

Atc.

'

(89)

the coolant-film

temperature difference


Atc, as might particularly be true for large reactors with long or small diameter At/ tubes and low q/A, the simplest basis of comparison is the pumping power required to give the same Atc. Replacing to in Eq. (88) by q/(C Atc), for turbulent flow,

» D.1 «S' *(Atcy Lp»c»»J

g.

Similarly,

for streamline flow,

p

=

L«g*

*L*1

T

unL grS(Atc)'

JLl

(on

Ip'c'i

Evidently, the smaller the physical-property groups in brackets in Eqs. (90) and (91), the better the coolant from this standpoint. Values for typical coolants are listed in Table 14. Water is by far the best at temperatures it (and the organics) can reach, and fused salts and metals above that. If in a comparison of coolants all are not in the same (streamline or turbulent) regime, P should be computed by the appropriate

equation for each. as might be 6.52 Coolant-film Temperature Difference At/. If At/ Ate, approached in an enriched power reactor, the pumping power for turbulent flow of nonmetals to obtain the same At/ and thus h by Table 3, item a, is

»

For liquid metals in turbulent flow in tubes, Table 4, item a, yields =

49,850L„SA" gtD.<>

*

ro.747>.<" Lp»cs"fc°-'

/ _

V

_7_V Nu/

'] J

1

A plot of h against Similar relations can be derived for other channel shapes. pumping power per unit of area P/A also is convenient for comparing coolants on the constant At/ basis for a specific geometry. ,0* For streamline flow in round pipes, Eq. (89) (increased by the ratio 4.36:3.66 from Table 6 for constant q/A) yields g.

LpVfc'J

?

to

o

.

I.

Units

X

X X XX

XXX X

are centimeters,

X

column.

2.24 5.55 2.86 8.96

XX

particular

10" 10' 10' 10'

I0< 10* 10» 10'

14.35 16.3 19.4

9. 12 14 11.83

17.35 84.50 25.0

8.50 4.74 4.84 Ill

10« 1010*

X 10'

0.262 151 1.898 3.45 3.21 160

351 82.4 438 132.4

7.30 7.54 3.08 6.38 2.83 8.51

580 358 730 496

28.9 31.1 22.6 29.6 39.8 41.2

10.75 14.10

2.45 5.51

40.20 136.8

seconds, and calories.

9.99 X 5.50 5.64 1.299

grams,

I0< 10' 10" 10"

XX X

in that

2.07 3.38 1.47 7.56

43.6 2.210 85.100 68.400 37.300 1.110

5.03

0.753

0.721

Thermal Turbulent M, (87) Any

Thermal Streamline M, (84) Any

Forced Streamline Me Mi (96) Any

I.

the coolant

61.60 44.90 268.0 119.7

X

X XX

*

The

I0« I0» 10" 10«

I0« 10" 10"

X

I.

2.77 0.990 1.47 2.49

188 4.73 13.89 13.61 15.39 8.92

7.11 31.5 17.79

10" 10"

1.40 8.97 1.258 1.039 X 1.042

17.86 36.2

10"

0.721

Forced Turbulent Mc + Mi (95) Nonmetal

3.44

Forced Streamline Mi (94) Any

Groups*

X

11.22 25.2 35.4 41.9 53.6 56.1

Forced Turbulent Ml (93) Metal

Physical-property

XX

0.0479 0.0927 0.0532 0.1278 0. 1413 0.1178

2.160 2.05 10« 625

0.0416 0.0788 0. 1552

0.875 2.95 4.54

11.670 130.800

0.01230 0.0339

1.582 1.847

19.79

Forced Turbulent Mi (92) Nonmetal

0.000922

(91) Any

Forced Streamline M.

Coolant

0.188

Forced Turbulent M, (90) Any

14.

X

I.

Btnaller the group, the better

HjO (saturated at 300°C) Organic liquids: Dowtherm (300°C) oil (300°C). Circo XXX Fused salts: NaOH (350°C) HTS (300°C) KClLiCl (400°C) Liquid metals: Na (300°C) Sn (300°C) Hg (300°C) Pb (500°C) Bi (300°C) K (300°C) Gases: H. (300"C. lOatmabs). CO> (SOCC, 10 atm abs) He (300°C, lOatmabs). Air (300°C, lOatmabs).

Equation Coolant type

Circulation Flow Constant

Table


+

7.

Sec. 9-3]


9-93

In streamline

flow this comparison is less appropriate, since At, and h are more on D than on w and constant D would be inappropriate to compare fluids at constant At, and h. Table 14 shows that sodium and then the other liquid metals are by far the best in this respect. 6.63 When neither Atc not At, predominates, both Hot-spot Temperature. should be considered. One comparison has been made"" based on the same Ate, At/, and total volume of cylindrical channels Vc, as well as channel length 2z2, and q. Different D, AP, and w are required for each coolant. For turbulent flow of nonmetals, dependent

p For streamline flow,101

=

A

Jgc

r_^Pr-o.4l

Vc>(Atc)*


Jgc

Alt LpV*

Ve»(At<)*

J

f95)

At, Lp»e«fcJ

Dividing Eq. (96) by Eq. (95) yields P.tr..miin<,/Pturbui.nt ■= 1.835 Pr°-4, which suggests that, if optimum design of a reactor yields coolant flow near the critical Re, it may be desirable to assure turbulent flow with nonmetallic coolants and streamline flow with From Table 14 water is by far the best in turbulent flow, but in streamline metals. flow sodium is better. Comparison based on simultaneous constant Ate and At, is unnecessarily restricted, since the permissible At, + At, or At, + At, + At, is the important consideration. Comparison on this basis for a cosine distribution would be made by computing w in Eq. (81) by trial and error for each coolant, then substituting each to into Eq. (88) or For this or any other flux distribution the relative ranking of the fluids will be (89).

intermediate between that for constant Atc and for constant At,; thus either water or sodium or NaOH might be best, depending on the design. By Eq. (84), the efficiency of a coolant to dissipate 6.64 Thermal Circulation Atc. heat with low coolant temperature rise is inversely proportional to the group (n/ffp'c)^. If turbulent flow is attained, by Eq. (87) the pertinent group is (m'-V/Jp'c1-*)0-"7. These groups are also given in Table 14 and show in this respect the moderate superi However, taking At, into account, the liquid metals would generally ority of water. be best, and certainly such would be the case at the high temperatures that might be reached in emergency cooling by this method. 6.65 Considerations apart from those so far discussed General Evaluations. cannot readily be made quantitative. However, a number of qualitative evaluations of coolants for various reactor designs have necessarily been made. For instance, Parkins'06 concludes that liquid metals are preferable for (nonboiling) heterogeneous power reactors because of the high temperatures and low decomposition obtainable. Those recommended are Na, Na 4- K, Pb, and Pb + 2.5 per cent Mg. Shimazaki,10' from cost estimates for complete power plants using graphite-moderated solid-fuel reactors, finds Pb + 2.5 per cent Mg, Bi 4- 44.5 per cent Pb, Bi, Na, and HTS salt in that order of decreasing economy, though all are satisfactory. From the standpoint of current know-how, Na was believed best. 6.6

Comparison of Heat-transfer

Surfaces

Different shapes or arrangements of strip, plate, gauze, or porous fuel elements or other heat-transfer surfaces can be compared and evaluated in a manner similar to that employed for coolants. This is best done in terms of the dimensionless correla tions of fr and Nu against Re. Details are available elsewhere." 7

SPECIAL STEADY- AND UNSTEADY-STATE CALCULATION AND TEST METHODS 7.1

Analytical Unsteady-state

Thermal Analysis

The analytical solution of Eq. (5) has been extensively developed,0, for zero or constant heating density H, constant physical properties, simple shapes or initial

9-94

FLUID AND HEAT FLOW

[Sec.

9

temperature distribut ions, and /or constant coolant h and I. These idealized solutions However, they art do not closely fit most unsteady-state reactor thermal problems. Generally the frequently adapted to solve actual problems with adequate accuracy. maximum temperature of the solid vs. time (to estimate incipient mechanical failure), the maximum surface temperature vs. time (to estimate incipient coolant boiling;), and the average temperature of the solid (to estimate thermal stress or heat storage) are the results sought. The principal solutions with nuclear-power applications are as follows:* Uniform initial temperature ti. Surface or coolant temperature suddenly change? to ti. H = 0. a. Maximum, minimum, and average temperature of an infinite slab are obtainable as a function of time 6 from Gurney-Luric," Groeber,5 Hottell," or Bachniann,M charts for constant h. b. Same for an infinite cylinder. '.*MW 108 c. Same for a solid sphere. d. For an infinite bar, a parallelepiped, or a finite cylinder, the local or the over-all mean value of (I — ti)/(U — ti) is obtainable by multiplying together the values for the separate two or three dimensions. e. Temperature in a semi-infinite solid.108110 Surface temperature changes linearly with time. 2. Uniform initial temperature U. 1.

H

= 0.

in a semi-infinite solid."110 Temperature change At in an infinite slab at distance x from the heated face ran be computed by the "method of reflections"110 by summing terms At' for a semiinfinite solid:

a. Temperature

LU C.

=

Al', +

- A
d.

For

At'tR+I

+ A<',r_,

-

At'tK+*

-

■■

an infinite

slab

- «_,.). - ^ ia do

a solid infinite cylinder, from «m»,

e.

+

The maximum (at 6 = x>) temperature difference across [obtainable by Eq. (9), since dt/dd will be uniform): = («™„

(97'

Eq. (20):

- U„)«

=

^ 4a £ de

(98'

Cases c and d at shorter 0.111 shapes, case Id above applies.

/. For finite

An irregular variation of surface temperature with time can be fitted by » At each change in slope, the change can be broken line of straight segments. considered as a new dt/dS starting then. The temperature changes at any location due to all previous straight segments are algebraically added.110 H •* 0. 3. A definite q/A is forced into the body, initially at ti. = infinite slab of thickneri one face of an 6 starts to enter q/A 0; a constant o. At the other being adiabatic. Reference 112 gives tn»< and /„„,,;

is

Plots of most of these cases sre given in Ref. 98.

at

=

0

=

0

9

b.

from q/A Same as case o, but q/A increases linearly with decreases linearly with 9."* in contact at constant c. Same as case a, but the unhealed face maintained at
0

"«

mi. 4.

h

I

/.',

g.

*


6.

<•*

with a coolant


Sec. 9-3]

9-95

d. Same as case a, but the unheated face is in contact with a body of known heat capacity with k and k = ■»."< e. At 6 = 0 a constant q/A starts to enter the inner face of a concentric cylindrical

shell.1"

/. An irregular variation of q/A with time can be handled as under case 2g above. For instance, an unheated "equalizing period" is analyzed by continuing the initial heating, later starting an additional equal negative heating period, and adding the two temperature changes anywhere algebraically. For an infinite bar, a parallelepiped, or a finite cylinder, the total local or mean temperature change is obtainable by adding the changes computed separately for each dimension. The heating density throughout a body at It suddenly changes from 0 to a constant value H. Heisler"8 gives curves for the temperature distribution throughout infinite slabs, infinite solid cylinders, and solid spheres for the cases of constant sur face temperature and constant coolant temperature and h. The heating density H throughout the body varies with time. a. For an infinite slab of thickness It, initial temperature tu cooled face kept at U, and linear uniform heating H = a6. Letting a(ir/2R)% = B, g.

4.

5.


t = U

+

ncpB'

y Li \L

1 ~

(e--">°

»*

ii-i

+ n'flfl

tm„ is at x = R, and on integrating to obtain 2[1

-

cos

-

tmv, sin

~1 lit J

1) sin

nrx/2R

(99)

is replaced by

for/2)]

nx b.

Case a, except that H = 2H.0 rep

c.

Hoe69.

j ri -(-!)■ L,

n-l

L

n

- e-" sin tgl

^+ be

nlB6

2R

J

tmu and <„, are obtained as in case a. Same as case b, but cladding and stationary coolant layers without heat genera tion are added.1"

The above solutions cover many reactor problems.*

Where heating or temperature

rise is stated, cooling or temperature fall can be read. Also, although uniform initial temperature
Numerical Design Methods

These

methods subdivide the volume of each solid and fluid by a rectangular or a as fine as necessary for the desired accuracy, and assume that the mass of each subdivision is concentrated in a point, or "node."11* The thermal resistance between adjacent nodes is that between the sections of the subdivisions which pass through the nodes normal to the "bar" joining the nodes, namely, the distance between the nodes divided by k times the cross section (the average cross section if it varies). In the interior of a body, nodes are placed at the Along surfaces half-sized subdivisions are generally used, centers of the subdivisions. with the nodes at surface locations of known or desired temperature. The heat balance for each node of unknown temperature is then written in terms of the data

radial and circumferential grid,

* For example. Ref. 110 combines methods 2b and 2y to obtain the temperature distribution and maximum thermal stress in a plate subjected to a complex temperature cycle of a reactor coolant.

FLUID AND HKAT FLOW

9-96

[Sec.

9

and of the surrounding unknown temperatures, and the equations solved simul Suitable relations are taneously or consecutively for the unknown temperatures. given below for node 0 in a two-dimensional rectangular grid. If conditions are not uniform along z, it should also be subdivided. In all cases Ax is in the direction between the two nodes under consideration and Ay (and Az) normal to that direction. In a square (or cubical) lattice they are, of course, equal and interchangeable. 7.21 Heat conducted from adjacent node n at distance Ax is at a rate Conduction. Aq„.o

=

(<„

-

t„) Az

Ax„/k„ +

Ay 2

= K„,oAt„.o

^—

(101!

Axo/fco

7.22 Convection and Radiation. Heating rate by convection from fluid at 1/ passing a wall node is Aq/,a = (
ut'l^Z + Ax/2fc0 Convective h by Arts. 3.2 to 5.4 and radiation h are directly additive if the same applies. 7.23 Forced Surface Heating. Heating rate at a given q/A is

A

|7.24

Volume Heating.

Heating rate at a given Aqv.o

7.26

Transport.

=

H

Az Az Ay =

(103)

9'

H

is

H'

Heating rate by a fluid flowing into a subdivision is = (tr — t
Aqr.o

c't AIt

The outgoing stream is not of concern; thus this term changes on reversal of flow. 7.26 Accumulation. Heat accumulated by node 0 is at a rate A?0

-

tt'o

-

to)pc Az

Ax

^

=

A9

(104

A6

I

where Ato is the rise in temperature during time increment A9. The over-all balance of heat rates for node 0 is Ago

= Aq,.o + 2(A?„,o

+

Aqf,0

+

AqA,o

+

AqT,o)

(lOo^l

7.27 Transient Problems. In these problems all the temperatures and other Accord quantities in Eq. (105) are known, except t'o or A/0 in the accumulation term. ingly the equation can be written for each node in the problem, and each V at the end of Afi directly computed. The process is then repeated using the I' values in place of This "iteration" process is the I values, and t" is computed for each node at 2 AO. continued until the desired total time interval is covered. For convenience, Eq. (105) is converted to i'o

- fa'

+

Co

+ {l I

-

[^(2K„.„ Leo

+ K,,a

h +^2(Xfc.i.) +c'T)]\ -II Co ,

+ =

B +

Boto

+ 2B„<„ +

B,lf +

BTtT

A6 — Co

K/.o t/ +

AS , — c't tr Co

1106'

The B coefficients defined above can be computed for each node, and the calculation of Eq. (106) carried out on a digital computer. If temperatures or flow vary suffi ciently, the B values can be modified during the process.

Sec. 9-3]


9-97

The number of times that Eq. (106) must be employed to solve a given problem is Large increments favor speed and economy, and proportional to (AO Ax Ay Az)~l. With a high-speed digital computer available, a small increments favor accuracy. single computation with a relatively fine grid is preferable; with hand calculation the use of two or three coarser grids and extrapolation to zero node spacing is preferable. If AO is selected too large, it will be found that successive values of U, t'o, t"o, etc., To prevent this "instability," Bo should not be oscillate more and more widely. negative at any node; thus AO should not exceed

S/f „,o

e'. + K/,0

-+- c't


anywhere. The "truncation error" due to the finite size of the time increments can be estimated by repeating the calculation with smaller increments; for any node 0 it cannot exceed a maximum of fltoui(A''o — A<0)/2 AO, where (At'o — A/0) is the largest change in suc cessive increments.119 Equation (106) simplifies considerably under some conditions. In particular, for a node in a square subdivision, completely surrounded by two, four, or six identical equidistant nodes, K»,n equals k Az. If there is no heat convection or transport and the maximum stable AO (AO™,) is employed, where

Afl Aflmax

(One-dimensional)

(Two-dimen-

(Tbree-dimen-

(Ax)« = — > 2a

(Ax)' —

(Ax)» —

then Bo, Bt, and BT all equal 0.

sional)

4a

If H'

sional)

'

(10/)

Da

also equals 0, Eq. (106) yields

I'O =

«n).v

(108)

Thus, merely averaging the surrounding

node temperatures U gives the central one Equation (107) is thus the most frequently used AO, on account of speed. A somewhat more conservative procedure consists of averaging to in equally with each of the m values of t„. In this case A0msx later.

to

and

AO

-

+

2 <«

''--rir 3a

Oct

,

^

(109) (110)

lot

An unknown steady-slate temperature distribution or heat flow is obtainable with these numerical methods by carrying out the transient calculation at the given condi tions until no further significant change with time occurs. While this generally requires many calculations, it may be the best method if the transient problem has already been set up on a high-speed computer.

At

0

_

steady state,

t0 = t'o

q' + H' 2K,.o + K,,o + c't

and Eq. (106) may be written

y

Li

K,.ot,

K„.ot„

2Kn.o + K,.0 + c't

2K"„.0

+ K/.o + c't

c'rh SK„.„ + A'/.o + c't

(m)

One such equation is obtained for each unknown temperature. A direct method to obtain the steady-state temperatures is to solve all the equations simultaneously. Though seldom used, this method is at times practical with a high-speed computer.

A

more rapid solution is obtained by "relaxation,"

or balancing

(or even

over

9-98

FLUID AND HEAT FLOW

[Sec.

9


balancing) each t ime the most unbalanced node. The temperature unbalance of each node is recorded, as well as its temperature. When a node has been balanced by changing its temperature, the corresponding change in the balance or unbalance of each neighboring node is added to its previous unbalance. The node with the largest unbalance is always balanced next."8 Figure 10 shows the boundary fits obtainable for a specific case with coarse rectangular and cylindrical "square" grids. For maximum speed it is desirable to solve the problem first on a coarse grid. The temperatures thus found around each square are averaged [or inserted in Eq. (Ill)], yielding a fairly reliable temperature for the center of the square. A grid containing twice as many points is obtained, diagonally to the first grid. This finer grid of tem peratures is then improved by iteration or relaxation, and a still finer one drawn and improved as before, if desired."8 This procedure is faster than going at once to the finest grid, since it provides reliable estimates of the temperatures and fewer circuits of computation about the shape are needed. It is also more accurate, as additional

SQUARE GRID

Fia. 10. Square grids for thermal analysis of LMFR fuel element. F. T. Miles, and O. E. Dwyer. Nucleonics, July, 1954.)

(From C. William*.

values of the desired temperatures and heat flows are provided at coarser grid sizes, permitting a graphical extrapolation to zero grid spacing. Other variations of these methods are also employed, including heavier weighting of to, balancing of whole groups of nodes in the early iterations, radial and triangular grids of nodes, etc.98 7.3

Graphical Methods

For one-dimensional cases, such as thin or flat fuel elements or shields, Eq. (108) or (109) can be readily carried out graphically by the Binder or Schmidt method. Ji To obtain l\ by averaging ta and U as per Eq. (108), a straight line is drawn between If the ta and tc in Fig. 11a to intersect the position of node b, and AS = (Ax)s/"2a. surface temperature is held constant or varies in a known manner with time, one node would be placed at the surface. If the surface temperature depends on coolant tem perature t/ and coefficient h, or on known heat velocity q/A, it is more convenient to place the surface between an inner "actual" and an outer "imaginary" node, as in Fig. 116 and c. If heat generation occurs, t he additional temperature rise t'

-t

=

(Ax)'

Sec. 9-3]


9-99

is added to each heated node at each time increment A9. If, h, g/A, H, and even k at the surface may be varied as time or temperatures advance. If two materials A and

B (i.e., fuel and cladding) are adjacent, the ratio Axa/Axb may be set = s/olaIcib so If this gives inconvenient spacings, Ax as to maintain A8 by Eq. (107) the same. may be kept constant and the design completed by weighting to other than zero in one


of the materials and using Eq. (106) for that material." The solid or hollow cylinder and sphere with radial temperature and geometrical symmetry are also best treated as one-dimensional cases for either numerical or For the latter the same methods of Fig. 11 are employed, except graphical solution. that equally spaced nodes at r„ and r<, on the radius of a cylinder are spaced on the For a sphere the spacing is as working plot1'0 distances proportional to (r„ + n,)-1. (r„ + rk)-«. Comparable to Eqs. (109) and (110), smaller Afl than A0mal can also be used graphi cally (method of Nessi and Nisollc'). However, the added effort is probably better

(o>

Fio.

(b)

(c)

Details of the Binder or Schmidt graphical method for one-dimensional heat con duction in a slab, (o) Central region with H = 0; (b) surface with convection cooling; (c) surface with known heat velocity. {From C. F. Bonilla, Ref. 98.) 11.

In general, if graphical analysis is carried out carefully expended into decreasing Ax. and on a large scale, it can give satisfactory results for one-dimensional cases more rapidly than the other methods. 7.4

Homogeneous Electrical Analogues (Steady State)

As was done in Art. 3.12 of Sec. 9-2 for streamline flow, in heat conduction and convection design problems thermal resistance may be simulated by electrical resist Merits of the electrical ance and temperature and heat flow by voltage and current. analogue over calculation, or performing a thermal experiment, are greater speed and economy, particularly in steady-state problems or when various similar cases are to be covered. Two-dimensional steady-state problems with isothermal and insulated edges and one or more homogeneous thermal conductors can be solved most conveniently with Teledeltos paper,* though soft metal foil, brass shim stock, trays of conducting solu tion, and semiconducting plastic sheeting have been used. Teledeltos grades available are L (approximately 1,000 to 2,000 ohms per square) and H (approximately 10,000 to 20,000 ohms per square). Crosswise resistance is some 12 per cent higher than lengthwise resistance along the rolled sheet; this can be approximately corrected for by a corresponding difference in t he x and y scales. The resistance R may be increased almost isotropically up to fivefold by punching out holes on a reasonably fine grid. * Available from the Western Union Telegraph Company.

9-100

FLUID AND HEAT FLOW

Equally spaced circles spacing)

[Sec. 9

(D/S

is the ratio of diameter to center

1

1.25

1.5

2

3

4

5

as follows are required

:

D/S

(90° grid)

0

0.39

0.51

0.65

0.79

0.86

0.90

1

D/S

(60° grid)

0

0.34

0.48

0 62

0.76

0.82

0.86

1

For conduction problems the shape is cut to

scale out of the paper, allowing an extra Du Pont 4922 silver in. at the isothermal edges to connect a good conductor. lacquer is generally used; it has a lengthwise resistance of only several ohms per foot for a thin J-^-in. strip and also negligible contact resistance (a fine copper wire can be Clean brass bars clamped incorporated to minimize the lengthwise resistance). against the paper, which is backed with rubber, are more convenient for making changes; contact resistance is about % in. of the paper, negligible in large analogue 14

sizes.

0.06" K -12

r >L« 9.6-^6"-*) — 1— -.vwifA-Q -• • • • ' * WfA^-O• • * •—

*_^vw-0

r"


—

SECTION

FOR FIG.b,c B I (a)

Fig. 12. Electrical analogues for two-dimensional steady-state heat conduction and con vection study of a long thin-strip stainless-clad fuel element, (a) Actual size and thermal data; (b) possible homogeneous analogue arrangement [central dots apply to (c)]; (c) possible lumped analogue arrangement [nodes are at dots in (fc), and Ax = Ay]. (From C. F. Bonilla, Rcf. 98.) Coolant film resistances are reprcsentable by additional paper slit into strips normal to the surface (strips of finite width, of course, depart from the true homogeneous To keep solid and film resistances in proportion, the strip length L analogue). is set equal to Sk/h, where S is the geometrical scale ratio of the analogue. For instance, in Fig. 12, an analogue for a very thin strip of U"6 with stainless cladding, L = (6/0.03) (12/3,000) ft = 9.6 in. If L is unreasonably long, additional resistors may be connected to each strip or some of the strips may be cut off at the base and the length of the rest shortened to give the same total resistance of the strips. Strips should be curled with string or small rubber bands to prevent contact at their edges. If different sheets of paper are attached together, a narrow strip of silver lacquer is If the line of glue is nonisothermal, it should be frequently cut across used as glue. with a razor to prevent conduction along it. Uniformly Unsteady-state problems require electrical capacity (see Art. 7.5). distributed capacity could in principle be obtained with a large grounded plate covered with thin dielectric and a conducting sheet on top; actually uniformity would be poor, and the size would be very large or time very short. Large capacitors can be attached at small points, but the electrical connection resistance is high; large electrodes decrease the resistance of the main sheet. Thus "lumped" or "network" or "passivenetwork" analogues (Art. 7.5) are preferable for unsteady state. Here connection Heat generation also requires electrodes as current lead-ins. resistance is not important, though points at which temperature (voltage) is sought should be roughly equidistant from two or more lead-ins rather than near one. In

Sec. 9-3]


9-101

Fig. 126 the fuel has been subdivided by short strips. Each rheostat r is adjusted until the currents i are distributed in proportion to the given heat-generation distribu tion (uniform for a small uniform strip fuel element such as this). If the resistances r are relatively high, the settings will not need trial-and-error readjustments to match Equal currents without repeated adjustment can also be the heat generation. obtained by using a constant-current ballast tube at rated current or a "beam-power" (6SJ7 or 6V6) vacuum tube, which is grid-controllable over a 10-fold range of current, roughly.

Points A and C

are the hottest surface and interior points, and B is the coldest. obtain the temperature rise A* of any such points above the coolant temperature, the voltage difference AE above the end of the strips is read by a potentiometer or Dividing the electrical (Ohm's) law into the thermal law, high-resistance voltmeter. for any analogous locations,

To


*L AE

= «

El

IB.

= 1

I

_L

kbR',

Q12)' y

where R'. is the ohms of a square of the paper used and 6 is the thickness (length of strip fuel element) of the section in Fig. 126 generating heat q. If the thickness of the Tjias were significant, a larger analogue could be made with the fuel region of the proper resistance per square and a two-dimensional array of current inputs over it. For two- and particularly three-dimensional problems, acidified strong CuS04 solu tion can be employed. The container has the desired shape to scale, and isothermal Use of alternating current avoids edges or surfaces are sheet-copper electrodes. Solid electrolytes that can be cut and melted, such as gelatin electrode polarization. in salted glycerin water, are also used. Fluid film resistances are more difficult to apply with these materials. 7.5

Lumped Component Electrical

Analogues (Steady and Unsteady State)

Replacing the conducting paper of region ABCD in Fig. 12 by a network of resistors is a further departure from the homogeneous thermal problem. However, this is outweighed by practical advantages in many cases, such as greater speed of setup if a resistor board is available, greater accuracy of resistors than sheets, and greater Portions of the problem are usually convenience for unsteady-state problems. "lumped" in any case, as in Fig. 126, and added "lumping" is not detrimental unless fine local temperature distribution in this region is desired. A resistor board for general use may have clips to hold cartridge resistors or a linear Up to three decades are or tapered rheostat permanently at each resistor position. also available in single plug-in resistor assemblies (Goodyear). Many nodes are For single problems provided, only the desired ones for any problem being connected. the network can be put together more quickly and economically by soldering the Good linear wire-wound proper resistors together and mounting them on plywood. In rheostats (General Radio) are everywhere within 1 per cent of their scale value. fixed resistors the lathe-cut carbon-film type (Aerovox) gives a good combination of cost, stability, and uniformity ( ± 1 per cent). In setting up a lumped-component analogue with constant fluid temperatures (i.e., small changes in stream temperatures or for a cross section normal to the stream or streams) the general methods of Art. 7.2, illustrated in Fig. 12c, are employed for the For convenience in obtaining surface temperatures, nodes are usually resistances. As many of the nodes as possible inside the conduction located along all surfaces. region are usually placed on a square grid, so the same size resistor can be used between them. If each central resistor in the square grid in Fig. 12c is arbitrarily selected as 100 ohms, each edge resistor, between surface nodes, is 200 ohms. To maintain the

proportion

(El)

m

(Eli)

=

(El)

FLUID AND HEAT FLOW

9-102

[SfiC. 9

each full surface film resistor equals R'.k/h Ax, or

X

100

X

= 640 ohms

(0.0075/12)

At corners B and D, 1,280-ohm resistors are used, except that the two in this case. Equation (112) gives any at B can evidently be replaced by one 640-ohm resistor. desired At, as before. For unsteady-state analyses, a condenser* is attached to each node with an electrical capacity proportional to the thermal capacity represented by the node. Thus, the capacitor for a central node would be proportional to bcp Ax Ay, or bcp(Ax)* with a A side node has half this capacity, and a corner node (B and D) onesquare grid. The other terminal of each capacitor is connected to the reference voltage, quarter. The dielectric wire F in this case. Capacitors up to 20 id are frequently employed. must be good to keep self-discharge low over the time of operation; leakage time con stants RC of 20,000 and 100,000 ohm-farads (seconds) at 25°C —still higher at lower Mylar dielectric (Good-All) yields even temperatures — are available (i.e., Sprague). higher RC. The scale of resistances and capacitors may be independently selected, but they then control the ratio of thermal time to equivalent time in the electrical analogue. The time-scale ratio is obtainable by dividing the heat-conduction equation (Eq. (5) with H and the parenthesis = 0] by the comparable relation dE/d.8 = VtE/C",R"„ The product C",R", can where C", and R", are the farads and ohms per unit length. be computed for any cross section (such as one transverse increment of the grid), They can also be C", and R'"e for one length-wise because C. and R. vary inversely. increment of the grid if the distance variables in V*E are changed to grid steps. Thus

Sfi

^

(Ax)'

(113)

C'",(R'".) 8,

is

t

is

is

8,

8,,

If can be calculated. impractically short For an estimated desired range of The capability or long, Ax, C'"„ and/or R'", may be altered as desired or convenient. proportional to the number n of such an unsteady-state analogue to subdivide space Its capability to subdivide time is, for of the largest capacitors (C""e)m„x available. the practical purposes, proportional to n(C"',)m*x(R'",)m*x/(8t)miu, where (R'"c)m*x highest resistor size available allowing sufficient sensitivity and (8c)mlB ls 'he minimum reliable response time of the electrical recorder, or indicator plus observer, Study of a transient in heat generation or in coolant temperature may be illustrated Considered as a slab, by the case of Fig. 12. Jfc

hR —

=

3000 12

X X

0.03 12

_ „oe = 0.625

8t

-

r

8,

it

8.

a

From a chart for case la, Art. 7.1, h'8/kcp = 0.010 for 90 per cent of the surface tem perature change uncompleted and 1.7 for 10 per cent of the midplane change uncom Taking C". as of 0.0036 to 0.60 sec. maximum reasonable range of pleted, or While this range would be 0.0082 to 1.37 sec. 20 tiF and R'", as 100 ohms yields would be simpler to increase R'", to say 10* to feasible with fast instrumentation, Rheostats 10» ohms, and Voltage could be increased as desired. proportionally. Indi might need adjustment during the experiment to keep the currents constant. A

is

*

purely resistive network111 can instead be used and step-wise subdivision of time, but the solution more time-consuming. This process amounts to solving Eq. (106) simultaneously for all nodes for one A0, then repeating as many times as necessary. In large complicated shapes the temperature may be desired at many points, for thermal-stress calculations. This may be accomplished with a reasonable number of condensers by first setting nodes over the whole object up to the available number of condensers and finding their temperature history. Then subdivisions are set up in turn with finer grids, and the boundary temperatures made to vary as

t


3,000

12

previously found.

Sec. 9-3]


eating or recording instruments currents appreciably.

9-103

should be of high enough resistance not to alter the

Lumped-component electrical analogue circuits can also be devised for steady-state transient analysis of complete systems, such as a reactor, pump, heat Such methods have not yet been developed in exchanger, turbine, and condenser. general terms. and even

7.6

Mass Transfer and Mass-transfer

Analogues

Mass transfer is the passage of molecules of a dissolved or vaporized constituent one phase to another, by diffusion and convection in the direction toward Normally the phases are insoluble or only slightly soluble in each other, equilibrium. and the substance being transferred may be the only, the major, or a minor con Nuclear applications include the dissolving of fuel or stituent of one of the phases. fertile elements by liquid-fuel or breeder-blanket solutions; the removal of converted fertile elements, residual fuel, or fission products by evaporation, sparging, or extrac tion of their solution by a second liquid or a solid; the dissolving or differential thermal In fact, a principal advantage of liquidsolubility corrosion of pipes and vessels; etc. fuel and homogeneous reactors is that by mass transfer the fission-product concentra tion can be kept low for safety and neutron economy and the fuel, fertile, and converted elements can be maintained at their most economical concentrations. Most nuclear mass-transfer processes employ at least one dilute solution. This makes mass transfer analogous to heat transfer with constant properties, since the molecular and eddy diffusion in the laminar boundary layer and core are comparable to the corresponding conductions and the physical properties of the dilute phase or phases (the main resistance to mass transfer) are substantially constant at the values for the pure phase or phases. Theoretical approaches can be based on velocity dis tribution such as Eqs. (31) to (33) of Sec. 9-2, Eq. (35) of Sec. 9-2 to yield t, and finally Agreement is good at the same Re between the integration of Eq. (37) of Sec. 9-2. analogous heat- and mass-transfer expressions over the whole practical range of Deissler has obtained theoretically, for Schmidt number Sc, n/Dvp, or 0.5 to 3,000. flow in a smooth pipe with Sc above about 50,


from

St„

= 0.079/fm Pr~«

St.v

= 0.079/fw Sc"?<

(114)

which is adequate for diffusion-controlled thermal corrosion of pipes by most liquid metals and fused salts. Also, Chilton and Colburn's empirical analysis has shown that jo and jn [see Table 3, Sec. 9-2, and Eq. (116)1 are roughly equal for any given

shape and Re value.124 Thus both analyses indicate that Tables 3 and 8 for nonmetals would hold for mass transfer in metals or nonmetals on replacing Nu by Sh (Sherwood number F'/u; see below) and Pr by Sc. At Re above about 10,000, jh is also roughly Thus the average mass-transfer coefficient F' can be estimated from equal to Jfj/V. pressure-drop as well as from heat-transfer data. Laminar mass transfer is evidently analogous to laminar heat transfer. Tables 5 Also freeto 7 can be applied by further substituting Dvp for k/c in Gz and Gz'. convection Table 9 applies; 0 At should be computed1" as Ap/p. In all cases, the total rate of mass transfer u>u is obtained by wu =

where A

(Ar)„

F 'A(Ac)„

=

F~

(Ap)

= surface area through which mass transfer occurs = average concentration difference causing mass transfer, mass

(115)

per unit volume F' = film mass-transfer coefficient, velocity units, i.e., lb/(sec) (ft1) (lb/ft ') = fps For a volatile solute, partial pressure p is more convenient. If one phase flows parallel or countercurrent to the other or has a constant concentration at its surface,

FLUID AND HEAT PLOW

9-104

[SEC.

9

=

F-,

j

(Sc')*

A (Pr)*

=

(116)

u'

~CP

\Pr

a' D

\Pr

/

=

/

F'

(1

The mass-transfer quantities in Eq. 16) are primed, and the heat-transfer quantities not primed. For air and naphthalene Sc' = 2.5. Expressing 7" in degrees Rankine, evaporation rate w' in mils (0.001 in.) of naphthalene per hour, and naph thalene partial pressure driving force Ap' in millimeters of Hg, F' = 0.0257 Aw'T'/Hp' on inlet -stream concentration (or temper Using fresh air and basing F' (and fph. the vapor tension p', of naphthalene in millimeters, where ature), Ap' is

h)

are

logp'.

- 11.450 -^|^

(117)

is

h

is

is

If true local F' and are desired, Ap' set equal to p', — p't, where p't, the local average partial pressure in the stream as calculated from upstream total vaporization given by the ideal-gas law as rate. p'& in millimeters 553.6WT' Q'M' where

W Q'

M'

= upstream rate of evaporation, = gas flow rate, cfs

lb/sec

— molecular weight (128.2 for naphthalene) 7" = temperature, °R

7.7

Similitude

Tests

For new and complicated shapes or assemblies of fuel elements, coolant channels, coefficients and temperature and flow distributions can be approxi For best results, mately estimated by methods in this and the previous chapter. etc., heat-transfer

driving forces. is

* These mass- and beat-transfer coefficients must be based on analogous ence between the surface and the approach concentrations or temperatures the surface to local-mean driving force common. is


In other cases the mean Ac integrated over the area the log mean Ac would apply. would be used. Because of the similarity in the correlations for heat and mass transfer, the latter In particular, heatcan be utilized as an analogue to yield information on the former. transfer coefficients for fuel elements or other objects of special shape can be obtained by casting or machining the object of a material that will vaporize or dissolve slowly. The object, with all pertinent surroundings to scale, is placed in a nonrecirculating stream of an appropriate fluid at a temperature yielding a suitable mass-transfer rate and at a velocity yielding the desired Re. The average coefficient* h for the test fluid or, more importantly, for the reactor fluid, if it is a different fluid, can be computed from the loss in weight of the object Even more simply, the ratio or from the gain in concentration in the mixed stream. between the local h (based on approach temperature) at different positions, or between h and h, is the ratio of the rate of vaporization or solution at those positions as meas For gases at 0 to about 50°C naphthalene is reliable and con ured mechanically. venient;' such a study was made of the STR fuel-element assembly.1" Camphor, For cold water as the acetamide, and other solids that sublime are also suitable.117 Heat-transfer fluid, 0-naphthol, benzoic acid, and cinnamic acid are useful."4 information for liquid metals is not obtainable in this way, since the electronic con tribution is absent in mass transfer and Sc values are not possible as low as the Pr values of liquid metals. The Chilton-Colburn relation yields

The differ but

the most convenient,

Sec. 9-3]

heat hemoval prom nuclear reactors

9-105

full-scale tests at working conditions are indicated. However, in many cases this would be quite difficult, expensive, and/or time-consuming. But fluid-flow or heattransfer characteristics for any given shape have regularly been expressed as relations Thus, if it is possible in a test to among a certain number of dimensionless ratios. control all but one of the dimensionless ratios to the working-condition values, the remaining ratio in the test will also have the same value as it would at working condi The desired ratio can be computed from the test data, and the unknown tions. quantity therein computed for the working conditions. The deviations from working conditions that are of interest are size, velocity, and The following cases are most fluid properties (different fluid and/or temperature). Arbitrarily, the primed quantities will refer to the test and unprimed encountered. D is any characteristic length. to the working conditions. Flow Distribution, Type of Flow, Pressure Drop. In a scale model at the 7.71 same Re as the working unit, the values of N/D, /f, cd, and any other such size and flow dimensionless ratios are the same as in the working unit. Then from Eq. (60), Sec. 9-2, the ratio AP/AP' for streamline or turbulent flow in a pipe, and also for flow Since the ratio AP/AP' is the through an orifice, is obtained as (w /u)')*(p' / p)(D' I D)*. same for all such standard cases, it is necessary and sufficient also for complex shapes to keep Re the same in the scale model as in the working unit to obtain similar flows.

Re-^'-^ Generated for wjivans (University of Florida) on 2015-09-23 02:54 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

p

£-!£ Dp

p

D

p

(118)

AP can

then be computed from AP' by the above ratio. Alternatively, the/ versus Re curve found with one liquid holds for all other liquids. For example, the pressure vs. sodium flow rate curve of an SIR fuel-element assembly was obtained as versus Re much more conveniently with water than with A full-scale model was used, and (as a precaution, though not apparently sodium. The neces required) actual velocity u by employing hot water so that p'/p' = pip. sary water temperatures for sodium at 600 and 900°F are 171.5 and 211°F, respec Similarly, flow distribution in the STR core at high Re was studied with a tively. full-scale model and air (instead of hot water). Also, a 1.5 scale model of the PWR thermal shield '" was tested with air.* 7.72 Turbulent h. In general, to obtain the correct Nu or St under any conditions, the problem is to obtain in the test the same Re and Pr as in the working unit. How ever, for nonmetallic fluids in turbulent flow it is experimentally well established (e.g., Table 3, item a) that ja
/

-

Thus only Re

-

need be kept the same [Eq. (118)],

V

\c>)

in which case

u'p'Kph')

(119)

For instance, average or local h may be desired over a fuel element too small for the installation of thermocouples and heating elements; in that case, a model several times larger than the working unit can be tested. A large model may also be used to permit Most frequently tests at lower tem testing at lower than actual working velocity. peratures or with more convenient fluids are desired; the product D'u' is set at Dupp'/p'p. With metallic fluids, Table 4 indicates that it is adequate to keep Pe the same for Nu to be the same in the test as in actual operation. Because of the low values of Pr for liquid metals, no other fluids can replace them in similitude tests, however.

*A one-fourth-size model of the PWR core yielded useful flow-distribution data with air,121 even though only one-sixty-seventh of the PWR Re was obtained. However, for accurate results the same This is difficult to achieve in a full-scale or smaller model because of the high pressures Re is needed. For instance. CO: at 1,200 p.si has been used (to test the LPK and /or velocities and powers required. Much denser gases, such as Freon C-318 (C*Fs, M' ■»200). UF« (Mf = 352), or very core design). cold air, might perhaps be employed at atmospheric pressure.

FLUID AND HEAT FLOW

9-106 7.73

Streamline

if Gz is the

same.

[Sec. 9

h. Tabic 5 shows that Nu in simple streamline flow is the Since in a scale model L' ID' is the same as LID,

cuoD

-it

c'u'p'D'

¥~

,

and

h

K'

=

„__,

D' k

(120)

Dk-

*t.

7.74 Conduction and Convection. If conduction is to be simulated as well as turbulent convection, it is necessary for h and k,/D to be proportional in the test and the working conditions, where k, is for the solid or every one of several solids. Thus, besides Eq. (118) there is the requirement that Nu, = h'D'/k', = hD/k,. As before, Eq. (119) will hold for nonmctals. Eliminating u/u' and h/h' among the three equations,

-

is

^L_qW At'

(122)

q'k,D

a

a

7.76 Forced Plus Free Convection of Nonmetals. In forced-convection film coefficient, free convection may be significant, as for fuel element during low flow or pump failure (see Table 10). Setting the Gr X Pr products equal and solving with Eqs. (118), (119), and (122) gives the requirement for q':

(118), and any At can be obtained from the corresponding

5

D' and u' must satisfy Eq.

-J(S" <&)"($)" is

is

6'

0

is

if

7.76 Unsteady State. If steady-state conditions arc properly simulated for the ta,/D* also simulated cooling of a solid as above described, the unsteady state the same. The unsteady-state fractional temperature change at thus the same >= ea.D',/a',Dt. as that in the test at

REFERENCES 1. 2.

Lane,

J.

A.: Proc. Univ. Reactors Con/., Oak Ridge Institute for Nuclear Science, 1954. 1954; Hilsen-

NBS-NACA Tables, National Bureau of Standards, Washington, D.C., rath, J., and Y. S. Touloukian, Trans. ASME, 76: 967-985 (1954).

Bromley, L. A.: Unit. Calif. Radn. Lab. Rept. UCRL-1862. Lindsay, A. L., and L. A. Bromley: Ind. Eng. Chem.. 41: 1508-1511 (1950). Sons, Inc., New York, 1949. 5. Jakob, M.: "Heat Transfer," vol. John Wiley 1952. Georgia Institute of Technology, 6. Orr, C: Ph.D. dissertation, 7. Deissler, R. G.. and C. S. Eian: NACA, RME52C05, 1952, and RME53G03. 1953. 8. Hougen, J. O., and E. L. Piret: Chem. Eng. Frogr., 47 295-304 (1951). 3.

:

&

4.

1,


A

is

it

Thus to simulate conduction and convection in a scale model of an irregular fuel element or assembly first necessary to pick the solid, fluid, and test temperature convenient procedure to plot the left-hand side range such that Eq. (121) holds. of Eq. (121) against temperature for each solid under consideration and the right-hand Every solid-fluid intersection yields a feasible pair of materials side for each fluid. and test temperature. When the test temperature and materials have been selected, any D' and u' can be used that obey Eq. (118). Actual temperature differences At from fluid to solid at any position are obtained from the corresponding At' by

Sec. 9-3]


9-107

Perry, J. H. (ed.)". "Chemical Engineers' Handbook," 3d ed., McGraw-Hill Book Company, Inc., New York, 1950. 10. Aladev, I. T.: Bull. USSR Acad. Sci., Tech. Sci. Div., 11: 1669 (1951). 11. Monrad, C. C, and J. F. Pelton: Trans. Am. Inst. Chem. Engrs., 38: 593 (1942). 12. Stein, R. P., and W. Begell: Jour. Am. Inst. Chem. Engrs., 4 (March, 1958). H., and S. Stemerding: "General Discussion on Heat Transfer," pp. 20113. Verschoor, 203, Institute of Mechanical Engineers, London, 1951. 14. Deissler, R. G.: ASM E Paper 53-S-41, 1953. 15. Dingee, D. A., et al.: Battelle Memorial Inst. Rept. 1026, 1955. 16. Miller, P., et al.: Nuclear Eng. Sci. Congr. Preprint 25, Cleveland. 1955. 17. Jackson, C. B.: "Liquid Metals Handbook," 3d ed., Government Printing Office, 1955. 18. Lubarsky, B., and S. J. Kaufman: NACA TN 3336, Washington, D.C., 1955. H., and W. B. Harrison: Chem. Eng. Progr. Symposium Ser., 9: 93 (1954). 19. Poppendiek, 20. U.S. Atomic Energy Commission: "Reactor Handbook," vol. 2, pp. 277-286, by R. D. Brooks and S. K. Freidlandcr, McGraw-Hill Book Company, Inc., New York, 9.

1955.

Sleicher, C. A., Jr., and M. Tribus: Proc. Heat Transfer of Fluid Mech. Inst., 1956, pp. 59-78. 22. Stein, R. P.: ABC Reactor Heat Transfer Conference, New York, Nov. 1, 1956. 23. Norris, R. H., and D. D. Streid: Trans. ASME, 63: 525 (1940). 24. Kays, W. M.: ASME Paper 54-A-151, 1954. 25. Deissler, R. G.: NACA, TN-2410, 1951. 26. Rickard, C. K., et al.: AEC Reactor Heat Transfer Conference, New York, Nov. 1, 1956. 27. MacDonald, W. C, and R. C. Quittcnton: Chem. Eng. Progr. Symposium Ser., 9: 59-67 (1954). 28. Douglas, W. J. M., and S. W. Churchill: Chem. Eng. Progr. Symposium Ser. Preprint 21.


16,

Louisville Meeting,

1955.

29. McAdams, W. H.: "Heat Transmission," 3d ed., McGraw-Hill

New York, 1954. 30. Dwyer, O. E., et al.: Proc. Berkeley,

31. 32. 33. 34. 35. 36. 37. 38. 39.

40. 41. 42. 43.

44. 45. 46.

1953 Conf. on Nuclear

Book Company, Inc.,

Eng., p. F-73, University of California,

1953.

McGoff, M. J., and J. W. Mausteller: MSA Memo Rept. 87, 1955. Tidball, R. A.: Chem. Eng. Progr. Symposium Ser., 5, 1953. Norris. R. H., and W. A. Spofford: Trans. ASME, 64: 489-496 (1942). Kays, W. M., et al.: "Gas Turbine Plant Heat Exchangers," American Society of Mechanical Engineers, 1951. Wilke, C. R., and O. A. Hougen: Trans. Am. Inst. Chem. Engrs., 41: 445-451 (1945). Leva, M.: Ind. Eng. Chem., 39: 857 (1947). Donohue, D. A.: Ind. Eng. Chem., 41: 2499-2511 (1949). Eckert, E. R. G.: "Introduction to the Transfer of Heat and Mass," McGraw-Hill Book Company, Inc., New York, 1950. U.S. Atomic Energy Commission: "Reactor Handbook," vol. 2, chap. 1, sec. 3.7, by A. F. Lietzke, T. M. Hallman, and E. M. Sparrow, McGraw-Hill Book Company, Inc., New York, 2d ed., 1958. Kyte, J. R., et al.: Chem. Eng. Progr., 49: 653 (1953). Moscicki, I., and J. Broder: Roczniki Chem., 6: 321 (1926). Hyman, S. C, C. F. Bonilla, and S. W. Ehrlich: NYO-564, 1951. Isakoff, L., and F. W. DeVries: M.S. thesis in chemical engineering, Columbia Univer sity, New York, 1951. Eckert, R. G., and A. J. Diaguila: Trans. ASME, 76 : 497-504 (1954). Pigford, R. L.: Chem. Eng. Progr. Symposium Ser., 17: 79-92 (1955). Sanders, A.: M.S. thesis in chemical engineering, Columbia University, New York, 1954.

47. Misra, B.: Ph.D. thesis in chemical engineering, Columbia University, 1955; Chem. Eng. Progr. Symposium Ser., 18: 7-21 (1956). 48. Carpenter, F. G., and A. P. Colburn: "General Discussion on Heat Transfer," pp. 20-26, Institution of Mechanical Engineers, London, 1951. 49. Seban, R. A.: Trans. ASME, 76: 299 303 (1954). 50. Rohsenow, W. M., et al.: ASME Papers 54-A-144 and 54-A-145, 1954. 51. Untermyer, S.: Nucleonics, 12(7): 43-47 (July, 1954). 52. Nakagawa, Y., and T. Yoshida: Chem. Eng. (Japan), 16: 74-82, 104-110 (1952). 53. Paschkis, V., and G. Stolz: Quenching Program, Columbia University, Mechanical Engineering Department, Mar. 1, 1955. 54. Ellion, M. E.: Memo 20-88, Jet Propulsion Laboratory, California Institute of Technology. 1954.

9-108 55. 56. 57. 58. 59. 60. 61.

FLUID AND HEAT FLOW

[Sec. 9

Pike, F. R., P. D. Miller, Jr., and K. O. Beatty, Jr.: Chem. Eng. Progr. Symposium Ser., 17: 13-19 (1955). Bonilla, C. F., and C. W. Perry: Trans. Am. Inst. Chem. Engrs., 37: 685-705 (1941). Kutateladze, S. S.: Oldelenie Tekknicheskikh Nauk, 4: 529-536 (1951). Cichelli, M. T., and C. F. Bonilla: Trans. Am. Inst. Chem. Engrs.. 41: 755-787 (1945). Robinson, D. B., and D. L. Katas: Chem. Eng. Progr., 47: 317-324 (1951). Styuschchin, N. G., and L. S. Sterman: Tech. Phys. (USSR), 31 : 448 (1951). Grady, J. J.: M.S. thesis in chemical engineering, Columbia University, New York,

J.

1950.

W. H.: Trans. ASMS, 74: 969-975 (1952). Hayworth, H. C., and N. Nicholaus: M.S. thesis in chemical engineering, Columbia University, New York, 1951. 64. Pramuk, F. S., and J. W. Westwater: Chem. Eng. Progr. Symposium Ser., 18: 79-83 62.

Rohsenow,

63.

65. 66. 67. 68. 69.

70. 71. 72. 73.


74.

(1956). Rohsenow, W., and P. Griffith: Chem. Eng. Progr. Symposium Ser., 18 : 47-49 (1956). Gessler, J. L.: M.S. thesis in chemical engineering, Johns Hopkins University, Balti more, 1948. Bromley, L. A.: Chem. Eng. Progr., 46: 221-227 (1950). Banchero, J. T., et at.: Chem. Eng. Progr. Symposium Ser., 17: 21 (1955). Lottes, P. A.: Proc. 1955 Con}. Nuclear Eng., p. A-l, University of California, Los Angeles. Bailev, R. V.: at Tulane University. Cook, W. H.: ANL. 1954. Armaeost, W. H., et al.: Trans. ASME, 76 : 715-748 (1954). Lyon, R. E., et al.: Chem. Eng. Progr. Symposium Ser., 17: 41-47 (1955).

Bonilla, C. F.,

et

al.: NYO-7638,

1956.

75. Piret, E. L., and H. S. Isbin: Chem. Eng. Progr., 50: 305-311 (1954). 76. Goldmann, K.: NDA Rept. 10-68, 1953. M. L., el al.: Report 54-77, Department of Engineering, University of 77. Greenfield, California, Los Angeles, 1954. 78. McAdams, W. H„ et al.: Ind. Eng. Chem., 41: 1945 (1949). 79. Clark, J. A., and W. M. Rohsenow: Trans. ASME, 76: 553 (1954). 80. Jens, W. H., and P. A. Lottes: ANL-4915, 1952. 81. Buchberg, H., et al.: Proc. Heat Transfer and Fluid Mechanics Inst., pp. 177-191.

Stanford University, Stanford, Calif.,

1951.

82. Gunther, F. C: Trans. ASME, 73: 115 (1951). 83. Bernath, L.: Du Pont Memo DPW-54-526, 1954; Chem. Eng. Progr. Symposium Ser.. 18: 1-6 (1956). 84. Weatherhead, R. T.: ANL, 1955. Institute of 85. Jeffery, R. W.: B.S. thesis in mechanical engineering, Massachusetts Technology, Cambridge, Mass., 1952. 86. Motte, E. I., and L. A. Bromley: UCRL-2511, 1954. 87. Carter, J. C: ANL-4766, 1952. 88. Beighley, C. M., and L. E. Dean: Jet Propulsion, Mav-June, 1954, pp. 180-186.

Mumm, J. F.: ANL, 1954. Jens, W. H.: Mech. Eng., 76: 981-986 (1954). 91. Roy, G. M., and C. A. Pursel: General Electric Company, 1954. 92. Armacost, W. H., et al.: Trans. ASME, 76: 715-748 (1954). 93. Nerad, A. J., and A. Howard: unpublished reports, General Electric Company, 1936. 94. Bromley, L. A., et al.: UCRL-1894, 1952. 95. Martin, W., et al.: AECU-2169, 1952. 96. Rosenthal, M. W., and R. L. Miller: ORNL-2294 (1957). 97. LeTourneau, B. W„ and R. E. Grimble: Nuclear Set. Eng., 1 : 359-369 (1956). Book Company, Inc., 98. Bonilla, C. F. (ed.): "Nuclear Engineering," McGraw-Hill New York, 1957. 99. Poppendiek, H. F., and L. D. Palmer: ORNL-1395, 1952; Chem. Eng. Progr. Sympo 89. 90.

100. 101. 102. 103. 104. 105.

sium Scr., No. 11, 1954. Poppendiek, H. F., nnd L. D. Palmer: ORNL-1701, 19.54. Hamilton, D. C, et al.: ORNL-1769, 1954. Carberry, J. J.: Chem. Eng., 60: 225-227 (June, 1953). Goodman, C. et al.: "The Science and Engineering of Nuclear Power," vol. 2, pp. 120-176, Addison-Wesley Publishing Company, Reading. Mass., 1949. Bonilla, C. F.: AEC Rept. M-4476, 1949. Buckley, P. S.: Chem. Eng., 57: 112-114 (September, 1950).

Sec. 9-3]


E.: F55-71, Proc. 1953 Con/, on Nuclear Eng., University of California, Berkeley, 1953. Shimazaki, T. T.: Chem. Eng. Progr. Symposium Ser., IS: 113-119 (1954). Carslaw, H. S., and J. C Jaeger: "Conduction of Heat in Solids," Oxford University Press, New York, 1947. Bnchmann, H.: "Tafeln iiber Abkuhlungsvorgange einfacher Korper," SpringerVerlag 01IG, Berlin, 193S. Fritz, R. J.: Trans. ASME, 76: 913-921 (1954). Russell, T. F.: First Report, pp. 149-187, Alloy Steels Research Committee, England. Newman, A. B.: Trans. Am. Inst. Chem. Engrs.. 30: 598-612 (1934). Newman, A. B., and L. Green: Trans. Electrochem. Soc., 66: 345 358 (1934). Beatty, K. O., et al.: Ind. Eng. Chem., 43 : 1527-1532 (1950). Rubin, T.: AEC Rept. Y-F-10-28, 1950. Heisler, M. P.: Trans. ASME, 78: 1187-1192 (1956). U.S. Atomic Energy Commission: "Reactor Handbook," vol. 2, Engineering, AECD3646, McGraw-Hill Book Company, Inc., New York, 1955. Dusinberre, G. M.: "Numerical Analysis of Heat Flow," McGraw-Hill Book Com pany, Inc., New York, 1949. Hellman, S. K., et al.: Trans. ASME, 78: 1155-1161 (1956). Patton, T. C: Ind. Eng. Chem., 36: 990 (1944). Liebmann, G.: Trans. ASME, 78: 655-665 (1956). Hazard, H. R., and J. M. Allen: BMI-1141, 1956. Deissler, R. G.: NACA TN-3145, 1954. Sherwood, T. K., and R. L. Pigford: "Absorption and Extraction," McGraw-Hill Book Company, Inc., New York, 1952. Bonilla, C. F.: Paper 122, vol. 9, p. 331, UN Conference on Atomic Energy, Geneva,

106. Parkins, W. 107. 108. 109. 110. 111. 112. 113. 1 14.

115. 116. 117. 118. 119. 120. 121. 122. 123. 124.


125.

9-109

1955.

126. DeBortoli, R. A., el al.: Nuclear Set. Eng., 1: 239-251 (1956). 127. Plewes. A. C, et al.: Chem. Eng. Progr., 60: 77-80 (1954).

9-4

THERMAL STRESS AND DISTORTION BY Robert Daane

Thermal stresses arc prevalent in reactors as a result of the nonuniform temperatures with the removal of heat generated in both the fissionable materials and other materials in the presence of irradiation. Sudden changes in the heat-generation rate or the coolant temperature or flow rate may give rise to large temperature differ ences and correspondingly large thermal stresses. In many reactor designs, it is necessary, in consideration of nuclear characteristics, strength, and corrosion resist ance, to have a conjunction of various materials with unmatched thermal-expansion coefficients which may lead to large thermal stresses at operating temperature. In addition to the thermal stresses, which may lead to fractures, the above thermal conditions will usually give rise to distortions of reactor parts. For example, fuelelement distortions may interfere with coolant flow or cause changes in reactivity. Distortions of control rods and other moving parts may interfere with their operation.


associated

NOMENCLATURE AND UNITS C = curvature of a bar resulting from uneven heating = modulus of elasticity, psi h = surface heat-transfer coefficient (heat-transfer rate per unit surface area per unit temperature difference between the surface and the bulk of the surrounding medium), Btu/(hr)(in.*)(°F) k = thermal conductivity, Btu/(hr)(°F)(in.) N = number of cycles of repeated thermal strain q = heat generation per unit volume, Btu/(hr)(in.3) T = temperature at a point, °F x, y, z = rectangular coordinates, in. a = coefficient of linear thermal expansion, (°F)~l t = unit strain At, — inelastic strain per half cycle of repeated thermal strain v = Poisson's ratio a = unit normal stress, tension when positive, psi a, b, d, I = characteristic dimensions of various thermally stressed objects, in.

E

1

THERMAL STRESS AND DISTORTION BY ELASTICITY THEORY

The theory of elasticity, based primarily on the assumption that stress is propor tional to strain, makes possible the calculation of thermal stresses in many important problems. Although elastic thermal stresses are of very limited significance, they are usually of some interest in view of the intractability, in most cases, of any more realistic analysis. 1.1

Order of Magnitude of Thermal Stress

The largest thermal strain that can occur in an unrestrained part having a fairly regular shape (i.e., without relatively thin sections or protrusions or sharply reentrant boundary surfaces in which stress concentrations may occur) is generally 9-110

Sec. 9-4]

thermal stress and distortion

t- KaAT

9-111 (1)

where AT is the maximum temperature difference in the part and K is a number less than 1 and greater than Then for elastic conditions the maximum thermal stress is approximately „

-

KEaUT

(2)

1 — v

The more irregular the geometry of


the part, the more in error these estimates may be. However, as an approximate rule of thumb, the estimate of thermal stress given by Eq. (2) may justify disposing of the thermal-stress problem without further analysis. The approximate rules expressed by Eqs. (1) and (2) do not apply to thermal-stress problems involving multiply connected parts or a conjunction of materials of unmatched coefficient of thermal expansion. For example, very large thermal strains, perhaps even large enough to cause rupture of ductile materials, may occur at bolted or riveted joints connecting fuel subassemblies of one material to a support structure and reactor vessel of another material as the ambient reactor temperature changes. Axial Temperature Gradient in Long Bar. Axial temperature gradients do 1.11 not, in general, give rise to very large thermal stresses in long cylinders or other long bars unless the gradient changes sharply with distance along the bar. In that case, the stress is generally less than

where (&T)d is the maximum temperature difference existing within a length d along the bar, in the neighborhood of the sharp change in gradient, and where d is a length of the same order of magnitude as a major cross-sectional dimension. 1.2

Some Thermal-stress

Formulas

Some formulas are given, as follows, for the maximum elastic thermal stresses occurring in some simple but frequently encountered shapes that are heated internally with a constant heat generation q per unit volume, are cooled uniformly on the surface, and are at steady-state temperature. Edge or end effects are not considered. These are actually explicit expressions of Eq. (2) for specific cases. 1.21 The maximum stress is at the boundary surface, and Long Solid Cylinder. the biaxial stresses are tre =
]

Eaqd* 32k(l

where

-

v)

fa)

d is the diameter of the cylinder and at and a, are, respectively, circumferential longitudinal stress. If the plate is cooled on one or both surfaces, 1.22 Flat Plate, Not Free to Bend. the maximum stress is at the colder with surface temperatures T\ and T2 (Ti < surface and is given by

and

--5orb,[S where

t is

the thickness of the plate and

+

(4)

(biaxial) stress in any direction in the

surface plane. 1.23 Flat Plate, Free to Bend. The maximum (biaxial) stress is at the surface of the plate and is the same at both surfaces regardless of any temperature difference bet ween surfaces: ... Eaql* — =
a?

12fc{l

1.24

drical

-

v)

(5) V

Hollow Tube. The maximum stress occurs at the inside or outside cylin Again, at surface of the tube wall, whichever is at the lower temperature.

9-112

FLUID AND HEAT FLOW

either surface,

"

a«

9

[SKC.

=
fh I k (* - - £S3 « - « fc^r. - beW +

+

1

■

and at the outside surface by

+»-»t?j-

- " r5; [& (" - " +

where o is the inside radius and


1.3

]

"

outside radius of the tube.

b is the

Thermal Distortion of Bars and Rods

Another frequently encountered problem in reactor design is the thermal distortion Many such members, of parts, which may interfere with their proper functioning. such as control rods and fuel rods or subassemblies, have approximately the shape of a long bar with constant cross section. If the temperature distribution in such members is constant in the longitudinal (2) direction or is a slowly varying function of distance along the z axis, then the distortion may be described as a combination of pure bending (i.e., curvature in a longitudinal plane), a longitudinal expansion, and a distortion of the cross-sectional shape. Considering the thermal bending of the bar as made up of two components of curvature C, and C» (the reciprocal radii of curvature in the xz and yz planes, respec tively), these components are given by Lx v

_ afTx dA 7

_

*

ajTydA

(8)

1

where the temperature T is a function of x and y, the integrations are to be made over the complete cross-sectional area, and Ix are the moments of inertia of the crosssectional area about the y and x axes, respectively, and the x and y axes are the If the principal centroidal rectangular coordinate axes for the cross-sectional area. temperature T is a reasonably slowly varying function of distance along the axis, the lateral displacement of any point Zi of the longitudinal centroidal axis of the bar with respect to a straight line tangent to that axis at a point 0 may be found by

u =

—

v =

—

/

I

Cxdzdz

/

/

Cvdzdz

Jo Jo Jo Jo

19

u and v are the x and ;/ components, respectively, of the lateral displacement. The longitudinal expansion w of the bar of length L is given by

where

L

2'.v dz

(10)

where 7',v is the temperature increase averaged over the cross-sectional area at any point z. The distortions of the cross-sectional shape may be calculated by first solving the plane strain problem (where t, = 0) for the given temperature distribution over the cross section and then superposing on those results the biaxial strains «' given by e', =

-

-v(aT„

+ xCx + yC,)

{W

If the bar is of a shape involving thin outstanding flanges or similar protrusions, and if the stress in such a protrusion is predominantly compressive, an additional


Sec. 9-4] Table

9-113

Properties Related to Thermal Stress Behavior of Some Reactor Materials*

1.

Material (alloys are given by weight %)

E, 10' psi

Yield strength,

k.

Htu/(hr) (°F)(in.)

X I0'«/°F

10' psi (0.2% unless other wise noted)

w

Reduction in area, %

Ultimate

strength, 10
Fuel and Fuel Alloys Uranium, as cast. . . U-5.8% Zr, induc tion-melted

U-90 %

melted

Zr, are-

U-5.2% Cr,

U-87.5 % Al,

cast. . .

forged

Plutonium, cast, a. . Thorium, forged bar, annealed

7.9

1.3

26.2

7.0

0.92

14.4

3.7

0.67

24

U.l

10

10.4

II

13 16 10 6

10

97.2

6

1.4

103 75.8

37 3

8.8

10.8

34

27 + 0.1 X temp (°F)

6.4

28

0.21

1.8

0.28

75 (com pressive)

0.30

25.6

56 166.2 121 122 22.5 125 (com pressive)

73

31.2

Reflector and Moderator Materials Beryllia, sintered, 2.8 g/cm1

.


Beryllium, vacuum cast, extruded at I290°F Graphite, electrode Boron carbide, B4C, molded

39 44

0.5-1.2

4.4-5.0

3. 8

0.34

6.4

6.8

0.05

1.0-2.2

7.4-9.7

0.25

2. 5

5. 8

15 26

30 1.3

38

0.5-2.4 10

1

Inert Fuel-element and Structural Materials 347 stainless annealed

steel,

Aluminum, 23-0 Zirconium, crystal Zircaloy II, arcmelted crystal bar. Nickel, annealed...

Molybdenum, arccast, hot-rolled . . . Structural steel 0.3 % carbon, annealed Titanium, forged

bar

29 10.3

13.2

9.2

0.75 10.7

0. 28 0.33

13.9

2.6-3. 1

0.75-0.84

0.33

14 32.3

2.6-3. 1 7.4

0.8 4.4

0.33

17

9.2

48 29

2.8 6.5

15

4.7

0.32

19

6.4

0.32

2.5

0 28

8.2

0.33

50 41-45 (elong. 2 in.)

35 5

10-18 45

8.5

10

(0.5%) 99.8 49 yield point 85

45 30-40 30 (elong. 1 in.) in 1 70

i

60.9 38

25

85 13

30-38 65 46 32 101 90

90

Control Materials Hafnium, annealed arc-mcltcd crys tal bar Cadmium, chill cost. Cobalt, cast

99.73%.

Silver, 1000 fine, annealed

Ag-30%

Cd, cast..

3.3

14 7-10

17.6

4.4

30 10.3

7.9 M.4

19.2

11.3

13.3

1.0

3.3

29.8

Lead, sand-cost

2

4

1.5

30 50 (elong. 2 in.) 92 75 (elong. 2 in.)

65 10

34 21.5 30

Materials

6 6

0. 1

6.5

3.6

0.28

16.3

1.6

0.4

* All data for this table were kindly provided

35-45

yield point

7. 1

Shield

Concrete, plain Portland Iron, hot-rolled ingot

0.3 0.3

0.575

0. 100. 18 26-32 !>'i,:ld z<^" point I

0.8

by Dr. H. A. Sutler, Btrttelle

65-78 100

42-48

1.7-1.9

Memorial Institute.

9-114

FLUID AND HEAT FLOW

[SEC. 9

buckling or twisting type of thermal distortion may occur. Distortions of the latter type are not readily calculated. However, an estimate of their probability of occur rence can be made as follows: In the thin protrusion in which the stress is predomi nantly compressive, the effective buckling stress is approximately given by wcompressive

:

E(cT

- «7',v - xC, - yCy)

(12)

This stress may then be compared with the critical stress*1 leading to elastic instability for a thin plate supported approximately in the same way as the thin flange is oriented with respect to the rest of the bar. 1.4

Material Properties

Mechanical and physical properties pertinent to thermal stress analysis are given for reactor materials in Table 1. Unless otherwise noted, the data are for room temperature. some important


1.5

Limitations

to the Significance of Thermal Elasticity Theory

Stresses Calculated by

Many reactor thermal-stress problems involve ductile materials for which the material behavior ceases to be elastic long before fracture occurs. For relatively simple shapes, the order of magnitude of the thermal strain will be approximately as given by Eq. (12) even when inelastic strain occurs. The stress, however, will be perhaps orders of magnitude less than that given by the theory of elasticity because of the very large inelastic strain, compared with the maximum elastic strain that a ductile material may experience with a relatively small change in the stress. Not only are the results of an elasticity analysis of thermal stresses inaccurate, but they are of very little significance as a criterion of danger of fracture in ductile materials. This is especially true in the case of thermal stresses, because they are self-equi Unlike stresses caused by sustained mechanical oads, where resultant librating. forces and moments must be balanced by definite average stress and stress moment resultants, thermal stresses in simply connected parts balance only one another. Therefore, it is in many cases more realistic to estimate the inelastic thermal stresses and strains using a model based on the theory of plasticity, even though the mathe matical difficulties encountered in such calculations make it necessary to use a very rough approximate model. 2

THERMAL-STRESS

FRACTURE

Thermal stresses alone or in combination with other stresses may lead to cracks in reactor parts. Such cracks may constitute failure in themselves, as in the case of fuel-element cladding, or may render the part, such as a pressure vessel, incapable of sustaining the loads imposed upon it.

Brittle Material

2.1

Only in the case of brittle-material fracture is the magnitude of the stress a direct measure of the likelihood of failure. In such cases, the elastic thermal stress has some For example, for flat plates of brittle materials subjected to thermal significance. shock, such as may occur as the result of a sudden abnormal change in coolant tem perature in a reactor, Manson* gives the following formulas for the maximum allow able sudden change (decrease) in surface temperature:

A? * Superscript numbers

refer to References

=

^ 3 25(1 Ea

~ v)

ah

nt end of subsection.

(13)

Sec. 9-4]


where a is the half thickness of the plate and small values of ah/k,

AT

and

=

~

Met

(1

at, is

-

9-115

the breaking stress, for relatively (14)

v)

for large values of ah/k. Ductile Material

2.2

In ductile materials, a single application of a thermal stress will not generally result in a fracture. Thermal-stress cracks may occur in ductile materials, however,


because of repeated application and removal or reversal of thermal stress. There may also be some possibility of brittle fracture of ductile materials by thermal stress if the conditions are conducive to intergranular failure, such as stress rupture or inter-

granular corrosion. 2.21 If a ductile material is subject to work hardening and Cyclic Plastic Strain. if the work-hardening effects are much more pronounced or stable at the lower of two temperature levels between which the material is cycled than they are at the higher temperature, Freudenthal' points out that after a few cycles of thermal stress, the material will tend toward an elastic behavior with very large residual stress at the lowtemperature part of the thermal cycle, which may lead to a brittle fracture. A possible cause of fracture in ductile materials by thermal stress is fatigue. The cyclic temperature differences occurring in reactors are usually large enough so that large thermal strains occur during each cycle. However, the number of cycles expected during the life of the reactor is usually small compared with the number of cycles for which most fatigue data are available. Therefore, it would be overconservative to limit thermal stresses to values usually regarded as allowable in problems involving fatigue. Manson* and Coffin* have proposed a criterion of fatigue failure that regards total plastic strain as a measure of damage, according to which the number of cycles of thermal stress that may be expected to cause failure is given by (15)

where K and n are constants determined from uniaxial low-cycle high-strain fatigue tests and Aep is the plastic strain developed in the material during each half cycle. Some values of K and n reported by Coffin are as follows: For type 347 stainless steel (annealed) with temperature cycles of about ± 200°F to + 5O0°F, with an average temperature of 660°F in all cases, K = 0.13 and n = 2.0. For the same material at a constant temperature of 660°F, K = 0.0518 and n = 2.33. For 24 ST aluminum, K = 0.313 and n 2.00. The total plastic strain Aep occurring during each half cycle can bo estimated, neglecting work hardening, by

-

where

the yield strength of the material and the effective total strain u is

V2

'(f,

-

<2)s

+ («i

- tiY +

(«•

-

ei)»

(17)

where «i, n, and t3 are the principal strains, computed by elastic stress analysis. The above approximation will be reasonably good for problems not involving large thermalstrain concentrations.

REFERENCES 1.

Roark, R. J.: "Formulas for Stress and Strain," 3d ed., pp. 312-318, McGraw-Hill Company, Inc., New York, 1954.

Book

9-116 2. 3. 4.

FLUID AND HEAT PLOW

[Sec. 9

Manson, S. S.: Behavior of Materials under Conditions of Thermal Stress, "Heat Transfer Symposium," chap. 2, University of Michigan, Ann Arbor, Mich., 1953. Freudenthal, A. M.: Effect of Rheological Behavior on Thermal Stresses, J. Appt. Phys., 35 (9): 1110-1117 (1954). Coffin, Jr., L. F.: Design Aspects of High Temperature Fatigue with Particular Reference to Thermal Stress, ASME Paper 54-A-252, December, 1954.


BIBLIOGRAPHY Timoshenko, S., and J. N. Goodier: "Theory of Elasticity," 2d ed., pp. 399-437, McGrawHill Book Company, Inc., New York, 1951. Soderberg, C. R.: Working Stresses, "Handbook of Experimental Stress Analysis," chap. 10, John Wiley & Sons, Inc., Now York, 1950. Prager, W.: "Theory of Perfectly Plastic Solids," Applied Mathematics Series, John Wiley & Sons, Inc., New York, 1951. Nadai, A.: "Theory of Flow and Fracture of Solids," McGraw-Hill Book Company, Inc., New York, 1950. Coffin, Jr., L. F.: A Study of the Effects of Cyclic Thermal Stresses on a Ductile Metal, Tram. ASME, 76: 931-950 (1954). Lui, S. I., J. S. Lynch, E. J. Kipling, and G. Sachs: Low Cycle Fatigue of Aluminum Alloy 24 ST in Direct Stress, Trams. Am. Inst. Met. Engrs., 176: 469 (1948). Thielsch, H.: Thermal Fatigue and Thermal Shock, Welding Research Council Bull. Ser. 10, April, 1952. (Pamphlet available through American Welding Society.) Weisberg, H.: Cyclic Heating Tests of Main Steam Piping Joints between Ferritic and Austenitic Steels — Sewaren Generating Station, Trans. ASME, 71: 643-649 (1949). Sully, A. H.: "Metallic Creep and Creep Resistant Alloys," pp. 79-123, Butterworth 4 Co. (Publishers) Ltd., London. 1949.

SECTION

10

REACTOR MATERIALS BY B. BOWEN, M.S., Group Loader, Solid State Physics, Atomics International, North American Aviation, Inc. H. A. SALLER, B.S. (deceased), formerly Assistant Technical Director, Batelle

DWAIN

Institute. International Nickel Company, B.Ch.E., M.S., Metallurgist, formerly Senior Metallurgist, Argonne National Laboratory. JOHN P. HOWE, B.S., Ph.D., Chief, Research, Atomics International, North American Aviation, Inc. SIDNEY SIEGEL, A.B., Ph.D., Technical Director, Atomics International, North American Aviation, Inc. VINCENT P. CALKINS, B.S., M.S., Ph.D., Manager, Applied Materials Research, Aircraft Nuclear Propulsion Department, General Electric Company. Memorial

ROGER SUTTON,

A.B., M.S., Ph.D., Group Leader, Nondestructive Testing, Argonne National Laboratory. RUSSEL W. DAYTON, Ch.E., M.S., Ph.D., Assistant Technical Director, Battelle Memorial Institute.


WARREN J. McGONNAGLE,

CONTENTS 10-1

THE METALLIC STATE BY DWAIN B. BOWEN

1 Structure of Metals and Alloys 2 Thermodynamics and Kinetics of Metals and Alloys 3 Deformation and Fracture

References and Bibliography

l'Aoe 10-2 10-9 10-13 10-16

PROPERTIES OF REACTOR MATERIALS BY H. A. SALLER

10-2

10-17 1 Fuels and Fertile Materials 2 Fuel Diluents and Cladding Materials. . 10-38 10-54 3 Solid Moderators 10-00 4 Structural Materials 10-09 5 Control and Absorber Materials 10-71 References

10-3

STRUCTURAL

PAOC 10-92 2 Experiments Using Nuclear Reactors. . 10-9G 3 Experiments Using Cyclotrons 4 Elementary Processes for Damage in 10-99 Solids in Macroscopic Produced 5 Changes 10-105 Properties of Materials. 10-124 References

RADIATION DAMAGE TO 10-5 LIQUIDS AND ORGANIC MATERIALS BY VINCENT P. CALKINS 1 Mechanism and Nature of Radiation Damage 2 Calculation of Radiation Energy Ab

sorption

3 Calculation of Radiation Damage References

MATERIALS

10-6

IN HIGH-TEMPERATURE WATER REACTOR SYSTEMS

BY ROGER SUTTON 10-73 10-74 10-82

1 Corrosion 2 Galling and Wear

References

RADIATION DAMAGE TO SOLIDS BY JOHN P. HOWE AND SIDNEY SIEGEL Nuclear Reactors ation

as Sources

of Radi

10-83

10-120 10-131 10-148

NONDESTRUCTIVE TESTING BY WARREN J. McGONNAGLE

1 Types of Nondestructive Teats 2 Physical and Chemical Properties 3 Thickness Measurements References

10-150 10-101 10-102 10-100

RECENT DEVELOPMENTS IN REACTOR MATERIALS

10-7

10-4

1

10-126

BY RUSSEL W. DAYTON 1 Fuel and Fertile Materials 2 Cladding Materials 3 Control Materials References

10-1

10-108 10-185 10-190 10-191

10-1

THE METALLIC STATE BY Dwain B. Bowen

1

STRUCTURE OF METALS AND ALLOYS


1.1

Electronic Structure

The chemical and metallurgical properties of 1.11 Electronic Configuration. matter are almost completely determined by their electronic binding and structure, the only nuclear characteristics usually of significance being charge and mass. (The most notable exception is the dependence of ortho and para hydrogen on nuclear The electrons are attracted to the nucleus by the classical coulomb field, but spin.) their density of packing about the nucleus is limited by quantum mechanical require In combination these effects lead to the atomic ments of exclusion and uncertainty. The atom with atomic number Z consists of a nucleus carrying Z shell structure. positive elementary charges surrounded by Z negative electrons. These electrons are grouped in shells that, starting at the innermost, are labeled K, L, M, etc., or are characterized by the principal quantum numbers n = 1, 2, 3, etc. The latter indicate The maximum number the number of nodes in the radial part of the wave function. of electrons each shell can contain is 2n', although each shell need not be full. These shells show a fine structure, the subshells, indicated by the letters a, p, d, and/or char The maximum acterized by the azimuthal quantum number I (0 for s, 1 for p, etc.). number of electrons in a subshell is 2(22 + 1). Thus the first shell (n = 1) can con tain 2 electrons, both in the s "subshell"; the second shell (n = 2) can contain 8 electrons, of which 2 are in the s subshell and 6 in the p subshell; the third shell (n ■» 3) can contain 18 electrons, of which 2 are in the * subshell, six in the p subshell, and 10 in the d subshell; the fourth shell (n = 4) can contain 32 electrons, of which 2 are in the subshell. Table 1 s subshell, 6 in the p subshell, 10 in the d subshell, and 14 in the shows the way in which the shells are probably filled, as deduced mainly from spectro scopic data. The regularity in filling shells as the atomic number is increased is responsible for the structure of the periodic table and the uniform variation of proper ties among the elements. The angular characteristics of bonding are largely deter mined by the azimuthal quantum number I and the magnetic quantum number m.

/

Example: Iron {Z = 26) has shells filled as follows: First shell, 2 electrons, completing the shell Second shell, 2 s electrons and 6 p electrons, completing the shell Third shell, 2 s electrons and 6 p electrons, completing the s and p subshells Fourth shell, 2 s electrons, completing the « subshell (note that chromium would have only 1 electron in this subshell) Third shell, 6 electrons of the d subshell Summarizing, the first and second shells are complete with 2 and 8 electrons, respec tively; the third and fourth are incomplete with 14 and 2 electrons, respectively. The relation between outer subshell structure and chemical similarity is evident in the vertical columns of Table 1 and in the horizontal rows of elements 58 to 71 and 91 to 96. As the atomic number Z is increased, the additional electrons find a place on the outer side of the cloud except for five sequences where inner shells are being filled (Table 1). An inner d subshell is being filled from elements 21 to 28, from elements 10-2

THE METALLIC

SEC. 10-1] Table 1.

Ground-state

Electron

STATE

Configurations : d'

d

: d'

10-3 of the Elements*

:
: d« : d'

: d'

d«

d>°

i»

2i 2»

U

Na :

B

C :

n: o

F

Ne

5

6 :

7 :

8

9

10

s

CI

A

16

17

18

Mi

11 i 12

Ip

p

Al

Si

13

14

is

4>°

4j, 3d

4. 25 ; Ga : Ge : Aa Se : Br ; Kr


4p

31 I 32 ; 33 i 34 : 35 i 36

Mo

5.. 4(2

42

Ga

In ; Sn J Sb I Te : I : Xe 49 ! 50 i 51 i 52 j 63 i 54

Sp

6j°

6.. 4/, 5d

fia

La* 57

Tl

: Pb : Bi ; Po j At '■Rn 81 i 82 ; 83 : 84 j 85 i 86

Op

7i° 7«. 5/, ftd

7..

Tht 90

7«" Co (58)

*V:

Pr

Nd

(59)

60

' (61)

Sm | 62

Eu 63

Tb

Dy

Ho

Er

(65)

(66)

(67)

(68)

Tm

1

69

/

f

64

r

/«

70

Lu

Gd

P

Yb

f

r

71

/»

/"

/"

/»

6d°

t5/:

6d

Pa (91)

U

92

Np

Pu

(93)

(94)

* From L. I. Schiff, "Quantum Mechanics," 1949.

York,

!■Am j (95)

Cm (96)

1st ed., p. 270,

McGraw-Hill Book Company, Inc.,

New-

REACTOK MATERIALS

10-4

[SEC.

10

Included in these groups are ferromagnetics, An inner /subshell is being filled from elements 57 to 71 zirconium, and molybdenum. and from elements 89 to 103. Included in the latter group are thorium, uranium, and This exceptional electronic structure makes reactor materials frequent plutonium. exceptions to the simpler theories. The forces holding the atoms of a molecule or a crystal 1.12 Atomic Forces. lattice in their equilibrium configuration are of the same basic nature that gives rise Classical coulomb forces of attraction act between all pairs to the atomic structure. of nuclei and electrons and of repulsion between all pairs of nuclei and between all It might seem, casually, that a balance between these forces pairs of electrons. alone could account for the bonding, but the quantum requirements on the electrons It is possible to write a system of differential are essential to an accurate description. equations that accurately describes the motion of all the electrons and nuclei, but they It is at this point are too complicated to be solved exactly for even the simplest solids. that various approximations are invented, one being suitable for one class of materials and another being suitable for another class. Inevitably there are materials that fall between the extremes in each class and are often only poorly described by either treat and simplifications have approximations ment. The principal and important acquired names, and the physical content of the approximation is often spoken of as a force. Thus, in an external electric field an atom will exhibit an induced dipole Furthermore, moment as the nucleus is pulled one way and the electrons the other. The dipole-dipole many molecules exhibit permanent electric-dipole moments. interaction between these polarizable or polar species is called a van der Waals force. It varies inversely as the fifth power of distance between atoms. For polarizable atoms the proportionality constant is independent of temperature; for polar atoms it The words exchange force are used to cover varies inversely as the temperature. several situations where the quantum mechanical requirement of antisymmetry of the These forces can be either attractive or repulsive electron wave function is involved. depending on whether the kinetic energy of the electrons is decreased (attraction) or A typical increased (repulsion) when the requirement of antisymmetry is imposed. example of repulsion occurs when atoms (in contact) are compressed to a smaller The uncertainty principle then requires that the electrons increase their volume. momentum and hence their kinetic energy. A typical example of attraction occurs when an antisymmetric electron wave function more effectively shields the coulomb repulsion of the nuclei than the unsymmetrized wave function does. Metallic crystals are bound predominantly by these exchange forces whereas the coulomb forces between cations and anions provide most of the binding in ionic crystals. In the presence of an external field the charge density for Valence Bonds. 1.13 unfilled atomic levels shows a bumpy structure that is specified by the azimuthal and When the atoms are combined into molecules magnetic quantum numbers I and m. or crystal lattices, the angular structure of these lobes is reflected in the structure of It is often possible to describe a particular angular the compound or crystal structure. configuration of atoms as a hybridization, or mixture of several of the available atomic wave functions. The final structure is then a balance between the requirements of space filling of electron density and the directional character of the most suitable The most familiar examples of hybridized and antisymmetrized wave function. valence bonding are found among the organic compounds, but its influence is also clearly evident in uranium and plutonium, where /and d hybrids are operative. It has a structure of atoms tightly Graphite is an example of s and p hybridization. bound into planes, with the planes weakly bound to each other. Each atom in a plane has three nearest neighbors that form part of a hexagonal network of atoms. These atoms may be considered bound to each other by hybrid wave functions of the type


39 to 46, and from elements 71 to 78.

-L

[+(%,) +

\/2

+(«&>)]

i

= 1, 2, 3

V73

where ^(2s) is the

2s

atomic wave function for carbon and f{
THE METALLIC STATE

Sec. 10-1]

10-5

function for carbon with its axis in a direction in joining the atom under consideration The wave functions given above are not to one of its three nearest neighbors. orthogonal, but they can be made go by writing them as

V3

\/l5

[Vs+(2*)

\Zl5

+ *(o-,2p)

3*(<722p)]

+iK
[V8*(2a)

The fourth electron provides conduction and fills 1.2

+

a p level with axis normal to the plane.

Crystal Structure

1.21 Crystal Systems. Crystals are classified into seven systems that are dis tinguished from one another by the length of and the angle between the crystallographic axes as follows (Fig. 1, Table 2).

Table 2.

The Crystal Systems Characteristics

Ortborhombic. Hexagonal

Three equal axes, mutually orthogonal a = 0 = 7 = 90° Three mutually orthogonal axes, of which only two are equal o = 6 * <• a 0 = 7 = 90° Three mutually orthogonal axes of different length

o=4-c

l/Mt Three

equal to these

-

a = 0 = 7 = 90° coplanar axes at 120°. a

Trigonal . . . Monoclinic . Triclinic. . . .

-

Fourth axis at right angles

0 = 90°

Cu, Ag, Au Fe, Ni, Th 0-Sn a uranium

n-Zr

y = 120°

but not orthogonal = 0 = 7 90° Three axes of any length, two of which are orthogonal y 90°

As, Bi, Sb (calcite)

Three axes of any length, each differently inclined

KiCn07

Three equal axes, all equally inclined,

a =

6-o

„

*

A

*

a* 0*

y

*

The orientation of crystallographic plan indices. The orientation of a plane is spec ified by the reciprocals of its intercepts on the crystallographic axes (and reduced to the three smallest integers having the These indices are enclosed same ratio). in parentheses (hkl). A group of equiva lent planes, such as the cube faces of a cubic crystal, is indicated by braces \hkl]. The indices for a direction, signi fied by brackets [hkl], are vector compo axes nents along the crystallographic A whose sum is in the desired direction. group of equivalent direction is indicated Only in the cubic by carets system are planes and directions with the same indices always perpendicular. Among the properties that vary signifi cantly with crystallographic orientation

.

CaSO»-2H,0 (gypsum)

90°

or directions can be represented by Miller

Zi

/

c

FlO.

I

Tetragonal

Examples

3fr

Cubic

op


System

1.

Crystal axes.

10-6

REACTOR

MATERIALS

[Sec.

10

are ease of grinding and polishing, resistance to chemical attack, clastic modulus, critical shear stress, thermal expansion, and thermal and electrical conductivity. 1.22 Close Packing. There are two structures that represent the closest packing

possible for hard spheres. These are the hexagonal close-packed and face-centered cubic structures. They can both be realized by the appropriate stacking of basal (or equivalent 11 1 1 1, octahedral) planes. The hexagonal close-packed structure is shown in Fig. 26, and the face-centered cubic structure in Fig. 2a. Through each atomic posi tion, the cubic structure has four equivalent close-packed planes | 111 |, the hexagonal structure only one — the basal plane. Usually the hexagonal structure is not ideal, the distance between basal planes not being correct for true packing of spheres.

0


(a)

Fio.

(cl

(t>)

Face-centered centered cubic structure. 2. (a)

cubic structure,

(6) close-packed

hexagonal

structure,

(c) body-

These structures are important for metal physics because so many metals exhibit them, and because it is often possible to think of more complicated structures as examples of distorted close packing. Alpha uranium, for example, can be considered as a deformed hexagonal close-packed structure, with 4 of the 12 nearest neighbors moved in to appreciably closer distances. Thus: Number of Neighbors and Distances in Uranium* ■I

»C. W. Jacobs and B. E. Warren.

J.

2

2

4

2.85 3.27 1 2.76 Am. Chem. Soc, 59: 2588 (1937).

4

3.36

1.23 The body-centered cubic structure is likewise important for its frequency of occurrence and simplicity. It is illustrated in Fig. 2c. 1.24 An atomic radius can be ascribed to an atom and is often of value in esti mating atomic misfit and strain in alloys (see Fig. 3). 1.3

Electronic Properties

1.31 Electron Mobility. Most of the electronic properties of monovalent and many properties of higher valence metals can be described by treating the electrons The results as a gas held within the bulk of the specimen by a square well potential. of this approximation are called the free-electron theory of metals. The electrons must obey Fermi-Dirac statistics, and this leads to a special distribution of energies among the electrons. The number per unit volume within an energy range dE is

where h is Planck's constant and m is the mass of the electron. The total density of electrons must be equal to the density of atoms times the valence, so the formula is valid up to a cutoff called the Fermi energy above which dN /dE quickly becomes


Sec. 10-1]

THE METALLIC

Fig. zero.

3.

Radii of Moms.

Z

STATE

= atomic

10-7

number.

The Fermi energy Wt is determined such that

where No is the total density of valence electrons. The Fermi energy for several metals is given in Table 3. These electrons are free to move through the crystal and under an applied electric field produce a current. Table

3.

Metal

lr\. volts

Li

4.7 9.0

Be

Na Mg Al

K

Ca

Ti V

Cr Fe

Co

Ni

Cu Zn

Rb

Source: J. C. Slater, "Introduction pany,

Inc., New York,

1939.

Fermi Energy U\ of Metals

3.1

4.5 5.6

Metal

Sr

Zr Mo Ru Hli

2.1

Pd

3.0 5.4 6.3 7.0 7.0 6.2 7.4 7.0 5.9

Ak CA

1.8

Wt, volts

2.5 4.5 5.9 6.4 6.3 6.1 5 5

4.7

Ca

1.6

Ba Ta W

2.3 5.2 5.8 6.3 6.3 6.0 5.6

Os

Ir Pt

An

to Chemical Physics," 1st ed., p. 477.

McGraw-Hill Book Com

10-8

MATERIALS

REACTOR

The conductivity

[Sec.

10

is given by a = No — m

t

the charge on the electron and f is called the relaxation time. It is the time during which an electron is accelerated by the electric field before being It is of the order of 10_1< sec scattered by a lattice imperfection or thermal oscillation. for most metals at room temperature. During this time the electrons responsible for conduction (those with energies near Eo) travel the order of 10 atomic diameters. The monovalent metals exhibit superior conductivity over those of higher valence because of the opportunity available in the latter for conduction electrons to be scattered into unfilled nonconducting levels, e.g., is to 3d transitions in iron. At room temperature and above, the principal source of electron scattering in a pure metal is by lattice oscillations and increases nearly linearly with temperature. The contribution from impurities and lattice imperfections is additive and inde Thus the resistivity can be written pendent of temperature. where

e is

average

P = Pthermal


This expresses what is known as Matthiessen's law. At low concentrations p.tnti«r. is proportional to the concentration of defects (the order of a few microhm centimeters for one lattice defect in 100 normal lattice atoms). 1.32 Brillouin Zones. The free-electron model is useful for some metals but fails to give a reliable picture of others and contains no hint of the semiconducting or The periodic electric field insulating properties of the semimetals and insulators. produced by the lattice must be included in the calculation to account for these Just as X rays can suffer Bragg reflection and for these conditions cannot properties. travel through the lattice, so electrons with certain critical de Broglie wavelengths are forbidden. The energies corresponding to these wavelengths give rise to the forbidden bands. The critical wavelengths are X

- -n

n =

1, 2, 3

where d is the atomic distance in the lattice. X =

. . .

The de Broglie wavelength is given by > V

where p is the momentum of the electron and h is Planck's constant. A three-dimensional lattice possesses different d values in different directions. In reciprocal lattice (momentum) space these equations define a set of planes that enclose To each differential These parts are called Brillouin zones. parts of the space. element of the zone there corresponds a possible electron state with momentum equal to the vector from the origin to the element. Each zone can contain just twice as many electrons as the body has atoms; thus the first Brillouin zone could be just filled by the valence electrons of a bivalent metal. The energy of possible electron states is continuous from the origin to the zone face but may be discontinuous across the If electrons only partially fill a zone, they act as face because of a forbidden band. if they were free ; if they just fill a zone and there is a large energy gap to the next zone, the material is an insulator; if the energy gap is small, the material is a semiconductor: if the zones overlap (negative energy gap over some regions of the zone face), one gets intermediate semimetal behavior. 1.4

Alloying

When different elements combine to form alloys, there are three ways in which the atoms may be represented in the lattice: (1) They may form compounds with a definite structure and periodicity among the constituents. (2) They may form a substitutional alloy in which the lattice structure of one of the constituents forms a framework into which atoms of the other constituents are fitted nearly at random

THE METALLIC

Sec. 10-1]

STATE

10-9

(there is almost always some short-range preference for either like or unlike nearest An intermediate situation exists for those alloys which form superneighbors). At elevated temperatures the constituents are arranged nearly at random lattices. How on the lattice sites, and this disorder may usually be quenched into the alloy. ever, when suitably heat-treated, the atoms arrange themselves in an ordered sequence on the same space lattice. (3) The lattice structure of one of the constituents mayform a framework into which the atoms of one or more of the other constituents are Several phases, e.g., a compound and a sub fitted into interstitial (nonlattice) sites. stitutional alloy, may exist in equilibrium in the same sample, the number of allowed phases being governed by the phase rule

P


P

+ V = C +

2

= number of phases where V = variance, e.g., temperature, pressure C = number of components This leads to a heterogeneous structure. If the metal is not in thermodynamic equilibrium, the phase rule may be violated and even greater heterogeneity occur. Usually two metals are able to form substitutional alloys over a broad composition range only if their atomic sizes (see Fig. 3) differ by less than 15 per cent, although limited solubility may occur (a fraction of 1 per cent) when the sizes are unfavorable. There is a tendency to form intermetallic compounds when the two constituents differ So-called electron compounds*'1 are formed markedly in electrochemical character. when the valence electron to atom ratio has particular values, alloys with the same value of the ratio having the same crystal type (see Table 4). In each of the structures there are just enough electrons to fill a sphere in the Brillouin zone that is tangent to

the surface. The only atoms that are small enough to be stable in interstitial positions are hydrogen, beryllium, boron, carbon, nitrogen, and oxygen (see Fig. 3). Table 4.

Crystal Structure of Electron Compounds

Electron atom ratio

Crystal atructure

1.4 1.5

Face-centered cubic Body-centered cubic or 0 manganese 7 brass Hexagonal close-packed

'K»

1.75

2

Examples

CmZnj. CiuAl, AgiZm, UCut CuZn, AiuSn, CuiAl, AuZn, AuCd, CuiSi. CoZni CusZm, CuuSns, Ag»Zn« CuZm, CuCdi, CuaSn, CuiSi, AuZni

AgiAl,

AuiAl,

THERMODYNAMICS AND KINETICS OF METALS AND ALLOYS 2.1

Lattice Oscillations

The principal contribution to the specific heat of metals arises from their elastic oscil At elevated temperatures, where there is equipartition of energy among the lations. available degrees of vibrational freedom, the internal energy increases linearly with temperature and the specific heat takes on the Dulong-Petit value of 5.96 cal/g-atom, corresponding to k per atom per degree of vibrational freedom, where A: is Boltzmann's constant and there are three degrees of freedom. The lowest frequency oscillations that can be sustained are those with wavelengths comparable to twice the dimensions of the specimen; the highest frequency oscillations are those with wavelengths com The latter correspond to frequencies parable to twice the interatomic separation. To each of the various modes of vibration of the order of 10" cps for most materials. there corresponds a quantum energy hv, where h is Planck's constant and v is the As the thermal energy per atom kT becomes smaller than the quantum frequency. energy for a particular oscillation, it becomes progressively less likely that oscillation will be excited. Thus, there are deviations of the specific heat from the constant value as the temperature is lowered. The Debye theory of specific heat satisfactorily * Superscript numbers

refer to References


10-10

REACTOR

MATERIALS

[Sec.

10

accounts for the main features of the data for nearly all the cubic materials and for many materials with less symmetric structures, in spite of the fact that the model on which it is based cannot be valid in detail. This circumstance can be attributed in part to the fact that near absolute zero of temperature the specific heat is most sensi tive to the distribution of normal modes (the basic assumption of Debye theory), but the modes that are excited in this region are the ones for which the assumptions of the Debye theory are most accurately obeyed. For temperatures greater than 0. 2 of the Debye temperature any reasonable assumption about the mode distribution will give a reasonably accurate reproduc tion of the specific heat. The cutoff fre-

L4x70(l/f,»)

+ (2/»,»)»

i

quency is where Vo is the atomic volume, waves, vi is the velocity of longitudinal and v, is the velocity of transverse waves. The specific heat is

0 6

08

T5

[5

L4

AV Jo

T6

X


where

Fig.

heat of a solid as a function according Debye's of temperature, to theory. C. Slater, "Introduction (From Physics," 1st ed„ p. 236, to Chemical McGraw-HUl Book Company, Inc., New York, 1939.) 4. Specific

J.

=

h,

kT

(e*

- l)1

A'„

-

kT

The Debye temperature is then defined 9d

A'o =

as

9d

T

C„ has been plotted in Fig. 4 as a function of reduced temperature, i.e., temperature divided by Debye temperature. The Debye temperature for a number of metals is given in Table 5.

Table 5.

Debye Temperature and Gruneisen's Substance

0d° abe

Ag

SS 391

Al

Au Be

C (diamond) Ca Cd Cr Ca Fe Ge

K Li Mg Mn Ma N:i Ni Pb Pt Tu

Ti

Th

V

Zn Zr

IN

1000 IS60 230 160 48S 315 420 290

99.5

328-430 290 S79 159 370 68 223 24S 350 142 310 235 280

2.40 2. 17

3.03 1.98 3.35 2.37

l.9< 1.60 1.34 1. 17 2. 15

2.42 1.57 1.25 1.88 2.73 2.54 1.75

1.62

2.68

Constant

10-11

STATE

THE METALLIC

SEC. 10-1]

Most There are a few materials for which the Debye theory is a poor model. The failure here is certainly notable of these among the reactor materials is graphite. By treating the oscillations in the basal due to the tremendous elastic anisotropy. plane and normal to the basal plane separately,* one can again get agreement with experiment. In addition to the lattice contribution to specific heat the valence electrons may also The exclusion principle prevents all but a small add a relatively smaller contribution. In those metals for which the freefraction of the total electrons from contributing. electron theory can be applied, the additional contribution is Lv

"_

Rr* kT W,

2

where Wi is the Fermi energy of the metal (see Table 3). In most cases this only amounts to about, 1 per cent of the total at room temperature. In a few metals the Dulong-l'etit value is exceeded by as much as 50 per cent. In uranium, for example, at 500°C the specific heat is about 9.5 cal/g-atom, or nearly This anomalous behavior is probably 60 per cent greater than the classical value. If the lowest due to the several possible configurations of the electrons in the/ band. configurations are separated by an energy comparable to kT, then thermal excitation of the electrons from one configuration to the next will contribute to the specific The anomalous contribution in uranium, for example, can be accounted for by heat. an band configurational energy difference of the order of Ho electron volt and a relative degeneracy of 30.

/


/

Thermal Expansion

2.2

The thermal expansion is dependent on the lattice oscillations in much the same way as the specific heat. In many metals the proportionality factor for the two properties is the same for all lattice frequencies, and the thermal expansion coefficient can be written

7XC. V

\c>7V„

where y = Gruneisen's constant = — d In v/d In V v = lattice (Debye) frequency V = volume = — (l/F)(dV/dp)r X = compressibility C, = specific heat This formula is applicable only to metals with small electronic specific heat; thus it should not be applied to uranium or ferromagnetic materials near their curie point. Gruneisen's constant has only a small range of values about 2. Typical values are given in Table 5. 2.S

In

Thermal

metals the heat is carried principally

Conductivity by the electrons.

In

this case the electrical

resistivity and the thermal conductivity are often related by the Wiedemann-Frams law for temperatures greater than the Debye temperature.

*£ where p

K

T

—

-^ -

2.45

X

10-»

watt-ohm/ (°C)»

electrical resistivity

= thermal conductivity = absolute temperature

k = Boltzmann's constant e = electronic charge For these metals the thermal and electrical conductivity are each limited by the In insulators and in poor number of conduction electrons and their scattering. electrical conductors the heat is carried, at least in part, by the lattice oscillations and

10-12

MATERIALS

REACTOR

can be written

K

[Sec.

10

= IvC,

where I is the average distance between lattice imperfections that perturb the oscilla tions, c is the group velocity of the waves, and C, is the specific heat. Again, the formula must be applied with reservation to materials with large electronic specific heats. The imperfections responsible for scattering lattice waves are usually larger than those which scatter electrons. Grain boundaries, precipitates or more generally two-phase regions, and voids are typical lattice wave scatterers. Stainless steels are generally poor thermal conductors because they contain a wide range of imperfections that are effective in scattering both the electrons and the lattice waves. Some metals exhibit appreciable conductions by both mechanisms. Thus in nickel at room tem perature about 90 per cent of the heat is carried by the electrons, the remainder by the lattice. 2.4 Allotropy

Allotropy is the existence of a substance in more than one physical form, as in the The thermodynamic requirement for equilibrium at any three phases of uranium. given temperature and pressure is that the Gibbs free energy

F

=

E

+

PV

- ST


be a minimum, where E = internal energy P = pressure

V S T

volume entropy = temperature The PV term should be generalized to include all elastic strains in the metal. In normal practice it is much smaller than the other two terms and affects the ultimate However, it can be extremely important in deter equilibrium only in a minor way. mining the path or manner in which the equilibrium is achieved or, if the minimum is a broad one, the direction and type of nonequilibrium behavior that is sustained when the transformation rates are slow. The largest strains arise from precipitates and anisotropic thermal expansion. (The or-(3 transformation in uranium is broadened 10 to 20°C by the latter.) The transformation may involve a complete rearrangement of the atoms, or the two structures may be related to each other, involving only a slight motion of the atoms in the unit cell. For complete rearrangement, the higher temperature phase usually Examples of has symmetry as great as or greater than the lower temperature phase. In steel the both types of behavior can be found for diffusionless transformations. low-temperature martcnsite phase is the complex one, while in uranium the highThe difference here may be resolved by temperature /3 phase is the complex one. The complex structures formed by deformation examining thermodynamic stability. of the unit cell that are observed at the higher temperatures are the thermodynamic equilibrium structures, while the complex structures that are observed at low tem peratures are metastable transition structures, although their approach to equilibrium may not be achieved during their useful service life. Thus, as a general rule, the thermodynamically stable structures that transform by atomic rearrangement have greater symmetry in their high-temperature form; those which transform by distortion of the unit cell (or several unit cells) have greater symmetry in their lower temperature form. The metastable structures that transform by distortion of the unit cell have greater symmetry in their higher temperature form. Transformation rates arc usually controlled either by diffusion or by deformation. In alloys the different phases that can exist together have different compositions, and precipitation cannot take place until segregation of atomic species has occurred. Precipitation may then occur as rapidly as nuclei of a critical size and the correct composition arc formed, or the original structure may be retained in the nuclei and they will continue to grow until an external or internal stress deforms the grain, result ing in the new structure. In this case diffusion controls the rate during heat treatment -

-

THE METALLIC STATE

Sec. 10-1]

10 13

By proper heat treat and deformation controls it during the actual transformation. ment the size of grains can be controlled; by straining in a preferred direction the orientation of the precipitate can be controlled. Diffusion

2.6

Diffusion in and through solids is governed by Fick's law. It states that the current density of foreign matter in a matrix is proportional to the gradient of concentration

i

=

-DvC

The diffusion constant D may be a function of the materials and their relative concen tration, the grain structure, and the temperature. If the body is porous, it is most likely that gaseous products will diffuse most rapidly through the pores, the diffusivity Within the bulk of the depending on their size, distribution, and connectivity. material diffusion may take place either by penetration directly through the grains (volume diffusion) or by diffusion along grain boundaries (grain-boundary diffusion). The former is likely to predominate at higher temperatures; the latter at lower tem In grain-boundary diffusion the significant geometry is associated with peratures. the two-dimensional grain. For plane geometry this leads to a concentration vs. penetration curve that is linear in In C versus x whereas the volume diffusion is linear in In C vermis x1 (provided D is independent of C). Fick's law can be written in the alternative form


— at

= V • (DVC)

The temperature variation of the diffusion constant

is given by a formula of the type

D = Doe-*'*T

The constant D0 is observed to have values from 10~* to 104 em'/sec in different This range of values covers both grain-boundary and volume diffusion. systems. The activation energy Q is the order of tens of thousands of calories per mole. 3

DEFORMATION AND FRACTURE 3.1

Deformation Geometry

Plastic flow in crystals

can occur either by slip or by twinning. Planes of low indices and close packing are preferred as slip planes; the lines of densest packing are the preferred slip directions. Thus,

Miller

Crystal typo

Slip plane

(111) (110), (112), or (123) (0001)

The will

Slip direction

[HO]

[III]

[1120]

lower the symmetry, the fewer slip planes a material has and the more brittle it tend to be. Among the family of slip planes, slip takes place on the plane of greatest resolved shear stress and along the most favorably oriented slip direction. In a cylindrical test specimen the relation between applied stress and critical shear stress (the minimum stress required to initiate slip) is given by

where r

t

= a cos

cos 9

critical shear stress a = applied stress > = angle between direction of a and normal to slip * = plane angle between a and slip direction =

10-14

REACTOR

MATERIALS

[Sec.

10

In unstrained single crystals the critical shear stress is the order of 0.1 kg/mm*. As slip takes place, sections of the deformed crystal rotate in such a way that a slip direction becomes more nearly parallel to the direction of applied stress. As a result of rotation of the crystal, other slip planes may become more favorable, and slip takes place on them. As progressive deformation by slip occurs, the consequent work hardening may result in required shear stresses of several kilograms per square millimeter. The separation between planes of active slip can be observed by careful electron There slip lines are separated by 200 to 800 A and are grouped into microscopy. bands of fewer than a dozen lines. For aluminum single crystals the band distribution is as follows.* to 1 ji at — 180°C, containing 1 to 2 lines 2 n at 20°C, containing 3 to 4 lines 4 n at 250°C, containing 5 to 6 lines 10 it at 500°C, containing 6 to 12 lines

In


all cases the distances of slip on a given plane (or line) is approximately 2000 A. The twin geometry is characterized by a twinning plane across which the lattice sites are mirror images of each other. This configuration is achieved by a simple shear, in which every plane of atoms shifts in the same direction an amount propor tional to its distance from the twinning plane. The characteristic twinning planes and directions are

Crystal type

3.2

Twinning plane

Twinning direction

(111) (112) (1012)

[ill!

Deformation

[U21

Mechanism

For an ideal perfect crystal, stress and strain should be completely reversible for strains up to the order of 50 per cent, with plastic deformation setting in only for stresses of the order of one-half the elastic modulus or the order of 1010 to 10" dynes/cm'. In real metals the elastic limit is exceeded for stresses and strains that are always several orders of magnitude (100 to 1,000 times) smaller than this. To account for this the dislocation was conceived. An extra plane of atoms is inserted part way into the lattice, forcing partial disregistry of the perfect periodicity, par The dislocation caused by the extra half plane is ticularly at the intruding edge. usually indicated pictorially as an inverted T, thus, _L. Under applied load, a stress concentration occurs at the line of disregistry, and the dislocation moves through the lattice, causing a shear displacement of one part of the crystal relative to the other. The plane generated by the moving dislocation line is the slip plane. There are two fundamental types of dislocations, the edge dislocation and the screw dislocation. In the latter, slip is parallel to the dislocation line rather than perpendicular as in the edge dislocation. Compound dislocations may be formed from segments of edp" and screw dislocations. The concentration of dislocations is specified by the number of dislocation lines intersecting a unit area of the crystals. In a good single crystal there will be the order of 10* to 10* per square centimeter; in a cold-worked metal the order of HI12 per square centimeter. The region of disregistry extends over about Work five atom distances. The energy stored is about 1 ev per atomic plane. hardening presumably occurs because of the mutual interference of dislocations with each other and with vacant lattice sites created by their own motion. In alloys this interference is magnified by impurity atoms or precipitate particles.

Sec. 10-1]

THE METALLIC 3.3

STATE

10-15

Grain Boundaries


One of the most pictorial and well-founded examples of dislocation theory is found Figure 5 illustrates this concept. in a representation of low angle grain boundaries. The phenomenon of polygonization, in which a cold-worked metal is partially annealed, may be represented as a transformation in which a random array of dislocations, which give rise to bent planes, line up to form a series of low angle subgrain boundaries and minimize the lattice strain.

Flo.

5.

Low angle grain boundary composed of an array of edge dislocations.

In polycrystalline samples strained at low temperatures the slip or twinning that is characteristic of single crystals takes place within each grain, but with varying magni In a sample with a gross elongation of 5 per cent the individual crystals exhibit tude. This gives rise to internal stresses elongations that vary between 3 and 14 per cent.4 between grains that are characteristic of the deformation applied, and lead to the Bauschinger effect wherein prior compression raises the yield stress for further com At high temperatures deformation in polycrystals pression but lowers it for tension. is usually achieved by relative weakening of the grain boundaries and viscous slip along them. Secondary or steady-state creep rate seems to be determined by the equilibrium subgrain structure characteristic of the test temperature.4 3.4

Calculated cohesions at concentration the cohesive

Rupture

cohesive tensions are of the order of 1,000 times larger than the observed This can be explained by the existence of flaws, leading to a rupture. of lines of force around the edge of the notch surrounding a flaw. When strength is higher than the slip strength, a material yields plastically.

10-16

REACTOR

MATERIALS

[Sec.

10

However, after sufficient plastic deformation the two strengths become equal, and the material breaks. Fatigue failure occurs when alternating stresses fragment the crystals to dimensions the order of 0.1 to 1 ix. The alternating stress may also accelerate diffusion of inter stitial impurities to precipitate in and embrittle the grain boundaries. Endurance limits are reached after 104 to 108 cycles at stresses from 5 to 30 kg/mm!.

REFERENCES

1. 2. 3.

Hume-Rothery, W.: J. Inst. Metals, 35: 295 (1926). W. Hume-Rothery and G. V. Raynor: "The Structure of Metals and Alloys," The Institute of Metals, London, 1936. Newell, Gordon F.: Specific Heat of Lamellar Crystals, Brown Univ. Tech. Reft., Contract NON562(08), Sept. 15, 1954. Heidenreich, R. D., and W. Schockley: J. Appl. Phys., 18: 1029 (1947). A. F. Brown:

Nature., 163: 962 (1949). Boas, W., and M. E. Hargreaves: Proc. Roy. Snc, A193: 89 (1948). 5. Ancker, B., T. H. Hazlett, and E. R. Parker: Relationship between Small Angle Disloca tion Boundaries and Creep, Unit. Calif. {Berkeley) Tech. Rept. 15 (AECU-2919), Minerals Research Laboratory, 1954. 4.

BIBLIOGRAPHY On Electron Theory and Molecular

Barrer, R. M.: "Diffusion

On Thermodynamics and Kinetics in and through Solids," The Macmillan Company, New York,

1941.

Lumsden, J.: "Thermodynamics of Alloys," The Institute of Metals, London, 1952. Mayer, J. E„ and M. G. Mayer: "Statistical Mechanics," John Wiley & Sons, Inc., New York, 1940. "Thermodynamics in Physical Metallurgy," American Society for Metals, Cleveland, 1950. On Mechanical Properties Cottrell, A. H.: "Dislocations and Plastic Flow in Crystals," Oxford University Press. New York, 1953. Elam, C. F.: "Distortion of Metal Crystals," Oxford University Press, New York, 1935. Read, W. T.: "Dislocations in Crystals," McGraw-Hill Book Company, Inc., New York, 1953.

W., J. H. Hollomon, and F. Seitz: "Imperfections in Nearly Perfect Crystals," Sons, Inc., New York, 1952. John Wiley

Shockloy,

On General Topics Barrett, C. S.: "Structure of Metals," McGraw-Hill Book Company, Inc., New York, 1943.

Boas, W.: "Physics of Metals and Alloys," John Wiley Sons, Inc., New York, 1947. Cottrell, A. H.: "Theoretical Structural Metallurgy," Longmans, Green Co., Inc., New &

&

York, 1948. Hume-Rothery. W., and G. V. Raynor: "The Structure of Metals and Alloys," The Institute of Metals, London, 1954. Kittel, C: "Introduction to Solid State Physics," John Wilev Sons, Inc., New York. &


Binding

C. A.: "Valence,"

Oxford University Press, New York. 1952. Hume-Rothery, W.: "Electrons, Atoms, Metals and Alloys," Philosophical Library, Inc., New York, 1948. Mott, N. F., and H. Jones: "The Theory of the Properties of Metals and Alloys," Oxford University Press, New York, 1936. Peierls, R. E.: "Quantum Theory of Solids," Oxford University Press, New York, 1955. Wilson, A. H.: "The Theory of Metals," Cambridge University Press, New York, 1954. Coulson,

1953.

Seitz,

F.: "The Modern Theory

1940.

"The Physics

of Solids,"

McGraw-Hill Book Company, Inc.,

New York,

of Metals," McGraw-Hill Book Company, Inc., New York, 1943. "Introduction to Chemical Physics," McGraw-Hill Book Company, Inc., New York, 1939. Zwikker, C: "Physical Properties of Solid Materials," Interscience Publishers, Inc., New York, 1954. Seitz, F.: Slater, J.

C:

10-2

PROPERTIES OF REACTOR MATERIALS BY (deceased)*

In presenting information on the properties of reactor materials, it has been found desirable to separate the various materials into groups according to their function in the reactor, for example, Fuels and Fertile Materials, Fuel Diluents and Cladding Materials, etc. For the more familiar materials, as aluminum or stainless steel, only The bibliography gives references that information pertinent to reactors is included. The lesser to detailed information on mining, fabrication, and related subjects. known materials such as uranium, thorium, and zirconium, which have been developed for reactor application, are covered in more detail, since information on these materials is not so generally available. A brief summary of important properties of reactor materials is given in Table 1. More details for each material will be found in the various articles. Summaries of corrosion resistance to liquid metals are given in Figs. 1 and 2. 1

1.1

FUELS AND FERTILE MATERIALS

Reactor Application and Special Requirements

The operation of a reactor is dependent upon fissionable material, and these mate rials are certainly the most important of all reactor materials. Only slightly less important in the long run are the fertile materials — those elements which can be is,

changed into fissionable material and thus increase our supply of reactor fuel. Fissionable materials may be incorporated in reactors in various forms: pure metal, alloy, or a compound, that UO2 or UC. The uranium may be in its natural state, may be enriched in the fissionable isotope uranium 235. The poor corrosion or necessary to protect and resistance and relatively low strength of uranium make The strengthen the basic material in order to overcome these limiting properties. must be treated similarly. properties of plutonium parallel those of uranium, and metallurgical point of The fertile materials thorium and uranium 238 are, from view, relatively inferior materials and must also be strengthened and protected by cladding or alloying. As nuclear properties dictate the select ion of fuels and fertile materials, their limiting physical and mechanical properties must be enhanced by various techniques. An ideal fuel or fertile material would possess good thermal conductivity, good strength and ductility, radiation stability, and corrosion resistance. Uranium, thorium, and plutonium fall far short of this goal. a

it

it

it

1.2

Uranium

Production. Abundance and Availability. Uranium makes up about 0.0004 the earth's crust and actually more abundant than some of the more found in familiar metals such as cadmium, bismuth, mercury, or silver. Uranium a variety of rock as large number of minerals. The highest concentrations occur in 1.21

is

is

per cent of

a


H. A. Sailer

• The publishers

They wish and the editor-in-chief deeply regret the untimely death of Dr. Sailer. of Battelle Memorial Institute for reviewing Dr. Sutler's contribution.

to thank Mr. Donald Loiier

10-17

,1

2.

.

. .

Mo8U. 6.24

3.21

25

Decomposes at 2500°C 1870

1870

6.24

650 2550 ±

SiC

BciC.

.

13.69

327°C, 10.06

25"C, 6.08 I00°C, 7.7

l0O°C,

370°C. 0.088

400°C, 0.060

30°C, 0.02

I8°C, 0.376 200°C, 0.19

100°C, 0.34

0.53

50°C, 0.05 <0.04

Materials

25-l00°C, 5.3 25-20O°C, 7.7 0-l700°C. :~ r*M 0- 500°C. •5.^ •

25-IOO°C, 11.54 40°C, 26

5.82 20-100°C, 5.2 0.6 20-l00°C, 23.8

13 29

23

27°C, 22

1700°C, 50

Annealed, 27 400"C, 15

30-38 65

10-18 45

Annealed,

17.7 63

8.07 50

14

30.6

56

strength) 125

(As cast)

(As cast)

(Compressive 75

22.4

28

Ultimate 10' psi

strength

S

1.74 2.2-2.8

1315

IOO°C, 6.07

6.31

Cladding

200,,C1 0.42 I00°C, 0.033

I00°C, 0.022

See Tabic 22-363-C, 9.3-10.8 20-l00°C. 20 I90-300°C, 6.66

30-100-C, 11.5

25-l25°C, 45.8

Yield* I01 psi

Tensile

I

. . .

Magnesium. Sintered BeOt.

1.847

660.2

1845 1820

2.694

± ±

6.50 6.55

and

0.063

100"C, 0.090

I00°C.

Materials

±

Beryllium

V

(28).

25 25

±

Aluminum

U

.

. Diluents

15.38

I00°C, 6.59

6.649

Fuels and Fertile

Coefficient of thermal expansion, 10"' per °C

Materials*

2

Zirconium Zircaloy

632 2500-2600

19.60 10.96

25

1690

11.71

±

-650 1840

133

7

6.72

I

alloy. alloy. .

(UOi)

10

±

O Al-16 weight % Zr-4 weight Vc 00

Plutonium Uranium Oxide

Thorium.

+

19.3

of Reactor

Thermal conductivity, calAsecHciujrC)

Properties

Specific heat, cal/(mole)(°C)t

General

!

Uranium.

Materials

Melting point, "C

I

Density, g/cro1

Table


1.

400°C, 23

Annealed, 6. 400°C, 39

44

10.3

13.8 14

10.6 14.4

13-16 21

10.6

24

Modulus of elasticity 10« psi

5

1

13.36

2130

1

(

* strength, and the modulus

15 6.16

19

vary directly

25-2227°C,

25-32l°C,

of elasticity

specified.

±

t I|H

Cal/0?H°C). Thermal conductivity,

321

Control

6.

Compiled by the author from various sources. At room temperature except where temperature 0.2 per cent offset.

7-7sa +

8.694

2622 10 1690 241

10.2 4.507 8.57

J

steel

stainless

Boron

4140 +

purity)

0.036'

with the density;

0.22

Material*

tt°C, 0.32 25oC, 0.41

0-l00°C,

100°C. 0.037

0. I2|

25-IOO°C, 0. I09| 100-C, 6.24 0-500°C, 6.61 0°C, 6.01

I00°C.

Materials

I00<>C,0.37-0.48

Structural

2.066

Moderators

85

147.6 67.5

58.4 40.8

expansion

86.4 10.3

67.3

67 85-100 1200°C, 14.7

29.2 14

32.5 1-10

48 15.5

29

0.5-1.2

appears unaffected.

100-120

0.500-2.4

Annealed.

of thermal

40 70-85

35

however, the coefficient

0-100°C, 5.9

20-100°C, 31.8

20-l00°C, 11.5 5.1 25»C, 8.

20-l00oC1 16.5

20-2SO°C. 1.9-4.0

5

to

steel (SAE

(commercial

±

Boron

Titanium

1395-1425

8.51

at 25-C

1427

Sublimes 3650

8.027

2.27

Solid


±

7.

REACTOR MATERIALS

10-20

TEMPERATURE

[Sec.

10

(CI

Ferrous Metals ArmcoIron Carbon Steel GrayCost Iron SAE 52100 U45Cr, LOCI Sicromo 5S (5Cr,0 5Mo, l.5Si Z-9Cr,0.5-IMo Steels 18-e StamiessSteels. Type 310S.S. (25Cr, 20Ni) Ferritlc StainlessSteel (l6Cr) Worthite(20 Cr,Si, Mo.Cu). Tool Steel (I8W,4C£,IV1 Invar[36NO 9Ni Steel Femico(28Ni, IBCo) Alntco5 (Ni.Co.Al.Cu) Hodfield's Mn Steel Nitralloy G (nitrided)__ 60Ft-20P Brazing AJJpy


Nickel and Nickel Alloys Nickel lnconel(l3Cf,6.5Fe) Nichrom*tl5Cr,25Fej. Monel(30Cu) Hastelloy*A,B,C,D Ni-Mn,NI-Mo,Ni-P grazing Alloys Copper and coppc Alloys Copper(OFHC andR deox.) Copper(electrolytic) Be Copper (2 Be) Al Bronze (5 to BAD CuproB Super Nickel(20 S 30 Ni) Brass (40Zn) Nickel Silver (17Zn,l8 NO Ambroc (5Zn, 20Ni) Tin Bronze( 10Sn) Everdur l3Si, IMn) TrodolloyNo.l (2.5CO,O.SBe)^ Refractory Metals Molybdenum Tantalum Wolfrom Chromium Other Metols Co and High Co Alloys 2S and 3S Aluminum 24S and 52S Aluminum Al-Si Eutectic(l2S0 _ Sb,Bi,Cd,Ca,Au,Pb,Se,Ag,S.Sn Magnesium, Ptand SI Graphite(CI Non-Metols AijOj (Sapphire & Alundum) Haenium& BG Aluminite BeOUery dense)_ _ MgO_ ThOj, ZrOi, Porceloin Pyrei.Vycor B OtherGlosses Dorhoid Asbesta Silicone Rubbers Teflon

TEMPERATURE (F)

m

-

considerfor long-timeuse — for short-timeuseonly — no structural posslblliti'M ZZZZ POOR ('" 1 UNKNOWN— no data for these temperature* GOOD

ESE3 LIM1TE0

Fig.

["Liquid 1. Resistance of materials to attack by sodium and sodium-potassium alloys. Metal Handbook" NAVEXOS P-73'3 (rev.), 2d ed. (revised), U.S. Atomic Energy Commission and Department of the Nary, January, 1954.]

properties of reactor materials

Sec. 10-2] LIQUID

10-21

IMS

METAL -

MATERIAL FERROUS METALS

"■■□HO

Purt Iron

■HDBHBMHHD

tor bon Steal (Low-Ca Mild-C)

□HDHDEinHnnn BB

Gray Cast Iron f* - ' £

_iWMH^M

EES

■■^□■□□■□HDl

□□■!□□□□□□□[

□HpngnnanHDnsnni

'~iOODBKBMEDDBB

Cr-Ni Austinitic Stainless [with Nb or Mo os req.l High-SpeedHigh-W Tool SteeUwithCr.Mo ondV) High-Nickel 5tee

£WBT

NON-FERROUS METALS

3DHDHHDE

Bi,Co,Cd,P6,Sb,Sn

IDHBHBHDDE

IB B«rylllum

ODHBHnnDBBBHH

Chromium

. or Be

□□ □■□□□□□□□I

Al

Manganese

mmmumm

□■□□0

Hostelloys A.B8C

□□□□HE □@D@E

□HPDDDHC

Molybdenum

□□□□□□□□□E

^■□□mdedh

Nichromeand conel^

wmnnmwn

BDBSDc!

HOBpuar □□□□□□HE HBppQpbpPC □□□□E gjg

Platinum,Gold& Silver Silicon Steilite NflRt* & Other Co-Cu Alloys with Wor Mo

nDDDBD

mmnomnnou\

8ross,Tln,Bronzt,Etc.


HH0BBQ

too JOOL_J

"□□□□□□DC

□□□□E

aBHDnnEMDL: □□□□HE anHBDHnHnL: □□□□HE

□□□§□£ □□□□HQ

aDHBaBnanDDnDDBu .11 DaDinannnnDH HnBHDEODBBHH HBHnnnnnni TddpuHp

NON-METALS Ai20j a BeO(dense) Grophite MgO (porous)

^□□□□□■□i !□□□□□□

Poreelama Other Silicate Refractories Pyrex Gloss

tio,

Ha

a ZrO,

Ouorfz(fused Si02) LIQUID

METAL—

^

p"j

pr^ pr^py^prj p-*^

^^□□□□□□□□□□□□H

BHBBBBBBBDBBDaL

iRKHftHW'

f REE OF RESISTANCE

GOOD LIMITED POOR UNKNOWN-

Considerfor long-timoum For short- timeuM only No structural potsibllitits No data for these temperature

• Materialsweretested'at the mettingpointsof thesemetals

Fio.

("Liquid Metal Hand 2. Corrosion resistance of various materials to liquid metals. (rev.), 2d ed., U.S. Atomic Energy Commission and Department of

book," NAVexos f-733

the Navy, January,

1954.)

REACTOR MATERIALS

10-22 Table

2.

Density, g/cm3 Density, g/cma

017.8 Reduction of Liquid, 665°C, by 500°C UOi H. 16.54 ± 0.08

Pressed and sin tered powder 18.9

Mallinckrodt

Thorium Theoretical

25°C.

all.

al9.8

71

0~ M.I

at I400°C

Cast 11.55-11.63

10.86

-0.

I6°C, 6.524 27'C, 6.649 227°C, 427°C, 627°C,

7.606

8.952

11.107 10.147 774(o)°C, 9.177 827°C, 9. 147 727°C, 8.337 827<>C,10.430

774WC,

Electrical resistivity, jiohm-cm

Plutonium

>I8.04 Wrought material 18.5-19.0

-I49.70°C, 5.639 -99.57°C, 6.053

Entropy

UOi

Theoretical 10.96

Melting point, °C . . . 1133 ± 1 Boiling point, °C. . 3900 Heat of fusion, 2.5 3.0 Heat of vaporization. 93 kcal/molo -249.28°C, 1.211 Specific beat, -20I.6I°C, 4.657 cal/(mole)(°C)

Enthalpy (Hr)

Uranium oxide,

Theoretical

al9.3

27°C,

Arc-melted iodide

powder, tapped 10.28 2500-2600

11.66 1690 + 10 >3000

632 ± 7

4.6 137

-258-C, 0.378 -253°C,

80.46 ± 0.34

0.968

-248°C,

198. 7°C. 6.61

2.270 5.150 2.425 2.637 3.365

-243°C,

-238°C. -233°C, -223°C, -I73°C, 6.958 -I23°C. 10.07 -73°C, 12.47 -11°C. 14.17 + 27°C,

15.38

Hr-Hm.,.

400°C, 1.680 600°C, 5.340 800°C, 9.250 I000°C, 13.280 UOOOC, 17.420 MOO'C, 21.620

12.052

I27°C, 13.941

227°C, 15.601 327°C. 17.056 427°C, 18.387 527°C, 19.646 627°C, 20.882 727-C, 22.760 827°C, 24.761 20°C, 29.4 X I0« I00°C, 35. 1 X 10" 200°C. 41.4 X 10" 300°C, 46.7 X I0» 400°C, 50.8 X I0« 500°C, 54. 1 X 10" 600°C, 56.3 X 10'

130

99.3°C, 6.59

20°C, 12.7

ohm-cm

X

10'

25°C. 200°C, 300°C. 400°C, 500°C,

150 116 110 102 120

25°C.

13.6 ±0.8

20°C,

18

Density 10.9 g/cm»

Thermal conduc tivity, cal/(sec) (cm)(°C)

Thermal expansion,

io-v°c

10

Physical Constants of Uranium, Uranium Oxide, Thorium, and Plutonium' Uranium


[SEC.

20°C, 0.057 I00°C, 0.061 200°C, 0.066 300°C, 0.071 400%:, 0.077 500°C. 0.082 600°C, 0.088 700°C, 0.094 725°C, 0.096

(See Table 3)

I00°C, 0.022 0.019 0.014

200°C, 400°C, 600°C, 800°C,

I00°C, 650°C.

0.090 0. 108

0.011 0.009

I000°C, 0.008

22-363°C.

9.3-10.8

' Compiled by the author from various sources.

Alpha Pu, cr55

0H

30-l00°C. 11.5 30-500°C.

11.9

30-l000°C, 12.5

Sec. 10-2]

PROPERTIES OF REACTOR

MATERIALS

10-23

igneous rocks with much lesser amounts found in sedimentary rocks (sandstones). Uranium has been found in sea water, river water, and living organisms. Pitchblende (U02) is the richest of the uranium ores but accounts for a relatively The bulk of the uranium now produced small amount of the uranium production. comes from ores containing relatively low amounts (about 0.25 per cent U3Os) of The uranium occurs in these deposits as uraninite (UO») or in combination uranium. and roscoelite with vanadium in a mixture of carnotite [K(UO«).(VO<)s'3HsO]


[H,K5(Mg,Fe)(Al,V)4(SiO,)„]. Most of the foreign production of uranium

comes from the pitchblende deposits at Great Bear Lake in Canada or the Belgian Congo in Africa and from low-grade The South African uraninite deposits at El Dorado in Canada and in South Africa. ore represents tailings from a gold-recovery operation involving the cyanide process. Domestically, uranium occurs as low-grade carnotite and roscoelite in Colorado and Utah. Less extensive deposits of uraninite occur in several other states. Extraction and Purification. The relatively high-grade pitchblende is frequently concentrated by mechanical processes prior to extraction of the uranium by chemical methods. Low-grade ores are treated directly by either an acid leach or an alkaline leach process. The final product of any of these methods is UjOg of varying degrees of purity. Metal Preparation. The production of metallic uranium is made difficult by its extreme chemical activity with the atmosphere, most of the ceramic materials, and other metals. Historically a number of techniques involving reduction of the oxides or halides with aluminum, carbon, and the alkali metals were investigated. Efforts are now concentrated on the reduction of the halides by the alkali metals. For some applications requiring metal of unusual purity an electrolytic process involving uranium halides is used. 1.22 A number of the physical constants for uranium, Physical Constants. uranium oxide (UO»), plutonium, and thorium are listed in Tables 2 and 3. More complete data on uranium may be found in the volume "Chemistry of Uranium," by Katz and Rabinowitz.1'*

Table 3.

Thermal Expansion of Uranium* Linear thermal expansion,

I0~V°C

From X-ray data, parallel

to axis 25 I25°C

25-325°C

25-650°C

-

21.7 26.5 36.7 (100) 1.5 -2.4 -9.3 (010) 23.2 23.9 34.2 (001) Volume 45.8 61.5 48.6 (100) 1. = 1,7-c (1 + 17.2 X 10 •<+ 30.8 X 10-V) 9.2 X IO-«i + 40.4 X 10 (010) li = l.j-c (I 67.5 X IO->««) l,7°c (1 + 25.1 X 10 'I 21.3 X I0 »f» + 57.5 X \0 '1') (001) 1. Volume V, Vn'c 0 + 45.0 X 10 •( 3.4 X 10 + 5.0 X I0"t>)

-

-

-

-

-

-

* Based on data of C. M. Schwartr and D. A. Vaughn,

Battelle Memorial Institute.

There

is another type of material that is fissionable about which much less is known. type includes compounds of uranium other than the oxide. Such elements are potentially useful in reactors as uranium-bearing material in a matrix of a strong, corrosion-resistant metal such as zirconium or stainless steel. Some properties of these compounds are listed in Table 4. 1.23 Transformations and Crystallography. Uranium metal exhibits three crystalline forms. The a phase, which exists up to 640°C, is orthorhombic with 30 atoms per unit; cell dimensions are a0 = 2.854 A, 60 — 5.867 A, and Co = 4.957 A. The 0 phase exists between 660 and 760°C. Its crystal structure is tetragonal with 30 atoms per cell. Cell dimensions are a<>= 60 = 10.590 A and Co »= 5.634 A. The

This

* Superscript numbers refer to References at end of subsection.

10-24

REACTOR

MATERIALS

[Sec.

10

y phase, which is tetragonal, exists from 760°C up to the melting point. Its structure is body-centered cubic with ao = 3.474 A. 1.24 Health Hazards. Uranium constitutes a health hazard both from whole Health-physics groups of the various body radiation and from inhalation of its dusts. laboratories and the Atomic Energy Commission are in a position to give competent

advice on this problem. 1.25 Mechanical Properties. Values for a number of mechanical properties of It will be noted that many of the results uranium are given in Tables 5 through 13. The technology of uranium is not sufficiently are based on a single experiment. mature to have built up a large number of accepted average values for the various properties. Table 4.

Properties of Uranium Compounds

Compound

UO»

Cubic

UC. UN UAli

fee bo tetragonal foe fee Simple cubic

UC


UA1,

UAL UiSi.t USi USU

raw

USi.

U.Fe UFe,

UCui U.Ni UNi UNi, UNii UBen U.Ti U.Mn UMn.

Density,

Structure

g/om>

10.96 13.63 11.68 14.32 8. 14-8. 38

6.8 5.7 ± 0.3

Orthorhombic Tetragonal Orthorhombic

12.20 10.4 9. lip

Hexagonal

Tetragonal Simple cubic

8.98a 8. IS 17.7 13.21 10.60 17.6

bet fee fco bet

Too complex for present identification

Hexagonal foe fee Hexagonal bet fco

13.46 11.31

4.37 15. 22 17.8 12.57

Decomposition °C

temperature,

~2I75 ~2250 2480 -2630 ± 50 1590 1350 730 1665 1575 1700 1610 1510 815 1235 1052 780 810 985 1295

-2027 ~870 726 1120

* Compiled from H. A. Sailer and F. A. Rough (eds.), "Compilation of U.S. and U.K. Uranium and Thorium Constitutional Diagrams," Battelle Memorial Institute, June 1. 1955. t These formulas have been identified by X-ray study but are frequently still referred to on con stitutional diagrams as UsSii and UiSij, respectively.

Table

6.

The Effect of Carbon Content on the Tensile Properties of Uranium* Carbon content,

ppm

60 210 360 550 630 710 820 1.250

Total impurities, ppm

260 410 525 650 815 1.000 925 1.800

0.2%

offset yield strength, I0> psi

20 22 28 29 28 28 25 25

Ultimate strength, 10" psi

52 53 59 72 75 70 62 74

Elongation in 2 in., %

8 8 8

II

17 12 9 10

t

Modulus, 10* psi

23 24 23 25 24 23 30 24

* Based on data from Los Alamoa Scientific Laboratory. t All specimens cut from cast plates and pulled in as -cast condition; specimens were standard 0-50>-in.

rounds.


Sec. 10-2] Table

Tensile Properties of Uranium at Elevated Temperatures*

6.

Test Speeiment

temperature,

°F

570°F rolled:

III0°F rolled:

0.2% offset yield strength, psi

Ultimate strength, psi

III. 100

Elongation, %

Room Room

43.300 24.550

Room Room

26.000 24.600

88.400 61.700

13.5

570

17.500

35.200

49.0

570 570

18.800 15.300

31.570 26.900

43.0 33.0

930 930

5.100 7.100

11.130 10,500

61.0 44.0

930

5.450

10.450

57.0

570°F rolled:

III0°F rolled:

63.800

570°F rolled:

III0°F rolled:


10-25

* Compiled from data of G. T. Muehlenkamp and A. D. Schwope, unpublished information. t All specimens were standard 0.505-in. rounds. X Alpha anneal is 12 hr at II I0°F, slow-cooled, Beta anneal is I 2 hr at 1290°F, slow-cooled.

6.8 8.5 6.0

Battelle Memorial Institute,

f

Table

7.

Low-temperature Tensile Properties of Commercial Uranium* in Various States of Heat Treatment!

Heat treatment

As oast

Teat tempera ture, °F

75

-50

-100 Aa ..-rolled (1020°F)

75

--50

100

a-rolled and 7-annealed ( 1560°F) a-rolled and 0-quenchcd (I335°F) a-rolled, 0-quenched (I335°F), and cr-annealed (I020°F)

75

-50

-100

75

-50

- 100 75

-50

- 100

0.2%

offset

yield

strength, 10' psi

Ultimate strength, 10>psi

Reduc tion in area,

28 28 27

56 47 42

10 4 3

31 40 37

96 109 87

10 4 4

26 27 17

57 43 36

10 4

36 40 42

84 92 88

32 33 37

90 83 89

%

Elonga tion in 2 in., %

Modulus of elas

ticity.

Poisson's

ratio

I0« psi

24 26 26

0.21

25 24 23

0.22

4 3 5 3 3

21 31 26

0. 18

12 6 5

24 27 30

0.24

5 4

12 5 6

12 5 6

22 31 27

0. 19

3 3

II

* All specimens were standard 0.505-in. rounds. t Based on unpublished data of S. J. Paprocki, Battelle Memorial Institute.

Melting and Casting. The chemical activity of uranium makes melting and casting by ordinary techniques impractical. In induction melting, the use of vacuum, inert atmosphere, or protective flux or slag with graphite crucibles yields ingots of good quality of almost any size and shape. For special applications where high purity is desirable or for some alloys, arc melting 1.26

Manufacture.

with the atmosphere

REACTOR MATERIALS

10-26 Table

Young's modulus,

29 1 29. 3 30 3 29.5

7-extrudcd

Cast Cast

Table

Test

Material

tempera ture, °F

Normal uranium, as cast (500 ppm carbon, 700 ppm total impurities)

-5 75 300 390 480 570

-5

impurities)

1,000 ppm total

300 390 480 570

* Data from Los Alamos iScientihc

Table 10.

0.5% offset yield strength. 10>psi

information.

85

III

86 81 60 48 90 87 63 67 42

Deforma tion at propor tional limit, in.

Deforma tion at

0.016 0.044 0.034 0.017 0.018

0.022

0.080 0 070 0.080 0 084 0.086 0.092

0. 174 0.083 0. 153 0. 176 0. 290 0 364

2 6 5 3 3.1 2.2 1 7 1.2

0.036 0.034 0.013 0.018 0.023

0. 133 0.081 0.077 0.089 0. 110

0 205 0. 142 0.205 0 264 0.427

1. 1 3.0 2.1 1.7 0 7

Eberbach (263-g load) Eberbach (645. 6-g load)

Deforma tion at 120.000 psi, in.

yield strength, in.

Modulus in com pression. I0« p«i

information.

Hardness of Alpha-rolled

Rockwell G

J. J.

0 22 0.21 0.25 0 22 0 24 0 23

Laboratory, unpublished

Laboratory, unpublished

Scale

5. McGraw-Hill t All readings

ratio

Compression Tests on Uranium*

9.

Normal uranium, or-rolled (600

Poisson's

11.9 12.0 12. 1 12. 1 12. 1 12. 1

30.0 29.7

* Data by Henry L. Laquer, Loa Alamos Scientific


Shear modulus, 10« psi

I0« psi

•y-extruded

ppm carbon,

10

Young's Modulus, Shear Modulus, and Poisson's Ratio for Uranium in Various Conditions of Fabrication*

8.

Condition

* From

[Sec.

Uranium

Transverse specimen

97 80. 5 220 240 250 260 250 240 270 260

*

t

Longitudinal specimen

98 83 240 250 260 280 320 300 320 300 290

Katz and E. Rabinowitih, "The Chemistry of Uranium," part I, NNES. Div. Book Company, Inc., New York, 1951.

VIII.

vol.

taken on the same sample.

is used successfully. Either an inert electrode (graphite, tungsten, or molybdenum) or a consumable uranium electrode can be utilized. This latter technique produces metal of unusual purity. Forming and Fabrication. Uranium metal is reasonably ductile and can be fabri Hot fabrication is normally carried out cated by standard metallurgical techniques. Fabrication in the 0 phase is possible but very difficult at a-range temperatures.

Sec. 10-2]


10-27

Gamma-phase uranium is too soft of the close temperature control required. In the a range and plastic for fabrication by techniques other than extrusion. fabrication by extrusion, forging, rolling, pressing, and swaging is accomplished Because of the reactivity of uranium with air, heating is performed in an readily. Uranium can be fabricated at room inert atmosphere or other protective media. because

Table 11.

Charpy Impact Strength of Uranium* Impact strength,

ft-lb

°F Forged t

a- rolled t

-290 -100


-75 -50 -25 0 50 75 100 150 200 300 390 480 570 610 615 645 750

5 6 8 10 10 12 14 14 15 18 25 32

As castf

0-qucnchedt

0-quenched, or-annealedt

?-annealedf

4

6

II

8 10 12 12 12 13 14 15 17 23 27

8 8 10

6 7 7 8 9 10 12 14 20 27 52

5 6 7 7 8 9

II

12 15 18 29

II

13 16 16 17 22 28 50

18 36 70 85

57 76 85

102 109

* Compiled from data of Eric Jette, Los Alamos Scientific Laboratory, and S. Memorial Institute, unpublished information. + Normal purity. $ Hitch-purity material: 240 ppm carbon, 333 ppm total impurities.

Table

12.

The Shear Strength of Uranium

J.

Paprocki, Battclle

*f

Shear strength,

I01 psi

Condition} >>i-in. diam

Aa cast Cold-swaged If Cold-swaged as above and annealed at I I00°F. . . Rolled at 11 I0°F Rolled 62}$ % at 11 I0°F and rolled 68°; at 615°F Rolled aa above and annealed at 11 I0°F

85 127 83 80 113 83

^-in. diam

sB-in.

84 114 88 79 95

diam

75 95 80 77

* Data of Eric Jette, Los Alamos Scientific Laboratory, unpublished information. t Normal uranium 430 ppm carbon, 613 ppm total impurities. X All samples tested at room temperature. Vii-in. specimen, 94 per cent; }i-in. specimen, 75 per cent: *6-in. specimen, 44 per cent.

i

Frequent annealing during temperature by drawing, rolling, swaging, or pressing. cold fabrication is desirable. A Powder Metallurgy. Uranium is amenable to powder-metallurgical techniques. good grade of uranium powder is produced by first treating the metal in hydrogen This hydride is then decomposed by heating at about 450°F to produce the hydride. Metal compacts of the desired shape can be produced in vacuum at around 900°F. Uranium compacts of good either by cold pressing and sintering or by hot pressing.

10-28

REACTOR

MATERIALS

[Sec.

10

quality can also be produced directly from the hydride by cold pressing and vacuum sintering or hot pressing in a vacuum. Joining. Uranium can be welded using the Heliarc process and the shielded-arc Again, the principal problem lies in the reactivity of consumable-electrode process. Brazing is made difficult both by the reaction with the uranium with the atmosphere. atmosphere and by the formation of brittle intermet allies between uranium and It is possible to join uranium by first plating the uranium with silver brazing alloys. or nickel and then joining the silver or nickel by conventional brazing methods. Table 13.

Rolled at II00°F, 10

hrat

10 hr at I290°F, 1110°F

Rolled at MOOT,

10 br at

Time, hr

Stress, psi

Annealing history

and

Creep of Uranium at 930°F*

MI0°F...

Total deformation, %

2.750 2.220 1.720 1.500

1.530 1.200 1.700 1.515

3.72

2.220 1.720

1.200 1.350

4.72

* Data of G. T. Muehlenkamp and F. R. Shober,

2.20

0.82 0.85 1.92

Minimum secondary creep rate,

Grain aiie after test, mm

%/hr

0.0025 0.0013 0.0003 0.00035 0.0018 0.0009

0. 12-0. 13 0. 10-0 II

0.045-0 05 0.05-0.055

Battelle Memorial Institute, unpublished

infor


mation.

Uranium can be machined by most of the conventional methods Machining. It is advisable to use heavy cuts and provided some special precautions are taken. A coolant must be used both as a health measure and in order to prevent high speeds. burning of the chips or even the machined piece. 1.27 Corrosion Behavior. Uranium has very poor corrosion resistance in either air or water and thus must be protected by means of cladding or plating for reactor Figures 3 through 6 show corrosion rates in air, steam, hydrogenapplication. saturated water, and distilled water. I00O

800i

0

2

4

6

U METAL REACTE0,

8

10

12

14

WT-PER CENT

Fm. 3. The oxidation of uranium in air for 4-hr periods from 68 to
50 10 20 30 40 METAL REACTED, WT PERCENT

FlQ. 4. The action of steam on uranium various temperatures.*

ai

Uranium and sodium in contact for long periods of time at 550°C show no evidence of reaction; however, uranium must be protected by means of cladding or plating when in contact with either air or water in reactor application. The poor corrosion resistance of uranium in reactor coolants Protective Techniques. After the use of extensive has encouraged the development of protective techniques. cleaning procedures, uranium can be electroplated with a number of metals, using conventional plating baths; however, protection by plating is often precarious. Uranium has been clad with a number of metals by means of roll cladding. * Reprinted from T. Wathen, Corrosion of Uranium Metal in Air and Steam turea, BR-223-A, May 13, 1943.

at Various

Sec. 10-2]


10-29

1.28 The need to improve corrosion resistance and strength of uranium Alloys. A large number of binary systems have has lead to considerable interest in alloying. been investigated in some detail.* Uranium alloys might be grouped as systems containing several intermetallic com pounds and those having no true compounds. uuu = \ This latter group may be subdivided further into systems showing limited solid solubil ity and those showing considerable solid


solubility. Many elements form

100

or more com Such systems have pounds with uranium. limited solid solubility in the terminal phases. Compound-forming elements whose diagrams are well defined include aluminum, beryllium, bismuth, carbon, cobalt, copper, gold, iron, • lead, manganese, mercury, nickel, silicon, and tin. Other elements, including boron, hy 10 drogen, oxygen, and nitrogen, form com pounds, but their constitutional diagrams are not completely defined. 0.1 Several elements when alloyed with urani E um do not produce intermetallic compounds or show any appreciable solid solubility in — the terminal phases. Such systems contain i i i i 1 i 1 1 eutectic, peritectic, or monotectic reactions. 2.5 3.0 19 2.0 and uranium-vanadium Uranium-chromium 1000 follow the simple eutectic system. A periTCK) tectoid reaction characterizes the uraniumFio. 5. The corrosion rate of uranium in tantalum and uranium-tungsten systems. hydrogen-saturated water versus l/7\* Thorium, silver, magnesium, and cerium Elements of another group when added to uranium form monotectics with uranium. produce alloys with extensive solid solubility at elevated temperatures but no true Both zirconium and titanium form this type of alloy. compound. intermetallic Molybdenum behaves similarly except that solid solubility is not complete at elevated one

400

300

200

100

25

50

75

TESTING

Fig.

6.

100

125

150

TIME. DAYS

Corrosion of uranium in aerated distilled water.*

The uranium-niobium system is similar but contains no intermediate temperatures. phase as found in the others. The intermediate phase found in uranium-zirconium, and uranium-titanium systems is formed from a mctastable 7 uranium-molybdenum, Since it is ductile, it is not considered a true intermetallic compound. phase. * Reprinted from W. A. Molliaon, et al., Corrosion of Tuballoy in Distilled Water. University of Chicago, CT-3055, June 23, 1945.

REACTOIi MATERIALS

10-30

[Sec. 10

A number of the compound-forming elements are useful in improving the mechanical Large additions of such properties of uranium when added in small quantities. The greatest improvements in highmerely form brittle compounds. elements These temperature mechanical properties are likely in the solid-solution systems. would include alloys with molybdenum, niobium, titanium, and chromium. Tensile Properties of Forged Uranium -Aluminum

Table 14.

Average modulus of elasticity, f 10t psi

Uranium content, weight

%

10.0 10.4 10.9 •11.3 11.3

0 12. 5 22.7 22.7 30.5

Tensile strength, 10» psi

0.2% offset yield strength, 10' psi

Elongation.

5.0

13.0 22. 5 18.6 23.5 26. 1

Alloys* Reduction

%

45 20 4 13 10.5

10.8 11.6 14.5 14.9

of area, %

34 7 14 11.5

" Preparation, Properties, * From H. A. Sailer, Alloys," and Cladding of Aluminum-Uranium Battelle Memorial Institute, published at the International Conference on Peaceful Uses of Atomic Energy. UN-562. USA, 1955. Single specimens were used except in the 22.7 per cent uranium alloy. t Estimated to be correct within ± 3 per cent.


Table 15.

Alloys*

Thermal Conductivity of Uranium-Aluminum

Temperature, °C

Thermal conductivity, cal/(sec)(cin)(°C).t of indicated uranium content, weight %

200 300 400

12.5

22.7

30.5

0.44 0.43 0.43

0.40 0.39 0.38

0.36 0.36 0.35

* From 11. A. Sailer, " Preparation, Properties, Alloys/* and Cladding of Aluminum-Uranium Battelle Memorial Institute, published at the International Conference on Peaceful Uses of Atomic Energy, UN-562. USA, 1955. X (2.419 X I0») = Btu/(hr)(ft)(°F). tCal/(sec)(cm)(°C)

Table 16.

Linear Expansion of Uranium -Aluminum Coefficient

Temperature

of linear thermal expansion, uranium content, weight %

10 V°C

°c

20-100 20-200 20^300 20-400 20-500 100-500

Alloys*

Ot

12.5

22.7

30.5

23.9 24.6 25.5 26.5 27.7

20.0 21. 1 22. 1 23. 1 23.5 24.4

20.0 21.2 21.9 22.5 22.7 23.2

19.4 20.8 21.3 21.6 22. 1 22.6

* Compiled hy the author from the "ASM Metals Handbook," of Rattelle Memorial Institute, 1952. t "Metals Handbook," 1948.

1948 edition,

and unpublished infor

mation

The greatest improvement in corrosion properties is found again in the solid-solu Small alloy additions, which produce supersaturated solutions of tion-type alloys. a uranium, are helpful initially, but these alloys may lose their enhanced corrosion resistance on aging. Somewhat larger alloy additions, which will produce a super-

Sec. 10-2]

properties op reactor materials

10-31

Unfortunately, such alloys may transform saturated y phase, are even more helpful. and lose their corrosion resistance. Stable alloys of the intermediate phase may offer the most promise for corrosion resistance. A few intermetallic compounds such as UjSi have shown excellent corrosion resist ance. The use of such compounds is limited considerably by their lack of fabricability. Table

17.

High-temperature

Temperature,

Alloy

Room

11.3 weight % uranium

Tenaile strength, 10' psi

13.0

300 570

7.5 2.5

Room Room

19.7 20. 1 14.1 14.7

300 300 570 570

Room 300 300 300 570 570

0.2% offset yield strength, 10' psi

Elongation,

Reduction in area.

5.0 3.5

452 65 90

11.3 11.5

28 28 31 27 31 34

9.2

10.0

8.6 8.8 19.0 14.0 13.7 13.3 8.1

1

8.6 7.2 7.0 6.5

30 34 35 35 48 42

8.2

57 60

59.5 58

* Data compiled by the author from various sources. t Tests in metal annealed at 700" K for I hr; 2S-0 values from "Alcoa Aluminum and Its Alloys." X Elongation in 2 in., others in 1 in.

Table

18.

Densities of Uranium-Zirconium

Alloys*

Density g/cm' Uranium content. weight %

0 2 4 6 8 10 12 14 16 60

Arc-melted and rolled alloys

6.54 6.64 6.72

Pressed and sintered alloys

6.59 6.76

6.83 6.91

6.99 7.07 7.16 7 20 10.40

7.24

It-

* II. A. Sailer and F. A. Rough, "The Alloys of Uranium," Battel Memorial Institute, International Conference on Peaceful Uses of Atomic Energy, UN-558, USA, 1955.

is

a

The selection of an alloying element as diluent for enriched uranium in a power based principally on nuclear rather than metallurgical reasons. This reactor shown by the interest in the low-cross-section metals zirconium, aluminum, and beryllium as diluents for thermal reactors. When nuclear properties are about equal, fabricability and physical properties become the important criteria. Important properties of the fuel-element alloys uranium-aluminum and uraniumzirconium are given in Tables 14 through 20. is


Properties of Uranium-Aluminum Alloys*-

REACTOR MATERIALS

10-32 Table 19.

Uranium-Zirconium Alloys at 370°C*

Tensile Properties of Arc-melted Zirconium content, weight %

0


Tensile strength, psi

31.400 48.400 77.000 122.800 135.200 145.200 115.300 96,100 97.800 86.400 77.900 64.200 17.300

21.300 35.100 50,500

2.4 5.5 11.5 20.5 31.4 41.7

100.800 124.000 112.000 97.000 68.100 63.800 64.800 72.000 53.200 8.500

50.5 60.6 70.4 80.0

89.4 100.0

10

[SEC.

Klongation in 2 in.,

Reduction in area, %

27.0

57.0 49.0 25.0 29.0 18.0

17.0 10.0

8.0 8.0 7.0 8.0

7.0 13.0 28.0 20.0

22.0 22.0 20.0

27.0

5.0 7.0t

22.0 Ot

II.

42.0

* H. A. Sailer and F. A. Rough, "The Alloys of Uranium," Hat telle Memorial Institute, International Conference on Peaceful Uses of Atomic Energy, UN-558, USA, 1955. t Specimens broke in what appeared to be rolling defects.


Table 20.

Thermal Conductivity

of Uranium-Zirconium Alloys*

Conductivity, watts/ (era (°C), by composition, we ight % uraniumt

°c 95

80

60

0.260 0.270 0.285

0.193

0. 106 0. 125 0. 148

0.068

0.306

0.254 0.279 0.307 0.338 0.370 0.406 0.443

100}

20 100 200 300 400 500 600 700 800 900

0.330

0.360 0.393 0.431 0.475

0.522

0.209 0.230

0.172 0.197 0.224 0.253

0.282 0.313 0.344

50

0.077 0.096 0. 118 0. 139 0. 167 0.200 0.240

0.053

0.066 0.084 0.104 0. 127 0. 153 0. 183

0.286

0.216 0.252

325

0.290

14

3

30

22

0.090 0. 100 0. 110 0. 120 0. 140 0.160 0. 180

0.097 0.106

0.110

0. 0. 0. 0.

0. 123

0. 140 0. 142 0. 144

0.130

0.147

0.210

117 130 144 161

0.187 0.220

0.116

0

0.208

0.198 0.188 0.182 0.179 0.178 0.178 0.185

0.250

0.290

* Compiled from data of H. W. Deem and R. A. Winn, Battelle Memorial Institute, unpublished information, 1955. t Arc-melted crystal bar zirconium. 24 hr at 575°C. All other compositions were treated I hr at % 100 per cent uranium heat-treated 800°C,

1 hr at 500°C, air-cooled.

1.3

Thorium

Abundance and Availability. While thorium makes up 0.0012 1.31 Production. per cent of the earth's crust, it is very widely distributed and much of it cannot be Actually, the workable deposits of uranium exceed those of thorium. mined. The common thorium mineral is monazite (CeLaDiJPOi-ThSiO,. Large deposits are Lesser deposits occur in Oregon, North and South found in India, Brazil, and Africa. Carolina, and Florida. Extraction and Purification. Several methods for extraction of thorium compounds from monazite sand are available. The sand may be treated with excess concentrated sulfuric acid to form a thick paste. This paste is slowly stirred into excess water and allowed to settle. The solution contains the bulk of the thorium and rare earths, perhaps as phosphate complexes. The thorium is then separated from this solution by partial neutralization, which causes fractional precipitation of thorium phosphate, or by a carbonate process based on the relative solubilities of the carbonates of thorium

Sec. 10-2]


MATERIALS

10-33

and the rare earths. There is a similar process based on the solubility of thorium oxalate. Still another process involved the digestion of monazite sand with caustic soda and solvent extraction of the thorium. Thorium, like uranium, is chemically active with the Preparation of the Metal. Therefore metal is prepared by reduction atmosphere and most ceramic materials. of the halides with alkali metals. High-purity thorium is produced either by the electrolysis of its halides or by the thermal decomposition of the iodide in the deBoer process. 1.32


Physical constants of thorium along with those for Physical Constants. uranium and plutonium may be found in Table 2. Transformations and Crystallography. Thorium has two crystalline forms. 1.33 The a phase, which exists up to 1380 ± 25°C, is face-centered cubic with a0 = 5.089 A. This constant increases to 5.161 A at 1200°C. The 0 phase is face-centered cubic with do = 4.12 A. Thorium, like uranium, is an a emitter and also the parent 1.34 Health Hazards. Considerable quantities of radioactive element of the radioactive thorium series. Radioactive products radon may be released during the grinding of thorium ores. may be released during the reduction and casting of primary metal. Information on proper techniques for handling thorium may be obtained from the AEC. Mechanical Properties. 1.36 The mechanical properties of thorium are influenced The effects of deliberate additions of carbon, greatly by minor amounts of impurities. oxygen, and nitrogen are shown in Table 21.

Table

21.

Effect of Carbon, Oxygen, and Nitrogen on Tensile Strength of Iodide Th* Analysis, weight

Carbon

Oxygen

%

Ultimate

Nitrogen

0.06 0. 10

0.24

0. 16

0.37

0.40

strength, 10>psi

Reduction of area. %

20 36 46 60 30 35 25

60 46 51 25 55 50 47

* Compiled from data of R. M. GoldhofI, H. R. Ogden, and R. I. Jaffee. "A Study of the StrenKthcning of Thorium by Alloying, Cold Work and Aging." BM1-776. November, 1952.

Room-temperature

properties of thorium are affected considerably by variations

in strain rate and cold work. Orientation in sheet material has only a slight effect. These data are summarized in Table 22. The variations of properties both in tension and compression with temperature are shown in Tables 23, 24, and 25. The Charpy impact strength of cast thorium varies from 1.5 to 10.5 depending on the amount of impurities present. While the relationship between tensile properties and impurity content is not fully established, the impact strength at room temperature Wrought thorium exhibits a transition tends to decrease with decreasing purity. from ductile to brittle fracture in the range from 210 to 390°F. Impact strength (foot-pounds)

increases

gradually from a value of

12

at

— 100°F

to 33 at 210°F.

It

then rises sharply to 84 at 390°F. This rise continues to a value of 95 at 450°F. The dynamic shear modulus of wrought and annealed thorium at room temperature is 4.62 X 10« psi. This value decreases gradually to 3.8 X 10« psi at 570°F. It then drops off sharply. Only exploratory-type tests have been run on the fatigue strength of thorium. Rotating-beam-type tests at room temperature on cast material gave an endurance limit of 12,000 to 12,500 psi for 5 X 10s cycles. Cold rolling increased this limit to

10-34

REACTOR MATERIALS Table 22.

10

The Effect of Cold Work on the Room-temperature Tensile Properties of Thorium* Propor tional limit,

Strain rate, in./(in.)(min)

Test direction!

[SEC.

psi

0.007 0.007 0.007

Parallel to RD Perpendicular to RDJ. . . . Parallel to KDJ Parallel to RD Parallel to KDJ Perpendicular to RDJ. . . . Perpendicular to KDJ. . . . Parallel to RDJ Parallel to RDJ

25.500 26.500 17.300 15.000 27.500 20.500 20.100

0.007 0.007 0.007 0.007

39.61 61

39.61


45.500 42.500 24.500 24.700 42.900 23.000 23.100 47.000 37.000 47.000

Ultimate

Reduction

Elonga

of area.

tion,

strength, psi

36

48.800 43,600

33.9 41.2

33.850 33.200 44.200 32.500 32.600 44.000 37.200 44.000

48.5 28 50 49 0 55.0 43

6.3

43.0

18.0001 5.0001 9.2006 2.000

Compiled from data of A. D. Schwope TeBts conducted in air. Tests conducted in argon. Sensitivity of strain measurement Sensitivity of strain measurement

§

and L. L. Marsh,

9.7 ±0.2 9.7 ±0.2 9.7 10.0 0.2 9.0 0.2 0.2 8.9

Poisson's ratio

0.25 0.27

± ±

10*

0.03 0.01

Jr., Battelle Memorial Institute.

in. /in. was in. /in. was 60

f

The Tensile Properties of As-cast Thorium*

Room 200 400 600 800 1000 1100

10' psi

17 10

3.5

Yield strength. 10' psi

25 17 12 10 5 6 8

•F

Proportional limit,

5 6

TeBt temperature,

3 3

Table 24.

23.900 26.200 30.500 26.000 13.800 17.000

17.0001 5,0001

Young*s modulus, psi

' ± ±

0.005 0.01 0.01 0.025 0.01 0.025


5 p

§ U %

t

*

75t 75t 75t 570t 570t

1

75t

psi

1

°F

Proportional limit,

Strain rate, in./(in.)(min)

1

Temperature.

Compression Properties of Hot-roiled and Annealed Thorium*

/t

Table 23.

Modulus of elasticity, 10* psi

22 21 19 17 15 14 13

I,

*

From G. C. Danielson et ah, Progress Report on an Investigation of Properties of of Its Alloys, Amet Laboratory Kept. 24, vol. 1951. These data are points taken from a line graph.

t


1

'-..

* Compiled from data of A. D. Schwope and L. L. Marsh, Jr., Battelle Memorial Institute. t The first four specimens listed were given 25 per cent cold work reduction; the remaining six 50 per cent. t Annealed 3 hr at 1350°F; grain size 0.010 mm. The tensile data were obtained by the was noticed. % At these strain rates, a yield phenomenon in. long, 0.250 in. wide, with a gauge use of test specimens J>2 in. long with a reduced section I length of in.

Th

and

Some


Sec. 10-2]

10-35

Bending fatigue testa on hot-rolled and annealed metal gave values of psi. psi for unnotched and 18,000 psi for notched samples at 107 cycles. There is a limited amount of information available on creep properties of thorium. is summarized in Table 26.

15,000 22,000

It

Table 25.

Test temperature, °F

The Tensile Properties of Hot-rolled Annealed Thorium* 0.2% offset yield strength. 10' psi

Proportional limit, 101 psi

Room 400 590 800 930

22 8 7

27 12

6

Ultimate strength, 10' psi

Modulus of

Elongation,

elasticity, 10" psi

%

10

II

37 23 22

9

40 32 38

9

17

8

50

Diamond pyramid hardness

90 78 66 60 40

* Compiled from data of A. D. Schwope and L. L. Marsh, Jr., Battel le Memorial Institute.

Table 26.

Stress, psi

Time,

°F

200 400 600

20.000 14.000 10.000

1.660 1.500 1.300

Temperature.


Creep Properties of Thorium*

hr

Total deformation,

Min creep rate,

%

%/hr

15.4 1.55 1.4

0.002t Nil 0.0001

* Compiled from data of A. D. Schwope and L. L. Marsh, Jr., Battelle Memorial Institute, Third-stage creep started after 1. 100 hr.

t

1.36 Manufacture. The melting and casting problem for Melting and Casting. thorium is quite similar to that for uranium except it is made even more difficult by the higher melting point of thorium. Techniques described for uranium are appli The selection of crucible material is somewhat more limited than in the case cable. Again arc melting, particularly with a consumable electrode, will of uranium. produce a very high quality product. Thorium is an extremely ductile material that can be Forming and Fabrication. fabricated by any conventional method to produce a wide variety of shapes. Hot working at about 600°C is generally recommended for the primary breakdown of cast The precautions suggested for the heating of uranium also apply to thorium. ingots. Fabrication to final size is easily done at room temperature. Annealing is not essen tial but may permit larger rates of reduction. Powder Metallurgy. Thorium powder of good quality is easily made by reacting the metal with hydrogen to produce the hydride. This hydride is then reduced by Powder metal objects can be made either by cold pressing and heating in a vacuum. sintering or by hot pressing. Joining. The welding characteristics of thorium are very erratic, probably as a result of variations in metal purity or welding conditions. Single-pass welds are easily made by the shielded-tungsten-arc process. Attempts to make a second pass over this weld invariably cause cracking. The addition of 2.5 atomic per cent of molybdenum, tungsten, or niobium greatly improves the welding characteristics of the metal. Thorium cannot be brazed with any of the usual brazes because of the formation of brittle intermetallics. A possible method is joining by first plating with silver or nickel and then brazing the silver or nickel. Thorium has machining characteristics similar to those of mild steel. Machining. It can be turned at rates of 150 to 175 fpm with high-speed tools. Coolants are

10-36

REACTOR

MATERIALS

[Sec.

10

The scale resulting from hot working is hard and likely desirable but not essential. Scale removal prior to machining is recommended. to dull the cutting edge of tools. Thorium chips are pyrophoric and should be removed frequently during machining. Thorium, like uranium, has relatively poor corrosion Corrosion Behavior. 1.37 in 7 shows the weight gain of thorium sheet in air Figure air and water. resistance In boiling distilled water, thorium becomes covered with at various temperatures. Table 27.

Alloy additions, atomic

%

0 0 2.5 Ag 1 Al 2.5 Au 5 B 5 Ba 2.5 Bi

I.C

2.5 C


SC 5C

5 Ca

5Ce 1 Co 5 Cr 5 Cu 2.5 Fe 1 Ga 1 In 5 In 5 Mg 5 Mn 5 Mo 5 N 5 Nb 2.5 Ni 5 P 2.5 Pb 5 8 5 Se 5 Si 2.5 Sn 5 Sr 1 Ta 5 Ti 2.5 Tl 5 V 5 W 1 Zr 1 Zr 5 Zr

The Effect of Alloying Additions on Yield-strength for Arc-melted Binary Thorium Alloys* 0.2-;

Heat treatmentt

2 2 2 2 2 2 2 2 1 1 2 2 1 1 1 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 | I 2 I | 2 1

offset yield strength, psi

29.000 19.400 21.200 24.600 20.700 20.200 21.000 24.800 27.200 57.600 52.100 29.400 19.600 31.000 28.800 32.800 23.100 23.800 20.300 22.500 35.600 19.400 30.800 23.000 23.600 14.700 23.800 21.500 24.000 14.000 12.600 20.500 23.000 22.100 31.800 10.900 24.200 20.900 31.800 23.500 45.000 18.000

Ultimate strength, psi

Elongation,

fi

in 1 in.

32 46

39.000 31.750 37.000 40.800 37.400 34.200 32.900 44.200 38.800 58.600 57.700 42.000 30.400 46,000 33.300 44.000 41.000 36.600 34.800 36.100 49.800 29.800 41.500 37.600 37.600 29.000 37.300 35.800 42.700 26.600 25.200 32.200 41.000 34.200 42.600 23.600 36.000 33.400 43.000 37.200 56.000 34.000

34 34 19 44

35 37 36 3 38 46 33 31 23 42 31 42 21 20 19 42 6 39 18 38 33 18 30 33 ' 12 23

Increment

VI

Reduction of area, %

Room temp.

46 49 31 43 37 38 50 32 48 19 27 36 38 41 10 44 43 37 48 38 40 51 50 48 45 37 42 42 47 24 50 35 50 54 50 48 56 31 31 25 43

* Compiled from R M. Goldhoff, H. R. Ogden. and R. L Jaffec, "A Study of Thorium information), (unpublished BMI-720. Dec. 26. 19 hr at 750°C; heat treatment 2. 2 hr at 850°C. t Heat treatment I

106 79 91 99 88 85 84 97 104 175 175 106

III

118 115 105 104 85 86 87 116 113 107 81 89 61 85 88 91 68 57 79 88 83 105 53 88 80 107 80 93 78

N

300T

60 4! 51 57 54 47 44 65 57 94 119 59 41

n

70 64 54 SO 55 58 85 87 64 <> 52 44 47 52 64 31 42 46 60 53 62

«

48 51 66 63 86 59

Alloys'

Rates of +0.03 to —0.006 mg/Ccm'Xhr) have an oxide but may lose or gain weight. been reported. The corrosion resistance of thorium in eutectic sodium-potassium alloy is excellent. w»s At 1100°F, the corrosion rate during a 6-day static test in sodium-potassium The corrosion rate increased with increasing oxygen content 0.006 mg/(cm*)(hr). Thorium is not attacked by lithium in 6 days at 1100*F in the sodium-potassium.


Sec. 10-2]

10-37

In pure bismuth at 1650°F, is completely disintegrated by gallium at 1100°F. severe attack occurs in 1 br. Under the same conditions, pure lead produces only slight attack in 1 hr. Lead at 1830°F dissolved a sample completely in 40 hr. Thorium can be Protective Techniques. electroplated with most of the common metals. Again, chemical pretreatment prior to plating is most important. Interest in alloying of Alloys. 1.38 thorium has been considerable, though not A so great as that shown for uranium. number of binary diagrams have been developed and are found in the compila tion by Sailer and Rough.1 The effect of various alloying elements on the yield strength and hardness of thorium is shown in Table 27. but

1.4

Plutonium

Plutonium offers another source of


fis

for reactor operation. While it has been considered primarily as a weapon material, its advancing tech nology will eventually make it available for use in reactors. Although minute amounts of plutonium have been found in uranium ores, its source is limited to that produced from uranium 238 in a reactor. Some of the physical constants of pluto nium are listed in Table 2. Additional on the allotropic forms of information TIME, HR plutonium is given in Table 28. Fio. 7. Corrosion of thorium in air at Plutonium is an a emitter and offers a (Compiled from data various temperatures. The major very serious health hazard. °f M- w- MalleU, BatteUe Memorial Inttidanger arises from the fact that plutonium tute- ™publx»hed information.) entcring the body is deposited primarily in the marrow of the bones and is then excreted very slowly. While plutonium may enter the body by ingestion, inhalation, or absorption through the skin, the principal Extreme precautions must be taken in handling the path is through inhalation. material in order to prevent the formation and inhalation of fine particles. Plutonium material

sionable

Table 28. Temperature Phase

Alpha.

ran tee of

stability, °C

.

Beta

Below 117 117-200

Properties

Crystal structure

Orthorhombio like uranium (doubtful) Unknown

of Plutonium

Density at 25°C, g/cm1

Metal*

Average linear expansion coefficient,

I0"V°C

Electrical resistivity, pohm-cm

Temperature coefficient

of

resistivity t

19.8

55

150 at 25°C

-29.7 X

17.8

35

110 at 200°C

Zero (approx)

36

110 at 300°C

Zero (approx)

102 at 400°C 120 at 500°C

+ 1.5 X 10* Zero (approx)

10-<

(complex)

Gamma . . . Delta Kpsilon

2(1(1 101)

Unknown

300-475 475-637

(complex) Fee Bee

16.0 16.4

-21 4

Cyril Stanley Smith, Properties of Plutonium Metal, Phy*. Rev., 94 (4) (May 15, 1954). t Values are in arbitrary units.

* From

REACTOR MATERIALS

10-38

[SEC.

10

metal also emits low-energy 7 radiation. Certain alloys of plutonium produce Health-physics groups of the various neutrons and must be treated accordingly. laboratories and the AEC are in a position to give competent advice on these problems. 2

FUEL DILUENTS AND CLADDING MATERIALS


2.1

General Requirements

Even if uranium and thorium were completely satisfactory from the viewpoint of corrosion resistance and strength, it would be necessary to clad or otherwise coat the fuel to prevent the entrance of fission products into the reactor coolant. Also, in the case of enriched reactors, it may be desirable to dilute the fuel with some low-crosssection material. These cladding and diluent materials must meet very severe requirements. These are as follows: 1. These materials must have adequate strength even under the adverse conditions of temperature and irradiation. 2. They must be corrosion resistant in the reactor coolant. 3. They must be compatible with fuel materials. 4. They must be physically stable under irradiation. 5. They must possess good heat-transfer characteristics. 6. They must have good nuclear properties such as low-capture cross section. 7. They must be reasonably fabricable. 8. They should be reasonable in cost. From a consideration of cross section alone, only aluminum, beryllium, magnesium, and zirconium, with a thermal cross section of less than 0.5 barn per atom, are attrac tive for thermal reactors. Of these four, magnesium is usually ruled out on the basis of strength and corrosion resistance. The remaining three, aluminum, beryllium, and zirconium, are the leading cladding and diluent materials for thermal reactors and are discussed in some detail. Some consideration is also given to the ceramic mate In fast reactors, a number of other materials rials BeO, BejC, SiC, and MoSii. become quite attractive. Primary among these is stainless steel, which is discussed in detail as a structural material. Stainless steel may also be considered for specialpurpose enriched thermal reactors. 2.2

Zirconium

The low thermal cross section, excellent corrosion resistance, reasonable strength, and good fabricability of zirconium cause it to be a leading cladding and diluent material for thermal reactors. Its position is further improved by the fact that The only real short corrosion resistance and strength can be improved by alloying. coming possessed by zirconium is its high cost. This situation should improve as production increases. Zirconium minerals are wide 2.21 Production. Abundance and Availability. It ranks ninth among the metals in abundance. spread over the earth's surface. The principal zirconium minerals are zircon (ZrSiO«), zircite (ZrOj), and baddeleyite Commercial sources occur in the United States (Florida and North Carolina), (ZrCh). All zirconium ores contain hafnium in amounts Brazil, Ceylon, India, and Australia. from j$ to 3 weight per cent. Some ores contain as much as 20 per cent. The removal of hafnium with its high cross section Extraction and Purification. from zirconium is essential before the metal can be used in nuclear reactors. This is accomplished by means of solvent extraction, which produces a pure hydrated zir The oxide is conium oxide. A modified Kroll process is used to produce metal. chlorinated either with carbon tetrachloride or by mixing with carbon, pelletizing, and The chloride is purified by distilla treating with chlorine at elevated temperatures. The product is tion and then reduced by molten magnesium in a closed container. a slightly porous sponge. In this Sponge zirconium can be further refined using the de Boer iodide process.


Sec. 10-2]

10-39

process, the sponge is converted to zirconium tetraiodide, which is then thermally Oxygen, nitrogen, and magnesium are decomposed on a heated zirconium filament. the significant elements removed in this process. Physical constants are given in Table 29, together 2.22 Physical Constants. with those of aluminum and beryllium for comparison.

Physical Constants of Aluminum,

Table 29.

Zirconium

cross

CmVg

Density, g/c

0.215 ± 0.008 0.0048

0.0090 ± 0.0005 0.00060

0. 18 ± 0.02 0.001 19

Calculated 2.694,

X-ray (25°C)

a, room temperature, 6. 50 a, 863°C, estimated 6.36 0, above 863°C, 6.40

measured


and Zirconium*

Beryllium

Aluminum

Thermal-neutron section: Barns/atom

Beryllium,

2.699

1.8477

± 0.0007

Melting point, °C Boiling point, °C Heat of fusion, cal/mole Heat of vaporisation, kcal/mole

660.2 2327 2550

1315 2970 2334

67.9

75.528

Specific heat, cal/(g)(°C)

I50°C, 0. 1367 0°C, 0.2079 100°C, 0.225 300°C, 0. 248 600°C, 0.277

-200°C, 0.02 I00°C, 0.20 0°C, 0.42

Coefficient of linear expansion per °C

-

20-IOO°C, 20-200°C, 20-400°C, 20-600°C,

1845 + 25°C 5500 125

I00°C, 200°C, 400°C, 600°C, 800°C,

-

- I89°C. 100°C,

I27°C, 0.0739 427°C, 0.0815 727°C, 0.0862 862°C,a, 0.0880 862°C,0, 0.0797 II27°C, 0.0797

0.51

0.57 0.64 0.69 0.74

23.8 X 10 • (99.28% extruded

24.7 X 10"' 26.7 X 10 * 28.7 X 10 '

Be) 11.54 X 10-' 25-300-C, 14.44 X 10"' 25-500°C, 15.95 X 10 ' 25-700°C, 17.23 X 10 "

25-100°C,

25-1000°C.

Electrical resistivity, pohm-cm

Room, 0.0692

-

0.64 1. 53

18.77

Hot-pressed,

X

10

4. 525

Extruded, 4.076

c»l/(«ec)(cm)(eC)

Vacuum-caatRoom, 100-C, 200°C, 400°C,

0.503 0.503 0.530 0.546

(0.04% Hf) -200°C, 7

-

100°C, 23 0°C, 40 100°C, 58 200°C, 73 300°C, 87 400°C, 100 500°C, 110 600°C, 119 700°C, 125 800°C, 128

0°C. 2.63 20oC, 2.66 100°C, 3.86 400°C, 8.0


25°C c axis, 6. 15 X 10-' a axis, 5.69 X lO"' Worked and a-annealed: thickness, 6 . 7 to 10.1 X io-» transverse, 4.8 to 6.3 X 10 " longitudinal, 4.6 to 5.9 X IO-«

extruded 0°C, 0.36 I00°C, 0.34 200°C, 0.32 300°C, 0.30 400°C, 0.28 500°C, 0.26 600"C, 0.24

Flakeextruded 0.355

0.33 0.305

0.28 0.26

50°C, 0.05 I00°C, 0.049 I50°C, 0.048 200°C, 0.047 250°C, 0.046

0.235 0.21

• Compiled by the author from U.S. Atomic Energy Commission, "Reactor Handbook," vol. 3General Properties of Materials, AECD-3647, McGraw-Hill Book Company, Inc., New York, 1955.

10-40

REACTOR MATERIALS

[Sec.

10

2.23 Mechanical Properties. Mechanical properties of zirconium and some of its alloys are given in Tables 30 to 33 and Figs. 8 to 12. It will be noted that the physical properties of zirconium, like those of uranium, reflect the lack of years of experience to give single acceptable values.

Table 30.

Tensile Strength of Various Types of Zirconium*

Type of zirconium and melting method

Sponge, inductionmelted in graphite crucibles Sponge, inductionmelted in graphite crucibles Sponge,

arc-melted


Iodide, arc-melted (0.004 weight % nitrogen) Iodide, arc-melted (0.002 weight % nitrogen)

0 2% offset

Tensile

Elongation in 2 in..

Vickers hard-

Fabrication history

Grain size.

Rolled at I560T. baked 30 min, air-

0.035-0.045

37.600

61.500

21

180

at

0.025

35.800

56.400

24

155

30%, annealed 1 hr at 1300°F Forged and rolled at 1830°F, cold-rolled 30%, annealed 1 hr

0.035

37.600

63.300

30

180

0.035

16.400

35.500

36

104

0.065

13,600

29.000

29

74

yield

strength, psi

cooled Forged and rolled

I830°F, cold-rolled

at BOOT Forged and rolled at I450°F, cold-rolled 30%, annealed 1 hr at I300°F Forged and rolled at

I450T,

Btrength, psi

ct

new

cold-roUed 30%, annealed 1 hr at I300°F

Iodide, arc-melted

Hot-rolled at MOOT, cold-rolled 66%, annealed M hr at

0.02

9.400

34.800

47

Iodide, arc-melted

Hot-rolled at MOOT, cold-rolled 10%, annealed 20 hr at

0.05

7.700

24.800

40 (in 3 in.)

II10T

73

I380T

* From B. Lustrminn and Frank Kerze, Jr. (eds.), "The Metallurgy of Zirconium," NNES, Div. McGraw-Hill Book Company, Inc., New York, 1955

VII-

vol. 4.

Table 31.

Tensile Properties

Tin

analysis. weight %


0 1. 1 1.6

2.0 2.5t 2.8

37,600 47,800 51.800 57.000 50.400 57.800

of Zirconium -Tin Alloys at Room Temperature* Ultimate tensile strength. psi

Elongation to max load.

63.300 68.800 69.400 72.200 61.200 67.800

* From A. D. Schwope and W. Chubb, "Mechanical Apr. 14. 1953 (unpublished information). t Specimens taken transverse to rolling direction. 2.24 Manufacture. sponge and an increased on the direct melting of the sponge to form an

13 14 12 12 ib

Total elongation in 2 in..

Reduction

%

of area. %

30 34 32 34 30 34

49 55 54 56 57 56

Properties of Zirconium-Tin Alloys," BMI-817.

Melting and Casting. With improved quality of zirconium interest in zirconium alloys, greater emphasis has been placet! This is accomplished quite successfully by pressing sponge. electrode. This electrode is then melted in a conventional


Sec. 10-2]

0.2

10-41

0.

601-

N0TE: ALLOY CONTENT IS WEIGHT PER CENT

50 in 0.

I !§30 Id or

,20-

UNALLOYED

ZR

02

TA

u

t- 10


100

200

300

400

600

500

TEMPERATURE. F Tensile strength vs. temperature for unalloyed high hafnium zirconium and alloys with high hafnium zirconium base. {Benjamin Lustman and Frank Kerte, Jr. (eda.), " The Metallurgy of Zirconium," NNES, Div. VII, vol. 4, McGraw-Hill Book Company, Inc. New York, 1955.]

Fio.

8.

70

note: 60

alloy content

is

weight per cent

S50 Q-

40

£30 I-

20 UNALLOYED

ZR

10

0

100

200

300

400

500

_i_ 600

-1_

700

TEMPERATURE. F Tensile strength vs. temperature for unalloyed low hafnium zirconium and alloys with low hafnium zirconium. [Benjamin Luttman and Frank Kerte, Jr. (eds.), "The Metallurgy of Zirconium," NNES, Div. VII, vol. 4, McGraw-Hill Book Company, Inc., New York, 1955.]

Fio.

9.

10-42

REACTOR 90

MATERIALS

[Sec.

10

MOLYBDENUM O -MOLYBDENUM

NIOBIUM

• -NIOBIUM

BO

• -TUNGSTEN a -TANTALUM e -COPPER e -HAFNIUM • -VANADIUM

70 TANTALUM

60 50 40 30

UUUWWWWW

\ V- HAFNIUM* V \ V \ RANGES FOR CRYSTAL -^REPORTED £

,\\\\\\\ \\\\\ \

1,7/

20

\\ \\ \\ \\ \\ \\ \\

\

BAR

NITROGEN

ZIRCONIUM

•-NITROGEN •-OXYGEN

100

O-TIN

• -ALUMINUM • -CHROMIUM • -IRON • -TITANIUM

90 BO

70

4{tZ^-~-

60


90 40

\\\\\\\\\ ^^REPORTED

30 20

RANGES FOR

\ \

^TITANiuM-"

(GRAPHITE MELTED SPONGE)

\\ \\ \\ \\ \\ \ CRYSTAL

BAR

ZIRCONIUM

10

ADDED ELEMENT,

Fig.

ATOMIC

PER

CENT

Room-temperature tensile strengths of various zirconium alloys. [Ben/ami* Luetman and Frank Kerze, Jr. (eds.), "The Metallurgy of Zirconium," NNES, Div. VII. vol. 4, McGraw-Hill Book Company, Inc., New York, 1955.] 10.

LEGEND

ZIRCONIUM-8% TITANIUM AT 930F ZIRCONIUM-4% TITANIUM-0.2 % NITROGEN

AT 930 F

ZIRCONIUM-4.8% TIN AT 930

10.

F

CRYSTAL BAR ZIRCONIUM AT 500F 3 47 STAINLESS STEEL AT 930F

RATE. PER CENT

PER HR

The creep properties of zirconium and some zirconium alloys compared with typ* 347 stainless steel. (Prepared by ft. W. Dayton el al. from data of BMI Progress Report* /*' months of October, 1951, and August, 1952, BMI-710 and BA//-769, Not. 1. 1951. «■* Fio.

11.

Sept. 1, 1952.)


Sec. 10-2]

10-43

water-cooled arc-melting furnace. The resulting ingot is of high quality. For alloys, a second melting is frequently used to ensure homogeneity. Zirconium is a very soft, ductile metal that can be Forming and Fabrication. fabricated by any conventional method to produce a variety of sizes and shapes. Hot fabrication is usually done at about 1450°F. There is some tendency for oxida tion during heating. This is troublesome if it is desired to fabricate to finished size. Table 32.

Tin analysis, weight

%

Tensile Properties of Zirconium-Tin Alloys at 260°C* 0.2% offset yield strength, psi

0

10.800 19.200 22.300 23.600 23.800 28.400

I.I

1.6 2.0

2.5t 2.8

Ultimate tensile strength, psi

26.800 34.800 35.600 35.000 31.600 36.600

Elongation to max load, %

20 22 19 18 16 17

* From A. D. Bchwope and W. Chubb, "Mechanical Properties Apr. 14. 1953 (unpublished information). t Specimens taken transverse to rolling direction.


Table 33.


%

52 46 45 46 44 42

70 76 76 73 73 73

of Zirconium-Tin Alloys," BMI-817,

Fatigue Properties of Zirconium and Some of Its Alloys* Test

Composition,

Total elongation in 2 in.,

weight

%

temper ature,

°C

Room Iodide Zr + 0.34 Oi Iodide Zr + 0.34 Oj Iodide Zr + 2.02/2.2 Sn Iodide Zr + 2.1/2.7 Sn Sponge Zr + 2.5 Sn + 0.14 Oi Iodide Zr + 2.5 Sn

400

Room 400 400 400 400 260

Fatigue

lii nit, psit

Flexure tensile strength, psi

Unnotched

33.300 12.800 93.700 22.400 21.100 21.800 31.500 27.6001

21.000 10.500 56.000 15.000 18.000 20.000 24.000 21.500

Fatigue Notched

8.000 6.500 12.000 6.000 8.000 8.000 8.000 19.000

ratio}

0.63

0.82 0 60 0.67 0.85 0.92 0.76

0.78

* From Benjamin Lustman and Frank Kerze, Jr. (eds.), "The Metallurgy of Zirconium," NNES vol. 4. McGraw-Hill Book Company, Inc., New York. 1955. t The fatigue limit is the stress below which a material can be cycled indefinitely without failure (endurance limit). Because of the exceedingly long time required to obtain the true fatigue limit, in this report the fatigue limit is based on a minimum of 1.5 X 10' cycles. X Fatigue ratio = (unnotched fatigue limit)/(tensilc strength). 5 Axial-load.

Div. VII.

Light

machining removes this contamination readily. Protective atmospheres during are also helpful. Some of the alloys, particularly those containing more than 3 per cent tin, are quite hard at elevated temperatures, and fabrication is difficult. Powder Metallurgy. The high ductility of zirconium metal makes it difficult to produce a powder by crushing. A good grade of powder can be produced, however, by heating zirconium metal at 1470°F in purified hydrogen. The hydride thus formed can be readily crushed to a fine powder. The powdered hydride is reduced to metal by heating in a vacuum. Powder compacts can be made by pressing at 20 to 60 psi and sintering at 2200 to 2370°F. Joining. Zirconium can be furnace brazed using either copper or silver provided Although these brazed joints a suitable inert atmosphere or vacuum is maintained. are strong, they are not corrosion resistant in water at elevated temperatures. In most respects, zirconium is welded very easily. The only problem lies in its

heating

10-44

MATERIALS

REACTOR

[Sec.

10

130 m

120 110 I00|

90 p.ryst Al -RAR eo "ZIRCONIUM +25^ 70 -TIN, AF*CMELTED -Y 60 ~ARC- MELTEt 50 CRYS TAL-BA ZIRC ONIUM 40 30

d

UJ m

o

20)

R

\

in

10

400

800

600

TEMPERATURE,

1000

1200

F

200

0

-200

'

Fia. 12. V-notch Charpy impact tests on arc-melted zirconium and zirconium-2.5 per cent tin alloy. (Reprinted from R. W. Dayton, et al., Hydrogen EmbriUlcment of Zir conium, BMI-7&7, August 22, 1952.) 1

1000

1 |

900 Fv

/

/

IELTED SHEET

700

li /

J

B00

X\ *7

&

1

*

&

500

w

«l

6

n-

J

600

8

E

All

<

//

400

h

200

/

/ / /

300

//

300

400

500

600

700

TEMPERATURE,

800

900

1000

C

200

Flo.

'

/

24

/

100

H R

NITRO GEN

[Benjamin Heat resistance of zirconium in air, oxygen, and nitrogen. and Frank Kerte, Jr. (eds.), "The Metallurgy of Zirconium," NNES, Din. VII, vol. 4. McGraw-Hill Book Company, Inc., New York, 1955.] 13.

it

affinity for nitrogen. This makes necessary to weld in an atmosphere of helium Welding accomplished by a variety of methods including heliarc, resistance, «eam flash, or spot techniques. Zirconium alloys such as Zircaloy (1.5 weight per cent tin, 0.10 to 0.15 weight per cent iron, 0.08 to 0.12 weight per cent chromium, and 0.04 to 0.06 weight per cent 2

is


GRAPH TE MEL TED SH EET

-ARC

Sec. 10-2]



10

EXPOSURE

TIME, doy»

10-45

1000

KX)

Flo. 14. Weight-gain time curves for the corrosion of crystal-bar zirconium in hightemperature water. [Benjamin Lustman and Frank Kerze, Jr. (eds.), " The Metallurgy of Zir conium." NNES, Dit. VII, vol. 4, McGraw-Hill Book Company. Inc., New York, 1955.] nitrogen) are much less sensitive to nitrogen pickup and can be welded under less exacting conditions than those required for the pure metal. The machining character Machining. IOO0 istics of zirconium resemble those of alumi num. Zirconium is soft and very ductile : and galls readily with materials rubbed against it. Some care must be taken to avoid overheating in order to prevent turn 1500 This may be accom ings from igniting. PSI 15PSI S 1 plished by use of generous clearance angles 100 and sharp tools. 2.25 Corrosion. The corrosion be havior of zirconium in air, oxygen, and ? nitrogen is shown in Fig. 13. The behavior z" of zirconium, Zircaloy 2, and other tin-base alloys in hot water and steam is shown in Figs. 14 through 17. 15PSI PSI 10 The corrosion behavior of zirconium in various liquid metals is as follows: :i5psi

jT /

-

//

Gallium— 840°F, bad attack Mercury— 200°F, good: 400°F,

y

■

fair

resistance

Lithium— 1830°F, ZrN tarnish film, 4 hr [the nitrogen probably resulted from con tamination of the lithium (ORNL-583)] Sodium potassium — 1110°F, negligible weight gain, 6 days Lead— 1830°F, Sn— 930°F, Pb-Bi-Sn— 930°F, resistant Bismuth — 1830°F, severe attack Magnesium— 1200°F, bad attack Calcium— 2370°F, resistant

100

TIME, hr

1000

Fio. 16. Corrosion of arc-melted zirco nium sponge in steam at 750°F at 15- and 1,500-psi pressure. [Benjamin Lusiman and Frank Kerze, Jr. (eds.), "The Metal lurgy of Zirconium," NNES, Din. VII, vol. 4, McGraw-Hill Book Company, Inc., New York, 1955.]

10-46

REACTOR MATERIALS

[Sec.

10

Protective Although the properties of zirconium are generally very Techniques. good, there are special applications where a thin coating of some other metal might be Zirconium can be roll clad only with titanium. Other metals form brittle useful. intermetallic compounds at the bond line. It is possible to electroplate zirconium

100

I

o

10

NUMBER OF SAMPLES


550F: 600F: 680F. 750F: 850F:

I

10

21 9

17-240 2-151 10

100

1000

EXPOSURE TIME, days

Fig.

Weight-gain time curves for the corrosion of Zirealoy 2 in high-temperature water [Benjamin Lustman and Frank Kerze, Jr. (ede.), " The Metallurgy of Zirco nium," NNES, Div. VII, vol. 4, McGraw-Hill Book Company, Inc., New York. 1955.] 16.

and steam.

500

600

700

ANNEALING

Fio.

800

900

1000

TEMPERATURE,

C

Effect of heat treatment on corrosion resistance of Zirealoy 2. [Benjamin Luttman and Frank Kerze, Jr. (eda.), "The Metallurgy of Zirconium," NNES, Div. I'll, tol. 4, McGraw-Hill Book Company, Inc., New York, 1955.] 17.

with either nickel or iron.

Such as-plated deposits are not strong but are much improved by a short diffusion anneal. 2.26 There has been an interest in zirconium alloys for two principal Alloys. reasons. One is the need to alloy in order to improve corrosion resistance in water. The other is the need to alloy in order to improve the high-temperature strength. Alloys for corrosion resistance were first based on the zirconium-tin binary system.


Sec. 10-2]

10-47

Additions of 2.5 to 5 weight per cent tin were found very helpful in this regard. Eventually, it was found that alloys containing small amounts of iron, chromium, and nickel in addition to tin were even better behaved. These latter additions also per mitted the tin content to be reduced markedly and thus improved fabricability. It has been found that additions of small amounts of tin, molybdenum, niobium, titanium, and aluminum greatly increase the short-time tensile strength at elevated Such alloys have strengths comparable to stainless steel at 930°F. temperatures. The creep strength of such alloys at this temperature is only about one-fourth that of stainless steel. A complete compilation of zirconium binary alloys may be found in "Zirconium and Its Alloys."3 2.3

Aluminum

The low thermal-neutron absorption cross section, low cost, ease of fabricability, and good corrosion resistance of aluminum make it a favored diluent and cladding Its broad usefulness is limited by a low material for low-temperature operations. melting point and a decrease in corrosion resistance and strength at moderately elevated temperatures. Since aluminum is a well-known metal, details of its production and fabrication will not be discussed. In general, it is available as a large number of alloys in a wide


Alloy

desig

nation

Min Max

I22J1

CGI 00 A 9^2 10.8 1.5 SN 122A 0.5 1.5 1.0 CN42A 3.5 4.5 0.8 G4A 0.1 0.4 G8A 0.2 0.8 G 1OA 0.2 0.3 SG70A 0.2 0.5 SGI00A 0.6 0.8

1421-1

2I«

218+ 220* 356{.«I 360t

max

II. 0

13.0 4.5 6.0 1.0 3.0 0 13.0

II.

0.6 0.3 0.3 0.2 6^5 7.5 9.0 10.0

Ti,

1.8

0.3 3.5 0.3 7.5 0. 1 9.5 0.1 0.2 0.3 0.4

4.5 8.5 10.6

0.4 0.6

Sn, max

Min Max

0. 1 0.3 0.05 0.3 0.5 0. 15 0.35 0. 1 0.9 1.3 0. 1 1.2

Others (

max

Min Max

M in Max

0.6 0.8 0. 1 0.8

S5A

Zn, max max

Mn.

Fe. max

SI2A

Ni

MK

Si

Cu

ASTM

13t

U321

Casting Alloys*

Chemical -composition Limits of Selected Aluminum

Table 34.

0.3 0.2 0.5

0.1 0.2 0.2 0 2 0.2 0.2

0.1 0.1 0.1 0.1 0.1

0.1 0.2

0.2 0.5

0.05 0. 1

i.i) 1.7

0.3 3.0 2 3 0. 1

0.1

0.5

0. 1

Each Total max max

o.s

0.2 0.15 0.3

bios 0.05 0. 15 0.05 0. 15 0.2 0.05 0.15 0 05

0 15

0.2

data by C. M. Craighead, Battclle Memorial Institute,

* Compiled from aluminum producers'

t Die-casting alloy.

X Sand-casting alloy. *j Permanent- mold alloy. J The remainder of the alloy is aluminum.

Chemical -composition Limits of Selected Wrought Aluminum

Table 36.

Cu

Si

Or

Mg

Others Zn, max

Fe. max

Alloy

t

0.20 0.20 Y.O 0.80 1.0 i. 5 4.5 0.40 0.50 0.50 3.8 4.9 0.30 0.10 1 t 0 40 0.80 0.70 0 15 0.40 0. 10 0.20 0.60 0.35

t

Ti, max

Min Max Min Max Min Max Min Max

Min Max

1100 3003 2017 2024 5052 6061 6063

Mn

0.60 0.70

* Compiled from aluminum producers' t 1.0 Si + Fe. t 0.4 5 Si + Fe.

0.05 1.5 1.0 0.20 0.90 1.2 0. 10 2.2 0. 15 0.80 0. 10 0.45

0. 10 0. 10

0.80

II 10 0.25 0. 10 0.25 2.8 0. 15 0.35 0. 10 1.2 0. 15 0.35 0.25 0. 15 0. 10 0. 10 0.10 0.90 1.8

data by C.

M.

Alloys*

Craighead.

Al, min Each

Total

max

max

0.05 0.05 0 05

0.15

0.05 0.05 0.05 0.05

0. 15 0. 15 0. 15

0.15 0.15 0.15

99.0 Remainder Remainder Remainder Remainder Remainder


10-48

REACTOR

Density. g/cm"

2.69 2.69 2.69 2.88

I3t

43

43t I22-T2 I22-T6I I22t I42-T2I I42-T57U I42-T77 I42-T61:

2.95 2.76 2 81 2 81 2.81 2.81 2.65 2.65 2.60 2.57 2.68 2.68 2.68

214

2I4T 2l8t 220-T4

356-T5I 356-T6 356-T7

2.68 2.68

356-T6J


360}

[SEC. 10

Physical Constants for Selected Aluminum Casting Alloys*

Table 36.

Alloy

MATERIALS

Approximate melting range,

°C

Electrical conductivity at 20°C. %

IACSt

Thermal conductivity at 20°C. eal/(sec)(cm)(°C)

io-«/°c

0.37

39 37 42 41 33 34 44 34 38 38 35 35 24 21 43 39 40 41 37

570-585 570-620 570-620 490-620 490-620 490-620 530-630 530-630 530-630 530-630 580-640 580-640 530-615 450-620 570-615 570-615 570-615 570-615 555-595

Coefficient of thermal expansion (20-300°C).

21.6 23.9

0.35

0.39 0.38 0.31

0.32

23.4

0 40

0.32

24.5

0 35 0.31 0.33

24.5 23.9

0.33 0.24

24. 1 24.5 21.4 21.4

0.21

0.40 0.36 0.37 0.38 0.35

* Compiled from aluminum producers' data by C. M. Craighead, Battelle Memorial Institute. Annealed Copper Standard. X Chill-cast samples, all others oast in green-sand molds. IT Annealed; while castings are not commonly annealed, similar effects on conductivities may result from the slower rate of cooling of thick sections as compared with thin ones and other variables in foundry practices; comparison of the values for as-cast and annealed specimens wilt show the extent to which variations may be expected, depending upon the differences in the production of different types of castings.

t IACS — International

Table 37.

Density, g/cm'

AUoy

EC-0 EC-HI

Physical Constants for Selected Wrought Aluminum Alloys*

9

II00-O

II00-HI8 3003-O 3003-H 12 3003- HM 3003-H 18 2017-O 2017-T4 2024-O 2024-T3 5052-O 5052-H38 606 l-O

606I-T4 6061-T6 6063-T42 6063-T5 6063-T6

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

70 70 71 71 73 73 73 73 79 79 77 77 68 68 70 70 70 70 70 70

Approximate melting range,

Electrical conductivity

°C

at 20°C, % IACSt

650 660 650-660 640-660 640-660 640-655 640-655 640-655 640-655 510-585 510-585 500-635 500-635 590-650 590-650 580-650 580-650 580-650 615-658 615-650 615-650

62 62 59 57 50 42 41 40 45 30 50 30 35 35 45 40 40 50 55 55

Thermal conductivity at 25°C, oal/(sec)(cm)(°C)

♦ Compiled from aluminum producers' data by C. M. Craighead, t IACS — International Annealed Copper Standard. t 20 to I00°C only.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

56 56 53 52 46 39 38 37 41 29 45 29 33 33 41 37 37 46 50 50

Coefficient of thermal expansion, (20-300°C)

io-«/°c

23 81 23. 8{ 25.6 25.6 25.0 25.0 25.0 25.0 25.0 25.0 24.7 24.7 25.7 25.7 25.4 25.4 25.4 25.2 25.2 25.2



Sec. 10-2] Table 38.

Tensile and Yield Strengths of Cast Aluminum Alloys*

AUoy 13 43 43t 43t

I22-T2t

122-T52J 122-T6lt 122-T65J

I22-T55IJ AI32-T55lt I42-T2lf I42-T6U I42-T77t I42-T57lt

I42-T57IJ

2l4f

218

220-T4f 356-T6t 356-T6{

356-T7f 356-T7J 356-T5lf 356-T7lt


360

10-49

Yield strength, Room 20 15 8 8 19 36 29 35 34 19 40 24 28 33 11 27 24 23 27 29 23 19 20 27

400°F

I01 psi 600°F

13

4

16

4

19 13 13

6 5 3

18

3

22 11

5 3

8

2

* Compiled by the author from aluminum producers' t Sand casting. X Permanent-mold casting.

Tensile strength, Room 400°F 35 28 18 23 25 32 36 43 34 25 42 27 29 36 24 40 41 30 36 31 30 24 27 38

I01 pai 600°F

21

8

23

8

24 22 21

10

22

8

28 18

9 9

13

4

II

8

data.

variety of shapes. Compositions of typical cast and wrought alloys are given in Tables 34 and 35. Detailed information can be obtained in handbooks prepared by the principal producers: Alcoa, Kaiser, and Reynolds. 2.31 Physical Constants. Pertinent physical constants of pure aluminum are listed in Table 29 along with those of zirconium and beryllium. Selected physical constants of alloys favorable for reactor use are given in Tables 36 and 37. 2.32 Mechanical Properties. Mechanical properties of selected aluminum alloys are given in Tables 38, 39, and 40 and in Figs. 18 and 19.

IOO

200

300

400

F TEMPERATURE. 18. Effect of temperature on stress to produce a creep rate of 0.001 per ccnt/hr for several 0.065-in. -thick wrought alloys. (Conttrucled from aluminum produccrt' data by C. M. Craighead ct al., Battelle Memorial Institute, Aug. 1, 1952.)

Fio.

REACTOR MATERIALS

10-50 Table 39.

[SEC. 10

Tensile and Yield Strengths of Wrought Aluminum Yield strength,

Tensile strength,

I0< pai

Alloys*

10s psi

AUoy Room

EC-O EC-HI

9

1100-O 1100-1112

II00-HI4 II 00-1116 IIO0-HI8 3003-O

3003-HI2 3003-HI4 3003-HI6 3003-HI8

II

20 17-0

20I7-T4 2024-O 2024-T3 2024-T4 2024-T36 5052-O 5052-H32


5 24 5 13 14 17 21 6 15 18 21 25

5052-H34 5052-H36 606 l-O 606I-T4 606I-T6 6063-T5 6063-T6 6063-T42 6063-T83 6063-T83I 6063-T832

46 10 50 50 56 14 27 30 34 8 20 40 25 30 14 35 30 40

400°F

600°F

3

1

6

i

3 4

i 2

9

3

8

3

II 20-23 20-23

Room

4 5 4 5

II

4

II

4 2 2 2

6 15 15

* Compiled by the author from aluminum producers'

13 27 13 15 17 20 24 16 18 21 25 29 27 68 25 70 70 75 29 35 35 39 18 35 45 30 35 20 40 30 45

400°F

600"F

7

3

9

3

7 8

3 4

14

6

17

4-5 6

17 23-28 23-28

8-9 8-9

18

7

25

8

18 19

3

data.

2.33 Corrosion. Aluminum possesses good corrosion resistance in water up to temperatures slightly above the boiling point of water. Recent work by Draley of ANL has shown that the addition of small amounts of acid to the water improves corrosion resistance markedly. Good corrosion behavior at temperatures around 200°C has been observed. This improvement in corrosion

□ ROOM TEMPERATURE (ROTATING

BEAM)

H300°F

■ 400°F I 5O0°F

3S-HI8 24S-T4 &S-T6 52S-0 I7S-T4 52S-H36 I42-T6 A355-T5I

Fio. 19. Comparison of the fatigue strengths of various aluminum alloys at 5 X 10' cycles as affected by temperature. Tested after stabilization periods of 0.5 to 2 hr at testing temperature. (Constructed from aluminum producers' data by C. M. Craighead et al., Battelle Memorial Institute, Aug. 1, 1952.)


Sec. 10-2]

10-51

Modulus of Elasticity of Aluminum Alloys in Tension*

Table 40.

Average modulus,

I06 psit

Alloy

1100, 3003. 6061, and 6063 5052 2017 2014 and 2024

75°F

300°F

500°F

10.0 10.2 10.5 10.6 10.3

9.5 9.7

8. 15

8.0 8.4 8.5 8.25

10.0 10. 1

9.8

* Compiled from aluminum producer*' data by C. M. Craighead, Battelle Memorial Institute. The modulus of elasticity is about 2 per cent higher in compression than in tension. For most calculations, a value of 10,300.000 psi can be used for all alloys in both tension and compression. The modulus of rigidity is approximately 3,900.000 psi, and Poisson's ratio is about one-third.

t

resistance in the pH range 5.5 to 6.5 makes aluminum reactor applications. 2.4

quite attractive

for power

Beryllium


The very low cross section, high melting point, and good strength of beryllium make it appear to be an excellent diluent or cladding material. However, in spite of a con siderable amount of research, beryllium has not gained the important position it was Its favorable properties have been more than offset by its once assumed to merit. high cost, low ductility, poor corrosion resistance, and extreme toxicity. While beryllium is a relatively new and unusual metal, its importance as a reactor material does not warrant any extended discussion on its abundance, melting fabrica tion, and the like. In general, beryllium shapes are made by powder metallurgy or a combination of Table 41. Test tempera ture, °C

Room

Directional

Direction of test

Longitudinalt Transverset Longitudinalt

400

600

800

No.

Ultimate

of tests

strength, psi

Elonga tion,

%

Longitudinal IT Transverse ■} Longitudinalf Transverset Longitudinalt Transverset

0 0

Longitudinalt Transverset

2 2

30.590 22.840

14.8 1.9


2 1

13.570 11.200

10.6 2.5

0 6


2 1

4.760 5.860

9.1

4.2

• From D. W. White and

J.

46.600 29.125 63.650 25.500 68.450 26.700

0.55

0.3 5.0 0.3 6.6 0.3

0.3

Mechanical No. of tests

Properties*

Ultimate strength.

Elonga tion,

psi

%

32.730 19.400 39.970 16.550

0.36 0.30

Reduc tion,

1.82

0.18

5^7 39.870 16.600

13.2 1. 1

8.0

37.870 14.250

13.4

8.6

9.6 0.3

27.670 14.500

13.4 1.4

32.8

7.2

25.850 15.980

20.8

23.2

1.2

0.3

9.270 6.900

25.6 9.6

24.0 8.9

I.I

1.7

1.7

Burke (eds.), "The Metal Beryllium," The American Society for Metals.

1955.

t

As extruded. X Annealed at 800°C for I hr. Annealed at 800°C for I hr plus

f

Reduc tion,

4 2 2 1 2 1 0 0

Transverse^

200

Effects on Beryllium

I000°C for 1 hi.

10-52

REACTOR

MATERIALS

[Sec.

10

powder metallurgy and conventional fabrication. By these techniques, a rather large assortment of sizes and shapes are available. Considerable information on beryllium is available in the book "The Metal Beryl lium," edited by D. W. White and J. Burke and published by the American Society for Metals. 2.41 Physical Constants. Physical constants of beryllium are listed in Table 29. 2.42 Mechanical Properties. Mechanical properties of beryllium are given in Tables 41 through 43. Table 42.

The Effect of Extrusion Temperature and Machining on the Tensile Properties of Beryllium* As extruded

As extrudec , machined. and annealed at 800°C

and

machined

Extrusion

Stresses

As extruded, machined. and etched

t f>m n/»rn111pa

°c


500 700 900 1060 1105 1208

Ultimate,

Elongation.

Ultimate.

Elongation.

Ultimate,

pai

%

psi

%

psi

%

73.100 60.150 44.500 36.500 36.900 34.800

0.28

51.550 47.550 45.050 52.133

3.35

72.200 59.750 51.150 49.600 54.300 51.300

0.31 0.31 1.45 2 57

* From D. W. White and

0.31

0.45 0.39 0 64

0.44

J. Burke

1955.

Table 43.

* From

M. C. Udy, H. L.

(unpublished

Shaw,

(eds.),

50.200

2.74 2 47

3.38

i. 80

Elongation

"The Metal Beryllium," The American Society

3.82 3 76 for Metal.

Impact Strength of Beryllium*

and F. W. Boulger,

'The Properties

of Beryllium."

BMI-T-H

information).

2.43 Corrosion. Beryllium is reasonably corrosion resistant in air at temper atures up to 750°F. At 1290°F, results are erratic, but some data show beryllium about equal to 18-8 stainless steel. The corrosion resistance of beryllium in hot water varies widely, depending In boiling water, extruded ingot appears superior upon the grade of metal tested. to extruded flake or powder process material. Several grades of beryllium showed some resist ance at 525°F but were badly at tacked at 600°F. In general, the corrosion resistance has been too erratic to warrant use of the metal in hot water. Cladding. Improvement in corrosion resistance can be achieved by cladding with aluminum, using conventional roll-cladding techniques. Although cladding with zirconium would be desirable for high-temperature water application, it does not

Sec. 10-2]

PROPERTIES

OF

reactor materials

10-53

layer is formed between beryllium and appear feasible, since a brittle-compound zirconium. The resistance of various types of beryllium to sodium and its alloys is shown in

Table

31. 2.5

Other Materials

Stainless steel is recently gaining prominence as a fuel diluent and cladding material in spite of its cross section. Its high strength and corrosion resistance tend to offset Stainless a cross section higher than that of zirconium, aluminum, or beryllium. steels are particularly attractive as a matrix material containing fuel in the form of a powdered uranium compound. Table 44.

Physical Properties of BeO, Be2C, SiC, and MoSij*


BeO

BeiC

SiC

MoSii

Density, g/cm'

Single-crystal, 3 025 Hot-pressed, 2.6-2.95 Slip castings after firing, 2. 2-2. 8

X-ray, 2.44 Crystals from BeO-C reaction, 2.30-2.40

Calculated 3 210 Green at 30°C, 3.218 Average values at 20*C, 3 10-3.30

X-ray, 6.24 Powder, hot-pressed at 2.400 psi and I7000C:6.20

Melting point, "C

2550 ± 25

No true melting point at atmospheric pressure— some melting observed above 2200

Decomposesat about 2500

1870

Specific hemt,cal/(g)(°C)

I00°C, 0.308 400°C, 0.420 800°C, 0.492

30-100°C(98%Be:C), 0.334 ± 0.013

32rC,0 251 627°C, 0.280 927°C, 0.299 I227°C, 0.316

Sintered beryUia den sity 2. 25 g/cm! lootrc, 80.000.000 I200°C, 4.000.000 1400-C, 250.000 1600°C, 35.000 I800T. 6.500 2000'■(',1.600

Impure in sintered shapes 30°C, 0.063 425°C, 0.047

Recrystallized, no bond

Density 2.87 g/cm* 200°C, 0. 19 400°C, 0. 13 600°C, 0.095 I000°C, 0.055 I400°C, 0.038

30°C, 0.0215

Recrystallized 400"C, 0.060 600*C, 0.052 800"C, 0.044 1000°C, 0.0385 I200°C, 0.032

25-50-C, 5.6 25-200T, 7.7 25-400°C, 9.5 25-600°C, 10.5

0-l700°C, 4.3-4.5

Electrical resistivity, ohm-cm

Thermal conductivity, csl/(sec)(cm)CC)

Coefficient of linear thermal 25-100(5 5 ± 1) expansion, per"C X 10 ■ 25-300(8.0 ± 0.6) 25-600(9.6 ± 0.8) 25-800(10.3 ± 0.9) 25-1000 (10.8 ± 1.0)

800^,6.5 1200°C, 2.5 1400oC, 1.7

Bonded Hot-pressedf 25T, 21.5 10% fireclay 400°C, 70 12.500 600°C, 103 4.200 900°C, 160 1.400 nOO'C, 193

370.6°C, 566.6°C, 683. 9°C, 710.0°C, 800.6°C, 823. 8°C.

0.0886 0.0803 0.0764 0.0767 0.0757 0.0757

0-I500°C, 5.1

* Compiled by the author from various sources, t Values in microhm-centimeters.

Since stainless steel has long been popular as a structural material, its properties arc covered in that section. Ceramic materials offer a type of diluent or cladding material with some very They possess very low cross sections and extremely good highattractive properties. They suffer from lack of temperature properties and are relatively inexpensive. thermal-shock resistance. Physical and chemical constants and selected mechanical properties of BeO, BesS, SiC, and MoSij are given in Tables 44 and 45.

10-54


Compressive, 10' psi

Hardness

Modulus of elas ticity, 10" psi


Modulus of rup ture, 10' psi

10

Mechanical Properties of BeO, Be2C, SiC, and MoSij*

ac

Be,C

BeOt Strength : Tensile, 10' psi

[SEC.

Density 2.7-2.8 g/cm' 400°C. 15.0 800°C. 13.5 I000°C, 10.5 1 IOQoC. 8.0 I200°C, 4.0 500°C, 71.0 800°C. 64.0 I000°C, 36.0 I232°C. 24.0 I600°C, 7. 1 Dense pure heryltia Moh value 9

400°C, 39 I000°C, 38 II00°C, 28 I200°C, 10 Minus 325 mesh. pressed and sin tered, \70VC: room temp. 4. 6

Knoop values: 50-kg load. 2550 25-kg load, 2740

MoSii

Self-bonded brick may be heated to 1700°C under a 50-psi compressive load without perma nent deformation

27°C, 22.0 980°C, 40.0 1090°C, 42.2 I200OC, 42.8 I3I5°C. 41.1

800°C, 5.93 1000-C, 8. 16 1300°C, 2.09 I350°C, 1.82 I500°C, 0.99 Knoop values 2100-2700

100-350

850-870 25°C. 59

400°C, 23 500°C, 23.8 700°C, 24.3 I000°C, 24.8 25°C, 8-10 I370°C, 7-14

25°G. 50.7 1090°C, 35.7 I315°C. 20.7

* Compiled by the author from various sources. t Tensile strength and elasticity specimens fabricated from Blip castings; rupture specimens pressed; other properties for sintered beryllia.

3

3.1

Rockwell A-80 to A- 90 Knoop. 1-kg load.

compressive

strength

and

SOLID MODERATORS Special Requirements

Moderating materials for thermal reactors must be capable of reducing neutron energy very rapidly. This requires a low atomic weight to give a large average logarithmic neutron energy loss per collision; moderators must also have a low absorp tion cross section and a high scattering cross section. Desirable properties other than nuclear would include reasonable cost, fabricability, good strength at operating temperatures, corrosion resistance to reactor coolant,

1.668

1.492

1318

00 TEMPERATURE,

Flo.

C

20. Variations of graphite bulk density with temperature. Baltelle Memorial Institute, from unpublished information. 1952.)

(Compiled

by

J.

A. Slyk,


Sec. 10-2]

10-55

Such properties are desirable, but they are thermal stability, and radiation stability. Selection of moderator materials is thus based not obtainable in any single material. on a compromise between nuclear and physical properties. From a nuclear consideration, only carbon, beryllium, oxygen, deuterium, and This, then, limits our selection of solid hydrogen show promise among the elements. moderators to graphite, beryllium and its compounds, oxides, carbides, and hydrides. Of these, graphite and beryllium or its compounds have received the most attention. 3.2

Graphite

Graphite has very good nuclear properties as a moderator and, in addition, is abundant, cheap, and easy to fabricate. It has good strength properties up to very high temperatures. Its chief handicap is its lack of corrosion resistance. It is the best known and most widely used solid moderator. Table 46.

Ash and Impurity Contents of Artificial Graphite* Chemical analysis,

Grade

B

AGXt Generated for wjivans (University of Florida) on 2015-09-23 02:54 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

C-18t

AGHT*

AGOTJ est

ppm

Ash, ppm

I.S00 1.200-2,000 700-1.700 350-700 350-700

Al

1.3 1.0

20 10 10

1.0-1.5 0.2-1.0

Ca

Ti

Fe

V

2.000 1.200 500 100-200

50 35 100 20-30

40 100 10 10 20

70 200 14 10 75 25

0. 15

* Compiled from

t

J

data of J. R. Huffman, ANL-4304, June, 1949. Typical commercial graphite. Normal reactor graphite.

The

abundance of carbon and the methods of manufacture of graphite are well no discussion here. It might be well to point out, however, that The raw materials are carefully reactor-grade graphite is always an artificial product. This is shown in selected in order to eliminate any high-cross-section impurities.

known and require

Table

46.

Physical constants of several grades of graphite arc 3.21 Physical Constants. given in Table 47. Information on the variation of density with temperature, thermal

750

IOOO

1250

1500

1750

2000

2250

TEMPERATURE, C

Fig.

(Compiled 21. Thermal-expansion coefficients of various graphites. BaUelle Memorial Institute, from unpublished information, 1952.)

by

J.

A. Slyh,

10-56

REACTOR MATERIALS

[SEC. 10

Table 47. Physical Constants of Graphite* Thermal-neutron-absorption cross section, milliharns/atom: AGOT (<0.5 ppm B) Density, g/cm* Calculated from lattice Constanta: Artificial Natural Helium displacement:! Artificial Natural Commercial electrodes Commercial blocks AGHT AGOT

2 27-2. 28 2. 27-2. 28 2. 10-2. 17 2 26 1. 55- 1. 90 I. 55-1 75

1.61-1.65 1.65-1.72

Sublimation temperature, °C Heat of vaporization, kcal/mole Heat of sublimation, kcal/mole Specific heat (C„), cal/(molc)(°C):

3650 ± 25 I 43 140- 170

25°C 127°C 327°C 527°C 727°C 927°C I127°C

2. 066 2.851 4. 03 4. 75 5. 14 5. 42

5.67

Electrical resistivity (parallel), Mohm-cm: Commercial electrodes Commercial blocks

850-1,250 850 1. 100

AGHT AGOT


4.8

800t 500-850 800 720

CS

GBF

compiled by J. A, Slyh, Battelle Memorial Institute, for U.S. Reactor Handbook." vol. 3. General Properties of Materials, AECD-3647. McGraw-Hill Book Company. Inc., New York, 1955. t An explanation of density measurements by helium displacement is given in R. P. Rassman and W. R. Smith, Density of Carbon Black by Helium Displacement, Godfrey L. Cabot, Inc., Boston. Mass., Jnd. Chrm. Eng., vol. 35 (1943). X Perpendicular: 1,050 to 1,250 ôhm-cm. * Revised by the author from data Energy Commission, "

Atomic

expansion coefficients, and thermal conductivities are shown in detail in Figs. 20 through 23. 3.22 Mechanical Properties. Tensile and creep properties of various types of graphite at several temperatures are given in Tables 48, 49, and 50 and Figs. 24 through 26.

900

1300

1700

2100

2500

2900

3300

TEMPERATURE, C

Fia.

22. Typical thermal conductivities of various graphites. (Compiled Battelle Memorial Institute, from unpublished information, 1952.)

by

J.

A. Slyh,



10-57

Tensile, Compressive, and Transverse Strengths of Carbon and Artificial Graphite at Room Temperature* Tensile strength,

psi

Compressive strength,

Transverse strength,

psi

psi

Type

Commercial electrodes: Carbon Porous carbon Graphite Porous graphite Commercial blocks: Carbon Graphite Commercial tubes: Impervious carbon. . . Impervious graphite. .

Trans verse

400-1.200 80-200 500-2.500 50-100 1.000 500-2,000

900-1.300

J.

2.000-10.000 3.000-900 3.000-4.000 300-500

verse

1.000-4.500 1.500-4.000 4.500 4.500 1.000-2.500

1.000-2.000

600 1952.

Modulus of Elasticity at Room Temperature for Carbon and Artificial Graphide* Modulus of elasticity, 10* psi

0.5-1.4 0.5-1.2

Carbon Graphite Commercial hlooks: Carbon Graphite Commercial tubes: Carbon Carbon, impervious Graphite Graphite, impervious Commercial electrodes and blocks, porous

1.1 0. 5-0. 8 1. 7-2. 1 2. 6-2. 9

1.3-1.4 2. 1-2. 3

0.12 1.4-2.1

AGOT

J.

Trans

800-6,000 200 600 3.000 5.000 1.000-5.000 150-250

10.000 10.000

Type of product Commercial electrodes:

by

verse

A. Slyh. Battelle Memorial Institute, unpublished information.

Table 49.

* Compiled

Longi tudinal

Trans

2.000 8.0(10 3.000-6.500

500-3.000 500-3.000

2.200

* Compiled by

Longi tudinal

2.000 2.500

AGOT AGHT


Longiluminal

A. Slyh, Battelle Memorial Institute, from various sources.

060

S

|

055

CONDUCTIVITY DETERMINED PARALLEL

I050 g

EXTRUSION

/l.75 056

M7C '__ 041

u*o

>■' >

§Q

035

8

Q3°

|

025

n.65 03: .

^1.55

029

cr

jr 0.20 019

0.15 140 1.45 150 1.55 160 1.65 170 1.75 ISO BULK DENSITY, g per cm'

Fig. by

J.

23. Thermal conductivity of graphite vs. bulk density at room temperature. A. Slyh, Battelle Memorial Institute, from unpublished information, 1952.)

(.Compiled

10-58

MATERIALS

REACTOR

[Sec. 10

184

2500

4200 3400 TENSILE STRENGTH,

5000 psi

Flo.


24. Variation of room-temperature tensile strength with density for commercial block graphite (EBP). (Reprinted from C. F. Malcolm and A. F. Groton, Tensile Strength of Type EBP Graphite at Elevated Temperature and Its Relation to Apparent Density at Room Tem perature, NAA-SR-67, March, 1950.)

2000

3000

4000

TEMPERATURE, F

Fiq. 25. Effect of temperature on the tensile strength of various graphites, (h from M. C. Udy and F. W. Boulger, The Properties of Graphite, BMJ-T-35, June 20, 1930.) 3.23 Corrosion Behavior. is given in Table 51. 3.3

The corrosion resistance of graphite in various media Beryllium

and Its Compounds

Beryllium and its compounds, such as the carbide or oxide, possess excellent nuclear properties as moderators. Although the metal itself lacks favorable high-temperature


Sec. 10-2]

MATERIALS

10-59

TEMPERATURE.F

Fio.

26. Effect of temperature on the elastic modulus of commercial electrode graphite. (Adapted from Data of L. Green, Jr., Measurement of Young's Modulus of Grade AVF Graphite at Elevated Temperatures, NAA-SR-98, Dec. 7, 1950.)

Table 60. Type of

Kind of

graphites

test

AGHTt AGHTt

Tensile

AGIITf


C-18t C-I8{

Tensilo

C-I8t C-IBt

Temper

stress, psi

Reduction

ature,

of area, %

4.380 2.200 2.800 2.800 5.300 4.000 2.800

4800 4800 4320 4800 4800 4300 4300

Maximum

Creep Creep Creep Creep Creep

Ductility of Graphite*

°F

Elongation, (2-in. effective gauge length), %

1.3 1.3

7 9 2 18 18

0.5 5-7 0.4-0.5

% 8 1

II

14 8 3

II

2. 7 1.1

Increase in volume,

4

* From R. E. Adams and H. R. Nelaon. NEPA, BMI-N-45. May, 1950. t Parallel to extrusion direction. X Perpendicular to direction of applied pressure.

Table 61.

Reaction of Graphite with Corrosive Agents*

A Kent

Gases:

Temperature, °F 840t and higher 1650-1830 3630 and higher 1650-3450

Potassium vapor.

Room Room Room Up to 5430 1470 and lower 1470 and higher

Up to 1470

1470 and lower 1470 and higher

Aqueous solutions: Dilute acid or alki Ji Strong acid (HNOi and HjSO^

KOH

(50 per cent

Alkali

hydroxides

Solids:

Sodium carbonate

Leal than hoiling 570 and higher 660 and higher

At fusion At fusion At fusion 2730 and higher 2730 and higher

Effect or product Produces

carbon

oxides

Methane (catalyst necessary) Acetylene (traces) ? Produces carbon fluorides Interlamellar compounds Interlamellar compounds Interlamellar compounds No effect Negligible reaction Increasingly severe attack No effect Negligible reaction Carbon monoxide

No attack Graphitic oxide Reacts with graphite No attack No attack Carbon monoxide Metal carbide Metal carbide and carbon oxides

• Compiled by J. A. Slyh, Battclle Memorial Institute, unpublished information, Attack at 660°F is reported; the weight loss is not measurable, but a slight permanent length is noted.

f

increase

in

properties, both the oxide and the carbide are very refractory and have excellent The high cost and extreme toxicity of beryllium and properties. high-temperature ita compounds are serious handicaps to its widespread usage. Details on the properties of beryllium and its leading compounds, oxide and carbide, have been covered in detail in Art. 2 under Fuel Diluents and Cladding Materials and will not be repeated here.

REACTOR MATERIALS

10-60 4

10

[Sec.

STRUCTURAL MATERIALS 4.1

Special Requirements

While there is considerable duplication of materials and properties required for Fuel Diluents and Cladding Materials and for Structural Materials, there are important differences between the two. Under Structural Materials are included all those parts which make up the actual reactor core with the exception of fuel elements, control rods, Table 62.

Composition of Wrought Stainless Steels* Composition,

AISI type

Mn

Si

(max)

(max)

%

Ni

C'r

Other t


Austenitic Steels 301 302 302B 303 304 304L 305 308 509 309S 110 31OS 314 316

3I6L 317 321 347

0.08-0.20 0.08-0.20 0.08-0 20 0.15 max 0.08 max 0.03 max 0. 12 max 0.08 max 0.20 max 0.08 max 0.25 max 0.08 max 0.25 max 0.10 max 0.03 max 0. 10 max 0.08 max 0.08 max

2.00 2.00 2.00 2.00 2.00 2.00

2.00 2.00 2.00

2.00 2.00 2.00 2.00 2.00 2.00

2.00 2.00 2 00

1.00 1.00

2.00-3.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1 50 1.50

1.50-3.00 1.00 1.00 1.00 1.00 1.00

16.00-18.00 17.00-19.00 17.00-19.00 17 00-19.00 18.00-20.00 18 00-20.00 17.00-19.00 19.00-21.00 22.0O-24.0O 22.00-24.00 24.00-26.00 24.00-26.00 23.00-26.00 16.00-18.00 16.00-18.00 18.00-20 00 17.00-19 00 17.00-19.00

6.00-8 00 8.00-10.00 8.00-10.00 8.00-10.00 8.00-11.00 8.00-11.00 10.00-13.00 10.00-12.00 12.00-15.00 12.00-15.00 19.00-22.00 19.00-22.00 19.00 10.00

22 00 14 00

10.00-14.00 11.00-14 00 8.00-11.00 9.00-12.00

1

Mo 2.00-3.00

Mo 1.75-2 50

Mo 3.00-4.00 Ti 5xC min Ch lOxC min

Martcnsitic Steels 403 410 416 420 431 4 40A 440B 440C 501 502

0. 15 max 0. 15 max 0.15 max . 15 min . 20 max

.60-0.75 .75-0.95 .95-1.20 . 10 min . 10 max

I 00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

11.50-13.00 11.50-13.50 12.00-14.00 12.00-14.00 15.00-17.00 16.00-18.00 16.00-18 00 16.00-16.00 4.00-6.00 4.00-6.00

1.25-2.50 Mo 0. 75 max

Mo

0. 75 max

Mo 0.75 max

Ferritic Steels 405 430 430F 446

0.08 max 0. 12 max 0. 12 max 0. 35 max

1.00 1.00 1.25 1.50

I 00 I Oil 1.00 1.00

11.50-13.50 14.00-18 00 14.00-18.00 23.00-27.00

Al 0.10-0

30

N 0 25 max

* From Taylor Lyman and Carl H. Gerlach (eds.). "Metals Handbook," 1954 supplement, American Society for Metals. t Other elements are as follows: Phosphorus is 0.045 per cent maximum and sulfur is 0.030 per cent maximum, t Austenitic steels. except for type 303, in which phosphorus or sulfur ur selenium is 0.07 per cent minimum, with zirconium or molybdenum 0.60 per cent maximum. Phosphorus is 0.040 per cent maximum and sulfur is 0.030 per cent maximum, II Alartensitic .steels. except for types 416 and 430F, in which phosphorus or selenium is 0.07 per cent minimum, with zir conium or molybdenum 0.60 per cent maximum. Type 414 has been omitted from this article because of lack of extensive use.

i^

?

03 '""'

g/em't.

of linear io-«/°c

thermal

ex

jiohm-cm.

310

20°C, 90 400°C 800°C IOO°C. 14.0 600°C. 16.5 I000°C. 19.5 100°C. 0.032 500°C, 0.040

1370-1425

7.92

Type

. . . 0. 12

0

Battelle

1480-1510 57 88

I 0.

§

. . .

.

Memorial

416

Steels*

Institute,

1952.

10. ll.5(500°C) 12.4(78700 060

1480-1510 57

7.73

Ferromagnetic

ll(700°C) 10.2 11.2 13.2 0.060 0.065

tic

Type

II

Eichen,

72 100 121 16.5 18.0 20.0 0.037 052

410

7.75

Type

Martens!

Stainless

0

and Erwin

347

7.98

Type

of Various

0.

Ferromagnetic

10 10.5 12.0 0.057 0.062

1450-1510 60

446

7.60

Type

1450-1480 67 98 119 8.8 10.0 11.2 0.050 058

Ferritic

430

7.70

Type

0

X

Compiled from steel producers' data by J. H. Jackson 62.43 - lb/ft'. Gram/em" :Cal/(sec)(cm)(°C) (2.419 10') - Btu/(hr)(ft)(°F). - Btu/(lb)(°F). UCal/(g)(°C) Can be slightly magnetic after cold working.

037 052

5 00

II

Nonmagnetic

16.0 17.2 19.7 0.035 0.050

1400-1425 72

321

1370-1400 76

Type

7.92

316

Constants

7.92

Type

Physical

Austenitic

63.

I

Thermal conductivity, cal/(cm)(sec)(°C)t Specific heat, cal/(cm)(°C)f Magnetism

Coefficient pansion,

range, °C Melting Electrical resistivity,

Density,

Constant

Table


0

X

X

*

t

I

X

I

7.58-7.75 7.60-7.98 [Wrought 1370-1510 45-79 88-104.5 115-121 8.8-17.0 10.0-18.0 11.2-20.0 0.032-0.087 0.040-0 080

[Cast

Range of all types of stainless steel

10-62

REACTOR

MATERIALS

[Sec.

10

This includes all supporting members, pressure vessel, tubes, and solid moderator. valves, pumps, and the like. Although in some cases a single material like stainless steel serves as both cladding and structural material, such cases are not the rule. The nuclear and physical properties considered desirable for Fuel Diluents and In the selec Cladding Materials are, in general, desirable for Structural Materials. tion of structural materials, more emphasis is placed on strength and corrosion resist ance than on nuclear properties. Aluminum, zirconium, and stainless steels have been most widely used as structural materials in thermal reactors. Of these, aluminum and zirconium have been described under Fuel Diluents and Cladding Materials. As less emphasis is placed on low thermal-neutron capture cross section and more emphasis is placed on high-temperature properties, a number of metals and alloys These include the metals molybdenum, titanium, and warrant consideration. niobium and alloys of the so-called "superalloy" type developed for gas-turbine application. As operating temperatures continue to rise, metals are no longer able to meet the Ceramics and cermets (ceramets) will be used for such applications. requirements. 4.2

Stainless Steels


The good corrosion resistance, reasonable cost, relatively high strength, and fairly low cross section of stainless steel have made it the leading structural material for Although much emphasis has been placed on Type 347 stainless reactor application. Table 64.

A1SI and ASTM designation, type No. 301 302 302B 303 304 305 308 309 310 316 317 318 321 330 347 403 405 410 414 416 416 420 420F 430 430F 431 440A 440C' 440K 442 446 501 502

Typical Room-temperature Physical Properties of Wrought Stainless Steels* Yield strength (0.2% offset).

Tensile strength. 10' psi Antoo 0 80 0 90 0 75 0 80 0 75 0 80 0 80 0 80 0 75 0 75 0 75 0 85 0 85 2 85 0 65 0 60 0 65 0 100 0 85 0 75 0 90 0 85 0 65 0 65 0 105 0 95 0 100 0 100 0 80 0 75 0 60 0 60 0

Hard ened

200 0 200 0 220 0 200 0 200 0 270 0 245 0

220 275 285 285

0 0 0 0

220 0 220 0

An nealed 40 0 30 0 35 0 30 0 30 0 25 0 30 0 30 0 35 0 30 0 30 0 30 0 30 0 40 0 35 0 38 0 32 0 38 0 65 0 50 0 40 0 50.0 50 0 35 0 35 0 90 0 55 0 60 0 60 0 45 0 45 0 25 0 25 0

* Compiled from steel producers' data by t Static determination.

J. II.

Hard ened

165 0 165 0 175 0 115.0 180 0 220 0 175.0

185 0 240 0 275 0 275.0

175.6 175 0

Elongation in 2 in.. % An nealed 50 0 50 0 40 0 40 0 50 0 50 0 40 0 40 0 40 0 40 0 40 0 40 0 40 0 46 0 40 0 20 0 20 0 20 0 15.0 15.0 20 0

ISO ISO 20 0 15.0 20 0 20 0 8.0 8.0 20 0 20 0 30 0 30 0

Hard ened

2 0 2 0 2 0 15 0 10 0 2 0 5.0

10 0 2 0

10 10 14.9 14 9

Reduc tion of area, % 60 0 60 0 60 0 50 0 60 0 60 0 50 0 50 0 50 0 50 0 50 0 50 0 50 0 72 0 50 0 50 0 50 0 50 0 50 0 40 0 50 0 40 0 40 0 40 0 40 0 60 0 40 0 35 0 35 0 40 0 40 0 70 0 70 0

Iiod impact, ft-Ib 85 85 80 85 60 85 70 80 80 70 70 70 80 98 80 85 25 85 60

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

60 0

70 0

65 0 85 0

Rockwell hardness An nealed

Hard ened

B90 B90 B90 B90 B80 B90 B95 B95 B90 B95 B95 B95 B95 B95 B95 B90 B95 C30 B90 B% BI00 HI00 B95 C28 BIOO BI05 BI05 B95 B95 B85 B85

Modulus of elasticity. 10*DSlt 29.0 29 0 29 0 29 0 29 0 29 0 29 0 29 0

300 29 0 29 0 29 0

290 C45

bis

C45 C30 C40 C55 CSO

C45 C55 CSO C56

C40 ('40

Jackson and Krwin Kichen. Uattelle Memorial Institute, 1952.

29 30 29 29 29 29

0 1 0 0 0 0

290 29 0 29 0 29 0

290 280 31 31 31 29 29

8

t 8 0 7

290 290

Sec. 10-2]


10-63

steel for many applications, a steel containing less nickel and chromium could be used. This would, of course, lead to savings both in dollars and in neutrons. Designations, compositions, and cross sections of a number of wrought stainless steels are given in Table 52. Cast stainless steels find little application in reactors and are therefore not covered here. In those cases where a cast material is desirable, complete information may be

found in the ASM "Metals Handbook." Since availability, melting, fabrication, machining, and the like of stainless steels are covered well in the literature, these items will not be discussed here. 4.21 Physical Constants. Physical constants of typical stainless steels are shown in Table 53. 4.22 Mechanical Properties. Mechanical properties of wrought stainless steels are given in Tables 54 and 55. Additional data are available in ASTM Special Tech nical Publication 52-A, 1950. 4.23 Corrosion. Stainless steels have been used a great deal as reactor containers for water at temperatures around 600°F. Although stainless steels develop a darktarnish film and sometimes a powdery red or black oxide under these conditions, they are in general very satisfactory. The oxidation resistance of stainless steels varies greatly with chromium content. Typical maximum operating temperatures are given in Table 55.


Table 56. AISI and ASTM designation

Oxidation Resistance of Stainless Steels* Max temp for continuous operation

with good resistance to scaling and oxidation,

AISI and ASTM designation

°F

301 302 303 304 305 308 309 310 316 317 318 321 347 403 405

* Compiled

1650 1650 1650 1650 1650 1700 2000 2100 1700 1700 1700 1700 1700 1250 1300

from various sources by

J.

Max temp for continuous operation

with good resistance to scaling and oxidation,

°F 410 414 416 418 420 420F 430 430F 431 440A 440C 440F 442 446

1200 1250 1250 1300 1200 1200 1500 1550 1600 1400 1400 1400 1800 2050

H. Jackson and W. K. Boyd, Battelle Memorial Institute. 1952.

In general, the chromium-iron and austenitic chromium-nickel alloys are not resistant to molten metals such as zinc, tin, solder, babbitt, copper, brasses, aluminum, bismuth, lithium, bismuth-lead, bismuth-indium, and lead. Molten lead and molten sodium under certain conditions are exceptions. Stainless steels, in general, are cor rosion resistant in sodium and sodium-potassium alloys at elevated temperatures The carbon con provided the oxygen content of the sodium is below 0.02 per cent. tent of the sodium must also be kept low. Types 304, 310, 316, 321, and 347 appear to resist potassium at temperatures of 1400 to 1600°F. Stainless steels are generally not resistant to molten caustics. 4.3

High-temperature

Alloys

A number of so-called superalloys containing nickel, chromium, cobalt, and iron Although many of these alloys have been developed for gas-turbine applications. have relatively high thermal cross sections, some are attractive for thermal reactor because of their excellent high-temperature strength and oxidation application

REACTOR MATERIALS

10-64

[Sec. 10

For intermediate or fast reactors, many of the alloys are attractive. resistance. Nominal compositions of a number of heat-resisting alloys are given in Table 56. Typical stress-rupture values are given in Table 57. Table 56. Alloy

|

Nominal Compositions of Heat-resisting Mn

C

Si

Cr

Ni

Co

Mo

S-588

Timken 19-9 19-9

DL DX

16-25-6..

0.08t 1.25 I.Ot 14.75 25.5 0. I5t 2.0t 0.75t 16.0 15.0 0 04 1.38 1.00 13.5 26.2 0.23 0 65 I. 16 23.2 12.3 0.07 1.50 0.50 12.5 15.0 0. 10 1.0 0.4 20.5 32 0.42 1.5 0.8 18.4 20 0. 10 1.35 0.70 16.0 25 9 0.30 I. 10 0.60 19.0 9 0 30 1.00 0.55 19.2

Cb

Ti

Al

*

Fe

Other

Rem

V 0.25 N 0. I5t

Alloys

Iron-Chromium-Nickel Wrought alloys: A-286 Croloy 15-I5N... Discaloy 24 II. R. Crown Max Haynea Alloy 88. Incoloy T

W

Alloys

1.25 1.55

3.9

2. 10 0.35t 1.40 1.05 1.61 0.

II

3 0

2.0

0.6

4.0 6.0

4.0

4 0

1.25 1.50

1.2 1.2

0.40 0.30 0.55

0 6

10

Rem

Rem Rem Rem Rem Rem Rem Rem

B 0 15 N 0 15

Rem


Col mlt- Nickel- Chromium -Iron Alloys Wrought alloys: K-42-B N-155 Refractaloy 26 Refructaloy 80

0.05 0.15 0.05

0.7 0.5 0.7 0.7 0. 10 0.6 0.7 0.42 1.25 0.4 0.38 1.20 0.4 0.31 0.9 O.S

S-590 S-816

V-36

0.7 1.5

18.0 21.0 18.0

20.0 20.5 20.0 25.0

43 0 22.0

20.0 20.0 3.0 37.0 20.0 3.0 20.0 30.0 10.0 20.0 20.0 4.0 20.0 43.0 4.0 20.0 42.0 4.0

2.5 5.0 4.0 4.0 2.6

1.0

2.5

0 2

13.0

2.8

0.2

18.0 14.0 24.0

4.0 4.0 2.2

Rem

N 0. 15

40t

3.0

Nickel-base Alloys Wrought alloys: Hastelloy X... . Iuco 700 Inco 739 Inconel Inconel W Inconel X Inconel X. 550. M-252 Nimonic 75. . . . Nimonic 80A. . . Nimonic 90 ... . Waspaloy Cast alloys: Hastelloy B . Hastelloy C

0.15

0. 15

1.0

0.12 0.05 0.08 0. I0t

0.4 0.70 0.50

I.Ot

22.0 15.0 0.2 15.0 0 20 15.5 0.25 15.0 0 3 15.0 0 3 15.0 0.65 19.0 0.6 20.0 0.50 20.0 0.40 20 0 0.75t 19.5

0. 10 0. 10

0.8 0.6

0.7 0.7

0 10 0 07 0 04

0 05 0.2

0.05 0.35 0.04 0.60 0 03 0.5 0.03 0.5

1.0 16.0

45.0 49.0 28.0 77.0 76.0 75.0 73.0 73.0

9.0 3.0

Rem

0.6 0.6

Rem 10.0 10.0 76 0 76 0 58.0 16.0 Rem 13.5 4 25 65 0 57 0

28.0 17.0

2.0

3.0

0.5

1.7

2 7

1.0

2 5

0.6

7 0

2.3 2 5 2 5 0 4

0 9 1. 1

0.87

0 06 1.0 1.4 2 50 1 25

2.3 2.3

10 6.5 6.5 5.0t 2.4 0.5 0.5 2 0t

4.0

5.0 5.0

5.0

1.0

Cobalt-base Alloys Wrought alloy: Haynea 25 (L605) Cast alloys: Haynes 21 Haynes 30(422-19) Haynes 3 I (X-40) Haynes 36 * From Taylor

I

0.25 0.40

0.60 0.60 0.60 0.60 0.60 0.60 1.2 0.50

0 40

0.40

50

20.0

10.0 51.0

27.0 24.0 25.0 19.0

17.0 51.0 10.0 55.0 10 0 54.0

3.0 62.0

Lyman and Carl H. Gerlach (eds.),

can Society for Metals.

t Maximum.

10

0 12

5 0

1.0

6.0 8.0 14.5

'Metals Handbook."

10 10 10

B 0 03

1954 Supplement. Ameri



Typical Stress-rupture

Properties

Stress.

Alloy

I200°F

100

100

S-588

Timkcn 19-9 19-9

DL DX

16-25-6. .

61.0

48.0

35.0

I 500°F

100

1.000

Iron-Chromium-Nickel Wrought alloys: A-286 Croloy I5-I5N. . . Discaloy 24 H.R Crown Max. Haynes Alloy 88. Incoloy T

29.5

21.5

59.0 33.0 41.0 45.0 52.0 52.5

32.0

20.0

18.5

13.5


66.0 50.0 80.0 48.0 60.0

42.0

31.0

25.5

20. 1

30.0 34.0 38.0 42.0

25.0 25.0

14.7 18.0 17 0 19 0

10.8 15.0 13.5 17.0

8-590

S-816 V-36

40.0 40.0 63.0 38.0 45.0

37.0 31.0 51.0 30.0 36.0 35.0

100

1.000

100

1.000

27.0 24.0 38.0 22.0 28.0 26.5

12.0

8.0

5.0

2 5

12.5 14.0 15.0

9.0

5.6

9 5 11.0

5 5

3 5 1 0 5 0

18.5

7.7 10.0

26. 1

Cobalt-Nickel-Chromium-Iron Wrought alloys: K-42-B N-155

13.8

41.0 24.5 49.0

28 0

1.000

I800°F

I 600°F

Alloys

18.0

55.0

Alloys*

of Heat-resisting

I0> psi to rupture in 100 sod 1.000 br

I350°F

.000

10-65

15 0 16.0

13.5

7.0 10.0

9.0 10.0

Alloys

17.5 18 0 27 0

22.0 24.0 23.0

II.

0 13.0 18.0 16.0 17.5 18.0

8.5

Nickcl-baso Alloys Wrought alloys: Hastclloy Alloy X. Inoo 700 Inco 739 Inconel Incond W Inconel X Inconel X, 550. M-252 Nimonic 80A Nimonic 90 Waspaloy Cast alloys: Hastelloy Alloy B . Hastclloy Alloy C

44.5 22.0 74.0 80.0 67.3 76. I

30.5

26 0

15.5

10.0

73.0

42.0 34.0 5.7

30.0

28.0

20 0

19.0

11.5 18.0

19.0 4. 7 I 18 16 14.

14.5

10 5

54.0

45.0 48.0

68 0

56.0 63 0

6.8 30.0 38 0

52^0

35^0

48.0 50.6

36.0 38 0

57 5

51.0 49.5

40.5 42.5

35.0 32.0

25 5 25.0

28.0 34.0 29.0 24.0

3.7 21.0

3.5

III)

2 7

1.6

4.8 9.0

2.3

10.2 10.0 8 5

28 0

18.0 15.5 17.9

31.5

20.0

19.5

18.5 19.0

12.7 14.5

13.2

9.2

Cobalt-base Alloys Wrought alloy: Haynes Alloy Cast alloys: Haynes Alloy Haynes Alloy Haynes Alloy Haynes Alloy

25 (L605) . .

70.0

58.0

43.0

33.0

23.0

17. 0

15.5

10.5

7.0

3.8

21 30 (422-19) 31 (X-40) . 36

51.0

44.2

22.0 28.6 28.4 29.0

14.2 21.7 23.4 25.5

13.2 14.8 18.0 18.5

9.4

7.0

46.0

2.0 36.0 33.0 41.5

16.7 15.8

55 .0

32.0 47.0 45.0 48.0

10.0 11.3 10.5

9.8 7.2

* From Taylor Lyman and Carl H. Gerlach (cds.). "Metals Handbook," Society for Metals.

21.0 23.0

7. 1

1954 Supplement, American

10-66

MATERIALS

REACTOR

[Sec.

10

Another type of alloy containing iron, chromium, and aluminum has excellent oxidation resistance at temperatures up to 2200°F but is not strong at these tempera tures. Such alloys might well serve as oxidation-resistant claddings for stronger, less corrosion-resistant materials. Typical compositions and properties are given in Tables 58 and 59. Table 68.

Mechanical Properties of Iron-Chromium-Aluminum Tem

Cr. %

Density, g/cm'

pera ture,

C %

Al. %

16-18 23-27 40-45 65-68 17 25 17 25 17 25 10

IS


25 10

lit

I8t

ll.5t

25t 251

4.5-18 4.5-7

Room Room Room Room

0.03-0.07 0.03-0.07 7.5-12.01 0.03-0 07 7.5-12. 5| 0 03-0.07 0 03 5.6 5.07

200 200 400 400 800 800

0 03 0.03 0.03 0 03 0 03

5.6 5.07

5.6 5.7

7.0-7 2 6.9-7.2 6.8-7.0 6.75-6.85

5 10 10 20 6 3 10 10 5

strength, I0> psi

10-'/°C

20 I00°C, 2O-IO0°C, 20 I00°C. 20-100°C,

14.5-15.0 14.5-15.0 16-17 16-17

Room Room Room Room Room Room Room Room

0-l000°C, 13.9

1200

0-I000°C. 15.3

Elonga

Tensile

Coefficient of thermal expansion,

°C

Alloys*

tion,

85-99 99-114 114-142

15-25 15-20

2-7

150- 170 160-1*0 240- 260 270-300

85 86. 5 79 81.5 10 10. 1 128 250 255

0-1000°C, 0-1000°C,

16.3 18.6 0-1000-C. 26.9 86

22 5 25 0

96.5

* Compiled by the author from various Annealed at 700°C. t Arc-melted, hot-rolled.

Brinell hard-

5.0

99. 8 128 100.5

18

sources,

t

Table 69.

Creep Properties of Iron-Chromium Stress to rupture, psi

Cr,

Al,

%

%

25t 25f 25t 25t 25t 25t 25t 25t

5 5 5 5 5 5 5 5

Other,

Temperature,

Alloys*

Stress to produce minimum creep, psi

op

%

5 Ta 10 Ta 2.5 Nb 5 Nb 5 W 5 Mo 1 Be

-Aluminum

2200 2200 2200 2200 2200 2200 2200 2200

0.5 hr

1 hr

10 hr

100 hr

695 1.100 1.335 825 1.115 960 800 880

550 950 1.160 700 930 730 650 775

270 500 720 350 460 370 350 440

100 310 420 240 290 270 >230 310

* Compiled by the author from various t Arc-melted, hot-rolled.

1.0 % /hr

0.1

%/hi

0.01 %/hl

400 550

330 420

240 320:

420 320 290

300 220

210

I75J

sources,

t Extrapolated.

4.4

Molybdenum,

Titanium, and Niobium

Among the elements, molybdenum, titanium, and niobium are promising structural materials of intermediate cross section. Titanium possesses excellent corrosion resistance in hot water and has good strength at moderate temperatures. Molyb denum and niobium have excellent high-temperature strength but are not corrosion Physical constants of molybdenum, titanium, and niobium are given in resistant. Table 60. Mechanical properties of these elements are given in Table 61. Hightcmpcrature properties of several molybdenum alloys are given in Table 62.


properties of reactor materials Physical Constants of Molybdenum,

Thermal-neutron cross section:

2.4 ± 0.2 0.0151

Cm«/g

2622 ± 10 0°C, 0.0589

....

Thermal conductivity, cal/(sec)(cm)(°C) Coefficient of linear thermal

expansion,


io-«/°c

Titanium

Niobium

5 6 ± 0 4

Cold-pressed bar, 6.0 Sintered bar, 9. 8 Arc-cast, 10.2 Specific heat, oal /(gm) (°C)

and Niobium*

Titanium,

Molybdenum

10-67

I00°C, 0.0650 475°C. 0.0750 27°C, 5.78 727°C, 23.9 II27°C, 35.2 I927°C, 59.5 2622°C. 81.4 0°C, 0.32 I473°C, 0.26 2173°C, 0.17 27'C, 5. 1

1 1 ± 0 1 0.0071 8 57

0.070

4 507

0-500°C, 0. 1386

1690

2415 0°C, 0 0647

20°C,

18°C

47. 8

13 1

25°C, 0.41

25°C,

8. 5

500°C, 5. 1 1000°C, 5.5 1500°C. 6.2 2000°C, 7. 2

•Compiled by the author from data found in U.S. Atomic Energy Commission, "React r Hand book," vol. 3, General Properties of Materials, AECD-3647, McGraw-Hill Book Company, Inc., Now York, 1955.

Table 61.

Mechanical Properties of Molybdenum, Titanium, and a Niobium-Titanium Alloy* Molybdenum

Tensile strength: Test temperature, °C Yield strength, 10> psi . . . Ultimate strength, 10>psi

38 40 67

Hardness, Vickers.

93 24 55

150 15 52

Titaniumf

200 12 47

Room temp (arc-cast, hot-rolled), 225 Room temp (an nealed at II50°C),

200-220 100°C, 210 300"C, 120 500°C, 59 600°C, 39 800°C, 13 900°C. 7

187

I00°C. 110

300°C, 400°C, 500°C, 600°C, 700°C, 100-hr creep rupture:

69 59 58 56 54

870 18.5

0.06

980J

12 5 0 5

1090 10 0

I * Compiled by the author from various sources. t Commercial grade. X 57-hr rupture time

Nb-IOw/oTi

1200

1200

14.7

15.6

Room temp (arc-

Room temp

Recrystallized

Arc-melted

melted sheet), 140 Room temp (an nealed sheet), 88 870°C (arcmelted), 63.2

(arc- melted), 187 870°C (arcmelted), 156

In sodium

Recrystallised, arccast, cold worked; tested in vacuum

Temperature, °C. Stress, I0> psi. . . Creep rate. % /hr

Niobium

93 200 315 538 90 73 56 17 102 84 66 22

Room temp,

Niobium,

with calcium

getter

II

425 20

980 15

0.15

Coldrolled 980 16.5

0.077

980 12.5

0.07

10-G8 Table 62.

REACTOR MATERIALS

[Sec. 10

Maximum Stress-rupture

Strengths Shown by Molybdenum-base Alloys at 1600, 1800, and 2000°F* Stress,

psi, to produoe

rupture at

Alloy I600°K

In

In 0.75% Cb 0.092% Zr 0.46% Ti 0.50% Ti

I 800° I"

1.000

100

2000'F

Hr

Hr

69.000 64,000 66.000 > 70.000

51.000 56.000 53.000 55.000

22.000 > 25.000 > 25. 500 > 25.000

60.000 63.000 63.000

39.000


In 10 Hr 0.75% Cb 0.092% Zr 0.46% Ti

75.000 74.000 76.000

46.000

•From Howard C. Cross and Ward F. Simmons, "Alloys and Their Properties for Elevated Tem perature Servico," Battclle Memorial Institute. note I: All 1.000-lir rupture strengths are estimated. note 2: All above values are for materials tested in stress-relieved condition. 4.6

Ceramics and Cermets

operat ion, it is necessary to depend on materials These may be either other than metals. RUPTURE IN 100 HOURS ceramics or cermets. xFS-26(54TiC,40Ni 60 Ceramic materials as a class have ex 6CrjCz) cellent high-temperature strength and OKI6IB — 50 *KI5IA(80TiCL20Ni) They suffer may be oxidation resistant. from low ductility and poor thermal shoek DTC-66K50 TiC,50 Inconel) resistance. The leading contenders for nuclear application, BeO, Be2C, MoSi., and SiC, have been covered in detail in Art. 2 under Fuel Diluents and Cladding Materials. Cermets, a combination of a refractory ceramic and a metallic binder, attempt to capture the worth-while properties of both metals and ceramics without retain ing the poor qualities of either. Such 2000 1800 1400 1600 materials have good high-temperature TEMPERATURE.F strength and oxidation resistance. Com Fio. 27. Rupture-strength curves for positions of leading cermets are given in (Reprinted Howard C. cermets. from Table 63. Rupture data for cermets are Alloys and Cross and Ward F. Simmons, shown in Fig. 27. Their Properties for Elevated Temperature A comparison of high-temperature Service, Battclle Memorial Institute.) strength of a large number of hightemperature materials is given in Fig 28.

For extremely high-temperature


Sec. 10-2]

Table 63.

10-69

Cermets'

Chemical composition,

%

Alloy

TiC FS-26 FS-27 K-151A

K-I52-B

K-162B TC-66-I

54.3 42.9 80 70 70 50

Ni

CnCi

Mo

40.0 50.0 20 30 25 Balance

5.7 7. 1

5 Xnconel

(infiltrated)


* From Howard C. Cross and Ward F. Simmons, perature Service," Battelle Memorial Institute.

300

500

"Alloys and Their Properties for Elevated Tem

1000 Temperature,

Kig.

1500

2000

F

(Reprinted from Howard C. 28. Stress vs. temperature curves for rupture in 1,000 hr. Cross and Ward F. Simmons, Alloys and Their Properties for Elevated Temperature Service, Battelle Memorial Institute.) 6

CONTROL

AND ABSORBER

MATERIALS

Materials to be used as absorbing controls in reactors nuclear and metallurgical properties. These include:

need to possess a number of

1. A high cross section for absorption of neutrons 2. Adequate strength 3. Low mass to permit rapid movement

4. Good corrosion resistance to reactor coolant 5. Chemical and dimensional stability under heat and irradiation 6. Availability, fabricability, and reasonable cost Boron and cadmium of the better known elements have high absorption cross sec In use, the boron is usually incorpo have been used for control materials.

tions and

10-70

MATERIALS

REACTOR Table 64.

[SEC. 10

Physical Constants of Boron, Cadmium, and Hafnium* Boron

Cadmium

Hafnium

Thermal-neutron cross section: 750 ± 10 42 Crystal, 2.33

Cm'/g

2400 ± 200 12.9 Cast, 8. 648

Amorphous, 2. 3

Heat of vaporization (25°C), Heat capacity, cal/(mole)(°C)t Temperature range, °C Klectrical resistivity, /iohm-cm


Linear coefficient of thermal ex pansion, I0"e/°C Thermal conductivity, cal/(sec)(cm)CC)

115 ± IS 0.39

Calculated. 13.36 Measured (crystal bar).

Cold-worked. 8.694

13.36 2130 ±

321

2000-2300 90.000 1. 54 + 440 X 25-927 Pressed powder I00°C, 13.20 320°C. 36 600°C, 0.08

I0T

26.750 5.31 + 2.94 X 10 25-321

T

6 00 + 0.524 X 25-2227

I8°C, 7.54

20-100°C, 31.8 Room temperature. 0.22

20-750°C, 8.3 B
10

T

I00°C, 47. 1 300-C, 75.0 600°C, 106.3

I00°C. 9. 82 300°C, 16.50

Crystalline

IJ

0-l000°C. 5.9

435°C, 0. 114

* Compiled by the author from data found in U.S. Atomic Energy Commission, "Reactor Hand book," vol. 3. General Properties of Matter, AECD-3647. McGraw-Hill Book Company, Inc., NewYork, 1955. t T in degrees Kelvin.

Table 66.

Mechanical Properties of Boron Steel. Boron Stainless Steel. Cadmium, and Hafnium* Boron

Boron steelt

Tensile strength, pai

Modulus of elasticity, 10« pai

Hardness

steelt

Annealed 2000°F, 1 hr W.Q. 0.2% yield: 67.300 0.2% yield: 58.400 Ultimate: 86.400 Ultimate: 147.600 Normalised I600°F, Annealed 2000°F, 1 1 hr A.C. hr W.Q. ; aged 16 lu 0.2% yield: 100.000 st 1470°F A.C. Ultimate: 136.000 0.2% yield: 136.500 Ultimate: 174.000 1550°F, 1 Annealed Annealed 2000°F, 1 hr F.C. hr W.Q. 32.5 29.2 Normalised I600°F, Annealed 2000°F, 1 1 hr A.C. hr W.Q. ; aged I6hr 32.5 at 1470°F A.C. 29.5 Annealed 1550°F, 1 Annealed 2000°F, 1 Annealed hr F.C.

I550°F,

stainless

1

hr F.C. Rockwell: 21. 5 C to 21.8 C Normalized I600°F, 1 hr A.C. Rockwell: 33 C to 36. 2 C

hr W.Q. Rockwell: 35. 4 C

Cadmium

Hafnium

99.95% cadmium, chill-cast and aged at room temperature Ultimate: 10,300 99.95% cadmium, cold-worked 70%,

Arc-melted crystal bar 0. 2 9b yield: 40.800 Ultimate: 67.500

aged at room tem perature Ultimate: 13,700 7. 1-10

14.0

l-in. chill casting Brinell: 21-23

Cold-swaged, an nealed at I000°C; cold-swaged

Annealed 2000°F, 1 hr W.Q. ; aged I6hr at I470°F A.C. Rockwell: 36.6 C

* Compiled by the author from various sources. tSAE 4140 + 1.25 per cent boron. X 17-7 stainless steel + 1.25 per cent boron.

F.C. ■»furnace cooled.

A.C. -» air cooled.

20

n

Sec. 10-2]


10-71

Cadmium may be protected by cladding or may rated in steel or otherwise contained. Recently hafnium, the high-cross-section itself be clad to a structural material. material removed from zirconium, has gained prominence as a control material. Hafnium has good strength and excellent corrosion resistance in water. For many Properties of boron, cadmium, and hafnium are applications it is an ideal material. given in Tables 64 and 65. A number of the rare earths have high cross sections and may eventually be useful Some of their properties are given in Table 66. as control materials. Table 66.

Element

Sc

Y

Lat

Ce

Pr

Nd Pm


Sml Eu Gd

Tb Dy Ho

Er Tm

Yb Lu

Physical and Chemical Constants of the Rare-earth

Thermal-ncutronabsorption cross section, barns /atom

Density. g/cmJ

13 ± 2 1.38 ± 0.14 8.9 ± 0.3 0.70 ± 0.08 11.2 ± 0.6 44 ± 2.0

3 016 4.472 6. 162 6.768 6.769 7.007

6500 ± 1000 4500 ± 500 44.000 ± 2000 44 ± 4 1100 ± 150 64 ± 3 166 ± 16 118 ± 6 36 ± 4 108 ± 5

7.540 5. 166 7.868 8. 253 8 556 8.799 9.058 9.318 6.959 9.849

Melting point, °C

Boiling

1550-1600 1500 920 804 935 1024 1300 1052

27501 3500t 4515 3200 3290 3450 3000t 1900t

point,

°K

73t 94t 81 79 79 69

I700t 1350 1400-1500 1400 1500 1500-1550 1550-1650 824 1650-1750

Heat of vaporiza tion, kcal /mole

3000t 2800t 2600 2600t 2900t 24001

I800f

70t 46t 42t 72t 70t 67 67t 70t

5lt

32. 5t 59t

2200t

Metals*

Heat of fusion, kcal /mole

3.8 4. 1

2.4 2.2 2.4 2.6 3.0 2.6 2 5

3.7 3.9 4. 1 4. 1 4. 1

4.4 2.2 4.6

Crystal structure at room temperature

Hep Hep Hop Fco

Hep Hep Rhom-hcp Bcc Hep Hep Hep Hep Hep

Hep Fco Hep

* Compiled by the author from U.S. Atomic Energy Commission, "Reactor Handbook," vol. 3. General Properties of Materials, AECD-3647, McGraw-Hill Book Company, Inc., New York, 1955. and unpublished information from Ames Laboratory, Iowa State College. t Estimated values. — 5.302 A; density — 6.190 g/cm*. The low-temperature X Fee La has a lattice parameter of ao transition occurs between 300 to 350°C. 8.996 A 23° 13'. ^ The true structure of Sm is rhombohedral; at

-

-

It is available in Boral4 is a mixture of 35-50 weight per cent B4C and aluminum. sheet form \i in. and \\ in. over-all thickness, including 0.020-in.-thick aluminum clad. The thermal conductivity is 25 Btu/(hr)(ft)(°F) at 200°F and 19 Btu/(hr)(ft)(°F) at 500°F. The density is 2.53 g/cm1 and specific heat 0.175 cal/g. The heat generation from the (n,a) reaction is 7.4 watts/ft* in a thermal-neutron flux of 10'° neutrons/ (cm*) (sec) — the heat generation is substantially independent of thickness, since both and the J4-in. sheets are substantially black to thermal neutrons. The cost the is $15 per sq ft for j-jj-in. sheet and $18 for >i-in. sheet. REFERENCES Katz, J. J., and Eugene Rabinowitz: "Chemistry of Uranium," NNES, Div. Villi vol. 5, chap. 4, McGraw-Hill Book Company, Inc., New York, 1951. (Unclassified.) 2. Sailer, H. A., and F. A. Rough: "Compilation of U.S. and U.K. Uranium and Thorium Constitutional Diagrams," BMI-1000, June 1, 1955. 3. Lustman, B., and Frank Kerze, Jr. (eds.): "The Metallurgy of Zirconium," NNES, Div. VII, vol. 4, McGraw-Hill Book Company, Inc., New York, 1955. 4. U.S. Atomic Energy Commission: "Reactor Handbook," vol. 1, Physics, p. 760, McGraw-Hill Book Company, Inc., New York, 1955. 1.

10-72

REACTOR MATERIALS

[SEC. 10

BIBLIOGRAPHY 1.

U.S. Atomic Energy Commission: "Reactor Handbook." vol. 3, sec. 1, General Properties of Materials, AECD-3647, McGraw-Hill Book Company, Inc., New York, 1955.

Lyman, Taylor (ed.): "Metals Handbook," American Society for Metals, 1948 ed. 3. Lyman, Taylor, and Carl II. Gerlach (eds.): "Metals Handbook," 1954 Supplement, American Society for Metals. 4. Sailer, H. A.: Metals for Reactor Core Construction, Science, 119 (3079): 4 (Jan. 1. 2.

1954). 5. Keeler, John 6.

R.: Thorium, in "Engineering Materials Handbook," C. L. Mantell (ed.), McGraw-Hill Book Company, Inc., New York, 1958. "Handbook on Titanium Metal," Titanium Metals Corporation, 4th ed., July 15,

1951. 7. Lyons, R. N., et id.: "Liquid Metals Handbook," U.S. Atomic Energy Commission, Department of the Navy, NAVEXos, June 1, 1950. 8. Edwards, J. D., F. C. Frary, and Zay Jefferies: "The Aluminum Industry," McGrawHill Book Company, Inc., New York, 1930. 9. White, D. W., and J. Burke (eds.): "The Metal Beryllium," American Society for Metals, 1955. 10. Howe, John P.: The Properties of Graphite, Am. Ceram. Soc., 35 (11) (1952). 11. Parke, R. M.: Molybdenum. A High Temperature Metal, Metal Progr., 60 (1): 81-96

J.


(1951). 12. Smith, C. Stanley: Properties of Plutonium Metal, Phys. Rev., 94 (4) (May 15, 1954). 13. "Zirconium and Zirconium Alloys," American Society for Metals. 1953.

10-3

STRUCTURAL MATERIALS IN HIGH -TEMPERATURE WATER REACTOR SYSTEMS BY


Roger Sutton The problem of selection of structural materials for use in high-temperature water reactors is more involved than similar selections for conventional use because of unique factors peculiar to nuclear reactors, such as radiation effects, as well as the conditions normally encountered in high-temperature water systems. The choice of materials is, as in conventional systems, based on considerations of corrosion, galling, wear, mechanical properties, and fabricability. Information concerning the two last items is available in Sec. 10-2. In all cases where trade names and proprietary designations are used, it is to be understood that such nomenclature is used to indicate the material actually tested. There is no implication of superiority of these products over equivalent grades produced by other manufactures. 1

CORROSION

Materials in reactor systems must generally be much more corrosion resistant than in nonnuclear systems, both because of transport of radioactive material to the external system and because of deposition of corrosion products on fuel elements other heat-transfer surfaces. 1.1

and

Water Conditions

Several conditions of service must be taken into consideration in the selection of use as guides, bearings, gears, shafts, seals, etc. The major ones are

materials for

1. Water temperature

2. 3. 4. 5.

Water velocity Gas content of the water Type and amount of dissolved or suspended solid matter in the water pH of the coolant

1.11

Inhibitors.

Corrosion inhibitors used in conventional fossil-fuel power-plant Organic inhibitors would be decomposed by irradiation. Inorganic inhibitors would become radioactive and seriously limit accessibility to the system for maintenance and repairs. Inhibitors would also prohibit use of ion-exchange systems. Irradiation Effects. Neutron irradiation causes dissociation of water. At 1.12 elevated temperatures the net rate of dissociation is decreased by an increase in the rate of the back reaction. See Art. 3.1 of Sec. 10-5. Dissolved Gases. With continuous degassification the gases resulting from 1.13 the net radiolytic dissociation of water will achieve a steady-state concentration. An experimental in-pile loop of AISI 347 stainless steel, when operated at 500°F with con tinuous degassing, maintained an oxygen concentration of 0.02 to 0.5 ml /liter (STP). Similar systems, operated without degassification, reached a state of pseudo-equi librium with a hydrogen content of several milliliters per liter (STP) as a result of the 10-73

systems cannot

be used in nuclear systems.

10-74

REACTOR MATERIALS

[SEC. 10

intensification of the recombination reaction from the high gamma intensity and the removal of oxygen from the system by corrosion processes prior to equilibrium. The initial corrosion could be almost entirely suppressed by initially pressurizing the The pseudo-equilibrium concentration of hydrogen is a func system with hydrogen. tion of neutron and y flux, water temperature, ionic impurities in the water, and, indirectly, the over-all corrosion rate of the system. Furthermore, oxygen gas has been injected into the coolant system up to at least 30 ml/liter (STP), and hydrogen The major effect of these gas has been injected up to over 500 ml/liter (STP). additives on corrosion rate is attained at approximately 3 ml/liter Os or 25 ml/liter H2. Hence it is possible to operate a high-temperature water reactor either degassed, under steady-state conditions in a closed system, or with the deliberate addition of gases, particularly hydrogen. 1.14 Other Control. In addition, through the use of filters, deionizing columns, and addition agents such as sodium hydroxide and sulfuric acid, it is possible to control, at least partially, the pH and purity of the coolant.


1.2

Selection Criteria

From the corrosion standpoint, tentative criteria for the selection of the materials to be used for the critical components of a high-temperature, high-pressure, watercooled, and moderated power reactor might be as follows: 1. Satisfactory for unrestricted service: Rate of weight change less than 20 mg/ (dm*)(month), no local attack, and thin adherent film. 2. Usable in special applications and limited quantities. Rate of weight change between 20 and 200 mg/(dm*) (month), no local attack, and no evidence of thick or loose films. 3. For use only where other conditions dictate absolute necessity. Rate of weight change over 200 mg/(dm8) (month), minor local attack, and no loose, flaky scale. 1.3

Corrosion Data

Data reported by S. C. Datzko1* are summarized in Table 1. The addition of over ml/liter of hydrogen permits the widest selection of materials under the conditions These conditions were: chosen for presentation in the table. 50

Temperature, °F

500 2,000 30 >500,000 approx 0.1 % of total flow

Pressure, psig Flow past specimens, fps Resistivity of water, ohm-cm Ion-exchanger bypass

" of t he " Since the data are from as-tested specimens, a weight loss indicates sloughing corrosion product, and materials showing significant weight loss are, in general, the least desirable. 2

GALLING AND WEAR 2.1

Galling

Parts subject to relative motion (as in bearings, seals, control drive mechanisms, and other components involving rotary or translatory motion) must function without conventional lubricants and must therefore be resistant to galling as well as to cor rosion and wear. Galling is the physical manifestation of the union of the lattices at the high spots If the lattices are widely of two contacting surfaces moving relative to one another. dissimilar, the effect may be negligible, although John M. Bailey and Douglas Godfrey* propose a mechanism of fretting wherein it is postulated that the occurrence and extent of adhesion or cold welding were indicated during the first few cycles of fretting of practically all material combinations. Copper and iron, for instance, showed If the metals are similar, and particu significant adhesion or cold welding to glass. larly if the materials involved arc soft, local welding will occur. If motion is to con * Superscript numbers

refer to References


high-temperature water reactor systems

Sec. 10-3]

10-75

tinue, these welds must be broken with resulting surface damage. The resultant "scoring" is termed galling, and the ultimate result will be increasing temperatures and pressures on the high spots, finally causing seizure. From the earliest stages in the development of the nuclear power plant the problem of corrosion was recog nized. As a result, early designs made generous use of 18-8 type stainless steels. However, these steels cannot be used for sliding surfaces, since even the very simplest mechanical parts may gall and seize when assembly is attempted or, at best, after a few cycles of operation. Since galling is inevitably related to wear, data available apply to both. Much of the wear encountered in these systems involving boundary lubrication is promoted by incipient galling. Table

1.

Corrosion of Metals in High-temperature

Water*

Weight change mg/(dm!)(month)

Hydrogen

Material

Oxygen Degassed

500 ml/

50 ml/

liter


liter

Zirconium Zirconium -2.5% Bn Zirconium — 5%. Sn

AISI

A1SI

AISI AISI AISI AISI AISI AISI AISI AISI

302 303 304L 304 316 347 304 (nitrided) 347 (nitrided) 410 (hardened) 44C (hardened)

PH PH PH (nitrided) USS-W (nitrided) 17-4 17-7 I 7-7

-

4 1 0 0 0 1 0 175

-40 -36

-41 -70 92 170 I

-

70-30 cupronickel

I nconel X

K

Monel Monel 20% Pd, 15% Pd.

.. (ace-hardened). (annealed) 30'; Cd, 50% A* 30% Cd, 55% Ag

10 7 10 3 2

0 2 0

-2

2 0 - 200 •250

-4

-

8 129 •168

-

2 I 1 12 12 3

-7

- 49

-74

-7

-40

-7

-10

-97

-3

- 10

-15 -300 -200

-3 -6

Inconel

K

liter

12 « 5

10

A-Nickel

0.3 ml/

liter

14

5

-5

Carpenter 20 Stellite 3 Stellite 19 Stellite 21 Stellite 25

Monel

25 ml/

20

- 12I -5

- 1002 - 150 - 18 -30

-6

-1300

-36 -33 200

95 38

2 -30

Temperature, 500°F; Water velocity, 30 fps. • Summarized from S. C. DaUko, " Corrosion_of 2.2

- 2100 2900

Metals in High Temperature Water," ANL-5354.

Wear

Wear is the displacement of material as a consequence of relative motion between The result is a change in clearance brought about by surface damage. surfaces. This interferes with the performance of the device and ultimately progresses to a point where the unit can no longer perform the function for which it was designed. 2 21 Abrasion. When relative motion is introduced, high temperatures are At the points where these high tempera developed at the points of actual contact.

10-76

REACTOR MATERIALS

[SEC. 10

With air or water present, tures are developed, chemical reactions are accelerated. Therefore, wear products in the form of oxides oxidation is the most usual reaction. are produced. Since metallic oxides are generally abrasive in character, there is an increase in wear rate. Also, general corrosion may interfere with clearances and may also produce products that result in added wear by abrasion and damage of surface finish. 2.22 Fatigue Cracking. Fatigue, i.e., failure of a material by rupture as a result of cyclic stressing, is a function of the degree of stress and the numbers of cycles of Because of other limitations, the stresses in reactor components are operation. usually so low that this mechanism of failure may be neglected. Plastic Deformation. This type of failure occurs when soft materials arc 2.23 overloaded with resultant undesirable changes of geometry. If the deformed material If the is the supporting medium of a harder, more brittle surface, spalling will result. material is homogeneous, deformation will eventually result in loss of function. 2.3

Wear Testing


Several factors are influential in determining the extent of wear and galling. most important are:

The

1. Environment, i.e., the medium in which the moving elements operate, its tem perature, and its pressure 2. Geometry, including surface finish 3. Load 4. Relative surface velocity

5.

Materials

6. Service conditions, e.g., whether continuous or intermittent

operation is involved

Piston-cylinder Tests. 2.31 R. C. Westphal and J. Clatter" have reported on the wear resistance of a number of materials combinations as determined by split The experimental data were obtained piston-cylinder tests involving linear motion. from tests performed under the following conditions: Water temperature, °F Reaiativity of water (min), ohm-cm Oxygen content, ml/liter STP

500 500,000

5-16

The last condition was maintained by pressurizing the system at 500°F (with argon

gas containing 5 per cent oxygen) to 100 psi above the saturated steam pressure of 665 psi. The stress was low (3 lb radial load applied to a split piston 0.747-in. diameter) in order to minimize seizure. Test duration was approximately 500,000

cycles. Table 2 is a partial summary of the results of these tests, arranged in ascending These particular materials combinations were selected to cover order of wear factor. the full range of wear-factor values observed and to illustrate the influence of several variables on the mechanisms of wear. Materials combinations were arbitrarily classified into five groups by the authors: values up to 25, excellent; from 25 to 50, good; from 50 to 100, fair; from 100 to 200, poor; and those over 200 unusable. It is seen that nitrided surfaces have the lowest wear factor of any observed. Unfortunately, materials in this group may have insufficient corrosion resistance in liigh-temperature, high-pressure water. Second in wear resistance is hard industrial chromium plate. It was found that it could be used successfully with a wide variety of mating materials. Corrosion resistance is assured by plating on a corrosion-resistant base of stainless steel, since If unit any porosity in the plate might otherwise cause eventual lifting of the plate. loads are significant, the use of corrosion-resistant base material of more strength and hardness than the usual AISI 300 types is indicated in order to provide a better resistance to deformation and subsequent cracking and spalling of the chromium stainless steels such as 17-4 PH have been used plate. Precipitation-hardening A danger in the use of chrome plate must be recognized, namely, the successfully.

Sec. 10-3]


10-77

possibility of flaking or scuffing. This can be minimized by careful control of the plating process. It will also be noted that excellent wear resistance was found with martensitic straight chromium stainless steels, particularly AISI 440 C, as compared with nickelThese straight chromium stainless steels are not chromium stainless-steel grades. Table 2.

Partial Summary of Piston-cylinder (First item ia the piston)


Materials combinations^

Wear Tests*

Wear factor mg wt loss /{lb. load) (million eyelet), oxygenated water

1. Nitrided type 347 ss-nitrided 347 ss 2. Nitrided chrome-plate-nitrided chromium 3. As-plated chrome-nitrided titanium 4. Metainic LT-l-nitrided Armco 17-4 PH 5. Honed chrome plate— Stellite 3 6. Type 440-C ss-type 440-C ss 7. Type 410 ss-Stellite 3 8. Honed chrome plate-Kentanium K- 151 9. Honed chrome plate-Armco 17-4 PH 10. Honed chrome platc-nitrided Armco 17-4 PH 11. Stellite l-type 440 C as 12. Stellite 12-honed chrome plate 13. Stellite 6-Stellite 6 14. Nitrided Armco 17-4 PH-nitrided titanium 15. U.S. Steel type W (SA)-Stellitc 3 16. Wall Colmonoy 6-Stellite 6 17. As-plated chrome-as-plated chrome 18. Armco 17-4 PH-honed chrome plate 19. Haynes 21 (SA)-Stcllit* 6 20. Carboloy 608 (chrome carbide)-honed chrome plate 21. Haynes 21 (SA)-Hayncs 21 (SA) 22. Stellite 3-honed chrome plate 23. Stellite 3-Wall Colmonoy 6 24. Armco 17-4 PH-Stellite 6 25. Honed chrome plate-Stellite 6 26. Stellite 3-SteUite 3 27. Armco 17-4 PH-Carboloy 608 (chrome carbide) 28. Metamic LT-I-Metamic LT-I 29. Stellite 3- Haynes 25 (CW) 30. KR-Monel-honed chrome plate 31. Stellite l-Stellite 6 32. Haynes 21 (PH)-Haynes 21 (SA) 33. Haynes 21 (SA) — honed chrome plate 34. Honed chrome plate-Haynes 25 (CW) 35. Stellite 3-Armco 17-4 PH 36. Haynes 21 (PH)-Haynes 21 (PH) 37. U.S. Steel type W-Stellite 3 38. Honed chrome plate-honed chrome plate 39. Honed chrome plate-Hastelloy D 40. Type 304 ss-as-plated chrome. . . 41. Armco 17-4 PH-Stcllite 3 42. KR-Monel-Stellite 3 43. Stellite 3-Hastelloy D 44. Armco 17-4 PH-Arroeo 17-4 PH 45. Everdur 1012-Haynes 21 (SA) 46. Stellite 3-Armco 17-4 (SA) 47. Everdur 1012-honed chrome plate 48. Type 304 ss-type 304 ss • R. C. Westphal and J. Clatter, WAPD-T-64. SA. solution annealed; PH, precipitation hardened; CW, cold-worked.

0.0 0.0 4.7 7.0 8.3 15

I*

16 20 22 23 25 27 28 31 32 34 35 39 41 51 55 61 62 65 71 72 76 77 81 88 98 102 105 115 120 130 135 140 150 150 170 300 460 660 700 830 3.200

t

of corrosion but are often usable in water. Metamic LT-1, which is a cermet consisting of chromium and alumina, exhibited excellent wear and corrosion resistance when run in combination with an abrasive resistant mating material (Table 2, No. 4). Against mating materials with slightly less abrasion resistance, the wear factor was found to be high. The cobalt-base alloys, as represented by the various Stellites, show excellent to very good resistance to wear and galling, both against themselves and in combination

suitable for

hydrogenated

use in oxygenated water because

REACTOR MATERIALS

10-78

[Sec. 10

This, coupled with their more than adequate corrosion with other types of material. resistance, makes these alloys among the most useful of those available. Hardness has a very pronounced yet inconsistent effect on Effect of Hardness. For example, nitrided surfaces, chrome plate, and Stellites have decreasing wear. The Effect of Corrosion on Wear in Piston-cylinder Tests*

Table 3.

Wear factor, mg wt loss/fib load) (million cycles) Materials combinational Oxygenated

water

8.3 Haynes 21 (SA)-Hayncs 21 (SA) Stellite 3-Wall Colmonoy 6. . . Stcllite 3-Hayncs 25 (CW) Stellite 3-Armco 17-4 PH U.S. Steel type W-Stellite 3


Armco 17-4 PH-Stellite 3

KR-Monel-Stcllite

3

Stellite 3-Haatelloy D Everdur 1012-Haynes 21 (SA)

• R. C. Westphal and J. Clatter, WAPD T-64. t SA, solution annealed; PH, precipitation hardened;

20 35 51 61 65 71 77 81 102 115 130 140 150 170 300 660 830

Hydrogenated water

0.3 13

II

19 60 24 34 47 1. 1 28 33

II

45 25 31 43 18 2. 2

CW, cold-worked.

However, many hardness values that correspond to decreasing wear resistance. contradicting examples can be cited, for instance, la. Armco type 17-4 PH vs. Stellite No. 3 (Re 54 to 60) had a wear factor almost three times greater than b. b. Armco type 17-4 PH vs. Stellite No. 6 (Re 40 to 46) (Table 2, Nos. 41 and 24). 2a. U.S. Steel type W vs. Stellite No. 3 with the type W material at maximum hardness (Re 42 to 45), had a wear factor of (130). b. If the hardness was reduced to Re 39 by overaging or was in the solution-annealed condition (Re 24), the wear factor was substantially reduced (31). 3a. With Armco type 17-4 PH stainless steel the effect of hardness was directlyThe wear factor for fully hardened material (Re 45) in combination with opposite. Stellite No. 3 was 115. 6. When the material was solution-annealed (Re 32) the wear factor rose to 700. 4a. Stellite No. 21, when run in combination with itself, resulted in the least wear when both elements were solution annealed. b. More wear when one element was hardened. c. Most wear when both elements were hardened. Effect of Surface Finish. Surface finish is another variable that affects the wear resistance and galling tendencies of different materials combinations. When the mating materials differ greatly in hardness, it is important that the harder of the two have a good finish; otherwise wear is accelerated by continuous cutting action on the softer material by the rough surface followed by abrasive wear. Under certain circumstances wear can be reduced by controlling the degree of roughness of at least one of the mating surfaces. As an example, as-plated chromium vs. as-plated chromium, with a surface roughness of 30 to 40 /iin., has a wear factor of 34, whereas the same materials combination with honed surfaces and 1 .0-pin. surface rough Light sandblasting of large bolt threads made of ness has a wear factor of 135. 1&-8 stainless steel was successful in preventing seizure, and sandblasted journals or

Sec. 10-3]


10-79

sleeve bearings, made of materials that ordinarily seized would operate, although the wear rates were still very high. The porous structure developed by powder metal lurgy techniques also presents an interrupted surface and may result in better antigalling characteristics. The effect of corrosion on wear is illustrated by comparing the Effect of Corrosion. wear factors listed in the two columns of Table 3. In the second column are the results of a number of tests, standard in every respect, except that Westphal and Glatter substituted a partial pressure of hydrogen for the oxygen-argon mixture. In almost every case a substantial reduction in wear factor is noted. The greatest change took place with those materials which are most susceptible to oxidation. The combination of Everdur No. 1012 (silicon bronze) and chrome plate has an extremely high wear factor (830) in oxygenated water and a very low wear factor (2.2) in hydroThere was an 88 per cent reduction in wear for the combination genated water. of Kll-monel and chrome plate. In contrast, the Stellite combinations, because they are more oxidation resistant, did not show improvements of this magnitude. The reductions ranged from a few up to a maximum of 50 per cent. Table 4.

Coefficients of Friction in Water at Room Temperature*


Materials combination

Armco type Armco type Armco type Initial Final Armco type

Initial

17-4 PH-Armco type 17-4 PH 17-4 PH-Stellite I 17-4 PH-aa-plated chromium:

17-4 PH-ground

chrome

plate:

Final Stellite 1-aa-plated chromium U.S. Steel type W (SA)-Stellite I U.S. Steel type W (SA)-Hayncs 21 U.S. Steel type W (SA)-as-plated chrome U.S. Steel type W (SA)-ground chrome plate U.S. Steel type W (SA)-Armco 17-4 PH Stellite 6-U.S. Steel type W (SA) Stellite 6-U.S. Steel type W (SA) sandblasted Powder materials applied to sandblasted U.S. Steel type W (SA) vs. Stellite 6 Lead Copper Teflon Silver Neolube

Tin

PbO (litharge) PbiCh (red lead) 2

PbCOiPb(OH)i

Static

Kinetic

0.45 0.25

0.45

0.24 0.50

0 17 0 50

0. 16

0. 14

0.62 0.29 0.35

0.62 0.22

0 21

0. 17 0. 16

0.27 0 20

0.79 0.32 0.38 0.14 0.20 0. 19

0.25 0.27 0.48 0.31 0.31

(white lead)

PbOi PbCrO* (chrome yellow) SnOi • R. C. Westphal and

J.

0.32 0 35 0.41

0.46

0 23

0.13 0.79 0 32

0.37 0. 13 0. 16 0. 18 0.21 0 23

0.44 0.22 0 25

0.26 0.27 0.31

0.39

Clatter, WAPD-T-64.

Some very interesting observations on friction were made by Effect of Friction. These data (Table 4) include measurements of static and Westphal and Glatter. In addition, the effective kinetic friction of several materials combinations in water. These materials were ness of various materials as solid-film lubricants is shown. applied in powdered form to a lightly sandblasted surface, since it was found that this condition promoted greater adherence. The introduction of lead or lead oxides onto lightly sandblasted surfaces of wear Neolube, a com test specimens greatly reduced the wear rate as well as the friction. mercially available product consisting of an alcohol suspension of colloidal graphite, was also effective in this respect, although not to the same degree as were the lead

10-80

MATERIALS

REACTOR

[Sec.

10

The most effective solid film lubricant of all was metallic lead operated compounds. in oxygen-free or hydrogenated water. Wear and friction are closely related phenomena. Any reduction in friction of a given material combination that is accomplished by any of the various methods mentioned is always accompanied by a corresponding reduction in wear rate according to the authors. Nevertheless, it does not necessarily follow that low wear rates will be found for all materials combinations having low coefficients of friction, nor are high friction coefficients always an indication of high wear. 2.32 Fretting Tests. Unpublished work by VV. K. Anderson and associates st Argonne National Laboratory yields data on fretting and wear of several materials Basically, the test consisted of oscillating balls, 0.135-in. diameter, combinations. against a face plate having a 2-in. radius of curvature at a speed of 4 cps over a distance of in. Contact between the balls and the face plate was maintained, as was a stress of 100,000 psi (Hertz formula), by spring loading. The tests were conducted with oil lubrication at 250°F. A summary of the results of these tests is presented in Table 5. Table

6.

Fretting Tests*

Material


Uall

410 410 nitridad 420 440C 440C nitrided 17-4 PH 17-7

PH

322 322 nitrided Stcllite 3

Stellite 6

J. Frank, W. Neisz, t NA, not acceptable;

*

Plate

410 420 410 410 420 440C 440C 17-4 PH 17-7 PH 17-7 PH Stellite 3 Stellite 6 17-4 PH 322 Stellite 3 17-7 PH 322 Stellite 6 17-7 PH 322 Stellite 3 17-7 PH

Resultt A D A

NA D

NA D

NA NA

D A NA

NA

A D D A

A D A A

NA

and D. N. Dunning, unpublished work, D, doubtful; A, acceptable,

2.33 Component Wear Tests. Gear and Bearing Wear Tests. A series of tests on gears and associated bearings were run in oxygenated water at 450 to 500°F. Two 3-in.-PD 1-in. face gears were driven on a common shaft at 100 rpm by an exter nal motor. Mating gears, on a parallel shaft, were spring loaded. The reaction stresses also produced calculable bearing loads. Results of the tests are summarized in Table 6.

Rack and Pinion Wear Tests. A test of oscillating bearings, guide bearings, and rack and pinion gears was established at Argonne National Laboratory under the direction of M. C. Shaw. A vertical shaft was positioned by two cylindrical guide bearings carrying a rack. The rack meshed with a 3-in.-PD 1-in. face pinion sup ported by two bearings on a horizontal shaft. The pinion was oscillated through 170° by an external crank and sector. Varying the frequency of cycling between 1 and 5 cps permitted actual load variation. Tests were conducted at 450 to 500°F in A summary of data is given in Table 7. oxygenated water.

Sec. 10-3] In


a private communication,

W.

with

are acceptable corrosion-wise

10-81

K. Anderson has compared a list of materials that the results of simulated component wear tests and

His findings selected combinations that were satisfactory from both standpoints. are summarized in Table 8. The general superiority of the cobalt-base alloys, as represented by the Stellites, is evident. Table

Gear and Bearing Wear Tests*

6.

Material drive gear

AISI AISI AISI

Cast iron Cast iron Cast iron


AISI AISI AISI AISI AISI AISI

316 316 316

26 68 96 26 48 96 34 26 48 34 96 20 10 30 20 20 20 92 20 20 10 20 20 20 20 92 92 20 105 96 56 20 20 20 56 20 30 20 20 20 20 20

AISI AISI

316 440C AISI 440C U.S. Steel type W AISI 440C Stellite 3 Stellite 3 Stellite 3 U.S. Steel type W, Cr plated U.S. Steel type W, nitrided AISI 420

Haatelloy C

AISI

440C

AISI AISI

440C 304 nitrided

Stellite 3 U.S. Steel type W U.S. Steel type W

Tantung Hastelloy Haatelloy Haatelloy

Tantung G Hastelloy C Stellite 3 Hastelloy C Cr plated

G C C C Cr plated 440 C

AISI

316

U.S. Steel type W Ampco 18 Stellite 3 AISI 304 nitrided S Monel

440C 440C 440C 440C 440C

Arapco IB Ampco 18 Anipco 16 S Monel S Monel S Monel S Monel K Monel

Resultsf

lb/in.

Cast Iron Cast Iron Cast Iron

316 316 316 316 316 316 U.S. Steel type W U.S. Steel type W U.S. Steel type W U.S. Steel type W U.S. Steel type W U.S. Steel type W, Cr plated U.S. Steel type W, nitrided AISI 420 Stellite 3 Stellite 3 Stellite 3 Stellite 3 Stellite 3 AISI 304 nitrided AISI 304 nitrided

AISI AISI AISI AISI AISI AISI

Tooth load.

Driven gear

440C S Monel

Ampco 18 440 C S Monel K Monel Ampco 18 S Monel

D D

NA

D D

NA NA

D D

NA NA D D D

NA

A

VG NA

A A D D NA) Wear O.K. NA J case chipped A

NA NA NA

A A G

VG A A D

NA NA A

NA NA NA

NA

Bearings Material Bearing

Balls

Races

Stellite 3 Stellite 3 Stellite 3

Stellite 3 Type W Type W

17-7

17-4

PH

PH

load,

11.

Results

Retainers

AISI AISI AISI AISI

304 304 304 304

20 30 20 16

VG NA D NA

* Data from tests performed at Argonne National Laboratory, under the direction of M. C. Shaw. D, doubtful; NA, not acceptable; VG, very good; A, acceptable; G, good.

t

REACTOR MATERIALS

10-82

Table 7.

[Sec. 10

Rack and Pinion Wear Tests* Gear and Rack Tests

Material pinion gear Stellite 3 Stcllite 3 Stellite 3 U.S. Steel W nitrided U.S. Steel W nitrided AISI 410 nitrided

AISI

440-C

Rack gear Stellite 3 Stellite 3 Stellite 3 U.S. Steel W nitrided U.S. Steel W nitrided AISI 410 nitrided

AISI

Tooth load, lb/in. 260 72 29 72 29 260 260

440-C

Ratincf NA G G

VG A

VG G

Pinion Bearings (Oscillating Type) Bearing type

Bearing load

Needle Needle Needle Needle Needle Needle Needle Spherical ball Sleeve

130 130 130 37 14.5 130 14.5 14.5 130

Bearing material

AISI

440-C AISI 440-C Cr plated SAE 52100 SAE 52100 SAE 52100 Armco 17-4 PH Armco 17-4 PH AISI 410 Nitrided Graphite 40

Rating D

NA NA A D

NA NA

NA NA


Guide BearingB Sliding Type Sleeve Sleeve Sleeve Sleeve

37.8

Linear ball Linear ball

t

Sleeve Sleeve

11.5 3.1 1.2

t 37.8 37.8

Graphite Graphite Graphite Graphite

C-18 C-18 *I4 if 14

440C 410 Nitrided Ampco 18

Lead Bronze

NA NA

A

A NA NA NA NA

* Data from tests performed at Argonne National Laboratory under the direction of M. C. Shaw, t NA, not acceptable; D. doubtful; A, acceptable; VG, very good. X Bearing load not reported.

Table 8.

Materials Combinations Suitable for Use in High -temperature Water Reactors* 17-4

* W.

K.

Anderson,

PH

vs. 17-4

as Bearings

PH

U.S. Steel W vs. Stellite 6 U.S. Steel W vs. Stellite 3 U.S. Steel W vs. Haynes 25 Stellite 3 vs. Haynes 25 Stellite 3 vs. U.S. Steel W Stellite 3 vs. 17-7 PH Stellite 3 vs. Stellite 3 Stellite 3 vs. Stellite 6 Stellite 6 vs. U.S. Steel W Stellite 6 vs. 17-4 PH Stellite 6 vs. Stellite 6 private communication.

REFERENCES 1.

Datzko, S. 1954.

C:

"Corrosion of Metals in High Temperature Water," ANL-5364, Oct. 4,

2. Bailey, John M., and Douglas Godfrey: Coefficient of Friction and Damage to Contact Area during Early Stages of Fretting, part II, Steel, Iron, Iron Oxide, and Glass Combi nations, NACA Tech. Note 3144, April, 1954. 3. Westphal, R. C, and J. Glatter: WAPD-T-64, Dec. 4, 1953. 4. De Paul, D. J. (ed.): "Corrosion and Wear Handbook for Water Cooled Reactors," Supt. Docs., U.S. Government Printing Office, Washington, D.C., 1958.

RADIATION DAMAGE TO SOLIDS

10-4

BY

The types of radiation that can affect the macroscopic properties of solids include charged particles, neutrons, and y rays. The nature, properties, and attenuation The characteristics for each of these radiation types are described in Sees. 4 and 7-3. effect produced by a given type of radiation depends markedly on the nature of the Metals are affected by corpuscular radiations, e.g., substance being irradiated. protons, neutrons, electrons, having energy and momentum high enough to produce In nonmetallic systems, the ionization and excitation pro atomic displacements. duced in the solid are of primary importance, so that all types of radiation, including The ambient temperature at which the expo 7 rays, can produce property changes. sure occurs has a marked effect on the magnitude of the changes produced. The major sources of charged corpuscular radiation are electronuclear machines, e.g., the cyclotron for protons and deuterons, the electrostatic generator for electrons. The major sources of neutrons are nuclear reactors. The most important sources of y radiation are reactors, materials activated in reactors, or fission products contained in or separated from spent reactor fuel elements. 1

NUCLEAR REACTORS AS SOURCES OF RADIATION

Nearly all reactors that have been constructed to date have been used to some extent for radiation-damage investigations, regardless of their principal purpose. However, the reactors that have been designed primarily for research or test purposes provide the most useful facilities for such investigations and include the Materials Testing Reactor, the Oak Ridge research reactors, the Brookhaven reactor, the Canadian NRX reactor, and the CP-5 reactor at the Argonne National Laboratory. Several additional reactors of advanced design are planned or under construction (see Of these the Engineering Test Reactor (ETR) was placed in operation Sec. 13-6). late in 1957 at somewhat below the design power. 1.1

The Materials Testing Reactor (MTR)

a

a

3

L

3

a

A

L

ft

9

ft

6

is,

The MTR is a water-moderated and cooled thermal reactor; it uses highly enriched uranium as its fuel. The reactor, when operating at the original design power level of 30 Mw, has an average thermal-neutron flux of 2 X 1014 neutrons/(cm2)(sec). The active core fills a region 40 by 70 by 60 cm and is surrounded by a beryllium The reflector that fills the remainder of a cylinder 3 ft high and 5 ft in diameter. beryllium reflector in turn, surrounded by graphite reflecting region approxi mately thick and high. Experimental test facilities are provided within the active core and in both reflecting regions.* The test positions within the core tank are provided by the insertion of an experi mental assembly in one of the lattice positions or by replacing beryllium reflector element in an Tests in position. positions can be carried out in assemblies that simulate fuel element, approximately by in. in square cross sections, or in alumi *


John P. Howe and Sidney Siegel

John R. Huffman: The Materials Testing Reactor. Nucleonici, 10-83

12: 21-26 (April,

1954).

10-84

REACTOR

MATERIALS

[Sec.

10

L pieces having an axial hole 1.375-in. ID. No leads can be brought out from Tests in .4 positions can be carried out in specially constructed substitute pieces. assemblies about 3 by 3 in. or within cored beryllium pieces having axial holes up to 1.375-in. ID. Electrical leads and fluid lines can be brought out from tests in .1 Table 1 gives pertinent data relating to these L and A test positions. positions. num

L

Table la. Data on Core Test Positions in MTR at 30-Mw Power All tent positions are vertical and have a usable length of about 20 in. The ambient temperature is about I00°F. The 7-ray flux in all L positions produces heating at a rate of about 14 watts/g. L-43 3.0

L-45 3.5

Z.-47

6

6

6

L-54 2.8

L-SS 2.8

L-57 2.5

1.3

1.0

1.0

£-41 1.8 X 10" 6 X 10"

Fast flux > 1 Mev

Facility

L-51 1.8 X 10" 1.0 X 10"

Fast flux > 1 Mev

L-52 2.0 1.3

,-53 2 5 1 3

3.0

Lit 2.3 1.0

i.a

t

I.I

0 8


Table lb. Data on Be Reflector Test Positions in MTR at 30-Mw Power All test positions are vertical and have a useful length of about 2 ft. The ambient temperature is about 100°F, but special cooling can be provided to A pieces. Values are given for one quadrant of the reflector; similar values exist in the other three quadrants.

Fast flux > 1 Mev Gamma-ray heating rate, watts/x Facility Fast flux > 1 Mev Gamma-ray heating rate, watts/g

,1-4

4-6

1. 25 X I0'< 5 X 10" 2

1.4 3 1.5

A-38 1.9 X 10" 10 X 10" 2.5

A-7 0.75 1.5 1.0

A-19

A- 40

1.4 5 1.25

1.0

0.8 0.9

4-43 0.75 0.8 0.75

A-l

0.75 1.0

0.9

A-41

0.6 0.5 0.4

.4-5 0.6 0.8 0.75

Ail 0.5 0.3 0.5

A large number of test facilities penetrate the shield and the reflector region and provide blind or through thimbles in which irradiations or in-pile tests can be carried out. Relevant information on these facilities appears in Table 2. The physical location of these facilities and contours of the thermal-neutron flux at the mid-plane of the reactor are shown in Fig. 1. 1.2

The Low-intensity

Test Reactor

(LITR)

The LITR was constructed at the Oak Ridge National Laboratory as a hydraulir mock-up of the MTR and is in many respects identical in physical configuration with the MTR. It operates at 3 Mw, with an average thermal-neutron flux in the core of 2.2 X 10" neutrons/(cm,)(sec). The experimental facilities include six beam hole>. four low-flux short tubes, and lattice or beryllium reflector positions similar to those in the MTR. The beryllium reflector pieces in the LITR are 2 by 2 in. in cross section. Table 36 Table 3a gives data on the thermal-neutron flux in these various facilities. gives information on the neutron energy spectrum at the inner end of the HB-2 hole; this spectrum is typical for a position near the surface of the active lattice in both the MTR and the LITR.* 1.3

The ORNL Graphite-moderated

Reactor (X-10)

The X-10 reactor at the Oak Ridge National Laboratory is graphite moderated and air cooled and uses natural uranium fuel. It operates at a power of 3.5 Mw, with an * Data on the Laboratory.

LITR

was compiled

by C. D. Cagle

and

J.

A. Cox of the Oak Ridge

Nation*.

6

6

6

ID

4

1

41

ID and ID

ID

2>, ID

HR-3.4 VH-1. 2.3.4

VG- 1.3.5.6

VG-7. 10-14. 16-20

.

Cd ratio (indium) 22

X

6 X

4 2.5 X 10"

009 0.5-0.

0

pebble graphite

X 10"

0.3 1.5

Air-cooled Water

Air-cooled Air-cooled

100 too

100 100

•

-36

from top into pebble graphite Slant from top into

X

Slant

4 X 2.2 1.9

Air-cooled

100 • •

36

X X

2-3

I

10" 10"

6 X

10"

s

10" 10"10"

X

1.2

Air-cooled Air-cooled •

24 20

12

X

5

ID

X

1

100

.

HR-1.2

0.009

Cd ratios 2.4 10"

10"

X

8

36

10"

Air-cooled

100

(all VG's)

(all VG's)

Air-cooled

100

•

ID

72 by 72

T-2

Not meas ured

10"

X X 008

5

4

-36

2.3

10"

X

10"

Air-cooled

100

'100

Cooling water avail able in facility cubicle Cooling water avail able in facility cubicle

Cooling facilities

100

•

VG-2.4

10 by 10

6

VG-9

9 10

X

2

10 by 16

0.1

10"

X

0

10" 10"

Temp, •F

•

HG-9

254

X 10"

X 5

ID

2.0

10

gamma, watts /g

Max

Facilities

10"

X 10"

10"

2.

Inclined down to tank wall Horizontal to tank wall Horizontal to tank waU Vertical from top to HG-9 to Pb Horizontal window Vertical through graphite Horizontal through tank to tank Horizontal Vertical from base ment in Be

to lattice

Horizontal

in Be

ID

flux

Irradiation

Max fast, neutrons/ (cm*) (sec)

3

4.5 X 10"

Max thermal, neutrons/ (cm*) (sec)

Estimated

MTR

1

DB- 1.2.3. 4.5.6 HG-5.6

to core

Horizontal

Orientation and penetration

3

in Be

Usable length, in.

5

ID

Cross section, in.

Dimensions

Table


2. =

Water can be provided These are hydraulic rab bits (standard shuttle sizes. 1J.fi by by in.)

VG used for piping (not available) Pneumatic rabbit facility

No coffin available for removal of shielding plugs

Beam-hole plugs usually water cooled

Beam-hole plugs usually water cooled

Remarks

3,

3

•

1

1

12

4K

lOJi

T-2-V-I.2

HT-!

HI-2.J

X

X ~I2 ~I2

ID

HG-J.4

1 X

ID

1.

1 X 1

8

10"

0>>

10"

10"

10"

X

HO- 1.2

2

through graphite N-S horizontal through graphite N-S horizontal through graphite

X 2

ID

9 12

1

Vertical reactor top to T-2 Horizontal through tank E-W horizontal

10"

(indium) I0«-I0»

Cd ratio (indium) I0«-I0* Cd ratio

Cd ratio (indium) X 10<

Max fast, neutrons/ (cm") (sec)

Max

Air-cooled 100

Air-cooled Air-cooled

100 100 100

0.003 0.003 0.003

Water-cooled 100

Air-cooled

Air-cooled

= 100

100

Air-cooled

« 100

Cooling facilities

2.5

X 10-'

10'

0.0001-0.01

Temp. °F

(Continued)

gamma, watte /g

Facilities.

5

4K

12

in T-2

X 10"

2.

Horizontal

10«

10"

X 10""-

Max thermal, neutrons/ (cm') (sec)

2 X

N-S horizontal across T-2

graphite

Vertical from top into permanent

Orientation and penetration

X

4X4

36

Usable length, in.

55

4X4

ID

Cross section, in.

flux

Irradiation

Estimated

MTR

2

T-2-H-3.4.5. 6.7.8

VG-23-26. 29-41.43 45-55. 57-62 T-2-H-I.2

Facility designation

Dimensions

Table


2 X 5

X 1

1

X

4

Used for reactor ments

Remarks

instru


10-87

RKACTOR

MATERIALS

[Sec. 10

KEY TO CONTOURS OF THERMAL NEUTRON FLUX CD 5 4

a 8

a a

> IOl4/cm2MC •

3

25 2 0 5

a) 025

a

Q Q

0 1

0.05 001

@ 0005 13 .a1

a- 5xl0"4 Q' io-4

a

I0"J io-' 19= io-'

Q


CORE FLUX DIAGRAM

Fig. lb.

Fio le.

average thermal-neutron flux of 5 X 10" neutrons/(cml)(sec). The moderator stack is nearly a cube 24 ft on edge. There are 1,248 horizontal fuel channels through the graphite, of square cross section large enough to accommodate a cylinder 1^4-in.

diameter on an 8-in. lattice spacing. These channels are cooled by air flowing through them. There are normally about 820 of these channels loaded with fuel, so that an appre ciable number of the remainder is available for experimental purposes. Leads can be brought out from the experimental piece to equipment near the reactor face. Three fuel channels contain special hollow fuel slugs that provide a high local fast-neutron flux. These assemblies can accommodate a sample 11 in. long by % in. in diameter. The exterior of these elements is cooled by air, but the ambient temperature in the central region is about 100°C. The neutron energy spectrum in these hollow slugs is given in Table 4. The principal experimental facilities are provided by a group of about 50 horizontal holes, 4 in. square, that penetrate the graphite at right angles to the fuel channels and Thermal -neutron Flux in

Table 3a. Beam

tube

Flux, neutrons/

(cm*) (sec). . . .

HB-2 3 X

10"

Test Facilities at 3-Mw Power Level

HB-4

HB-5

V-l

7.8 X 10"

7.6 X 10"

4

Core position Flux, neutrons/(cin!)(sec) .

Table 36.

LITR

C-42 3. I X 10"

X

C-44 3.3 X

10»

0>-

5 X

energy, Mev

Thermal >0.6 >2.6 >4.0 >6.2 >8.1

LITR

V-4

2.4 X 10" ! 2.6 X 10"

10"

C-46 3.4 X 10"

Neutron Spectrum at Inner End of HB-2 in Neutron

I V-3

V-2

C-48 3.4 X 10"

C-38 3 X 10"

at 1-Mw Power Level

Flux, neutrons /(cm*) (sec) II X 10" 7.0 X 10" 2.0 X 10" 0.7 X 10" 0.25 X 10" 0.07 X 10"

radiation damage to solids

Sec. 10-4] Table 4.

10-89

Neutron Energy Spectrum in X-10 Hollow Slugs

Knergy, Mev

Thermal 3.8 X 10"

>. 1.6 X 10"

>2

6.5 X 10'°

>4 1.4 X I0'»

>6 3 .5 X 10'

lie between them. In all except a few special cases these holes have no special cooling provided and operate at about 135°C. Twenty-two of these holes extend completely through the reactor; the remainder go from the shield to a position near the reactor center. The maximum thermal flux in these holes depends on their location in the reactor and varies from about 4 X 10" to 1.1 X 10" noutrons/(cm!)(sec) from peripheral to central holes. One of the 4-in.-square holes is The cadmium ratio is about unity. provided with a water-cooled, enriched uranium converter cylinder, which can accom modate a sample up to 2 in. in diameter and 10 in. long. The fast flux in this facility is about 10", and the temperature 25°C* 1.4

The BNL Research Reactor


The research reactor at. the Brookhaven National Laboratory is graphite moderated and air cooled and operates at a power near 25 Mw. The maximum thermal flux The over-all dimensions of the reactor produced is 5 X 10" neutrons/(cm*)(sec).

One of the shield are approximately 60 (length) by 40 (width) by 40 ft (height). 40-ft-square ends is reserved for the operational loading and unloading of the fuel The other five faces of the box arc research areas where all the experi elements. mental facilities are located. A pattern of 30 horizontal beam holes extends through the graphite core and the opposite shielding walls in the direction perpendicular to the fuel channels. These holes, 4-in. squares in the graphite, are spaced 3 ft apart horizontally and 4 ft apart vertically. They are used for either neutron-beam or in-pile-irradiation types of experiments. The bottom face of the reactor is accessible by means of two tunnels built under The larger tunnel is 2 ft square in cross section and has neath the graphite structure. an exposure window 3 by 2 ft in area. The smaller tunnel is 1 ft square in cross section A small flatcar moves on rails in the bottom of and has a window 3 by 1 ft in area. each tunnel and can be loaded through openings in the floor of the building on both sides of the reactor. These tunnels were built to provide for the irradiation of largeLeads can be taken out of the tunnels from the size instruments or equipment. experiments to monitoring or recording meters in the reactor room. The face opposite the loading and unloading face is used primarily for the produc tion of radioisotopes and to provide service irradiations of special targets. Eleven special tubes, parallel to the fuel channels, terminate at this face; some of them are used to convey samples pneumatically into and out of the core. Others can be used manually to insert samples of special nature, with or without monitoring leads connected to them. The fuel channels are also accessible through this face. A wide variety of devices is built into the reactor for doing irradiations of special ized character at controlled ambient temperatures. The graphite and air tcmpcraTwo of the 4-in.-square beam holes t ures in the reactor range between 50 and 200°C. have been provided with water-cooled irradiation chambers to yield a controlled tem perature of 30°C. These are built so that samples can be inserted or removed while the reactor is in operation. A third such beam hole has a chamber cooled by liquid A fourth beam hole has a nitrogen to provide an ambient temperature of — 190°C. furnace installed to provide elevated temperatures up to 500°C. Detailed information on the flux level, the dimensions, and the temperature of some of these facilities is given in Table 5. The neutron energy spectrum in one of the 4-in. horizontal beam holes is shown in Fig. 2, where the total flux above any given • Data on the X-10 graphite

National Laboratory.

reactor

were compiled

by C. D. Cagle and J. A. Cox of the Oak Ridge

10-90

REACTOR Table

5.

Data on Facilities in

Reactor facility

Water-cooled holes

MATERIALS

Max aiie, in.

2X2X7


OD, 6 long 15 X 24 X 24 6 X 12 X 24 12 X 12 X 24 3Ji X J?« cross section 2H OD 2 OD, 24 long

IK

X

7

10

Reactor Ambient

Thermal flux

Containers are J< ID, 2H long Carriers are l>< ID, 4 long l« OD X 4

H

High-temperature facility

BNL

[SEC.

Choice,

I0««-4

X

temperature,

•c

10"

Up to 4 X 10" J X 10"" 1.5 X 10" 4 X 10" 2 X I0»

150-200 50-75 30

-190

10"

Up to 5 X 10" Up to 5 X 10" Up to 4 X 10" 1.2 X 10"

JO 175 175 ran 50-175 50-75 125-500

NEUTRON ENERGY ,EV

Fio.

2.

Neutron flux-cnergy

distribution hole (E-25 Brookhaven reactor).

energy is plotted against that energy. reactors generally.* 1.6

This

is probably

representative of graphite

The NRX Reactor at the Chalk River, Canada, Laboratories

The NRX reactor is D20-moderated and uses natural uranium fuel. The power level is near 40 Mw, and the maximum thermal flux is 6 X 10" neutrons/Ccm'Hsec). The moderator temperature is maintained below 120°F; the graphite reflector tem perature below 320°F. * Data on the BNL graphite Laboratory.

reactor

were compiled

by Marvin

Fox of the Brookhaven National

RADIATION DAMAGE

Sec. 10-4]

10-91

TO SOLIDS

Experimental facilities include a central thimble, lattice positions, tubes at the periphery of the DaO-moderated core, and horizontal beam holes that penetrate the shield and reflector. Relevant data for these experimental positions are given in Table 6.* Table

6.

Data on Experimental Facilities in NRX Reactor Thermal flux. neutrons/(em3) (see)

Facility

6 X 10" ~5 X 10" ~5 X 10" (fast flux) ~1 X 10" 2 X 10" to 2 X 10" <1 X 10"

5K" diam 2H" diani

5' long, y long,

>S" diam or less 5' long, diam 1.6" long, 0.62" diam Twelve 4" diam, three 12" diam

IJi"

Air or Air or Water Air or Air Air or

water water water water

The CP5 Research Reactor at Argonne National Laboratory

1.6


Cooling

Useful dimensiona

The CP5 reactor is D20-moderated and uses enriched uranium fuel. The reactor tank is about G% ft in diameter, and the reactor is a right cylinder about 4 ft in height and diameter. The tank is pierced by a large number of horizontal beam holes, and vertical thimbles extend downward through the tank cover. The reactor operates at 1-Mw power level and provides a maximum thermal flux within the core of 2 X 10" neutrons/(cm*)(sec). facilities available in the reactor are described in Table 7.

Experimental

Table 7.

Data on Facilities in the CPS Reactor Fluxes,

neutrons/fern') (sec) Internal size

Number

Facility

provided

Thermal

; 3 A 10 7 2 1 4 4

Sri. hII rabbit

1.7

The

Armour

research

2 1 1 2 2 1 2 1 1

X X X X X X X X X

Epithermal

10" 10" 10" 10" 10" 10" 10" 10" 10"

1 2 5 1 1 5 1 S 5

X X X X X X X X X

10" 10" 10" 10" 10" 10" 10" 10"

10"

of hole, in.

1. 4 and 1. 0 diam 0. 5 diam 1. 4 and 1. 0 diam 1. 4 and 1. 0 diam 4.75 diam 5. 75 diam 0. 5 diam 3. 75 diam 11.0 and 3. 75 diam

Armour Research Reactor

reactor was designed and built for the Armour

Research

The maximum thermal-neutron flux at the Foundation by Atomics International. center of the core is 10" neutrons/(cm')(sec) at an operating power of 50 kw. It is a homogeneous

solution-type

(water boiler) reactor and has a graphite reflector.

It

contains a 5-ft-square thermal column, seven 4-in. beam holes, two 3-in. beam holes, one 2-in. pneumatic tube, one lJ-^-in. pneumatic tube, one lj^-in. tube passing through the center of the core, and y exposure facilities in the subpile room that utilize the

The thermal column has nine 4-in. -square remov gaseous fission-product activity. able graphite stringers. Access to the thermal column is gained through two 6-in.diameter holes and two 6-in.-square holes and through the 5J.£-ft square thermal * Data on the NRX Chalk River, Ontario;

research reactor were compiled see also Ref. 1.

by W. B. Lewis,

Atomic Energy of Canada,

Ltd.,

10-92

REACTOR

column door opening.

MATERIALS

[Sec.

10

The y facilities in the subpile room consist of four 4-in. expo tubes, and a 6- by 18-in. rectangular exposure port.

sure tubes, two 8-in. exposure 2

EXPERIMENTS USING NUCLEAR REACTORS

Properties of materials and material systems are measured during and after irradi ation either to determine their behavior and utility in reactor application or to learn and describe the basic phenomena produced in material systems by irradiation. In-pile measurements are made (1) to observe those changes which may undergo transients or which obtain a steady state while under irradiation but which may change subsequently, for example, creep and thermal conductivity at elevated temperature, and (2) to determine and control the conditions t hat prevail around or in the specimens being studied. Poxtirradiation examination and measurements are made for those properties which do not change or recover at the temperatures of handling, storage, and measurement after irradiation. Reactor positions are selected on the basis of the characteristics given in the fore going article and the demands of the experiment, for example, (1) fast-neutron flux to produce damage by knock-ons or recoil displacements, (2) slow-neutron flux for damage to fissile materials and other substances having atoms of high-neutronabsorpt ion cross section, and (3) 7-radiation field for damage by ionization or excitation.


2.1

Experimental

Conditions

These may be provided, measured, and controlled as follows: 2.11 Dosage may be estimated from standard measurements of flux of each kind of radiation that are made for experimental positions for each reactor presently in use. Values are given for a standard power level, and if the operating data that give power vs. time are used, an integrated flux may be obtained for the experiment in question. More accurate values may be obtained by flux monitors suitably incorporated in an Procedures for flux monitoring are given later. Total energy input into experiment. selected materials placed in selected positions in the X-10 reactor at Oak Ridge has been measured calorimetrically When the material placed in the by Richardson.8 calorimeter is varied, the relative energy input due to fast neutrons and y radiations Such devices and measurements may be used as standards for has been determined. dose in limited applications. The exposure or dose given to materials containing fissile atoms is often determined by quantitative analysis of a standard fission product. Chemical dosimetry1 can also be used but is more adaptable to studies of covalent materials. 2.12 Thermometry may be accomplished most conveniently by means of thermo For proper insulation see Art. 2.31 on electrical resistance. Resistance couples. thermometers appear to be satisfactory, although no extensive calibration of them has been made. Gas thermometers made of quartz or metal and filled with helium have been used but are bulky. Excellent thermal contact of the thermometer with the Also the specimen is required because of radiation heating of the sensing element. generation and flow of hoat in the specimen must be taken into account in locating the thermometer and in ascertaining the representative temperature value for the experiment. Temperature control is carried out using standard laboratory procedures based on Heating to supplement radiation heating is done elec signals from thermocouples. Whenever trically, and cooling is usually accomplished with flowing gas or water. The response time and lati possible, the cooling system of the reactor itself is tised. tude of the control system and of the auxiliary heating and cooling must be designed to fit the expected variation in power level in the reactor used. 2.2

Specimens, Specimen Holders, and Containers

These must be designed not only to fit the reactor position to be used but also to facilitate handling and measuring upon removal from the reactor; for example, speci

Sec. 10-4]

RADIATION DAMAGE

TO SOLIDS

10-93

mens for tension tests are often machined prior to insertion in the reactor and used for electrical-resistivity measurements while in the reactor and subsequently, as well When as for the determination of stress-strain curves after removal from the reactor. such preparation is not possible, hot laboratory techniques (see Sec. 7-4) are used for preparation and examination after reactor exposure. 2.3

In-pile Measurements

In-pile measurements in reactors

are made as follows. Electrical resistance is usually determined by bridge methods, which compen sate for the resistance of the leads into the reactor. In all electrical measurements attention must be paid to the insulation. At best, organic insulation is not useful for exposures above about 1018 nvt; however, ceramic-type insulation is generally accepta ble. Asbestos properly supported serves well for thermocouples; Fiberglas also serves but is somewhat less reliable. The ceramic materials magnesia or alumina appear Braid appears suitable for support in to be entirely suitable if properly supported. most instances; however, for the most rugged construction, sheathed lines analogous to calrods in construction are desirable. Vacuumtight seals for electrodes present considerable trouble in that glass-to-metal seals can be used only in low flux. Ceramic bushings (e.g., porcelain) soldered to metal by the titanium or zirconium hydride method* are perhaps the most dependable. 2.32 Pressure measurements are accomplished using gas leads made of metal or of fused silica which may be brought out of the reactor to conventional piezometers. When it is necessary to confine gas to small volumes within the reactor, metal dia phragms with electrical contacts to permit null-point determinations have been used. An external pressure which balances that to be measured is read by standard means outside the reactor. 2.33 Measurements of displacement or strain may be made using differential If they are properly insulated and supported, no change in calibration transformers. of the transformer occurs. Resistance strain gauges change their calibration by 5 to 10 per cent at 10" nvt owing partially to changes in the mounting cement and partially to the changes in the resistivity of the wire used in the gauge. Pneumatic gauges depending on change in pressure drop across an orifice have also been used for measure ment of displacement.


2.31

2.4

Postirradiation

Operations and Measurements

Insertion, removal, and handling of apparatus must be done according to the means specified at each site and according to the reactor position. 2.41 Level of radioactivity to be expected may be estimated if the composition is known in some detail by using tabulated cross sections and decay constants together with neutron flux, exposure time, and cooling time. Since the composition of many commercial materials is not known in detail, it is more convenient to use results of actual exposures such as those given in Table 8. Handling of irradiated materials is always subject to monitoring by a health-physics organization. 2.42 Shielding during operation and upon removal of the apparatus from the reactor is also specified by local practice (see instructions for reactor to be used). The radioactivity to be expected upon removal is estimated beforehand, and precautions are taken to see that shielding during removal, handling, and transport is adequate. Handling, Provisions for shielding at various activity levels are described elsewhere. examination, and measurement of properties after removal from the reactor are Items specific to radiationaccomplished by means of remotely operated equipment. damage studies are as follows (more general techniques are given in Sec. 7-4). 2.43 Metallography is done by carrying out the standard steps remotely in a hot cell. Sectioning is usually accomplished with an abrasive saw, shear, or machine tool, Normally these are standard depending on the shape, size, and nature of the material. tools whose controls are extended through the cell wall, motorized, or operated with Mounting may be performed in a standard mount the general-purpose manipulator.

8. The

Gamma Radioactivity

1.

= saturated value of component activity given in table ™ irradiation time 1/14 = half-life for component activity Plot each component activity on semilog paper by drawing

.

a

j

A

U

S555 2222 X XX 078

XXX

X

555 222

t•

A 10' 10«

232

1

10> 10*

X3 81 6.

>

222X 1a4 S 17

XX

XX X 2713

555 5 2222 55 22

1953. Radiation C. D. Bopp, "Gamma Induced in Engineering ORNL-1371. Materials," May For thin specimens, hence no self-absorption. 1.25-Mev average energy. with Co"1, two photons per disintegration, Ion chamber calibrated Hoiiren and analysis not known, probably commercial uloctriuut copper. AKC aourae.

100 days 70 days

5.7 5. 5.

X 10' 10* 10" I0»

4 3 2,

36 XX 1324

X

X

15 hr 15 hr

days days days days

X

None None

30 120 100 60

10» 10' 10»

.7 3.3 2.4 years 20 days 5. years

5.

1

10' 10« I0» 10'

.

I0> I0»

3

10' 10' 10'

0 X8

33

years 5.3 years

years years

3

1.2 8.3

X 4

hr hr

8 10« 10' X 10* I0»

8.

10' I0>

10* I0>

A.

10"

Flux

5

1.360 15.400

7

X

X

Cof Zr» Graphite:

XX 45 days 45 days 45 days

5.

X

Hi

(g), Neutron

years 115 days

(sec)

5.

3.1

10' 10' 10' 10'

A.

Photons?/

3

5.

hr hr hr hr

10' 10' 10'

25 days 25 days 25 days

10" 10» 10"

X 9

X

years years 60 days 60 days

5.

CO90

3

15 15 13 20

10' 10' X 10' X 10'

days days days days

45 45 25 25

10' I0« I0» 10»

XX 575 3312

X 9 1.5 3.5

Equivalent

XXX

years

years

3 5

hr hr hr hr

A.

AJJft

A,(fi).

10"

10*

X

A.

permit the estima

X

15.380 4.510 4.370 4.700

Activities

33

hr hr hr

hr hr hr hr

A.

Component

5545

X XX 739 X

4.380 4.500 4.280

4.870 4,700 1.850 1.850

Irradiation time, hr

of the

T

reactor

XX 51 1.

Steel: 304 347 430 Ail 28 52S

Armco iron 1015 steel

2. 3. 4. Values

ji

Saturation

0.693(,\

Reactor*

graphite- moderated

in

slope — 0.693/1,-^) and intercept plot the composite curve.

line with

as selected points on the time axis and Add the components (sum over Read off values of the total activity at the desired cooling time.

straight

'-exp(-^)]

-

(

Material

where

t

[/

Exposed

listed in the 01i.NL

Materials

material

of Commercial

The following tabulated component activities determined by exposing the commercial of the material. The procedure for estimation is: tion of the radioactivity Compute the value of each component activity for the material from the relation

Table


a

8

7

3. 5

XX

•

i }1 |


Sec. 10-4]

RADIATION DAMAGE

TO SOLIDS

10-95

ing press appropriately serviced and controlled. Grinding may be done in two general ways: (1) changeable, graded papers on a rotating wheel against which the mounted specimen is moved by a mechanical arm or (2) a lapping machine on which the speci mens, properly weighted, are ground. Polishing may be carried out either electrolytically or mechanically, depending upon the material and desired data. Electropolishing is readily accomplished in a commercial unit or a homemade device, but Mechanical polishing each must provide for changing reagents and current control. is technically difficult but has been successfully done on standard polishing wheels. The wheels are modified to allow changing of cloths remotely and accurate alignment with a mechanically moved specimen holder. Cleaning in a solvent bath agitated The equipment is commercially available and ultrasonically has proved successful. The specimen is held with the general-purpose manipulators needs little modification. or by a fixed manipulator. Etching with standard techniques is easy and requires only provisions for handling, introducing, and disposing of reagents. Cathodic Photo etching is much more difficult, and no commercial equipment is available. micrography is accomplished using a standard metallograph either mounted inside the cell with all controls operated remotely and the image brought outside optically or mounted outside with shielded stage, some controls operated remotely, and standard Both methods involve expensive instrument modification, and the choice is optics. usually dictated by cell-wall thickness and the activities involved. 2.44 Microhardness can be measured through the use of a commercially available instrument* that is adapted for remote operation by the factory. Only the optics need be modified, as all other adjustments are electrically actuated. The instrument may be inside or outside the cell, as in the case of the metallograph. 2.46 Testing. Materials testing procedures employ the standard test instruments that are easily adaptable to remote operation by the use of elongated controls through the cell wall, motorizing, or general-purpose manipulators. Some instruments, such as tensile machines, may be purchased directly as remotely operable units. Wherever possible, the test specimens should be machined prior to irradiation in order to avoid the expensive and slow process of remote machining. 2.46 Machining may be done with standard machine tools whose controls are made remote in the usual manner. For some cases these tools must be modified to suit special requirements through addition of air chucks and other screw-machine actuating devices, turrets, hydraulic or air table traveling devices, etc. Lathes, milling machines, drill presses, hydraulic hack saws, grinders, shears, etc., have all been successfully modified and remotely controlled at a relatively low cost. 2.6

Decontamination

Decontamination

of equipment, specimens, and hot cells often follows the comple

tion of a study. It is important to confine contamination to as small an area as possible and to cover all exposed surfaces of the equipment with a disposable material. Chips from machine tools, grindings, dissolved material, aerosols, lubricating fluids, etc., should be caught in disposable or easily cleanable containers. These procedures lessen the hazard and difficulty of decontamination and permit reuse of equipment that does not require drastic decontamination. Disposal of radioactivated materials must be planned in advance. Procedures are described in Art. 10 of Sec. 11. 2.6

Reactor Safety

Coordination with reactor operations and safety is an extremely important aspect of the design of every in-pile experiment. If it is possible for the temperature of any given experiment to rise dangerously should the supply of cooling fluid fail, and if it is possible for this failure to occur, it is customary to interlock the temperature-control apparatus for the experiment with the control for the reactor so that the reactor is * Wilson Mechanical Instrument Division of American Chain and Cable Company, Incorporated, Park Avenue, New York 17, N.Y.

230

10-96

REACTOR

MATERIALS

[Sec.

10

scrammed at a given temperature level. Some experiments require that the schedule of operation be controlled by, for example, the temperature rise in the experiment in question. 3

EXPERIMENTS USING CYCLOTRONS

Radiation-damage investigations have been carried out using charged particle protons, deuterons, or a particles from cyclotrons.4 This method is particu larly useful for fundamental studies of radiation effects on solids and for exploratory investigations preceding more elaborate reactor irradiations. The merits of the cyclotron technique are principally: beams,

Accurate knowledge of integrated exposure in samples 3. Well-defined volume in which radiation is present, so that auxiliary apparatus not activated 4. High radiation intensities available in large cyclotrons 1.

2. Low induced activity

is


The principal disadvantages of this type of experiment are: 1. Small cross-sectional area of cyclotron beam, permitting use of only very small samples 2. Restricted range of charged particles in solid materials, i.e., very small irradiated volume and high density of heat production 3. Nonuniform damage because of diminishing energy of impinging particle as it penetrates the sample

3.1

Dosage

Two principal means for measuring

the integrated beam to which the sample has during an experiment are commonly used, the beam-current integrator and the activated foil counting. The integrator circuit permits t he beam current to be measured during the exposure. In the circuit generally used, the beam current charges a capacitor, the resulting voltage on which is amplified by a highly stable d-c amplifier and used to trigger a gas been exposed

SEAM CURRENT

RELAY R, DISCHARGES INTEGRATING CAPACITATORS

THYRATRON VOLTAGE DETECTOR 8 RELAY CONTROL

Fio.

3. Beam-current

integrator.

Sec. 10-4]

RADIATION DAMAGE

10-97

TO SOLIDS

discharge tube. The counting rate of this tube is then a measure of the current; the A circuit diagram total number of discharges a measure of the integrated current. of such an integrator is shown in Fig. 3. The integrated beam intensity may also be measured, subsequent to the completion of an experiment, by activating a suitable thin foil placed in front of the sample. The activity measured is due to Copper foils 0.001 in. thick are conveniently used. 0 decay of Zn", which is produced in copper by protons, deuterons, or a particles. The total sample exposure may be determined by counting t he whole monitoring foil ; el!mm in addition, sectioning the foil into strips per , VERTICAL FIDUCIAL MARK mits the determination of the beam profile -TV across the sample. Typical beam distribu VERTICAL STRIP are shown in tions Fig. 4a and b for an SOi — BOTTOM OF caupi r

C

profile

4a.

6

8

/

16

I

30MBARQ MENT. OPENING

t

-TARGE1

\

14

//

i

Vertical

6

10 12 DISTANCE IN mm

4

1

.1

— —

§

J

Fig.

0

-2

r

-TA WET BOMBARC OPEfJING

0

10

/

20

HORIZONTAL DISTANCE IN mm

beam-distribution

Fio.

4b.

Horizontal beam-distribution profile.

1

is

is

is

irradiation with 37-Mev a particles. As evident from these curves, the sample area which less than cm1. With accompanying reduction in uniformly irradiated ,

average intensity, this area may be increased by oscillating the beam across a layer of the target or, vice versa, by suitably moving the target in the beam. 3.2

Experiments at Low Temperatures

Because of the short range of the charged particles in their passage through solids,

is

is

is

the power density in the sample quite large, several kilowatts per cubic centimeter. It not generally possible to immerse the sample in a heat-transfer liquid because of the short particle range; a table of particle range vs. energy for different materials and different particles given in Table 9. The sample may be cooled and maintained at the desired low temperature by keep in close metallic contact with a support that itself cooled by suitable liquid, ing e.g., liquid nitrogen. This method allows the sample to be situated within the cyclo

tron

vacuum systems, permitting

a

is

it

the use of the undeflected beam,

which may be

larger in intensity and area than an external beam. However, the close metallic contact required for good conduction of the heat from the target to the cooled support may make impossible to make measurements of various types, e.g., electrical con ductivity, creep, etc., during the exposures. The sample may also be maintained at desired temperature by means of forced convection cooling with a cold gas, helium, cooled with liquid nitrogen. Very high gas velocities at the sample surface are required to obtain good cooling conditions with suitable only where this method, and desired to keep the sample as a whole at some fixed tem below some given temperature rather than attempt to maintain perature. The gas cooling method, however, permits greater flexibility in the type of it

is

it

is

it

a

it


.STRIP NO USED FOR VERTICAL DISTRIBUTION

-H

—

J/

30

2

||s

-1

_-

/ /

40

,

50

<

■+2inm FOIL CUTS

1

J

60

HORIZONTAL FIDUCIAL MARK STRIP USED FOR VERTICAL PROFILE VERTICAL FIDUCIAL MARK

\

70 01 <5 X

TOP OFQAUDl

//

BO

measurement that can be made during the irradiation and can be more readily adapted annealing experiments in situ.

to

Particle Range vs. Energy

Table 9.

Pb

[Sec. 10

REACTOR MATERIALS

10-98

,.=

10 Mev = 20 30 = 40

-

11.3 g/om'

Ag

3.5 X 10"' cm

p =

= = = =

10. 2 19.6

30.0

Alpha Particles

10.5 g/cma

Cu

p = 8.96 g/cm'

= 2. 1

2.4 X 10-i cm

-

7.7 14.7 24.0

-

6.5

X

Al

p — 2. 7

-

g/cm*

5.9 X = 18.5 = 43 = 60

10-» cm

13 21

IO-> cm

Deutcrons Pb

p = 11. 3 g/em"

AK

p = |0. 5 K/cm*

Cu

-

e/tm'

Al

-

= 17 X 10-" cm = 50 = 95 155

10 Mev = 20 X I0~J cm 60 20 30 =115 40 185

p = 8.96

= 37 X = 120 250 390

10.3 X 10"" cm = 42 = 85 140

-

p = 2.7

g/cm' I0> cm

-

-

Protons


Pb

p = 11. 3 g/em"

Ag

p = 10.5 g/cm'

-

10 Mev = 30 X 10"' cm = 88 20 185 30 40 300

24 X 10-' om = 79 = 160 = 265

-

3.3

Cu

p = 8.96 g/cm'

-

21 X 10-> cm = 70 = 140 = 230

Al

p = 2. 7

g/cm'

= 59 X 10 220 440 740

-

'

cm

Experiments at High Temperatures

In studies where the sample must be kept at a controlled, elevated temperature during the exposure, it is generally necessary to establish a balance among three factors: (1) the heat generated by the beam, (2) any additional heat supplied to attain At the desired high temperature, and (3) cooling losses to the ambient surroundings. moderate temperatures, e.g., below 500°C, the tests can be carried out in a suitable inert atmosphere, but near 1000°C high vacuum is generally required to reduce the heat losses. Rapid and erratic beam fluctuations, the high beam-power density in the target, and the small heat capacity of the targets generally used can result in temperature changes exceeding 125°C/sec. The control system for the additional heat supplied For conductive materials, the to the sample must be extremely rapid in response. best choice is electrical-resistance heating in the sample itself, electronically controlled by means of thermocouples attached to the sample. Special attention must be paid Forces on the sample due to the to problems peculiar to cyclotron irradiations. fluctuating current through it in the presence of the cyclotron magnetic field may Also, proper beam integration requires a high resistance disturb the experiment. between the sample and electrical ground, and this complicates the control circuits. 3.4

Induced Radioactivity

This may be estimated crudely, using the principal elements in the target materials, the tabulated reactions and cross sections for those materials, and the decay constants for the activated products produced. Because of the small size of the specimen, handling is a simple problem compared with that for experiments with reactorsHealthCooling times of a few hours permit handling of specimens with short tongs. physics monitoring must accompany the work, however.

Sec. 10-4]

RADIATION DAMAGE 3.5

10-99

TO SOLIDS

Use of Electrons

The comparatively low momentum of electrons having energies even in the several Mev range results in only minor changes in the physical properties of solids, except for The principal molecular materials in which ionization is of primary importance. application of electron irradiation has been in the determination of the threshold In the energy for atomic displacements and in irradiations of molecular materials. former case, an accurately known, stable, and well-defined value for the electron Such conditions are well satisfied with accelerating energy is of primary importance. a belt-charged type of electrostatic generator, the Van de Graaff machine, which can readily supply beam currents of 100 â at energies in the 1- to 2-Mev range. Radioactivity of specimens is customarily no problem, since only a few nucleides are activated by electrons.

ELEMENTARY PROCESSES

4

FOR DAMAGE IN SOLIDS


Damage to ionic, metallic, and valence crystals is due to processes that displace one or more atoms at a time from their normal sites in the solid, producing structural changes. Primary defects produced by radiation may rearrange themselves, returning the original or equivalent structure or combining with themselves and with other defects. In conducting solids, atoms are displaced as recoils or knock-on atoms by In nonconducting solids, similar knock-on close collisions with energetic particles. events occur and in addition defects are produced by ionization and excitation. Atom Displacements

4.1

4.11 Threshold Energy. A threshold energy e0 imparted to an atom (in a collision process) will move it sufficiently far from its original site that return is impossible, provided the temperature is sufficiently Threshold energies have been de low. termined by fast-electron irradiation for a few solids and are given in Table 10. It is to be expected that this threshold energy is a function of direction of dis placement of the atom and possibly, to some extent, the temperature. 4.12 Single atom displacements occur if the energy given to the primary knockon is bet ween «o and 3eo. 4.13 Multiple Atom Displacements. the energy imparted to a primary knock-on atom exceeds three times the threshold energy, the probability of its Flo. 5. Number of atom displacements, producing secondary displacements be g(x), as a function of x = (energy in excess comes significant and will increase with of threshold) /(threshold energy). The average number of total energy. displacements produced by a given primary, including the primary itself, has been worked out.' The results, also given by Scitz and Koehler,7 are shown in Fig. 5. The expressions, if x ■ t/e0 — I, are

If

g(x)

-

1,

0 < x < 1; g(x)

-

1

+ In

x, 1 < x < 2; g(x)

= 0.56(1

+ x), x >

2

where g{x) is the number of displacements by given primary knock-on of energy (including itself). 4.14

The

total number

of

displacements

produced by each

e

type of primary

particle may be estimated using the methods of collision theory. The cross section a for a collision in which the atom Fast Electrons. energy than e0, when the electron energy is E, is as follows: Define

receives

more

REACTOR MATERIALS

10-100

[Sec.

10

imoM

My

(»i + y =

m0 c

to V

2nioC

0

rest mass of electron

velocity of light velocity of elect ron/e Table 10.

Threshold

Displacement Energies

Solid

«o, ev

Electron enerKy. Mev

Graphite Ni Cu

24.7 ± 0.9 34 ± 1 25 ± 1 30

0. 12 ± 0.005*

Ge


-A-rr eo

— «■-(*)'

»

where

E

«m . mo

0.58t 0.49J

0.63«J

* D. T. Eggen, "Energy Required for Atomic Displacements in Graphite Determined by Electron Bombardment," NAA-SR-69, Apr. 10. 1950 (secret as of September. 1950). t H. W. Kenworthy, NAA-SR-report. to be issued. Energy for Radiation Damage in Copper. Displacement t D. T. Eggen and M. J. Laubenstein, Phy$. Rev., 91: 238(A) (1953). 1 E. E. Klonti, AECU-2267. 1952. use and for purposes of illustration, assume that M = 2.25Z atomic Deviations from these values may be interpolated units and «0 = 25 ev. inearly within the accuracy of the estimates. Then

For convenience in mass

a = 2.5

X 10-"2«

— P*

|(y

- - p In 1)

F The fraction F of the atoms displaced 9xicr6i

=

1)

y]\

lO'V in the target of a solid of atomic number Z is shown in Fig. 6. Secondary displacements do not occur in the range considered. Cyclotron Particles. Using the defini tions and assumptions given above, the cross section for all collisions of a charged particle with an atom in a solid in which an atom receives energy in excess of 25

r\ 75 MEV

5 VIEV

- - In

by 1017 particles/cm*

,l 5MEV u.

+ O.O23Z0[2(y«

= 4.44 â-hr/em* is a convenient dose, let

of 10" particles/cm*

Since an irradiation

y

•«-.125 ME

ev is -^t075 SMEV 20

Fia.

6.

"V^ 40

6a

80

W.hZJZm X

100

lO'11 cm'

Z\, TO, and E are the atomic num ber, mass in atomic mass units, and energy in electron volts of the incident particle For 10-Mev protons at a dose of is the atomic number of the atom in the solid. Fractions of atoms

displaced

per

where

fast electron.

and Zj 10" part ides /cm1, the fraction of primary displacements is

F

= 1.15

X

10-»Z,

Sec. 10-4]

RADIATION DAMAGE

10-101

TO SOLIDS

Interpolation or extrapolation of the cross sections to selected values of the particle mass, charge, or energy may be made by means of the following functions where k is a scale factor for each of the variables as given.

o(m,Z,,Z,,E)

= ke{.m,Zi,Zi,kE)

=

- c(km,Z,,Z2,E)

k

"

—
The energy given atoms by cyclotron particles will often exceed spectrum of the primary knock-ons or displacements is dn

X W-uNtQZSZSm MEi1

6.55

dt = 0

where

and

t < to

.

~-

k

3«0.

o(m,ZhkZt,E) The energy

.

t > cm

e = energy given to the primary knock-on e„ = AmM /(m + M)* = maximum energy of a knock-on

incident particles per cm' are in electron volts. Using this spectrum, the average number of displacements per primary collision is as given by Seitz,7 Q

■» total


and the energy units

-

t

0.885 + 0.561 In

total

4

The fraction of all atoms displaced may be estimated by combining this number with the fraction of primary displacements above. The limitations on these estimates are discussed by Seitz. Fast Neutrons. The cross section for displacement collisions of fast neutrons is just the experimentally determined scattering cross section a,. The energy spectrum is dn

—

=

QNgtr,

dt

= 0

Using this together with 1-Mev neutrons is

4.16

e0 < t < €m

tm

e <
and

c > tm

the g(x) above, the total fraction of atoms displaced by 10"

FM

- F p(z.v.)

FM

= 4.5

X

10"
(m + Af)!

The mean free path between collisions of the primary knock-on with station

ary atoms depends on the interaction potential between two atoms. The potential function used by Seitz* gives several atom distances. Another expression for the interaction potential chosen by Brinkman,' which gives reasonable numerical values

of

the potential at distances of approach about the size of an atom, leads to a much free path, equal approximately to one lattice spacing in most solids from energies of the order of the displacement energy up to another threshold energy, the magnitude of which depends on the properties of the atoms in the solid in question.

smaller mean

4.2

Models of Structural Changes Produced by Displacements

vacancy pairs are thought to be the structural defect resulting the single atom with energy «0 < t < 3«o. It is reasonably established that the vacant site with some relaxation of neighboring atoms

4.21

Interstitialcy

from the displacement of

well

* Manuscript

reviewed

by private communication.


10-102

KEACTOII MATERIALS

[Sec.

10

However, the arrangement of atoms in the neighborhood represents physical reality. of the displaced atom is less clear.9 In open structures it may occupy a true inter stitial site and thus be correctly described by its term. For all crystals the location of the displaced atom will be called an interstitial in spite of this question. If the mean free path of several atom distances is correct, the displacements estimated above arc nearly independent and remain at low temperatures as interstitial-vacancy pairs except for those removed by radiation thermal spikes and other recovery processes.10 Graphite provides a good example of the case in which single displacements pre dominate. A count of displaced atoms by neutron diffraction" agrees with the calculation on this basis. 4.22 Closely spaced collisions produce atomic rearrangements that differ in Since important ways from those produced by the widely separated collisions. collision of the primary knock-on with its immediate neighbor is highly probable, double, triple, or multiple vacancies are produced with high probability, with the appropriate number of interstitial atoms more or less surrounding the region of the vacancies. Moreover, Brinkman8 shows that at some critical value of the number of secondary displacements produced by a primary knock-on the region becomes unstable. Displacement spikes result from the above instability promoted by the quasi-thermal excitation of the region as a result of the collision processes, which will induce region? above a critical size to collapse. Thus, mixing and disordering of a small region result directly from the displacements. In this picture the number of permanently displaced atoms that are produced by the collision processes involving neutrons with medium and high mass number solids is less than that in the picture employing the longer mean free path. Cyclotron particles produce a significant number of displace ment spikes, but these do not affect the estimate of the total point defects very greatly. Brinkman gives relationships that permit estimating the type and number of multiple displacements. Kinchin and Pease" estimate that a large number of second Replacement Spikes. ary collisions are sufficiently energetic to push an atom out of the site while the colliding atom merely occupies the vacancy. This process, analogous to interstitialcy migra tion, can play an important role in the rearrangement of atoms in some multieomponent systems, where the atoms are about the same size and mass. Thermal spike is a term describing the situation in a small region in which a rapidly Brooks" has shown that the large moving particle is slowed down and stopped. amount of the energy of the particle which is dissipated to the electrons flows away by electronic conduction without achieving equilibrium with the lattice for at least 100 atom distances. A smaller fraction of the energy, that which produced knock-ons discussed above, is dissipated directly to the lattice and is conducted away by lattice waves or by moving atoms. A small fraction of the energy is stored by displacing Although the duration of the spike is too short atoms from their normal lattice sites. to permit equilibrium to be established in any given region, it is nevertheless a useful approximation to think of the region as having a high temperature and to use the picture that thermally activated processes take place to the extent permitted by the time. A theoretical description of the thermal spike has been given by Brooks. 4.23 Fission Recoil Effects. Because of their mass and energy, fission recoils Probably few displace a large number of atoms from their normal sites in the solid. of these displaced atoms are interstitial-vacancy pairs. The thermal-spike and displacement-spike pictures mentioned above are probably most suitable for describing changes induced by fission recoils. 4.24 Foreign atoms introduced at the end of the fission recoil range produce another type of damage. Table 11 lists most of the fission-product elements, their yield, and volume. Changes in density may be estimated (1) on the oversimplified assumption that the volume of the atoms is additive or (2) on the assumption that elasticity theory may bo used to compute crystal lattice distortions around foreign atoms. Lattice stresses around atoms as large as krypton, xenon, and cesium (the daughter of Xem) are probably relieved by the association of lattice vacancies with the region.


Sec. 10-4]

Table 11. Element

Fission-product

Fission yield

Y„

Yields and Atomic Volumes

Atomic volume

%

0.4

Se

Kr

33.7 16. 1 14.2

10. 5 29.4

Zr

Nb Mo Tc Ru Rh Pd

9.4 8.6 8.4 8.5 9.0

IB. 4

6.2 12.4

3.7 1.7

KiFVlOO

0.71 1.15 1. 17 2.86 1.69 4. 18

55.8

8.5

Y

Fi, cm'/mole

18.5 28

4. 1 2. 1

Rb Sr

10-103

1.73

0.53 1.04 0.31 0. 15

Sb

Te

2.0

Xe Ca

1.2 20.5 18.7

I

la

Ba

6.6 6.2

Ce

17.0

Pr


Eu Br 2r<

-

2.6 2.7 0.2 0.2 198.3

7.70 13. 10 2.59 1.40

20.4

3.48

21.6 21.0 21 21.5 21.8

17.3

Sm

0.41 0.31

70 39.3 22.6

5.7

Nd Pm

U">

20.4

25.8 37.6

1.23

27.6

-

3.64 0.55 0.58 0.04 0.06

Vol of tisaion product from 1mole of U = Y i V,/ 100 -- 49.97cm« Vol of 1 mole of U = 12.35 cm'

Table 12 Vacancies

Ap*

Cu

Ag

Au

1.3

1.5

1.5

Interstitials Ap«

1.4t

5.0J * Microhm-cm Blatt.

t

per atom per cent defect.

j Jongcnburgcr.

4.26 Changes in properties that result from the processes and structures men tioned above are given in following articles. Some of the elementary relations between structures and properties may be summarized as follows: Electrical resistimty incre ments due to scattering of conduction electrons by interstitials and vacancies have been calculated by Jongenburger and by Blatt (see Table 12). The changes in residual resistivity estimated from these values together with the cross sections given for cyclotron particles or electrons are five to fifteen times greater than the observed changes. Qualitative descriptions of other property changes may be given in terms of the structural defects: energy stored, due to interstitial-vacancy pairs; changes in mechanical properties, due to tying down or evolution of motion of dislocations by defects; disordering, by displacement and thermal spikes and by motion of vacancies;

other

reactions, activated in the region of the spike.

10-104

REACTOR

MATERIALS

[Sec.

10

Mechanisms for Removal of Damage, Recovery

4.3

It has been Thermally activated movement of defects permits their removal. estimated on theoretical grounds14 that the activation energy for motion of interstitials is between 0.1 and 0.2 ev in a typical face-centered-cubic crystalline solid, copper. The activation energy for motion of a vacancy is significantly higher in most solids. An unequivocal assignment of activation energy and frequency factors for movement of these primary defects in solids is impossible at the present time. In Table 13 are Table 13. Solid

Graphite

Al

Ni


Cu Ag Au

Type of irradiation

Temperature Ranges for Recovery

Electron* Proton t

I03°K

Fast neutron!

I50°C !20°Kt

Fast neutron 5 Electron** Electron** Deuterontt Fast neutronft Deuterontt Deuterontt

Temperature range for significant recovery electric resistivity

Temperature of irradiation

10°K 20°C

30°C 80° K

I0°K

80° K

I0°K

80° K 10°K

I0°K

of

80-1 10°K; 15O-30O°K; no studies at higher temperatures 15O-300°K; no studies at higher temperatures I80-330°C; 1000-I400°C; 1800°C I000-M00°C; I800°C 180-330°C;

I50-300°K 180-330°C;

I000-I400°C;

1800°C

2I3°K 33°K; 273°K 20°C;

200°C

33°K; 40-280°K (very gradual); 200°C 243°K; 200°C 30°K (sharp); 30-200°K (gradual); 20O-230°K I0-230°K (gradual); 230-260°K (rapid)

* S. B. Austerman and J. E. Hove, Irradiation of Graphite at Liquid Helium Temperatures, Pit**. Rev., 100: 1214-1215 (1955). t G. Hennig and J. E. Hove, Interpretation of Radiation Damage to Graphite, P/751. Proc. Inter*. Con/, on Peaceful Veet of Atomic Energy, August, 1955. t W. P. Fatherly, Bull. Am. Phys. Soc, SO: 8 (1955); also T. J. Newbext, ibid. % W. K. Woods, L. P. Bupp, and J. F. Fletcher, "Irradiation Damage to Artificial Graphite," P/746. \ A. W. McRcynolds, W. Augustyniak, M. McKeown, and D. B. Rosenblatt, Neutron Irradiation Effects in Cu and Al at 80°K, Phys. Rev., 98: 418-425 (1955). ** C. J. Meechan and J. A. Brinkman, Electrical Resistivity Study of Lattice Defects Introduced iD

Copper by 1.25 Mev Electron Irradiation at 80°K, Phys. Rec, 10S: I 193 (1956). J. W. Corbett, J. M. Denny, M. D. Fiske, R. M. Walker, Electrical Irradiation of Copper Nesr I0°K, Phys. Rev., 108: 954-964 (1957). A. Sosin, C. J. Meechan, Defect Production and Migration in Copper and Nickel, NAA-SR-20*! Nov. 15. 1957. " tt F. Seits and J. S. Koehler, The Theory of Lattice Displacements Produced During Irradiation." P/749.

given the temperature ranges in which recovery of changes in electrical resistivity occurs and which must be related to thermally activated motion of primary defects It may be noted that complete recovery observed in several solids studied to date. requires heating to temperatures associated with the usual annealing of metals. The case of graphite may be somewhat clearer than that for metals in that single interstitial atoms lie between the planes and move at approximately 80 to 100'K The possible reactions of the primary defects in Vacant sites move near 1800°C. are discussed later. graphite After apparently complete removal of interstitial-vacancy pairs, defects remain that These defects are thought to be either jogs increase the resistance of metals to flow. in dislocations or aggregates of interstitials or vacancies forming the equivalent of stacking faults or both. Fission-fragment damage anneals only partially at temperatures below those at which atomic diffusion occurs. Depending on the activation energy for diffusion in the material in which the fission products find themselves, there will be a tempera ture range in which the atoms are sufficiently mobile that further diffusion to metastaCertain of the fission-product atoms, par ble or stable configurations is possible. The ticularly the inert gases, are far too large to fit in ordinary metallic lattice.

RADIATION DAMAGE

Sec. 10-4]

10-105

TO SOLIDS

initial lattice strains arc probably relieved by diffusion of vacancies to the region of the inert-gas atom; subsequently, the region of vacant sites along with the gas atom may diffuse out of the crystal, first to grain boundaries and then along or with grain While this is proceeding, if the concentration of fissionboundaries out of the solid. product atoms is sufficiently high, agglomeration and bubble formation are possible. At temperatures too low for diffusion, fission-product atoms will influence the physical and mechanical properties of solids in all the ways generally pictured for the effects of impurities. Because of the large mismatch in atom size, large effects at very low concentration may be expected. 4.4

Ionic Crystals


Displacements occur in ionic crystals in the way described above. In addition, two other sources of point defects exist. Vacancies may evaporate from jogs in dislocations,15 the energy being supplied by excitons" or by thermal activation. Interstitial atoms may be produced17 by ionization of an anion by y radiation or fast electrons followed by movement of the anion itself under coulomb repulsion or of a cation pushed by the anion. The cross section for the process is approximately 10-(ts+») cm^ where n is the number of electrons removed. The interaction of crystal defects with one another and with electrons and holes is covered in the references cited. u-" 6

PRODUCED IN MACROSCOPIC PROPERTIES OF MATERIALS

CHANGES

6.1

The

Nonfissionable

Metals

of radiation on nonfissionable metals are generally relatively small as the engineering applications of the materials are concerned. The results observed, however, are of considerable interest in basic studies on the properties of imperfect materials. Mechanical Properties. The effects on mechanical properties require some 5.11 consideration in the practical application of metals to structures in a strong radiation field. The changes in electrical properties have been extensively explored also but are probably of lesser engineering significance. The mechanical properties investigated have been of considerable variety. Table 14 summarizes the results on numerous metals.*18 The changes in electrical properties of metals have 6.12 Electrical Properties. been studied primarily as a tool for the fundamental investigation of radiationdamage phenomena. In general, the resistivity near room temperature in samples exposed at such temperatures changes very little, less than 1 per cent. At lower tem peratures of exposure and measurement, significant changes are observed, and the study of these changes and their annealing behavior at higher temperatures has been of considerable importance to the subject. Other electrical or magnetic properties have been investigated in special instances, including magnetic permeability and thermoelectric power. Table 15 summarizes results on the electrical resistivity changes observed in various metals. The thermal conductivity of metals generally follows the behavior of the electrical conductivity. Measurements have been made in a few materials, e.g., Inconel and stainless steels, at elevated temperatures in the range 300 to 800°C with no changes

far

effects

as

observed.

The thermoelectric power of several irradiation. The data appear in Table 6.13

Several * These

given

materials has been studied1' as a function of Hi.

Effects of Radiation on Reactions in Solid Metals (Metallurgical Processes). major types of reactions in solid metals (metallurgical processes) induced or

data have been collected in Kef. 18.

from

numerous

unpublished

AEC

reports.

A useful summary

is

10-106

REACTOR Table 14.

Change in Mechanical Property •

Metal Aluminum, high-purity annealed

MATERIALS

Offset yield strength

[SEC.

Properties of Metals and Alloys

Exposuret

Change observed

Remarks,

2 X 10", cm' thermal neutrons, 5 X IO"/cm' fast neutrons

+ 100%

Exposed at about 50°C for all prop erties given

Ultimate strength Fractional elongation Aluminum, high-purity half hard

Offset yield strength Ultimate strength Fractional elongation

+60%

-67% 2 X IO"/cm" thermal neutrons, 5 X IO»/cm' fast neutrons

+ 10% +6% 0

Exposed at about 50°C for all prop erties given

Aluminum

Creep rate

~IOu/(cm')(sec) thermal neutrons

Negligible

Tested at 450*C 27 kg/cm>

Aluminum

Creep rate

~IO»/(cm*)(sec) 9-Mev protons

Negligible

Tested between 170-335*0. 45-180 kg cm"

Aluminum alloys:


10

Exposed at 70*Cfor all propertiesgiven

IO»/cm* fast neutrons

Type 61SO

Yield strength Tensile strength

+290% + 150%

Type A MS

Yield strength Tensile strength

+6% +6%

Type 52SO, 61ST6

See above

~ 50- 100% for both Y.S and T.S

Beryllium

Various

~3 X IO«°/cm' fast neutrons

Negligible

Copper Ann. OFHC

Hardness

2 X IO'»/cm» rut neutrons 6 X IO"/cm' fast neutrons

41

+53 Brinell numbers

Exposed at 8CTC

2 X 10'Vcm' fast neutrons lO'VCcni'Msce). I6-Mev deuterons

0.4-2

Exposed at 40"C, measuredat 2ffC At 260°C, 700 kg/cm"

9 X 10" neutrons/cm' thermal neutrons, 2X10" neutrons/cm' fast neutrons

RolB — Ru80

Exposed at 30T, alloy originally is

Quenched for 2 hr at 285°C

RolOO-RaKB

Exposed at ~3TC alloy near maxi mum hiirdacsi originalb'

Quenched for 18 hr at 4O0°C

Ro67

Copper, single crystal*

Critical shearing stress Creep

Copper-2% Be alloy: Quenched from 800"C

Copper-manganese:

Hardness—Rockwell G

DP11 hardness

-

57 DPH

0kg/mm>

Negligible

-

Ro85

1.4 X 10'Vcm'fast neutrons

17.4% Mn

+ 11 6 DPH numbers

1.35% Zn 34% Zn

1.5 X IO»/cm» fart neutrons

Exposed at ~30T. alloy original!;, overageduSemaut All exposedat ~30*C

+ 13 7 DPH numbers + 16 8 OPH numbers

Hardness

Exposed at 60T

Mate

1.0% Mn 5 8% Mn

Copper-line:

Exposed at ~30T

26-60RF 71 Rf

64

-

Exposed at

-JOT

Exposed at J0*C. annealed 1 hr si 7S0°C before exposure

Sec. 10-4] Table 14. Metal

radiation damage to solids Change in Mechanical Properties of Metals and Alloys. Property-

Exposuref

(Continued)

Change observed

Remarket

Rp

Exposed at 30°C, cold-rolled before exposure

Hardness (Con/.)

1 5X IO-»/cra>fast neutrons (Con(.)

100-101

Iran

Hardness

2 X 10" /cm' fast neutrons

84-90RB

Exposed at 40°C

Iron alloys: Type 304 SS

Hardness

1.3 X lOu/cra' thermal neutrons, lO'Vcm* fast neutrons 2 X 10»/«n> fast neutrons

Negligible

200°C during exposure

Insignificant

Irradiated at I00°C, testedat l.700cpm

34% Zn (C
Fatigue strength

Type 316 SS

Hardness

5 X 10'Vcra' fart neutrons

83

Ultimate strength Elongation at rupture Type 347 88

Hardness Yield point Ultimate tensile strength Hardness


10-107

Hardness Elongation at rupture Elongation at rupture Creep rate

6 X 10'Vcm' fart neutrons

Exposed at 25°C

75

-

-15% 93 RB

Exposed at I60°C

2300 — 5400 kg/cm' 7300 kg/cm'

6001. 1 X l0>'/cm>36-Mev a particles 6.7 X 10" /cm' 36-Mev a particles 6 X l0>»/cm'fast neutrons 7 X IO"/cm' fast ~IO''/(cm')(sec) neutrons

1 X 10'Vcm' fast neutrons

SA-285 pressure vessel Tensile strength (2% ■toe! offset) Elongation at rupture

4 X 10"/cm' fast neutrons

Nickel

98 RB +90%

SA-212 pressure vessel Fatigue endurancelimit steel

Molybdenum

-

fast

190 — 230 DPH

Exposed at -70°C, measuredat 20°C

190-260 DPH 57 -34%

Exposed at 160°C

Negligible

Exposed at 280°C

Decreased 25%

Tested in reactor at ~70O°C under 8.000-psistress

48.000

-

42.000 pei

+60%

12-3%

Sample without notch exposedat 40*C Exposed under load of 1.000 kg/cm'

Exposed at SO'C

Hardness Yield strength (2% offset) Fracture strength Ductile-to-brittle transi tion temperature

ICVera* fast neutrons

Hardness Ultimate strength Elongation at rupture

IO»°/cm' fast neutrons

43 96 Rb 4400 6600 kg/cm' 47 26%

Exposed at 30°C

Hardness

2 X IO»/cm« fast neutrons

+ 10 DPH numbers

Exposed at Tcfl-C

Yield strength (2% offset) Elongation Creep rate

+35 Brinell numbers

--

100.000—140,000 psi 220.000 160.000psi

-30

+70°C

--

-

17.000

~IO»/(cm')(sec) thermal and fast neu trons

24.000psi

Negligible change Rate decreasedby fac tor of 20

Tested in reactor at 260T in stress range 1,000-2,500 kg/cm'

• Creep rates are "in-beam" or "in-pile"; other properties are before-and-after measurements. t In "Exposurev and "Remarks" columns conditions are not repeated. Initial conditions govern all entries for a given metal in "Metal," "Property," and "Change observed" columns until a changed condition is indicated by a new entry.

REACTOR MATERIALS

10-108

Change

Exposure

Metal

4

+ 40%

X 10'Vcm'fast

+ 10% + 0.7% Negligible

I00°C,

Exposed at measured

neutrons

at

I70°C Exposed at 50°C. 190°C measured at Exposed at 50°C. measured at + 20°C Exposed at 50°C

-

AISi alloy (2.3% Si, in solution)

4.8 X 10'Vcm' fast

Copper OFHC

2.2 X I0"/om« (36-Mev a particles) 1.5 X 10'Vcm' fast neutrons

+ 30%

X 10'Vcm'fast neutrons 1.7 X 10'Vcm'fast neutrons

+ 0.39%

Exposed at ~50°C

+ 0.01%

Exposed at ~50°C

1.5 X

+ 0.3%

Exposed at ~50°C

-0.65%

Exposed at ~50°C

-1.8%

Exposed at ~50°C

Negligible

Exposed at ~50°C

neutrons

CuSn alloys: 1% 8n

+ 0.2%

1.7

5.4% Sn CuZn alloys: 1.35% Zn

10'Vcm' fast neutrons 1.5 X 10'Vcm'fast neutrons 1.5 X 10'Vcm'fast neutrons 1.7 X 10'Vcm' fast neutrons

6.3% Zn 12.6% Zn Generated for wjivans (University of Florida) on 2015-09-23 02:55 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Remarks

observed

2.2 X 10'Vcm' (36-Mev a particles)

Aluminum, 2S

Iron Iron alloys: Type 304 SS Type 347 88

10'Vcm' fast neutrons 7 X 10'Vcm' (I8-Mev

Type 347 SS

deuterons) 25 X 10'Vcm'fast

Type 322W SS

25 X

4 X

+ 5% + 4%

neutrons 10'Vcm'fast neutrons 2 X 10'Vcm'fast neutrons 2.2 X 10"/om« (36-Mev particles) 1.4 X 10'Vcm' thermal neutrons (~2 X 10'Vcm' fast) 25 X 10'Vcm' fast

Nickel, typo A Zirconium

X lO^/cm* 35-Mev a particles 2 X 10'Vcm1 9-Mev protons 6 X I01* neutrons/cma fast-neutron flux 1.7

Chrome!

vs. Alumel

Exposed at 280°C, measured at ~20°C Exposed and measured near

-

150°C

measured

Negligible + 60% + 6% Negligible

at 25°C

Exposed at 260°C, measured at 25°C Exposed at ~50°C Exposed at

100°C. measured at — 170°C Exposed at ~30°C, measured at 20°C

Exposed at 260°C, measured

Thermoelectric

Exposure

Metal

Exposed at — 100°C. measured at — 170°C Exposed at 60°C. measured at 20" '

Exposed at 260°C.

neutrons

Table 16.

10

Electrical Resistivity

Table 16.

Aluminum, high-purity annealed

[Sec.

at 25°C

Power

Thermoelectric power vs. unirradiated metal

-0.03

ftv/°C

-0.3Mv/°C Negligible

Re murks

-

Exposed, measured at 120°C Measured near 30°C Exposed, measured near 100*0


Sec. 10-4]

radiation

damage

to solids

10-109

affected by radiation have been studied extensively: order-disorder transformations, precipitation from solid solution, phase change from austenitic to ferritic structure, and diffusion. In the alloy CuiAu, neutron exposures have been observed to Order-Disorder. In a somewhat disorder initially well-ordered specimens at temperatures near 50°C.20 higher temperature range, radiation has been observed to accelerate the ordering of initially disordered specimens." Similar disordering effects have been observed in NijMn from measurements of changes in the magnetic properties." In the alloy 0 brass, which normally exists only in the ordered states below its transition temperatures, exposure to 4 X 1017/cm2 of 36-Mev a particles at — 150°C This is much greater than is observed produced a 100 per cent increase in resistivity." This change has in other brasses and anneals out rapidly below room temperature. been interpreted as disordering of the 0-brass structure at low temperatures. All these results can be described qualitatively in terms of enhanced local diffusion The exact mechanism required to account for the rapidity of the on a microscale. effects is not yet clear. Several hypotheses, described as "thermal," "displacement," or "replacement" spikes, have been proposed.*'" The system that has been studied most exten Precipitation from Solid Solution. sively is copper containing about 2 per cent beryllium." The changes in electrical resistance during and subsequent to exposure, the changes in hardness after varying exposures, and the subsequent alteration of these changes during annealing were In summary, a correspondence between effects produced by neutron investigated. irradiation and by low-temperature (~100°C) aging was observed. The results suggest that radiation can produce precipitate nuclei and that these form because of the accelerated microdiffusion which takes place in the presence of radiation. The decrease in resistivity observed in type 322W stainless steel (Table 15) has also been attributed to precipitation induced by radiation. Phase Transformation. Austenitic stainless steels, alloys of iron, chromium, and nickel, are metastable only at ordinary temperatures; the stable state is a mixture of The ferritic phase is ferromagnetic, and magnetic measure austenitic and ferritic. ments have been used as a sensitive means for detecting the presence of transformed A slight increase in ferritic content after neutron bombardment of type 347 material. but of a magnitude probably below practical stainless steel has been observed, significance. The effects of radiation on the several processes discussed in earlier Diffusion. paragraphs suggest that microdiffusion is accelerated in metals when they are exposed Direct experiments on this phenomenon have not yielded to particle radiation. positive results, however. Detailed experiments on the effect of radiation on the diffusion rate in the coppergold, copper-nickel, and lead-tin systems and on self-diffusion in silver have been made. Copper-gold and copper-nickel samples were exposed in a fast-neutron flux of ~5 X 10" neutrons /(cm') (sec) in the temperature range 280 to 380°C, with no observable change in diffusion rate compared with control samples at the same temperature.** Lead-tin samples, exposed in a much lower flux, also gave negative results. Silver self-diffusion was studied using cyclotron beams of 3 X 10" neutrons/ The temperature range was 700 to 850°C. No (cm1) (sec) of 10-Mev protons. detectable change was observed."

"

5.2

Moderator

Materials

Graphite. Information concerning changes in the following properties of 6.21 graphite is of direct use in the design and operation of reactors: (1) linear dimensions, (2) thermal conductivity, (3) enthalpy (stored energy), and (4) Young's modulus. Of engineering importance but not of direct use are changes in (1) strength and (2) •Revised by D. S. BillinRton in Ref. 24.

10-110

REACTOR

MATERIALS

[Sec. 10

Studies aimed at understanding and describing radiation damage machinability. have made use of measurements of the following quantities, in addition to those above, as a function of irradiation dose and temperature and of temperature and time after irradiation: (1) lattice parameters; (2) electromagnetic coefficients, (a) electrical resistivity, (6) thermoelectric power, (c) Hall coefficient, (
Nomenclature

of Graphites

Designation

Killer

Binder

Manufacturing process

CSF CS-GBF KC KS WSF WS-GBF TS-GBF

Cleves* Cleves Kendallt Kendall

Standard | Standard Chicago** Standard Standard Standard Standard

Acheson

Whitingf Whiting

Texasf

GBF

Acheson Acheson Acheson

GBF GBF

* Petroleum coke (mid-continent crudes) from the Gulf Oil Company refinery at Cleves, Ohio, Petroleum coke (Pennsylvania crudes) from the Kendall Oil Company refinery at Bradford, Pa. X Petroleum coke (mid-continent crudes) from the Standard Oil Company (Indiana) refinery at


f

Whiting, Ind. i No. 2 medium-hard coal-tar manufactured by the Barrett Company. U Petroleum coke (mid-continent crudes) from the Texas Company refinery at Lock port, BX ** Barret-Chicago No. 7HO coal-tar pitch manufactured by the Barrett Company.

Table 18.

Thermal Conductivity of Unirradiated Graphite Parallel and Transverse to Direction of Extrusion Grade

Parallel conductivity, cal/(cm)(sec)(°C)

Transverse conductivity, cal/(cm)(sec)(°C)

KC CSF TS-GBF

0.43 0.40 0.20

0.27 0.26 0.18

Graphite Types. Table 1730 lists the trade designation of several types of graphite used and studied in connection with reactors. Methods of blending, forming, baking, The properties of a graphite graphitizing, and purification are given by Currie." body appear to be controlled largely by (1) crystallite size, influenced by source material and graphitizing temperature; (2) crystallite orientation, determined largely by forming methods; (3) amount and nature of intercrystalline binder material, probably governed by the source material used for binder, by temperature, and by time of graphitization; and (4) density. These factors may vary somewhat through out a body. Dimensional changes arise primarily because graphite crystallites expand along their c axis ([001] direction) and are influenced secondarily by preferred orientation. Figures 7 and 830 show length changes of several graphites in directions parallel and The temperature variation of the physical perpendicular to the forming direction. expansion during irradiation is shown in Fig. 9.'° Thermal conductivity and its reciprocal, thermal resistivity, are shown in Table 18" and Figs. 10, 11, and 12 for throe types of graphite. Stored energy or the increase in enthalpy in irradiated graphite has been measured by two methods: (1) determination of specific heat by comparison in a differential calorimeter with unirradiated specimens and (2) accurate determination of heat of

Sec. 10-4]

RADIATION DAMAGE

10-111

TO SOLIDS

The first method gives data of the type shown in Fig. 13, which are combustion. typical of well-graphitized material. This method supplemented by the second gives the data shown in Fig. 14. Increasing the temperature of irradiation lowers the amount of energy stored at constant irradiation as shown in Fig. 15. Young's modulus varies with irradiation in the manner shown in Fig. 16. io I

1

1

1

1

.

1

.


1900 2000 MWD/T 4 2 6 8 10 14 12 COOLED TEST HOLE EXPOSURE NVT

Fig.

7.

16 IO"20

Length increase of several graphites in directions

parallel

to forming direction.

•O*

bj

MWD/T 0

2

4

6

8

COOLED HOLE

Kio.

8.

10

12

EXPOSURE

14

NVT

16

18

20

22

X I0"Z°

Length changes of several graphites in directions perpendicular

to forming direction.

The recovery of the above properties upon heating is illustrated by the data on dimensions plotted in F"ig. 17. Complete restoration of properties does not occur short of 1800 to 2000°C. Radiation recovery of all properties can also occur. Figure 1 8 shows the behavior of the dimensions of graphite specimens that may be taken as illustrative of other properties. The mechanisms concerned in the radiation damage of graphite and a model of damaged graphite involve the following points: (1) Carbon atoms are displaced predominantly one at a time by primary and secondary collisions; a few collisions may displace pairs of atoms. The fraction of displaced atoms per megawatt-day per ton Ls 3.5 to 7 X 10_J. (2) A number of displaced atoms remain relatively close to a

REACTOR MATERIALS

10-112

[Sec.

10


vacancy and are called close pairs. (3) Displaced atoms occupy interplanar positions and cause the extension of crystallites in the c direction. (4) Well-separated inter stitial-vacancy pairs accept electrons and serve as scatterers for electrical conduction. As electrons are trapped, holes are created. At first electrical conductivity is lowered,

EXPOSURE NVT

Fio.

X 10

9. Temperature variation of the physical expansion

of graphite during irradiation.

iaor

J L_J L 0123436 I

Fio.

10.

exposure.

1500

1000

MWD/T

I 7

I 8

I 9

L

I 10 II

12 13 14 COOLED TEST HOLE EXPOSURE NVT X 10"*°

Thermal resistivity of graphite parallel to the axis of extrusion as

a

function of

but finally the increase in carriers balances the increase in scattering. Close pairs do not become charged. (5) Interstitial atoms are immobile below 80°K, but above this temperature (a) close pairs may recombine and (6) charged interstitials may separate because of coulomb repulsion. Paramagnetic resonance, which is due to unpaired electrons on these interstitials, sharpens as a result of this separation. (6)

Sec. 10-4]

RADIATION

DAMAGE

10-113

TO SOLIDS

SêiZ.

150

^TSGBF

jS

125

A

10 75

/

<

5

50

25


0

500 0

4

2

COOLED

1500

1000

MWO/T 6

8

10

2000 12

14

.20

TEST HOLE EXPOSURE NVT X 10" transverse to the axis of extrusion

11. Thermal resistivity of graphite exposure.

Fio.

50

100

ISO

EXPOSURE TEMPERATURE

Fio.

12.

Effect of exposure temperature

as a function of

200

"C

on conductivity of graphite.

REACTOR MATERIALS


10-114

300

200

100

[Sec. 10

400

ANNEALING TEMPERATURE °C

Fig.

13.

Effect of annealing

temperature

on stored energy in graphite.

600

OS Fig.

14.

Stored

energy

1000

MWD/T

COOLED TEST HOLE remaining in graphite

2000

10

15

EXPOSURE after

NVT

annealing

20 X I0"20

as a function of exposure.

Sec. 10-4]

RADIATION DAMAGE

DO

200

MO

10-115

TO SOLIDS

400

500

600

ANNEALING TEMPERATURE °C

Flo.

15. Release of stored energy in graphite

by annealing.


Above room temperature single, charged interstitials may combine to form cules or ions releasing energy. (7) Hot

C2 mole

atoms also may combine with vacancies before loosing their recoil energy and ac quiring a charge. (8) Arrays of charged interstitials, Cj ions, and clumps of atoms bend the graphite planar layers and scat ter the lattice waves that conduct heat, thus reducing thermal conductivity pro gressively. (9) At higher temperature, reintegration and migration of interstitial Q M.WD/f , , , complexes into planes and to crystalline I 3 4 2 0 1 boundaries and discontinuities, respec COOLED TEST HOLE EXPOSURE NVT 1 10"20 tively, take place. (10) Only at tem Fro. 16. Effect of irradiation on Young's peratures above 1800°C do vacancies modulus of elasticity of graphite. diffuse to boundaries. Mechanical properties of beryllia do not change more Beryllium Oxide. 6.22 than 10 per cent after irradiation up to about 220 Mwd/ton.

2000

ANNEALING

Fig.

17.

TEMPERATURE, "C

Restoration or recovery of graphite dimensions

by annealing.

10-116

REACTOR MATERIALS

[Sec. 10

003

2 g

z

2

X

UJ


tf

EXPOSURE

Fio.

18.

(ARBITRARY

UNITS)

Recovery due to additional radiation at increased temperature. Table 19.

Thermal Resistivity

of Beryllium Oxide Transient Flow of Heat (Ratio of thermal resistivity after irradiation to that before)

Mwd/ton

Density, 2.7 g/cm'

54 109 219

2.9 g/cma

1. 19 1.54 1.51

1.33 1.46 1.62

Steady-state Flow of Heat Thermal resistivity, ("C) (em) /watt Temperature of irradiation, °C Unirradiated

Irradiated 54 Mwd/ton

0.82 0.65 0.59

0.61 0.51

200 300 350

0.47

conduclirity falls off slightly as shown in Table 19. of the change of thermal resistivity is complete after annealing at about 800°C for 1 hr. The thermal

Recovery

5.3

Reactor Fuels

Solid fuel materials for nuclear reactors may be discussed conveniently and char acteristically under the following headings: (1) metallic-uranium-base materials, (2)


Sec. 10-4]

10-117

dispersions of uranium or uranium compounds in ductile metallic matrices, (3) Materials on which information is available will be listed under ceramic materials. these headings. 6.31 Units of Dose or Exposure. The chief effects of radiation in fuel materials The are a function of the number of fissions, other variables being held constant. per cent of all atoms fissioned in the materials is called burnup, bu. It is related approximately to other units of exposure as follows: 1 % 6m

1

Mwd/Ct*

~

8.8

=k

an nvt of 3.44

X

10s

Mwd/ton (metric uranium)

X

10"

(BNL

graphite-moderated reactor)

Uranium-base Fuels — Unalloyed a Uranium. Dimensional changes often 5.32 At constant tem termed growth accompany burnup in single crystals of a uranium. perature the growth rate appears to obey the relation 1

dl

I d(bu)

0

Values of G are listed in Table 20. Table

20.

Radiation Growth of Alpha Uranium Crystals* Cryttlallographic direction

-420

[1001 [010] [001]


Growth coefficient G ± 20

+420 ± 20 0 ± 20

Temperature: I00°C approximately Range of applicability: 0 to ~0. I % bu in Uranium and Its Alloys, P/745, Proc. Intern, *s. H. Paine and J. H. Kittle, Irradiation Effects " Progress in Nuclear Energy," vol. 4, part 1. Pcrgamon Con/. m Peaceful lUee of Atomic Energy; also, Press, London (to be published).

Polycrysialline a uranium grows at rates that increase with preferred orientation Since (actually the detailed texture) and decrease slightly with increasing grain size. highly textured specimens grow as much as twice as fast as single crystals under identical conditions, the phenomena can not be merely growth of individual grains subject to the restraints imposed by adja cent grains. The dominant variable, tex ture, is a highly complex quantity. Figure 19 shows how the growth coefficient varies with the textures produced by rolling at PREVIOUSLY 200 ROLLED AT Textures produced by the above 300°C. 600°C AND A domi- G BETA QUENCHED rolling are discussed by Foote.35 100 nant component of the texture places the pole for the (010) plane, i.e., the [0101 direc tion, in the rolling direction. Recryslallization at approximately 600°C leaves the 1 (010) pole concentration about the same 15 but shifts the high (110) pole concentra 5 10 20 25 30 0 tions from the rolling direction and near PER CENT REDUCTION IN AREA 60° from the rolling direction to more ran Fio. 19. Effect on G of rolling beta The distribution of dom orientations. quenched alpha uranium at 300° C. These changes (100) poles is also shifted. phis the change in grain boundaries result in lower growth coefficients. Table 21 shows the effect of recrystallization on the growth coefficient as well. Thermal cycling textured uranium between, for example, Tu = 500°C and

// //

T, = 100°C also causes

growth in the direction in which the metal was worked (for example,

* Central metric ton.

10-118 Table 21.

REACTOR

MATERIALS

[SEC. 10

G for Round-rolled Uranium Rods Worked and Heat-treated Rolling temperature,

300 400 500 600 640

°C

Subsequent

heat treatment

As rolled 2 hr at 630°C As rolled 2 hr at 600°C As rolled 2 hr at 600°C As rolled 2 hr at 575°C As rolled 2 hr at 575°C

as Shown

a

475 204 321 163 4 6

-22

This growth varies with grain size and texture in ways analogous to those rolled). given above. Changes in length follow the relation


where n is the number of cycles and t refers to time ; Gt and G are both plotted in Fig. 20

for cold-worked and recrystallized uranium. The temperature variation of radiation growth G is known qualitatively, is approxi mately zero at 500°C,36 appears to be a maximum near 150 to 200°C,"" and has perhaps 20 per cent of the 100°C IRRADIATION 400 value at -100 to -120°C." Surface roughening occurs in coarse grained uranium upon either irradiation 300 or thermal cycling because of the relative displacement of adjacent grains. Mechanisms for radiation growth have ^200 been suggested along the following two lines: 1. Microscopic deformation3*"-** in and 100 around a thermal spike, the known laws for macroscopic deformation governing A and leading to flow as given above. 20 40 60 difficulty with the mechanism is that dis PER CENT REDUCTION IN AREA locations cannot move the requisite dis Fig. 20. Comparison of thermal cycling tance during the lifetime of a thermal spike, 10~u sec. and irradiation on growth of uranium. 2. Anisotropic diffusion of vacancies and interstitials produced by displacements, the interstitials moving predominantly be tween the somewhat loosely spaced corrugated planes parallel to (010) and aggregat ing to form platelets, thus causing growth in the [010] with concomitant diffusion of The nature vacancies predominantly in the [100] causing shrinkage in this direction. of coupling between grains to enhance growth in textured uncrystallized specimens is not known. Mechanisms for thermal cycling grmcth involve the following: The expansion coeffi 1. The differential thermal expansion of adjacent grains. cients in the three crystallographic directions are: Direction 1100]

[0I0| [010]

Coefficient of thermal expansion

36.7

-9.3

34.2

X

10*. 25-650°C


Sec. 10-4]

RADIATION

DAMAGE

10-119

TO SOLIDS

2. Flow and relaxation of stresses at grain boundaries in the temperature range above 350°C. 3. Yield by slip (assisted by twinning) under the stresses due to differential expan sion in the temperature range in which grain boundaries are rigid. 4. The existence of two major components of the texture in worked uranium. Other pairs of deformation mechanisms may take part,38 but their existence has not been demonstrated experimentally as has the above pair."-*' In the article on Density Changes. elementary processes, it was shown that changes in density should be about 2.5 to Figure 21 shows 3.5 times the burnup. the results of measurements that fall in It may be noted that the band shown. the minimum amount found corresponds reasonably well to the volume taken up 10 .02 .04 .06 .08 Some void forma by fission products. PER CENT ATOM BURNUP tion evidently occurs in some specimens Fig. 21. Decrease in density of un due (1) probably to cracks at grain with burnup. boundaries and occlusions formed as the crystals deform and strain one another, (2) to diffusion of vacancies into the stress fields caused by the large fission products. Data on agglomeration of fission gases and bubble formation are not available as yet. Mechanical properties of a uranium change drastically after small burnup. Meas urements are illustrated in Table 22.

Table

22.

Effect of Burnup on Some of the Mechanical Ultimate strength

Properties

Yield strength

of a

Uranium

% elongation

Young's Conditions Psi X 10""

Control*."

0.035 % burnup at 120°C. . Same irradiated, annealed 15 hr at 400°C in vacuo . Same irradiated, tension tested at 285°C Cast uranium18 high carbon: After * Standard tension 0 range.

% change

104 76

-27

65

-38

71

-32

78.2 29.8

-62

test specimens

Psi X

10""

% change

33

modulus, psi X I0"«

l-in. gauge

17

change

117

0.36

-97.9

52

58

0.54

-96.8

70

112

0.7

-95.9

71.5

of ^4-in. diameter prepared

25 28

12

from material quenched

from the

Konobeevsky, Provdyuk, and Kutaitsev40 report a 4 per cent Electrical Resistivity. increase in the resistivity of rolled foil at exposures to nvt = 10»° at 80°C. Activation energies for recovery are given in Table 23. Royal41 reports a 1 per cent change after 2 X 10~« per cent bu. Table 23.

Activation Energy for Recovery of Electrical

Resistivity

of

Irradiated Uranium Temperature, 171 209 218

°C

Activation

energy, kcal

45.3 52.5

62.7

Fuels — Alloyed a Uranium. A number of elements have Uranium-base been added to uranium in small cmantities up to approximately 4 atom per cent in order to modify its properties." 6.33

10-120

REACTOR MATERIALS

[Sec. 10

The most interesting systems are provided by those addition elements which show some solubility in the p and/or y phases but lower solubility in the a phase and hence The completely 6table room-temperature modify the transformations on cooling. state produced by annealing, aging, or tempering is a uranium containing a rejected phase. Only a few data are available on the properties of these alloys after irradiation. Uranium-Zirconium. As in the case of a uranium, dimensional changes depend on preferred orientation. Quantitative comparison with pure a uranium is not possible at the present time. If properly heat-treated, these alloys contain little or no pre ferred orientation and hence do not change dimensions to any great extent. Density changes correspond to the lower line of the band shown in Fig. 21. Thermal conduc tivity changes approximately 5 per cent at 0.1 per cent burnup. Uranium-Chromium. Dimensional and density changes in these alloys appear to be similar to the uranium-zirconium alloys in the range of a few hundredths of a per cent burnup. Uranium-Aluminum. Englander" reports that a 3 per cent uranium-aluminum alloy solution heat-treated around 1090°C, quenched to room temperature, and tempered at 480°C shows little or no di mensional changes under irradiation or thermal cycling. Uranium-base Fuels — Alloyed 0 5.34 4 Uranium. Chromium and molybdenum Fhave been used to make metastable fiGenerated for wjivans (University of Florida) on 2015-09-23 02:55 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

<

I 5

^*I5 O O

<

z

C fe

I 3 I 2

o

0

1 4 2 3 5 6(xlO-3) FRACTION OF TOTAL ATOMS FISSIONED

22. Change of electrical resistivity of aluminum-uranium alloys with burnup.

Fio.

l«

r 172* PRECISION 1 5 7. t^r 1 5 7 • >
JS

i okC.

0

1 1

4 2 3 5 6UK>-FRACTION OF TOTAL ATOMS FISSIONED

Fiq. 23. Change of thermal resistivity aluminum-uranium alloys with burnup.

of

Tucker and Senio* uranium alloys by quenching the /3 field to room temperature. have shown that the metastable alloy does not change to a uranium under irradiation. As may be seen from Ref. 41, 5.35 Uranium-base Fuels— Alloyed y Uranium. alloys containing in the neighborhood of 8 weight per cent molybdenum or niobium may be quenched from the 7 field to room temperature without transformation. On tempering or aging at 350°C or above in the a field, the y phase decomposes into a uranium and an e prime phase. Konobeevsky, Provdyuk, and Kutaitsev'0 show that such an alloy containing 9 per cent molybdenum is transformed to the y structure by an irradiation of the order of 1019 nvt at 50°C. 6.36 Dispersions of Uranium and Uranium Compounds in Ductile Metallic Matrices. If fissile materials are dispersed as discrete particles throughout a metallic matrix, damage by fission recoils and atoms, may be confined to the region of the The damage to the matrix then becomes primarily that due to fast neutrons particle. discussed in foregoing articles. Aluminum-Uranium Alloys. Research reactors utilize alloys containing approxi mately 2 atomic per cent uranium in the form of intermotallic compound UAI4 dis Measurements of the properties of several alloys persed in commercial aluminum. containing 5 to 17 weight per cent uranium have been determined by Siegel and Billington" and are shown in Figs. 22 to 25. Density changes in uranium-aluminum •Page 11 of Ref. 41.

Sec. 10-4]

RADIATION DAMAGE

10-121

TO SOLIDS

alloys are approximately 2}$ per cent at 1 per cent burnup and within experimental error are linear with burnup. Bodies consisting of aluminum in which particles of Aluminurn-\JOi Mixtures."'** alloy UO2 are dispersed behave in a fashion analogous to the aluminum-uranium mentioned above. Effects on the aluminum itself decrease with increasing particle size of the UOi. Beryllium-Uranium Beryllium containing 0.5, 1.0, and 3 per cent Alloys.10 uranium dispersed throughout the beryllium as UBeu showed changes in electrical resistivity from 1.5 to 9 per cent after 0.06 per cent burnup, the change being approximately linear in uranium concen tration. Changes in dimension and den sity were not outside experimental error of measurement. Zirconium-Uranium Alloys. Dilute al100 .

E

eo

*

15


— — 0

I

Z

3

4

6(xl0_i)

5

' I 0'

FRACTION OF TOTAL ATOMS FISSIONED

Flo.

of hardness of aluminumalloys with burnup.

24. Change

uranium

ELECTRICAL RESISTANCE RATIO HARDNESS

100 200 300 400

20

£

500 600

ANNEALING TEMPERATURE

(°C)

Fio. 25. Effect of annealing on electrical resistivity and hardness of irradiated aluminum-uranium alloys.

loys of uranium in zirconium consist of the two phases nearly pure zirconium in which is dispersed nearly pure uranium. In the range of a few hundredths of a per cent burnup, little or no change occurs in physical properties, and density changes proceed linearly with burnup at approximately the theoretical rate. Sleel-XjQt Mixtures. Weber44 gives the following changes in hardness dispersions of UO2 in austenitic stainless steel:

UOi

irradiation, DPH

Hardness after irradiation, DPH

Particle siie UO,,

24 24

225-250 175-260

475-500 330-440

3 15-44

Volume

6.37

Ceramic

%

Hardness

Materials.

before

Beryllium

Oxide-Uranium

Oxide.

After

burnup

of

5 X 10-4 per cent, the following changes were noted in beryllia bodies containing 2 per cent U()2 mixed with BcO, density 2.4 to 2.8 g/cma, UOi particle size approxi

mately

+1

cent.

2fi\ (1) thermal resistivity increases by a factor of 5 to 6; (2) linear dimensions, per cent; (3) elastic modulus, —30 per cent; (4) compressive strength, —30 per

The broadening of X-ray diffraction lines from U3Os41 shows that Uranium Oxide. fragmentation of the crystal occurs at a burnup of approximately 3 X 10~* per cent.

UOs

is relatively unaffected. Test bodies have been made by (1) impregnating Cfraphite-Uranium Oxide. graphite with a uranium salt and baking to give UiOs, (2) mixing UOj with graphite and a resin and baking at 1370 to 1400°C. The results of irradiation are shown in Tables 24 and 25.

10-122

REACTOR MATERIALS

[Sec.

10

Effect of Irradiation on the Properties Table 24. of Impregnated Graphite-U308 Bodies (Hanford Irradiation Facility)

Fractional change in property Mg of U'»/g of graphite

0 12 0

II

0 115

Dose,

Mwd/t*

27.3 27.3 56.6 56.6 186.3 186.3

Diameter

Length

+ 0.021

-0.007

+ 0.040 + 0.003 + 0.080

-0.025 -0.0008 -0.045

Electrical resistivity

+ 2.93 + 2.79 + 3.14 + 2.83 + 4.07 + 3.21

Temperature of unimpregnated < temperature impregnated < 100°C. AGOT-KC graphite cut parallel to direction of extrusion. * Equivalent to nvt = 8.2 1 X 10IB thermal neutrons/cm".

Table 25.

Effect of Irradiation on the Properties Dosage;

1.85 ± 0.05 X

of

Molded Graphite-UO* Bodies

1018neutrons/cm1

Temperature: 66-88°C


Fractional change in property

UOi particle

Type sample

size

NoO NoU

Transverse cut AGOT

586 334 94 486 334 94

Young's modulus

+ + + +

0.44 0.37 0.34 0.38

+ 0.41 + 0.35 + 0.33 + 0.30

Electrical resistivity

+ 0.39

+ 0.28 + 0.30 + 0.32 + 0.34 + 0.21 + 0.21

+ 0.28


-0.22 -0. II -0.8

-0. -0.

14 10

-0. -0.

13 13

-0.09

Annealing of specimens used to obtain data for Table 25 was done for as long as 22 hr at 800°C with little change. 5.4

Semiconductors46

These materials are of special importance because of their increasing use in solidstate electronic devices, as rectifiers and amplifiers. The electronic properties of semiconductors are dependent on the concentration of carriers, which arise from the presence of impurities or lattice defects. Radiation with massive particles thus produces very large and very rapid changes in the electrical properties of this class of materials. In n-type germanium the conductivity decreases from 0.27 to 0.004 (ohm-cm)-1 after 1.5 X 1016/cm2 of fast neutrons. The conductivity, after passing this minimum value equivalent to the intrinsic conductivity of the semiconductor, then increases with further irradiation, reaching 0.03 (ohm-cm)"1 at 3 X lO'Vcm*. In p-type germanium the conductivity increases monotonically, changing from 0.04 to 0.06 (ohm-cm)-' after 3 X lO^/cm5 fast neutrons. The behavior of germanium is quite complex, depending on the exposure tempera ture and the initial state of the material. In silicon, the conductivity decreases with irradiation in both n- and p-type material. The rate of decrease is much greater in p type than in n type, however.

Sec. 10-4]

radiation

damage

to solids

10-123

Data for p- and n-typc samples appear below: Electrical Resistance of Silicon Specimen Exposure. . . . P type, ohma

n type, ohms

0 30 23

3 X 10" 50 25

1.5 X 190 31

10"

3 X 10" 16.000 32

4.5 X 10" 68.000 37

These data are typical, but since the properties of semiconductors depend so mark edly on their initial state of purity and heat treatment, their applicability to particular For this reason, practical devices such as diodes and tran samples is questionable. sistors must be individually investigated at the present state of knowledge. 5.6

Ionic Crystals


In these materials, of which the alkali halides are most typical, the principal changes that have been investigated are the optical properties and the ionic conductivity. The increase in optical absorption is produced by both 7-ray and particle radiation. The data on this phenomenon are very extensive and have been reviewed by Seitz." These optical effects are electronic in origin; electrons are excited by the incident radiation and are trapped near lattice defects that already exist in the crystal. Such trapped electrons or combinations of them have characteristic absorption spectra and annealing or bleaching behavior. The data on ionic conductivity changes are meager and ambiguous. The con ductivity of KC1 was lower subsequent to irradiations with y rays and short exposures in a reactor."'4' The conductivity of NaCl was similarly decreased by a factor of 30, subsequent to exposure to 1016/cm' of 350- Mev protons at 50°C. Prolonged exposure in a reactor, however, produces an increase in conductivity in these materials by a factor of 10. 6.6

Covalent Crystals

Quartz, zircon, and diamond are typical examples of covalent crystals and have been investigated extensively.47 The most striking effect in these crystals is the decrease in density produced by radiation. At an exposure of 2 X 10,0/cm» of fast neutrons in a reactor, the density of quartz At about 5 X W/cm2, the material is no longer crystalline, decreases by 15 per cent. In spite of these large changes in hence is optically isotropic and nonpiczoelectric. density, macroscopic defects, e.g., cracks or bubbles, were not observed. On annealing at 950°C, the sample reverts from the amorphous to a polycrystalline a quartz structure; in addition, macroscopic channels about 0.05 mm in diameter, 5 mm long, are formed. In the case of zircon, exposure to 3 X 10so/cm1 produces a density decrease of In these crystals long-range structural order is preserved, but mosaic 4.2 per cent. crystalline blocks are created. The annealing of those changes barely begins at 1000°C and is incomplete even at 1600°C. In the case of diamond, the lattice expansion observed is tabulated below: sure, cm* fast neutrons 5 X 10" 10" 1.5 X 10"

IX

2.0 X 10" 4 X 10" 6 X 10"

Density decrease 1.5

2.5 3.0 3.2 3.7 3.7

A high degree of long-range order persists in these heavily exposed crystals. X-ray The crystals become diffraction patterns continue to indicate a well-ordered lattice. opaque after lO'Vcm2, and annealing at 1600°C, which produces a 70 per cent recovery in the density decrease, docs not appreciably affect this opacity.

10-124

REACTOR MATERIALS 5.7

[Sec. 10

Glasses48

The principal effect of radiation on glass is the increase in optical absorption. This is of practical importance for windows and optical devices. The darkening is clue to the same phenomenon observed in ionic crystals; the radi ation excites electrons, which are trapped in vacancies and thus have typical absorp Special glasses containing cerium have been developed that are radia tion spectra. The cerium ion serves as an electron trap, preventing the trapping in tion resistant. vacancies and the format ion of F centers. Typical data on darkening of glass protected by the addition of cerium and unpro tected glass, having an index of refraction 1.517 and a dispersion of 64.5, are as follows: Per Cent Transmission Unprotected

Pro tec ted Exposure, r


X

-

450 mn 500 550 600 700

0

I0«

4.7 X 10"

0

89 91 91 91 92

85 88 89 89 91

58 68 71 74

91 91 92 92 92

10'

3.4 15 31 44 68

Acknowledgments Sincere thanks are due J. Brinkman for indispensable suggestions on methods of displaying the fundamental processes of radiation effects, J. Douic for collecting information on methods of examining radioactive materials, and J. Bastion for check ing manuscript and proof.

REFERENCES Gilbert, F. W.: Nucleonics, 10: 6 (1952). Richardson, D. M.: " Calorimetric Measurement of Radiation Energy Dissipated by Various Materials Placed in the Oak Ridge Pile," OHNL-129, Oct. 1, 1949. 3. Proctor, B. E., and S. A. Goldblith: "Dosimetry Research of the Cobalt-60 Source," AEC-NYO-3339, Sept. 1, 1952. D. V. Cormack, R. W. Hummel, H. E. Johns, and J. W. T. Spinks: Irradiation of Ferrous Ammonium Sulfate Solutions: Energy Absorp tion and Ionization Calculations for Cobalt-60 and Betatron Radiation, J. Chem. Phys., 22 : 6-12 (1954). C. S. Hochanadel and J. A. Ghormley : A Calorimetric Calibration of Gamma Ray Actinometers, J. Chem. Phya., 21 : 880-885 (1953). 4. Pearsall, C. S., and P. K. Zingeser: "Metal to Nonmetallic Brazing," NP-841. Apr. 5, 1.

2.

1949.

Yockey, H. P., et ol.: Cyclotron Techniques for Radiation Damage Studies, Rer. Set. Instr.. 24 (10): 1011-1019 (October, 1954). 6. Snyder, W. L., and J. Neufeld: Phys. Rev.. 97: 1636 (1955). 7. Seitz, F., and J. S. Koehler: The Theory of Lattice Displacements Produced during Irradiation, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, August, 1955. " 8. Brinkman, J. A. : The Production of Atomic Displacements by High Energy Particles," NAA-SR-1459, September, 1955; On the Nature of Radiation Damage in Metals J. Appl. Phys., 26: 961-970 (1954). 9. Lomer, W. M., and A. H. Cottrell: Annealing of Point Defects in Metals and Alloys, Phil. Mag., 46: 711-719 (1955). 10. Cooper, H. G., J. S. Koehler, and J. W. Marx: Irradiation Effects in Cu, Ag, and Au Near 10°K, Phys. Rev., 97: 599 (1955). 11. Dienes, G. J.: Theoretical Aspects of Radiation Damage in Metals, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/750, August, 1955. 12. Kinchin, G. H., and R. S. Pease: /. Nuclear Energy, 1: 200 (1955). 13. Brooks, Harvey: TID 10633, The Thermal Spike Picture of Radiation Damage, Knolls Atomic Power Laboratory Report. 5.

Sec. 10-4] 14.

Seitz, Seitz,

RADIATION DAMAGE

TO SOLIDS

10-125

F.: Revs. Mod. Phys., 18: 384 (1946). F.: Color Centers in Alkali Halide Crystals,

Revs. Mod. Phys., 26: 7 (1954). 16. Kittle, Charles: "Introduction to Solid State Physics," pp. 319-320, John Wiley & Sons, Inc., New York, 1953. 17. Varley, J. II. O.: A New Interpretation of Irradiation Induced Phenomena in Alkali Halides, /. Nuclear Energy, 1: 130-143 (1954). 18. Sutton, C. R., and D. O. Lesser: Iron Age, 174: 97, 128 (1954). 19. Andrew, A., and C. R. Davidson: Induced Thermoelectric Potential from Radiation Damage, Phys. Rev., 89 (4): 87(5-877 (1953). 20. Siegcl, S.: Phys. Rev., 76: 1823 (1949). 21. Blewitt. T. H., and R. R. Coltmann: Acta Met., 2: 549 (1954). Appl. Phys., 36 : 344 (1954). 22. Aronin, L. R.: Appl. Phys., 24: 229 (1953). 23. Bowman, F. E., and R. R. Eggleston: 24. Billington, D. S.: Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/744, August, 15.

J.

J.

1955.

25. 26. 27.

28.


29. 30. 31.

Murray, G. T., and W. E. Taylor: Acta Met., 2 : 52 (1954). J. W. Cleland el al.: Phys. Rev., 91: 238 (1953). Reynolds, M. B., el al.: J. Metals, 7: 555 (1955). Martin, A. B., R. D. Johnson, and F. Asaro: Diffusion of Gold into Copper, Appl. Phys., 26 (3): 364-369 (1954). P. Duwez and R. D. Johnson: "The Effect of Irradia tion on Diffusion in Copper-Gold and Copper-Nickel Powder Techniques," NAA-SR168, March, 1952. Martin, A. B., and R. D. Johnson: The Effect of Cyclotron Bombardment on Selfdiffusion in Silver, J. Appl. Phys., 23 (11): 1245-1254 (1952). Hennig, G., and J. E. Hove: "Interpretation of Radiation Damage to Graphite," Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/751, August, 1955. Woods, W. K., L. P. Bupp, and J. F. Fletcher: "Irradiation Damage to Artificial Graphite," Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/746, August, 1955. Currie, L. M., V. C. Hamister, and H. G. McPherson: "The Production and Properties of Graphite for Reactors," Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/534,

J.

1955.

Eatherly, W. Paine, S. H., Intern. Conf. 34. "Progress in

32. 33.

P.: Bull. Am. Phys. Soc, 30: 8 (1955). T. J. Newbert, ibid. and J. H. Kittel: Irradiation Effects in Uranium and Its Alloys, Proc. on Peaceful Uses of Atomic Energy, P/745, August, 1955. Nuclear Energy," vol. 1, part 1, Pergamon Press, London (to be pub

lished). 35. Foote, F. G.: Physical Metallurgy of Uranium, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/555, August, 1955. 36. Pugh, S. F.: Damage Occurring in Uranium during Burnup, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/443, August, 1955. 37. a. TID 7502, "Growth of Irradiated Uranium," pp. 564-571. b. AERE M/R 2004, The Effects of Radiation of Some Enriched Fuels. 38. Chiswick, H. H., and L. R. Kelman: Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, August, 1955. Metals, 4: 651-656 (1952). 39. Burke. J. E., and A. M. Turkalo: 40. Konobeevsky, S. T., N. F. Provdyuk, and V. I. Kutaitsev: Effects of Irradiation on Structure and Properties of Fissionable Materials, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, August, 1955. 41. Billington, D. S.: Radiation Damage in Reactor Materials, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, August, 1955. 42. Englandcr: Private communication August, 1955. 43. Cunningham, J. E., and E. J. Boyle: MTR Type Fuel Elements, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/953, August, 1955. 44. Weber, C. E.: Dispersion Type Fuel Elements, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/561, August, 1955. J. W. Cleland 45. "Semiconducting Materials," Academic Press, Inc., New York, 1951. and J. II. Crawford: Phys. Rev., 95: 1177 (1954). 46. Nelson, C. M., et al.: Phys. Rev., 90: 364 (1953). 47. Crawford, J. H., and M. C. Wittcls: Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/753, August, 1955. 48. Kreidl, N. J., and J. R. Hensler: NYO-3780, Bausch and Lomb Optical Company, Rochester, N.Y., 1954.

J.

10-5

RADIATION DAMAGE TO LIQUIDS AND ORGANIC MATERIALS BY Vincent P. Calkins

Radiation damage can

be defined as any adverse change occurring in the physical or properties of a material or system as a result of exposure to radiation. "Radiation damage" is a relative term: What is damage to a material or system as far as one person is concerned could constitute an improvement in properties and be very beneficial from the viewpoint of a second person. For example, the evolution of a gaseous hydrocarbon from an organic fluid being irradiated could represent an explo sive hazard, an unwanted change in the viscosity of the liquid, or a new and very desirable method for synthesizing this hydrocarbon. However, changes in the properties of a material or system associated with a power plant are viewed with suspicion and are considered generally to be possible sources of damage. Experimental radiation-damage data on the following general classes of material are compiled and evaluated: water, aqueous solutions, fused salts, organic fluids, and These data are then used as a organic solids in the form of elastomers and plastics. basis for evaluating radiation damage to various electrical, electronic, and mechanical systems as a function of the specific materials these systems employ.


chemical

1

MECHANISM AND NATURE OF RADIATION DAMAGE

Although the fundamentals of the ionizing properties of radiation and the mecha nism of radiation absorption are discussed elsewhere, a summary statement of the effects of various types of radiation on liquids and organic materials is pertinent at this point. 1.1

Fast Neutrons

Fast neutrons react with the nuclei of the atoms of an irradiated medium, and an appreciable fraction of the energy of the neutron is transferred to each struck nucleus. The nucleus is ejected as a recoil ion, the neutron being scattered or reflected to strike other nuclei until it is degraded to a thermal-energy state. The ejected nuclei in the form of recoil ions traverse the medium and interact with orbital electrons, thus Eventually, the recoil ions become producing molecular excitation and ionization. neutralized, although they still possess energies considerably above the thermal level. As fast-moving neutral particles, they are thermalized in two ways: (1) by still losing an important part of their remaining energy by causing electronic excitation and (2) by losing the other part of this energy by undergoing elastic collisions with other atoms of the medium, thereby suffering losses in momentum. About 95 to 98 per cent of the energy of a fast neutron is transferred to a medium via electronic excitation and ionization, and only 2 to 5 per cent by loss in momentum. Thus, since organic compounds and covalent materials are not already ionized and are usually highly susceptible to electronic excitation and ionization, the main bulk (95 to 98 per cent) of the deposited energy of the fast neutrons is instrumental in The causing considerable damage to these materials by excitation and ionization. 2 to 5 per cent energy of the fast neutrons deposited by momentum losses also causes damage in these nonionic materials by dislocation of atoms; this is commonly called 10-126

Sec. 10-5]

RADIATION DAMAGE

10-127

TO LIQUIDS

Conversely, since ionic materials such as many inorganic Wigner-type damage.* compounds are already ionized and since metals can be considered as being in an ionized state, these materials are not very susceptible to further electronic excitation and ionization, and the bulk of the energy deposited by fast neutrons in these mate rials, therefore, has little effect. t Some YVigncr-type damage does occur, but since this damage can be caused by only a very small fraction of the total fast-neutron energy deposited, only slight damage to metals and ionic materials frequently occurs at fast-neutron dosages one thousand-fold and more greater than those dosages at which considerable damage occurs to nonionic materials such as organics. Gamma Radiation

pair

(e+,e~).

tion

is

In

y

a

is

y

jS,

Gamma radiation reacts with matter primarily by the photoelectric effect, Compton effect, and pair production — depending upon the atomic number of the constituents of the irradiated medium and upon the energy of the attendant y radiation. For materials containing elements of low atomic number (27 or less) such as hydrogen, carbon, or aluminum, the photoelectric effect predominates for y energies below 0.1 Mev, the Compton effect for energies from 0.1 to 10 Mev, and pair production For materials containing elements of high atomic number (82 or above 10 Mev. greater) such as lead, the photoelectric effect predominates for y energies below 0.5 Mev, the Compton effect from 1.0 to 4.0 Mev, and pair production above 4 Mev. In the photoelectric effect, the y ray interacts with the entire atom and transfers all the energy of the photon in one encounter. This energy is received by a single atomic electron, which is usually in the K or L shell, and this electron is ejected from the atom with an energy of E — he — the energy of the incident photon minus the binding energy of the electron in the atom. In contrast to the photoelectric effect, the 7-ray photon in the Compton effect collides elast ically with a single electron, the incident photon giving up a portion of its energy to the electron, which recoils and In accordance with flies off at an angle to the direction of the incident photon. in the current theory, pair production caused by the complete absorption of region of the nuclear coulomb field with the resulting formation of a positron-electron

y

brief, the net result of the photoelectric effect, Compton effect, and pair produc to produce either an ejected electron or an electron pair that traverses the medium, interacts with orbital electrons, and causes damage by electron excitation and ionization. Since the subsequent loss in charge of the ejected electrons and electron pairs does not result in a heavy neutralized recoil ion, no transfer of energy by radiation will cause very little loss in momentum occurs. Therefore, in general, damage to ionic compounds but can cause considerable damage to nonionic or covalent compounds. 1.3

Thermal Neutrons

Thermal neutrons

a

is

y

is

is

are captured by nuclei, and damage to the surrounding material due primarily to the secondary radiation evolved. The main damage to organic materials by slow or thermal neutrons due to ionization of the organic media by the 2.2-Mev rays that are given off when the neutrons are captured by hydrogen atoms. With chloro-organics such as neoprene, the n-/3 reaction also an important factor For organics due to the high thermal-neutron absorption cross section of chlorine. major portion of the damage produced by possessing a high nitrogen content,

is

*

"Covalent compounds — which include the common gases and liquids and organic materials — con formed from a group of individual atoms held together by shared sist of molecules each of which The molecules themselves are bound together electron bonds which give rise to strong exchange forces. by relatively weak van der Waals forces, i.e., by the attraction due to electric fields." Ionic compounds — which include salts and oxides — consist of a combination of highly electro positive and electronegative elements in the forms of ions held in a crystal lattice. The lattice of ions no actual union is held together by electrostatic forces in accordance with Coulomb's law, and there in the crystal to give molecules, although all crystals can be considered as consisting of huge molecules limited only by the capacity of the crystal for growth. the sixe of which crystal lattice with the ions being Metals can be considered as consisting of positive ions in a

is

is

t


1.2

immersed

in an electron

gas.

REACTOR MATERIALS

10-128

[SEC. 10

thermal neutrons may be due to the N14(n,p)C14 reaction, the damage being caused For organic or other covalent of the media by the ejected proton. materials containing lithium or boron, the primary damage should result from dense ionization due to the a particles given off by capture of thermal neutrons by either lithium or boron. For ionic compounds and metals, the extent of the n-y reaction in any material will depend upon the capture cross section of the constituents of the material, but the ionization produced by the generated 7s would cause very little damage in this type of material. For those metals and ionic compounds containing lithium and boron, very little damage would be caused by ionization by the generated a particles, but once the a particles have lost their charge, some Wigner-type damage would be caused The degree of such damage would by the neutralized but energetic a particles.* depend upon the lithium or boron content and the degree of orientation or order in the crystal lattice of the matrix material, but generally, the degree of such damage would be small even at considerably high thermal-neutron dosages.

by ionization


1.4

Fission Products

Fission products are generated as ions and are usually contained by the fuel matrix Since the fuel matrix is ordinarily ionic in nature, damage by of a nuclear reactor. fission ions would not be caused by ionization but by displacements or dislocations of the matrix atoms by collision with the neutralized fission ions. Exposure of covalent or organic materials to fission products would result in considerable damage caused by direct electronic excitation and ionization of the organic media by the fission ions. Also, a build-up of fission-product impurities in the fuel matrix due to prolonged fissioning could eventually affect the physical and mechanical properties of the fuel system by ordinary metallurgical contamination. 1.5

Other Particles

Beta radiation consists of electrons emitted from nuclei, and damage is 1.61 caused by the same general mechanism as the recoil (Compton) electrons produced by 7 radiation. Protons, deuterons, and a particles are often produced for irradiation testing 1.62 purposes by the use of particle accelerators such as the cyclotron or Van de Graaff. Protons cause damage in organic and ionic materials by the same mechanism as fast neutrons and also thermal neutrons in certain reactions such as the Nu(n,p)C" Deuterons, being essentially heavy protons, cause damage by a similar reaction. The highly energetic a particles produced in the cyclotron cause damage mechanism. by the same mechanism as that for thermal neutrons involving neutron a reactions. 1.63 High-energy electrons and X rays are readily produced by the betatron and the Van de Graaff, and the mechanism of damage is similar to that for jS and 7 radiation. 1.6

Damage as a Function of Energy Absorption

The amount of radiation damage suffered by any material as a result of exposure to radiation should be some function of the actual energy absorbed or deposited per unit * As in the case of the recoil ions ejected as a result of the collisions of fast neutrons with the nuclei of atoms of the irradiated material, or particles and fission ions arc positively charged and lose most of their energy by interaction with the orbital electrons of the atoms of a nonmetallic medium or by interaction with the so-called free electrons of a metallic medium, thereby causing excitation and ionization. As positively charged ions, these particles do not cause many primary displacements of the atomB in the lattice of an irradiated medium by collision with the nuclei of these atoms because of the coulombic repulsion forces between the charged ion and the charged atomic nuclei. As these ionic particles approach the end of their trajectory, they lose their charge and become neutralized, although they still possess energies considerably above the thermal level of the medium. Therefore, as fast-moving neutralized particles they are not subject to coulombic repulsion forces, and even though in this Btate they still lose an important portion of their energy by causing electronic excitation and ionization of the medium, they can now cause and do cause actual displacement of the atoms in the lattices of the irradiated material by billiard-hall-type collisions.

RADIATION DAMAGE

Sec. 10-5]

TO LIQUIDS

10-129


Since y rays, X rays, electrons, ft rays, and slow neutrons in the mass of material.*-1 case of the n-7 reaction produce damage by the same general mechanism, equivalent Fast damage to the materials might be anticipated for the same energy deposition. neutrons, protons, a rays, fission products, and slow neutrons in both the n-p and the n-a reactions produce damage by other mechanisms, and hence a different damage vs. energy relationship might be expected. The difference in damage to water by X rays and electrons and to water by a rays

A considerably higher concentration of hydro has received considerable attention.1 gen and hydrogen peroxide has been formed with a rays than with X rays or electrons. Many different mechanisms of reactions have been postulated to account for this It has also been found (see Table 1) that, on an equivalent energy phenomenon. basis, irradiation of borated water by a pile flux of fast neutrons, slow neutrons, and y rays produced considerably more hydrogen than irradiation of nonborated water by the same pile flux. Despite the difference in results obtained by irradiating water either directly or indirectly with a rays from the results obtained by irradiation of water with X rays and electrons, an analytical survey of a large variety of experimental data reveals that y rays, electrons, and combined pile radiation on an equivalent energy basis do produce equivalent damage to materials such as organics, even though damage is produced by different mechanisms. Certainly, for covalent materials such as organics, the generalization that equivalent damage is produced for equivalent energy absorption is a good rule of thumb, and for data examined to date for these materials, this rule has been found to be good within For those cases where pile fluxes are most accurately known, this rule a factor of 2. has been found to apply even closer than within a factor of 2. Recent unpublished and preliminary results on the effects of radiation on organic materials containing boron indicate that the gross damage occurring for y irradiation and for pile irradiation is equivalent — within a factor of 2 — for equivalent energy absorption, although a more quantitative nnalyssis of the products of each irradiation seems to indicate that there is a difference in their composition. 2

CALCULATION OF RADIATION ENERGY ABSORPTION

To relate radiation damage to energy deposition or absorption, a definition of an This need has been satisfied by the absorbed dose of ionizing radiation is needed. Commission on Radiological Units at the recommendation of the International Seventh International Congress of Radiology, Copenhagen, Denmark, July, 1953.' This Commission recommended that the absorbed dose of any ionizing radiation be defined as the amount of energy imparted to matter by ionizing particles per unit mass of irradiated material at the place of interest, and this dose should be expressed in rads. The rad is the unit of an absorbed dose and is equivalent to 100 ergs per gram of material. Since it is frequently easier to measure the absorption of ionizing energy in air rather than in the material actually being irradiated, a general relationship between Therefore, although it is not strictly valid, roentgen and rad is very convenient. 1 rad can be considered as being roughly equivalent to 1.2 roentgens. Accurate calculations of the 95 to 98 per cent energy transferred to a medium from radiative particles through the mechanism of electron excitation and ionization can be made only with considerable difficulty, but approximately, calculations of this energy transfer can usually be readily made. It is nearly impossible, even with involved mathematical equations, to make either very accurate or approximate calculations of the 2 to 5 per cent energy transferred from neutralized heavy particles to a medium through the mechanism of loss in momentum. Since covalent materials such as the organics are highly susceptible to molecular electron excitation and ionization, and since radiative particles lose most of their energy in this manner, methods for approximate calculations of this type of absorbing energy for these Listed below are methods for making materials are of considerable importance. * Superscript numbers

refer to References


10-130

REACTOR MATERIALS

[SEC. 10

approximate calculations of the energy absorbed by a material from each of the principal components of a normal pile radiation, namely, fast neutrons, >'s, and thermal neutrons. Absorption Energy from Fast Neutrons

2.1

The amount of energy E, absorbed per gram of an organic material from a fastneutron dosage
(j>n

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from a

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115

Absorption Energy from y

2.2

b.o.oi-i.i

I

jjf».»i-i.o = ^.o.oi-i.c.o.m-i.o

where

Radiation

y

photon dosage

Eay =

can be approximated

by the

yllyEy is

is

it

E

a

2

is

y

y

y

is

y

is

where Ea~< the energy absorbed per gram of material from radiation and Ey the The term n-, the energy absorption coefficient average energy of the photons. radiation of energy ,. and (in square centimeters per gram) of the material for the fraction of energy dissipated by a narrow beam of rays in traversing an absorber. It obtained by multiplying the probability of each interaction process by the probable fraction of the photon energy actually dissipated in the absorber as a result of the process. The energy -absorption coefficients of various materials are given as ii*/p in Table In many cases where the py value for particular material being con of Sec. 7-3. sidered cannot be found, the ny value for an analogous material can be readily sub stituted. The value, for water can generally be used for ordinary organic materials. 2.3

Absorption Energy from Thermal Neutrons is

the main source of energy deposition in organics For the H(n,y)D reaction, which by thermal neutrons, the energy absorbed per gram can be crudely approximated by the following equation: *

The deposition of enerKy in an ordinary organic compound by fast neutrons and thermal neutrons due primarily to its hydrogen content, since hydrogen has a much larger scattering and absorption cross section than carbon.

is


a

is

is

is

is

is

J.£

where p is the density of material, the fraction of the initial energy of the neutron transferred to the hydrogen* atom per collision, and 2. the macroscopic scattering cross section of hydrogen in the material being considered; 2, equal to a.Na where a, the microscopic scattering cross section for hydrogen, and Nh equal to the num ber of hydrogen atoms per cubic centimeter of material. To determine more exactly, but still not rigorously, the total energy absorbed from fast-neutron dosage, the energies absorbed from neutrons over each small incremental energy range would have to be calculated, using the dose and scattering cross section corresponding to each energy range. These energies would then need to be summed to obtain the total energy Ealot*1. On the basis that 0.01-Mev neutrons are the lowest energetic neutrons above thermal that contribute any appreciable energy to the medium through the scattering process, the total energy absorbed would be

radiation

Sec. 10-5]

damage

to liquids

10-131

PHy

Ey

= 0„ P

n

is the thermal-neutron dose, p is the density of the material, Ey is the average where energy of the photons generated by the n-y reaction, and Zo is the macroscopic absorption cross section of hydrogen in the material being considered; S« is equal to cqjVh, where a, is the microscopic absorption cross section for hydrogen and JVh is the number of hydrogen atoms per cubic centimeter.


2.4

Average Energy in Breaking a Bond

The average energy required to break a chemical bond is approximately 25 ev, and the average energy difference between successive electronic levels is about 5 ev. It is obvious that the Mev energies possessed by ordinary pile fast neutrons and y photons constitute large sources of energy that can be transferred via secondary processes and Although cause extensive elect ronic excitation and ionization in a chemical compound. the average energy of a thermal neutron at normal temperatures is 0.025 ev and is insufficient to cause electronic excitation or ionization, the capture of these low-energy neutrons by matrix nuclei results in secondary radiation from n->, n-y, and n-a reactions which involves energy in the Mev range. Two different methods have been frequently used in the past to relate the rate of In reaction of liquids and gases as a function of the irradiation energy involved. gases, the ion-pair yield has been used as a reaction rate measure, the energy required to produce an ion pair varying in different gases from approximately 28 to 38 ev. This energy range is approximately 18 to 22 ev greater than the ionization potential range of the gases, and this difference represents the energy necessary for the formation of electronically excited states. In liquids, since it is difficult to measure the number of ions produced by radiation, it has been frequently customary to determine the rate of reaction by the amount of The term G has been used for this pur material reacting for a given energy input. pose, and it is defined as the number of molecules reacting per 100 ev of energy For general covalent materials, it is useful to remember that the normal absorbed. value of G is approximately 4. In this article, the changes in physical and chemical properties of the various materials listed have been related to the amount of energy absorbed per gram weight of irradiated material. 3 3.1

CALCULATION OF RADIATION DAMAGE

Radiation Damage to Water, Aqueous Solutions, and Fused Salts

Excellent reviews of the effects of radiation on water and aqueous solutions and many These reviews have appeared under the general other materials have been written. title of Radiation Chemistry in the "Annual Reviews of Physical Chemistry" from 1950 through 1954 4 and in the same "Annual Review" for 1955 under the title of These reviews are concerned with the funda Radiation and Hot Atom Chemistry. mental aspects of the effects of radiation on various materials. The general mechanism of the effects of radiation on water has received rather It is generally agreed that the decomposition exhaustive treatment in these reviews. of pure water proceeds to a limited extent and then comes to a standstill because of the The main initial decomposition reactions of irradiation are back reaction.

H,0

-» H + OH H2 + H,02

2H,0 —

Although the H«Oj formed decomposes to oxygen in

(1) (2) a secondary reaction, the back

10-132

REACTOR MATERIALS

[Sec.

reaction involving the combination of the reaction products to re-form water can formulated as follows: H + H„02 -» H20 + OH OH + H2 -> H2Q + H


Net reaction: H2 + H202-> 2H20

10 be

(3) (4) (5)

The decomposition of irradiated aqueous solutions proceeds according to Eqs. (1) and (2), but the presence of solutes protects the initial decomposition products by Thus, hydrogen can destroying the free radicals by oxidation-reduction reactions. evolve and the hydrogen peroxide can decompose. Since solute molecules vary con siderably in specificity of reaction, appreciable differences in results can be obtained when various aqueous solutions are irradiated. The data in Table 1 are concerned primarily with the engineering aspects of the radiation damage to water, aqueous solutions, and fused salts. As can be seen from this table, the reaction of water with pile radiation is reversible, and relatively low However, when borated water is irradiated, an equilibrium pressures are obtained. irreversible reaction occurs and the gas evolution is a linear function of the combined energy absorption from neutron scattering and neutron capture minus the -^-energy In other words, the -/-energy absorption appears to favor the back absorption. reaction, whereas the energy absorbed from neutron scattering and the n-a reaction with boron appears to favor the forward reaction. Even without the presence of boron, antifreeze solutions will tend to decompose extensively under radiation; the presence of boron simply accelerates this effect. As would be expected from the mechanism of reaction, fused salts are not detectably affected by the pile radiation. In the liquid state, they are not even affected by that portion of the fast-neutron energy (2 to 5 per cent) deposited in the liquid by momen tum losses, since the ions are then in a less ordered state than they would be in the solid form. A fairly extensive review of the unclassified data available on the decomposition of water by radiation has been made.6 This review is divided into four main parts: — an approximate picture of the gross process of radiation(1) General Considerations induced decomposition of water, (2) The Mechanism— the role of ionization and the principal chemical reactions involved, (3) Major Factors — the major variables and their effects upon decomposition, and (4) Practical Considerations — general observa tions on decomposition that are of interest to the reactor designer, general rules for minimizing decomposition, and specific decomposition data obtained from in-reactc-r experiments and from full-sized reactors are presented. Four major conclusions have been made in this review: " Under the 1. average neutron-gamma flux levels existing in cooling water or moderating water of present reactors, both the instantaneous decomposition rates and the maximum concentrations of decomposition products are very close to zero for initially gas-free, relatively pure water." Relatively pure water is defined in Monson's review in a very approximate manner as having a minimum specific resistance of 200,000 ohm-cm and a maximum gas content of 0.1 ml/liter. 2. "In general, increased temperature results in reduced decomposition rates and Above 300°F, the recom equilibrium concentrations of decomposition products." bination rate is high and essentially independent of temperature. 3. "In general, the presence of ionic impurities results in increased decomposition rates and equilibrium concentrations of decomposition products, some impurities At low tem producing slight increases and others producing very large increases." peratures, solutions of some ionic impurities such as KBr, KI, and CuS04 may produce partial pressures of 1,500 psi under radiation conditions that produce only a partial pressure of less than 10 psi for relatively pure water. At high temperature, i.e., above 400°F, exploratory work has shown that certain impurities strongly catalyze the backward reaction. Such impurities are copper, rhodium, palladium, platinum, silver, and iodine; and tin, iron, and titanium to a lesser extent. by excess hydrogen that 4. "At low temperature, decomposition is so repressed no decomposition occurs at average neutron and gamma irradiation levels

6

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10-134

REACTOR

MATERIALS

[Sec.

10

in present reactors if hydrogen equivalent to 5 to 10 psi partial pressure is initially dissolved (and maintained) in otherwise relatively pure water." The same statement is essentially true for water used at high temperature. In general, the reactor designer should consider the following steps in attempting to minimize water decomposition: (1) Initially remove all the oxygen economically feasible, and dissolve a small amount of hydrogen in the water before subjecting it to irradiation; (2) maintain a partial pressure of a few pounds per square inch of hydrogen (or deuterium for heavy water) in the system; and (3) maintain the water relatively pure (10s ohm-cm or higher) by installation of a bypass, mixed-bed, ion-exchange system.


3.2

Radiation Damage to Organic Fluids

In a nuclear power plant, organic liquids, in general, can be used in many different ways, such as for greases, lubricants, coolants or heat-transfer media, moderating Two of the most striking effects occurring in an organic materials, and allied usages. liquid as the result of exposure to radiation are gas evolution and changes in viscosity. The viscosity usually increases upon continual exposure to radiation until the liquid has polymerized into a solid form. Several broad generalizations can be made regarding the effect of radiation on organic fluids. The gas evolution is not dependent upon the dosage rate, but it is a linear function of the total dose of energy applied. The per cent viscosity increase is also a function of the radiation dosage rather than the dosage rate, and a marked threshold of reaction appears to exist. The viscosity gradually increases initially with radiation dosage until a definite amount of radiation has been absorbed; at this point there is an exponential increase in viscosity with small incremental amounts of radiation energy. Generally, there is considerable difference between the behavior of aromatic compounds and other organic compounds under radiation. Aromatic* are far more resistant to radiation damage than aliphatics, the radiative energy absorbed presumably being taken up primarily by excitation of the shielded system of r elec trons in the aromatic nucleus.* In a more definitive manner, the difference in susceptibility of various aromatic structures to radiation damage can be considered and explained on the basis of their ability to assume several different electronic configurations, that is, on their ability to The higher the resonance energy of the aromatic structure, the more resonate. resistant it should be to radiation. The resonance energy of a compound being defined as the difference between the experimentally determined energy of formation and the calculated energy of formation as determined by adding up the individual bond energies, the following values in terms of kilocalories per mole are obtained in order of decreasing resonance energy for various organic materials: anthracene, 105; naphthalene, 75; quinoline, 69; stilbene, 54; indole, 54; diphenyl, 47; styrene, 46; pyridine, 43; and benzene, 39. The values given for diphenyl, styrene, and stilbene include 8, 7, and 15 kcal/mole of resonance energy, respectively, in addition to the 39 kcal/mole for the benzene ring. The experimental radiation-damage data obtained to date on various homologues of benzene, naphthalene, and diphenyl are in good general agreement with the rule that "the higher the resonance energy of the aromatic structure being tested, the greater the resistance to radiation damage." This is a good general rule to remember in predicting the effects of radiation on a new aromatic-type material, and this rule should be of assistance in synthesizing or developing a new material to meet certain radiation-resistant requirements. By using the general law that equivalent damage to organic materials is obtained for equivalent energy absorption, the engineer can utilize the experimental data presented in Table 2 (also Tables 3 and 4) to estimate the radiation stability of a given material under any mixed pile flux. For example, the life of a material such as * If the symbol sigma (tr) is used to represent the pair of electrons that goes to make up the single bond between the carbon-to-carbon atoms and the carbon-to-hydrogen atoms of an aromatic nucleus, then pi (*■)can be arbitrarily used to represent the pair of electrons that goes to make up the secoad

However, each pair of these pi («-) electrons should be bond between the carbon to carbon atoms. considered as being distributed over the entire aromatic ring structure and should not be considered as belonging to any particular carbon-to-carbon atoms.

radiation

Sec. 10-5]

damage

to liquids

10-135

hexadecane (item 4, Table 2) subjected to a radiation field of 1010 thermal neutrons/ and 10a y photons/ (ems) (see) is estimated (cmJ)(sec), 10' fast ncutrons/(cm!)(see), as follows:

2.82

10" (th.n.)/(cm»)(scc) X 10" (th.n.)/(cm1)(see)

X

1.0

X

10»

rads =

0.4 rad/sec

(f.n.)/(cm')(sec) X 10" (f.n.)/(cm2)(sec)

X

1.0

X

10»

rads =

4.8 rads/sec

X

1.0

X

10«

rads = 51

10' 2.05

10" 7/(cm»)(sec) 1.95

X

1018

7/(cm')(sec)

Total

rads/sec

= 56.2 rads/sec

From Table 2, a dosage of 5.0 X 10* rads for hexadecane causes an increase in the 100°F viscosity of 135 per cent and an increase in the 210° viscosity of 55 per cent. Therefore, the time necessary to increase the viscosity to these values is 5.0

X10» rads

56.2 rads/sec

=89xl0agec


= approx. 2,470

hr

If the 135 and 55 per cent increases in viscosity can be tolerated, hexadecane would Closer have a useful life of at least 2,470 hr under the assumed pile flux conditions. scrutiny of the table will reveal that an asymptotic increase in viscosity of hexadecane occurs at 20.0 X 108 rads; this rad dosage is equivalent to 9,880 hr of operation under the assumed pile flux conditions. Much of the data in Table 2 have been adapted from the experimental results originally obtained by Bolt and Carroll of the California Research Corporation.' These experimental results have been modified to conform to the particular format used in this table. 3.3

Radiation

Damage to Elastomers

Many different usages can be found for elastomers in a nuclear power plant, e.g., as gaskets, flexible connecting tubes, hoses, and elasticlike electrical insulating material and for many other allied functions that necessitate the use of a material which is elastic enough to be pressed to fit a given contour or which has a large amount of vibration resistance. For such uses in a nuclear power plant, elastomers could be subjected to various degrees of radiation flux, depending upon their site of application in the power plant. Although elastomers are of a specialized nature, they are organic solids, and as such, The effects of radiation on they are subject to considerable damage by radiation. five different physical properties of eight well-recognized elastomers are given in Table 3. As organics, the amount of damage is proportional to the energy absorbed Since the atoms and molecules of organic solids are in general in a by each material. far more ordered and restrained state with less degrees of freedom than those of organic liquids, it would be expected that damage to solids would occur at lower Comparison of the data in Tables 2 and 3 verifies radiation dosages than for liquids. this observation. Examination of the data in Table 3 shows that the tensile strength, set-at-break, and the compression set values decreased initially for some of the elastomers and In considering such changes, it should be increased initially for other elastomers. borne in mind that the effects of irradiation on elastomers would necessarily depend upon the curing states of these materials, the curing states in turn being dependent As a corollary to this statement, it should be upon the fabrication processes used. likewise true that ionizing radiation might prove very beneficial as a means to develop and control desirable curing processes for such materials. The service life of any of the elastomers listed in Table 3 can be estimated for any mixed pile flux conditions in the same manner as previously described for organic

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5SSS

ioi-i

iflilllflllliil 'r

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10-139 S-S S~

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=a

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at •<

g-

8 715 502 1300 AR

Solid

z

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— S *4 150 MO 56.6 240

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3.28

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5

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f f

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o

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06

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3.25

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in

i

s

s

§ a

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140 190-350 140 140

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a

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g —

If

PS

s

a

s -r

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0.93

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CD

S

m

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o

R

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papA

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■■«£

No.

0.3» 0.3

9.9 34

1.7 0.3

i 57 42

123 4 56 7 2.8' 5.4

k

5.2 7.3

36 ND'

0.4 ND' 0.6'

14 3.1 ND>

1.6 0.7 4.3

h

17" 4.9 2.2 *» 0.3"

1*

0.6° 0.

2.6

1.8

27 10 2.1 0.8"

10 3.4

31 50

0.7 0.4 0.4"

1.3

4.8 0.6

-25%

-75%

3.2 8.7°

2.1'

7.7 5.6

1.5 6.0 4.4

22 7.5 4.2° 37 12°

6.0

5.5 0.12

Thermal neutrons. 10" neutrons/ cm1

Particle

31

110 21

-75%

Strain at 400 psi

1.

5.0

17

1.9

17

1.9

1

28

II

-25%

Compression set

10' rads

1

Hycar PA- 21 Thiokol ST

Hycar OR- 15 Silastic 7-170 Butyl (GR-I50)

(GRS-50)

15

2.6 1.8

25

3.4 1.8

-75%

-25%

Set-at-break*

- 25 % -75%

Elongation6

65 120

13 6.3

- 25 % -75%

Tensile strength

II

BunaS

Natural rubber Neoprene W

Specific material tested*

properties,

to Elastomers

3.6 0.01

3.1 3.8 2.2

2.9

2.5 5.0

Fast neutrons, 10" neutrons/ cm*

energy

Roent gens, 10'

1.1 1.1 I.I I.I I.I 1.1 I.I I.I

Pho tons,* 10" 7/cm*

2.0 2.1 2.1 2.1 2.1 2.1 2.1 2.2

to

Gamma

dosages each equivalent rads 1X10'

a

** a

X

X

a

X

t

in water-cooled Those materials were irradiated The reactions are irreversible. Since facility of graphite reactor using water temperatures of 75 to I05°F. as such was not measured. Some evolution does accompany the changes in the properties shown in the table wore the primary subject of study, gas evolution to these changes but could also present a corrosion evolved from problem, e.g., when chlorine physical properties noted, and such evolution not only contributes neoprene rubber. Percentage elongation was calculated by dividing the change in distance at the moment of rupture between a set of gauge marks on the specimen by the original distance between the marks and multiplying by dividing the per cent elongation value of the irradi by 100. The per cent change shown in the table was calculated ated specimen by the value of the unirradiated specimen and multiplying by 100. in the same as that the broken pieces of the specimen were carefully fitted set-at-break was determined elongation percentage general way except Percentage The together and the change in distance between the set of gauge marks was measured 10 min after rupture of the specimen instead of at the moment of rupture. allows the material some time for restoration. delay in the measurement > 1.0 Mev. Fast neutron •Average energy = 1.0 Mev. minimum value of minus 58 per cent at 100 The tensile strength decreased to minus 25 per cent value down to 10* rads; then as brittlencsa increased, the 300 X I0; rad asymptote. and approached tensile strength increased exponentially This dosage gave gave a negative change. positive change in the value of the property measured; all dosages not so annotated maximum of plus 42 per cent at 30 to increase from the plus 25 per cent change value up to I07 rads, then decreased to The tensile strength continued and approached 107 rads, then increased exponentially 300 I07 rad asymptote. value of minus 37 per cent at 120 minimum of minus 60 per cent at 40 X I0T rads, then increased exponentially minimum and The tensile strength decreased from the minus 25 per cent value down to 300 X 10' rad asymptote. approached N.D. — not determined. 07 rads, then decreased exponentially and maximum of plus 48 per cent at 1.3 The set-at-break continued to increase from the plus 25 per cent change to I07 raid asymptote. approached I0T rads, then decreased expo maximum of plus 50 per cent at 0.7 The set-at-break continued to increase from the plus 25 per cent change valuo up to 107 rads. nentially to the minus 25 per cent change value at 2.6 " The compression sot increased from the plus 25 per cent change value up to I0T rads, decreased to minimum maximum of plus 35 per cent at value of and approached a 300 107 rad asymptote. 107 rads, and then increased exponuntially minus 65 per cent at 50

Rubber rubber Chloroprene Butadiene ityrene rubber Butadiene acrylonitrile rubber Silicone rubber Iaobutylene rubber Polyacrylonitrile rubber Polyaulfide rubber

Olasa of material

Damage

75% decrease in listed

Radiation

Dosagea to give 25 and

Table


3.

r

is

a

a

g

6

e

rf

/

c *

X

X

1 X

a

a

■

'*

X 2

a

a

X

a

a

a

a

a

X

a

a

X

X

6

a

1

Sec. 10-5]

RADIATION DAMAGE

TO LIQUIDS

10-141

However, the nature of the curing state of the elastomer in question would liquids. need to be considered in order to evaluate its radiation stability. Most of the radiation-damage-rad-dosage

and particle-equivalence

data given in

Tables 3 and 4 are based upon the experimental results of Sisman and Bopp ;''B these published results were previously presented primarily in graphical form by plotting the changes in physical properties as a function of the accumulated radiation dose in terms of thermal-neutron flux times the irradiation time (nvt). 3.4

Radiation Damage to Plastics

There are multitudinous

applications for plastics in various parts of a nuclear power an electrical system involving motors and selsyns, plastics are used as brush holders, grease seals, insulating tapes, spacers, slot insulation, shaft insulation, electrical-wire insulation, and end punchings. Data have been compiled in Table 4 on the effects of radiation on five different As in the case of elastomers, the plastics exhibit radiation properties for 38 plastics. damage at lower dosage levels than do the organic liquids, this higher susceptibility to radiation again being explained on the basis that the atoms of a solid possess less Since plastics are covalent materials, the degrees of freedom than those of a liquid. generalization can be applied, and the service equivalent-cnergy-equivalent-damagc life of these materials under a mixed pile flux can be calculated in a manner analogous to that described previously for organic liquids. plant.


3.5

For example, in that portion of


Systems

In Table 5, an attempt has been made to utilize the experimental radiation-damage data compiled in the preceding tables, particularly Tables 2, 3 and 4. These data have been applied to various systems of a nuclear power plant in order to analyze and evaluate the effects that radiation-induced changes in individual materials might have Since electrical, electronic, and on the systems in which such materials are employed. mechanical systems constitute a major portion of a nuclear power plant in which organic materials might be widely used, these three systems have been chosen for analysis and evaluation of their probable radiation stability. The various electrical, electronic, and mechanical systems are listed in the second column of Table 5, and the corresponding radiation-susceptible components of each of these systems are listed in the third column. In the fourth column, some of the more common materials that are actually used in the construction of these com ponents are listed, and the most radiation-susceptible properties of the materials used in these components are listed in the fifth column. In the sixth column, a dosage level is listed that would give a 25 per cent change in the various properties listed for each particular material. The dosage level listed is based upon experimental data recorded in preceding tables, and the 25 per cent change that has been selected as a conservative value should lie safely below that per cent change associated with an exponential rise in damage with small additional increments of radiation. In the seventh and last column, references are given to preceding tables for respective materials in order to calculate the thermal-neutron, fast-neutron, and 7-radiation dosages that are equiva lent to the rad dosages given in the sixth column. This table is based in part on the premise that the stability of a given system is a function of its most radiation-susceptible materials. Certainly, the operational stability of a system might be endangered by any change appreciably greater than a 25 per cent change in any of the physical properties of one or more of its materials of construction, but the system may still be operable, depending upon the vital function of the affected material. For most materials, the 25 per cent change in one or more physical properties repre sents a good engineering reference value, but this is not true for certain electronic materials, particularly semiconductors. Not only can changes in semiconductor properties occur frequently at very low neutron and 7 dosages,* but in addition, small * Experimental data indicate that even a single photon can sometimes apparently affect the proper

ties of

a semiconductor.

Class of material

2

345

6789

? > 1.4

link.

65

>500

2.4

230

>500

210* >600

6.3

3 0

110

>400

>300 >400

>300 >400

>400

100

ND >500

ND 2I0« ND

ND

>300

>300

1

Ill,

2104

Karbate

> >400

>

>400

100

>

0.3

>500

>400

100

2.4

>500

>400

100

5.0

4.0

4.4

4.0

4.0

4.6

4.7

4.6

4.6

2.

2.

i.\

2.1

2.

2.2

2.1

2.1

2.3

2.1

2.2

2.3

1

IS

41

100

>

Havcg

100

>70

53

4.7

4.6

70

17

>

17

Bakelite

>300

240

3.5

0.85

8.7

1.7

1

>80

70

it

17

15

5.5 3.5

2.3 3.8

9.8

200 3.7

5.5

4.2

43 3^5

4.

5.

16

>80

>70

53

>300

240

Bakelite

1 0*

9.8

3.7

1

II.

200

40

1 2

i

70

>500

>500

>500

>500

1

15

17

/

5^0 4.

88 41

]

4.0*

2.5

0.4 0.6

5.2

22 7

10 2

14

>ll.l>

8.7 3.2

1.0

70

17

Catalin

II 2. 0.4 1.3

54 2 354 0.71

energy

PhoRoent tons,e gens, 10" 10'r ■y/cm1

Gamma

to

2.0

4.2 4. 4.4 4.8

0. 86

4.4

Fast* neutrons. 10" neutrons/ cm'

dosages each enuivalcnt 10' rads"

1

2

ND >5.0«

4.2 5.4 4.2 10

2.3

-ii) 5.4 3.4

I

1.34 2.0 1.6

J

>4.0"

II

430 14

240 4.7 2.5

301

Thermal neutrons. 10" neutrons/ cm1

Particle

1

13

1.7

17

Fluoroethene

' >500 >I3 >4.0*

94

301

94

-75%

-75% -25%

Impact strength

-25%

Shear strength

10' rads

121 22

I.I

I.I

I.I

I.I

I.I

1.2

I.I

1.2

1.2

'.11'.11i!i

12

1.0

3.3

1.0

Lucite

> 140

120 >4.(M

53 3.3

6.4

53

6.4

>500

>500

>500

>500

>500

>500

Duralon Melmac and Plaskon Melamine

>4.0* 2.3' ND >5.0'i

4.2 1.6 4.2 1.3

2.3 0.4 0.4 0.5

4.2 5.4 4.2 4.0

2.3 2. 0.4 1.0

>500 >I3 >4 0"

450 14 3.4

240 4.7 1.6

Tenite 11 Pyraline Forticel Ethocel R-2

21

26

8.24

170 450 14

24 240 4.7 2.5

-75%

-25%

-75%

-25%

Elastic modulus

decrease,

to Plastics

. .

I

II

Damage

-75%

Elongation

Cibanite Ameroid Plastacele

-25%

Tensile strength

5 8

CR-39

Specific material tested8

Radiation

1 1 . . 1 1

1(1

,

Ally] diglycol car bonate Aniline formal dehyde, unfilled Casein Cellulose acetate Cellulose acetate bntyrate Cellulose nitrate Cellulose propionate Ethyl cellulose Furan. asbestos, and carbon-black filler Melamine form aldehyde, cellulose filler Methyl methacrylate Monochlorotrifluoroethylene Phenol formalde hyde, unfilled Phenol formalde hyde, asbestos fab ric laminate filler Phenol formalde hyde, asbestos fiber filler Phenol formalde hyde, asbestos filler I'lunol formalde hyde, graphite 1,11. I'liwml formalde hyde, linen fabric laminate filler

,

I1

1.

|

I—

No.

4.

Dosages to give 25 and 75%

Table


X 1

1 1

1.

1

2.

420 0.2

3.0 >24 420

>24 67

20 0.4

420

20

67

3.0

>300

60

4.3

0.31 0.09

110 >300

0.027 53

5.2

5.9

8.9

28 3.0 4.7 5.8 3.6

3.5

1.1

. .

1.1 1.1

1.2

. . . .

2

2

2. 2.

1

a

The reactions are irreversible. Since water-cooled Those materials were irradiated in facility of a graphite reactor using water temperatures of 75 to I05°F. Some evolution will occur, and in the plastics con the properties shown in the table were the primary subject of study, gas evolution as such was not measured. the evolved halide some corrosion problems. taining halides, gas may present Fast neutrons Mev. Mev. Average energy = This dosage gives gave negative change. positive change in the value of the property measured; all dosages not so annotated to approach 07 rads, then increased exponentially The elastic modulus decreased from the minus 25 per cent change to a minimum of minus 63 per cent at 00 I07 rad asymptote. the 200 The elastic modulus increased from the plus 25 per cent change to maximum at plus 27 per cent and then leveled off until X I0T rads had been dosage of received by the specimen. I07 rad asymptote. The impact strength decreased exponentially from the minus 25 per cent change to approach the 400 The shear strength decreased exponentially from the minus 25 per cent change to approach the 30 10' rad asymptote.

Vinylite

20

67

220

51

10 3.3 1.4 0.068 4.0

2

3.0

220

53

0.04 ND >500 ND ND

4.5

>500 0.06 ND >500 ND ND

3.5 3.5

2

38

4.3

80

220

14

1 1

plaakon

0.04 ND >500 ND ND

>500

4.

Beetle urea

>2.0 ND >500 ND 85

>2.0 ND >500 ND

5.5 ND >500 ND ND

>500

4.5

>500 >600

6.0

41221 22222

Saran

Teflon Butracite film Polectron Geon 2046 Formvar

>500

6.7

ND

4.9 2.2

I.I

2. 1 1 1 1 1 1 111

37

32 33 34 35 36

>500

>500

>500 0.02 54 >500 83 >I2«

>500

5.5 200 >500 370 >I900

1.0 >600

3.6 2.7

>500 37

3.0 3.0 4.7

4.6 3.3

4.6

.

0.003 10 >500 12 9.71

>500

>500 0.01 1.9 >500 180 25

41 IC

>500 >600

ND

0.81 0.81 6.2

5.7 500 180

5

Styron

>500 >600

ND

360 9.0

5.0 4.0

40 ND

ii ii ii ii 2 I1 1 2 22 2 . .

31

>600i >600

>600* >600

7.6 >600

0.5 >600

300* >600

100' >600

Styron 475 Amphenol

>

.

29 30

ND

1

ND

\*

ND

65

19

360 I30* >500" 94*

360*

>500 220"!

V 1'

29

12

>500 1000

360 800

Plaskon alkyd Polythene

>500' ND 0.

150 >500* 0. 2*

0.09*

5.4

36

1 11 002 29 2 22 2 22 21

film

>500 37

360 9.0

>500* >500* 0.2*

2.2 ND

>500< 0.4 ND 0.5 0.3' 9.0

36

6

.

Mylar

21 14 13

0.9 0.4 0.7

470* I60* 0. I
0.8

40 44

ND

2.2 4.0

>300 ND

25 ND

ND

>300

70

ij iiii

28

40 ND

2.2 ND

2.2 ND

36

0.8

110 40 ND

2.5

Micarta

1

Bakelitc Experimental plastic Q-817 Nylon KM- 0001 Nylon FM-3003 Selectron 5038

11 1. 1.

26 27

22 23 24 25

21

20

formalde hyde, paper lami nate filler Phenol formalde filler hyde, paper Polyalphamethyl styrene Polyamide Polyamide Polyester Polyester, mineral filler Polyethylene Polyethylene terephthalate Polystyrene, black pigment filler Polystyrene Polystyrene, white pigment filler Polytetrafluoroethylene Polyvinyl butyrate Polyvinyl carbazole Polyvinyl chloride Polyvinyl formal Vinyl-vinylidene chloride Urea formaldehyde, cellulose pulp pig ment Vinyl chloride acetate

Phenol


19

1. 1

5

a

a

1

•

I

1

X

6 ed •

X

X

a

>

a

X

'

c k

10-144 Table

MATERIALS

REACTOR 6.

[SEC. 10


Systems

Dosage

No.

Type of system:

Radiationsusceptible components

Most radiation

Actual materials

susceptible properties"

used

levels to give 25 % change, 10' rada

Refer ence for particle-

rad

desage eq ui valents»

Electrical 1

Disconnects

Base insulation

2 3 • 4 5

Fuse holders

Gasket (pressure Base insulation

Light bulbs Magnetic amplifiers

9 10


II

Glass envelope Core damping

fluid Core insulation Intcrwinding insulation Potting com

12

pound

Terminal board

13 14 15 16 17

Wire insulation Motors and selsyns

Brush holders

Grease seal

Insulating tapes

28 29

31

I'.nd punchings and slot insula

tion Shaft insulation, etc.

E, SAB, C. 8AF

1.3

3-3

Natural rubber Linen-filled phenolformald Paper-filled phenolformald

SAB E. I

2.6 0.3

3-1 4- 18

E, I

0.8

4-19

Glass Silicone

oil

0. 1 >50

4-16

2-1*

E. T, I

Mylar Silastic

E. T C

Paper-filled

E, I

0.8

4-19

phenolformald Unfilled phenol

E

1.0

4-U

E E

9 7 0.003 0.3

4-35 4-31 4-18

E. I

0.8

4-19,

C SAB. C E, T. I I E. T, I (cloth) E SAB. C E (Teflon)

0.6

3-2 3-4 4 6*4-22* 4-8* 4-34 34-3 1

formald Formvar Teflon Linen-filled

phenolformald Neoprene

Buna N Felt Leather

Rubberized cloth Polyvinyl chloride Natural rubber Teflon-coated fiberglas Fibergloa and silicone resin Acetate cloth Buna N treated fiberglas Teflon-coated fiberglas Fiberglas amine

mel-

Mylar Teflon Mica with shellac Mica with mylar

36

Paper-filled phenolformald Asbestos- filled phenolformald

38 3')

Light transmission' Viscosity

100

Nylon Kraft paper

32 33 34 V,

37

4-20

T, E, I

Paper-filled

27

30

2.2

1

E,

I

0.4

4-22

0 5

4-8"

12

0.4

4-27 3-5*

phenolformald

18 19 20 21 22 23 24 25 26

T. E, S. I

Asbestos- filled phenolformald Buna S

type)

6 7 B

Paper-filled phenolformald

Kel F

Fiberglas with j silicone rubber

Pliability

0 4 0.5

0.4 0.5 12 2 6 0.003 >50

16

2-IV

E, I SAB, C (Buna N)

0 4

M

E (Teflon)

0 003

4-31

T, E

6.4

4-10

E, T

12 0.003

4-27 4-3! 4-2> 4-27 4-l
E

I

0.4

E, T (Mylar) E. I

12

E, I

53

E, I

C

0.8

1.7

0.4

4 4

4-14 4-12

radiation damage to liquids

Sec. 10-5] Table 5.

Radiation Damage to Electrical, Electronic, and Mechanical Systems. (Continued)

RadiationType of system

No.

10-145

susceptible components

Most radiation

Actual materials

susceptible properties"

used

Dosage levels to give 25 % change. 10' rads

Refer ence for

particlerad dosage

equiva lents''

40 41 42 43 44 43 46

Wire insulation

Fibergloa

and silicone resin

47 41 49 SO Relays and switches 51

Nylon

Neoprene Paper

Base insulation and shell

Paper-filled phcnolformald Linen-filled phenolformald Asbestos-filled

52 53


Fish paper Varnished cam bric Polyvinyl chloride Cellulose acetate Teflon Formvar

54

Thermocouples

Wire insulation

55

Transformers

Wire insulation Oils

56 57 58 59 60

Gaskets Base insulation

phenolformald Unfilled phenol formald See Motors (Wire Insulation) See Motors (Wire Insulation) Chlorinated hydrocarbons Dichlorobiphenyl Buna N rubber Hycar PA Paper-filled phenolformald Paperboard

61

E, T, I E. T. I

E

E. I E E Pliability E, I C E, T, E, 1

I

E, I T, E. I E

0.5 0.5 12 1.6 0.003

9.7 >50

4-84-84-34 4-4 4-31 4-35 2-13-

0.4 0.6 0.5 0.8

4-23 3-2 4-8-1 4-8'

0.3

4-18 4-15

> 100 1.0

4-13

Viscosity

> 100

2-37

Viscosity SAB, C SAB E, 1

> 100

0.4 0.3 0.8

2-37 3-4 3-7 4-19

E, T. 1

0.5

4-8"

E, T. I E, I

0 5 9 0

4-84-26-

lilectronic 62 63 64

Capacitors

Dielectric Base insulation

65 Coaxial cables 66 67 68 69 70 Connectors 71

Wire insulation

73

Ion chamber

Base insulator

75 76 77 78 79

Printed

circuits

Paraffin See Relays (Base Insulation) Polystyrene Polythene Rayon Polyvinyl chloride Neoprene

Base insulation

72

74

Paper

Base insulation

Polystyrene Polyethylene Paper-filled phenolformald Asbestos-filled phenolfurmald Unfilled phenol

formald Polystyrene Polyethylene Teflon Methyl methacrylate Urea formalde hyde

T, E. EM, S, I K, I

K, I E, I

>500 9 1 12 0 >600 9 0

I

53

E, E

I

C T, E. EM, 8. I

E, E

T, E, EM, S, I

E

1

T, E T, E, S

0 6 6 0 8

4-30 4-26 4-4< 4-34 3-2 4-29 4-26 4-19 4-14

1 0

4-13

600 9 0 0 003 1 0

4-29 4-26 4-31 4-11

3 0

4-37

REACTOR MATERIALS

10-146 Table

6.

Radiation

[Sec. 10

Damage to Electrical, Electronic, Systems. (Continued)

and Mechanical

Refer

No.

Type of system

Radiation-

Most radiation

susceptible components

susceptible properties*

Dosage levels to give 2S% change, HF rads

ence for

particlerad dosage

equiva lents*

Printed circuits (continued) 80 81 82 83

Resistors


84 85 86

Thermistors

87

Transducers

88 89

Vacuum tubes

90

Base insulation (continued)

Melamine form aldehyde Paper- filled phenolformald Conductor binder Epoxy resin Paper-filled Base insulation phenolformald Asbestos- filled phenolformald Paperboard Wire insulation See Electric Motors (Wire) Wire insulation See Electric Motors (Wire) Damping fluid Silicone oil Paper-filled Tube bases phenolic Unfilled phenolic

T, E

6.4

4 10

E, I

0.8

4-19

S

E,

100

I

0.8

T. E, I Et T.

I

Viscosity E, I

MOO 0.5

4-19 4-15 4-8*

0.8

2-13* 4-19

E

1.0

4-13

>50

Mechanical O rings

Neoprene

C

0.6

3-2

92 93

Shaft seals Fan belts

Leather Cloth-backed

I

0.4 0.5

4-2> 4-fr"

94 95 96 97

Lubricant Antifreese

SAE-20

>100 0 5

2-51 1-3* 4-8*

0.5

4-V

C C E, T, I E, T. I C E, T, I (cloth)

0.6 0.5 0.5 0.6 0.5

3-2 3-8 4-8* 4-84 3-2 4-84

E, SAB. C. SAF

1.3

3-3

C C C C

0.6

3-2 3-8 3-5 3-2

91

Combustion engines

neoprene

Gaskets

Cork Cork-impregnated cloth Neoprene

98 99 100 101 102 103

Filter Hoses

Thiokol Cloth Paper Neoprene

Cloth-backed

104 105

Prestone

Fluid discon

Starter generator

neoprene See Electric

O rings

Motors Buna S

nects 106 107 108 109 110

III

112 113 114 115 116 117 118

Gaskets Gauges

Hydraulic

Diaphragm

Neoprene

Thiokol Silicone rubber Neoprene

E, T, I

Viscosity Viscosity E, T, I E, T, I

I

Hydraulic fluid

Leather Di(2-Ethylhexyl)

Seals, 0 rings

Nylon

I

Neoprene Buna S

C E, SAB. C, SAF SAB, C C E. SAB, C, SAF

system

scbacatc

Buna N

HoseB

Neoprene

Buna S Buna N

Viscosity

SAB

>02

0.1

0.1

0.4 0.6 0 4 >50

0.4 0.6 1.3

0.4 0.6 1.3

0.4

4-2> 2-12*

4-22 3-2 3-3 3-4 3-2 3-3 3-4

Sec. 10-5] Table 6.

RADIATION Radiation

DAMAGE TO LIQUIDS

Damage to Electrical, Electronic, Systems. (Continued)

10-147

and Mechanical

Refer

No.

Type of system

Radiationsusceptible components

Most radiation

Actual materials

susceptible properties'1

used

Dosage levels to give 25 C/G change, 10' rads

ence for

particlerad dosage

equiva lents'

Hydraulic system 119

(continued)

Cloth-backed ncoprene

120 121 122 125 124 125 126 127 128

Cylinder cups

Leather Neoprene

Couplings Pneumatic systems Pressure switches

Pumps

Leather

Neoprene (Similar to Hydra ulic System except hydraulic fluid is replaced by air) Neoprene Diaphragm

Packings, shaft

Nylon Leather

E, T,

I

0.5

4- 6*

c

0.4 0.6 0.4 0.6

4-22' 3-2 4-22' 3-2

c

0.6

3-2

I

0.4

4-22

(Cloth)

I

C

I

Neoprene

I c

Natural rubber Buna N Thiokol Silicone rubber Teflon Cotton Rayon Paper Leather Kel F Butyl rubber

SAB SAB, C C C E E, T, I E, I E, T, I I E. I C

0 4

0.6

4-22' 3-2

129 130 131 132 133 134 135 136 137 138 139 MO

Valves

Packing, seats,

2 6 0 4 0.1

0.4 0.003 0.5 1.6

0.5 0.4 1.7

0.4

3-1 3-4 3-8 3-5 4-31

4-8' 4-4" 4- 6* 4-22' 4-12 3-6

See pumps

gaskets • T = tensile strength, E — elongation, I = impact strength, SAB — set-atS ™ shear Btrength, brcak, C » compression set, SAF = strain at 400 psi, and EM — elastic modulus. ' Reference is made to table number and item, e.g., 4-19 refers to item 19 of Table 4. • Decrease in foot-candle output. ' The equivalent dosage for this material has been estimated on the basis of chemical and physical similarity to a material actually tested. • This is one of several materials which may be used in this application.

It

is

is,

For percentage changes in such properties can render a given instrument useless. these materials, the 25 per cent change represents a very high upper limit value; a threshold value is frequently needed, i.e., that radiation dosage radiation-damage at which the semiconductor properties just begin to change. Based upon the premise that a system is no more stable under radiation than its most radiation-susceptible materials, the service life of each of the systems listed in Table 5 can be estimated for any pile flux conditions in the same manner as previously In evaluating the behavior of electronic components described for organic liquids. under radiation conditions, it should be remembered that although radiation damage in general, a function of total dosage and not dosage rate, to organic-type materials still entirely possible that the ionization produced in an electrical insulation system during irradiation will cause the resistivity of radiation-sensitive insulating could therefore be materials to be changed in situ as a function of dosage rate. possible for a given electrical insulation material or Bystem to receive the same total dosage under different fluxes or dosage rates and yet exhibit somewhat different radiation-induced changes.

it


seals, gaskets

10-148

REACTOR

MATERIALS

[Sec. 10

In general, the service life of systems employing the more radiation-susceptible organic materials can be improved by fivefold to tenfold by using more aromatic-type organics, and the service life of systems employing even the most radiation-resistant organics can be improved a thousandfold and more by judicious replacement of the The replacement of organic covalent organic materials with ionic-type materials. compounds with inorganic ionic compounds in many component applications, such a-s electrical insulation for very thin wire, constitutes a major development undertaking. This must be done if such systems are to be used for However, the path is clear. appreciable periods of time in high-radiation fields.

REFERENCES 1.

Calkins, V. P.: Radiation Effects on Reactor Materials: Nonmetals, 9-12 (September,

2. 3. 4.

IS

(9):

Allen, A. O.: Radiation Chemistry of Aqueous Solutions, J. Phys. <& Colloid Chem.. 12 (1): 11 (January, 1954). Rollefson, G. K., and R. E. Powell (eds.): "Annual Review of Physical Chemistry," vols. 1-5, Annual Reviews, Inc., Stanford, Calif., 1950-1954. U.S. Atomic Energy Commission: "Reactor Handbook," vol. 2, Engineering, H. O. Monson, Water Decomposition, McGraw-Hill Book Company, Inc., New York, pp. 175191,


M ucleonict,

1954).

1955.

6.

Bolt, R. O., and J. G. Carroll: NEPA-1844, Apr. 30, 1951; TID-5094, July 1, 1952; TID-5132, Jan. 9, 1953; TID-5136, Feb. 28, 1953, California Research Corporation.

7.

(Classified.) Sisman, O., and C. D. Bopp: "Physical Properties of Irradiated Plastics," ORNL-92S. U.S. Atomic Energy Commission Technical Information Service, Oak Ridge, Tenn..

June 8.

29,

1951.

Sisman, O., and C. O. Bopp: "Radiation Stability of Plastics and Elastomers," ORXL1373 (supplement to ORNL-928), U.S. Atomic Energy Commission Technical Informstion Service, Oak Ridge, Tenn., Feb. 1, 1954.

BIBLIOGRAPHY Allen, A. O. : Yields of Free H and OH in the Irradiation of Water, Radittfion Research, 1: 85 (February, 1954). Calkins, V. P.: Radiation Damage to Nonmetallic Materials, Chcm, Eng. Progr. Symposium Ser. No. 11, 60: 28-42, American Institute of Chemical Engineers, New York. 1954. Goodman, Clark (ed.): "The Science and Engineering of Nuclear Power," vols. 1 and 2. Addison- Wesley Publishing Company, Cambridge, Mass., 1947-1949. Javitz, A. E.: Impact of High Energy Radiation on Dielectrics, Elec. Mfg., 55 (6): 86-104 (June, 1955). Rollefson, G. K., and R. E. Powell (eds.): "Annual Review of Physical Chemistry," vals. 1-5, Annual Reviews, Inc., Stanford, Calif., 1950-1954.

10-6

NONDESTRUCTIVE

TESTING

BY


Warren

J.

McGonnagle

Nondestructive testing is a general name given to all test methods that permit test The emphasis in this ing or inspection of material without damage to the material. discussion will be on the application of nondestructive testing to nuclear-reactor components. This article will not include a discussion of proof tests, coupon tests, or such mechan ical tests as hardness, creep, and tensile strength. The essential parts of most nondestructive tests are (1) application of a testing or inspection medium, (2) modification of the testing or inspection medium by defects or variations in the metallurgical structure of the material, (3) detection of this change by a suitable detector, (-1) conversion of this change into a form suitable for interpretion, (5) interpretation of the information obtained. Nearly every form of energy has been utilized in nondestructive testing. Likewise, nearly every property of the materials to be inspected has been made the basis for some method of nondestructive testing. However, these test methods can be divided into the following rather well-defined groups: visual, pressure and leak, penetrant, thermal, radiography, acoustic, magnetic, electrical, and electromagnetic. A number of defects are considered of special significance in reactor design, con struction, and operation. Among these are the following: lack of bonding (liquidmetal or metal-metal bonds) between cladding and core, thick or thin spots in the clad, cracks and holes in the cladding, laminations, inclusions, defects in closure welds such as inclusions and lack of penetration, porosity, pipe seams and laps in the core material. Metallurgical variables such as grain size, grain orientation, and heating treatment; chemical variables such as variation in carbon content ; and certain physical properties Some are often of special significance in reactor design, construction, and operation. of these properties or variations in properties can be found by use of nondestructive

tests.

about the selection and application of nondestructive tests are not since each actual case is a specific problem and undoubtedly has some features about it that make it unique and different from other applications. Some of the factors that must be considered when searching for or selecting the best test method are the reason for making the nondestructive tests; type of material or materials to be tested; the method of fabricating the material; kinds, location, and size of possible or expected defects; the sensitivity and resolution required; how standards or specifications are to be determined ; thickness, shape, and surface condi tion of the object to be tested; and how destructive tests are to be made. An under standing of the basic principles of each type of test is essential if a complete under standing of the possibilities, advantages, and limitations of each method is to be Generalizations

of too much value,

appreciated.

In

the following articles eight of the nine test methods previously mentioned will be The magnetic method will not be discussed, since in general the materials to the reactor engineer are largely nonmagnetic. 10-149

discussed. of interest

10-150

REACTOR MATERIALS 1

[SEC. 10

TYPES OF NONDESTRUCTIVE TESTS 1.1

Visual Tests


Visual inspection is the most widely used nondestructive test. It is simple, easy to apply, quickly carried out, and usually low in cost. In visual nondestructive tests the specimen to be tested is illuminated with visible light and then examined with the eye or by light-sensitive devices such as photoelectric cells. The human eye has excellent visual preception. The acuity and contrast sensi tivity decrease rapidly as the energy level of illumination decreases. Consequently, it is of prime importance to have adequate lighting. The tendency toward ocular fatigue is accelerated by glare and efforts to sec at low levels of illumination. For observation at very low light levels the human eye should be dark-adapted in advance. The length of time that a human inspector is permitted to work should be limited in order to avoid errors due to fatigue and loss of visual reliability and discrimination. Optical aids such as lenses, mirrors, microscopes, and telescopes provide a means of compensating for the limits of visual acuity of the human eye by enlarging small discontinuities in materials. Enlarging projectors and comparators provide a means for improving viewing conditions for rapid inspection of small precision parts. Boroscopes permit direct visual inspection of the interior of hollow tubes, chambers, and other internal surfaces. Photoelectric and other light-sensitive systems can often be used to replace direct visual inspection and compensate for errors due to operator fatigue. 1.2

Pressure and Leak Tests

In pressure and leak tests defects are revealed by the flow of gases or liquids into or through the defects. The simplest test of this type is a hydrostatic test in which the pressure within the test specimen (hollow) is made greater than the external pressure. The commonest test of this type is to use air or water pressure and detect leaks through the walls by However, this is not sufficiently sensitive for most escape of air or water seepage. The presence of leaks may also be revealed by nuclear engineering applications. changes in pressure or by bubble formation when the object is covered with a soap solution or immersed in water, kerosene, or similar liquid. Pressure and leak tests can be made more sensitive by using fluorescent liquids and penetrants, radioactive gas or liquids, or gas such as Freon, ammonia, sulfur dioxide, and helium. Halogen-sensitive leak detectors can be used with any gas that contains In the commercial halide torch, the presence chlorine, fluorine, bromine, or iodine. of halide gas is indicated by the color change of a flame. The flame also acts as an aspirator for drawing the gas to be tested into the flame. The General Electric Such detectors halogen-gas detector can detect halogen gas concentration of 1 ppm. can locate a leak so small that only J.{0o oz of halogen gas will pass through the opening A fan is used to draw an air sample over the halogen-sensitive element, in a year. which is a heated platinum surface. One precaution that must be observed is to keep the area in which the testing is being carried out free of vapors from halogen compounds. Helium gas used in connection with commercially available mass spectrometer Commercial helium-leak helium-leak detectors gives a very sensitive leak test. detectors can detect the presence of 1 part of helium in 200,000 parts of air or are capable of detecting helium leakage rates of the order of 1 cm* (NTP) per year. A mass spectrograph is a device for separating or sorting atoms having different In the mass spectrograph, molecules of the gas sample are ionized by colli masses. The ion beam is accelerated by a difference in potential and sions with electrons. then passed through a magnetic field. A separation of ions takes place in the mag netic field, because ions of different mass travel in circular arcs having different radii The radius of the path R taken by a particular ion is given by the of curvature. expression R = KVM /H, where V is the velocity of the ion, M the mass of an ion, H

Sec. 10-6]

NONDESTRUCTIVE

TESTING

10-151

is the magnetic field intensity, and K a constant. Ions having equal mass will all emerge from the magnetic field at a certain position; ions of a different mass will The ions can then be collected and measured. Figure 1 emerge at different positions. In the case of the helium-leak shows a schematic diagram of a mass spectrometer. detector the magnetic field is set so that only helium ions are collected and measured. With the use of the proper magnetic field, ions of other mass can be collected. In some types of mass spectrographs, it is possible to vary the magnetic field and thus collect and measure a number of different kinds of ions.

Analyzing Magnet

Collector Generated for wjivans (University of Florida) on 2015-09-23 02:55 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Ground

Plote Suppressor Collecting

Slit

Focus

/

Plates

Source Magnet

Ionization Chamber

rp — Filoment Source Mognet

Fio.

1. Schematic

diagram

of mass spectrometer.

(General Electric

Company.)

Dirt, oil, and scale may seal small holes, so it is necessary to have clean test speci mens when the helium-leak detector techniques are used. A technique that can be used with a hollow vessel is first to evacuate the vessel and then connect it to the A fine stream of helium is moved over the surface of the helium-leak detector. vessel. Helium will be drawn into any hole through the walls and will register on A variation of this technique is to surround the test specimen with a the leak detector. helium atmosphere by enclosing the test specimen in some sort of gastight chamber, such as a plastic bag. A second technique is to build up an internal helium gas pres sure and scan the surface with a "sniffer" connected to the helium-leak detector. A third technique can be used for detecting the presence of minute holes, cracks, fissures, etc., in cladding and in end-weld closures of fuel elements or similar objectB.

10-152

REACTOR

MATERIALS

[Sec.

10

When this technique is applied, the test specimen is first placed in a heliumpressurizing vessel and exposed to a helium atmosphere. The specimen is removed from the pressurizing vessel and transferred to a second chamber, which is connected to a vacuum pump and helium-leak detector. This chamber is first evacuated to a pressure of approximately 150 m, then the throttle valve on the leak detector is opened, and the relative quantity of helium released is read on the leak-indicating meter. It has been found Figure 2 shows the helium-leak test apparatus for this technique. possible to detect the presence of one fuel element that has an excessive response in a small group wit h normal response, which makes it possible to test a limited number of fuel elements at a time. Dirt, oil, and scale on the test object cause large response when this technique is used, but samples that have had an acid dip are sufficiently clean for testing. A large open cavity (such as made by a machine tool) does not

TO

10 SLUG


BASKET

TEST CHAMBER DETAIL B

ATMOSPHERE

SEALED VALVES TO VACUUM

FLEXIBLE LINE

VACUUM

PUMP

TO ATMOSPHERE

SLUG BASKET

DETAIL A PRESSURIZING

VESSEL Fig.

2.

TO 50 PSI HELIUM SUPPLY

Helium-leak-test apparatus.

produce a noticeable response because helium is not retained long enough after the fuel element is removed from the helium pressurizing vessel. 1.3

Penetrant Inspection

Penetrant inspection is a quick and easily applied nondestructive testing method for the detection of surface discontinuities. methods are Penetrant-inspection applicable to all metals as well as glazed ceramics, plastics, and other nonporous materials. Defects that can be found by penetrant inspection include surface cracks, porosity, cold shuts, fatigue cracks, grinding and heat-treat cracks, seams, forging laps, bursts, and lack of bond between joined metals, through leaks in welds, and through leaks in tubing. The basic principle utilized in penetrant inspection is that the penetrant is drawn into surface discontinuities by capillary action. The presence of the penetrant in the discontinuities is revealed by the use of a developer. There are a number of penetrants commercially available. Among these are Dy-Chek, Zyglo, Flaw-Findr, Spotcheck, Met-L-Chek, and Chek-Spek. The technique consists basically of (1) cleaning the surface — surfaces to be inspected must be free of grease, oil, scale, or other foreign materials that would clog surface openings; (2) applying the penetrant —the penetrant can be applied to the part by


Sec. 10-6]

NONDESTKUCTIVE

TESTING

10-153

brush, spray, or immersion; the required penetration time may vary from 1 to 10 min or longer depending on the penetrant, material, and type of defect; (3) removing the excess penetrant — after the penetrant has been in contact with the surface long enough for the penetrant to enter the discontinuities in sufficient quantity, the excess pene trant remaining on the surface is removed with a suitable solvent; (4) applying the developer — the developer is used to draw the penetrant from the discontinuities by a blotting action and spread it on the surface of the piece for a short distance on each side of the defect; (5) inspection and inter pretation — the most important part of the Figure 3 shows the principle procedure. of penetrant inspection. Penetrants can be divided into two types, depending on the way the defect indications Some of the commercial are to be viewed. THEPENETRANTCANSEEP WHENDEVLLOPERDRIESIT DRAWS PENETRANT OUTOFCRACKONTO ONLYINTOFLAWSWHICH penetrants are to be viewed under white SURFACE REACHTHE SURFACE ONEACHSlOEOFIT, light, while others are to be viewed under THE SIZEOFTHE INCREASING INOICATION ultraviolet light (black light— 3600 A). (b) Dy-Chek is an example of the former; Fio. 3. Principle of penetrant inspection. When Zyglo is an example of the latter. Dy-Chek is used, discontinuities are indi cated by red dots or lines on a white background, and when Zyglo is used, the discontinuities are indicated as glowing yellow-green dots or lines against a dark background. Partek, or filtered particle, inspection is used for locating invisible cracks in porous materials. This method depends on the unequal absorption into a porous surface of a liquid containing fine particles in suspension. Cracks absorb the liquid to a greater extent than the rest of the test specimen, and the suspended particles are filtered out and concentrated at the defect. This type of inspection has found wide application in revealing cracks in unfired ceramic materials that will open up during firing. The primary limitation of penetrant inspection is that only defects which are on the surface or are open to the surface are revealed. Another disadvantage of penetrant inspection is that all defects regardless of depth give almost identical results. 1.4

Thermal

Tests

The basic principle of thermal tests is. to apply heat to the test specimen and measure the resulting temperature conditions. The heat may be applied by direct thermal contact with a heat source, by electric current heating, or by induction heat The resulting temperature indications may be detected by use of temperatureing. indicating substances such as Teinpilstik, Tempilaq, Thermocolor, temperaturesensitive phosphors, infrared detectors, thermocouples, bolometers, and resistance thermometers. One thermal test that has been widely used in testing reactor components is the In the frost test the test specimen is sprayed with a coat of so-called "frost test." acenaphthene (mp 200°F), a frosty-appearing compound, and then the surface is Areas of poor heat transfer become hotter than the rest of the heated by induction. If the amount of heat applied is properly adjusted, the coating will melt surface. on poorly bonded areas, giving the typical patterns shown in Fig. 4. Commercial induction units can be used to supply the necessary heat. For cylindrical fuel For flat fuel elements a rectangularelements a single-turn inductor can be used. shaped inductor can be used. The frost test has limitations in its ability to find defects. Defects must present a thermal resistance sufficiently great to make the surface temperature over the defect A defect 0.1 to 0.2 in. across may rise above the melting point of the acenaphthene. offer only a small thermal resistance, though it may be 1 in. long. Likewise, large nonbonded areas in which there are small well-bonded areas may present little in the

REACTOR MATERIALS


10-154

[Sec. 10

way of thermal resistance. This test can locate defects of approximately 0.1-in.' area, which has a minimum width or length of 0.2 in. Cases have been found in which even a large defect did not present a sufficient thermal resistance. This test has not been found entirely dependable. Smaller thermal resistance can be de tected by increasing the applied power; however, above a critical power level, false indications may occur because the fuel element is off center as it passes through the indicator or the uranium is eccentrically positioned in the clad. A more sensitive thermal test is the thermographic method in which a tem The perature-sensitive phosphor is used. phosphor is so temperature-sensitive close to a transition temperature that localized in temperature reveal small increases This themselves as relatively dark areas. method has been used for testing for lack The of bond in a metal laminate.*-1 disadvantages of the use of this phosphor are that the test object must be viewed under ultraviolet light and the color An induction change is not permanent. heater can be used to supply the heat energy to the phosphor-coated sample. Fio. 4. Typical frost test results. Photocells can be arranged to detect the color changes. Lead sulfide cells and other infrared detecting cells can be used as detectors for Information on infrared detecting cells and their properties can thermal radiation. be found in the bibliography at the end of this article. One technique is to incorpo rate the temperature-sensitive cells in a bridge circuit. The amount of bridge unbalance can then be amplified to the desired value for recording. The amount of bridge unbalance will show the difference x RAY SOURCE in the thermal radiation originating from different parts of the sample. The stability of the operation of such a detector can be improved by periodically interrupting the the radiant energy and then amplifying intermittent signal.

,

1.6

Radiography

test is a nondestructive Radiography method that uses penetrating radiation such The principle of X-ray as X rays or 7 rays. film radiography is shown in Fig. 5, the in tensity of the radiation being modified by passage through the test object. The radiation emerging from the test ob ject will be of uniform intensity if there are no variations in the thickness or density of X RAY IMAGE the specimen. If the test object does have Fig. 5. Principle of X-ray film radi such variations, the intensity of the emer ography. gent radiation will vary. The radiographic process consists in making a record of these variations in intensity and interpreting the record in terms of thickness and density variations in the object. A radiograph * Superscript numbers refer to References


Sec. 10-6]

NONDESTRUCTIVE TESTING

10-155

a

:

/.

is

a

it

y

X

it

is

it,

is

should accurately record the size, shape, and location of any defects that are present. It is generally considered that the radiographic technique is capable of detecting a A " penetrometer," con variation of 2 per cent of the thickness of the test object. sisting of a small flat piece of the same material as the test object and of a thickness about equal to 2 per cent of the object thickness, with various size holes drilled often placed on the object during the exposure as a check of the quality through of the job. If the image of the penetrometer, including images of the holes, appears on the developed film, the radiograph considered satisfactory. The basic physics underlying the radiographic technique has been treated in detail in the bibliography at the end of this article. has been found that the intensity of a homogeneous beam of Experimentally rays or rays decreases as way that for small passes through material in such thickness Ax, the change in intensity Al proportional to the thickness and to the incident intensity This information can be summarized in the following expression

Al

-j

=

-/i Ax is

/

p

is a

where constant of proportionality and the negative sign used to indicate that the intensity has decreased. The above expression can be put into the exponential form

where

- 7oe-«*

= = x = = =

/o e

is

is

y

7

n

= exp

/pi — \pi

- —pi Pi

/J \

It

7-

(

/

:

6)

is

a

is

M|

is

is

it

a

it


/

intensity of rays that have passed through material intensity of rays impinging on material thickness of material base of Napierian logarithms linear absorption coefficient In the above expression, the linear absorption coefficient varies with the absorbing In general, the absorption coefficient of materials for X rays and material. rays rays to a greater increases with density; for example, steel reduces the intensity of extent than aluminum. The tabulated absorption coefficients given in the literature are usually mass absorption coefficients rather than linear absorption coefficients. The mass absorp tion coefficient equal to the linear absorption coefficient divided by the density of material. The mass absorption values are independent of the state of the material and of the state of aggregation. If the mass not absorption coefficient of a material known, can be calculated empirically.1 Two dissimilar materials can be detected in radiography because of difference in their absorption of the X rays. Generally stated, to preserve sensitivity, best to use the softest radiation that gives the required penetration. For locating small inhomogeneities in material, the maximum contrast obtainable on the radiograph p. The maximum contrast obtained desired. h-nS P2 p2~ p7p'^Xj by having the greatest possible difference in the ratio of the intensity of the X-ray Flo. 6. Differential X-ray absorption, beam transmitted through section free of defects to the intensity transmitted through a section of equal thickness containing The ratio of these intensities an exponential function of the difference of defects. Thin the product of the mass absorption coefficients and the density of each phase. ratio is given by the following expression (see Fig. pt

xj

REACTOR MATERIALS

10-156

[Sec. 10

where in/pi is the mass absorption coefficient of one material, m/pi is the mass absorp tion coefficient of the other material, pi is the density of one material, o? is the density of the other material, and i2 is the thickness of the foreign material. Many sources of radiation are available today. Table 1 gives a number of these sources and specifies the thickness range (of aluminum and steel) in which each source finds its greatest usefulness. Table 1 Practical thickness limits, in. Source

50-kv X rays 100-kv X rays

2.000-kv X rays 24-Mev Betatron


Cobalt 60

Steel

Aluminum

0.005-0.025 0.040-0.190 Up to 1 Up to 2.00

o.o su o ion 0.200-1.00 Up to 5.00

1-5 1-8 2-20

1-8 1-12

2-6 2-6

When an X-ray beam strikes a film, usually less than 1 per cent of the X-ray energy The use of calcium tungstate and lead screens makes it is absorbed by the film. The possible to utilize the wasted energy and, consequently, reduce exposure time. calcium tungstate screen intensifies because of the blue fluorescent light it emits when The lead screen intensifies because of the secondary electrons struck by X rays. ejected by the X rays. . X-Ray Beam

Al

Flo.

7. Schematic

diagram

Con Wall

of end-weld closure and experimental

arrangement.

One limitation of the radiographic method of nondestructive testing is the detection of a thin, flat discontinuity, such as a crack. If it is oriented so that the radiation passes along one of its larger dimensions, it may be detected, but. if the defect is oriented so that the radiation passes only through the smallest dimension in traversing the specimen, a record could be expected only if the width were at least a few per cent of the total specimen thickness. In some cases, it may be advisable to make more than one exposure, irradiating the object from different directions. Many of the radiographic techniques that are used in industry also find application in testing of reactor components. Details of these techniques can be found in the bibliography at the end of this article. Information and characteristics on X-ray film can be obtained from catalogues issued by the various film manufacturers.

Sec. 10-6]


10-157

One technique of special interest in testing for defects in closure welds of fuel Since the region being radiographed varies in thick elements is illustrated in Fig. 7. A fast film was used to record the inner edge ness, a double-film technique was used. of the weld, where the intensity was low, and a slow film was used to record the outer To determine the sensitivity of this technique samples of the same size and edge. Holes of various sizes and shape as the component to be tested should be made. Some of the holes should be filled with a depths can be drilled into these samples. lead oxide paste or similar material, and others left unfilled. These filled and unfilled holes then serve as voids and inclusions of known dimensions. A clad uranium fuel element can be inspected by making use of the 0.18-Mev y ray accompanying the a decay of uranium 235. X-ray film is placed on each side of a flat fuel element or wrapped around a cylindrical element. This will show if (1) the uranium is eccentrically displaced toward one side, (2) the core is thin or thick, or (3) a combination of (1) and (2) exists. It will also show defects provided the defects are greater than 2 per cent of the total thickness of the specimen. Instead of a film, a scintillation counter can be used as a detector for the y rays. The use of scintillation counters increases the sensitivity of the method. It should be emphasized that radiography and autoradiography bond and fusion bonds.


1.6

will not differentiate between metal-to-metal

Acoustic Methods

The technique of striking the test object with a hammer and listening for the char acteristic sound has been used since ancient times. Gross defects in castings and other structures can often be found by this technique. Measuring the rate at which the vibrations decay can be used as a qualitative inspection technique. The rate of decay can be measured by simple electronic timers or an oscilloscope. Measuring the rate of decay can also be used for other purposes, which will be discussed in the article on measuring physical and chemical properties. Ultrasonic waves can be used for detecting flaws in materials that are capable of transmitting elastic vibrations. Quartz crystals or barium titanate transducers are used to convert the electrical oscillations of a radio-frequency generator into mechanical (ultrasonic) vibrations and for the reverse process of converting mechanical vibrations into electrical signals. The physical laws of acoustics that apply at audible frequencies also apply at ultrasonic frequencies. However, certain phenomena that occur in the ultrasonic range are not usually observed in the audible range. Because of their short wavelength, ultrasonic waves tend to travel in straight lines. Ultrasonic waves are not reflected effectively by objects having dimensions less than the wavelength of the ultrasonic waves. An ultrasonic beam impinging on an interface between two media is partly reflected and partly transmitted in accordance with well-known laws. The characteristic that determines the amount of reflection is known as the acoustic impedance pV and is the product of the density of the media p and the velocity V of the waves in the media. The ratio of the reflected energy R to the incident energy Ho is given by 21 = Ro

feih

Uvi

~

g?I?V

+piVi)

The subscripts 1 and 2 indicate the two media. From this expression, it can be seen that at an air-solid interface nearly 100 per cent of the ultrasonic energy is reflected. Consequently, it is necessary to use a liquid couplant such as water, oil, or glycerin to transmit the acoustic energy from the transducer into the sample. This equation also shows why ultrasonic techniques are good for locating nonbonded areas. In order to obtain the maximum sensitivity and resolution, the ultrasonic beam should have as small a cross section as possible. There are three ways in which this can be done: by the use of small transducers, by the use of absorption masks, and by Absorption masks are made by covering the transmitting trans the use of lenses. ducer with an absorbing material through which a hole has been drilled. The prin ciple of physical optics carries over into the field of image formation with acoustic

10-158

REACTOR MATERIALS

[SfiC. 10

Velocity and Acoustic Impedance of Some Materials

Table 2.

Velocity Acoustic impedance g/(cm')(aec) X I0«

Material Cm/sec X 10-•

Aluminum Copper Nickel

6.22 4.30 4.60 5.60

Steel

5.81

Uranium Zirconium Polystyrene Mercury Water Oil (transformer) Air

3.37 4.62 2.67

Bran

Fps

20,700 14.300 15.300 16.600 19.400 11.200 15.400 8.900 4.900 4.750 4.650 1.100

1.46 1.43 1.39 0.331

1.70 3.61 4. II

4.98 4.76 6.27 3.10 0.294 1.930 0. 143

0.128 0.0000413


& 1 t A

I

1

INITIAL PULSE

§

ELAPSED

I

TIME

BACK REFLECTION

I

rz

T A

. Flo.

8.

Principle of ultrasonic reflection

technique.

Sec. 10-6]


10-159


Acoustic lenses concentrate the lenses and can be used in designing acoustic lenses. ultrasonic energy as well as limit the cross section of the beam. The higher the frequency, the smaller the defect that can be located; however, as the frequency is increased, the penetration of the beam is decreased and consequently The decreased a practical limit is set to the frequency that can be successfully used. penetration is due to the scattering of the waves by the grains of the metal. This effect is particularly large when the wavelength of the waves approaches the grain size in the metal. Velocities and acoustic impedances of longitudinal waves in materials that may be of interest and use to the reactor engineer are given in Table 2. Velocities of longi tudinal waves for materials not included in the table can be computed from the relationship V — [(a — l)E/(a + 1)(1 — 2a)p]M, where E is Young's modulus, a Velocity of shear waves is about Poisson's ratio, and p density of the material. 48 per cent that of longitudinal waves and is given by the following relationship: V = (G/p)H, where G is the modulus of rigidity. Three different ultrasonic techniques are in common use today: (1) pulse echo, (2) transmission, and (3) resonance. In the pulse-echo technique, which is shown in Fig. 8, a pulsed ultrasonic beam is sent into the sample. The beam is reflected from the opposite side of the sample as well as from In the echo that is displayed on any discontinuity or defect. an oscilloscope screen, a defect can be recognized by the rela In a variation of this technique the tive position of its echo. pulsed ultrasonic beam is projected into the sample at an oblique angle to the surface. The beam travels in a zigzag R R path between top and bottom of the sample, and the echoes 9. Principle of Fio. are picked up as in the above technique. ultrasonic transmis In the transmission technique shown in Fig. 9, a continuous sion technique. wave or a pulsed ultrasonic beam travels through the sample The energy lost by and the transmitted energy is received by a second transducer the beam in traveling through the sample reveals existing flaws and their extent. If continuous ultrasonic waves are used, it is necessary to prevent the occurrence of standing waves. In the resonance test, a mechanically driven tuning condenser sweeps the frequency The oscillator in turn drives the of an electronic oscillator through its tuning range. transducer. There are momentary increases in the energy drawn by the transducer whenever the frequency of the oscillator coincides with a thickness resonant frequency of the specimen. The increase in energy is indicated on a cathode-ray tube. A change in resonant frequency that cannot be accounted for as a change in thickness is usually an indication of a defect. This technique has been used for locating lack of bond in roll bonded material. However, it will not locate as small an unbonded area as the pulse-echo or transmission technique. 1.7

Electrical Methods

1.71 Electrode Methods. Electrical-resistance methods of nondestructive testing are based on the principle that the electrical resistivity in the immediate neighborhood of a defect differs from that in solid metal. With a metal sample of uniform dimen sions carrying electric current, there is a fixed potential difference between any two points at a fixed distance apart. The presence of a defect will increase the potential Clean, scale-free bare metal surfaces and pressure difference between the two points. contacts are required to avoid contact overheating or burning of the sample. Surface contact resistance is an important factor in such tests, and the surface contact resist ance may cause a greater variation in potential difference than the presence of a defect. Special electrodes have been designed to reduce the disturbing effect of surface contact resistance*'1 to a negligible factor. Alternating or direct current can * Superscript numbers refer to References at end of subsection.

10-160

REACTOR

MATERIALS

[Sec. 10

The alternating current gives better sensitivity for be used in electrical tests. surface cracks, while direct current gives good results in showing the presence of Both the electrodes can be placed on the same side of the test object, deeper defects. or they can be placed on opposite sides of the test object. One technique that can be used is to make the test object, part of a Wheatstone bridge. The initial balance of the bridge is obtained by using a sample that is known to be free of defects. In the technique described,1 the electrodes are placed on either side of the tesi The voltage (millivolt sample. Current (60 cycle) from 20 to 25 amp is used. range) developed across the sample is amplified, rectified, and observed visually on a By using this method, it is possible to detect cathode-ray oscilloscope or recorded. laminations and cracks and to differentiate between metal-to-metal bonds and fusion bonds. 1.72 Methods. Nondestructive testing using eddy currents is Eddy-current based on the principle that when a coil carrying alternating current is brought near a metal specimen, eddy currents are induced in the metal. The induced eddy currents are confined to the surface layers of the specimen. The depth at which the current has decreased to 1/e (or 37 per cent) of its value at the surface is called the "depth of penetration" and is given by the following:

S

where S = depth of penetration, in. = frequency, cps ii = permeability ( = 1 for nonmagnetic materials) p = resistivity, ohm-cm The above expression applies strictly to a plane solid. However, it may be extended to conductors of other shapes as long as S is much smaller than the radius of curvature of the surface. The eddy currents induced in the metal set up a magnetic field that opposes the original magnetic field. The impedance of the exciting coil or any pickup coils in close proximity to the sample is affected by the eddy currents in the metal. If the path of the eddy currents is distorted by the presence of a defect, as shown in Fig. 10, the apparent impedance of the coil is This change in impedance can changed. be measured and used to give an indica tion of defects or differences in physical structure. The impedance due to the presence of the induced eddy currents can be resolved into two components, an inductive com The ponent and a resistive component. inductive component depends primarily on the space relationship of coil and speci men. The resistive component depends The imped primarily on conductivity. ance diagram for the case in which the coil surrounds a solid metal rod is shown The dashed curves indicate in Fig. 11. Fio. 10. Principle of eddy-current test how the impedance changes when speci ing, cylindrical test specimen. mens of the same conductivity but different diameters are placed in a coil. The solid curves indicate how the impedance changes when specimens of fixed diameter but different conductivity are placed in the coil. Hochschild3 gives curves showing the effect of conductivity variations, diameter variations, and various types of flaws on the impedance of the test coil. With the use of the impedance diagram and associated curves, the selection of the proper test frequency can be calculated in terms of the dimensionless quantity Ka

/


= 1,980 (

Sec. 10-6] where K = (
NONDESTRUCTIVE

TESTING

10-161

o = radius of test sample conductivity of sample ii = permeability of sample = frequency Two types of coils can be used: a concentric-coil pickup and a point probe. The concentric-coil type of pickup encircles the sample and "looks" at the integrated resistivity of an annulus of the sample having a width equal to the effective width of the coil. The point probe "looks" at an area essentially equal to the area of the The configuration of the exciting and pickup coils can be varied to fit different probe. Changes in test conditions and samples. geometry such as change of sample diam eter, distance between probe and sample, and shape of the sample have a marked effect on the impedance of the test coil. A balancing technique is sometimes used in which there are two identical coils. One coil surrounds or rests on a standard ; the other surrounds or rests on the sample The difference between to be inspected. the outputs of the two coils is compared, and the difference in signal amplified. A number of eddy-current instruments They differ are available commercially. somewhat in circuitry, sensitivity, and ease of data presentation, although the fundamental principles of operation are the same. One commercial eddy-current is the Cyclograph. testing instrument The basic circuit for the Cyclograph con sists of an oscillator that drives the test The frequency of oscillation is de coil. termined by the resonant frequency of the O O.I 0.2 OJ 0.4 0JLo With the Cyclograph, only the test coil. Fig. 11. Normalized impedance plane resistive component of the impedance is curvi !3. Because of this fact, it is pos measured. sible for a change in the diameter of the sample to diminish or add to a resistivity change, giving a false indication. During the last few years the Forster Institute in Germany has developed a number Some of these instruments indicate of specialized eddy-current testing instruments. the magnitude of both the resistive and inductive components and allow the separation Consequently, it is possible to study of diameter changes from resistivity changes. the effect of various defects on one of the components independently of the other The Metals Comparator compares the impedance of a standard sample component. with that of the test object. It will indicate differences in conductivity but is rather insensitive to cracks and similar types of flaws. The Probolog is essentially an It has been used largely for measuring the effect of corrosion in inductance bridge. pipes. *• =


/

2

PHYSICAL AND CHEMICAL PROPERTIES

The nondestructive testing methods and techniques discussed can be used to measure physical and chemical properties of reactor materials. Elastic constants of solids can be determined acoustically by measuring the velocity of wave propagation and applying the equations for velocity given in Art. 1.6. Elastics constants can also be measured by exciting a suitably sized and shaped Relationships4 exist among specimen in transverse, longitudinal, or torsional modes. frequency, density, and elastic constants. Eddy-current methods can be used for detecting internal stresses. In the Hoch


10-162

REACTOR MATERIALS

[SEC.

10

schild3 reference the use of eddy-current techniques for detecting internal stresses is discussed. X-ray diffraction also provides a way to determine stress nondest rm The theory and techniques of this method can be found in standard X-ray tively. texts. rod are scattered by t In Ultrasonic waves traveling through a multicrystalline The amount of scattering increases as the size of the grain anl grains of the metal. If the loss in energy is measured wavelength of the wave become more nearly equal.6 as a function of frequency for samples having known grain size, it is possible to deter mine the grain size of samples of the same material. Eddy-current methods can he used as a method of measuring grain size, since there is a change in resistivity as the grain size changes. Preferred grain orientation often occurs during rolling, extrusion, and drawing of This orientation causes a directional variation in the mechanical properties metals. of the metal. The orientation can be analyzed by X-ray diffraction. The grain orientation can also be measured by ultrasonic techniques. The velocity of propaga tion of the ultrasonic waves will vary, depending on the direction of propagation. Changes in chemical composition of metals result in changes in resistivity. The changes may be determined by measuring the conductivity of the material either In In the d-c method conductivity of direct currents or by eddy-current techniques. the test sample is compared with the conductivity of a standard of the same size and In the eddy-current methods the resistance change is measured or compared shape. with a standard sample. When two metallurgically dissimilar metals in contact are moved against each other, a voltage is generated. The voltage generated depends upon the metals and may vary from a fraction of a microvolt to several millivolts. This phenomenon is referred to as the triboelectric effect. All metal alloys can be arranged in triboelertric series so that a metal which precedes another in the series will be found to be Heat treatment of most alloys pro positive with respect to the succeeding metal. duces chemical changes. The triboelectric effect can be used to distinguish among alloys that come from the same melt but have had different heat treatment. 3

THICKNESS MEASUREMENTS

Methods will be described for measuring the thickness of metallic and nonmetallio sheet material and metallic and nonmetallic coatings on various kinds of base materials. 3.1

Radiation

Absorption

X rays, y rays, /S particles, and a particles are absorbed as they pass through matter. Measurement of the radiation absorbed provides a basis for determining the thick ness of material. The type and energy of the radiation used depend on the nature and thickness of the material to be measured. The material to be gauged is inserted between the source of radiation and a suitable radiation detector. The output of the detector is calibrated in terms of the thickness of the material. X rays have the advantage that they can be adjusted within broad limits to the nature and thickness of the work by varying the anode voltage. For example, 0.4-mii aluminum foil can be measured with 5-kv X rays and 2-in. aluminum plates with An accuracy of + 1 per cent can be achieved over the whole range of thickness. 250 kv. As mentioned in Art. 1.5, the absorption of homogeneous X rays and y rays by material is given by the following relationship: /

=

Itf-i"

Berman and Harris8 have discussed the techniques in the precision measurement of It can be shown that to obtain maximum sensitivity materials by 7-ray transmission. for a fixed source intensity, the linear absorption coefficient should be equal to the reciprocal of the thickness. Gamma-ray sources are available from the United States Atomic Energy Commission that permit selecting the energy which most nearly fulfills the above condition.

nondestructive

Sec. 10-6]

testing

10-163

Table 3. List of Useful Beta Emitters (Available from Oak Kidge National Laboratory) Radioisotope

Yttrium

Maximum energy,

2. 54 days 14 . 3 days 61 days 53 days 13.7 days 4.0 years 3.08 X 10» years 2. 1 X I0» years 163 days 2 . 6 years 87 . 1 days 5.568 years 12.5 years

90

Thallium

Half-life

204

Sulfur 35

Mev

2. 18 1.712 1.537 1.50

0.93 0.765 0.71

0.29 0.254 0.223

0.167 0. 155

0.018

List of Useful Gamma Emitters (Available from Oak Ridge National Laboratory)

Table 4.


Radioisotope

Silver 110 Iron 59.

Cesium 137* (Ba 137) * Cesium 137. with a half-life of 37 years, barium 137.

Half-life

Maximum energy,

60 days 270 days 5 . 3 years 46 . 3 days 1 17 days 5 . 3 years 5 . 4 years 250 days 19.5 days 85 days 2 . 3 years 35 days 65 days 37 years (Csl") decays

to barium 137.

Mev

2.06 1.516 1.33 1.3 1.237

1.2 1.2 1.11 1.081

0.89 0.791

0.758 0.708 0.662 The 0.662 Mev is emitted by

Most X-ray gauging systems use two X-ray beams in a balanced-bridge or null This is done to avoid the necessity for extremely close regulation of anode system. Gamma-ray sources require longer time per measurement voltage and beam current. because of their low intensity. In the range of 1 to 50 mil of steel, soft X rays must be used; in the range from 50 mil to 2 in. of steel, X rays or soft y rays must be used; and in the range above 2 in. of steel, hard y rays must be used. Alpha particles have a range of several centimeters in gases but have a very short The range in aluminum for the most energetic o particles is of the range in solids. A few sheets of paper will completely stop most a particles. order of 0.05 mm. However, for measuring thin sheet foil such as capacitor paper or very thin sheet When an ionization chamber is used as the materials, an a gauge can be used. detector, 1 per cent change in weight can be detected. Beta rays are high-speed electrons that are emitted during the disintegration process The f) rays emitted from a radioactive source are of certain radioactive materials. Each ft emitter has a not monoenergetic but have a continuous energy distribution. It is customary in the literature to give only the maximum definite maximum energy. energy of the ft particle. Because of their small mass, ft particles do not travel in straight lines through an The absorption curve for a pure ft absorber but pursue a more or less random path.

10-164

REACTOR MATERIALS

[Sec. 10

The intensity falls off sharply for emitter looks somewhat like an exponential curve. small thicknesses of an absorber, since the low-energy 0 particles are rapidly absorbed. Thereafter the curve is fairly well represented by the exponential / = Ioe~i", where /* is the apparent linear absorption coefficient and x the thickness of the absorber (see The apparent mass absorption coefficient m/p has been found experimentally Fig. 12). to be nearly independent of the atomic weight of the absorber. 1.0 .9 .8

i

K

\

\

.6

\\

i

\\

\

\ \^

i

\

v

^


0

10

20

30

— »— .

i

40

Fia.

12.

1

ZIRCONIU STAINLES S

MILS

V

i

ALUMINUA

i

50

60

70

80

90

Absorption curve for

100 /3

110

120

130

140

150

160

170

180

190

rays emitted by uranium.

Beta-particle absorption depends essentially on the weight per unit area of the An estimate of the maximum thickness that can be measured absorbing material. using /3-ray transmission is given by the empirical relationship tP

where

-

0.546£„

-

0.16

t = p =

thickness of absorbing material, cm density of substance, g/cm3 E„ = maximum energy of (3 particle, Mev Beta-ray transmission can be used for measuring the thickness of steel in the range from 1 to 50 mil. 3.2

Back Scattering

X Rays and y Rays. When X rays or y rays impinge on material, scattered rays and y rays are emitted. Scattered radiation that leaves the absorber through the same surface as the entering primary radiation is called back-scattered radiation. The intensity and energy of the back-scattered radiation are dependent upon the scattering angle. Expressions for calculating the energy can be found in any standard reference on X rays. The amount of the back-scattered radiation depends on the The back-scattered radiation increases with scattering material and the thickness. increasing thickness up to a certain thickness that is determined by the energy of the X-ray or -y-ray source. 3.22 Beta Rays. Whenever 0 rays impinge on material, a fraction of the electrons The intensity of the back-scattered radiation may be reflected or back scattered. increases with increasing thickness of material until the thickness equals approxi A further increase in thickness does not mately one-fifth the range of the electrons. increase the intensity of the back-scattered radiation. The amount of back-scattered radiation increases with the atomic number of the material. Beta-ray back scattering can be used to measure the thickness of sheet material or coatings that do not exceed the limits set in the above paragraph. Zumwalt7 discusses the performance and sensitivity of /3 gauges.

X

3.21


Sec. 10-6] 3.3

10-165

X-ray Fluorescence and Reflection

When material is irradiated with photons of sufficient energy, the energy is absorbed This phenomenon is known as fluorescence. and partially reradiated. The fluores cent or reradiated radiation is monochromatic, having a wavelength that is character istic of the fluorescing material. This technique can be used to measure the thickness of plating up to about 0.001 in. The reflection of X rays has been used to determine the thickness of thin coatings. The X rays pass through the coating and are reflected by the base material. The intensity of the X rays is reduced by absorption due to the double transmission through the coating. This method can be used in measuring the thickness of metal films in the range from 10-' to 10_* cm.* 3.4

Autoradiography

In the case of reactor fuel elements another technique is available for measuring Uranium is radioactive. Two of the coating thickness, namely, autoradiography. emitted radiations have been used to measure the cladding thickness of fuel elements: the 2.3- Mev 0 particle emitted by a daughter of U"8 and the 0.18-Mev y ray emitted


by U!".

One way to measure the intensity of the radiation transmitted through the cladding is to use X-ray film placed in contact with the fuel element. This requires several hours exposure time as well as development and measurement of film density. A scintillation counter can also be used to measure the intensity of the transmitted radiation. Sodium iodide scintillator crystals can be used to measure y radiation, and anthracene crystals can be used to detect 0 particles. For the measurement of zirconium and stainless cladding, from 0 to 20 mils in thickness, a 10 per cent change in counting rate per mil is obtained measuring the For thickness greater than 30 mils the sensitivity decreases. For (3 absorption. aluminum cladding, the sensitivity is approximately one-third that for zirconium and stainless steel. The sensitivity to changes in cladding thickness is much less when the absorption of the y radiation by the cladding is measured. 3.6

Ultrasonics

The ultrasonic The principle of

resonance technique can be used to measure the thickness of metal. the resonance technique was described in Art. 1.6 on acoustic tests. Resonance occurs when the thickness of the metal is an integral number of half At resonance the relationslup between thickness t and wavelength X is wavelengths. n\/2 = t, where n is an integral. Also fx = V, where is the resonant frequency and V velocity; therefore I = nV/2f. Resonance will also occur at other frequencies that are harmonics of the fundamental. Consequently, in the case where the thickness of the medium is entirely unknown, it may not be obvious if the fundamental or which of the harmonics is in thickness resonance with the test sample. In this case, if the velocity of sound in the medium is known, the thickness is given by t = V/(2 Af), where A/ is the difference in two adjacent frequencies at which resonance occurs. In some of the commercial instruments the oscilloscope or a meter is calibrated direct ly in terms of thickness of steel. The instruments can be adapted for measuring other materials if the velocity of sound in the metal is known.

/

3.6

The

Magnetic Attraction (Magne-Gage)

operation of the Magne-Gage and similar devices

is based

on the magnetic

attraction of a small permanent magnet to the material being measured and/or its backing. The magnet is suspended freely from a horizontal lever and is counter

10-166

REACTOK

MATERIALS

[Sec. 10

The spring is wound by rotating a graduated dial until balanced by a spiral spring. the permanent magnet is freed from the specimen. The dial reading is compared with a calibrated curve to give an accurate measurement of thickness. Curved surfaces as well as plane surfaces can be measured.


3.7

Electrical Resistance

The resistance method can be used for measuring the wall thickness of plates, sheets, and tubes from one side only. The electric current is introduced in the test object by means of a pair of current electrodes. The potential drop is measured by means of an independent set of potential electrodes. In a relatively thin plate the resistance between two points on the surface is inversely proportional to the thickness of the plate and the total current flowing is proportional to the thickness. Thick nesses of mild steel up to in. can be measured with an accuracy of approximately A modification of this method has been made in which the results do not 5 per cent. depend upon the conductivity of the metal examined.* Eddy-current methods have been used for the measurement of the thickness of both metallic and nonmctallic coatings on metals. Coating thickness of oil, oxide, paint, or other nonconductors may be determined by measuring the change in impedance when the test coil is placed on the sample. The impedance change is due to the fact that intensity of the induced eddy currents varies with the distance between the coil and the base metal. The impedance change can be measured in various ways. One method that is used measures the change in frequency caused by the inductance change in the probe coil. Brenner10 describes an eddy-current thickness gauge that can be used for measuring coatings on metals provided the conductivities of the coating and base metal differ sufficiently. Measurements can be made on combinations of materials regardless of whether they are magnetic or nonmagnetic. 3.8

Pulsed Eddy Current

Pulsed eddy currents have been used for measuring thickness of metallic coatings. An electromagnetic field is applied at the surface of the clad metal by means of a probe coil. Echoes are produced whenever the wave encounters a sudden discontinuity The same probe coil is used for sending the such as that caused by an interface. The echo received from the electromagnetic radiation and for receiving the echoes. The echo front surface of the sample is balanced out by means of a bridge circuit. from the coating-base-metal interface contains information concerning the thickness This echo has a positive peak followed by a small negative peak. of the coating. The amplitude length or crossing point between the positive and negative peak can be used to determine the thickness of clad. The magnitude of the reflection between the clad and base metal is determined by the permeability and conductivity of the two metals. Forster has described an eddy-current method in which the flat conductor is placed between an exciting and a pickup coil. The presence of the metal between the coils alters the potential of the pickup coil. The change is determined by the frequency of the eddy currents, the kind of material, and the thickness of the material.

REFERENCES McGonnagle, W. J., J. H. Monawcck, and W. G. Marburger: Methods of Bond Testing. Non-Destructive Tenting, 13: 17-22 (1955). 2. Victoreen, J. A.: The Calculation of X-ray Mass Absorption Coefficients, J- Appl Phys., SO: 1141-1147 (1949). 3. Hochschild, R.: The Theory of Eddy Current Testing, Xon-Dcst motive Testing. It: 35-44 (1954). 4. Den Hartog, J. P.: "Mechanical Vibrations," 4th ed., p. 429, McGraw-Hill Book 1.

5.

Company, Inc., New York, 195C. Worlton, D. L.: Nondestructive Grain Size Measurements, 13: 24-26 (1955).

Non-Deatructire

Te*tin§.

Sec. 10-6]

nondestructive

testing

10-167

A. I., and J. N. Harris: Precision Measurements of Uniformity of Materials by Gamma Ray Transmission, Rev. Set. Instr., 25: 21-29 (1954). 7. Zumwalt, L. R.: The Best Performance from Beta Gages, Nucleonics, 13: 55-58 (1954). 8. Friedman, H., and L. S. Bicks: Thickness Measurements of Thin Coatings by X-ray Absorption, Rev. Set. Instr., 17: 99-101 (1946). 9. Thornton, B. M.: A Direct Current Method of Determining the Thickness of Metal Walls from One Side Only, Proc. Inst. Mech. Engrs., 140: 387-392 (1938). 10. Brenner, A., and J. Garcia-Rivera: An Electronic Thickness Gage, Plating, November, 1953, pp. 2-8. 6. Berraan,

BIBLIOGRAPHY Carlin, Benson: "Ultrasonics," McGraw-Hill Book Company, Inc., New York, 1949. Carlin, J. R.: Radioactive Thickness Gage for Moving Material, Electronics, 22: 110

113


(1949).

Clapp, C. W., and 8. Berstein: Noncontaining Beta-Ray Thickness Gage, Gen. Elec. Rev., 63:31-34 (1950). Clarke, E., J. R. Carlin, and W. E. Barbour: Measuring the Thickness of Thin Coatings with Radiation Backseat tering, Elec. Eng., 70: 35-37 (1951). Clausen, J. R.: "Practical Radiography for Industry," Reinhold Publishing Corporation, New York, 1952. Eastman Kodak Company: "Radiography in Modern Industry." Fine, M. E. : Dynamic Methods for Determining the Elastic Constants and Their Tempera ture Variation in Metals, ASTM-STP, 129: 43-67 (1952). Forster, F.: Theoretical Principles of Contactless Determination of Thickness and Con ductance of Metallic Coatings, Foils and Sheet, Z. Metallkunde, 45: 197-199 (1954). Forster, F.: "Quick and Precise Routine Determination of the Thickness of Steel Sheet from One Side," Report 26, Institute Dr. Forster, Rcutlingen (Germany), 1953. Hanstock, R. F.: "The Nondestructive Testing of Metals," The Institute of Metals, London, 1951. Harris, J. N., and L. R. Megill: Techniques Used in Measuring Uniformity of Metals with Gamma Radiation, Non-Destructive Testing, 11: 9-13 (1953). Jones, Clark R.: Performance of Detectors for Visible and Infrared Radiation, "Advances in Electronics," vol. 5, Academic Press, Inc., New York, 1953. Lewis, D. M.: "Magnetic and Electrical Methods of Nondestructive Testing," George Allen & Unwin, Ltd., London, 1951. Spronel, Wayne T.: "X-Rays in Practice," McGraw-Hill Book Company, Inc., New York, 1946.

Waidelich, D. L.: Coating Thickness Measurements by Pulsed Eddy Currents, Proc. Nat. Electronics Con/., Feb. 12, 1955.

10-7

RECENT DEVELOPMENTS IN REACTOR MATERIALS BY


Russell W. Dayton Much information on reactor materials has become available since Mr. Sailer 's contribution (Sec. 10-2) was written. This additional information consists largely of detailed data on new materials rather than revisions of data on old material?: hence it seemed best to add the present section to incorporate these new data instead of revising Mr. Sailer's section, which remains quite serviceable. The two sections do not appreciably overlap. The most important recent developments have concerned fuel materials. New alloys of uranium with zirconium, niobium, molybdenum, and silicon have been studied intensively, and significant results are available for uranium-plutonium alloys. Even more striking has been the development of ceramic fuels. Dense uranium oxide is finding increasing use as a reactor fuel, and applications of other ceramic fuels and dispersions of ceramic fuels in metal matrices are now being studied intensively. The properties of these new reactor fuels, and of new cladding and control materials, will be discussed in this section. 1

FUEL AND FERTILE MATERIALS 1.1

Uranium Alloys

Unalloyed uranium has poor corrosion resistance in oxidizing media, and poor When irradiated at low temperatures uranium will resistance to radiation damage. warp, grow bumps, and elongate after brief exposures, although no serious changes in volume will result. These defects can be greatly improved by 0-phase heat-treat ment, or by adding small amounts of alloy such as 2 weight per cent zirconium. At higher temperatures (over 400 to 500°C) volume swelling becomes a problem as a result of the nucleation and growth of fission-gas bubbles in the metal. More over, it is a problem that cannot be solved by adding small amounts of alloying elements, because to achieve a significant reduction in volume swelling such element* must be added in quantities that may result in important changes in crystal structure. Neither is heat-treatment effective at elevated temperatures. Improved corrosion resistance and reduction of fission-gas swelling are the primary In thermal-neutron reactors, a factors motivating the development of fuel alloys. satisfactory means of reducing swelling has been difficult to achieve because the amounts of alloying materials required to control it seriously penalize the neutron economy of the reactor. This conflict of requirements has made it necessary to seek a balance between the reliability of highly alloyed uranium and the good neutron economy of nearly pure uranium. Alloys which in varying degrees exhibit this balance are described below. Uranium alloys with small amounts of alloying additions are not described, since their properties and fabrication are not importantly different from those of unalloyed uranium. " 1.11 Uranium alloyed with 4 to weight per cent Alpha-phase Uranium Alloys. zirconium and niobium responds much differently to heat-treatment than dorrOn quenching from the uranium with smaller amounts of these alloying materials. 7-pha8e region, an a phase having a distorted structure forms by a martensitic trans 10-168

Sec. 10-7]

recent developments in reactor materials 10-169

This structure is thought to be a supersaturated a phase. On aging, occurs, and on overaging, a second phase is precipitated. A quenched uranium-5 weight per cent zirconium alloy, having a hardness of 534 VHN, has 1,000 times better corrosion resistance in boiling water than unalloyed uranium. This material also has much better resistance to radiation damage than low-alloy uranium, but does not show as good a resistance to either radiation or corrosion as do certain uranium alloys with larger amounts of alloy. An alloy of this distorted-** type, uranium-5 weight per cent zirconium-1.5 weight per cent niobium, was selected for the Experimental Boiling Water Reactor (EBWR) at Argonne.1'* This material is used in the heat-treated condition, since heat-treat ment can improve resistance to both corrosion and radiation damage. Transformations. The best heat-treatment for corrosion resistance is a y quench from approximately 800°C in the as-cast condition. Overaging is detrimental to the corrosion resistance. The best treatment for resistance to radiation damage is to heat the material to 800°C, then reduce the temperature to 650°C, hold for 24 hr, and quench. If the treatment for radiation stability is followed by an 800°C heattreatment for corrosion resistance, dimensional stability is lost. Conversely, irradia tion decreases corrosion resistance of 7-quenched material. Thus, the requirement for corrosion resistance conflicts with that for resistance to radiation damage.1 In the EBWR, the material is used as treated for radiation stability. Although the corrosion resistance is therefore not high in this condition, corrosion failure through a defect is still much slower than for unalloyed uranium, and there is time for corrective measures to be taken. The progress of the corrosion failure of a defective EBWR plate is given in Table 1. formation.

Corrosion of EBWR-type Fuel Plate with Known Defect in 660°F Degassed Water

Period

Time, hr

i

6

Table 1.

2 8 25 24 24 24

7

24

Appearance

of defect

Hole filled with oxide .'is-in. ruptured ring around hole 1

îa-in. -diameter

hole

s-in.-diameter hole. Small bulge on opposite side of plate 'ii-in.-diameter hole. Cladding broken through on opposite side Swelling enlarged around hole. Ring on opposite No change in diameter. cracked half way around Enlarged blister ruptured in addition J-i

2 3 4

i

Source: C. R. Breden, "The Experimental Boiling Water Reactor," Appendix C, Properties Material*, ANL-5607 (May, 1957).

side

of Fuel

is

Production of the EBWR fuel plate as follows.' The uranium Manufacture. induction-melted in a graphite crucible coated with a layer of magnesium drconate and a second layer of thoria. The alloy superheated to 1550°C to dis«jlve the alloying elements, cooled to 1350°C for outgassing, heated to 1450°C, and of such size >ottom-poured in multiple-cavity graphite molds. Each casting hat, after machining, can be used as the core blank for the roll-bonding process. These core blanks are enclosed in Zircaloy-2 jackets built up from strip comwnents, and the assembly then edge-welded using the inert-gas-shielded tungstentrc-welding process. The "pack" evacuated through a hole left in the end, which s then sealed by welding a pin in the hole. These packs are jacketed in mild steel, irid roll-bonded, using a total of 80 per cent reduction of thickness achieved in 12 The plates are then heated to 810°C, isothermally quenched to 030°C, and Misses. tged at that temperature for 24 hr. After decanning, the plates are machined to ength and width, and their surfaces are wet-abrasive-blasted and pickled. Gamma-phase Alloys. Alloying uranium with 10 to 20 weight per cent of 1.12 aolybdenum or niobium permits the phase to be retained on quenching from an Jevated temperature. Such alloys have far greater resistance to corrosion and y

is

is

it

* Superscript

numbers

refer to References


a

is

is

illoy

is


hardening

10-170

REACTOR MATERIALS

[SEC. 10

Table 2. Physical Properties of Uranium Alloys For y phase of uranium-molybdenum and uranium-niobium unless indicated U-Mo (9-

15 weight % Mo) 1 > 570°C. Ua + t< 570°C

-

Crystal struoture.

phase

at

room

tem

perature

A; Ua orthorhombic (very limited solubility of Nb) ; Nb bee. ao - 3.3 A, limited solu bility of U

>bcc, ao «■ 3.40

7 retained in 8 weight % Mo cooled in 1 hr from 900°C. and 12 weight % Mo cooled in I day from 900°C

y retained

Thermal conductivity (I0-100°C), cal/(sec)(cm)(°C)

0.034 for 8 weight % Mo 0.033 for 12 weight % Mo

Thermal coefficient of expansion (100-400°C), I0"«/°C

13.8 ± 0.5 for 8 weight % Mo 13.0 ± 0.5 for 12 weight % Mo

Nb)

%

y > 600°C. U
■yboc,ao 3.396 A for 12 weight % Mo 17a orthorhombic (very limited Mo); c ordered solubility of tetragonal ao = 3.427A, eo = 9.834, 2.871 r/n

-

Mctastablo

U-Nb (6-20 weight

by the *low cooling

Electrical resistivity, /* ohm-cm. . . 73 at 25°C. 65 after 42 days at 400°C Specific

heat

(300-400°C),


(mol)CC)

oal/

7-8 for 12 weight % Mo

Melting point, °C

1150 for 12 weight % Mo

Density, g/cm*

17.5 for 8 weight % Mo 16.9 for 12 weight % Mo


180 for 6 weight % Mo, 300 for 12 weight % Mo; 630 and 500 maxi mum, respectively, depending on

-1300 for 10 weight

heat-treatment

U-Si (3.8 weight % Si) . < 930°C, U + UiSh > 930°C Crystal struoture Metastable

phase

< tetragonal, 8.681 A

at room

tem

perature

ao — 6.018

e thremodynamically 930°C

A, eo —

stable

below

% Nb

180 for 8 weight % Nb. 350 for I 2 weight % Nb; 600 and 575 maximum, respectively, depending on heat-treat ment

Zr. 45 to 55 weight % i < 593, 0 > 593°C

Possibly boo, ao

-

10.69 A

f thermodynamieally below

stable

593-C

Thermal conductivity (10-1 00°C) , cal/(8ec)(cm)CC)

0.036 at 25°C 0.042 at 65°C

0.014 at 35°C

Thermal coefficient of expansion (I00-400°C), 10 '/°C

13.8 ± 0.5

9-11

Electrical resistivity, /i ohm-cm. .

56 at 25°C

164 at room temperature

Molting point, °C

V

~I500

Density, g/cm*

15.5 + 0.1

97


220 ± 10

-•250 annealed ^400 maximum

Source: R. K. McGeary, Alloys without Protective Cladding, WAPD-127. Part I (Apr.

25. 1955).

Sec. 10-7]


radiation damage than do a-phase uranium alloys, and they have been considered for use in both water-cooled and sodium-cooled reactors because of their good resist ance to radiation damage. A Zircaloy-clad fuel element containing a core of these alloys with an intentional defect in the cladding will resist failure for hundreds of days in 650°F water.3 The 7 phase of these alloys is unstable at the temperatures of use in prcssurizedwater reactors. If transformation occurs during which a-phase uranium and a second phase are formed, resistance to corrosion and radiation damage is adversely affected. Fortunately, radiation stabilizes the y phase, so that corrosion resistance is better in reactor use than in out-of-pile testing.4 Physical Properties. Some of the physical properties of the y phase of U-Mo and U-Nb alloys are given in Table 2. alloy, initial transformation is In the uranium-molybdenum Transformation. detected in a uranium- 12 weight per cent molybdenum alloy in a minimum time of 50 hr at a temperature of 450°C.5 For a uranium-10 weight per cent niobium alloy, initial transformation, as detected by resistivity change, is detected in a minimum of 20 min at a temperature of 500°C.' Mechanical Properties. The tensile properties of a uranium-12 weight per cent molybdenum alloy in various conditions of heat-treatment are given in Table 3. The tensile properties for this alloy over a range of temperatures are given in Table 4. Impact properties are given in Table 5.


Table

3.

Tensile Properties of Uranium-12 Weight % Molybdenum 0.05 In./min Strain Rate 0.5 % yield strength, 1.000 pel

Condition

At 25'C

1. 900°C/24 hr/WQ 2. (1) + 500°C/72 day« 1. 900°C/24

Tensile strength. 1.000 psi

166 156

Tested at

Reduction

Total %

of area, %

9.0

41.0

elongation,

166 157

10.4

107 100

4.0

32

At 3I6°C

hr/WQ

106 99 At 260°C 114 102 113

2. 900°C/24 hr/WQ 3. (2) + 500°C/40 hr

3.8

120 104 116

6. 1

118 103 104

6.8 5.0

6.8 5.2

At 3I6°C 108 100 102

2. 900°C/24 hr/WQ 3. (2) + 500°C/3 hr

23.3 32

5. 5

15.4 50 37 14.8 33 45

Source: R. K. McGeary, Alloys without Protective Cladding, WAPD-127. Part I (Apr.

25. 1955).

Table 4. Effect of Test Temperature on Tensile Properties of Uranium-12 Weight % Molybdenum Alloy Annealed 24 hr at 900°C and Water-quenched and -tested at 0.06 In./min Strain Rate Temperature, °C

0.5% yield strength, 1.000 psi

-65 -31

- 15 0 260 316

189 179 102 100

Tensile

Total

strength, 1.000 psi

elongation,

226 167 189 180 104 103

0 0

%

3.6 4.9

6.8 5.0


0 0 36

7.8 50 33

Socbce: R. K. McGeary. Alloys without Protective Cladding, WAPD-127. Part I (Apr.

25. 1955).

REACTOR MATERIALS

10-172 Table

6.

of U-12 Weight

Impact Properties

Test temp.,

Condition

hrs/WQ 900°C/24 hrs/WQ 900°C/24 hrs/WQ 900°C/24 hrs/WQ As-extruded

°C

9O0°C/24

..

[SEC. 10 % Mo Alloy

Absorbed energy, ft lb

2.0 2.6 8.9

25 66 66 71 25

Fracture

Brittle Brittle

14.5

5.7

Brittle

Source: R. K. McGeary, "Development and Properties of Uranium-base Alloys Corrosion Resistant in High Temperature Water," WAPD-127 (Apr. 25, 1955).

Fabrication. These alloys, particularly the niobium one, have high melting points, so that large amounts of carbon, which has a deleterious effect on corrosion resistance, Coating are picked up from an uncoated graphite crucible in induction melting. Tungsten the crucible with a BeO wash reduces contamination to a permissible level. or consumable-electrode arc melting avoids contamination, but introduces a form of macroscopic inhomogeneity which adversely affects performance.7 The microscopic inhomogeneity of the cast structure can be removed by high-temperature homogenization heat-treatments, with beneficial effects on corrosion resistance, as shown in Table 6.


Table 6.

Effect of Homogenizing

at 900°C

on the Corrosion Life of U-12 Weight %

Mo Extrusions

Unclad samples in 650°F water Corrosion life, dnys

As -extruded

24 hr at 900°C,

21-28 14-21 21-28 14-21 14-21 14-28 7-21 28-42

35-42 35-49 28-42 28-35 42-58 28-42 28-42 42-56

21

39

WQ

" Development and Properties of Uranium-base Alloys Corrosion Resistant Source: M. W. Burkart, in High Temperature Water," WAPD-127. Part III (July, 1956).

Extrusion cladding is one of the best means The alloys can be clad with Zircaloy. Speed is desirable because the diffusion zone of cladding, because of its speed. between the two alloys contains some layers having poor corrosion resistance (sec Table 7), and by minimizing the time during which the two alloys are in intimate contact under temperatures at which diffusion is rapid the thickness of these layers is minimized. 1.13 Zirconium containing 45 to 55 weight per cent Uranium-Zirconium Alloys. uranium forms a single-phase alloy, the < phase, which has good mechanical and The alloy is of physical properties, and good resistance to radiation and corrosion. reactors, though the low volume fraction of possible interest in natural-uranium Alloys with less uranium uranium which it contains is a serious handicap for this use. are useful in reactors fueled with enriched uranium. Ph/sical Properties. Some of the physical properties of the e-phase alloy are given in

Table

2.

Mechanical Properties. The mechanical properties of alloys of zirconium with 10, 22, and 41 weight per cent uranium in various heat-treated conditions are given in Tables 8, 9, and 10, respectively.


Sec. 10-7] Table 7.

Analyses of Layers Present in the Interface Formed between Zircaloy and Uranium-12 Weight % Molybdenum Analysis, weight

%

Layer

1 2 3 4

Remarks Mo

Sn

0. IS 3. 1

1.45 1.3 0,65

7.4 9.2

J

0.5 <0.5

12. 3

6 7 8 9

<0.5 <0.5

12. 5 12.8 12.3

U

2. 1 18.6

45.8

59. 1

78.0 'ii'.'i 85.3 89.0

Zr

Layer adjacent

94.0 84.0 43.0 35.0

to Zircaloy 2

Some corrosion attack on this layer after 72 hour Corrosion failure in \4 hr Identified by X ray as ZrMoi Corrosion failure in this layer near ZrMot layer

18.0

i's

15

Core alloy

Source: R. W. Dayton, "The Metallurgy and Fabrication of Uranium-Alloy Fuel ElementB," Conference (August, 1957). Table 8.

Mechanical

Properties of Zirconium-10


Room- temperature

Heattreatment

2


1 2 3 4

Treatment I : Treatment 2: f u rnace-cooled . Treatment 3: water-quenched. Treatment 4: f urnace-cooled .

77.800 54.500 88.100 40.800

Ultimate

Paris

Weight % Uranium Alloy

tensile properties

Elongation to

strength, psi

maximum %

109.200 79.900 105.600 71.800

4 13 4 19

load,

Total elongation in I in.,

8 17 8 32

Hardness

Reduction area, %

DPH

13 20 10 32

221 203 242 180

59 56 60 51

Hot-rolled at 700°C ( 1300°F) and air-cooled. Hot-rolled at 700°C (I300°F) and air-cooled;

reheated

to 700°C (I300°F) for 4 hr and

Hot-rolled at 700°C (I300°F) and air-cooled;

reheated

to 900°C (1650°F)

Hot-rolled at 700°C (I300°F) and air-cooled;

reheated to 800°C (I470°F) for I hr and

Source: Frank A. (Aug. 19. 1955).

Rough,

"An Evaluation

of Data on Zirconium-Uranium

Alloys."

for 1 hr and

BMI-1030

Fabrication. Cold-hearth arc melting is the only satisfactory melting method for Tungsten-electrode arc melting can be used, but consumable-electrode these alloys. melting is faster, cheaper, and gives a sounder ingot. To prepare these alloys, a con sumable electrode can be formed of the right proportions of uranium and zirconium.' Forging and rolling can be performed at 760 to 900°C without difficulty. Cold fabrication is more difficult for 20 to 40 weight per cent alloys, and frequent annealing

is required.' The alloys are easy to clad with Zircaloy by extrusion, roll bonding, or pressure With the alloys containing larger amounts of uranium, the bond line may bonding. contain brittle layers if cooling from the bonding temperature has been rapid. This occurs because alloys with about 20 weight per cent uranium, which exist in the bond

line

when higher uranium alloys are clad, become quite brittle when rapidly cooled. Corrosion Resistance. In 600°F water, zirconium-uranium alloys with 45 to 55 weight per cent uranium corrode at the rate of 0.05 to 0.10 mg/(cm2)(hr), 100 times faster than zirconium but far less rapidly than uranium. With more than 55 weight per cent uranium, or if oxygen or nitrogen exceed a few thousand ppm, a uranium separates, with adverse effects on corrosion resistance.

10-174

REACTOH MATERIALS

Table 9.

Heattreatment

1 2 3 4 5 5. 400°C 5. 450°C 5. 500°C

Mechanical Test

tem perature

Properties of Zirconium

22 Weight

Elongation

Total elongation in 1 in..

0.2 % offset yield

strength,

Room Room Room Room Room (750°F)

(840°F) (930°F)

[SEC. 10

Ultimate strength,

pei

102.800 94.000 *

psi

to maxi mum load,

5 12 *

120.200 101.900 83.400 106.600 103.500 67.300 59.200 48.000

81.400 59.700 43.700 41.000 36.900

16 17 0 3 23 24 29 48

2 21 14 10 3

Alloy

% Uranium

Hardness

Reduction of area.

23 25 0 2 24 37 34 69

DPH

Ra

240 238 387 274 245

61 61 72 65 60

Treatment I: Hot-rolled at 675°C (1250°F) and furnace-cooled. Treatment 2: Hot-rolled at 675°C (I250°F) and furnace-cooled;


reheated to 575°C (1065°F) for 24 hr and furnace-cooled. Treatment 3: Hot-rolled at 675°C (I250°F), heated to 800°C (I470°F) for I hr, and water-quenched. Treatment 4: Hot-rolled at 675°C (I250°F) heated to 800°C (1470°F) for I hr and air-cooled in a Vycor capsule. Treatment 5: Hot-rolled at 675°C (1250°F) to 800°C (M70°F) for 1 hr and furnace-cooled at a rat* of 100°C (l80°F)/hr.

Source: Frank A. Rough, "An Evaluation of Data on Zirconium-Uranium (Aug. 19. 1955). * Specimens failed in reduced section in brittle manner.

Table 10.

Heattreatment

1 2 3 4 s

Mechanical


111.600 83.600 80.200 53.600

Properties

of Zirconium-41

Weight

Room-temperature tensile properties

Ultimate strength,

Elongation to maximum load,

psi

92.000 144.400 133.000 126.200 73.400

%

3 4 3 1

Total elongation in 1 in.,

BM1-1030

% Uranium Alloy

Hardness

Reduction

rr

of area, %

0 4 5 4 2

0 1 5 4 5

Treatment I: Hot-rolled at 760°C (MOOT) and furnace-cooled; reheated and water-quenched. Treatment 2: Hot-rolled at 760°C (MOOT) and furnace-cooled; reheated and air-cooled in a Vycor capsule. Treatment 3: Hot-rolled at 760°C (MOOT) and furnace-cooled. Treatment 4: Hot rolled at 760°C ( 1400°F) and furnace-cooled ; reheated and furnace-cooled. from Treatment 5: Hot-rolled at 900°C (I650T) and water-quenched (I070T) for 24 hr and furnace-cooled. Source; Frank A. Rough, "An Evaluation (Aug. 19. 1955).

Alloys,"

DPH

.',■.

61 65 65 64 60

227 325 317 287 243

to 700°C

(MOOT) for

to 700°C

( I 300°F)

to 575°C ( 1070°F) 900°C;

of Data on Zirconium-Uranium

reheated

Alloys,"

4 hr

for 4 hr for 24 hr to 575°C

BMI-1030

With less than 45 weight per cent uranium, a zirconium separates, and when there is enough of this phase, corrosion resistance of the alloys increases, approaching that of zirconium. When clad with Zircaloy and corrosion-tested with a defect in the cladding, cor rosion progresses slowly. Specimens have been tested for over a year, with only a small pimple appearing at the site of the defect. The corrosion properties of these alloys have been discussed in detail.* 1.14 Uranium-3.8 Weight Per Cent Silicon Alloy. This is an interesting material The because of the large amount of uranium and the low cross section of silicon. material undergoes marked volume change and loses corrosion resistance on brief

Sec. 10-7]


exposure to radiation, changes can he found.

and will prove unsatisfactory

unless ways to prevent such

The physical properties of the alloy are given in Table 2. The e phase has approximately the composition U3S1. It is the Constitution. stable phase at temperatures below 930°C (the peritectoid temperature) and is formed by annealing the cast structure (which consists of uranium plus U3Si2) below The best epsilonizing treatment is exposure for 1 week at a that temperature. temperature of 800°C, which proves satisfactory if the carbon content is less than With higher carbon content, epsilonizing must be done at lower tempera 300 ppm. tures (since carbon reduces the peritectoid temperature by 100°C for 1,000 ppm).


Physical Properties.

This causes the reaction to become sluggish and makes epsilonizing impractical. There is no appreciable solubility of the e phase for either uranium Fabrication. or U3S1J, so that to achieve the desirable characteristics of this phase, it is necessary that the composition be nearly exact and homogeneity high. This requires very Induction careful melting practice, with precautions against carbon contamination. The < phase has little ductility, melting in BeO-coated graphite crucibles is best.7 However, the as-cast and although it can be coextruded with a Zircaloy cladding. Coexepsilonized material has better corrosion resistance than the fabricated alloy. truding this alloy with a Zircaloy cladding is unsatisfactory because the interface Coextrusion was found to give a diffusion has poor corrosion resistance in water. layer consisting of a layer of ZrSij next to the cladding and a layer of a uranium next This bond had extremely poor corrosion resistance and was unsatis to the U*Si. Furthermore, the extrusion produced stringers of a uranium in the fuel, so factory. that its corrosion resistance was not so good as that of the cast alloy. The best fuel elements of UsSi clad with zirconium were obtained by brazing with Some samples produced in this way an aluminum-6 weight per cent copper alloy. had fairly good corrosion life when tested with intentional defects in the cladding; however, difficulties were experienced with the formation of brittle layers of UAlj.10 1.16 Uranium-Fissium and Uranium-Plutonium-Fissium Alloys. If a radiated fuel is reprocessed by a particular type of pyrometallurgical processing11 which removes the major fission products, certain other fission products remain with the uranium and will build up to equilibrium levels. Table 11.

Range of Fissium Composition for U"5 Loading Element

Molybdenum

Weight %

0. 1-0.2

1.6-3.4 0.5-1.0 1.2-2.6 0.2-0.5

0. 1-0. 1

Source: L. R. Kelman, "Fast Reactor Fuel Development at FGF-73. Paria Conference (November, 1957).

Argonne

National Laboratory," ANL-

products have been called "fissium," and have the com The molybdenum and ruthenium of this mixture have a beneficial effect on the radiation stability of uranium and uranium-plutonium alloys, and also suppress the undesirable f phase of the latter alloy. These fissium alloys are being investigated for use in fast breeder reactors. The compositions of the alloys receiving most attention are given in Table 12. Physical Properties. The available data on the physical properties of these alloys are given in Tables 13 and 14. With more than 15 weight per cent plutonium in uranium, the f Constitution. phase appears. With even small amounts of this phase the alloys have poor proper ties, because they contain microcracks. They are fragile, crack on handling or Fissium suppresses the f phase because of the thermal cycling, and are pyrophoric. These remaining fission

position given in Table

11.

10-176


[Sec. 10

Typical Reference Fissium Alloys UMS fissium, weight %

Element

PuM9 fissium, weight ch

95.0

69 2

0.2 2.5

0 5

....

Plutonium Zirconium Ruthenium Rhodium Palladium

20.0

1.5

2.8 4.3

0.3 0.5

0 7 2 5

Sourcb: L. R. Kelman, "Fast Reactor Fuel Development at Argonne National Laboratory," FGF-73. Paris Conference (November, 1957).

Table 13.

Thermal Expansion

Coefficients and Densities Thermal expansion


U-20 Pu-5.4 Fs U-20 Pu-10.8 Fs

Coef., 10-VC

25-438 20-100 20-500 25-530 25 300 20-100 20-500 20-100 20-500

U-20 Pu-5 Mo TJ-15 Pu U-5 Fs U

of Castings Density.

Alloy, weight % Temp, range, °C

17.5 12.4 15.2 20 13 13.7 16 14. 5 18.5

g/cm"

17.6 16.6 17.7 18.8 17.95 19. 1

Source: L. R. Kelman, "Fast Reactor Fuel Development at Argonne National Laboratory." FGF-73. Paris Conference (November, 1957).

Table

14.

ANL-

ANL*

Comparison of the Thermal Conductivity of Uranium-6 Weight % Fissium with Unalloyed Uranium cal/(cm»)(sec)(°C)

U-5

weight

% fissium

Temp, °C

Uranium

BMI 20 100 200 300 400 500 600 700 800 900

0.026 0.034 0.044 0.053 0.062 0.071

0.080 0.088 0.097 0.125

ANL

0.040 0.048 0.056 0.063 0.071

0.066 0.068 0.069 0.074 0.081

0.088 0.096 0. 105

0.115 0.127

Source: L. R. Kelman, "Fast Reactor Fuel Development at Argonne National Laboratory," FGF-73. Paris Conference (November, 1957).

ANL-

Zirconium stabilizes the f phase, but presence of molybdenum and ruthenium. fortunately is a minor constituent of fissium." Fabrication. The alloys are melted in an oxide crucible, and precision-injectioncast into 0.144-in.-diameter 14-in.-long pins. The fuel is used in this form in a stainless-steel tube, with the space between the tube and pin filled with sodium to provide a thermal bond.11 The alloys are corrosion-resistant in sodium. Corrosion Resistance.

Sec. 10-7] recent developments in reactor materials 10-177


1.2

Uranium Compounds

1.21 Uranium Dioxide. Dense uranium dioxide has been found to have out standing properties for use in a pressurized-water reactor. The parasitic neutroncapture cross section is extremely low, and corrosion resistance in pressurized water and resistance to radiation damage are high.18 The major disadvantages of this material are high fabrication costs, low uranium content (only half as much uranium per unit volume as uranium metal), and low thermal conductivity. Because the thermal conductivity is low, temperature gradients are high in a fuel element of reasonable size, so that thermal fracture occurs. For this reason, the jacketing material must be thick enough to provide the structural strength required by the fuel element." These disadvantages are, however, outweighed by the advantages, and uranium dioxide has been selected as the fuel for a number of United States reactors. It is being studied intensively in the United States, the United Kingdom, Canada, France, Sweden, and other countries. Production of U02 Powders. The ability of uranium dioxide to retain fission prod ucts depends on achieving high sintered density, over 90 per cent of theoretical To achieve this density normal uranium dioxide, as produced by the density. denitration of uranyl nitrate hexahydrate (UNH) and reduction of UO3 to U02, must be pressed at high pressures (around 200,000 psi) and sintered at high temperatures (1700°C).13 As a result, fabrication costs are high when such material is used, and production methods yielding a readily sinterable oxide are being studied intensively. A high-pressed density is also desirable to minimize shrinkage during sintering, Table 15 gives the so that finishing operations on the sintered part are unnecessary. pressed density of a number of U()2 preparations.

Table 16.

Pressing Characteristics of Various

UO_>

Powders

Density, % theoretical Powder*

O/U ratio 227.000 psi

265.000 psi

1 8 3 1

69. 3

70.9

65.9

67.6

65.2

67. 1

63.0

65.7

189.000 psi

MCW, MA MCW,

- 325 sieve

fraction

as-received

and oxidized

NL'RS

2.02 2.02 2.02 2. 10 2. II 2. 17 2.04

67. 66 67. 72.

Belle and B. Lustman, "Propertiea of UOi," WAPD-184 * Kor description of materials see footnote to Table 17.

Source: J.

68.8 69.3

(September,

70.7

71.4 69.6 69.3 68.3 1957).

Oxide of improved sinterability results from: 1. Decreased particle size of U()2 (see Tables 16 and 18).1S " 2. Wet-ball milling of UO> from uranyl nitrate, before reduction to U()2 (see Tables 17 and 18). 3. Using UOj reduced from ammonium diuranate (ADU) (see Tables 17 and 18). Watson" says that the conditions under which the ADU is precipitated constitute the critical factor in achieving good sinterability, and he gives optimum conditions for precipitation when the starting material is uranyl nitrate. Murray 4. The use of U02 with more than the stoichiometric amount of oxygen. et al" report densities of 10 g/cm3 from sintering Springfields UOj.h at 1400°C, The MCW oxide was improved by oxida compared with 8 g/cm3 for MCW UOi.o«. tion, but was still less sinterable than the Springfields oxide. The Springfields oxide could be hot-pressed at 800°C and 10 tsi to nearly theoretical density. The composition of MCW U02 produced from ammonium diuranate when the starting material is uranium hexafluoride is given in Table 19."

REACTOR MATERIALS

10-178

Effect of Sieve Particle Size on Sintered Density (1750°C,

Table 16.

As-pressed density, % of theoretical

Powder

Observed sintered density, % of theoretical

56. 1 55 9 56.4

MCW, + 325 sieve fraction MCW, 325 sieve fraction MCW, wet-ball-milled

-

57.9

Table 17.

Highest


NAIR-2 MA

NAIR-I

as-received

NOH

Corrected * sintered density, % of theoretical

84 4 84.5

86.4 97.4

86.0

(September,

1957).

Hi,

Density reached in 8 hr in Ht at I700°C

65%TDG

Pressed density

Sintered density

Dependence of sintered density on pressed density

Relative rank based

Surface area

'»n

Density

Good Slight cracks

55% 65%

98.9%

57 5%

96 4%

Slight

5

3

Good Good

55% 65%

98.7%

55 3%

95 4%

Slight

1-2

1 2

Good Good

55% 65%

98. 1 %

55 1%

96 4%

None

3

t.

Bad

55% 65%

97.9%

55 1%

96 8%

Slight

1-2

I-I

Good

55% 65%

94.7%

55 3%

79 2%

Marked

4

3

Good Good

55% 65%

92.7%

55 0%

78. 5%

Medium

7

7

Flaky

55% 65%

79.2%

55 2%

58 1%

Marked

6

4

Cracks

MCW,

sintered density, 64 hr I700°C,

(based on pressed density)

Cracked

NHOL

hr)

Sintering Characteristics of U02 Powders

Compact condition after sintering

Powder

H2, 48

84. 5 84 4

Source: J. Belle and B. Lustman, "Properties of UOi," WAPD-184 * Corrected for the difference in the as-pressed density.

NURS*

[SEC. 10

Ciood

Source: J. Belle and B. Lustman, "Properties of UOt," WAPD-184 (September, 1957). * NOH — High pressure steam oxidation of U metal (72 hr at 343°C, 2.200 psig). NHOL = Steam oxidation of uranium from uranium hydride (4 hr at 400°C, 5 psig) heated 15 hr at

700°C in hydrogen.

-

MA Wet-bnll-milled MCW UOi. NURS = MCW UOi wet-ball-milled NAIR-1 = Air ignition (500°C) of

in dilute HiSOi, urea-precipitated

NAIR-2

urea-precipitated

reduction to UOj. reduction

MCW

Air ignition

to UOi. =» Mallinkrodt

of uranyl nitrate.

(500°C)

of

hydrogen reduction of UOi to UOi. ammonium diuranate to UiOt, hydrogen ammonium

Chemical Works UOi hydrogen-reduced

diuranate to UiO*. hy-drotren

from UOi, prepared by the dt-iiitration

5. The sintering atmosphere is important; a hydrogen atmosphere gives poor sintering.13 " Sintering in steam, or steam -inert-gas mixtures, with heating and cooling in hydrogen, has given good results, as seen in Table 20. British experience suggests the use of nonstoichiometric oxide and inert-gas sintering. u Physical Properties. Some new measurements of physical properties should be noted. The melting point is given as 2750°C,M compared to an earlier value 100 to 150°U lower. Further measurements of thermal conductivity have been made, and extended

Sec. 10-7]

recent developments in reactor materials 10-179 Table 18.

Properties of Uranium Dioxide Powders

Preparation

O/U ratio

Pyrohydrolysia and denitration of UNH to UOi, fol lowed by high temperature reduction to UOi with

2.01-2.03

Tapped bulk (g/cc)

Average particle size microns

Density of fired pellets ( % of theoretical)

4. 95

4.05

88.7

density

hydrogen

Dry milling

of

above

uranium rods

in rubber lined

mill with

90.2

2. 10

Wet milling of UOi in rubber lined mill with uranium rods before reduction to UOi

2.01-2.04

Micronized

2.01-2.05 2.02-2. 10

UOi prepared as in first entry above

Hydrolysis and precipitation of ADU,* followed by high temperature reduction to UOi with hydrogen

1.48

93.2

2 74

0.48

92.8

2. 18

3.00

89.9

Source: C. D. Harrington, "Preparation and Properties of Uranium Dioxide Powder," TID-7546, Book 2. p. 369 Paris Fuel Elements Conference (March, 1958). * See Table 19.


Table 19.

Composition of ADU-type UO. Composition

Ni

UOi.om-UOi.dh

70 ppm 100-300 60 30

F

Fe

Cr B Cu

0.2

Si

Ag Sn

Pb Mean particle size

10 SO 0. 1 1 2 1. 5-5.0 g/cm' 0. 5-2,1

Source: C. D, Harrington. "Preparation and Properties of Uranium Dioxide Powder," TID-7546, Book 2, p. 369 Paris Fuel Elements Conference (March, 1958). Table 20.

Effect of Atmosphere on Sintering of UOj Atmosphere*

Steam

84 62 98 43 98 98 98

Source: J.

1400-1500°Ct

H, + 16 HiO only Hi + 38 HiO only He + 2 HiO 1400-1 500°Ct He + 57 HiO 1400-l500°Ct He + 2 HiO only N, + 2 HiO 1400-1500'Ct Ni + 2 H.O only

Sintered density (% theoretical)

81.2 86.9 97.7 82.2 85.8 85.8 88.4 97. 3

97.7 86. 3

95.6 82.9

Belle and B. Lustman. "Properties of UOi," WAPD-184 * Soaking time I hr at 1500°C in indicated atmosphere, t Hi UP to H00°C and down from 1500°C.

O/U

ratio

2.00 2.00 2. 01 2. 10

2.00

2.00 2.01 2.00 2. 00

2.09 2.01 2. 11 (September,

1957).

10-180

REACTOR

MATERIALS

[SEC.

10

The later values are considerably lower, as shown in to a higher temperature. Table 21. Thermal conductivity three times greater has been achieved by incor No other data are available porating 10 per cent of Nb or Mo fibers in the body. on such material. The modulus of elasticity decreases from 26.4 X 10* psi at room temperature to 24 X 10" psi at 700°C. Few mechanical properties for UOj have been measured. Mechanical Properties. Values for the modulus Harrington14 gives the tensile strength as about 5,000 psi. of rupture at various temperatures are given in Table 22. Table 21.

The Thermal Conductivity of Uranium Dioxide Thermal conductivity, cal/(sec)(°C)(cm)

Temp., "C

Kinucry.

5% porosity

200 500 1000 1700

Kingery, porosity

26.7%

0.0176 0.0110 0.0076

Armour,

Armour,

5% porosity

25.5 % porosity

0.0124 0.0084 0.0054 0.0042

0.0094 0. 0066

0.0140 0.0090 0.0060

0.0032


Source: graph in J. Belle and B. Lustman, "Properties of UO«." WAPD-184

(September. 1957).

Mean Modulus of Rupture of U02 Specimens at I 750°C for one hour at various temperatures and surface conditions

Table 22. Sintered

Mean modulus

Test temperature,

°C

Specimens

of rupture,

psi

ground with following abrasives

As-Bintered #400 AltO.

A.O. 303- li

Cerium oxide

Room temperature

12.500

14.850 14.800 13.900

15.800

14.700

500

13,400

16,150 12.950

*

•

1000

12.800

14.500

*

•

Source:

J.

* Specimen

Belle and B. Lustman, "Properties of UOi," WAPD-184 (September, during heuting of the testing apparatus. failed spontaneously

1957).

Fabrication of Fuel Elements, Structural support and fission-product containment are necessary for uranium dioxide. For the PVVR fuel element, these functions arc fulfilled by a Zircaloy-2 can. The fuel is MCW U02, which is pressed, sintered, and ground to size on the diameter and ends, then slipped into tubes which are closed by welded-on end caps. Another jacketing material receiving attention is stainless steel, chosen in spite of its higher neutron-absorption cross section, because of its lower cost. The development of more readily sinterable uranium oxides will permit loweroperations — like extrusion and slip casting — and help Eliminating grinding operations on the U02 bodies will also reduce

pressure fabrication

to reduce costs, and the use of plate-type oxide fuel, now being studied, may permit this step to be taken. Another fabrication operation now being studied is to pack loose UOs in a tube which is sealed, and swaged to densify the U02. cost.

Sec. 10-7] recent developments in reactor materials 10-181 Belle and Lustman" report that U02 is stable in degassed steam and in Corrosion. water which is neutral or has a high pH. If dissolved oxygen is present in the water, a hydrated oxide forms, with swelling. In air at elevated temperatures, U02 is unstable, being converted to U»0«. In breeder reactors, mixtures of 1.22 Dense Ceramic Fuels Containing U02. UOi with Th02 or Pu02 are potential fuels. In enriched reactors, dilution of the UOt with other oxides such as Zr02 and A1203 may be of interest. Some work has been done on these materials, but only a little information is so far available. 1.23 Th02-U02. A complete series of solid solutions forms between ThO. and For reactor fuels, the greatest interest will be in the low-urania end of this U02. solid-solution range. The thoria-urania solid solution can be formed either by co-precipitating from a solution of the desired composition or by mixing the powders of the oxides, followed by Good results have been obtained17 by using a mixture of pressing and sintering. ThO. and U5Os, pressing at 13,000 to 15,000 psi, and firing at 1700 to 1750°C. The density achieved was 97 per cent of theoretical, which is higher than is achieved under the same conditions when U02 is used in the unfired mixture instead of Ui08. This fuel resists oxidation at 1750°C in air, for compositions containing up to It is also resistant to water corrosion and to attack by 30 weight per cent urania. NaK or molten lead. Resistance to radiation damage is good, as would be expected, for the crystal structure is the same as that for U02, which has good resistance to


radiation damage. This fuel can be jacketed to produce fuel

elements by the techniques used for U02. has also been jacketed in aluminum extrusions consisting of tubes joined together by ribs, for the Borax IV loading." The extrusions were loaded with fuel pellets and sealed after being partially filled with lead to improve heat transfer, so that assintered pellets could be used rather than ground ones. 1.24 U02-Pu02. Urania and plutonia also form a complete series of solid solu tions. It appears difficult to form the solid solution from sintering the mixed oxides. The solid solution can be formed from co-precipitated materials followed by calcina tion in hydrogen at 800 to 1000°C.17 Sintering information on this material is not As-pressed material of 65 per cent of theoretical density has been available. irradiated as a fuel element in a stainless-steel tube. During radiation, the oxide sintered, forming a central void. When the fuel element was lead filled, this sintering did not occur," indicating the efficacy of the lead in improving heat transfer. Little has been done with other oxides. With Zr02, 1.25 Other Oxide Fuels. extensive solid solution occurs, and such material may find use in reactors containing enriched uranium as a dilute solution of U02 in Zr02. Stabilized Zr02 containing Work CaO would be required. Limited experience is available on other oxides. has been done on U02 in BeO, MgO, and AljOi." In all these systems, U02 is Since the U02 will bo present in small amounts, the dispersed as a separate phase. fabrication conditions will resemble those of the diluent. Resistance to corrosion and radiation damage and to fission-product release will be sensitive functions of the particle size of the U02 and the permeability of the matrix. Other Uranium Compounds. 1.26 Table 23 is an up-to-date listing of the proper ties of a number of high-melting-point uranium compounds, of possible interest as reactor compounds.1' One disadvantage of most of these materials is that they do not have appreciable solid solubility for either of their constituents, so that if exact stoichiometry is not Achieving exact stoichiometry of achieved, lower-melting phases will be present. these high-melting compounds is difficult and will be lost during use because of the consumption of uranium atoms. Thus, the high melting point of some of these materials may be illusory, and unachievable in practice. An exception to this is the system UC-UC2. These materials form a complete series of solid solutions at elevated temperatures, and no low-melting eutectics. For this reason, low-melting phases can be avoided by working in the UC-UC2 region. Compositions near UC are interesting potential fuels and are being studied inten

It

REACTOR MATERIALS

10-182

Properties of Uranium Compounds

Table 23.


Com pound

Melt ing point, °C

X-ray Crystal structure

UBi

> 1500 Hexagonal a « 3. 12 c 3.96

UB.

> 1500 Tetragonal a c

-

Fcco

-

UBen

2025

UC

2375 Fee a = 4.961

UCi

2475

-

10.256

-

1590 Fee a

UN

2630 Fee a — 4. 880

r.si

7.811

Would

9.38

7.94

Would

4.373

3.06

13.63

12.97

11.68

10.61

930 Bet a 6.029 c = 8.697

15.58

14.99

-

Orthorhotnbic a

-

-

5.66

6-7.66 c

~ 1600 Bet a =

0.028

Should

0.080

Decomposes in hot water; of interest in gas-cooled and liquid-metal-cooled reactors

3.98

Hexagonal a <- 3.66 c = 4.07

1500 Cubic a = 4.03

-

have good resist ance to high temperature oxidation and corrosion ; provides some moderation

with water and air. but probably not with Ht

Reacts

Should have good thermal good conductivity and corrosion resistance Probably will cally reactive

be

chemi

Stable in boiling water, but probably has poor oxida tion resistance 0.041 at 50°C

5.469

Source: E. Epremian, "Uranium (November, 1957).

12.20

11.31

10.40

9.30

Resistant

to hot water: in air. Fair irradiation stability

8.98

7.27

9.25

7.48

8.15

6.02

Resistance to oxidation in air and wet corrosion in with increasing creases silioon content (or the series of uranium silicidea

3.91

c = 13.74

2750 Cubic a

require separated probably would oxidise readily in air or steam, but stable in Hi isotope;

oxidises

~1600

~I600

require separated probably would oxidise readily in air or steam, but stable in Hi isotope;

6.62

9.58

-

USi

UOi

11.75

10.87

Bee a

Remarks

12.82

5. 484

>2000

Thermal conductivity, eal/ (sec)(cm)(°C)

g/cm'

13.52

Tetragonal o =- 7.329 e 3.900

U8ii

con tent,

14.32

~I660

USli

sity,

Ura nium

g/cm1

8. 14

UiSit

U>.Si,

den

5.999

UA1.

US

7.075 3.979

Bet a = 3. 524 c

[SEC. 10

10.96

9.66

0.014

Kxcellent material; good irradiation stability, etc. chemical stability, Poor thermal conductivity

Compounds for New High Temperature Fuels," Paris Conference

Sec. 10-7] recent developments in reactok materials 10-183 The desirable char sively in the United States, the United Kingdom, and France. acteristics of uranium monocarbide, in addition to high melting point, include high The cor uranium density and a thermal conductivity as good as that of uranium. rosion resistance of uranium carbides in water is poor; so the materials are being considered for use in gases to which they are inert and in liquid metals. Uranium carbide bodies have been prepared by pressing and sintering the ele mental materials, by melting and casting, and by pressing and sintering the pre formed compounds. Near-theoretical density has been achieved by the first two methods, but high density is much harder to achieve by the latter method owing to the refractory nature of the carbide. Complete series of solid solutions form in the systems: UC-ThC, UCi-ThCj, UC-ZrC, UC-TaC, and UC-NbC. PuC should behave the same as UC." Such materials may be of interest for the future.


1.3

Dispersion Fuels

By Enriched fuel can be used as a dispersed phase in a metal or ceramic matrix. choosing a fuel-particle size that is large compared to the fission recoil range, most of the radiation damage can be confined to the fuel particles and their immediate vicinity, leaving the matrix unaffected. In this way, very high fuel burnup can be achieved, even at high temperatures, if a good high-temperature material is used as the matrix. Dispersions are normally made by powder techniques to achieve the desired particle sizes, usually followed by hot fabrication for dispersions in metals. Strictly speaking, some alloys like the U-Al alloys are dispersions, since the uranium is present as particles of UAU dispersed in an Al matrix. Normally, however, the UAU particles are so finely dispersed in cast and worked U-Al alloys that radiation damage occurs uniformly. Therefore such materials lack the advantages which dispersions can possess.

Certain precautions are necessary

to produce a dispersion of good characteristics.

First, the dispersed particles should be large compared to the powder of the matrix.18 Otherwise the dispersed phase will collect along planes of the dispersion during fabrication, causing transverse weakness and splitting. Second, any deforma tion must be made when the matrix is so plastic that the dispersed particles will not Finally, the dispersed be crushed, since crushing also leads to planes of weakness. phase and the matrix must be compatible at the temperature of fabrication and of use. Solid-phase reaction occurs with some combinations of materials, with adverse effects. The amount of dispersed phase must not be excessive. Normally, 30 to 35 volume per cent is the maximum, though dispersions containing up to 50 to CO volume per cent of the dispersed phase have occasionally been made. Some of the types of dispersions which have been investigated are described below, with such information about them as is available. 1.31 U02 in Al. Reaction occurs between U02 and Al at elevated temperatures. Difficulties from this source can be avoided by (1) minimizing the time at high tem perature during fabrication, (2) using a nonreactive type of U02,10 and (3) by using the dispersion at low temperature. UC does not react with Al, and hence might make a better dispersion. 1.32 U02 in Stainless Steel. This type of dispersion has been investigated because of the good high-temperature properties of a stainless-steel matrix. The properties of the material arc strongly dependent on the fabrication technique. Suitable conditions for making a 26 weight per cent dispersion of U02 in 347 stainless UOj particles and 10 to 149ji stainlesssteel have been given" as follows. 44 to steel particles are blended and pressed at 33 to 50 tsi. These are sintered at 1175 to 1350°C for 4 hr to 90 per cent density, then rolled at 1150°C to 90 per cent reduc Cold reduction must be avoided, since it crushes the U02, forming stringers tion. so that the transverse strength of the dispersion is low. The material can also be produced by hot extrusion, or by a cold-binder process

10-184

REACTOR

MATERIALS

[Sec. 10

in which a plastic mixture is extruded at low pressure and densification is achieved by sintering. The mechanical properties of some dispersions of this type are given in Tables 24, 25, 26, and 27. 1.33 Dispersions of U02 in Other Metals. Dispersions of U05 in Nichrome, molybdenum, and niobium also have been made.,° UOj reacts with zirconium at fabrication temperatures, and good dispersions cannot be made. Uranium monocarbide does not react unduly with zirconium, and satisfactory dispersions can be made. Over 50 volume per cent of uranium metal has been dispersed in magnesium. Still another type of potential interest consists of a dispersion of a fissionable in a fertile material. Possible types include uranium or plutonium compounds in thorium or uranium alloys. Several dispersions of this type have been prepared. Many types of dispersions are possible, and only those of greatest interest have been All of the dispersions can also be made with thorium or plutonium mentioned. compounds substituted for the uranium ones. 1.34 Dispersion in Ceramic Materials. Some dispersions of this type have been discussed in Art. 1.22. Table 24.

Mechanical

Properties of U02-Iron-base Dispersions Tested in Longitudinal Direction

Room temperature


Weight

ub.

25*

30f 30f

Ulti

540°C

mate tensile strength, pai

Yield

Elonga

strength, psi

tion,

Stainless steel Stainless Bteel

59.500

32.700

57.000

45.000

Iron

58.500

Matrix material

%

10.0

Ulti

700°C

mate tensile strength, psi

Yield

Elonga

strength, psi

tion,

36,800

24.300

32.000

Ulti

strength, psi

5.0

23,800

14.000

11.0

20.000

Source: J. E. Cunningham, R. J. Beaver, and R. C. Waugh, "Fuel Dispersions Components for Power Reactors," Paris Conference (1957). * KAPL data on unclad extruded dispersions, t BMI data on 6-6-6 mil clad fuel sheets.

Table 25.

Transverse

Yield

mate tensile strength, psi

Elonga tion. •4

9 0 13 0

in Stainless Steel

Tensile Strength of UOj-Iron-base Fuel Dispersions Ultimate tensile strength

Weight %

UOj

Fuel matrix

Room temperature

20* 25* 30t 35* 371 40* 45* 50*

Iron Iron Austenitic Austenitic Austenitic Austenitic Austenitic Austenitic

stainless stainless stainless stainless stainless stainless

33.500 28.500 26.000 16.000 14,500

540°C

650°C

760-C

13.000

12.000

8.500

11.000 8.000 4.000

Source: J. E. Cunningham. R. J. Beaver, and R. C. Waugh, "Fuel Dispersions in Stainless Steel Components for Power Reactors," Paris Conference (1957). * SUP data,

t BMI data. 1 ORNL data.

Sec. 10-7] recent developments in reactor materials 10-185 Table 26.

Evaluation

of Bend Ductility

of Fuel Sheets Containing

in Iron-base

Weight % VO,

20t 25+ 30t 30t 40t

«ot 20t 30t 40t 60t

nt

m

37J 371 371

UOj* particle

Matrix material

size, it

<44 <44 <44 <44 <44 <44 <44 <44 <44 <44 74-82 74-82 44-53 44-53 <44

Iron Iron Iron Iron Iron Iron Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel Stainless steel

Matrix

Clad-core-clad

Bend radius, in.

Remarks

0 0156 0. 0156 0.0156 0.0313 0.0156 0.0313 0. 0313 (min) 0.0313 0. 0938 (min) 0. 1250 0.010 0.029 0.010 0.029 0.029

Mo cracking No cracking Cracking No cracking Cracking No oracking No cracking Cracking No cracking Cracking Cracking No cracking Cracking No cracking Cracking

Source: J. E. Cunningham, R. J, Beaver, and R. C. Waugh, Components for Power Reactors," Paris Conference (1957). * Original size prior to fabrication.

UOi Dispersed

"Fuel

thickness of fuel sheet, mils

Dispersions

4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 4-12-4 6-8-6 6-8-6 6-8-6 6-8-6 6-8-6 in Stainless

Steel

BMI data. t ORNI. date.


+

Table 27.

Creep Rate at 700°C of 4-12-4 Mil Clad Fuel Sheets Containing 30 Weight % UO, in Stainless Steel Stress, psi

Creep rate,

% per hr

6,000

0.004

9.000 12.000 15.000 18.000

0.03 0. 18 1.3

4.0

Source: J. E. Cunningham, R. J. Beaver, and R. C. Waugh, Components for Power Reactors," Paris Conference (1957).

"Fuel

Dispersions

in Stainless

Steel

Only one other type need be considered —a dispersion of a uranium compound in graphite. Uranium oxide can be the dispersed phase; this reacts with graphite at 1800°C,18 forming uranium monocarbide. Uranium carbide also can be the dis persed phase. The dispersion can be made by impregnating graphite with a solution of uranyl nitrate (followed by heating to drive off water) and decomposing the nitrate to the oxide or, at higher temperatures, to the carbide. It can also be made by mixing Dispersions graphite, pitch, and uranium oxide or carbide, pressing, and sintering. with up to 50 weight per cent uranium have been made. 2

CLADDING MATERIALS

The information given below constitutes the major recent developments in cladding materials. 2.1

Aluminum

The only major development with aluminum cladding alloys is the effort their corrosion resistance in water at temperatures above 400°F, thereby

to improve to provide

10-186

MATERIALS

REACTOR

[Sec.

10

a cheaper competitor for zirconium alloys. Improvement has resulted from alloying with nickel and iron, but such alloys are still inferior to zirconium alloys. Additions to water are being studied as a means of reducing corrosion. Improve ments have been found with several additives."

Zirconium

2.2


Large increases in the production of zirconium, and decreases of cost, have occurred recently. Present production capacity is said to be 6 million lb per year. The price of sponge, zirconium is below $8 per lb," which makes it inexpensive for use in reac tors, because of its low neutron-absorption cross section. Zircaloy 2 (1.2 to 1.7 weight per cent tin, 0.07 to 0.2 weight per cent iron, 0.05 to 0.15 weight per cent chromium, and 0.03 to 0.06 weight per cent nickel) is now a com mercial material, with good and reliable corrosion resistance in water. Some trouble with corrosion "stringers" in sheet Zircaloy 2 has been experienced, but such stringers can be avoided by consumable-electrode arc melting in a vacuum, which suggests that they are caused by volatile impurities or possibly gas bubbles. Work has been done on another zirconium alloy, Zircaloy 3 (0.25 weight per cent iron and 0.25 weight per cent tin). The technology of this material is not so well It is a softer material developed as that of Zircaloy 2, and it is not yet in wide use. and equally corrosion-resistant. The short-time mechanical properties of annealed Zircaloy 2 are given in Table 28. The creep strength of zirconium and Zircaloy 2 at 500°C is given in Table 29. Table 28.

Tensile Properties of Zircaloy

0.2% yield Temp., °C strength, psi

27 425 500

Table 29.

Elongation,

Tensile strength,

42.000 16.200 13.800

2"

psi

%

30 37 42

55.000 20.000 19.500

The Creep Strength of Zircaloy

2 at

600°C"

1 % total deformation at 500°C, psi

Stress to produce

Time, hr Zirconium

100 1.000 10.000

4.800 2.400 900

2.3

Zirouloy

2

9.200 7.700 5.600

Niobium

In addition to its use as an alloying material with uranium, niobium has received increasing attention as a cladding material, and is said" to have been chosen by the United Kingdom as the material for cladding the fuel in the British fast breeder at Dounreay. The cost of high-purity ductile niobium is currently over $200 per lb. This cost would decrease markedly with increased production, but would not become very low, because increased use would require the introduction of low-grade ores which must be treated by expensive methods.*3 Mechanical Properties. Strength at high temperatures, with a lower neutron


Sec. 10-7]


absorption cross section than most high-temperature materials, is the major advan tage of niobium. At room temperature, pure niobium is not especially strong, having a tensile However, it retains its strength to high temperatures, having strength of 35,000 psi. a tensile strength of 14,800 psi at 2200°F, with a tensile elongation of 15 per cent. The 100-hr creep strength of pure niobium is 20,500 psi at 1600°F, and 14,500 psi at 1800°F. Room-temperature strength can be increased by alloying with tantalum, tungsten, and molybdenum, although this higher strength is not retained at high temperature. Fabrication. Niobium of high purity is easily fabricated. Vacuum-sintered niobium bars can be cold-worked into sheet or rod. Cold reductions of 98 per cent are possible because of the low rate of work hardening. Ingots can be made by arc Niobium can be resistance-welded under water melting in a cold-hearth arc furnace. and arc-welded in a protective atmosphere. Annealing is performed in a vacuum at 1100°C. Corrosion Resistance. Niobium is resistant to attack by sodium and NaK to It is thought that the major corrosion at this and 1250°F and probably higher. higher temperatures results from oxygen dissolved in these liquid metals. In aqueous media, niobium is as corrosion-resistant as tantalum. However, under such conditions, other considerations favor use of zirconium. In nitrogen at 900°F, surface embrittlement occurs at a rate of less than 1 mil in An adherent nitride film occurs, with slow nitrogen diffusion into the 1,000 hr. metal. The resistance of niobium to oxidation is poor. Above 500°C in oxygen, the metal At 1400°C a runaway reaction occurs in forms a porous scale and oxidizes linearly. The attack is not confined to a surface film. Oxygen dissolves and pure oxygen. diffuses in the niobium, embrittling it. Alloying niobium with titanium, molybdenum, vanadium, chromium, and aluminum improves its oxidation resistance. The most oxidation-resistant niobium-base alloy contains 23 weight per cent chromium, 15 weight per cent silicon, and 10 weight per This oxidizes at 10 times the rate of nickel-20 weight per cent cent aluminum. chromium in air at 1000°C. Table 30.

Control-rod

Effectiveness*

for Various Materials Absorptiveness,

Material

Effectiveness

relative

to hafnium

3.0 weight

Hafnium

% boron-10

in stainless

steelj. .

27.2 2. 40 9. 3

0.97 weight % boron-10 in stainless steelli 15 weight % EuiOi in Btainless steel!

Indium Silver Cadmium

8.7 weight

Tantalum

8.7 3.68 1.85

% gadolinium

in titanium

2.7 weight % SmiOi in stainless Haynes 25

77.5

steely. . . .

Titanium Zircaloy 2 2.S aluminum

31. 1

0.60 3.20 0.96 0. 16

1.12 1.00

0.98 0.96 0.93 0.88 0.80 0.79 0.71 0.71

0.68 0.24 0. 05 0. 02

Source: Russell W. Dayton, "The Effectiveness of Control-Hod Materials," BMI-II96 (June 24, 1957). * Determined by R. R. Eggleaton and a Critical Experiment (iroup of Hettis Laboratory on samples 2

X

f

2

X

0.2 in.

Macroscopic thermal-absorption cross section times thickness. X Dispersion of minus 100-rnesh particles of 90 % enriched boron.

*f Alloy. % Dispersion.

155 IS7

151 153

149 152

46.000 70.000 160.000

4.600 9.000 420

0.478 0.522

148 157

5.500 66.000 140

29

1.00

0. 138 0.266

190 58 197

2.250 20.800

0.042 0.958

123

1400

95

152

0.25

7.26

118

n,t

n.y

n.T

",T

».y

n.y

'1.7

1.1

n,a

".<*

2.58

0.46 2.46

1.000

11.000 3.000

16.000 15.000

2.600

5.9

0.096 8.2

30.000

1.46

0. 178

7.200

630 12.500

16.6 12

7.0001

section. barns

4.500

Cross

resonance

Materials

1.26

132

Energy, ev

Major

Control

1.0, 3.4. 7.2, other 3.8. 9.0. other

0.86.

22.6. 47.8

3.86. 9.10

42. 45. 52 31. 41. 57. 72, 86

Energy of other major resonances, ev

5

64

63

62

133

113 115

113 0.

55

49

48

3.64

11.0

3.37

106.7

3.29

Absorbing reaction

of Possible

5.

Neutron-

044

0.015

136

0.006

0.014

0.063

0.009

0.096

0. 122

0.151

006

grams neutrons per em'f

absorbing

0.

Gadolinium

Cadmium

62 30 84

0.519 0.481

107 109

150

1.00

103

45 47

37

1.00

59

755 4.010

71 947

27

5 0. 188

0.075

macroscopic cross section, (cm"')

Thermal

Properties

t

00

3 10

6

O

Atomic no.

Physical

Thermal cross section, barns

Selected

Isotope fraction abundance

31.

0

Cobalt

Element

A hsorhing tuass number

Table


0

0 0

2.500

380

15.45

78

30.1

1

is

0. 0.

is

is

150 500

30.000

4.91 n.y

23.3 34.0

5.400 5.000 3.800

5.6 0.654 1.30 n.y

n.y

3.500 370

13.000

4.28

2.18 4.40

n,y

»,T

6.000 10,000 1.100 139

2.36 7.80 5.69 73.9

2.600

0.46

n.y n.y

7.500

other

61.5

Neutron-

0.014

0.098

116

0. 110

090

0.090

absorbing capacity, grams neutrons per (m'H

the value given divided

10.4. 14.0. 24.1. 39.3

numerous

6.2. 9.5

1.8. 16

Energy of other major resonances, ev

cross section

section, barns

",y

Cross

resonance

(Continued)

Considered too costly for use. The isotopic resonance that for the natural element. The resonance cross section, even when given for the isotope, by the isotopic abundance. 94 This resonance scattering. cm' of the material can absorb. denned as the number of grams of neutrons which Neutron-absorbing capacity

100 168

98

1.00

197

198 199

430 960 130

385 0.615

191 193

5.57

18

4.71

5.

80

79

77

84 100 63

0.371 0.629

21.3

0.

185 187

1.00

181

105 380 75 65 13

5.48

35

Energy, ev

Major

Materials.

0.

Gold*

Iridium*

75

73

184 271 138 354

166

1.100 2.600

0.282

Control

Absorbing reaction

of Possible

Thermal macroscopic cross section, (cm"')

Properties

Isotope fraction abundance

0. 0. 0. 0.

177 178 179 180

164

Absorbing mass number

Physical

Thermal cross section, barns

Selected

1.

Rhenium*

Tantalum

72

68

66

Atomic no.

31.

5 5

Hafnium

Element

Table


t

0.

is

%

*

t tt

REACTOR MATERIALS

10-190 3

[Sec. 10

CONTROL MATERIALS

A control material works by absorbing neutrons; hence a material which absorbs more neutrons than another is both more effective and provides greater flexibility Thus, in a thermal reactor, where most neutrons are at thermal in reactor design. It is not energy, a material which absorbs only thermal neutrons is quite effective. as effective, however, as one that absorbs neutrons at epithermal as well as thermal energies.

The effectiveness of various absorbing materials has been measured and is given in Table 30." This table shows that materials like cadmium and gadolinium which The are black to thermal neutrons are not necessarily the best neutron absorbers. table also shows that materials which are not black to thermal neutrons but will also absorb epithermal neutrons — such as hafnium, europium, silver, or indium — may be (because of epithermal capture), or boron (because of l/v absorption) considerably more effective. Another important characteristic of control materials is retention of effectiveness in spite of burnout. If the absorbing isotope is present in small amounts and is Table

32.

Composition

weight %

Alloys

Strength, pai Teat

Redurt.

Elong

temp.,

of area

°F

0.2%

Ultimate

in 2 in., %

70 600

16.300 9.800

42.000 17.500

57 24

4.9

70 600

10.500 9.600

42.800 17.600

67 34

9.8

14.8

70 600

8.600 7,700

33.600 20.400

60 75

72.0

7.7

20. 1

70 600

8.900 7.700

32.100 17.100

82 30

2557

68.0

22.6

29.5

70 600

7.000 9.100

32.000 22.200

80 57

2555

78.7

13.3

4.9

2.97

70 600

21.900 18.000

51.000 19.500

41 17

2567

76.3

12. 1

9.4

2.00

70 600

22,800 15.700

53.200 17.600

50 35

2.00

70 600

9.300 10.300

37.600 16.900

75 40 75 50

Ag

In

2545

85.3

14.8

2533

81.6

13.6

2565

75.3

2563

Alloy

Cd

Al

Sn

yield

2571

70.4

7.6

2577

68.7

24. 1

5.6

1.49

70 600

10.700 11.500

40.800 17.100

2583

63.5

2.4

33. 1

0.50

70 600

7.800 7.700

35.100 12.900

20.1

HAH-5

74.6

19.8

4.9

0.56

70 600

25.200 17.100

42.800 17.600

H&H-7

69.6

19.8

9.8

0.53

70 600

19.200 13.000

36.300 13.700

U 26 30

32 3 15 0

0

So usee: I. Cohen, E. F. Losco. and J. D. Eichenberg, "Metallurgical Design and Properties of Silver-Indium-Cadmium Alloys for PWR Control Rods," presented at Nuclear Engineering Science 1958, reprinted by permission of The American Institute of Chemical Engineers. Conference, •All samples (gauge dimensions 2" 0.5" X 0.225") were cold-rolled 20% and annealed hr at 500°C except for H&H-5 and H&H-7. The latter were annealed hr at 600°C. I

I

X


Tensile Properties of Annealed* Silver-base

Sec. 10-7]



transmuted to a low-cross-section isotope, the control material will lose effectiveness This is the case with cadmium and gadolinium. This characteristic and rapidly. others of importance in the use of control materials are given in Table 31. A final general characteristic required of control materials is resistance to radia tion damage. As seen in Table 31, most absorbers capture neutrons by an n,y reac tion, and radiation damage is not severe. Boron and lithium, however, capture neutrons by an n,a reaction. This, in the case of boron, leads to the creation of an atom of helium and one of lithium for each B'° atom destroyed, and these foreign atoms are the cause of an even more severe radiation-damage problem than results from the fission products in a fuel material. No thoroughly satisfactory solution has been found to this problem for high-burnup control rods containing boron. It can be expected that control materials of high effectiveness will be developed from combinations of materials, each chosen to absorb neutrons of a given energy. Rare-earth metals, which after all are not so rare, offer possibilities of this kind. Other material combinations also are possible. One alloy of high effectiveness has been developed. This is an alloy of silver, with 15 weight per cent indium and 5 weight per cent cadmium." The cadmium supplies blackness to thermal neutrons, the silver and indium have moderately strong thermal capture and strong epithermal resonance capture. This alloy is relatively cheap, is made of common materials, and has nearly the effectiveness of hafnium. The mechanical properties of this material are given in Table 32. It is evident that its strength decreases markedly at elevated temperatures, as would be expected in view of its low recrystallization temperature. Other properties of this alloy are as follows:" Thermal Modulus Thermal Density Melting

In

conductivity of elasticity expansion point

0.18 cal/(sec)(°C)(cm) 9.86 X 10* psi 22.5 X 10-«/°C 10.17 g/cm*

at 300°C

775-825°C

G00°F water, the alloy corrodes at the rate of 0.18 mil per

year."

References 1.

2.

3.

4.

5. 6.

"The Experimental Boiling Water Reactor," Appendix C, Properties of Fuel Materials, ANL-5607 (May, 1957). Machery, R. E., C. H. Bean, J. R. Lingren, and J. J. Carson, Jr.: "The Experimental Boiling Water Reactor," Appendix D, Production of Fuel Subassemblies, ANL-5607 (May, 1957). Cohen, I., and E. F. Losco: "Development and Properties of Uranium-base Alloys Corrosion Resistant in High Temperature Water," Part II, Alloys with Protective Cladding, WAPD-127 (September, 1955). Bleiberg, M. L., J. D. Eichenberg, It. H. Fillnow, and L. J. Jones: "Development and Properties of Uranium-base Alloys Corrosion Resistant in High Temperature Water," Part IV, Radiation Stability of Uranium-Base Alloys, WAPD-127 (May, 1957). Van Thyne, R. J., and D. J. McPherson: Transformation Kinetics of UraniumMolybdenum Alloys, ASM Trans., 49: 598 (1957). Van Thyne, It. J., and D. J. McPherson: Transformation Kinetics of UraniumNiobium and Ternary Uranium-Molybdenum-Base Alloys, ASM Trans., 49: 576 Breden, C. R.:

(1957).

Dayton, R. W.: "The Metallurgy and Fabrication of Uranium-Allov Fuel Elements," TID-7546, p. 302 (March 1958).* 8. Kaufmann, A. R., J. L. Klein, Loewenstein, and H. F. Sawyer: "Zirconium Cladding of Uranium and Uranium Alloys by Co-Extrusion," NMI-TJ-8 (Oct. 4, 1957).* 9. Rough, F. A.: "An Evaluation of Data on Zirconium-Uranium Alloys," BMI-1030 (Aug. 19, 1955). " Development of UjSi e-Phase Alloys for Use in Pressurized Water 10. Wolf, R. A., el at.: Reactor," WAPD-155 (July 15, 1956). 7.

* This paper has been publiahed by the Atomic Energy Commiasion, aa document TIP-754fi, aa a compilation of all papera presented at the Conference on Fuel Element Technology, held in Paris, France, Nov. 18-23, 1957.

10-192 11. 12. 13. 14. 15.

REACTOR

Kelman, L. R.: "Fast Reactor Fuel ANL-FGF-73 (November, 1957).*

MATERIALS

Development

at Argonne

[Sec. 10 National Laboratory,"

D., P. W. Frank, R. J. Kisiel, B. Lust man, and K. H. Vogel: " Effects of Irradiation on Bulk UOs," WAPD-183 (October, 1957).* Belle, J., and B. Lustman: "Properties of UOj," WAPD-184 (September, 1957).* Harrington, C. D.: "Preparation and Properties of Uranium Dioxide Powder," TID7546, p. 369 (March, 1958).* Watson, L. C: "The Production of UOj For Ceramic Fuels," CRL-45 (November, Eichenberg,

J.

1967).* 16. 17. 18. 19.

20. 21. 22.

23.


24.

Murray, P., S. F. Pugh, and J. Williams: " Uranium Dioxide as a Reactor Fuel," TID7546, p. 432 (March, 1958).* Handwerk, J. H.: "Ceramic Fuel Elements in the Systems: ThOi-UOi and UOs-PuOj."

ANL-FGF-67

(November,

1957).*

Williams, J.: "Dispersion-type Fuel Elements Based on Fissile Ceramics," M/L-l, TID-7546, p. 554 (March, 1958).* Epremian, E.: "Uranium Compounds for New High Temperature Fuels," TID-7546. p. 549 (March, 1958).* Kopelman, Bernard: " Recent Developments in Dispersion Type Fuel Elements," TID7546, p. 231 (March, 1958).* Cunningham, J. E., R. J. Beaver, and R. C. Waugh: "Fuel Dispersions in Stainless Steel Components for Power Reactors," TID-7546, p. 243 (March, 1958).* Dickerson, R. F.: Battelle Memorial Institute, Columbus, Ohio (private communica tion, March, 1958). Klopp, W. D.: Battelle Memorial Institute, Columbus, Ohio (private communication March, 1958). Dayton, Russell W.: "The Effectiveness of Control-Rod Materials," BMI-1196 (June

24, 1957). 25. Cohen, I., E. F. Losco, and

J. D. Eichenberg: "Metallurgical Design and Properties of Alloys for PWR Control Hods," presented at Nuclear Silver-Indium-Cadmium Engineering & Science Conference, 1958, Chicago, 111.

•This

paper has been published

compilation of all papers presented France, Nov. 18-23. 1957.

as document TID-754fi. as a by the Atomic Energy Commission, on Fuel Element Technology, held in Parie. at the Conference

SECTION

11

CHEMISTRY AND CHEMICAL ENGINEERING BY

STEPHEN LAWROSKI, B.S., M.S., Ph.D.,

Director, Chemical Engineering Division,

Argonne National Laboratory.

LESLIE BURRIS, JR., B.S., M.S.,

Associate Chemical Engineer, Argonne National

Laboratory.

M.S.Ch.E., Ph.D., Associate A. RODGER, B.S.Ch.E., B.S.Met.E., Director, Chemical Engineering Division, Argonne National Laboratory.

WALTON


CONTENTS PAGE 1 2 3 4 5 6

Introduction 11-2 Nuclear Reactor Dependence 11-3 Chemistry of Important Elements. , . 11-34 Analytical Chemistry 1 1-50 11-57 Basic Methods of Separations Filtration, Centriftigation, and Evapo ration 11-103

7 Auxiliary Operations 8 Special Problems 9 Special Design Problems Raised by

Radioactivity

10 Waste Disposal References

11-1

PAGE 11-106 11-114 11-133 11-140

11-146

CHEMISTRY AND CHEMICAL ENGINEERING BY Stephen Lawroski, Leslie Burris,

1


1.1

Jr.,

and Walton A. Rodger

INTRODUCTION

General Statement of Problem of Separations Processes

By common usage in nuclear engineering, the term separations processes has come to mean those chemical processes which are applied to irradiated nuclear reactor fuels for the recovery and purification of the contained fertile materials and of the partially The ultimate disposition of the usual used or newly produced fissionable materials. by-products (i.e., the fission products, fuel element diluents and cladding, and chemi cals added in processing) represents the final phase of separations processing operations. In the consumption of nuclear fuels in a reactor core or in the production of new nuclear fuels by transmutation in a reactor core or blanket, the operation under the present status of reactor technology is normally interrupted after only a small fraction of the desired reaction can take place, whether it be fission or transmutation. As a result, several cycles of these incomplete operations in a reactor and intervening recovery and purification operations are necessary to obtain what would correspond Because the separations to a complete utilization of fissionable or fertile constituents. cycle must be repeated, it must be capable of high yields and low cost per cycle. This becomes very essential in the over-all complex of producing economic power from the fission process. In addition, the process should accomplish this with a minimum of holdup time to avoid excessive and, hence, costly inventories of fissionable materials. Since some of the metals often included as cladding or diluents in fuel fabrication are appreciably expensive, it is obviously desirable to have the process also recover these materials in an economic manner. The required degree of purification of fissionable or fertile materials from fission products may vary over a very substantial range. For example, in the case of homo geneous reactors or in the case of heterogeneous reactors where remote fabrication of fuel may be practicable, it is generally conceded that removal of the important fission On the other hand, when it is necessary products by a factor of about 10 will suffice. to handle the fuel directly for fabrication purposes, the required decontamination from fission products may be as much as 107-fold or more. The fission process results in the formation of about 35 different elements which may be present in a total of approximately 200 isotopes. Thus, the task of the separations process is to eliminate or, at least, to remove substantially a large number of elements and at the same time to maintain a high degree of recovery of the fissionable or fertile constituent. There are industrial examples of conventional chemical processes where thorough purification and essentially complete recovery are necessary and, indeed, are achieved at low unit cost. Comparable results would be obtained in reprocessing irradiated reactor fuels were it not for two unique and complicating factors: (1) the enormous amounts of radioactivity associated with the materials to be processed and (2) the criticality feature of fissionable materials. The presence of high levels of radioactivity gives rise to very substantially increased capital and operating costs because of (1) the need for inventory storage while cooling 11-2

Art.

nuclear reactor dependence

2]

11-3

to decay short-lived fission products so as to reduce heat- and radiation-damage effects during reprocessing, (2) the massive shielding required, (3) the need for ex traordinary provisions for operation and maintenance of processing equipment, and (4) the special handling and disposal problems associated with the waste products. The critical mass problem also contributes to high over-all costs because of manda The safeguards that are usually practiced consist of one or more of tory safeguards. the following features: (1) batch-size control, (2) equipment geometry control, (3) concentration controls on fissionable materials, (4) judicious incorporation of neutroncapturing materials into equipment, and (5) continuous-type operations throughout the process. There is one other complication which leads to high reprocessing costs of irradiated reactor fuels at the present time. Specifically, it is the fact that significant standard ization of reactor fuels has not yet been achieved with respect to shapes and com positions. This leads to reduced reprocessing plant capacity, excessive equipment, and extra operating personnel for nonroutinized procedures. The situation is expected to improve when fuel-fabrication technology is more thoroughly developed.


1.2

Applicability

of Basic Principles of Chemistry and Chemical Engineering

Just as in other fields of chemical processing technology, the basic principles of chemistry and chemical engineering can be successfully applied to the problems of separations processing found in the nuclear energy field. Since the validity of these principles can be thus extended so usefully, discussions of those principles and tech niques most frequently used by chemists and chemical engineers concerned with reprocessing of irradiated reactor fuels are included in appropriate parts of this section. These discussions are necessarily brief; as a result they are less useful to the expert than to one who is or expects to be engaged principally in other activities of nuclear engineering and still wishes to understand the language of the separationsFor those desirous of more complete trea process chemist and chemical engineer. tises, references are given to more comprehensive texts on chemistry and chemical engineering. Virtually all the known types of fundamental methods for achieving separations among constituents in a mixture have been investigated for application to recovery and purification operations for irradiated reactor fuels. Very frequently the concen trations of substances dealt with are extremely small. As a result many of the ideal ized laws of chemistry can be exploited with considerable confidence by the chemist or chemical engineer to predict or estimate the performance of a given process scheme from a limited amount of experimentally obtained data. This fact has often proved to be very valuable in the early stages of the development of a separations method when appropriate materials for experimental work have been scarce or when the presence of high levels of radioactivity has retarded the speed with which experimental work could be conducted. For the reader's convenience a periodic chart of the elements (Table 1) and a table of physical properties of the elements (Table 2) have been included. 2

NUCLEAR REACTOR DEPENDENCE

t

The chemical and chemical engineering problems peculiar to the nuclear energy field come about as the result of two conditions: (1) the need to handle materials present in very low concentration and (2) the necessity of dealing with very large quantities of radioactive materials. Both conditions are the result of and are widely influenced by the operation of a nuclear reactor, and in general, few separations problems exist completely divorced from reactor operation. For a more complete treatment of important nuclear reactions in reactor operation, the reader is referred to Sec. 4 of this handbook. A few of the reactions most important in influencing chemical separations will be discussed briefly here. t The authors are indebted to I. G. Dillon, Chemical Engineering Division, Argonnc National Labora tory, for assistance in the preparation of this article.

5

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Germanium Gold Hafnium Helium Holmium

5

-452.0

5380

±

2

67

10

3760

0.04 ±

1760 - 20 1945.4 0.0 00" -456.5 0.4/

18

1.25

0.073 0.031

0.079

0

85.60

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7.9

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72.60 197.2 178.6 4.003 164.94

152.0 19.00 223 156.9 69.72

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0.065092

53 73 14

92 4.8

4.7-6.0-

13.3-

02

32 79 72

87 64 31

63

9

Fr Gd Ga

9.

Eu

7.

Europium Fluorine Francium Gadolinium Gallium

2

8.5616-

±± ±± ± 4700

6.8 38.9 136 105

29.2 23.8

0.042 0.052 0.116 0. 099

0

4380 30 1981.4 ±0.2

1275 -30.5 4500 5250

0.309 0.070 0.055 0.1490.165

0.068 0.52 0.034

0.082

II

8.57 8.96

20

90

136 1409 2625 8730

2980 5020 2590

1130-

10.9 0098

128

0

19 8.9

MOO 82 -150 3430 2723

44

48 36

92.91 63.54 242 162.46 167.2

9 6 19

550

609.6 ±0.2 1560 35 6700 180

+ 19.0 ±0.4

4200

1300 35 2340 70 520.3 ±0.2

1497-

470 22.5

68.9 12

0.049 125

2620 -302.4

Coefficient of linear thermal expansion, near 68°F,

6

Copper Curium Dysprosium Erbium

41 29 96 66 68

I Cb(Nb) Cu Cm Dy Er

. . .

Columbium

c 140.13 132.91 35.457 52.01 58.94

I

58 55 17 24 27

2 3. ±±

Ce C» CI Cr Co

3

12 8.65 55 2.22

I0~>

170

0.215

3740

±0.2 ±0.4

1166.9 -308.9

2900° 1220.4 ±0.2

±

Cerium Cesium Chlorine Chromium Cobalt

10.82 79.916 112.41 40.08 t2.-e+e-

5.73 3.5 1.82 80

9

35 48 20 T—

74.91 211 137.36 9.02 209.00

6.62 1.6626

2.699

±±

4

(niobium)

33 85 56

As At Ba Be Bi

227.05 26.97 241 121.76 39.944

Boiling point, °F

Molting point, °F

Thermal conductivity, near 68°F Btu/(hr)(ft')(°F/ft)

for Metals.

Heat of fusion, Btu/lb

Society

Specifio heat at 68'F, Btu/(lb)(°F)

Some of the Elements Physical Properties 1948, by permission of the publisher. The American

Density at 68°F (20°C), g/cm>

Handbook,"

Table

Atomic weight (1947)

"Metals

0

Br (M C'a

89 13 95 SI 18

Atomic number

5

Ac Al Am Sb

A

4 67

^ Boron i_» Bromine Cadmium *" Calcium Carbon (graphite)

Symbol

from

X

Arsenic Astatine Barium Beryllium Bismuth

Actinium Aluminum Americium Antimony Argon

Element

Reprinted


2. 0.

6 2

1

±

3

'

X

0

1

1 226 222 186 102 Bi

94 84 19 59 91 88 86 75 45 J7

Os

N

,

.

O 51 91 46

05

110

1300 -96 5740 JJ7I 102

3.0 40« 20 12 44 1.51

90

1100" 145 1700 54000.86 6.63

4900 350 -361.8 2829 111.4 0.2 3224.3 ±1.8

160 115 4.9 126

0.25 0. IIS 0.033 061

?18

-7,1

26.

49

5.9 61.6

1

±± ±

Igg

0.033

177 048

0.031 0.218 0.058177 0.032

9900 -297.4 7200 536 7970

1420

0. 105 0.247

4950 -320.4

0.045 133 11.2

11.3 286

117

25.6

27.0

Heat of fusion, Btu/lb

0.045 0.031 0.79

0.

0.057 0.052 0.031

3.45

Specific heat at 68°F, Btu/(lb)(°F)

(Continued)

00

n 11 Re Rh Rb

2

2

Ra

10-'

± ±± ±

Pr Pa

10->

2 02

Pu Po

Pt

8

Pd

2651 -346.0 ±0.2

X

8.90 1649

1540 70 -415.5 ±0.5

10'

X

22.5 1.3318 12.0 1.82 21.45

1.

190. 16.0000 106.7 30.98 195.23

2030 3900 675 8670

20 0.04 90

-410.8

3160 2500

0.2

X ±

0

Kadi um Radon Rhenium Hhodium Kubiillum

239 210 39 096 140 92 231

46 15 78

Np Ni

3 7.05 0.8387

1. 7 ±

0 9

Plutonium Polonium Potassium. Praseodymium Protactinium

76

Nd Ne

K

3 9.7474 43 13 55 10.2

0 1202 2273 -37.97 4760

4960 -242

0.9

2802 -251 1519 621.3 367

-422.9

7.87 488 10' 6. 15 11.34 53

X ±±±±

2

(yellow)

.

60 10 93 28

Lu Me Mn Hg Mo

P

I 4

9

Osmium Oxygen Palladium Phosphorus Platinum

144. 27 20. 183 237 58.69 14.008

71 12 25 80 42

Fe Kr La Pb Li

±0.2 361 9600

-434.6 313.5 ±0.2 237 4449

10">

Boiling point, °F

Elements.

±±±±±

7.31 4.93 22.5

08375

of the

Melting point, °F

Properties

±±

9 0

Neodymium. Neon Neptunium Nickel Nitrogen

174.99 24.32 54 93 200.61 95.95

26 36 57 82

7

Ir

1

55.85 83.7 138.92 207.21 6.940

61 49 53 77

Physical

Density at 68°F (20°C), g/cm'

Some

0080 147 114.76 126.92 193.

Atomic weiKht (1947)

1.

Il(I'm) In

H

5

Lutecium

Atomic number

1 25

^ Magnesium >-» Manganese Mercury Molybdenum

Symbol

2.

0

Iron Krypton Lanthanum Lead Lithium

Hydrogen Illinium (promethium) Indium Iodine Iridium

Element

Table


X

II

38

41

0.0143 41

53 0.0145

0.027

5. 15

92

20 41

44 0.0051

14 0. 252 34

0.098

Thermal conductivity, near 680F Btu/(hr)(ft»)(°F/ft)

1

6 !

i

52

t±

4

.

99 127.61 159. 204.39 232. 12 169. 118.70 47.90 183.92 238.07 50.95 131.3 173.04 88.92 65.38 91. 22

69 50 22 74 92 23 54 70 39 30 40

Tm Sn Ti W

V

270

20

U ±

' 90

2700 350 787.03 3350 45

3150 - 170

13

26

0.047 0.031 0.034 0.054 0. 126 0.032 0.028 0. 120

0915 0.066

2530 2660

4120

6150 -162.4 1663

10.700

13

0. 0560.295 0. 176 0. 175 0.036-

4010 1638 2520 832.3

0

8

±

Y ±

X

5. 7.

I

Computed. For polycryBtalline zinc; in single crystals, varies from 34 (parallel to hexagonal axis) to to hexagonal axis). (perpendicular From 68 to 40°F. At 36 atm pressure. From 68 to 2I2°F. At 30 atm pressure. At minus 80°F. Sublimes. From 68 to 392°F. From 68 to 382°F. - At 32°F. "At 104-F. From 32 to 2I2°F. Monoclinic sulfur. At 77°F. Handbook." 1948, by permission of the publisher, The American Reprinted from "Metals Society for Metals.

43 36

79

1

Zn Zr

4 6.0 495 10"' 7.015.51 133' 6.5

±±

Xe Yb

±±± ±

449.4 ±0.2 3300 180 6170 35 2065

4900" 840 621 572 3300

95

9.357.298 4.54 19.3 19.0

6.24 8.33« 11.85 11.5

45 49 45 16

0.084 0. 162"

1260 4200 607

0.057-

8850

1 0

Vanadium Xenon Ytterbium Yttrium Zinc. .; Zirconium

S

(wolfram)

1 1 2

43 52 65 81 90

1760.9 0.0 207.9 ±0.4 1420 20 246.2 0.9« 5425 90 ±±±

Tc Te Tb Tl Th

10.49 0.97 2.6 2.07 16.6 ±

Ta

107.880 22.997 87.63 32.066 180. 88

35

180

4500 >2370 2190 428 2605 ±±

38 16 73

12.2 7.7 2.5 4.81 2.33

9

±

Thulium Tin Titanium Tungsten l_t Uranium

47

Ag Na Sr

101.7 150.43 45. 10 78.96 28.06

5 7

Technetium Tellurium Terbium Thallium Thorium

44 62 21 34 14

Ru Sm Sc Se Si

1 1

Silver Sodium Strontium Sulfur (yellow) Tantalum

Ruthenium Samarium Scandium Selenium Silicon


9

13 4.7 2.4

39

65

0.03

22.

4.3

16 6.2-

116 15.5

9.3

3.4

36 3.6

10. 9« 39

21 1.6-

22.5

0. 15 31.5

242 77.5

48

5.1

10

3

«b

'

1

<*• 1 »

' '•

• «

'

CHEMISTRY AND CHEMICAL ENGINEERING

11-8

2.1

[SEC. 11

Important Nuclear Reactions

is

it,

2.11 Fission. The basic reaction of the nuclear energy program is fission. Cer tain isotopes, Um, Pu23S, and U!", are capable of undergoing this process which con sists of the incorporation of a neutron followed immediately by the spontaneous rupture of the nucleus into two smaller major fragments plus one or more neutrons. In this process some mass is converted into energy. The fact that the initiator of the reaction, a neutron, is a multiple product of leads to the possibility of a chain reac tion. Fission of U2,s illustrated thus: »

92U"6 + 0n' —

immediate ■» light fission fission

[»jU"'l

fragment + heavy fission fragment + >ltf»l + heat

This reaction, then, provides the neutrons to

keep the reactor operating, the energy

is

a

is

is

0

is

.

/

a

is

is

„Ba"°

12.8 days

17La'«—4-» stable Ce 40.5 or

3

is

/3

2.14 Formation of Heavy Isotopes. Separations processes are further complicated by a series of side reactions that lead to highly active isotopes of uranium, plutonium. and higher transuranic elements. These reactions consist of neutron absorption or a decay (n,y) and neutron absorption and emission (ra,2n) interspersed with The pattern often very complex. Many of the isotopes so formed have schemes. undesirable nuclear properties. Since isotopes of one element cannot be chemically separated from one another, the quantity of these heavy isotopes present represents an absolute limit to the decontamination that may be obtained with any devised Table shows some of these heavy isotopes, their source, and separations process.

K

decay characteristics. 2.15 Electron Capture. Occasionally an unstable nucleus, in particular one with shell. an excess of protons, will acquire stability by taking up an electron from the This called K-electrcm capture (EC). The vacancy in the K shell will be filled with rays an electron from higher level accompanied by the emission of characteristic of the A' shell of the product. An example

„Ru"

+ _,«•(«) — 4aTc" +

7

is

a

X

is

The emitted rays are characteristic of the «Tc*7 atom. 2.16 In some cases possible for the nucleus of an isotope Isomeric Transition. to exist in two energy states, the higher being due generally to higher angular decay or (2) momentum. then possible by two routes, either (1) direct Decay passage to the ground state with the emission of a ray which may or may not be 7

/S

is

a

is

it

7


a

is

is

is

a

y

a

for exploitation, and the fission products, most of them radioactive, which pose the majority of the separations problems indigenous to the nuclear energy industry. 2.12 Neutron Absorption. Most isotopes accept a neutron into their nuclei without undergoing fission. Rather, emitted, leaving an photon of energy unstable nucleus that will lose its excess energy by process more moderate than fission. This process termed (71,7). Its result to cause nearly every material exposed to the neutron flux inside reactor to become highly radioactive. This true of structural materials, coolants (particularly impurities therein), alloying diluents in fuels, and moderators. 2.13 Beta Decay. One of the most common methods whereby unstable nuclei formed by neutron absorption lose their energy decay. by a process known as This consists of the expulsion of an electron from the nucleus and results in an increase of one in the number of protons in this nucleus. In short, the nucleus converted to one of an isotope of the next higher element in the periodic table. Beta decay statistical process occurring at a definite first-order rate such that the half-life for any particular species can be determined accurately. The daughter product of this form of radioactive decay often itself unstable, and continued disintegration by this or other nuclear processes will continue until stable product formed. This may be illustrated by the decay scheme of one of the more important fission isotopes, Ba'*>.

Art.

nuclear reactor dependence

2]

Table

11-9

Formation of Some Heavy Isotopes

3.

Energy
Type of

Method of production

Half-life

decay

Particles

rjta U»« U"«

U"'

V—

P-

1.2

Daughter Pu"'

Daughter Am2*1 Daughter U>" Daughter Am«« Np"> (n.y)

Np"«

0.07

(~60%) 0.36 (~40%) 0.51

0-

IT*

a

0-

0.715,

Pu'» Pu'»

a a

Pu»»

Daughter Np'" Daughter Np»" Daughter Cm1" Pu"»(n,T)

PUM1

Pu"» (n,T)

5.75 5.49 (76)% 5.45 (24%) 5. 162 (76%) 5. 118 (24%) 0 0.021 a 4.91 4.88 0.39 5.48

Am"'

Pu"'

(n,T)

Pu"2 (n,7) Daughter Pu"'

Ara'»»

Am"'

(n,-,)

Am"'

Am"'

(n,y)

Am'"

Am"» (n,y) Daughter Pu"1

Cm"'

Daughter Am"'

0 (99 + %)

(~I0'%) a

0-

a 0 (60) % EC (20) %t

IT (20';,)* a (~l %) 0 (~99 %)

a

a

Cm"" (n,y)

a

Cm""

Cm"» (n.-r)

a

IT, isomeric t EC, electron *

Soft 0.044 (65) 0. 102 (3) 0.93, 1.03 (small) 8 from 0.049 to 0.435

0.654,

0.44. 0.33

a

Cm""

0.06

~0.27 (45%)

Daughter U»»

Pu"«

0. 150

4.77 1.26 (55%)

0-

a

74 years 1.6 X 10' years 2.4 X 10' years 6. 75 days

0.027 to 0.43

Np»»

Pu»«

~0.05 9 from

a 0.245

Np»'"

-0.05

4.8

0-

(n,7) Np>» (r>,2n)

~0.06

a

a

U"> (n.2n)

U"» Np"« Np»>


Daughter Pu'« Daughter Pa"> U»s (n,7)

Gamma

0.628

~0.045

X

0.0496 0. 100. 0. 145

0.095. 0. 12 0.026. 0.033. 0.043

0.035.0.038.0.053

0 = 0.593 5. 171 5.267 5 225 6.066 6.110 5.732 5.777 5.985 5.798 5.755

rays

(-3%)

(84%) (13%) (~26%) (~74%) (13 %) (81 %)

(6%) (75%) (25%)

0.038.

0.058

0.075

23. 5 min 22 hr

6.3 X 10-3scc 2. 2 X 10« years 2. 10 days 2 . 3 days 2. 7 years 90 yoars 6.300 years 13 years

~5 X 10s years 5 hr 470 years 16.01 hr

~I00 years 8.8 X 10* years

0.044

163 days

0.226 0.278

~ 100 years 17. 9 years

transUion (sec Art. 2.16). capture (see Art. 2.15).

a somewhat less energetic /3 particle. This process is called An example is isomeric transition (IT), and the excited state is called metastable (m). the decay of Kr"™:

followed by the emission of

80 %0-

2.2

The is

Fission Yields

fission process (see Sec. 4 of this handbook) is random as far as a single event concerned but can be predicted statistically with some accuracy. Each event

11-10


[SEC.

11

produces two fragments the lighter of which will have the greatest probability of a Known mass of ~100 while the heavier has the highest probability of mass of ~140. fission products extend from mass number 72 through 160 and the elements zinc up The statistical probability of obtaining any into the rare earths, probably terbium. particular mass number from the thermal fission of Um is shown in Fig. 1. The yield is expressed as the per cent of fission events leading to the mass number indicated. Such a yield curve depends on the energy of neutrons and on the fissionable isotope. Since two particles are formed per event, the yield curve sums to 200 per cent. The particles born in fission are nearly all highly radioactive, and many of them decay very rapidly through a series of one to a dozen daughter products until a stable nuclide is formed. The first two or three of these steps are often so rapid that their In Table 4 are existence is only postulated and has not been proved experimentally. shown all the presently known fission chains. Starting with the first member of the chain whose half-life is 2 hr or greater, the follow ing information is given for this and each subsequent member:

4.

Name of element Mass number Atomic number Fission yield

5.

Half-life

8.

Energy of emit ted particles and Complete decay scheme References

1. 2.

7. (>.

photons

4

4

The data given in Table are insofar im possible current as of March, 1957. are indi Abbreviations used in Table cated below:

y d h

s

= second m = minute (half-lives) m = metastable (mass numbers) = hour = day = year

y

Numbers in parentheses after energy refer to the number of emissions of particular energy per 100 photons emitted. = independent fission yield Parentheses around a half-life indicate uncertainty in determination. ~> indicates that transition not certain. possible but existence

that

is

is

i

2.3

Laws of Radioactive Decay

is

it

1.

is

a

of

t

4

1.

The char The distribution of products born in fission has been shown in Fig. acteristics of the various fission products and their decay schemes have been detaili'd in Table 4. will show hat the range of half-lives encountered is Perusal of Table very large — from seconds to many centuries. Therefore, the actual composition mixed fission products in reactor after start-up and in spent fuel or waste streams after discharge from the reactor a constantly changing function of time. During operation of a reactor, in each instant a quantity of mixed fission product is formed whose composition born, each fission indicated in Fig. As soon as The accumulation product begins to decay according to its own particular half-life. of any particular fission product over a differential time element di can be expressed by the equation is


3.

Art.

NUCLEAR

2]

11-11

REACTOR DEPENDENCE dn

— dl

=

_

Rii

M

(1)

235

where n = weight of isotope X = decay constant = 0.693/^s (lyi = half-life in convenient time units) y = fission yield M = atomic weight of species considered R = rate of fission of U"6 in weight/time (time units consistent with those used for X) The solution of Eq. (1) when n = 0 and ( = 0 is

-* X

where R. = Ry M /235

At

* =

x

(1

-

(2)

e~*<)

,

R.

(3)

x


This means that after a time which varies with the half-life of the species, each fission product will approach a limiting concentration /fi/A//235X. Those with short half-lives reach this value very soon; those with long half-lives take much longer. To illustrate the accumulation of fission products within a reactor, Eq. (2) is solved for a number of representative fission products and the results plotted in Fig. 2 for the

0

Fio.

2.

50

100 DAYS

150 200 250 PILE OPERATION

300

350

Accumulation of fission products within a reactor during operation.

a

I Jt

*

*

-3

»rt r^ V

f

Jfi/ vJS

<3

/ *

QO

■8

A

-3

soa

ft

a

fc.j5

73 to

ssmsassa


s

to

E3 = «

— 5. ci S s S3S2« w»iO r«t■*©

V

B t>

5esss§sEs£ ooooooo'oo

—

& 3

o

** S *^"^

5

§

*

I n

6s &>iv to Co e «*«— 6s ^

w *_ w ^- —

•«

K

t*t S

ess. sic

.59 ■3*

R

>5

5

o

7

x

Z

o x

•s

o X

JS

2-

a

s<

<

I

I" s

A

t^

— -36 11-12

<**

r

83 84

85m

36

36

36

n1

U* Stable

y

85

10.57

0.15 (
0.855

0.54

0.15.0.305

032.0.0093

(678)Se

Br—

\X

Kr |2or„

^-4?* Kr —

ty 10.57

*

m Br — ^ 4.36

Ratable

Rb

â. itable Kr

H4mKr

(~2 m) Se — .-30 m Br — *■(table Kr

25 m Se

.2.3hBr

liable Kr

1

37

0.3

4.36h

Stable

Stable

114m

11.051

35.7

citable Br

T

Kr

0.94

fi

1.5

I.I

0.48

0.48

2.3

Stable

0

Kr —

Kr

^Kr

1

t h

85

83m

36

IO-»

0.48

X

1.2.3.4. 5.6

I.ZS.$.

1.1 %*. $.4

1.2. ».I0

1.13.$.

1.13.5

1.2.3.$

6

36

83

35

3.5

0.82(30) 1.04(45) 1.31(28) 1.47(19)

h

80%

82

36

Kr

0.55(70) 0.61(40) 0.69(30) 0.77(100)

Î7m8e^

56.5 m 8a

itable Br

l.*3.fc

3.

No

0.46 .

35.7

«IOm)Aa

Stable Se

6.5XI0«y8.'^

m) Aa— »-3.9m8e

Aa

6

3.5 X 10-' (0

Not

(82 m) Ge — citable

1.2.3

4.

Kt

82

35

Stable

0.133

35

Stable

81

34 0.25

0.16

I0«y

0.27

(9

Br

6.5

Stable

Stable

79

35

0.04

IO-<

h

0.04

79

34

8X

X

k

82

75

33

Stable

0.02

l.4(~30%) 4.1 (~T0%)

Ce — *- 90 m Aa — »- liable Be

«

8»

78

32

34

90 m

0.02

Prewnt

2.

Br

78

~0.9

k 1

*

1

1

8e

71

33

Ilk

0.02


6

3

b

Li

Kr

Element

89 89 90

38

39

38

2

3.01(1) 3.24(1) 3.52(1) J.ttKI) 4.87(1)

55

Br"

Br»-*-(instant)

~33sKr— No No

0.61 2.28

h

'1

- stable Sr

stable

stable

«- 64.8

15.4 m Rb —

53_d_8r

Rb — - stable Sr

»- (2.74m) Rb— »- 28 y8r—

18 m Kr —

10"

- stable Sr

-!7.8mRb-

►stable Sr

Y h

9

5

Y

Stable

64.8

28

Br— *4.51 No

1.463

y

Stable

-

Kr

Kr

Br — »-78 m Kr— —4-5 98% 55 No

d

53

»-insUnt

Stable Kr -« — !8.7dRb <0.002%

4.51

7

Stable

~2%

neutron

7

10"

Decay chain

. neutron emission

15.5s Br— «-2.77hKr-

7

0.275

0.53 1.08

0.908(63) 1.39 (6) 1.85 (100) 2.11 (5) 2.68 (II)

1.55(40) 0.166(20) 0.191 (100)2.19(50) 0.36 (14) 2.40(100) 0.845 (65)

(Continued)

■!9.5dRb— Stable Kr "<0.002%

Data.

7

s

5.9

Nuclear

Gamma energy, Mer

and

3.

5.9

4.8

7 2 4.8

7

Y

90

87

38

4-5

y

s

X

T

40

87

37

0.23(1 to 5%) 0.71 (~14%) 1.80 (~86%)

2.5(13%) 3.6 (9%) 5.3 (78%)

0.52(68%) 0.9 (12%) 2.8 (20%)

Beta energy, Mer

Schemes

X

Sr

Rb

Stable

18.

d 7

y

1.2.3.5

Ref.

1.2.3.4.

1.2.3.9

1.2.3,5. 6.8.9

1.2.3,5.

1.2.3.4. 5.6.9

6

90

86

38

Stable

17.8 m

2.77

Stable

Halflife

Decay

s

9

39

86

37 3.IXI0-»(0

3.7

88

38

Sr

Rbn

3.7

88

]

37

3.7

yield

Fusion'

Product

h

Rb

88

86

Mass number

Fission

s

36

36

Atomic number

Table


4.

6 10 Zr

I

M«

92 92 93 93 93m

39

40

Y♦

39

40

41

U-Nb

T

1.4%

1

1

t

Nb

t

Mo

95 96

42

40

95

41

6.4

6.4

0.07 0.16(99%) 0.93(1%)

d

d

Stable

Stable

35

90h

0.364(54)% 0.396(43%) (0.883) (3%)

65

0.77

0.22

0.722 0.754

Kr

SUble Zr

Rb— W

I.I

Sr

-datable

etable Mo

90hNb

m Sr

16. m Y— ■tableZr

•table Nb

>^"13% XI0»Zr— »-4.2yNb

Zr

aUbleZr

5.83 dY^

\60%

m Sr-*-

^M5dNb-"J'' 98.6%

*-65dZrCT

1.4%

(ehort) Rb -*■

Kr— »-(ihort)

lO.SmY—

1

95*

95

40

Stable

0.0085.0.023

No

(uiwbT

Jt

(1.67m) Rb— .-9

(3.) Kr-*-80aRb-*-2.6hSr-»-3.6h

• .4

41

6.4

94

40

Stable

4.2y

I.IXWy

0.063

0.7

0.6

0.55(90%) 1.5 (10%) 1.3. 2.7.3.6

3.1

1.22

0.33

9.8a Kr-*

>

Nb

6.5

93 6.5

2.2

41

j

Nb

10

Stable

3.5

2.6

h

6.5

6.5

6.1

6.1

6.1

Stable

0.55

Notf

0.551.0.64.0.66. 0.747. 1.025. 1.413

2

1

Zr

92

38

5.9

58.3d

h 7

2

5

98.6%

91

40

5.9

m

0.62 (7%) 1.09(33%) 1.36(29%) 2.03 (4%) 2.67(26%)

10

Zr

1

3t •

33%

91

39

2.4

9.7h

51

Y h

Zr—

91m

39

5.9

1

Zr

91

38

40% 51 m

h 7

Y

«7%

Sr


Y 1.2.3.5.

1.2.3.4 5.6.10

1.2.3.5.

1.2.3.4. 5.6

1.2.3.5. 6.10

1.2.3.4. 5.6.10

6

6

101

102 103

44

44

44

M 103 104

45

45

44

>»Rli

1

Rh

Stable

Stable

Stable

1.8

Stable

Stable

57 m

9 2 29

39.8

d

2.9

4.2

5.0

6.5

0

No/3

0.217 (~99%) 698(~l%)

0.04

0.498

No

041.0.140. 0.181.0.372, 740. 780) 0.14

0.66(100) 02(1)

0.747

7

103m

100

42

Stable

0.29

102(1) 1.15(2) 72«1)

0.58

—

90%\».

-10%^*' Mo <

Stable Ru

<5%

57 m fill

— *■stable Ru

^ •Ublei' Rh

>95%^r 39.8dRu^^

12m Mo — »-<25sTc

stable Ru

^ stable Mo

— •- stable Ru

yS^ 2.12XIO*yTc

04hTc

14.6 m Mo — »-l4.0mTc

Stable Mo

68

Decay chain

<5%k72.1mNb-^'^

>95% »-60sNb

(Continued)

Stable Mo

l7hZr.

Data.

h

Ru

» 99

99

0.45(14%) 0.87 (—1%) 1.23(85%) No0

1.267

1.91

Nuclear

Gamma energy, Mev

and

T

<5%

>95%

♦

44

2.12X105y

1 6

Ru

43

6.04

0.6

Schemes

Beta energy, Mev

Decay

(0

♦-Tc

68h

Stable

h

99m

6.1

5.9

Product

6

1.2.3.5.

1.2.3.5. 6.9

Ref.

1.2.

1.2.3.5.

1.2.3.5.

1.2.3.5.

1.2.3.6

6

Ru

1

Tc

-10%

0

6

Ru

N

6

Mo

99

42

b

43

98

42

\ 0

h" 90% 03

97

42

N

1

6

(-■

Stable

97

b

►N

60a

h

Mo

74 m

6.2

97m

41

17

<5%

>95%

6.2

97

40

—

Halflife

41

Element

Fissionyield

Fission

Miu number

4.

Atomic number

Table


2 6

r 2 1

4.

J. 6

0

-

11-17

2

bob o d

sis o

»-2

Sob

r-" 3

1.

0.618(100) 10 (8) 1.39 (20) 162 (9) 1.83 (6) 2.11 (9) 2.51 (4) 2.79 (2)

f P3

~l (15%) 2.7 (20%) 3.5 (40%) 4.1(25%)

Not

j=

2

t ■*

M IS

9

° 0.2

«x

b

2

V

1

m

5

d

$

1

i

/

'1

•

a

s

&5

a—

2

i

s

t

t

2

'

2

a

/f

i

*

£

■?

1.2.3.6.

1.2.3.6

1.2.3.5.

1.3.4.5

1.2.3

6

*

• 1

S5

!f

j

s

s

1

a

1

~

*

3

i

d

§ § §

d8

1

V

* *

sis s

d

d

*

o

* *

t

m

2

r—S ' d

2

e S

S t

Ji

!f

e 1.77(1) l.96«l) 2.!0(<1) 2.37(1) 2.66«1)

t

&

*

*

o 0.513(100) 0.624(53) 0.87 (3) 1.045(8) 1.14 «1) 1.55 (3)

3

1 t 9

*

e

$

o

s

o

I

ob 2.0 (3%) 2.44(12%) 3.1 (11%) 3.53 (68%) Othen (6%)

No

7

8

S3

i

se 0.039

0.31

S t o

*

8

« » «

S

a -s

n

8

«

a

a JS •A

a

.05

9

•a 0.13

7

*\

105

105m

S

0.726

-ST-*"!

* *

105

5

5


4

8

*m—

>,

b

*-3

t

In

1% 0.0008

6 X

J 0

0.161

6XI0»yIn

43dCd

50 m Cd

l.lhln-

Stable Cd

»-4.5 53hCd—

m Ag — »- stable Cd

In

Cd— *-sUble

In

Ag — >-stable Cd

Decay chain

stable Cd

3h Agr-«-5.l

3

161(86) 0.565(100)

161.0 311

(Continued)

22 m Pd — »-7.5d

2 0.267.0 281.0.331, 0.43.0.84. 1.27. 1.55.2.00

0.485 (100) 0.935 (73) 1.30 (31)

25.0 34. 0.44.0.87

0.230(10)0.490(220) 0.260(30)0.523(430) 0.263 (3) 0.335

Not

Not

0.243 (~ll) 0.340 (~89)

Data.

MdSn

, stable Sn

,

1.616(30%) 1.772(70%) 0.740

Nuclear

h

~l.6.~3.0

1.61 (-98%) 0.7 (~2%)

d

43

0.63

(58%) 0.5» (42%) 0.83

I0»y

Stable

4.5

53

and Gamma energy, Mer

0

5m

0.011

IIS

49

0.011

113a

49 113

0.011

115

48

1 1

50

0.01

114

I.

II

III

,

Sn

0.01

113

48

h

-94

—6%

Stable

5XI0-'

113

49

h

1b

Stable

5XI0-*

113m

48

Stable

5.3

0.01

Schemes

5

"1

51

Stable

0.018

III 113

47 0.570

2.0

y

7.5

h

_t

Cd

Decay Beta energy, Mev 0.70(8%) 0.80(1%) 1.04(91%)

Product

d

0.018

yield

Halflife

Fission

y

Cd

1

Cd

47

lumber

Fission"

Table

4.

Atomic Dumber


I'

0.

I

t -{ 0.0012

121 121 122 123

SO

50

t

Sn —

1

*?.

T«

125m 125 126 126 127

127m 127 127

52

52

52

52

52

S

53

d h

SUble

9.3

105

0.326 0.113(1) 0.377(4) 0.175(19) 0.427(100) 0.205 «1) 0.463(31) 0.214 «1) 0.595(88) 0.320 «1) 0.637(23) 0.11.0.0355

0.9. ~0.4 0.058 0.456 0.185 0.563 0.240 0.674 0.417 0.764 0.0835 Not

0.51(7)1.17.2.04. 0.12 (29%) 0.30 (45%) 0.444(12%) 0.612(14%)

~l 0.86(50%) 1.11(20%) 1.57(30%)

0.7

18%/"'

105dTe

—84%^^-^

93hSbr^^

1.5 hSn

50 m Sn— ■»9_h_Sb— ^.Uble

P.SmlSn"^

9.4dSn_

SUble Sn

9.3 hTe—

'

0.25

93

9h SUble

SUble

1.90

0.4 (-5%) 2.37 (-95%)

39.5 mSc^

0.153

1.26

Sb

i

0.2S

9.8 m.

h

O.0S6

0.25

0.1

023

0.01

0

1

84%

+

L-.-Te

SSd

0.004

125

SI

1 y

Te

2.0

0.023

125

SO

51

9.4d

125

50

SUble

39.5 m

SI

0.012

0.02

0.015

124

123

50

^^sUble

58dTe

1.2.3.4 1.2.3.4.

1.2.3.5.

I.Z3

1.2.3.5

1.2.3.5.

1.2.3.5. 6.9

1.2.3.5

»-»UbleI

Te

.UbleTe

1.2.3.5. 6.9

1.2,3.5

I

81%

P1

»-» CD

18%

j

51

SO

136dSn^

Not

1.42

SUble Sn

No-,

0.383

27.5hSn-*»Uble8b

^SubleSn

In^

,(~250d)8n

— »-«UbleSn

SUble Sn

17.5 m

4.5 m

6

Sn

SUble

136

SUble

SUble

27.5

SUble

d

0.015

0.014

0.01

SUble

0.065.0.024

6

123

0.013

120

50

0.01

h

50

0.014

119

so

Subl« ~250d

In

1

So

119m

so

51

<0.0I

118

JO 0.01


9n

6

1

I

129 129 128 130 131a 131

131

I3lnt 131 132 132

53

54

52

52

52

52

53

54

54

52

53

>

4.4

4.4

4.4

Btabfa

2.4

77h

Stable

~12d

1

fa

+ 2.9

0.03

8d

(20%)

0.29 (10%) 0.69 (4%) 0.989(15%) 1.453(71%) 0.150

No

1.67

1

0.73 (15%) 0.9 (20%) 1.16(23%) 53(24%) 2.12(18%)

0.22

0.815 (0.7%). 0.608 (87%). 0.335 (9.3%) 0.25(2.8%)

0.47 (52%) 0.98 (4.6)% 0.57 (17%) 2.46 (4.7%) 1.35 (15%) 2.14 (60%) 69(25%)

1

2.9

25 m

Schemes and Nuclear

0.308 0.788 0.106

0

0.528 (25) 624(6) 0.673 (94) 0.777 (80) 0.96 (20) 1.16 (8) 1.40 (II) 1.96 (~5) 2.2 (2)

0.23

Te

74 iTe

hSb— ratable

~76*\

8b

~24«/

85%

m8b^

15%

/ 25 u Ta

X

Te

8d

2mSb-«-77hTc-»-2.4hI-«-ataMeXe

23.1

30

1%^

99%^

atableXa

~l2dXa

l.7XI0'yI-*-atabk)Xe

/

Decay chain 33.5 dTe

(Continued)

12 m 8b— a-atableTa

I.I

4.6

Data.

0.145 (100) 0.950 (~4) 0.450(24) 1.140 (~8) 0.595 (~6) 0.08.0.163.0.284, 0.364.0.637.0.722 0.164

0.177

0.435(9). 1.08(0.7) 0.0268 0.475 ~0.77 1.12 0.038

0.165 0.534

Gamma energy. Mev

>

Xa

? 2.9

30

SUble

Stable

SUble

l.7XI0»y

h

*-

5 0

T.

Tt

1 Ii

Xe

1.0

74 m

d

1.0

33.5

Decay Beta energy, Mev

4 h

h

132

0.44

129

52

0.34

Product

a

1^

54

2.0

129m

52

8b-j~24% T.

4.6

Halffife

Fission

k

~76%

1.0

yield

129

1

B—

4.

FIbuoo*

Man number

Atomic number

Table


SI

4

Ref.

I.2.J.4

1.2. Mi 5.6.8

1.13.5

I.Z3.S

1.2.3.4. 5.6.8.

9

135

54

I

1

t 137 138

56

Ba

137

55

56

136

56

2

0.523(92%) 1.17 (8%) 0.66

5.9

22

■

tXe

eXe

m Te

— *-!7 m Xe— »-32.9 m

92%

m

8%^

J>r

C

52.

9.2 hXe

*-3.9mXe-*-27yCsCT

»- .table Ba

-6% »im

neutron

*-rtab

~94%

1

Stable

Stable

63 m

27y

Stable

12.9

I2.9d

lWbetO.06 and 2.49

22.1"'

Stable 0.341 (92.6%) 0.657 (7.4%)

86.1

8b

"

I0<

C. -»-

5.27 dXe-* •table C

rtable Ba

.table Ba

-,2.60 mBa

.table Xe

—

I

5.8

5.5

5.9

0.0062

0.0062

[ d

5%U.

136*

55

0.4

<

Stable

50

■]

Stable

/ 3.0

5

137m

136

54

6.3

No*

»,v

68h

97.6%\»,

15.6 mXe

20. 5h

2.3 dXe

I

56

(136)

54

7.6

Stable

0.21

-6

4.

^95%

134

54

135

2IXI0«y

0.250(100)0.37(0.1) 0.60(3)

0.52

1.8.1.27

/

2d >Te

X

Sb^*

4

1.2.3.4. 5.6.8

1.2. 3.6

1.2.3.

1.2.3.5.

I.Z3.4. 5.6.8

1.13.4,

1.2.3.5

6

CH—

! 56

6.2

0.548 (-5%) 0.910 (-95%)

0

9.2k

5

6.2

6 15m

(35%) 1.0(40%) 1.4(25%)

0.081

4.4 m

4%/^

X

j J.

Xe

135m

54

68

h

I

135

2.1

135

53

Stable

d

6.5

0.345

0.232

63 »T.

' y

55

6.0

133

55

d

5.27

0.53 (94)

I.

6.5

h 0.85(5)1.4(1)

i

2.3

0.4 (~9%) 1.3 (-91%)

J.

30%

133

54

20.5

2

0.16

1331

S3

5-4

t a 133 6.5


4.

•

I

1 ■C + 6.2

6.2

6.1

6.1

142 143

143 143 144

144

58 58

59 60 51

;

P1

r 2

6

|

0

17

HubU

5m

290

No-r

0.232 0.294 493 0.565 0.722 0.861

695 (12) 2.185 (5.9) 1.48(2.3)

0.081(17) 0.0I2(T) 039(<7) 0.100K2) 042(7) 0.134(29) 054KI)

000

1,1

d

Stable

0.0574 0.351 0.668 1.10

La

Xe — »- (ahort) Ca —

-

m Ba — *~ La — V 32.8 dCe— ratable Pr

**-9.5mC» 85 m Ba— *- aUble La

stable Ce

9-

~1

a

— »-l3.7dPr—

(ahort) La-*-290

Ce-*-l7.5

abatable Nd

m Pr-»-atable Nd

Xe-*- (ahort) Ca-*-(ahort) Ba — »-

— »-33hCe

Xe— »- (ahort) Ce— »- [<0.5 m) Ba — «-19 m La

~l m Ca — *-6 m Ba — »-74 ml* — >-stable Ce

a

92(1.5%) 12(98 5%)

0.175(34%) 0.309(76%)

k

I3.7d

-0.22 (6%) 0.50 (12%) 0.74 (5%) 1.125(40%) 1.40 (37%) 0.932

33

Stable

0.145

(1.3-1.6?)

1

59

5.9

141

59

32.8d

0.9 (~5%) 2.43 (-95%, 0.442 (67%) 0.581 (33%)

0

Nd

6.0

141

58

3.7h

»-4l »Xe—

2.7al—

a

Ce —

6.0

141

57

Stable

7. 1

—

I

Pr

6.0

139

57

40.5

»l2.8dBa

Decay chain

8 1

C—

1

h 7. 3

Ce

I. Stable

»-66aCe—

(Continued)

l6aXe—

Data.

0.0296.0.132, 0.162.0.304,0.436 0.537 0.328 (40) 0.438(6) 0.490 (50) 0.815(46) 1.60 (100) 50 (1) 3.00 (0.04)

2

u~

6.3

140

58

0.42(16%) 0.86 (12%) 1.15(20%) 1.36(30%) 1.62(14%) 2.20 (8%)

40.5

Nuclear

Gamma energy, Mev

and

h

La

6.3

140

57

Schemes

Beta energy, Mev

Decay

0.48 (40%) 1.022 (60%)

Product

h

12

Halflife

Fission

d

Ce

6.3

140

yield

Fiaaion*

56

Atomic Maaa number number

Table


4. 8

1.2,3.4. 5.6.8.

Ref.

1.2.3.4. 5.6.8.

1.2.3. 5.9

I.2.3.S

1.2.3.5

1.2.3

9 9

d

0

J

t=

P

s

11-23 £

b

b

■R M

o js

&

o oo

dd

If

o

L

*o

*

0.07 (68) O.$30«l) 0.100(100) 0.600«l) 0.170(0)

V

a

a

Jt

1

t

X ft

\ t 6

R

£ t

%

1

l

t

1

1.2.3,5.

5

1.2.3.5.

1.2.3.5

1.2.3.5.

1.2.3.5. 6.8

1.2,3.5, 6.8

1.2.3.

6

d

1

0.064 0.335(100) 0.100 0.440(40) 0.167(70) 0.650(18) 0.230(42) 0.700(26) 0.275(55) 1.50 0.019

Z

1

z M-l US

3

b

£

b

_

3

—

0

b

0 1

-e

5

A

£ t

t

t -o

3

|

o

5-

X

■a if*

II f

Z

K

.03.0 097.0 .112. 0.114,0.124, 0.188.0 198. 0.211.0.226, 0.24.0 266. 424.0.538,0.65 0.285. ~l.3

sO

;

« -;

1

-a

t

t n

2

3

1

•S s

J!

?

?

0.092 (550) 0.410 (30) 0.165 (15)0.440(35) 0.280 (25) 0.530 (250) 0.32(45)0.690 (15)

1

6

3

3

H

rf

5

N

3

id

w

5

X '7 \

^

3

5

i

I .£

S

s

s Z

s

s

a


6 4,'

M

9

7. 5,

3 M jg

H

oo

sa

rS

dd

ft

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•£>

^ 3

ii i

3

£

II

3

3

-

-

■

5 s

3

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3

pi-

11-24 5(5-" ~

8

3 S

li ■

3

3

r

£=> .2

» — ^

■

a ca . »o

S.r

j 55

111"-

-S

« &

! 77 ■ »

sill-;

2

H m

+ S

9? 8

lis

?

i 'ff

6

a ■S =o

I*.j

9

4

« tff «*>

£

K

I

Is

J

»

J

8«

1

•g

".

3

F

S

3


13. •I

k o

=

1

9 M

t

t i

*n

odd

IP a

es

111

Ji

f2|

as*

* -= r

k >-

C 5|osj|ci Tm\

'IS.**-

s. ti e 3 . = = =Z

Art.

2]

NUCLEAR REACTOR

DEPENDENCE

11-25


case where the production rate R is equal to 1 kg/day of total Uut burnup. This is Figure 2 equivalent to the operation of 900 Mw (heat) of installed reactor capacity. assumes no destruction of the accumulating fission products within the pile by neutron In the case of species with large absorption cross sections, particularly absorption. Xe1" and Sm14*, a correction must be made for this factor. The correction results in a lower equilibrium value for the species than that calculated. The simple equation (2) as plotted in Fig. 2 is accurate only for the first significant member of a decay chain or after long periods of time for any member all of whose progenitors have

240 COOLED

180 DAYS

Fio.

3. Decay

420

of mixed fission products.

reached their equilibrium value. In the case of intermediate times the effect of decay chains must be taken into account. To calculate the isotopic distribution of fission products in an operating reactor it is necessary to consider:

L Time

of irradiation

2. Relative fission yield of each isotope along a decay chain 3. Half-lives of the isotopes along the decay chain

Once the fuel has been discharged from the reactor, only decay equations are In general, the second decay is to a stable or very long lived product, so employed. the only equations needed are dA

A

„

4

(4)

= Ave-***

(6)

11-26

CHEMISTRY

CHEMICAL ENGINEERING

AND

[Sec.

11

dB

and

(6)

ill

-*,. + (B%

âA„ Xa — Xa.

H

_

V

x^° Xs —

\a')

e~x*

(7)

where At, and Bo are the amounts of A and B present at discharge as calculated In Table 5 is shown the distribution of isotopes from the previously (see Fig. 2). thermal fission of uranium 235 irradiated 135 days and cooled for various periods. A recently declassified report'f gives similar data for fast fission of uranium 235 and plutonium 239. 48

l/^-BS

40

80

160

120

DAYS

Fia.

DAYS

200 240 COOLED

RRADIATED)

280

320

360

4. Fission-product heat generation.

C

is

is,

Often it is sufficient to know how the gross activity of a discharged fuel will decay. As a rough rule of thumb the effective half-life of discharged reactor fuel is approxi mately equal to its age, or somewhat more accurately the effective half-life varies That a fuel which has been cooled for 60 days will have an approximately as l~l%. effective half-life of approximately 60 days. The relationship between the weight of fission product present and the number of disintegrations taking place per unit time given by the equation (expressed in curies) = 1.64

X

10'«VF

C

= curies = 3.7 X 1010 dis/sec = weight of fission product, kg X, = decay constant, sec-1 MW = molecular weight

where

t


(THERMAL FISSION

W

Superscript numbers

refer to references

at end of section.

'V*

MW

(8)

Art.

2]

NUCLEAR

REACTOR DEPENDENCE

11-27

There is plotted in Fig. 3 the per rent of total curies remaining during cooling of a 235 during 135 days' irradiation.

quantity of mixed fission product formed by the thermal fission of uranium 2.4

Heating Effects

The energy released by the radioactive disintegration of the fission products is manifested in several ways. Some causes chemical decomposition, some results in 100

100

9

80

7

(THERMAL FISSION U23S

- 135

DAYS

60

PILE IRRADIATED)

40 30

20

PlO 3 S

9 y 7

i

I


s a.

!5 2

o

3 2

£ 3

I

0.6 m

Q6 0.4 0.3

0.2 0.1

50 DAYS COOLED

Fig.

500

100

1000

5. Fission-product ft heating.

the formation of ion pairs, a little causes (y,n) reactions, but most of it is converted into heat. This heat production is of considerable importance in determining cooling loads in reactors during shutdown, in freshly discharged fuel elements, and in the Radioactive disintegrations are equated to heating by the follow storage of wastes. ing equation:

~ hr

=

k\„NEF

= 1.517

X 10-"X„.VAT

where k = factor to convert Mev to Btu = 1.517 X 10~" \n = decay constant, hr-1 N = atoms (particles) of radioactive material = atoms isotope in question E = energy of radioactive material per particle, Mevf F = fraction of total particles having energy E t For

0 particles it U assumed

that this energy is 0.3 to 0.4 of maximum

(9)

Um

fissioned X yield of

Mev.

y

d

4.800

8.290 800 4.800 1.310 900 6. 100 6. 500 6. 500 400 710

9. 285 4.800 4.800 0.315 5.900 6. 100 6. 500 500 400 >l 715

9.453 4.782 0.033 4.815 165 867 6. 100 500 500 067 400 ll v/.»

9.491 757 0.076 833

10. 130 4. 195 0.865

10.556 780 402

11.064 284 019 5.303 0.052 3.881 6. 100 6. 500 500 470 400 ii. wr

404

2.950 2.425 5.375 0.046 3.475 6. 100 6. 500 500 885 400 II 90-.

560 900

5.460

0.075 5.035 6. 100 500 6. 500 178 400 11 7H«

152 5,824 100 500 6. 500 134 400 Jl 610

6

0

y

66

6

66

X

d

Y

1 II

0.064 4.498 6. 100 500 500 I.79B 400 II 560

3. 1. 5.

040 000 100 500 500 402 400 ,*1

4

5

91,

d 22

5.060

6.400

3.700 4.590

700 5.585

700 018 5.735

3.700 043 5.748

3.700 0.605 5.825

3.700 1.020 5.836

3.700 1.516 5.848

3.700 1.850 854

3.700 2.240 5.860 . 800

1 1 3. 2

182

1.500 2.700 3.700

3.479 2.920

2.808

2.790

2.730

2.717

2.707

2.700

2.700

0 3 6. 6. 6 3 6

5 1 1

4.

0. 5

"V

"

1.500

0. 779 2.700

100 680 220 2.700

2. 5.

108 2.700

5.

0 2.

0

4

Stable Stable Stable I.I I0« Stable 65 Htal.U,

0 1.

0.090 2.700

.

0. 030 2.700

1 1 2. 5. 0.017 2.700

2. 5.

90 91 92 93 94

1.

007 700

2.

Zr

1

2.700

2 5

0

6 3J

500 400 son

556 6 6 in

900 900 100

TTsoo-"

1

Stable 57

1.

3. 0

2.700

2.

3.

0

89 91

1.

2.

Stable Stable 53 28

1.

87 88 89 90

4.960

5.072

090 5. 150

163

173

180

180

2. 4.

Sr

10'°

0. 480 100

0. 480 100 0.721 100 0.480 100 1.280 100

0. 480 100 1.392 100

0. 480 100 1.410 100

0. 480 100 1.470 100

0.480 100 1.483 100

0. 480 100 493 100

480 100 1.500 2. 100

0. 480 100 1.500 100

3

Stable 6.2

0. 173

0. 133

0. 133 0. 133

0. 133

0. 133

0.133

0. 133

0. 133

1.

401

0.040 0. 133 0. 133

0. 133

0. 133

0. 133

0. 133

0. 133

0. 133

0. 133 0. 133

0. 133

1.

2.

85 87

Stable Stable 10. 6y Stable

0.359

0.399

0.399

0.399

0.399

0.399

0.399

0.399

0.399

1

Rb

83 84 85 86

080 0.250

0

Kr

X

Stable Stable

0.009 0.020

0.009 0.020 0.040 0.080 250

0.009 0.020 0.040 0.080 0. 250 0.009 0.020 0.040 0.080 0.250

009 0.020 0.040 0.080 0.250

0.009 020 040 080 0.250

009 0.020 0.040 0.080 0. 250

0.009 0.020 0.040 0.080 0.250

0.009 0.020 0.040 0.080 0. 250

0.009 020 0.040 0.080 0.250 0. 399

CO 10 years yearn

year

300 days

0

79 81

cooling

-236

100 days

atoms fissioned after

of Uranium

60 days

Atoms/100

Fission

0

Br

Thermal

irradiated)

from

0

30 days

days

days

Spectrum (135

000

days

0

10'y

5.

Stable .Stable 6.5 X Stable Stable

Half-life

Product

5 1

77 78 79 80 82

Mans No.

Fission

1

Se

Isotope

Table


2

0

6

66 06

06

1. 6

6

626

626

y

SUble SUble

SUble Suble 7.5d

103

105 106 107 108 110

107 109 111

III 112 113 114 115 116

115

117 118 119 120 .22

Rh

Pd

Ag

Cd

In

8a

6.245 6.200 5.900 6.500 24.845 6.100 5.000 4.200 0.020 1.800 0.167

6. 107 6.200 5.900 6.500 24.707 6.100 5.000 4.200 0.060 1.800 0.189

6.500 22.777 6.100 5.000 4.200 0. 196 1.800 0.276

5.500 21.859 6.100 5.000 4.200 0.391 1.800 0.298

6.500 21.150 6.100 5.000 4.200 0.661 1.800 0.315 976

20.627 6.095 5.666 4.200 0.858 1.800 0.325

19.440 5.918 5.000 4.200 113 1.800 0.334

2.880 0.900 0.213 0.200 0.080 0.020 1.413 0.028 0.028 0.018 0.011 0.010 0.010 0.010 0.059 0.011 0.010 0.010 0.010 0.010 0.013

0.900 191 0.200 0.080 0.020 1.391 0.028 0.028 0.018 011 0.010 0.010 0.010 0.059 0.011 0.010 0.010 0.010 0.010 013

2.704 0.900 0.104 200 0.080 0.020

1.304 0.028 0.028 0.018 011 0.010 0.010 0.010 0.059 0.011

0.900 0.082 0.200 0.080 0.020 1.282 0.028 0.028 0.018 0.01! 0.010 0.010 0.010 0.059 011

0.900 0.065 0.200 0.080 0.020 1.265 0.028 0.028 0.018 011 0.010 0.010

0.010 0.059 011

2.042 0.900 055 0.200 0.080 0.020

1.255 0.028 0.003 0.031 015 011 0.010 0.010 0.001 0.010

0.057

0.900 0.046 0.200 0.080 0.020 1.246 0.028 0.014 0.042 0.004 011 0.010 0.010 0.002 010

0.047

Suble Suble

7 X

1

8 00

0

0

d 0 0.010 0.010 0.010 0.010 0.013

0.010 0.010 0.010 0.010 0.013

0

0.010 0.010 0.010 0.010 0.013

0

0.010

0.010 0.010 0.010 0.010 0.013

0

0.009

0 0

0.010 0.010 0.010 0.010 0.013

0

1.800

5.000 4.200 1.800

5.000 4.200 1.800 0.083

2.900 0.900 0.380 0.080 0.020 1.380 0.200 0.028 0.228 0.018 Oil 0.010 0.010 0.010 0.059

0.900 0.380 0.200 0.080 0.020 1.580 0.028 0.028 0.018 0.011 010 0.010 0.010 0.059 0.011

0.900 0.297 0.200 080 0.200 1.497 0.028 0.028 0.018 011 0.010 0.010 0.010 0.059

0.010 0.010 0.010 0.010 0.013

0.011

0.010 0.010 0.010 0.010 0.013

0.010 0.010 0.010 0.010 0.013

Oil

17.100

000 2.900 2.900

0

Suble SUble SUble SUble SUble

0.

0

SUble

6.100 5.000 4.200

6.100

6.100

11.083

6.500 25.000

6.500 25.000

6.500 24.991

6.400 6.200 5.900

6.400 6.200 5.900

6.500

6.500

6.391 6.200 5.900

0.009

0.009

0

SUble SUble SUble Stable 43 Suble

1. 1

I0«y

11.187

. 249

2.840

11.472

689

2.509

2.239

12.183

10'y

0.088

0.159

1.045 4.177 6.200 5.900

1.343 3.259 6.200 5.900

1.380 2.550 6.200 5.900

1.493 2.022 6.200 5.900 0.005 6.500

2.340 0.658 6.200 5.900 0.182 6.500

0.088

0.159

1.045

1.343

I.3S0

1.493

2.340

1.787

8 12.447

d

SUble

SUble SUble Stable 39. SUble I.Oy

2.1

X 1. 1

M

99 101 102 103 104 106

Ru

h 1 1

M

99

To

Stable Suble SUble 68 SUble

95 97 98 99 100

Mo

Suble 35

93 95

Nb


d 1.

1. 1

0

0

0

Q

— o — «rt r*.

d

sss

EB^ss

d o o

♦ r«.

TTO

000

ooo
h 9 h 0.045

0.020 0.073 0.014 015 0.009

0.020 0.073 0.014 0.015 0.012

0.041

0.042

0.014 0.015 0.013

0.073

«A O

r^CNNONO NOOO

>0 #» «** —

— o-ooo^v o so oonoo^ o ooomoo o-

d 2

oooono

ssssss

555^5

h

-

•3 r.

11-30 N

ddooNO

X

b

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00O OO*

oo

fN*

5.

if II oo fN — fN

OO OO

oo

o— r>.sO— fN

T -ooaooo oooooo

—

^Vbislb

O — OO

.NC>000 .•iô-om

toeNo •* m

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—

NtON-C

-OOfAOOO

.M*ON-fl

^TnOOO — fN

665

ooo oo« lAfNUC

665

£6

ill

y

1

X

J.

555

*\ •0 -o

?

o

ooo 00
IN W\
o ac .« -O

6. 500 200 750

—

o

65

* ■>

O oor-* N u-tfNIS, »J

ooo

O >£> B C

N c *

>o "> oo

m >OsOM-N K>

S

S

:S

tU

s

6. 500

22 200

* •

oo oo

S

-OnO* r%

21. 200

7.

— fN

600 6.300

000 900 400 600 6. 300

006 00*

900 400

o ooo o oo-o fN u-NfNls,

:S

2 4.

009 OOt

1.250

250

1.248

2 4.

S

0. 250

0. 248 1.000

0. 250

623

023 0. 100 0. 500 000

fN

7.

.1

02

1.

O fN c — fN

=

0 02

O—

oo

!

0

OO

500 200 840

N

NO

oo mo

492 200 870

— «0

21 200

029

n o—

no

21. 188

0.014 015

oot

wo

5

0.073

0.020

I

OO

r-J .0

7.600 6.300

10 years

N

s

0

OO OO

900 400

ON

2.613

OO

iÔ

2.613

oo OO

2.620

*-o

2.623

0.010 0. 100 003 500 000

ino«oo< OONOOO o— oôo 0.007 100 0.013 500 000

r>»

0.005 0. 100 0.018 0.500 2.000

0

—O O—

0.0)1 0. 100 002 0. 500 2.000

years

year

R

1.247

046

0.014 0.015 0.016

0.073

0.020

100 days

0. 247 1.000

O 0. 237 1.000

Noooooo ooooo 0. 232 1.000 0.002

o o o o o o 0. 227 1.000 0.020

— *■ f*\

2.627

0 i>» IA fN —

0.004 0. 100 0.023 0. 500 000

0.047

014 0.014 0.018

0

oo

>no

124

Hlat.l..

ooo ooo 0.074

001 020

60 days

cooling

fissioned after

r*

o—

I0>

014 014 019

—o

o

o

NO NO

2.

Stable

oo

O rA O

2.

Stable Stable Stable 3d Stablo Stable

30 days

0.074

oo

0.001 020

ON

-T..- ■O

0

o

55

10'

0 0

OOO OOO

Stable 1.7 8d 2.4h 20

5 1 00

b

days

100 atoms

oo f\ O O

o o o

K\ O O

0

Stable Stable 90 Stable Stable 77

days

Atoms/

2

N

93

i i

j

Stable Stable

No. 0) 13 O O ca

136 Stable 9.4

Mass


1

o n o o

0

.

OO

•—

'o

— ,-

o

• o

o e .e

o

O o

.c O

fN

y

Stable 12. Sd

Stable 40

Stable 33 Stable 33 290

Stable 13.

Stable Stable Stable Stable II. 3d Stable Stable

2.6y 54 27

Stable Stable 73 Stable Stable

Stable Stable 17y 14

Stable Stable Stable

138 140

139 140

140 141 142 143 144

141 143

143 144 145 146 147 148 150

147 149 151

147 149 151 152 154

151 153 155 156

155 156 137

La

Co

Pr

Nd

Pm

Sm

Eu

Gd

5.800 0.032 5.872 6.000 0.005 6.005 6.263 0.645 5.900 4.470 17.278 5.355 0.043 5.398 6. 157 1.630 4.200 3.300 0.008 1.800 0.740

5.800 0. 168 5.998 6.000 0.079 6.079 6.053 1.200 5.900 4.810 17.963 4.800 0.200 5.000 6.000 1.290 4.200 3.300 0.500 1.800 0.470

5.800 0.381 6.206 6.000 0.057 6.057 5.862 1.650 5.900 5.000 18.412 4.350 0.430 4.780 5.770 100 4.200 3.300 0. 125 1.800 0.740

5.800 0.855 6.675 6.000 0.013 6.013 5.432 2.270 5.900 0.091 190

h

d 18.883 3.730 0.908 4.638 S.

5.

hd

d

7

16.526

12.200 6.000 000 200 6. 100 4.200 300 1.800 0.740 22 340

12.201 6.000 6.000 6.200 6.099 4.200 3.300 1.800 0.740 22.339 0.176

13.068 6.000 6.000 200 5.232 4.200 3.300 1.800 0.740 21.472 1.461

14.321 5.999 5.999 6.200 3.980 4.200 3.300 1.800 0.740 20. 220 1.909

14.684 5.996 5.996 6.200 3.620 4.200 300

1.800 0.740 19.860 1.997

730 0.006

5.736 6. 194 2.040 4.200 3.300 0.001 1.800 740

01

1 2.

hh 0. 176

0. 150 0.027 0.001 0.178 0.004 0.012 0.007 023

0.150 0.027 0.002 0. 179 0.004 011 0.007

0

y

d 0

0.025

0.005 0.013 0.007

0.025

0.005 0.013 0.007

0. 150 0.026

0.150 0.026

2.327

2.

0 176

2.406

2.355

2.338 0.137 1.300 500 0.300 090

2.209

0. 176 2.424 1.300 0.455 0.300 0.090

1.461 1.139 1.300 0.491 0.300 090

1.909 0.691 1.300 0.495 0.300 0.090

1.997 0.603 300 0.496 0.300 0.090

0.031 0.013 0.007

0.030 0.013 0.007 050 0.018 0.013 0.007 038

0.012 0.013 0.007 032

0.011 0.013 0.007 031 0.026

051

0.650

0. 196

0. 172 0.174

0. 174 0.006 0.013 0.007

0. 176

4.290 0.500 0.150

4.569 0.045 0. 150 0.001

3.320 0.009 0.150 0.013

2.896 0.005 0. 150 0.019

2.789 0.004 0. ISO 0.020

2.473

0.300 0.090

2.600 1.300

0.001 0. 150 0.025

0.284 1.300 0.499 0.300 0.090

2.315

2.376 0.216 1.300 0.500 0.300 0.090

2.385 0.165 1.300 0.500 0.300 0.090

0.

0. 114 1.269 0.494 0.300 0.090

2.315

2.376

2.385

18.275

17.835

17.380

2.238

1. 17.035

0 1

0?022

5.900 0.001

5.900 0.868

2.480

4.060

3

0

267

5.900

6.300

6.300

6.300 0.001 5.900

6.300 0.004 5.900

3

172 0.031 0.006

6.000 6.300

6.000

6.000

6.000

6.000

6.000 6.296 0.270 5.900

6

16.465

6.000

6.000

6.000

6.000

6.000

6.000

5

201 0.910 4.200 3.300 314 800 0.740

17.900

6.940

6.080

5.950

5.930

5.864

120

5.800

5.800

5.800

5.800

5.800

5.800 0.004


2.

6

6 0

0

0

0

0

0


11-32

[Sec.

11

Table 6. Beta Heating by Individual Fission Products (Thermal fission Um — 135 days' pile irradiation) Basis: 1 g total fission product, Btu/hr Isotope

Sr" Y« Zr»

Nb"

Ru>" Rh"» Rh>" [111

Te'"

I»i

I in

Xe>" Ba"»

La'" Ce"' Ce>"

prm Pr'« Ce»«

Nd'« Pm'»


Pm'« Total. .

5 days

2 3 0 0 0 0 0 0 0 2 0 0 1 3 1 0 1 2 0 0 0 0

520 520 890 304 245 025 129 536 160 350 067 692 798 960 363 294 939 294 198 629 024 140

24 077

10 days

2 3 0 0 0 0 0 0 0 0 0 0 1 3 1 0 1 2 0 0 0 0

355 300 845 310 224 003 128 349 050 808 001 361 373 083 232 023 508 265 195 463 024 031

18 931

15 days

30 days

100 days

300 days

365 days

730 day.

727 100 325 219 047

0.053 0.097 0.043 0.034 0.014

0 0 0 0 0

0 116 0 005

0 108

0.074

0 06S

026 467 049 810

0 092 0 209 0 431

0 Oil 0 026 0 187

0.003

0 001

547 152 186 136 025

0 2 0 0 0

1. 100 0.095

0 948 0 081

0 388 0.033

0.021

0 020

0 015

1.534

1.223

0.473

2 3 0 0 0

203 100 796 309 205

1 2 0 0 0

0 0 0 0

126 229 017 279

0 123 0 063 0 001

0 1 2 1

187 046 360 109

0 0 1 0

1 2 0 0 0 0

170 235 192 340 025 006

0 2 0 0 0

15 934

60 days

815 580 677 305 158

11 120

1 0 0 0

226 800 485 271 094

118 000 173 022 025

6 967

0 1 0 0 0

0 1 0 0 0

016 815 156 002 024

4 763

022 042 022 017 005

0 003 0 002

0.032

Table 7. Gamma Heating by Individual Fission Products (Thermal fission Um— 135 days' pile irradiation) Basis: 1 g total fission product, Btu/hr Isotope

Zr" Nb"

5 days

100 days

300 days

365 days

730 dsn

2 295 3 140

1 539 2 532

0. 182 0 390

0.090 0. 195

• •17 0 "24

0. 530

0 265

0.081

0.027

0 0 0 0 0

0.041

0.041

0 044

0 353

0 0SI

10 days

15 days

30 days

60 days

4 005 3 590

3 766 3 580

3 200 3 538

0 0 i 0 0

0 0 0 0 0

4 3 0 0 1 0 t 0 0 0 0

215 520 045 043 592 332 382 010 900 074 083

0 1 14 0 0 0

042 860 140 778 541 095

0 042 1 418 II 000 0 702 0 398 0 021

Total. . . 29 652 Ba'"-La1M 16.000

19 553 9 508 7 346

12 594 4 226 6 738

7 119 0 840 s 435

4 588 0 103 4 071

0. 696

7 735

23 636 12 418 7 595

0 572

0.285

0 041

Total. . . 23 735

20 013

16 854

10.964

6 275

4 174

0 572

0.285

0.041

Nb"" Nb" Mo"

Ta"°> Ru'«"

Sn'«

I"'

Till Xe"« Ce'»'-

Ba'3,ra

Ba"° La'« Ce>«

Nd'« Pm'« Zr»-Nb»

0 0 1 0 0

461 096 268 007 585

0 043

131 027 161 005 383

003 001 896 002 107

0 022

0 003

0 1 8 0 0 0

0 0 3 0 0

042 083 425 631 293 004

Assumption: 100 per cent conversion

041 483 743 460 117

0. 009

0. 0. 0 0. 0

041 095 745 245 019

of y energy to heat.

041

Oil 092 106 002

0.002

Art.

NUCLEAR REACTOR

2]

DEPENDENCE

11-33


Equation (9) gives the result if all the energy is converted to heat. It is necessary in practical cases to make some estimate of the percentage of the disintegrations that will be converted to heat. The geometry of the problem largely dictates this estimate. In most cases it is assumed that all the 0 energy and some fraction of the y energy will be converted into heat. All data presented in this section, however, are calculated assuming 100 per cent absorption for both 0 and 7 energies. In Fig. 4 is shown the heat generation per gram of total fission product in Btu per hour as a function of

2

3

4 5

— 1— r"

" 1 1 1 4 3 5

7 9K33

DAYS COOLED

Fio. cooling time. heating, and

6.

Fission-product y heating.

In Tables 6 and 7 are the contributions of individual species to 0 and y these tables are graphically represented in Figs. 5 and G.

BIBLIOGRAPHY Coryell, C. D.,

and N. Sugnrman: "Radiochemical Studies: The Fission Products," part National Nuclear Energy Series. Div. IV, vol. 9, McGraw-Hill Book Company, Inc., New York, 1951. Gl&sstone, S.: "Textbook of Physical Chemistry," 2d cd., D. Van Nostrand Company, Inc., Princeton, N.J., 1946. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J.. 1952. Goodman, C. (ed.): "The Science and Engineering of Nuclear Power," vol. I, AddisonWesley Publishing Company, Reading, Mass., 1947. Hollander, J. M., I. Perlman, and G. T. Seaborg: Tables of Isotopes, Revs. Mod. Phys., 26(2): 469-651 (April, 1953).

III,

11-34


[SEC.

11

Hughes, D. J.: "Pile Neutron Research," Addison- Wesley Publishing Company, Reading. Mass., 1953. Pollard, E., and W. L. Davidson: "Applied Nuclear Physics," John Wiley & Sons, Inc..

3

New York, 1944. Rutherford, E. R., J. Chadwick, and C. D. Ellis: "Radiations from Radioactive Sub stances," Cambridge University Press, New York, 1930. Sons, Inc., Soodak, H., and E. C. Campbell: "Elementary Pile Theory," John Wiley New York, 1950.

CHEMISTRY OF IMPORTANT ELEMENTS Uranium

3.1

is

is

in

Uranium (at. wt 238.07) was first discovered by Klaproth in pitchblende (UjOs) It considered an extremely 1789 but was first isolated as metal by Peligot in 1840. con rare metal but actually makes up 0.0004 per cent of the earth's crust, which The siderably more than such metals as cadmium, bismuth, mercury, and silver. most important known sources are the pitchblende deposits of the Great Bear Lake

is

it

is 8.

Table 8.

a

is

Radioactive Decay Constants of Natural and Artificial Uranium Isotopes Natural Isotope*

Radiation

(UI)

4.76 4.52

2.

238

a a

a

4.21

4.51

35 8.91 7.07

X X 8.52 X

10' years I0« years I0« years I0« years 10" years

Amount in natural uranium, %

005 0.71

0

Energy, Mev

Type

234 (UII) 235 (AcU)

Half-life

X

No.

X

Man

99 28

Artificial Isotopes 228

a (80%)

6.72

229

a (~20%) K (~80 %) a

6.42

58 min

5.85

20. days 4. days 70 years 1.6 X I0» years 6. days 23 min

a a, y, y. y. e-

9

5 8

f

.3 rain

2 8

K

4 5 8 3

230 231 232 233 237 239

K (20%)

5.

I.

Source: J. J. Katz and E. Rabinowitch, "The Chemistry of Uranium," part The Element. If Primary and Related Compounds, p. National Nuclear Energy Series. Div, VIII, vol. McGrswHill Book Company. Inc., New York, 1951. 4.


A

is

in

it

region of Canada, the Belgian Congo, and Czechoslovakia. Principal sources in the United States are in Colorado and Utah, where occurs as carnotite and roscoelite. It also occurs in various rocks and minerals and in sea water, and has been found river water and living organisms. Uranium a radioactive element and as found in nature contains three radioactive large number of other uranium isotopes have been isotopes, U884, U835, and U"8. synthetically prepared. The radioactive decay constants of the natural and artificial uranium isotopes are given in Table 3.11 Uranium Metal. Uranium a malleable, ductile metal in the pure state. It When freshly cut, capable of taking high polish. lustrous white but

Art.

chemistry of important elements

3]

It is not hard enough to scratch glass. Some darkens in air. physical constants of uranium are given in Table 9. Uranium is highly reactive, 3.12 Compounds of Uranium. with nonmetallic elements except the rare gases and also forming pounds with many metals (iron, lead, mercury, aluminum, etc). electromotive series is believed to be near that of beryllium.

11-35 of the important

combining readily intermetallic com Its position in the Uranium shows a

Table 9. Physical Constants of Uranium Density at 25°C, g/cm': High-purity directionally solidified 19.05 ± 0.02 Wrought material 18.7-19.08 1132 ± I Melting point. °C 3818 Boiling point (extrapolated from vapor-preasure measurements), °C 640-676. ^668 (a-P) phase transition, °C ($-,) phase transition. °C 755-785. ~774 4.7 Heat of fusion, kg-cal/g atom (estimate) 106.7 ± 0.1 Heat of vaporization, kg-cal/g atom 116 ± 0.1 Heat of sublimation at 0°K, kg-cal/g atom 0.674 Heat of (a-0) phase transition (668°C), kg-c:il/g atom 1. 131 Heat of (0-y) phase transition (774°C), kg-cal/g atom Enthalpy, cal/g atom: 1521.4 ± 1.6 H.t...«x At 298-935°K: Hi-

-

Hniiu

3.I5T +

=

At 935-1045°K; Generated for wjivans (University of Florida) on 2015-09-23 02:56 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Specific

Hr

-

X I0-»T»

Hunts

-

10.38T

Hmux

-

9.107"

-

0.80 X

lOT*'

- 3525 - 1026

-

1046

(a phase)

(0 phase) (y phase)

heat, cal/(g atom)(°C), at 300-940°K:

C, C, C,

-

3.15 + 8.44 X 10-T + 0.80 X 10.38 iff phase) (y phase) 9.10

Entropy, cal/(°K)(g atom):

At 94I-I047<,K: At

-

Ht

At I045-1300°K:

4.22

1047-1 I70°K:

-

St

-

Siimut

11.99 ± 0.02

Sjti.im = 23.362 log T

-

21.062 log T Sr Sui.iut (a-0) phase transition ifi-y) phase transition Fusion Thermal conductivity, cal/(sec)(cm)(°C):

At At At

30°C 100-225-C 225-450°C Vapor pressure,

Volume expansion

coefficient,

cm'/lcm'J^C):

25-IOO°C 25-250°C 25-475°C 25-625°C Modulus of elasticity at room temp, psi Shear modulus at room temp, psi Poisson's ratio Modulus of resilience, in. -IV) /in.' Bulk modulus at room temp, psi Yield strength (approx) at room temp, psi: 0. 1 % offset

0.2% offset References:

(a phase)

(a phase) 58.770

(0 phase)

50.745

(t

phase) 0.7 17 1.081 3.3 0.060 0.063 + 8 X 10-»« 0.072 + 3 X 10"»(«

mm Hg. at I630-1970°K: log p (mm Hg)

At At At At

-

I0T>

2330 ± 21 = + (8.583

-

-

100) 200)

± 0.01 1)

39.70 X 46.60 X 54.20 X 59.30 X 25.5 X 10.6 X 0.20 0.176 14.2 X

10~«

10« 10~«

I0«

108 10fl

10*

27.000 33,000

F. G. Foote, "Physical Metallurgy of Uranium," Paper A/Conf. 8/P/555 presented at International Conference on the Peaceful Uses of Atomic Energy at Geneva, Switzerland, August, 1955. U.S. Atomic Energy Commission, "The Reactor Handbook," vol. 3. AECD-3647. McGraw-Hill Book Company, Inc., New York, 1955. J. J. Kats and E. Rabinowitch, "The Chemistry of Uranium," National Nuclear Energy Series, Div. VIII, vol. 5. McGraw-Hill Book Company, Inc., New York, 1951.

11-36

CHEMISTRY

AND


[SEC.

11

Trivalent number of valence states, the common valences being +3, +4, and +6. uranium is a powerful reducing agent, being oxidized to tetravalent uranium by a Important binary compounds include the hydride, few minutes' exposure to air. It is not possible to give all the reactions oxides, carbides, nitrides, and the halides. by which these compounds may be prepared,* but a number of the more important preparations reactions are given in Table 10. Table 10. Element

Oxygen

Important Reactions of Uranium

Reaction products

Some methods

UO.

U.O. UOi Hydrogen. . . .

UH,

Fluorine

UF. UF.

Chlorine

UCli


UC1. UC1. UC1. Bromine

UBr. UBr.

Iodine

UIi Ul4

Carbon Nitrogen

with Nonmetals

UC, UCi UNi.ti (UiNi + 2UNi) UN UN.

of preparation

Low-temperature oxidation of U UOi + H, at 600°C High-temperature oxidation of U Ignition of uranyl nitrate or UOr2HiO Decomposes to UiOs about 450°C Reaction occurs above 250°C Decomposes at temperatures > 450°C Useful as intermediate and preparation of powdered uranium U + HF (dry) at elevated temperatures UOi + HF -* UF( (insoluble in water) UF4 + F. at 275°C or U + F. + heat Highly volatile Reduction of UCU with Hi UHi + HC1 at 250-300°C UOi + HC1 at room temperature UO. + CCU at 450°C UCU + CI. at 550°C 2CCU — UCU + UCU UCU + CI. UH. + 3HBr at 300°C UBr( + Hi at 700°C U + Br. at 650°C UBr. + Br. at 300°C U + I. at 130°C initially, then at 570°C U + I. at I30°C at high partial pressure of iodine Decomposes to UIi at comparatively low temperature Powdered U + C at 800-l200°C U -f- N. at temperatures less than 1300°C U + N, above 1300°C U 4* N. at high nitrogen pressures

The common oxides of uranium are UO2, UaOi, and UO3. Of these, UO» is dark brown in color and is called the "brown oxide"; U,08, sometimes called the "black oxide," has a color that may vary from black to olive green; and UOi has a color varying from red to yellow depending on the crystalline modification. Uranium Hydride. Uranium hydride, a pyrophoric dark-gray powder, serves as the starting material for the preparation of many compounds, a typical reaction being

UH, +

3HC1 — UC1, + 3H,

Finely divided (and therefore exceedingly reactive) uranium metal may be prepared by decomposition of the hydride. Uranium Tetrafluoride. Uranium tetrafluoride is often used as the salt from which uranium metal is obtained by reduction with a more reactive metal such as calcium :

UF,

+ 2Ca + heat -» U + CaF,

Other uranium compounds can be similarly reduced to metal. Uranium Hexafluoride. The hexafluoride is an important compound because of its volatility. It is the compound employed in the gaseous diffusion plants for isotope separation. The triple point of UF« is 64.052°C, at which point the vapor pressure is 1,134 mm of mercury. The temperature at which the sublimation pressure reaches 760 mm of mercury ("boiling point") is 56.5°C. With the exception of UFi and

Art.


3]

11-37

UF<, which are regarded as nonvolatile, the other halides of uranium are also volatile but much less so than UF,. Uranyl Compounds. Two other important classes of uranium compounds are the uranyl compounds [such as UOtFj, UOj(NOi)s, etc.] and the uranates, which are more The commonly encountered as diuranates (such as KJU2O7, potassium diuranate). trioxide is amphoteric, forming uranyl compounds with acids and uranates with bases :


or

UO, + 2HC1 -» UO,Cl, + H,0 2UOj + 2KOH -> K2U2O7 + HsO U03 + 2KOH -► K5UO« + HjO

(uranyl chloride) (potassium diuranate) (potassium uranate)

The more common uranyl compounds are the halides, nitrate, sulfate, and acetate. These are soluble in water. The uranyl ion exhibits a green-yellow fluorescence, weak in an aqueous solution, but strong in the crystalline state. Dissolution of Uranium and Its Oxides. Uranium and its oxides may be con veniently dissolved in nitric acid, heat being required to initiate the reaction in the case of massive uranium. The products are uranyl nitrate and, if metal is being dissolved, NO and NO,. Generally about 5 moles of nitric acid per mole of uranium are required, although the stoichiometric requirement is 2. Uranyl nitrate is of particular importance because it can be extracted from a concentrated nitrate solution by many organic solvents (see Art. 5.3). Dissolution of uranium in hydrochloric acid proceeds rapidly, but some insoluble black residues appear unless oxidizing agents such as hydrogen peroxide or ammonium Dissolution The reaction product is uranium tetrachloride. persulfate are present. may be similarly effected with hydrobromic or hydriodic acid, but the rate is much slower. Uranium does not dissolve in hydrofluoric acid, presumably because of the formation of an insoluble protective coating of UF4. Electrolytic oxidation in sulfuric acid may also be used to dissolve uranium. Dilute sulfuric acid has little effect alone, but with oxidizing agents slow dissolution occurs. A slow reaction occurs with hot concentrated sulfuric acid, resulting in the formation of U (IV) sulfate. Cold 85 per cent phosphoric acid attacks uranium slowly, but the reaction becomes fairly rapid if the acid is concentrated by boiling, yielding U (IV) phosphate. Double Sails and Complexes. Another important series of uranium compounds are the double salts and complexes. For instance, UF4 will form the double salt KUFS with KF. The carbonate complexes formed by the action of ammonium or alkali carbonates or bicarbonates on uranates arc well known, having been used for many Use is made of the solubility of years in the analysis of uranium-bearing materials. ammonium diuranate in ammonium carbonate in recovery and separation reactions. 3.2

Plutonium

The entire knowledge of plutonium has evolved from the study of artificially For all practical purposes it does not exist in nature, although produced isotopes. after its synthetic preparation, Peppard' and co-workers found 7 ± 0.7 parts of plu tonium in 10" parts of pitchblende ore concentrates, where it is formed by the occa sional capture by U"8 of neutrons from spontaneous fission. Plutonium 238 was discovered by Seaborg, McMillan, and Wahl at the University of California in 1940. Plutonium 239, a fissionable and the most important isotope, was discovered in 1941 by Seaborg, Segre, Kennedy, and Lawrence. It is formed in nuclear reactors by the scheme U"« (n,7) U»«

— -— > 23.5 inin

Np"»

— -—

» 2.33 day8

Pu>"

It spontaneously fissions Pu13* decays by a emission with a half-life of 24,400 years. 35 fissions per gram-hour (half-life by this mode of decay = 6 X 10"

at the rate of years).


11-38

[SEC.

11

Several other plutonium isotopes (Pu"°, Pu'41, Pu*", Pu84') are formed in reactors, and isotopes of lower atomic weight may be formed by cyclotron bombardment of various uranium isotopes with helium ions. The radioactive properties of those isotopes produced in nuclear reactors are given in Table 11. Some of the important chemical and physical constants of plutonium are given in

Table

12.

Table 11.

Radioactive Properties of Plutonium Isotopes Produced in Reactors Radiation Half-life

Isotope

Type

Pu"« Pu>" Pu1"

Pu"'

Pu"« Pu»"

a T a y a y 0

a 0 y

Energy,

5.495 (76%), 0.044. 0.1 01. 5.150 (69%), 0.384. 0.124. 5.162 (76%), 0.047 0.0205 4.898 (80%), 0.56 0.085. 0.092.

Mev and %

5.453 0.149 5.137 0.100. 5.118

(24%), 5.351 (0.15%)

89.59 ± 0.37 years

(13%,), 5.100 (18%) 0.0385, 0.0520 (24%), 5.014 (0.1%)

24,360 ± 100 years 6,600 years 13 ± 0.2 years

~9 X 10' years

4.854 (20%)

4.98 hr

0.107. 0.160


Source: G. T. Seaborg and J. J. Kats, "The Actinide Elements," National Nuclear Energy Series, Div. IV, vol. 14A. McGraw-Hill Book Company, Inc., New York, 1954.

Table 12.

Physical and Chemical Constants of Plutonium

Density, g/cm1: Alpha Pu, 25°C Liquid, 665°C Melting point, °C Boiling point, °C Heat of vaporization, kcal/mole Vapor pressure, mm of Hg:

logP-

* "> + T (°K)

<17-587

19.60 ± 0.02 16.54 ± 0.08 632 ± 7 3235 ± 19 80.46 ± 0.34

(7.875 ± 0.047)

of linear expansion per °C: 48.39 X I0 » + 0.09539 X 10-«< (°C) 14 X 101 6 X I01 Shear modulus, psi 0.28 Poisson's ratio

Coefficient

-

« Pu(_uo to ioo°c> Young's modulus, psi

Plutonium Metal. The metal is highly reactive, and in moist air oxidation Water vapor produces rapidly, particularly at elevated temperatures. nonadherent corrosion products, while dry oxygen appears to inhibit the corrosion through the formation of a protective oxide coating. The metal is usually stored in dry air and at a low temperature to inhibit corrosion. 3.22 Valence and Compounds of Plutonium. Plutonium is readily soluble in all concentrations of hydrochloric acid, 72 per cent perchloric acid, 85 per cent phosphoric acid, and concentrated trichloracetic acid. Owing to formation of a protective coat ing, the metal is not visibly attacked by nitric or concentrated sulfuric acid. How ever, plutonium as produced in uranium metal in a nuclear reactor is easily dissolved in nitric acid. Moderate concentrations of sulfuric acid react slowly with plutonium. Plutonium has five oxidation states (+2, +3, +4, +5, +6) in addition to the metallic state and therefore exhibits an intricate and interesting chemical behavior. However, unlike In many respects its chemistry is similar to that of uranium. Some uranium compounds, plutonium compounds and ions exhibit no fluorescence. of its characteristics in these valence states are as follows: Divalent. Compounds are semimetallic in character and require strong reducing conditions for formation. 3.21

proceeds

Art.

3]

CHEMISTRY OF IMPORTANT ELEMENTS

11-39


Many trivalent plutonium compounds are known (anhydrous oxides and Trivalent. In aqueous solution plutonium is present as +3 ion and is relatively stable halides). to oxidation by air under moderately acidic conditions. Solutions are blue in color. Telravalenl. The dioxide of plutonium, formed by ignition of most plutonium compounds in air, is very stable to decomposition or further oxidation. However, the halides, if formed, decompose to the +3 halide and halogen. Only PuF, has been identified. Pu+* ions exist (alone or as complex ions) in solutions of 0.1 M H+ or greater. It is uncomplexed in perchloric or dilute hydrochloric solutions but is complexed by nitrate, by chloride in high concentrations, and very strongly by sulfate ion. A very important property of Pu+< in aqueous solution is its instability with respect to self-oxidation and reduction. occurs to +3 and +6 states, Disproportionation the extent depending on the complex and acid concentration. Pu+4 polymers are formed by heating in dilute hydrochloric or nitric solution or occasionally on dissolving Pu+4 hydroxide and result in a colloidal material. The solutions are brilliant green in color. The colloid is readily adsorbed on paper, sand glass, cotton, and other surfaces that assume a negative charge when immersed in water. The polymer may be destroyed slowly by heating in strong acid. In solution, pentavalent plutonium is believed to exist as Pu02+ ions. Pentavalent. There is no evidence of complex formation. Pentavalent plutonium in solution disproportionates to Pu+4 and Pu+e, but at low plutonium and acid concentrations, pentavalent plutonium may be the predominant species in solution, since the dis proportionation reaction is slow. The solutions are colorless. Table 13. Element

Fluorine

Some Important Plutonium Compounds and Methods of Preparation Compound

Conditions of formation

PuH.

Metal + Hi at room tem

PuH,

Metal + excess Hi at room or slightly elevated temperature Treatment of PuO«, PuF», or dried salts with HF-Hi mix ture at 550-600°C; Pu+" (aq) + HF PuO. + HF + Ot at 550-

PuFi

PuF«

perature

600°C

PuF, + F, at Chlorine

PuCh

400°C

Pu or PuOj + BrFi Pu + Cli at 450°C PuOi + Ch or

PuCL PuOj Hydroxide

MgPuO<

Oxidation of Pu(IV) to Pu(VI) aq and ppt with Ca(OH)i or

PuN PuC, PuiC, PutCOOi

Pu(COi)i

Plutonyl compounds . . PuO.(NO.)>

Starting compound for reduc tion to metal by more reactive metal. Insoluble in HiO

»*

800°C Nut stable at room temperature Analogous to chlorine Ignition of hydroxide or Pu Baits. Direct oxidation of Pu

Ppt of Pu+« aq with OH" Ppt of Pu*« with OH-

CaPu04

Insoluble in HiO

Soluble in HiO

Pu(OH)i Pu(OH)4

Characteristics

Difficult to dissolve. Warm 10 M HNOa + 0.005 M HF slowly dissolves Readily oxidized in air Tends to polymerize, giving difficultly soluble composition Slightly soluble in HjO

Mg(OH), NHi + Pu at 1000°C PuCU + NH. at 800-900°C

Pu vapor + CO Ppt with COr»

Pu+t exists as plutonyl ion in acidic aq solutions in absence of ooroplexing ions

Insoluble in HtO

Pu(COi)i NatCOi

dissolves

in exoess

11-40


[SEC. 11

Hexavalent. The chemistry of hexavalent plutonium is similar to that of hexavalent uranium. The trioxide has never been prepared, but a number of plutonyl and plutonate salts have been prepared. The plutonyl ion PuOj+* is believed to exist in solution in absence of complexing ions. The solutions are brown in color. Plutonium hydroxide and the sodium, potassium, and ammonium plutonates exhibit much greater solubility than the corresponding uranium compounds. Plutonium in solution in the trivalent state does not have a strong tendency to form complex ions, but in the tetravalent state it exhibits a strong tendency for complex formation [Pu+< + NO,"-» PuNO,+» + NO,"— Pu(NO,)2+»]. In the pentavalent state there is no evidence of complex formation, but in the hexavalent state complexes with the plutonyl ion are known. As in the case with uranium and thorium, plutonium forms intermetallic binary compounds with many elements (aluminum, carbon, lead, magnesium, tin, uranium, and others). A number of the more important preparation reactions for plutonium compounds are given in Table 13. 3.3

Thorium (at. wt "thorite" (ThSiOi).

232.12,

at. no. 90)

Thorium was first discovered by Berzelius in 1828 in a single valence state of 4 in

For all practical purposes, it has


compounds with other elements. Thorium is of principal interest because this element is a potential source of the fissionable uranium isotope U"a formed in a reactor by the following series of nuclear reactions:

Th»» (n,y) Th»» In principle,

—-—> Pa"» 23.3 min

-

> U"1 27.42 days

all the world's thorium can be converted to this fissionable isotope, giving an amount several hundred times that of the naturally occurring Um. Thorium is present in the earth's crust at an average concentration of approximately 12 ppm. It is widely distributed, being found on all continents, but few economically workable deposits are known. The principal industrial source of thorium is monazite sand, which consists largely of the orthophosphates of the rare earths and contains from 3 to 9 per cent of thorium oxide and variable amounts of siliceous matter, iron, aluminum, magnesium, and other elements. The principal monazite sand deposits are in Brazil, India, and southwest Africa. No large thorium deposits have been found in the United States. In the separation of thorium from monazite sand, the sand is first digested with concentrated sulfuric acid until the solid is completely disintegrated. The resulting thick paste is then slurried into water, and the insoluble matter allowed to settle. Thorium and rare earths remain in solution, from which thorium phosphate and some rare-earth phosphates are precipitated by partial neutralization. This thorium phosphate concentrate (80 per cent thorium phosphate) is generally redissolved, and the elements precipitated as the oxalates. Subsequent separation of thorium from rare earths is accomplished by fractional crystallization, utilizing the lesser solubility of thorium sulfate than rare-earth sulfates or the greater solubility of thorium in sodium carbonate or ammonium oxalate solutions. Recently, solvent-extraction methods have been developed for separating thorium nitrate from rare-earth nitrates. Natural thorium consists of Th"2 in 100 per cent abundance. The radioactive constants for the important thorium isotopes, both natural and synthetic, are given in Table 14. 3.31 Thorium Metal. Thorium metal when freshly prepared is silvery white in color but will turn a dark gray color on exposure to air. Its physical properties are greatly influenced by the degree of contamination with its oxide, the purest specimens The machining properties are com containing several tenths of a per cent of ThOj. parable to those of mild steel, and it may be extruded, rolled, forged, and swaged. The metal has a relatively low activity level and may be handled without protective clothing, although considerable quantities of highly radioactive radon gas may be

Art.

chemistry

3]

Table 14.

of important elements

11-41

Radioactive Constants of Thorium Isotopes Decay

mechanism

Isotope

Half-life Type

Energy,

Amounts in thorium,

natural

%

Mev

Natural:

Thm Th.., Thm Th..,

a a a

»-

Th„,

a 0~

Th...

Synthetic:

Th„,

a a a a

Thm

Th„. Th... Thm Th.„

00-

6.03-5.65 5.4 4.61 (25%), 4.68 (75%) 0.09 (45%), 0.21 (11%), 6.30 (44%) 3.98 0.20 (80%), 0.1 1 (20%)

18.6 days 1.9 years 8.0 X 10' years 25.6 hr 1.39 X 10'° years 24.1 days

7.55 7.13 6.57 (90%); electron ~4.9 1.23

Short Short capture

(10Tc)

100

8 ± 0.5 min 2.340 years 23.3 min

Short

Source: G. T. Seaborg and J. J. Katz. "The Actinide Elements," National Nuclear Energy Div. IV, vol. I4A, McGraw-Hill Book Company, Inc.. New York, 1954. Table 16.

Physical and Chemical Constants of Thorium


Density,

g/cm1 Melting point, "C Boiling point, °C Heat of fusion, kg-cal/mole Heat of vaporization, kg-cal/mole Specific heat. cal/(mole)(°C): At 99.3-C At I98.7°C Entropy at 25°C, cal/(mole)(°C)

Thermal conductivity, cal/(sec)(cm)(°C):* At I00°C At 650°C Coefficient of thermal expansion per °C: At 30-l00°C At 30-500°C

At

Series.

30-500°C

11.71 1690 ± 10 >3000 4.6 130 6.59 6.61 13.6 ± 0.8 0 090 0.108 11.5 X 10 • 11.9 X I0~« 12.5 X 10 " 18 None known to exist

Electrical resistivity at 20°C, >iohm-cmt Allotropic transformation Crystallography : Crystal structure Face-centered cubic Lattice constants, oo. at 26.2°C. A 5.0871 ± 0.0002 Atomic diameter 3.59 12 Coordination number Shear modulus at room temperature, psi 46 X 10' Young's modulus at room temperature, psi 9-10 X I0a Poisson's ratio 0.265 Source: U.S. Atomic Energy Commission, "The Reactor Handbook," vol. 3, AECD-3647. McGrawHill Book Company, Inc., New York, 1955. * Cal/(scc)(cm)(°C) X 241.9 Btu/(hr)(ft)(°F). t Superconducting below l.7°K.

-

released in grinding, melting, reduction, and casting operations. Some of the impor tant physical and chemical constants of thorium are given in Table 15. One of three processes is generally used for the preparation of thorium metal: (1) reduction of thorium tetrafluoride or other suitable compounds with a more reactive metal; (2) thermal decomposition of the vapor of a thorium halide on an electrically heated filament (the de Boer process), suitable only for the preparation of small quantities (<300 g) of moderately pure metal; and (3) electrolysis of a molten potas sium fluoride solution of thorium fluoride. Currently, nearly all thorium metal is produced by reduction of thorium tetrafluoride with calcium using zinc chloride as a The zinc is subsequently removed by heating the metal in a graphite crucible booster. in vacuum at 1100°C.

11-42


[Sec.

11

Because of its high melting point and reactivity, thorium is a difficult metal to melt. must be melted in a vacuum or an inert atmosphere in order to avoid oxygen or Beryllia and zirconia are the only refractories found satis nitrogen contamination. Thorium alloys readily with many metals and forms factory as crucible materials.


It

numerous intermetallic compounds. 3.32 Chemical Compounds. Thorium is attacked by water at 150°C and very rapidly at 300°C. It may be smoothly converted to thoria by superheated steam at 850°C. Thorium turnings ignite and burn brilliantly when heated in air, while The powdered metal is often massive metal is converted more slowly to the oxides. pyrophoric. The metal does not dissolve rapidly in any of the common acids except hydro chloric. However, a satisfactory rale of solution in nitric acid is realized if a small quantity of chloride or fluoride ion is present to catalyze the reaction. Nitric acid However, it may be used to dissolve passifies the surface of massive thorium. thorium, its alloys, and its oxides if fluoride ion is present. The heated metal reacts with oxygen, chlorine, bromine, hydrogen sulfide, nitrogen, A few of the more important compounds are dis hydrogen, and other materials. cussed below. Thorium reacts with hydrogen at 450 to 500°C to form a lower hydride Hydride. As with uranium, the ThH2 and a higher hydride believed to be ThH4 or ThH8.76. hydride serves as an intermediate for making powdered metal or for synthesis of a series of other compounds by reactions with hydrogen halides or hydrogen sulfide to form the corresponding halide or sulfide. Thoria, ThO«, the common oxide which is sometimes called the most refrac Oxide. tory substance known, may be prepared by direct oxidation of the metal or by ignition of the hydroxide, nitrate, sulfate, or organic salts. It is a white powder whose properties depend on the source, temperature of formation, and subsequent heat treatments. Fluoride. The addition of fluoride to soluble thorium salts results in the precipita tion of ThF4-8H20. Hydrates of the chloride, bromide, and iodide are soluble. The bromide shows a greater tendency for hydrolysis than the chloride, and the iodide is readily hydrolyzed. Nitrides and Carbides. Thorium nitrides, having the formulas ThN, Th2N3, and Th,N4, have been reported. The action of ammonia on thorium hydride at 1000°C yields Th2N2, a product stable to 1730°C but readily hydrolyzable by water and moist air. The two known thorium carbides ThC and ThCj are also easily decomposed by water. Nitrate. Thorium nitrate, probably the most common salt of thorium, may be formed by direct dissolution of metal in nitric acid containing fluoride ion as a catalyst or by dissolution of the hydroxide or carbonate in nitric acid. The several thorium phosphates, e.g., the normal phosphates ThrPhosphates. (P04)4-4H20 and the pyrophosphates ThP207-2H20, are important because of thenThey have low solubilities in use in separation of thorium from monazite sand. water and are soluble in acid with difficulty. Thorium iodate is insoluble in strong acids, a fact utilized in Iodate and Oxalate. The oxalate is also insoluble in strong making separation from rare-earth iodates. acids. Thorium hydroxide will precipitate on addition of alkalies to thorium solutions. Both the hydroxide and the oxalate may be dissolved in solutions of alkali carbonates The oxalate is also soluble in ammonium oxalates. through complex formations. Thorium also forms a wide variety of double salts Double Salts and Complexes. and complexes. and 2KFExamples of double salts are NaNOa-Th(NOj)4-9H20 ThF4-4H20. In general, the solution chemistry of thorium resembles that of Solution Chemistry. It is more electropositive than the quadrivalent cations of uranium and cerium. most other metals, being comparable to magnesium. Its analytical chemistry is rendered more difficult than that of uranium by its single valence state and the color less nature of its ions.

Art.

CHEMISTRY OF IMPORTANT ELEMENTS

3] 3.4

Other Elements of the Heavy-element

Transition

11-43 Series

There is considerable chemical similarity between the rare-earth group and the heavy-element transition group beginning with actinium. This group is generally designated as the aciinide series in analogy with the use of the term lanthanide series proposed for the rare-earth transition group. The half-lives become progressively shorter as the atomic number increases. Except thorium and protactinium, all have a known +3 oxidation state, this state becoming increasingly stable with increase in Many of the chemical properties of the heavy-element transition atomic number. group have been predicted by Seaborg4'» by analogy with corresponding elements of the rare-earth transition group. Many of these elements are formed in a nuclear reactor, and it is necessary to take account of their gradual accumulation in reactor fuel. Some of the newly created heavy-element isotopes may act as neutron poisons, and some are fissionable. After discharge, purification from these elements may be desired in the separations process, and proper account must be taken of their possible interference in analytical work. 3.41 Actinium. Actinium, the first member of the actinide series (at. no. 89), was discovered in 1899 by Debierne in pitchblende residues.6'7 It is much less abundant than radium, 1 ton of pitchblende containing 0.4 g of radium and only 0.0001 g of actinium. The longest lived isotope (the one discovered by Debierne) is the 21-year


Ac"7. The metal

is highly electropositive and chemically similar to lanthanum, its rareearth analogue. Only the trivalent oxidation state has been identified. The hydroxide, fluoride, phosphate, oxalate, and double salt with potassium sulfate are relatively insoluble in water. 3.42 Protactinium. The existence of an undiscovered element with an atomic number of 91 was predicted by Mendeleeff in 1871.* This element, named protac tinium, because it is the precursor of actinium, was discovered by Hahn and Meitner'-" and independently by Soddy and Cranston,"'1* both in 1917. The isotope Pa"1 (half-life 34,300 years) is found in pitchblende ore to the extent of about 200 mg/ton. This isotope is of some interest because it is fissionable with neutrons of less than 1 Mev. Two other isotopes are known in nature, 6.7-hr Pa"4 (UZ) and 1-min Pa"4 (UXj). The isotope Pa2" is a useful tracer and a precursor of U"'. It is formed in the following sequence of reactions:

Th"» (n.y) Th"»

—— » 23 min

Pa"»

27.4 day

»

U"»

1.6X10*

years

> Th"»

The metal, which may be prepared by dissociation of a pentahalide on a hot tungsten filament in vacuum, oxidizes slowly in air. The most stable oxidation state is +5, and only meager evidence has been obtained for the existence of a +4 state. coprecipitates from acid solutions with thorium oxalate or iodate, zirconium iodate or phosphate, manganese dioxide, cerium (IV) fluoride, and potassium cerium (IV) fluoride. In the absence of fluoride and in alkaline solutions, it will carry quantita tively on almost any precipitate. 3.43 Neptunium. Neptunium (at. no. 93), the first transuranic element syn thesized, was discovered in 1940 by McMillan and Abelson,14 who prepared the mass 239 isotope by slow neutron bombardment of U"*. Later Np"7 was prepared by fast neutron bombardment of U"s (n,2n reaction) by Wahl and Seaborg.64'" This isotope is of considerable interest because of its long half-life, and gram quantities have been prepared by nuclear reactor irradiation. The metal (melting point of about 640°C) is silvery and does not tarnish rapidly in air. In aqueous solution, stable oxidation states are +3, +4, +5, and +6. In the +6 state (obtained with strong oxidizing agents), its properties are similar to hexavalent uranium, and it exists as neptunyl ion in acid solution. In basic media a dark-brown Neptuniuan hexafluoride has been prepared precipitate, Np02(OH)2'xHjO, forms. and is the only known hexahalide. The +5 sta* e is stable in acidic aqueous media.

It

11-44


[SfiC. 11

In

sulfuric acid it disproportionates to the +4 and +6 states because both of these states are complexed by sulfate ion. The hydroxide is insoluble. Many stable compounds in the tetravalent state are known. Aqueous solutions are yellow-green in color. The hydroxide, oxalate, phosphate, and ammonium and The tetrahalides and oxyhalides have been potassium fluorides are insoluble. identified. Compounds of trivalent neptunium are not readily prepared in aqueous solution because of rapid atmospheric oxidation. However, they have been prepared by electrolytic reduction of the +4 states in dilute hydrochloric acid media at a mercury cathode. The trihalides have been synthesized. Americium. 3.44 The discovery of americium (at. no. 95) was made in 1944 by James, and Morgan, who prepared the 241 mass isotope by bombard Seaborg,t•6°•1,•17 ing uranium with high-energy a particles

U"8

(,a,n)

Pu"1

—-— » Am"1 10 years

-—

► 500 years

Np"»


It

may also be formed from plutonium 239 in a nuclear reactor by successive neutron In addition to the 500-year Am"1, the 1.5-hr Am1", 500-hr Am"0, 16-hr capture. Am24', and 400-year Am5" are known. Americium is a ductile, malleable metal with a density of about 12. Chemically, it is similar to the rare-earth metals europium, neodymium, and samarium, and it may be efficiently carried on rare-earth precipitates (fluorides, hydroxides, oxalates, etc.). The most stable oxidation state is +3, but it also exists in the +6, and there is evidence of a +5 valence state. In the hexavalent state, it is extractable into diethyl ether and hexone from nitric acid solutions containing a salting agent. 3.45 Curium. Curium, element 96, was first synthesized and identified in 1944 It was prepared by bombarding plutonium by Seaborg,t6°'16'17 James, and Ghiorso. 239 with high energy a particles, an (a,n) reaction giving 150-day Cm1". The only stable oxidation state known is +3. Very powerful reducing and oxi dizing agents have failed to produce higher or lower valence states. Chemically similar to gadolinium, it is carried on rare-earth precipitates and has the same insoluble compounds as do the rare earths. 3.46 Berkelium. Berkelium, element 97, was discovered by Thompson, Ghiorso, and Seaborg.11'19 It was prepared by (a,n) or (<*,2n) reactions on Am'41. It is analogous to terbium and has +3 and +4 oxidation states. 3.47 Californium. Californium, element 98, was prepared by a bombardment of Its discovery was announced in 1950 by Thompson, Street, 150-day curium 242. Since its half-life is only 45 min, relatively little is known Ghiorso, and Seaborg.20-21 of its chemistry except through comparison with the corresponding rare earth dys Only the +3 oxidation state is known. prosium. 3.48 Elements 99 and 100. (Einsteinium and Fermium) Discovery of elements Presumably their chemical properties will be 99 and 100 has also been announced. similar, respectively, to holmium and erbium. 3.5

The Fission Products

The fission products comprise some 35 elements, from zinc (at. no. 30) to gadolinium Many of the fission products exist in several isotopic forms. Approxi (at. no. 74). mately 200 isotopes of these 35 elements have been identified. The relative importance of a fission-product isotope in separations processes will The chemis vary, depending upon its fission yield, half-life, and chemical behavior. try of the fission products is discussed briefly below without regard to their relative importance in a particular separations procedure unless their fission yield is so low that they will not be important in any reparations procedure. Sec Ref. 56, p. 1525. X See Ref. 56, p. 1554.

t


Art.

3]


11-45

Classification of the fission products is made according to the groups in which they appear in the periodic table. Two inert gases, krypton and xenon, are born 3.61 Group O — The Rare Gases. in fission. The fission yield of xenon is very high, being 20 to 25 atoms per 100 atoms fissioned, while that of krypton is low, being about 1 atom per 100 atoms fissioned. Most of the xenon isotopes formed are stable. The longest lived radioactive isotope of xenon from fission is 5.3-day Xe133, which is created to the extent of only 0.3 atom per 100 atoms fissioned, and this will disappear after a short cooling period. The fission isotopes of krypton are also stable, except for the 9.4-year Kr", which is also created to the extent of 0.3 atom per 100 atoms fissioned. These gases are not , therefore, a major source of radioactivity and because of their chemical inertness arc not a significant health hazard. Ordinarily they are The gases are evolved on dissolution or melting of uranium. disposed of in the off -gases, but, if desired, they may be absorbed on activated charcoal The boiling points at 1 atm are 121.3 and 163.9°K cooled in liquid air to -100°C." for krypton and xenon, respectively ; the melting points are 104 and 133°K, respectively. 3.52 Of this group rubidium and Group I — Rubidium, Cesium, and Silver. cesium, members of the highly electropositive alkali metal group (subgroup A), are the only important fission products. Cesium is by far the most important, having a Rubidium, on the other hand, has a very high fission yield and long-lived isotopes. Invariably these elements have an oxidation state of +1 in their low fission yield. compounds and give up their outer electrons so easily that they are extremely reactive They decompose water vigorously (yielding hydrogen, metal ions, and chemically. hydroxyl ions) and in fact take fire almost instantly if thrown into water. Both are relatively volatile metals, melting at slightly above room temperature and boiling at 679 °C (Rb) and 690°C (Cs). Most of the salts of these metals are soluble. However, the chloroplatinates are insoluble in water. 3.63 Group II — Barium, Strontium, Zinc, and Cadmium. Barium and strontium are the only fission elements of importance in this group. Zinc and cadmium are of little importance because of their low fission yield. Barium and strontium are members of the alkaline-earth metal group. They are highly metallic, highly electropositive elements with a valence of +2. Both metals dissolve readily in acids and will burn when heated in air, oxygen, or carbon dioxide. At low temperatures, oxidation is slow owing to formation of a protective oxide film. Barium and strontium react readily with water, releasing hydrogen and forming metal and hydroxyl ions. The carbonates and sulfates are very insoluble. The fluorides and the oxalate of strontium are also insoluble. The sulfides, however, are soluble in water. Gallium and 3.64 Group III — Gallium, Indium, Yttrium, and the Rare Earths. indium, representatives of the A subgroup, are of negligible importance because of their low fission yield. Yttrium, of the B subgroup, behaves chemically like the rare earths, and it is convenient to consider these together. The elements with atomic numbers 57 through 71 form a remarkably similar group of elements known as the rare earths. These elements all form +3 ions resembling those of scandium and yttrium. The discovery of all the rare-earth elements followed the discovery around 1800 of two earthlike oxides given the names yttria and ceria. These oxides contain traces of all the rare-earth elements, but ceria consists largely of the oxides of elements of atomic numbers 57 to 62, while yttria consists largely of the remaining rare-earth oxides (63 to 71). Hence, these groups are designated as the cerium subgroup and the yttrium subgroup. Although yttrium itself is important, rare-earth elements of the yttrium subgroup are not important because of their low fission yields. While the elements all have valences of +3, cerium can also assume a valence of +4. Tetravalent cerium is a powerful oxidizing agent. It is more difficult to form tetravalent compounds of praseodymium and neodymium, but tetravalent compounds have been prepared and are even more powerful oxidizing agents than tetravalent

cerium. Except for the end elements of this group, lanthanum and lutetium, the rare earths

11-46

CHEMISTRY

AND


[Sec.

11

The metals are difficult to prepare because of their highly are highly magnetic. The most satisfactory method is by electrolytic reduction electropositive nature. A highly pyrophoric alloy of the rare earths of the cerium of the oxide or fluoride. It consists of group, obtained from monazite sand residues, is called Misch metal. 60 to 70 per cent cerium, the remainder being largely lanthanum. Important solubility relations of the cerium and yttrium subgroups are summarized in Table 16. Table 16. Properties of Rare-earth Compounds Yttrium group, yttrium and elements

Cerium group, elements

Hydroxides Carbonates

Slightly soluble in water Not soluble in water or (NHi)tCOi solution Not soluble in water or (NH<)»C»04 solution

Oxalates

Fluorides Potassium sulfates,


Nitrates Basic nitrates Double nitrates Phosphates RPO4 Formates

57-62

KiR(SO0a

Not soluble Not soluble in KsSOt solu tion Soluble in water; less solu ble in HNOi Slightly soluble Easily crystallised Not soluble Slightly soluble

" Source: W. M. Latimer and J. H. Hildebrand, Reference millan Company, New York, 1940. Used wtik permission.

63-71

Slightly soluble in water Not soluble in water; solu ble in (NHi)iCOi solution Not soluble in water; solu ble in (NH4)tCi04 solu tion Not soluble Soluble in K9SO4 solution Soluble in water; less solu ble in HNOi, especially

Gd(NO,),

Slightly soluble Not readily crystallised Not soluble Moderately soluble

Book of Inorganic Chemistry," The Mae-

3.66 Germanium and tin in the A Group IV — Germanium, Tin, and Zirconium. subgroup are of little importance, but zirconium in the B subgroup is very important in process chemistry. Zirconium is a member of the group IV series of metals (titanium, zirconium, hafnium, thorium, germanium, tin, and lead), all of which have a characteristic +4 oxidation state. Despite its electropositive character, zirconium metal is not readily soluble in nitric acid. Both hydrofluoric and moderately concentrated boiling sulfuric acids dissolve zirconium at substantial ratesv The dioxide is amphoteric, forming with hydroxides and carbonates insoluble zirconates (NasZrOj, CaZrOj) and with acids, salts [Zr(S04)j, Zr(NOj)4, etc.]. The silicate, known as zircon, occurs naturally and is valued as a jewel. Highly stable carbidep, nitrides, silicides, and the dioxide are formed at high temperatures, and the preparation of the metal is very difficult. Prior to the interest in zirconium as a reactor structural material, available zirconium metal always contained substantial amounts of hafnium, a metal with a high thermal neutron cross section. Efficient methods have Prince been developed for \ eliminating the hafnium impurity. The slightly soluble fluorozirconates, such as K»ZrF«, are u$ed in analytical separa Evaporation of the halide from a water solution leads Ho formation of a basic tions. V halide, ZrOClj (zirconyl chloride). Zirconium (and niobium) tends to form radioactive colloids in solution. These colloids tend to be adsorbed on surfaces (container walls, dust particles, etc), and they also contribute to erratic behavior of zirconium in aqueous separations processes. Consequently, there is frequent difficulty in decontamination frlpm this element and in analyses for determining its concentration. 3.66 Group V — Arsenic, Antimony, and Niobium. Arsenic and antimony of the However, niobium 95 of A subgroup are of little importance because of low yield. the B subgroup is important as the daughter of zirconium 95; it is not born directly in fission. Niobium is a seminoble, steellike metal with a high melting point (1950°C). In the

/

Art.

3]


11-47

majority of its compounds, niobium exists in the +5 oxidation state, lower oxidation However, it is sometimes trivalent and occasionally di- and It is a very tetravalent and exhibits a marked tendency to form oxy compounds. passive metal and when pure does not dissolve in sulfuric, hydrochloric, or nitric acid or aqua regia. It is attacked by these acids if impurities are present, but salts such as nitrates are not formed, and it is probably converted to a hydrated form of Nb206. Hydrofluoric acid dissolves the pure metal slowly. There are four oxides of niobium, NbO, NbjOj, NbOs, and Nbs06. Heating the metal or a lower oxide in air (also the sulfide, carbide, nitride, and others) will yield the pentoxide, which is the oxide of metaniobic acid (HNbOj) and by far the most Metaniobic acid is an insoluble white amorphous precipitate that important oxide. varies in composition, since it is a more or less hydrated form of NbsO». In addition to the metaniobates there are orthoniobates (NbO<)_,1 pyroniobates (NbjOi)-4, and perniobates (NbOi)-1. Niobium reacts with the halogens (or the pentoxide with the hydrogen halides) to Niobium trichloride may be prepared by heating the pentagive the pentahalides. chloride at red heat. The trichloride is the only metallic chloride that will reduce carbon dioxide, the products of the reaction being NbOCl3 (niobium oxychloride) and The oxychloride is decomposed violently by water, yielding niobic carbon monoxide. An analogous compound is the oxyfluoride. Both of the oxyhalides form acid. double salts with alkali halides (KsNbOFs). Other important compounds are the nitride Nb>N6, the sulfides NbS and NbiSj, the hydride NbH. In radioactive tracer solutions (such as those resulting from the dissolution of irradiated fuel in nitric acid), niobium, like zirconium, frequently is present as a radioactive colloid. Hence, these elements do not act as true solutes but tend to be adsorbed on surfaces (container walls, dust particles, silica, etc.). The formation of colloids also leads to erratic behavior and considerable difficulty in the decontamina tion of uranium and plutonium from these elements and in the analyses for determining the concentration of these elements. Tellurium in the A 8.57 Group VI — Selenium, Tellurium, and Molybdenum. subgroup and molybdenum in B subgroup are the only elements of importance in this


states being less stable.

group. Tellurium is a member of the Group VI series of elements, which comprises oxygen, sulfur, selenium, and tellurium. All have six outer or valence electrons and, conse quently, tend to form compounds in which they have an oxidation state of —2 (H2O, H2S, H»Se, HjTe), the tendency being strongest for oxygen and weakest for tellurium. While oxygen and sulfur are distinctly nonmetals, selenium and tellurium, especially Tellurium, however, is the poorest electrical the latter, show metallic characteristics. conductor of any of the metals. In addition to the valence state of —2, tellurium also forms compounds with more electronegative elements, in which it has valences of +2, +4, and +6, the last being the most important. Thus, tellurium forms the oxides TeO, TeOj, and Te03. The weak telluric acid HjTeO* is derived from the trioxide. It is a good oxidizing agent, being converted to TeOj. Similarly, tellurous acid and tellurites are derived from the dioxide. Both tellurous and telluric acids are slightly soluble in water. Other important insoluble compounds are the sulfide TeS2 and the iodide Tel<. Molybdenum is somewhat similar chemically to chromium, tungsten, and uranium. In compounds, it has is a tough, ductile metal with a high melting point (2620°C). oxidation states of +2, +3, +4, +5, and +6, but the +2 and +4 states are not stable in water solution. Because of the large number of possible oxidation states, it Molybdic acid is derived from the trioxide, and has a relatively complex chemistry. For instance, ammo the molybdate salts of this acid tend to form polymolybdates. The ammonium nium molybdate has the composition (NH«)«Mo7024-4H20. is analytically important, being insoluble in phosphomolybdate (NH<)JP04-12Mo03 water but soluble in ammonia and alkalis. The +2 halides are insoluble in water but dissolve in alkali. The +2 compounds are slowly oxidized by water.

It


11-48


[SEC.

11

Reduction of molybdic acid gives +3 molybdenum compounds, which are generally The hydroxide Mo(OH)j is insoluble in water or excess olive green in solution. alkali and on ignition is converted to Moid. The sulfide MojS, and phosphate axe only slightly soluble. Other important Molybdenum sulfide, MoSj, is the principal ore of molybdenum. tetravalent compounds are the oxide and the halides. The tetravalent compounds are not soluble in water. Compounds of the +5 state are formed by reducing molybdates in acid solution with moderately strong reducing agents which produce the ions MoO+* or MoO+1. The hydroxide MoO(OH)3 is insoluble. Molybdenum forms complexes in all valence states such as (MoaCl4)(OH)», (+2); KaMoCL, (+3); K,Mo(CN)8, (+4); (MoOClt)-«, (+5); and the polymolybdates, (+6), previously discussed. Iodine is the only important 3.68 Group VII — Bromine and Iodine. group During dissolution of the spent fuel member because bromine has a low fission yield. much of the iodine is evolved in the elemental state as a vapor and therefore creates an airborne radiation hazard. It is easily Iodine, the heaviest of the halogens, is a solid at room temperature. The halogen elements generally have a valence of — 1 and a melted and vaporized. Therefore, they are very powerful oxidizing agents, the high electron affinity. strength decreasing from fluorine, the most powerful oxidizing agent known, to iodine. Iodine dissolves in many organic solvents such as carbon tetrachloride, chloroform, It is also slightly soluble in water and alcohol, and hexane, giving a violet solution. The halogens above iodine will liberate oxygen from water giving a brown solution. In the case of iodine the reaction is reversed, and (yielding the hydrogen halides). A competing reaction, oxygen will liberate iodine from a hydrogen iodide solution. hydrolysis of the halogen giving HCIO, HBrO, or HIO, also takes place less com pletely with increasing atomic weight of the halogen. Iodine is the most electropositive of the halogens and can, therefore, form com These are low-melting, pounds with the more negative halogens (IC1, IC1», IF7, IF6). highly volatile compounds. Hydrogen iodide is a colorless gas with a pungent odor. It is extremely soluble in water, giving the strongly acidic hydriodic acid. The iodide ion, being a relatively large ion, frequently forms very insoluble salts (such as Hgl) and stable complex ions. Powerful oxidizing agents will oxidize iodide ion or iodine to iodates, which on The iodates may be further oxidized careful dehydration yield the pentoxide I20». Both the iodates and periodates are, in turn, powerful oxidizing to periodates. agents. 3.59

Group VIII — (Transition Group) — Ruthenium, Rhodium, and Palladium. Three metals of the platinum metal group are created in fission: ruthenium, palla All have densities of approximately 12 g/cm*. Ruthenium dium, and rhodium. has a gray color like that of iron; palladium and rhodium are more like silver in Rhodium is one of the whitest of all metals. The metals are untarnished appearance. in dry air. Palladium, like platinum, has the property of acting as a remarkable catalyst in many gas reactions. Spongy palladium is also capable of adsorbing large quantities of hydrogen. The metals Some of the more important reactions are summarized in Table 17. are relatively inert to strong oxidizing agents and consequently are not dissolved by many reagents. Ruthenium is the only element that exhibits all possible positive valences. Conse quently, its chemistry is exceedingly complex. In the lower oxidation states it is Compounds basic in nature, while in the higher oxidation states it tends to be acidic. of the higher states are powerful oxidizing agents. In alkaline solution the metal ta&y be oxidized to the ruthenite RuOt-1, ruthenate RuO»_1, perruthenate Ru04~, or the tetroxide RuOl; depending upon the strength of the oxidizing agent. The tetroxide is volatile, boiling at around 100°C, and advantage is taken of this to separate ruthe nium from other fission products in analytical procedures. The tetroxide is soluble in

Art.


3]

Table 17. Reagent

Boiling aqua regia

11-49

Reactions of Platinum Metals Rhodium

Ruthenium

Dissolves slowly

(H,RuCli)

Very slowly solu ble

(HiRhCU)

Palladium

Soluble

(HtPdCl.)

Hot cone nitric

Insoluble

Insoluble

Boiling with H.SO«

Insoluble

Insoluble

Fusion with KOH + KNOi

KiRuOt —soluble in

RhO.

Slowly soluble [Pd(NOi),] Slowly soluble (PdSO.) PdO

Forms soluble

PdSO.

water

Fusion with KHSO«

No reaction

Oxygen

RuO. at 700-I200°C some RuOi

may be

formed

Chlorine

KtRuCli in

presence

of KC1

K.Rh.(SO.). RhjOi slowly below

RhCli

1 150°C

PdO slowly at 700°C

PdCb


Principal source: B. S. Hopkins, "Chapters in the Chemistry of Less Familiar Elements,1 Publishing Co., Champaign, 111., 1939.

Stipes

alkalies, giving perruthenates (RuOi-). These are powerful oxidizing agents and may be reduced by organic substances, yielding the finely divided metal. The dioxide is formed when the metal is heated in air. It unites with other metal oxides to give ruthenites (BaRu02). The sulfate may be formed by oxidizing RuSj with nitric acid or by heating the tetroxide with sulfuric acid. While the tetra chloride has not been prepared, such complex compounds as K2RuCl6 (potassium or K2RuCl6OH form when potassium ruthenate (K2RuO<) is dis ruthenichloride) solved in cold dilute hydrochloride acid. The hydrous oxide Ru202 may be precipitated by alkalies from solutions of the The trichloride also forms an intense red complex with ammonia trichlorides. [Ru(NHa)((OH)Cl]Cl known as ruthenium red. Complex chlorides also form with alkali or nitric oxide, e.g., RuCl8NO-H20 and K2RuCl5-H20. Several species of ruthenium compounds are formed on the dissolution of irradiated fuel material, some of which are organic solvent-soluble. This has made ruthenium one of the most difficult elements to remove in solvent extraction processes for the decontamination of irradiated plutonium and uranium. Rhodium also can exist in numerous oxidation states (1, 2, 3, 4, and 6), but the +3 and +4 states are the most important. Its salts are generally trivalent. It is almost wholly basic in character. The oxide Rh203 forms when the metal is heated in air below 1150°C, and the hydrated oxides may be precipitated by alkalies. These tend to be soluble in excess alkali, probably owing to formation of rhodites. The trichloride is insoluble in water and oxides, but its hydrate is soluble. It forms double salts with many alkali chlorides, such as K2RhCl5. It is analogous to cobalt in forming coordination compounds of the types (Rh(NHs)6]Cl3, [Rh(NHj)6Cl]Cl2, K2[Rh(CN),], and K,[Rh(N02)«]. The principal oxidation states of palladium are +2 and +4. The palladous salts are easily oxidized to palladic compounds, which in turn are fairly good oxidizing The +2 compounds, such as PdCU, decompose to the metal on heating. agents. Like the other metals of the platinum family, palladium has a marked ability to form coordinated, complex compounds, such as K2(PdCl<), Agj[PdCl2-(OH)2], PdCl2-2CO, and K2[Pd(N02)4].

BIBLIOGRAPHY Katz, J. J.,

and E. Rabinowitch: "The Chemistry of Uranium," National Nuclear Energy Div. VIII, vol. 5, McGraw-Hill Book Company, Inc., New York, 1951. Scaborg, G. T., and J. J. Katz: "The Actinide Elements," National Nuclear Energy Series, Div. IV, vol. 14A, McGraw-Hill Book Company, Inc., New York, 1954. 8eries,

11-50


[SEC.

11

Rodden, C. J.: "Analytical Chemistry of the Manhattan Project," National Nuclear Energy Series, Div. VIII, vol. 1, McGraw-Hill Book Company, Inc., New York, 1950. Hollander, J. M., I. Perlman, and G. T. Seaborg: Tables of Isotopes, Revs. Mod. Pkyt., 26(2): 469-651 (April, 1953). Latimer, W. M., and J. H. Hildebrand: "Reference Book of Inorganic Chemistry," The Macmillan Company, New York, 1940. Hopkins, B. S.: "Chapters in the Chemistry of Less Familiar Elements," Stipes Publishing Co., Champaign, 111., 1939. 4

ANALYTICAL CHEMISTRY

t

It is This article is devoted to a discussion of radiochemical analytical principles. beyond the scope of this article to give detailed analytical procedures for fissionproduct elements or for reactor materials such as uranium, plutonium, and thorium. Analytical procedures for uranium and thorium are discussed in detail in "Analytical Chemistry of the Manhattan Project," National Nuclear Energy Series, Div. IV, Vol. 9. Plutonium and its compounds are discussed in "The Transuranium Ele ments," Div. IV, Vol. 14B, and "The Actinide Elements," Div. IV, Vol. 14A, of the National Nuclear Energy Series. 4.1

Radiochemical

Analytical Methods

Radiochemical analytical methods are based on the detection and counting of the radiations emitted by the radioactive species present. Analytical deter minations of two types are generally made: (1) relative concentration of a particular radioelement or radioisotope and (2) relative concentration of a particular type of activity. Examples of the latter are determinations made for gross a, 0, or y activities to establish the gross activity level of a sample. Radiochemical analytical methods for a particular radioelement are based on the additional fact that the chemical behavior of the radioelement is the same as that of other inactive isotopes of the same element which are present. In fact, this characteristic and the ability to detect the presence of a radioactive isotope by the energetic radiation it emits when it decays arc the bases for nearly all the applications of radioactivity to chemistry. For example, the use of radioactive tracers in the study of material processes, chemicai reactions, diffusion, analytical chemistry, etc., is based on the fact that the radioacftve atoms have the same fate as the nonradioactive isotopes of the same elements vhich are present in the same chemical form. The great majority of radioanalyses performed on process solutions are foi a re's" In process work, many of the originalactlv^ tively few active fission-product species. fission-product species will have disappeared by decay in the cooling time al?wet' between discharge of the irradiated uranium and processing. In purification f°°" esses, fission-product concentrations relative to uranium or plutonium may be redi^ The fission elements for which analyses are generally m^'by factors of 107 or more. are strontium, yttrium, zirconium, niobium, ruthenium, iodine, tellurium, cesiu"' barium, cerium, and total rare earths. The total rare-earth analysis includes yttriu)


energetic

also.

The concentration of a fission-product element in solution may often be very low. j a 1 M solution of irradiated uranium in which 0.01 per cent of the| atoms fissioned in 135 days, the approximate concentrations of some of the radioactive species after 60 days' cooling are shown in Table 18. At these low concentrations, isolation of a particular species in an analytical pro cedure with the required radiochemical purity in a reproducible fashion would be For example, the adsorption of a particular impossible by conventional procedures. species on walls or on other precipitates would occur to such an extent that results would be meaningless. To increase the concentration of a particular element to a value suitable for normal analytical procedures, inactive salts of that element are added to the solution. This inactive material is called a carrier. Not only is a ,

For example, in

t The authors arc indebted to R. P. Larsen, Chemical Engineering Division, Argonne National Laboratory, for assistance in the preparation of this article.

J I

Art.

analytical chemistry

4]

Solution:

1

11-51

Table 18. Illustrative Fission-product Concentrations A/ U. 0.01% atoms fissioned, 135 days' irradiation, 60 days' cooled Concentration

■

Isotope

it

g/om'

8r" Sr»

Zr" Nb"

Eu'M

Ru'"

Te'"

Iiii

Ca1"

Ba'» Ce'«


Ce'"

10-'

4 X 10-' 2 X 10' 10-' 5 X I0 » 4 X 10-' 1.6 X 10 » 8 X 10-" 5 X 10-' 2 X 10-'

4.X 10' 2.5 X 10'

io-* 4 X 2 X

10* 10-'

io-«

5 X 10-' 4 X 10-' 2 X 10-'

io-»

6.6 X 10-' 3 X I0 > 5 X 10-' 3.5 X 10"'

carrier necessary for handling the radioelement under consideration, but additions of carriers of the other radioelements are often necessary to dilute the effect of the These are adsorption or coprecipitation of one element on the precipitate of another. called holdback carriers. For carrying to occur, the carrier atoms and radioactive atoms of an element must be present in the same chemical state. In many cases, the exchange between carrier and radioactive atoms occurs immediately. Difficulties in obtaining exchange are encountered when the radioelement exists in more than one oxidation state and/or more than one ionic or molecular form whose chemical reactions are particularly In such cases exchange can be accelerated by heating the solution or carrying slow. out a series of oxidation-reduction cycles. The use of a strong complexing agent will often bring about exchange by the formation of a particular complex ion. In many analyses a chemical-yield factor is employed. After the various steps of a chemical procedure have been performed, the ratio of the amount of carrier present to the amount originally added establishes the yield. The radiochemical yield is the same as the chemical yield when interchange of carrier and radioactive atoms has occurred. The radioactivity of the final sample preparation can then be adjusted to a 100 per cent yield of carrier. In a radioanalytical procedure, it is necessary to exclude radiations from other radioactive isotopes present initially. This means that (1) the material finally counted must be of high radioactive purity or (2) it must be possible to eliminate radiations due to other radioactive contaminants (such as by the use of absorbers to cut out weak radiations) or (3) it must be possible to determine the counting rate for a particular isotope by using decay or absorption curves or by using spectrometers to discriminate radiations of a particular energy. These methods be described more fully below. Radiochemical Separations. 4.11 High radioactive purity is generally achieved using conventional chemical separation techniques in conjunction with inactive carrier elements. High chemical purity of the final preparation to be counted is generally not required. The factor by which a particular fission product must be freed from other radioelements will be dependent on its chemical concentration and half-life and on the energies and amounts of the other radiations present. Required decontamination factors may vary between 104 and 10'°. The conventional separa techniques are precipitation, ion exchange, solvent extraction, volatilization, and eloc trodeposition. Precipitation plays by far the most important role precipitation. (See Art. 5.2.) radiochemical analytical separations. Two types of precipitation are employed: (1 ) the precipitation of the radioelement under consideration and (2) the precipitation the contaminants. In both cases carriers are required. Such precipitations may

will

by

tion

in of

11-52

CHEMISTRY J^ND CHEMICAL ENGINEERING

11

be repeated as often as necessary to achieve the maximum separation. Precipitate separations are generally effected \by centrifugation. Some precipitates such as ferric hydroxide or manganese dioxide are very effective in carrying down certain contaminating activities present in solution. These are When scavenger precipitations are made, carriers of the contami called scavengers. Y nants are not used. The following zirconium fissioii-product and plutonium analyses illustrate the use of precipitation reactions. Zirconium is usu/ally analyzed by the barium fluozirconate method." Zirconium. After the addition of a known anhount of zirconium carrier to the sample, hydrofluoric acid is added and the solution alflpwed to stand for several minutes to effect exchange. Two successive lanthanum fluorrVde scavenging precipitations give essentially com activities. The addition of excess plete removal of rare-earth and /alkaline-earth barium nitrate results in the precipitation of barium fluozirconate, which serves to separate zirconium from other /fission activities. The corresponding niobium com pound is fairly soluble, but itsf separation is improved by the addition of niobium holdback carrier before the bartium fluozirconate precipitation. Further purification may be effected by dissolving) the barium fluozirconate in a boric acid-nitric acid mixture and reprecipitating the barium fluozirconate with the addition of hydro fluoric acid, barium nitrate, i£nd more niobium holdback carrier. To establish a chemical yield, the precipitate is redissolved, the barium is removed by a barium sulfate precipitation, and the Zirconium precipitated with cupferron.t This is ignited to zirconium oxide in a tared porcelain crucible and weighed to determine the chemical The precipitate is transferred to a small test tube and assayed for y activity yield. in a well-type scintillation counter (see Art. 4.12). One of the radioanalyses for plutonium is based on the principle of Plutonium. Because there is no inactive plutonium quantitative carrying on a precipitate. isotope that can be added as a plutonium carrier, carrying of plutonium must be effected on precipitates of other cations (see Art. 5.2). Lanthanum or cerium is used as the carrier and is precipitated as the fluoride. Both trivalent and tetravalent If other a emitters such as americium plutonium are quantitatively coprecipitated. or curium are present, they will also carry along with the plutonium. Consequently, separation of these from plutonium must be effected by some other means such as solvent extraction or ion exchange. A slurry of the final product precipitate (LaFs or CeFj) is transferred to a flat stainless-steel planchet, the edges of which were previously coated with a dilute solution of Zapon lacquer to prevent the solution from flowing over the edge. The This plate is then slurry is dried, ignited, and finally fixed with very dilute collodion. assayed for a activity. Ion Exchange. Ion exchange has proved fairly useful for fission-product separa tions, notably rare-earth separations. As described in Art. 5.5, conditions can be chosen for the selective absorption or elution of particular elements. Strikingly clean separations can be achieved. Volatilization. Volatilization is a practical separation technique if the radioelement itself (such as the noble gases or iodine) or a compound of the radioelement is volatile. Radioruthenium is separated from other activities by distillation of ruthenium tetroxide from very strong oxidizing media. Solvent Extraction (see Art. 5.3). Solvent extraction is frequently employed in analytical schemes. Generally, the solvent has both high selectivity and high extractive power for a particular component, and batch extraction is employed. An example is the separation of radioiodine by carbon tetrachloride. Solvent extractions are sometimes used to effect complete isolation of a particular component if batch extractions are slow. These consist of continuous extraction of a component by a solvent which overflows to a boiler from which the solvent is continuously vaporized, being subsequently condensed and returned to the extractor by gravity. The desired component is thus transferred essentially quantitatively to the boiler. L'ranyl nitrate is often extracted with diethyl ether in this manner. t Ammonium salt of nitrosophenyl hydroxylamine, CsHiNNOONH4.

f


[SeC.

Art.

4]

Ekctrodeposition.

ANALYTICAL Electrodeposition

material of interest or impurities. techniques.

It

CHEMISTRY

11-53

may be used to plate out either the active is less frequently used than other separation

Counting Techniques. f Radioactivity is detected and determined by means of 4.12 the ionization caused by the charged particles emitted in radioactive decay. The passage of a charged particle through matter results in the formation of ions through inelastic collision. Since many ion pairs may be formed by the passage of a single charged particle, the detection of a single charged particle is often possible with relatively simple apparatus. Two fundamentally different approaches are employed in the determination of radio activity. In one approach an effect proportional to the total ionization is measured, which gives the integrated effect of many particles. This is the principle involved in the use of the electroscope and the electrometer. In the other approach the effects of the bursts of ionization due to single particles are de tected, as in the Geiger tube, pulse ionization chamber, or cloud chamber, and in photo The graphic track and scintillation methods. IMPULSE RECORDER

SCALER

LINEAR AMPLIFIER


HIGH -GEOMETRY STANDARD ALPHA- COUNTING CHAMBER

A, B, C, D, E,

HIGH-VOLTAGE ELECTRODE COLLECTING ELECTRODE GUARD RING BOTTOM PLATE BREECH THREAD F, PARAFFIN INSULATION WIPING CONTACT G, HIGH-VOLTAGE H, HIGH -VOLTAGE INLET INSULATOR AMPHENOL

HIGH VOLTAGE

SUPPLY

J,

Fio.

ber.

7. High-geometry a-counting cham

8. Block diagram showing circuit com ponents for the determination of or activity .

Fio.

method chosen to measure the ionization depends upon the type of radiation involved and upon its intensity. Alpha radiation, which consists of charged helium nuclei emitted during radioactive disintegration, is characterized by short-range (several centimeters in air) highly ionizing particles. Several thousand ion pairs are produced in each millimeter of air through which This amount is much greater than for an average a typical alpha particle has passed. beta particle (a high-speed electron emitted in decay) or for a gamma ray (an electromagnetic radiation analogous in properties to an X ray), which produce 5 to 25 pairs and less than By appropriate means the ionization due to each typo 1 pair per millimeter, respectively. of ray in a mixture may be detected and determined separately. Alpha activity is conveniently determined by means of the pulse ionization chamber which consists of a pair of horizontal plates about 8 mm apart, with a potential of about The sample is placed on the lower electrode A, and 1.0O0 volts across the gap (see Fig. 7). the ions formed by the alpha particles are swept onto the electrodes by the electric field, This voltage pulse, which is of the order of thus causing a fluctuation in the voltage. 100 to 200 uv, is amplified electronically to the order of volts necessary for easy detection. Since the pulse frequency is often too rapid to be handled by an ordinary mechanical recorder, an electronic device called a "scaling circuit" is used to eliminate all but a known The fraction of the pulse to reduce the pulse frequency to a convenient magnitude. characteristics of the counting system are so arranged that the smuller pulses due to beta 24-18 A diagram and gamma rays do not trip the circuit, alpha rays alone being recorded. matic sketch of a complete system for counting alpha particles is shown in Fig. 8. In the

t Much of the following article is quoted from C. J. Rodden, "Analytical hattan Project," pp. 665-671, National Nuclear Energy Series, Div. VIII, Company, Inc., New York, 1950.

Chemistry of the I, McGraw-Hill

vol.

Man Book

9

11-54


[SEC.

11

ordinary "standard" alpha chamber the efficiency of detection is very close to 50 per cent, Highly active samples may exceed the which is somewhat modified by backscattering. capacity of the instrument (about 30,000 counts per minute). A lower efficiency is obtained

by the use of a low-geometry counter, either the air-screen type, in which the lower electrode is a wire screen and the sample is placed beneath it to yield an efficiency or "geometry" of about 10 per cent, or the vacuum type, in which the sample is at a considerable distance from the chamber, with the particles traveling in vacuum to the chamber and entering through a thin mica window. With the latter arrangement the geometry can be made very low without serious loss of accuracy. The method of detecting alpha disintegrations by visual observation of scintillations on a zinc sulfide screen is of historical interest only.


Beta-activity measurements are usually accomplished by a Geiger-Muller counter, which is doubtless the most convenient method of measuring beta activity.

*»

The Geiger counting tube consists essentially of two electrodes, one usually a fine wire and the other a concentric metal cylinder about an inch in diameter. The space between the electrodes is filled with a suitable gaseous mixture at a few centimeters pressure, and a potential of about 1,000 volts is applied across the electrodes. The voltage and the gas pressure are so adjusted that in the absence of ions in the gas no discharge takes place. A beta particle passing through the tube and producing one or more ion pairs initiates a cascade of ionization. A comparatively large surge of current results. The electronic "quenching" circuit to which the tube is attached is designed so that after a very short period of current flow (about 10~4 to 10-6 sec) the current is interrupted and the tube is recharged to enable it to detect another par ticle. The pulse is amplified, and the num LEAD TO ■ELECTROMETER ber of pulses is scaled down by a scaling CIRCUIT circuit to allow high counting rates to be INSULATOR handled by a mechanical recorder. Suit WSULAT0R HIGH-VOLTAGE LEAD able circuits and apparatus have been INSULATOR

described.""'0-"

Geiger tubes are made in a variety of The two most common types are the forms. bell-shaped thin-windowed tube which is de sirable if soft radiations are to be studied but which requires a relatively constant and not too high room temperature for proper opera tion, and the traditional glass-enclosed type. The latter is more rugged and, though less sensitive to soft radiations than the thinwindow type, is probably more satisfactory for general analytical work. 'THIN ALUMINUM WINDOWS Beta activity may also be measured by an electronic circuit in which the current pro Fio. 9. Ionization chamber for the measure duced in an ionization chamber is measured. ment of /^-emitting radioactive samples. Figure 9 shows the design of one such cham This apparatus, frequently called the "FP54" after the electrometer tube used in the ber. The photographic darkening amplifier, is used principally for high-activity measurement. technique is used for certain special applications. Since gamma rays are not charged particles, they are not detectable directly by ordinary Instead, electrons ejected from matter by the passage of highly pene counting devices. The ordinary Geiger trating gamma rays can cause ionization, which is then detected. tube may be used to detect gamma radiation in the presence of beta activity if a suitable absorber, such as a metal plate, is interposed between the tube and the sample to filter out all beta particles.'5-'7 The intensity of the gamma radiation is but slightly diminished, and the electrons ejected from the absorber and the tube walls are counted, the value given being proportional to the gamma activity.

The efficiency of a Geiger tube for counting y rays is only about one-fiftieth that for 0 rays. "Ionization chambers, particularly when filled with gas under pressure, are also used for determining gamma activity."'8-" Scintillation counters have come to be almost exclusively used for the measurement of 7 activity. In a scintillation counter, the y rays that are interrupted by a special crystal are converted to pulses of visible light, whfth are detected photoelectrically The photoelectric pulse is amplified and registered. Resolving times of less than

•

Art.

4]

ANALYTICAL

CHEMISTRY

11-55

10~8 sec are obtainable. Some of the crystals that are satisfactory are thalliumactivated sodium iodide, anthracene, stilbene, and a series of plastic phosphors. Scintillation counters have relatively high efficiencies and can be used for detection and measurement of 7 rays with energies from a few thousand electron volts to well into the Mev region. The counting effi ciency of a crystal varies with the energy HANDLE For of the 7 ray and crystal thickness. example, for a l-in.-thick crystal, the counting efficiency may vary from 100 per cent for a 0.15-Mev 7 to 50 per cent for a 0.5-Mev 7. Figure 10 is a schematic diagram of a 7 Aqueous samples of scintillation counter. LEAD TO ' the purified 7 activity are pipetted into a ABSORB HARD small glass tube, diluted to a standard BETA RAYS ' / / height, and placed in the well of an acti


(-ALUMINUM CAN

vated crystal. The simultaneous detection of hard 0 activity is prevented by the use of a lead absorber. 4.13 Measurement of Sample Activity. Measurement of the absolute number of PHOTOMULTIPLIER disintegrations is difficult, particularly for 0 TUBE and 7 emitters, but these are required for Fio. 10. Scintillation assembly for count such cases as fission-yield determinations. ing of 7 activity. Such measurements require knowledge of counting efficiency and geometry and various correction factors for absorption of the radiation (by itself or intervening material) and for scattering of radiation. Fortunately, for most work only relative sample activities are needed, and for this 300

W-BETA ABSORBER

TIME (LINEAR SCALE)

Kio.

curve.

11.

Resolution of complex

decay

Fio.

12.

COMPONENT

THICKNESS Img/crn llLINEAR SCALE)

Absorption curve for

/3 ray

in pres

ence of a 7 ray.

The purpose it is only necessary to count samples under the same conditions. measurement of sample activities under the same conditions requires adherence to the following techniques: 1. Employment of the same method of mounting samples 2. Use of uniform amount of precipitate or material to be counted 3. Use of same counting geomy try, which often requires use of a particular counter Discrimination of Particular Radiations. 4.14 Sometimes it is more practical to discriminate between the emanations of a particular element and those of others

11-56

CHEMISTRY

AND CHEMICAL

ENGINEERING

[SEC.

11

present than it is to separate the element completely. Thus, with a 7-ray spec trometer, it is possible to count a 7 ray of a particular energy in the presence of other If a particular radioelement is the only one present that has a 7 ray of 7 or /3 rays. that particular energy, there is no need to achieve high radiochemical purity of the Alpha pulse analyzers are used in a similar way to material prepared for counting. count a disintegrations of a particular energy in the presence of a particles having This technique cannot be used with /3 rays because they are different energies. emitted with a continuous energy spectrum. Gamma rays can also be counted in the presence of 0 radiation by using absorbers of sufficient thickness to cut out the j3 radiation. This method can also be used to count a very energetic /3 in the presence of weak /3's, but correction must be made for the proportion of the energetic 0 that is absorbed. Absorption or decay curves as illustrated in Figs. 1 1 and 12 are two other methods various radiations present in a sample being counted. of discriminating These methods are often used to establish the radioactive purity of a sample. In principle, In practice, any number of separate activities can be resolved by these methods. however, experimental uncertainties in the counting data limit the use of these methods to systems of two or three components. 4.2

Activation Analyses


The presence of many elements may

be detected and their concentration established by measurement and identification of the radioactivity induced by activation with For the method to be feasible, the element to be neutrons, protons, or a particles. detected must yield products with reasonable half-lives and measurable radiations, there must be only one source of a particular isotope, and it must be possible to separate the activity from other activities that might be simultaneously induced. Chemical separations may or may not be necessary for the determination of an induced are simul radioactivity, depending on whether or not interfering radioactivities taneously produced or absorption in the sample is excessive and/or unknown. If a chemical separation is not necessary, the activation method has the advantage of preserving the original condition of the sample. Identification of the radioactivity is made by measurements of the half-life of the Such methods may be extremely isotope produced and energy of the radiations. sensitive for the detection of impurities. The method may be made quantitative by standardization, If standards are not using samples of known concentrations. available, the quantitative determination of the concentration of an element requires knowledge of the cross section of the isotope undergoing the desired reaction, the flux of the bombarding particles, the efficiency of the detection instruments, and the yield of the chemical separation if one is necessary. The method is Neutron activation analyses are becoming increasingly popular. well discussed in Refs. 40 and 41.

4.3

Isotope Dilution Methods

of Analysis

The isotope dilution method of analysis is useful when there is no known quantita tive procedure for analysis of a component or when the yield of pure component after In such cases, the use of a radioactive isotope of isolation from others is unknown. the unknown element as a tracer will often allow determination of the yield and there fore the amount of unknown generally present. The quantity of unknown present may be calculated from knowledge of the weight and specific activity of the tracer If the weight added and the weight and specific activity of the final mixture isolated. of the tracer added is very small, then

A, where N* = number of unknown atoms originally present Nr = number of unknown atoms in recovered compound Aj and A, = specific activity of tracer and of element in recovered compound

Art.

5]

BASIC

METHODS

OF SEPARATIONS

11-57

The procedure is applicable even if the unknown element is radioactive provided it In this method has a known specific activity that is different from the tracer isotope. of analysis, the tracer must be in the same chemical form or be exchangeable with and only with the element it is to trace. 4.4

Radiometric Analyses

A radiometric analysis involves the determination of the weight of an element by Thus, measuring its radioactivity, the specific activity of the element being known. the weight of natural uranium in a mixture may be found by isolating the uranium and counting the a activity. A so-called radiometric analysis for an element involves the measurement of the radioactivity of a tagged element of known specific activity in chemical combination For example, the concentration of silver ion may be with the unknown element. determined by precipitation of silver iodide with radioactive iodide atoms. As little as 10 ppm of silver has been detected in this manner by absorption of silver iodide on ferric hydroxide.


BIBLIOGRAPHY

Cook, G. B., and J. F. Duncan: "Modern Radiochemical Practice," Oxford University Press, New York, 1952. Coryell, C. D., and N. Sugarman: "Radiochemical Studies: The Fission Products," parts I, II, and III, National Nuclear Energy Series, Div. IV, vol. 9, McGraw-Hill Book Company, Inc., New York, 1951. Friedlander, G., and J. W. Kennedy: "Introduction to Radiochemistry," John Wiley & Sons, Inc., New York, 1949. Jordan, W. II.: Determination of Nuclear Particles, Ann. Rev. Nuclear Sci., p. 207, 1952. Rodden, C. J.: "Analytical Chemistry of the Manhattan Project," National Nuclear Energy Series, Div. VIII, vol. 1, McGraw-Hill Book Company, Inc., New York, 1950. Seaborg, G. T., J. J. Katz, and W. M. Manning: "The Transuranium Elements," National Nuclear Energy Series, Div. IV, vol. 14B, McGraw-Hill Book Company, Inc., New York, 1951. Seaborg, G. T., and J. J. Katz: "The Actinide Elements," National Nuclear Energy Series, Div. IV, vol. 14A. McGraw-Hill Book Company, Inc., New York, 1954. Wahl, A. C, and N. A. Bonner: "Radioactivity Applied to Chemistry," John Wiley & Sons, Inc., New York, 1951. 6

BASIC METHODS OF SEPARATIONS 6.1

Reactor reasons :

Requirements and Specifications

fuel materials have to be reprocessed

for one or more of the following

1. To remove accumulated fission-product poisons, which interfere with the neutron economy of the reactor 2. To restore the heat transfer and structural properties of solid fuel elements, which deteriorate owing to the destruction of the solid lattice structure by the fission

process 3.

To recover bred fissionable material

The general separations problem

faced is to process a spent fuel so as to Recover any unconsumed fertile and fissionable material fed. 2. Recover any new fissionable material that may have been bred into the fuel. 3. Separate both items 1 and 2 from the fission products to the degree dictated by the subsequent utilization. This is generally called decontamination. It is some times, but not always, necessary to separate item 1 from 2. 4. Convert the fissionable material to the desired chemical and physical forms. 5. Provide for proper handling and disposal of the fission-product wastes. Both the chemical and physical features of a processing scheme are profoundly affected by the character of the fuel. The broad spectrum of fission products has already been indicated (see Table 4). Since it is necessary to separate the valuable 1.


11-58


[Sec.

11

material from all these fission-product elements, the chemistry of selected processes becomes very complex (see Art. 3). Processes considered to date fall into two general classes: (1) those which provide essentially complete decontamination from fission products and (2) those which If it is necessary to perform manual opera provide only partial decontamination. tions, such as metallurgical refabrication, upon the recovered fuel, or if rcenrichment in a gaseous diffusion plant is required, then a process capable of giving high decon If, on the other hand, it is only necessary to refabricate fuel tamination is required. to restore proper heat transfer and structural properties, then a process giving partial decontamination may be used. It is axiomatic in this case that remote methods of fuel refabrication must be used. The requirements of specific processes vary, but in the case of complete decon tamination processes, the product specification is usually based upon the amount of radioactivity contributed by the pure product itself. A common specification is that the amount of radioactivity contributed by residual fission products shall not exceed that contributed by the product isotopes in equilibrium with their daughters. This generally requires separation from fission products by factors of 10s to 10*. This factor is referred to as the decontamination factor. In addition limits are set on the nonradioactive contaminants such as those which will act as neutron poisons upon The limits on impurities, such as sodium, calcium, reintroduction into a reactor. phosphorus, silicon, iron, aluminum, and molybdenum, are generally less than 100 ppm. Since fissionable materials are expensive, product losses must be kept as low as Recoveries of greater than 95 per cent are required and 98 per cent or better possible. are usually obtained. Processes with this performance have been or can be developed using precipitation (Art. 5.2), solvent extraction (Art. 5.3), or fractional distillation (Art. 5.4). In the case of processes directed toward the restoration of structure and of heattransfer properties, very great relaxation of the decontamination specification is In some processes now under development, decontamination factors of only possible. 10 to 100 are sought. The largest single effort in this area is the development of pyrometallurgical processes, which are discussed in Art. 5.6. 6.2

Precipitation

By A. A. Jonke* and R. C. Vogel* Precipitation methods have long been used, both on a laboratory scale and on a plant scale, for the separation of elements or for the removal of impurities from a desired product. As a means of separation, decontamination, and recovery of reactor The original separations fuels, precipitation has been extensively investigated. process set up at Hanford, the bismuth phosphate process, was a precipitation process. Precipitation is also useful as an auxiliary step in conjunction with other separation methods. In separations-process work the problem of separating very minute amounts of material from a large bulk of substrate occurs. For example, in the extraction of plutonium from spent fuel, a difficulty arises because of the small concentrations of The plutonium and fission products present as compared with that of uranium. volumes of solution, at least in the early stages of the separation process, are so large that removal of the trace amounts of plutonium or of fission products by direct The problem can be solved by coprecipitating precipitation would not be practical. the trace constituent on a suitable carrier. A carrier is a substance added to a solution of trace element in an appreciable quantity so that when it is precipitated the trace element will follow the added constituent. In the processes that will be discussed here, it is generally necessary to add a carrier precipitant even though the solubility product of the trace element may be exceeded. Otherwise the amount of precipitate would be so small as to be imperceptible. * Argonne



Art.

5]

BASIC METHODS

OF SEPARATIONS

11-59

Besides the bismuth phosphate process, an example of a carrying procedure is the isolation of radium by carrying on BaSO(. In some cases precipitates carry down active substances where the chemical similarities are not so obvious. Thus, many In fact, ferric hydroxide active species are carried by ferric hydroxide precipitates. carries so many species it is often referred to as a scavenging agent. 5.21 On the basis of a number of observations, Fajans" Theory of Precipitation. in 1913 formulated the following principle: the lower the solubility of the compound formed by the radioelement (as positive ion) with the negative ion of the precipitate, the greater the amount of radioelement carried with the precipitate. Thus, Bi+' is carried by BaCOj and Fe(OH)>( but it is not carried by BaS04 or PbS04. Exceptions to this rule are not uncommon; lead is not precipitated with Hgl* or with cupric fumarate, although both Pblj and lead fumarate are rather insoluble. Clearly, factors in addition to that expressed in Fajans' rule are important in this phenomenon. It is usually possible to differentiate experimentally between two general types of carrying processes: (1) incorporation of the tracer into the crystal lattice of the precipitate and (2) adsorption of the tracer on surfaces of the precipitate during or When incorporation into the lattice occurs, the ratio of the after its formation. fraction of tracer to the fraction of carrier removed from solution is relatively inde pendent of precipitating conditions such as the presence or absence of an excess of the precipitant or of highly charged ions. If the tracer is carried by means of an adsorp When a tion process, this ratio is very dependent upon precipitating conditions. tracer is incorporated into the lattice, the distribution of the tracer is continuous, though not necessarily homogeneous, whereas if the tracer is adsorbed, its distribution is discontinuous, the tracer being distributed over the surface of the crystals or in some cases only on certain planes of the crystal. In 1926 Hahn">" proposed a classification of carrying phenomena to distinguish He described four cases of true coprecipitation from those of surface adsorption. These are principal types of carrying. 1. 2. 3. 4.

Isomorphous replacement Surface adsorption Anomalous mixed crystals Internal adsorption

If the carrying ion and the ion carried form isomor Isomorphous Replacement. phous crystalline compounds with the precipitating ion, coprecipitation of this type is to be expected. The radioactive element is distributed throughout the precipitate, as may be shown by a radioautograph technique, and the mechanism is simply one of replacement at normal ion sites in the crystal lattice. This true copre cipitation is not much affected by conditions during precipitation such as acidity, Washing the order of addition of reagents, rate of crystallization, and temperature. precipitate cannot remove the coprecipitated substance. The precipitation of radium with barium salts is an example of this class of carrying. Freshly formed precipitate crystals with large surface areas Surface Adsorption. may be capable of adsorbing radioelements effectively. For this type of carrying the Fajans rule is significant, but in addition the surface charge on the precipitate Because relative to the ionic charge of the tracer substance is an important factor. significant adsorption occurs only when these charges are of opposite sign, experi mental factors affecting the surface charge of the precipitate strongly influence the This type of carrying is recognized by sensitivity to such factors as acidity, carrying. order of addition of reagents, and physical state of subdivision of the precipitate. In many instances an appreciable part of the adsorbed activity may be washed off or The carrying of lead at tracer concentra displaced by another ion of similar charge. tions by CaS04 or AgBr and of radium by Ag2Cr04 are examples. Anomalous Mixed Crystals. A type of carrying that superficially resembles iso morphous replacement is observed in a number of instances when the isomorphism An example is the limited carrying of lead by BaClj-2HjO. is unexpected and unlikely. These crystals are monoclinic, but PbClj in macroscopic amounts crystallizes in the


11-60


[SEC.

11

rhombic system. The capacity of the BaCli for Pb+J is about 0.1 mol per cent lead in the crystals. A possible explanation is that a limited replacement of lead atoms by barium atoms can occur, leading to solid solution formation with an upper limit of lead content that is quite low. Internal Adsorption. Some cases of carrying do not fit into any of the three types already discussed and are characterized by a scattered distribution of tracer within the Thus, when lead is carried by precipitate crystals as shown by radioautographs. BaBr2-2H20, the small amount of lead is distributed in spots scattered through the The carrying of Pu+< by BiP04 is believed to be of this type. BaBr2-2H20 crystals. Process Chemistry. 5.22 Plutonium exhibits three stable valence states in solu Of these, the last two are important tion: trivalent, tetravalent, and hexavalent. in the precipitation method of separation. Tetravalent plutonium can be quantita tively converted to the hexavalent form by several common oxidants such as bismuthate, chromate, permanganate, and eerie ions. Hexavalent plutonium can be reduced to tetravalent plutonium by many reagents, including ferrous, sulfite, nitrite, and oxalate ions; hydrogen peroxide; and hydroxylamine. Tetravalent plutonium forms a rather large number of insoluble compounds includ ing the oxalate, phosphate, hydroxide, alkali fluorides (such as K2PuF6), arsenate, The hexavalent iodate, chromate, ferrocyanide, and several organic derivatives. The principal insoluble Pu+t form, however, forms very few insoluble compounds. compounds are various basic diplutonates (such as BaPu207, K2PU2O7, and CaPusOj.l and sodium plutonyl acetate. Uranium exhibits two valence states in solution, tetravalent as uranous, U+4, and hexavalent as uranyl, U02+*. The uranyl ion is quite stable in solution and is reduced to the tetravalent state with difficulty. Hence, only the hexavalent state of uranium Uranyl ion forms is generally important in the processing scheme considered here. a number of salts insoluble in alkaline solution but very few acid-insoluble compounds. Although many different fission products are produced in the fission reaction, only a few are important in the separations process. Because of their short half-lives the majority of the approximately 35 individual fission-product elements need not be considered. Those which are of major importance are the following: zirconium, niobium, cerium, ruthenium, lanthanum, yttrium, barium, strontium, and praseo Other important fission products (such as I2, Xe, and Kr) are gases and dymium. are mostly evolved during metal dissolution prior to the main separation process. Because many of the fission products present have variable oxidation states, decon tamination is not completed in one cycle. 5.23 Definitions. As stated earlier, a carrier is a substance added in substantial amounts to enable a trace element to be precipitated more or less quantitatively. Carrier precipitates should be amenable to ready dissolution. Plutonium (or fission products) can be precipitated with a suitable carrier, and the resulting macroscopic precipitate can be removed in the usual way, by nitration or centrifugation. The term product precipitation refers to the precipitation of plutonium together with the carrier while the fission products remain largely in solution. The term by-product precipitation refers to precipitation of carrier and impurities while plutonium remains in solution. A large number of compounds are available that will carry tetravalent plutonium from solution, including bismuth phosphate, lanthanum fluoride, zirconium phosphate, uranous and thorium iodates, hypophosphates and oxalates, metallic hydroxides, and cerous fluoride. The carriers used for tetravalent plutonium do not carry hexavalent As expected from the plutonium to any appreciable extent under process conditions. relatively small number of insoluble plutonyl compounds, very few carriers for hexavalent plutonium are known, except for sodium uranyl acetate. Scavenging is a term that has been applied to the formation in a process solution of a The precipitate designed to carry down fission-product activity but not plutonium. term is usually reserved for precipitates other than the by-product precipitate, which The usually has the same chemical composition as the plutonium carrier precipitate. use of scavengers is generally advantageous for the removal of certain fission elements that may otherwise be difficult to remove. Inactive eerie and zirconium ions, both

Art.

5]

BASIC

METHODS

OF SEPARATIONS

11-61


of which form insoluble phosphates, result in appreciably greater removal of phos phate-insoluble fission products, particularly zirconium and niobium, in the bismuth phosphate process. Many precipitates, especially when freshly formed, have large surface areas and may adsorb appreciable amounts of various trace elements. Manganese dioxide and ferric hydroxide are examples of such precipitates. A material called a holdback carrier may be used to prevent carrying of trace elements. This is achieved by adding to the process solution a fairly large amount By occupying most of the of a substance that is well adsorbed by the precipitate. available surface, this added substance greatly decreases the amount of trace element that is carried. As an example, the presence of zirconium appreciably decreases the proportion of protactinium carried by LaFj from a hydrofluoric acid solution. The coprecipitation of a trace element can sometimes be reduced by the addition of a complexing agent that increases the solubility of the element in the solution. Such a substance, called a solubilizing agent, may be added to reduce the carrying of For example, the certain fission elements during the precipitation of plutonium. presence of fluosilicate ion increases the solubility of zirconium and niobium phos phates during product precipitation in the bismuth phosphate process. Bismuth Phosphate Process. It should be noted that the initial develop 5.24 ment work on a plutonium separation process, which led to the bismuth phosphate The basic process process, was done at a time when no one had seen any plutonium. information was obtained, of necessity, using minute quantities of the element made The design and operation of a processing plant handling by cyclotron bombardment. tremendous levels of radioactivity were without precedent at that time. As a result it was imperative that the process be adaptable to simple equipment that would require a minimum of maintainence and that the limits of control be not too stringent. Furthermore, the press of time made it necessary to select a process that could be developed expediently. Although BiPC>4 and Pua(P04)< do not form isomorphous crystals, the carrying of Pu+4 by BiP04 is virtually quantitative under proper conditions. It is believed that Pu+* is carried by internal adsorption, wherein the plutonium is adsorbed on exposed crystal surfaces and is covered over by the next layer of newly forming bismuth phosphate. A schematic flow sheet for the bismuth phosphate process is shown in Fig. 13. Six BiPO< precipitations are used — three product precipitations and three by-product These are followed by two LaFj precipitations — one being a by-product precipitations. precipitation and the other a product precipitation. 1. The Extraction Step. This step is employed following the dissolution of spent Precipitation of reactor fuel in nitric acid to extract plutonium from uranium. plutonium with BiP04 is carried out from a slightly acid uranyl nitrate solution to which sulfuric acid has been added to complex the uranium and prevent precipitation of uranyl phosphate. The need for high recovery of plutonium demands that all the plutonium in the extraction solution be in the tetravalent state. Sodium nitrite is particularly suited for this purpose, since it not only reduces Pu+e to Pu+4 but also oxidizes Pu+' to Pu+4. Precipitation is carried out by the addition of BiO(N03) solution followed by phosphoric acid. The precipitate is centrifuged and washed with water. Many factors affect the plutonium recovery in the extraction step. They are (a) concentrations of carrier and precipitating ion, (6) concentrations of uranium and acid, (c) concentrations of certain interfering impurities, (d) amount of agitation and length of digestion time, and (c) temperature. The plutonium-bearing precipitate resulting from the extraction step is redissolved The resulting solution in nitric acid to prepare feed for the decontamination cycles. contains plutonium, carrier, and some fission products, but the uranium and the bulk of the fission products have been removed. A decontamination factor of 5 to 8 is obtained for 7 activity in this step. 2. Decontamination Cycles. There are two decontamination cycles, each of which is comprised of a by-product precipitation from an oxidized solution and a product The purpose of t hese steps is to make additional precipitation from a reduced solution.

11-62

CHEMISTRY

AND


[Sec.

11

A secondary purpose is to achieve separation of fission products from the plutonium. Volume reduction during some concentration of the product with respect to carrier. the decontamination cycles is accomplished by reducing the concentration of carrier in the product solution, thereby carrying the plutonium on successively smaller amounts of precipitate. Sodium bismuthate is used to oxidize plutonium prior to by-product precipitations. Small amounts of eerie and zirconium scavengers are added to aid in the removal of EXTRACTION STEP EXTRACTION FEED

1ST DECONTAMINATION

BIPO,

B1PO4

REDUCTION

BY-PRODUCT PPTN

-WOXIDATIONy


BiP0«

BiPO, BY-PRODUCT PPTN

J OXIDATION

LaF3 REDUCTION

PRODUCT

PPTN

Fig.

13.

CYCLE

BiPO,

—{ REDUCTION

BY-PRODUCT PPTN

CONCENTRATION

PRODUCT

PPTN

2ND DECONTAMINATION

•[oxidationV

CYCLE

PRODUCT

PPTN

STEP

LoFj REOXIDATION

BY-PRODUCT

PPTN

L0F3 METATHESIS

Bismuth-phosphate separations

PRODUCT ISOLATION

process.

phosphate-insoluble fission products, particularly zirconium and niobium. A small amount of HjPO< is then added to precipitate the carrier and scavengers, leaving the plutonium in solution. Prereduction of plutonium for product precipitations in the decontamination step is accomplished by ferrous ion, which reduces plutonium rapidly and exerts a beneficial effect on decontamination. The ferrous ion is oxidized to ferric ion, which acts as a holdback carrier. Ferric ion is carried on BiPO« precipitates and selectively replaces a portion of the fission elements normally carried. Silicofluoride, (SiF6)~2, is also added prior to product precipitation to act as a solubilizing agent. It improves decontamination through its ability to form phosphatesoluble complexes with zirconium and niobium. This prevents the carrying of these activities with the BiPO( precipitate. At the conclusion of the decontamination cycles the over-all decontamination factor for 7 activity is approximately 10'.

Art.

5]

BASIC

METHODS

OF SEPARATIONS

11-63

The main purpose of the concentration step is the reduction 3. Concentration Step. both of the volume of solution and of the weight of carrier associated with the plutonium. This step begins with a by-product precipitation that eliminates the LaFj is bismuth carrier prior to introducing the new carrier, lanthanum fluoride. capable of carrying plutonium quantitatively with smaller ratios of carrier to plutonium than is possible with BiP04. This change to LaFj carrier removes most of the remaining fission products from the plutonium and accomplishes considerable reduc tion of both the volume of solution and the carrier-product weight ratio. Two LaF3 precipitations are carried out in the concentration step —a by-product precipitation from an oxidized solution and a product precipitation from a reduced The over-all decontamination at the end of the concentration step is at solution. least 107. The individual and cumulative decontamination factors for the BiPO, process are summarized in Table 19. The product precipitate LaFj of the concentration step is converted to La(OH), by metathesizingf it with strong KOH solution:

LaF, + 3KOH — La(OH), + 3KF Then La(OH)j 4.

is dissolved in nitric acid. In the isolation step, The Isolation Step.

plutonium

is separated from carrier


and other salt impurities by direct precipitation of an insoluble plutonium salt. The plutonium concentration in the solution prior to precipitation is sufficiently large Plutonium compounds that are suitable for isolation pur that no carrier is needed. poses include the peroxide, the iodate, the potassium double fluoride, and the oxalate. Since the peroxide molecule contains essentially no anion, the preferred precipitant is hydrogen peroxide. The supernates contain appreciable quantities of plutonium, and it is necessary to recycle these solutions for recovery of the plutonium. In the Other Applications of Precipitation. 1. Uranium Recovery Process. 5.26 bismuth phosphate process, the uranium (which is partially depleted in uranium 235) is not decontaminated, and the material is simply stored in underground tanks. Since a considerable amount of uranium has been thus accumulated, a precipitation process has been developed for the recovery and decontamination of the stored uranium. Uranyl ammonium phosphate (UAP) can be precipitated from solutions containing uranium and phosphate by the addition of ammonium ion. The precipitation of the uranium is essentially quantitative at low acid concentration and low temperature. The separation of fission products from the uranium by the UAP method is based on the crystallization of the uranyl salt under conditions in which the impurities remain in The solubility of UAP decreases sharply as the temperature is lowered. solution. In practice, therefore, crystallization is attained by slowly cooling the solution. It is extremely important that the mother liquid be well removed from the crystals if good decontamination is to be achieved. The separation process may be repeated by dissolving the crystals in hot acid, followed by cooling in the presence of ammonium ion. Successive cycles improve the decontamination. 2. Preextraction Scavenging. In conjunction with solvent-extraction separation processes, scavenging may be used as a head-end treatment for the removal of certain A scavenging fission products that are difficult to remove by solvent extraction. treatment may afford a simplification of the solvent-extraction process such as a reduction of the number of cycles required. Many scavenging agents have been investigated for this purpose. These include true precipitates, such as manganese dioxide, as well as solids having good adsorption properties, such as diatomaceous earth, activated charcoal, and bentonite clay. Niobium and zirconium and, to a lesser extent, ruthenium may be partially removed by scavenging treatments. The use of scavenging techniques for the treatment of radioactive wastes has also been investigated extensively on a laboratory scale (see Art. 10).

t A double decomposition reaction in which one reaotant and one produot are insoluble. in a change in the precipitating anion.

It

results


11-64

Table

19.

Ethers: Diethyl ether Isopropyl ether Dibutyl ether Diethyl Celloeolve Dibutyl CelloBolve Phenylcellosolve Dibutyl Carbitol Penta-ether (dibu toxytetraethy leneglycol) Alcohols: Methyl isobutyl carbinol 2-ethyl butanol 2-ethylhexanediol-l,3 Ketones: Diethyl ketone


Hexone

Methyl n-amyl ketone . Methyl n-hexyl ketone. , Acetophcnone Cyclopentanone

of Uranium by Pure Solvents

Extraction

Solvent

11

[SEC-

Diet* ratio,

Formula

Molet ratio

(X

10")

Solu bility in

HK), wt %

(CHj)sCHOCH(CHi)i

C4H.OC4H. CiH»OCjH*OC.H« C«H»OCiH40C*Hf C«H60CtH*OH C4H»0(C«H40)iC*Ht C4H»0(CiH«0)4C«Hi

CHiCHsCH(CH»)CHjOH CH,CH!CH(CaH.)CH»OH CiHtCH(OH)CH(C»Hi)CHiOH

0.013 0.01

2.62

0.09

6.74

(C»Hj)*CO

0.08 0.04 0.02

(CH,)CHCH»COCHi CHiCO(CHi)4CHt CH«CO(CHi)iCHi CHjCOCeH* CHr—CHi

0.01

\=o / CHi-CH.

0.61

0

2.7 0.3 1.3

18.4

0 6 10 8

1.4 4.8

0.43 4.2

4.6 117

2.0 0.43

2.2

0.55

1.67

1.10 1.07 18.7

19

3.4

2.58

0.05 0.80

solvent, wt %

1.3 0.57

0.09 <0.009 <0.009 <0.009 <0.009 0.29 0.01 0.96 <0.009 <0.009 0.09 9.3! 0.54 57.4

C«H6OC«H.

Solu

bility of H.0 in

2.66

I

5

I

Cyclohexanone.

v-Methylcyclohexanone.

CH.

CH,

CHi p-Methylcyclohexanonc Mesityloxido Esters: Ethylacetoacetate Glycol diacetate Diethyl maleate Tributyl phosphate Others: Dimethylsulfolane

\ \ /

CHi-CHi

CH.— CH. CH.— CH. CH

CH.— CH

CH.

\

C=0

0.53

/

C=0

0.24

CH.— CH.

CH.— CH. (CH.).C:CHCOCH.

/C=0

CH.COCH.COOC.H. CH.COOCH.CH.OCOCH, HCCOOC.Hi

0.30

14.3

0.24

10.3

0.

II

0.25 0.07

4.82

2.4

2 8

3 4

11.6 16.4 1.3

4 9 7 0 2. I

HCCOOC.H.

(n-butyl). PO»

0.04

CH.

CHr-CH 1

Nitropropane.

/

o

/

\

1.68

0.64 13. 3-15.6

CH.— CH

CH.(CH.).NO.

-CH. o

<0.009 <0.009

Source: C. N. Stover, H. W. Crandall, D. C. Stewart, and P. C. Mayer, "Survey Uranium Extraction without Salting Agents," UCRL-576. Jan 13, 1950. * Aqueous phase initially 0.5 M in uranyl nitrate and 0.34 M in nitric acid. . Moles U/cm' of organic phase Moles pure 8olvent/craa of organic phase

of Solvents for

Art.

5]

basic methods of separations

11-65

Alloy Fuels. A precipitation 3. Recovery of Uranium from Uranium-Zirconium process has been developed for the separation of enriched uranium from reactor fuels In this process that contain large proportions of zirconium as an alloying element. a bulk separation of uranium from zirconium is effected by the coprecipitation of VF, with LaFi. Some decontamination from fission products also occurs in the precipitation step, but the principal decontamination is subsequently achieved by other methods such as solvent extraction. The process consists of dissolving the alloy in hydrofluoric acid, one of the few After dissolution, the uranium reagents capable of dissolving zirconium rapidly. is in the tetravalent form (if no oxidizing agent is present) and is partially soluble in the acid solution. To recover the uranium quantitatively it is necessary to use a Lanthanum ion is technique similar to that used for plutonium. coprecipitation added to the solution, forming a precipitate of LaFi which acts as a carrier for tetra Two successive precipitations with lanthanum are required for valent uranium. satisfactory uranium recovery. The precipitate, after centrifuging and washing, can be redissolved in an acidic The uranium solution preparatory to further decontamination by solvent extraction. is oxidized to the hexavalent state, and a complexing agent is used to combine with the fluoride ion present in the solution. Uranium can then be extracted by the usual solvents, such as hexone (methyl isobutyl ketone) or TBP (tributyl phosphate).


5.3

Solvent Extraction

Solvent extraction is an attractive proved method for recovering and purifying a wide variety of reactor fuel materials. Because no solid phases are involved, it is It is one of the several readily adapted to continuous and remote-control operations. methods by which mass is transferred from one phase to another and is governed by the same general principles that apply to any diffusional process. In such a process, the rate of transfer of mass is proportional to a driving force (which is the difference in chemical potential of the component in each phase), the area through which the transfer is occurring, and inversely proportional to a resistance. The main resistance to material transfer lies in the stagnant films on either side of the Material transfer occurs through these films only by interface between phases. diffusion, whereas in the main bulk of each phase, material transfer occurs by mixing or turbulence or, as it is sometimes called, eddy diffusion. Solvent extraction may be used to remove a constituent or solute from a mixture of To do so, another liquid must be found that several present in some liquid medium. is immiscible with the first and possesses the ability to act selectively on the mixture of solutes originally present. When two immiscible phases are brought together, any solute will distribute itself between the two phases and, in time, an equilibrium dis tribution of solute concentrations will be established (i.e., the chemical potentials of Other solutes may also be present and the solute in both phases will become equal). will distribute themselves in different ratios between the two phases. If the distribu tion of one solute is appreciably different from that of another, this fact may be utilized in separating the two solutes. Extraction operations generally involve four steps: 1. Mixing of the two phases so that the solutes can distribute 2. Separation of the phases

between the phases

3. Recovery of the solute from the extract 4. Recovery of the solvent, if desired, for reuse 6.31 Definitions. In separations process usage the extract is defined as that phase which a desired component is being transferred and the raffinale is that phase However, the term waste is from which the desired component is being transferred. often used instead of raffinale. Appropriate adjectives are used to describe the character of the extract and raffinate or waste streams. Thus, it is possible to have an organic extract or an aqueous extract and similarly an aqueous waste or an organic waste even though the organic phase is to be recovered and purified for reuse by distillation or by a suitable washing procedure.

into

11-66


[SEC.

11

The term stripping refers to the return of an extracted component into an aqueous The word scrubbing is used to describe an operation in which a dissolved prod uct element is purified from contaminants by contacting the solution with an immis cible solvent that preferentially extracts the contaminants. Thus, fission products " associated with uranium in an organic solvent may be scrubbed or "washed out of the organic phase by contacting this phase with an aqueous phase of suitable composition. A salting agent is a salt or an acid the anion of which is the same as that of the solutes. Through the mass-action effect of this common ion, it is possible to vary within limits the distributions of solutes among phases by varying the salting agent concentration. The mixing and settling operations constitute a unit or extraction stage. If equi librium concentrations of solute are attained in each phase, the stage is defined as an ideal or a theoretical stage. The nearness to which equilibrium is approached is & In vertical contacting equipment the efficiency measure of the efficiency of a stage. is often expressed as HETS (height equivalent to a theoretical stage). The functions of extraction equipment are to bring the two phases into intimate contact, separate the phases, and recover solvent and products from the separated The contacting and separation of phases may be accomplished in a variety phases. of equipment types such as mixer-settlers, sieve-plate or packed towers, and cen Such equipment is described more fully in Art. 5.35. trifugal contactors. Liquid-liquid extraction can be simple or fractional (depending on whether one component or multicomponent separation is made), single or multistage contact (depend ing on the number of stages), and batch or continuous. 5.32 Solvent Extraction Principles. Development of an extraction process and design of extraction equipment require the following basic information:


phase.

Phase equilibrium concepts and data Stoichiometric or material balances 3. Diffusion-rate equations 1.

2.

These principles will be developed for encountered case of completely immiscible In the ideal case 1. Phase Equilibria. pletely immiscible phases, the distribution

the relatively simple but more frequently solvents. for a solute partitioned between two com law holds and

y = mx

where y = solute concentration in extract phase x = solute concentration in raffinate phase m = distribution coefficient, ratio of concentration in one phase to that in the other at equilibrium However, m usually varies with concentration and other factors such as pH, tem Therefore, an equilibrium curve must be deter perature, acidity, and salt content. mined experimentally under conditions those existing in the approximating contactor. Balances, a. Successive 2. Material Batch Extraction. If the equilibrium data for the distribution of a solute be tween two immiscible solvents may be represented by a curve such as that in Xj X| Xo Fig. 14, then the amount of solute ex X, LB OF SOLUTE PER LB OF SOLVENT A tracted is Fio. 14. Graphical representation of suc cessive batch extraction with fresh solvent. yo (Xl ~ (10 Vi

-

where R

E

—

-e

weight of solvent A

= weight of solvent B x = weight of solute per unit weight of A y = weight of solute per unit weight of B 0, 1, 2 — initial concentration and concentrations

extraction.

after

first,

">

second,

.

Art.

OF SEPARATIONS

BASIC METHODS

5]

11-67

Equation (10) is depicted graphically in Fig. 14 with j/o = 0. It is not necessary, however, that the ratio R/E be constant in each extraction. If the equilibrium line is a straight line (y = mi) and equal quantities of fresh solvent are used, then the concentration of unextracted solute in solvent A after n extractions is *"

=

(«T^)"*»

(11)

If the separation of phases is incomplete and the total quantity of solvent present per extraction (fresh solvent plus hold-up) is kept constant, then Eq. (11) becomes (R + QmEy-iRxo Q)mE]» [R + (1

-

(12)

where Q is the fraction of solvent held up per extraction. Multistaoe Contact. A schematic diagram of the flow 6. Countercurrent If flow is continuous, then each stage must used in this method is given in Fig. 15. RAFFINATE

RAFFINATE

FEED FINAL EXTRACT

1

2

EXTRACT

Fig. Generated for wjivans (University of Florida) on 2015-09-23 02:56 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

FINAL

STAGE

STAGE

15.

STAGE

.

3

EXTRACT

RAFFINATE .SOLVENT

Countercurrent multistage contact.

If not, then the phases could be mixed and consist of a separate mixer and settler. settled on a batch basis before transfer to the next stage. A material balance around a stage or any group of stages will have the form or

-

Rx. + Ey. R(xt xi)

= =

- y.)

Rn + Eyi E(yi

where R and E again refer to weights of solvents A and B and the subscripts refer to entering and leaving concentra this subscript is a Frequently, tions. number indicating the stage from which the particular phase came. Thus if there were n stages, the equation would be Rx0 + Eyn+l = Rx, + Eyx

e

and I

(13)

Solution of this equation for r/„+I gives a line with slope R/E, which is commonly It is simply a termed the operating line. This line is de material balance line. picted together with an equilibrium line

X| X2 X, LB OF SOLUTE

Xo PER LB OF SOLVENT A

Fig. 16. Graphical representation tercurrent multistage extraction.

of coun

on Fig. 16. The number of ideal stages required may be found by stepping between the oper ating and equilibrium lines as shown in Fig. 16. The step points on the equilibrium line represent the concentrations in the phases leaving the stage, while the points on the operating line correspond to concentrations of entering raffinate and leaving extract (for the nth stage, for example, this would be xn-i, y», where the subscript represents the stage from which the solution came). If the equilibrium line is a straight line, y = mx, and the extracting solvent contains no solute, then x P — 1 M = — = fraction unextracted = (14)' v x, P»+l 1

-

where n = number of stages P = Em/R = extraction factor operating line

ratio of slope of equilibrium

line to slope of

11-68

CHEMISTRY AND CHEMICAL

ENGINEERING

[SEC.

11

If the extraction factor P equals unity, it can be shown that the fraction unextracted If the extraction factor P is less than 1, the fraction unextracted equals l/(n + 1). approaches and cannot be less than 1 — P. If the extracting solvent is not solute-free but contains to lb of solute per pound of E, Eq.

(14) becomes

"

—

p.+l

_ i

True countercurrent operation is c. Continuous Countercurrent Operation. approached in extraction equipment in which the two liquid phases pass continuously in opposite directions. For example, a packed tower may be operated completely filled with either the light or heavy phase, which then moves slowly through the For economic reasons, con tower, passing countercurrent to the dispersed phase. tinuous countercurrent extraction is often preferred in plant-scale operations. The equations or the methods of stepping-off stages developed for multistage countercurrent extraction are frequently used to determine the number of stages in a batch countercurrent or continuous differential extraction system, such as an extrac tion column in which the phases flow countercurrent to each other. In such cases, the height of a theoretical stage (HETS) is defined as a length of equipment such that the solute concentrations in the exit phases correspond to equilibrium. 3. Rate Equations. Liquid extraction rate equations are based on the two-film theory and on the assumption that the interphase transfer rate is proportional to a driving force expressed as concentrations of the diffusing substance. It is assumed that equilibrium exists at the interface and that the resistances of the two films are additive. Under these conditions, the rate of solute transfer is given by

N where

N

dA =

-

M*

= rate of transfer of solute,

A

= phase contact area = film coefficients for kg fca,

dA = ks{yi

x<)

- y) dA

(16)

lb-moles/(hr)(ft*)

rafnnate

and

extract

phases,

lb-moles/(hr)(ft')

(lb-mole/ft')

x, y = concentrations

of solute in main body of rafhnate and extract phases, lb-mole/ft» Xi, ;/, = concentrations of solute in raffinate and extract phases in equilibrium at interface. Since it is difficult to isolate individual film coefficients, the over-all coefficients Kr and Ke are usually considered in terms of an over-all driving force, i.e., concentrations between the two main streams. Since solute concentrations, as such, in different solvents cannot be subtracted to get over-all driving forces, it is necessary to take the over-all driving force as the difference between the actual bulk concentration in one phase and the concentration corresponding to equilibrium with the main body of the other phase. N dA = Kr(x y) dA (17) i*) dA = KB(.y*

-

-

i

Ke

is

y

J_=J_ Kr

fc«

1

J__J__m kg kg

,

is,

where i* is that concentration in the (or R) phase which would be in equilibrium with a concentration of y in the E phase and y* is that concentration of y which would be in equilibrium with x. Such driving forces can be estimated at any point from the equilibrium and operating lines. = mi, x — x* Where the distribution law holds, that proportional to y* — and y

rnkg

is

it

is

A

is

It can be seen that, depending on the equilibrium slope m, the over-all K may approach an individual film in which case that film said to be controlling. In most types of equipment the interfacial contact area not known, so k,


M

Art.


5]

convenient to express the unknown transfer coefficient, giving

Na dV

=

Kga(x

11-69

area per cubic foot of equipment

- x*) dV

KEa(y*

=

-

a with the

dV

y)

(18)

where Kga is the capacity coefficient in pound-moles per (hour) (cubic foot) (poundFor large values of m (i.e., high solute mole per cubic foot) and V is column volume. solubility), Kga is apt to be controlling, and for small values of m (i.e., low solute In such cases, the experimental data should solubility), Kga may be controlling. be expressed in terms of the controlling phase. For constant flow rates of extract E and raffinate R measured in cubic feet per hour per square foot, a material balance gives

SEdy

=

S

=

(19)

f" -

R dx x* KRa(x x*)

h>

-

Edy

J*

Jvt Kga(y*

-

y)

-Z-J

Na dV

Therefore

where S is the column cross section.

Tower height 6

=

["'

SRdx

(20)

—

Nog =

or

i*

- -

Jv

—

y

x

y

/

/«

I

Nog =

(21)

where Nog and Nog are the numbers of transfer units based on the raffinate and is

Z

is

extract phases, respectively. It necessary to specify the phase on which the num ber of transfer units based. The tower height then equals HTUogNog and HTUogNog, where HTUor and HTUoe arc the heights of a transfer unit based on the

raffinate and extract phases, respectively.

-£Kga

and

the general case, the number of over-all

x*

versus x

or

-

versus

those systems where the distribution law holds (straight equilibrium line) and the pure, an algebraic conversion can be made to give the transfer phase being extracted.

extracting solvent units based on the

is

For

—

HTUog

transfer units must be obtained by

y

x

graphical integration of

=

y

In

Thus,

y

- HTUor

-PKga

_

2.3 log |[1

-

1

N0g

where

(1/P)](l/Af) + 1/PJ

- (i/P)

(22)

extraction factor = mE/R unextracted The number of stages given by

HETS (HTU)og

Graphs are available for solving

=

-

(1/F))(l/Af) +1/P)

2.3 log

- (1/P)

"

these equations.44

2.3P log

P

-

(23)

P

Therefore,

log |[1

1

v

1

= fraction

P

—

M

is

P


is

y)

These integrals may be found graphically by plotting l/(x — x*) (from the differ ence between x on the operating line and x* on the equilibrium line) versus x or versus y. 1/(.V* — The difficulty of a given separation indicated by

(24)


11-70


[SEC.

11

In the use of such equations, the literature lists a variety of nomenclatures, so the proper form for direction of the diffusing component, e.g., extracting or stripping, must be considered. Factors Affecting Solute Distribution. 6.33 In fuel-processing work, the value of the distribution or partition coefficient is frequently designated as which is the ratio of solute concentration in the organic phase to that in the aqueous phase. The distribution ratio is influenced by such factors as salting strength, pH, temperature, and concentration and oxidation state of solute. Choice of solvent is particularly important in devising a separation process. A good solvent is one for which the distribution coefficient for the desired component is large. However, it is also important that the solvent be selective for the particular com ponent (or components) in preference to others that may be present. Ideally, it Other factors should be immiscible or essentially immiscible with the other phase. affecting the choice of solvent are chemical and radiation stability, flammability or ease of purification and recovery, density, viscosity, flash point, vapor pressure, availability, cost, corrosiveness, and toxicity. The relative binding of solute molecules by solvent affects the distribution coeffi cient. Organic compounds, such as ethers, ketones, or alcohols, are good solvents for uranium and plutonium. The functional group is the oxygen atom, and the ability oi these organic compounds to extract uranium or plutonium is a function of the electro A larger proportion of oxygen increases the extractive negativity of the oxygen atom. power of the solvent; increasing molecular weight within a homologous series decreases it. Such solvents as methyl isobutyl ketone (hexone), diethyl ether, diisopropyl ether, pentaether, dibutyl carbitol (DBC), and tributyl phosphate (TBP) are good A list solvents for uranium and plutonium and have been investigated extensively. of solvents and some of their properties are given in Table 19. Tributyl phosphate represents an unusual case because it has a strong, specific solvent action resulting from the formation of a definite coordination complex: UO,+' (aq) + 2NO," (aq) + 2TBP (org)

^ U02(N03).-2TBP

(org)

TBP

is usually diluted with an inert organic diluent such as kerosene or carbon tetra chloride to improve such characteristics as density and viscosity, to reduce emulsification tendencies, and to make it easier to strip the uranium from the TBP with water. The distribution ratios for uranium and plutonium are a function of the TBP concen tration in the organic solvent phase (see Table 20). In the case of hexone, the uranyl ion (which transfers with three to four water molecules of hydration) is solvated by two molecules of hexone. A common ion such as nitrate can, through a mass action or salting effect, increase the transfer of uranyl ion to the organic phase: UOs+«(.,,,

+ 2NO-,(.q)

+ 4H20

^ UO.CNO,),-^^,^ ^ UOJ(NO,)r4H,0(ore,

Adjustment of the salting strength is widely used to control the distribution of uranium Frequently used salting agents are nitric acid, aqueous and organic phases. aluminum nitrate, ammonium nitrate, and uranyl nitrate (self-salting). The hydrogen-ion concentration may control ionic or molecular species and thus influence the extractability of a particular element. For example, plutonium may polymerize in water at high pH. A moderate deficiency of acid is sometimes employed in order to restrict extraction of fission products. The extraction of uranium or plutonium may also be adversely affected, but that of the fission products is affected to a greater degree, so that the net An result is greater separation of uranium and plutonium from fission products. "acid-deficient" condition is one in which a portion of the normal acidity resulting from hydrolysis of uranyl or aluminum nitrates has been neutralized (generally with sodium hydroxide). The residual acidity is such that the pH is still between 1 and 3. Because of reaction with the solvent, the amount of nitric acid that can be tolerated as salting agent is limited for most of the organic solvents. The effects of salting strength, nitric acid concentration, solvent concentration, and between

Art. Table

BASIC

5] 20.

METHODS

OF SEPARATIONS

11-71

Effect of Certain Process Variables on Uranium Distribution Coefficient (Solvent: tributyl phosphate diluted with carbon tetrachloride)

Aq uranium concentration

(JO

TBP

concentration

HNOi

(M)

concentration

(M)

Uranium diat eoeff (org/aq)*

Effect of TBP Concentration 0. 1 0. 1 0. 1

0. 1

0.2 0.4

4 4 «

0.33

0 1 2 5

0. 1 1.0 1.3 1.4

0.70 1.45

Effeot of HNOi Concentration 0. 0. 0. 0.

0.4 0.4 0.4 0.4

1 1 1 1

Effect of Aq Uranium Concentration 0.4 0.4 0.4

0.01

0.05


0. 10

* Molarity

7.4 2.4

5 5 5

1.4

units.

A decrease in solute concentration are illustrated in Table 20 for tributyl phosphate. uranium distribution coefficient Ea", as the aqueous uranium concentration is increased, is experienced with TBP systems. The distribution coefficient is more dependent on the per cent saturation of the organic phase than on the aqueous uranium concen tration. As saturation of the organic phase is approached, the concentration of uranium in the organic phase becomes relatively insensitive to increase in the aqueous With other solvents such as hexone, organic-phase uranium uranium concentration. In concentrations seldom exceed a small fraction of the saturation concentration. these cases, the distribution coefficient will either remain the same or increase slightly as the aqueous phase uranium concentration is increased. Since there may be heat effects in hydration or solvation, it may be necessary to heat or cool an extraction apparatus to maintain the desired extraction coefficients. The effect of hydration is illustrated by an equation such as the following: [UO,(NO,)2-6H,0]M ^ [UO,(NO,)riH,OU

+ (6

- i)H,0

+ heat

The change in organic solubility with oxidation state is one of the bases for the In general, the separation of the heavy-element salts of plutonium and uranium. higher the oxidation state, the more organic-soluble is the metal ion. Uranium in the hexavalent state is comparatively stable, so it is always the valence state of plutonium that is varied. The distribution coefficient of plutonium as a function of oxidation state is shown in Table 21. Table 21.

Distribution

Coefficient of Plutonium

as Function of Oxidation Distribution coefficient {org/aq) of Hexone and 0.5 M HNOi tat'd with NH1NO1

Pu valence

III

IV

VI

State

0.14

(i.e.,

Pud*1)

1.7

7.8

For specific removal of a given ion, a process called chelation has been demonstrated Chelates are molecular on a small scale and finds use in analytical separations. Derivatives of diketones, particularly TTA (thenoyl complexes of specific geometry.

11-72


trifluoroacetone), are excellent for removing solution according to the reaction Pu+<(.„, + 4HK«,riI)

—

tetravalent

PuK<(or8) +

plutonium

[SEC. from

11

aqueous

4H+

in which HK represents the chelation agent TTA. The extraction is very dependent on the acidity, and hence great selectivity is possible for various ions. Those ions appreciably extracted from 0.5 M nitric acid are Pu (IV), Zr (IV), Np (IV), Ce (IV), Fe (III), and Sn (IV). The organic solvent for TTAf may be benzene, o-dichlorobenzene, and others. The plutonium may be stripped from the aqueous phase by the use of stronger nitric acid or by reducing to the trivalent state, in which state it is organic-insoluble. 6.34 Development of a Flow Sheet. Because of the above factors that affect the solute distribution coefficient, the equilibrium line rarely is a straight line through the origin (i.e., the ideal distribution law y = rax generally does not apply). For example, the distribution coefficient of plutonium is generally affected greatly by the uranium concentration and will change as the uranium is extracted from the aqueous phase. EXTRACTION


(ol

AO

Fio.

17.

CONC

Typical extraction

(b)

STRIPPING

AO

CONC

and stripping diagrams.

Therefore, data are generally needed under actual operating conditions. Such dais may be obtained by sampling coexisting phases in a contacting device and equi librating these phases, or by simulating the countercurrent extraction by countercurrent batch extractions in the laboratory. The operating line, however, is a straight line for immiscible solvents and is deter mined by a material balance from the ratio of phase flow and a single point (generally If aqueous-phase concentrations are plotted along the abscissa an exit condition). and organic-phase concentrations along the ordinate, then solute is extracted from the aqueous phase into the organic phase if the operating line lies below the equilibrium curve. Conversely, solute is stripped from the organic phase into the aqueous phase if the operating line lies above the equilibrium curve. In nuclear engineering termi nology the term extraction always indicates solute transfer into an organic phase: the term stripping always indicates solute transfer into an aqueous phase. Illustrative diagrams for simple countercurrent extraction and stripping operations are given in Fig. 17. If a given number of theoretical stages is known to be present, then for a particular flow ratio and entrance stream concentrations, the operating line must be so located (by trial and error) that the number of steps taken between the operating and equi librium line equals the number of known stages. Conversely, if a certain transfer of

tTTA

is

The plutonium chelate is probably

Art.

BASIC

5]

METHODS

OF SEPARATIONS

11-73

solute from one phase to another is desired, the number of stages required to effect this transfer may be determined for various flow ratios. In this latter case, there is a maximum flow ratio (aqueous/organic) for an extraction operation and a minimum flow ratio (aqueous/organic) for a stripping operation that can be used to effect the desired solute transfer. These flow ratios are those which produce an intersection of the operating line and equilibrium line at an operating-line terminal (entering and leaving streams are in equilibrium) or are those at which the In both cases an infinite number operating line and equilibrium line become tangent. of stages is required to produce the given conditions. These principles are illustrated in Fig. 18 for an extraction operation. Two operating lines are shown, a practical operating line and one requiring an infinite number of stages to achieve the specified The maximum slope of this latter operating line, which equals the extraction. maximum feasible aqueous/organic flow ratio, is that of a line that produces tangency ILLUSTRATIVE EQUILIBRIUM CURVES A AND B


OPERATING LINE REQUIRES INFINITE STAGES. SLOPE EQUALS MAXIMUM FEASIBLE AO/ORG FLOW RATIO

PRACTICAL

OPERATING LINE

.AO FEED CONC

AO

Fio.

18.

CONC

Determination of maximum

flow ratio (aqueous/organic)

for extraction

section.

with the equilibrium curve, as with curve A, or intersects the equilibrium line at the feed concentration, as with curve B. Compound columns employ a center feed and can be treated as having an enriching and exhausting section. In many fuel-recovery operations, the compound column (see Fig. 19) is considered to be made up of an extract ion (or stripping) and a scrubbing section, the scrubbing section being that in which unwanted components are further Simple columns are generally used for separated from the desired component. stripping operations (see Fig. 19). In the compound column, there is an operating line for each section. The equations of these lines when solute is introduced only in the aqueous feed are: V

= y = -Br

for the extraction section +

the symbols being defined in Fig.

yp)

19.

for the scrubbing section

(25) (26)


11-74

[Sec.

11

A material balance around the entire column gives (R + W)xr + Eyr

- Rx,

(27)

If the equations of the operating lines are solved simultaneously, the locus of the points of intersection will lie on the vertical line x = xp, as shown in Fig. 20. This Thus, if it is desired to design fact is of great convenience in constructing flow sheets. a flow sheet for recovering uranium from a solution initially containing 0.5 lb oi SIMPLE STRIPPING

COMPOUND EXTRACTION COLUMN

COLUMN AO STRIP

ORG

RAFF

ORG PRODUCT OR EXTRACT E Ibs/hr

AQ SCRUB W Ibs/hr

/"""SOLUTION

CONC =yp

AO FEED R Ibs/hr CONC =xp

AO PRODUCT

ORG


FEED

Flo.

SOLUTION 19. Simple

and compound

AO RAFFINATE (R +W) lbs /hr CONC » »r

EXTRACT ANlf E Ibs/hr countercurrent

extraction

schemes.

EQUILIBRIUM CURVE

0.005 AQUEOUS

Fia.

20.

SOLUTION CONC,

LB OF SOLUTE

LB OF SOLUTE-FREE AQUEOUS

Illustration of flow-sheet design.

uranium per pound of water with a loss of 1 per cent, scrub and operating lines are located on the equilibrium diagram to conform with the known number of stages &n
Art.

BASIC METHODS

5]

OF SEPARATIONS

11-75

i ion of a salting agent or a change in temperature. This may be sufficient to cause refluxing, that is, a change from stripping of a component in the first few stages to extraction of the component in the latter stages. The solute concentration will then go through a maximum in one of the middle stages. Extraction of most fission products by an organic solvent is low. For this reason, a good separation of the desired constituents (uranium, plutonium) is generally effected in the extraction operation. Most of the fission products remain in the aqueous raffinate stream, generally termed the aqueous waste, while the uranium and plu tonium are transferred into the organic stream. The scrub section of a compound column is used to enhance the separation of fission products from the desired con factor. The distribution ratio stituents, i.e., to improve the decontamination (organic /aqueous) for a particular fission product is generally very low (0.01 or less). Therefore, on a linear scale its equilibrium line is indistinguishable from the x axis

10

-

100

o

=>

Q

o

O.

(-

z

=!

o

.**

"i/i


> u.

o°

O-o

id in

5

a.

o

0.01

s

0.001

z

C0NC IN AQUEOUS

Fio.

PHASE.

u. —

O

a. UJ a.

WT OF FISSION PRODUCT WT OF SOLUTE-FREE AOUEOUS

21. Idealized fission-product operating diagram.

unless the y scale is greatly expanded. An idealized fission-product operating diagram as it would appear using logarithmic scales (convenient to portray high degrees of decontamination) is shown in Fig. 21. It is readily apparent that if the low distribution coefficients for fission products prevailed throughout, the concentration of fission products in the organic stream Considerable decon could be reduced to essentially zero in a very few scrub stages. tamination of product elements from fission products is effected by the initial extrac The use of a few scrubbing stages serves to enhance this separation from fission tion. The value of adding more scrubbing stages diminishes rapidly for some products. fission-product species, and a point is soon reached where the addition of another scrub stage is not worth while. It is not unusual for the concentrations of some fission products (relative to a However, product element) to be reduced by factors of the order of 10* or more. removals of a few fission products fall far short of theoretical predictions for several reasons : 1. Some fission products exist as radiocolloids that are well adsorbed by particles and on various surfaces. Consequently, their behavior cannot be predicted.


11-76

CHEMISTRY

AND


[SEC.

11

2. Several chemical species of a particular fission product may exist, one or two of which may be highly organic-soluble. 3. Some fission products seem to extract almost irreversibly into the organic phase. The residual radioactivity remaining with the product elements is usually accounted The identity of these is somewhat dependent on for by a few of the fission products. the process. Ruthenium, which can exist in several valence states and chemical species, some of which are known to be organic soluble, is a particularly troublesome element in solvent-extraction Zirconium and niobium also purification methods. Some decontamination belong in the category of fission products difficult to remove. from these fission products is effected when the product element is stripped from the organic phase by water or dilute nitric acid. Additional solvent-extraction cycles serve to provide adequate purification from Again, considerable purification is effected in the initial these fission products. extraction of the product element into the organic solvent and is increased by scrub bing of the organic product phase. It is not understood why reextraction of a product element provides high decontamination factors in view of the fact that these elements were not completely scrubbed out in the preceding extraction cycle. One possible explanation is that the time elapsed and operations conducted between extraction cycles (solution adjustments, heating, oxidation) allow equilibrium of the species of a fission product to be reestablished, i.e., allow new, less solvent-soluble species of a fission product to form. Another possibility is that particularly troublesome species of the fission products may have remained in the organic phase in the preceding Whatever the reason, an effective means of securing the requisite stripping operation. product purification is the use of multiple solvent-extraction cycles. 5.35 Solvent-extraction Solvent-extraction Equipment. operations (contacting and separation of phases) can be carried out in various types of equipment such as mixer-settlers; spray, perforated-plate, or packed towers; and centrifugal contactors. Separate equipment may be provided for the contacting and the separation, or both The function of the contacting operations may be carried out in a single apparatus. equipment is to provide a large interfacial area between the two phases and good phase Generally, the better the contacting, the more difficult the phase separation. For processing spent reactor fuel, it is desirable that the separation is apt to be. volume of solution held up be small in order to limit radiation-damage effects, par Solution holdup and equipment dimensions may also be ticularly to the solvent. influenced by criticality considerations. The contacting equipment most generally used in fuel separations processing are Information available up to mixer-settlers, packed columns, and pulsed columns. 1950 on commercial extraction equipment, including a review of the patent literature, has been summarized by Morello and Poffenberger." One of the simplest types of extraction equipment is a tank equipped with an agitator for mixing the phases. The phases may also be mixed by simply passing them together through pumps, through If the operation is batch, then the long baffled pipes, or through jets, nozzles, etc. tank can also function as the settler. If the operation is continuous, then a separate Several designs settling vessel must be provided for continuous separation of phases. of continuous multistage mixer-settlers have been reviewed by Davis el at.** Figure 22 is a sketch of a continuous multistage mixer-settler used in pilot-plant This is a modification of a mixer-settler used as long ago as 1904 and operation.47 The necessary hydraulic head for countercurrent flow whose designers are unknown. Successful application has been made by gravity is obtained by inclining the unit. of a new type mixer-settler, called a "pump-mix" mixer-settler/* that operates The hydraulic head for countercurrent flow, mixing intensity, and horizontally. interface control is effected by the impeller (see Fig. 23). Various other types of countercurrent extraction equipment may be regarded as a series of mixer-settlers. The Scheibel column,*' for instance, consists of a vertical column in which are located at regular intervals small turbine agitators, operated from a single concentrically located shaft that runs through the column. On either side of the mixing section is a short packed section in which the phases are coalesced. Agitated and packed sections alternate through the column.

Art.

BASIC

5]

OF SEPARATIONS

METHODS

MIXER

L

H

U

H

4--

IM

M

11-77

M

SETTLER


H

H

L

LIGHT

LIGHT

HEAVY

HEAVY

LEGEND:

Flo.

H— HEAVY PHASE

— LIGHT PHASE M — MIXED PHASES L

22. Interior flow diagram

ii

mm MIXED

Mi Fig.

23.

for multistage

i

1 11 1 1 1 LIGHT PHASE

r

L

HDVY PHASE

HEAVY

box extractor

unit.

MM I'll II "

LIGHT

I I

phAsh

flli

ff>HASEa

PHASE

Elevation view of pump-mix mixer-settler

HEAVY PHASE

showing three adjacent

stages.

Another type of vertical mixer-settler unit is the Fenske contactor,50 in which the ascending and descending phases are mixed in a stage by means of two small, hori zontal, rapidly reciprocating, perforated plates. A small settling volume is provided All of the agitators are mounted beyond the agitator for separation of the phases. on a single shaft. The mixer-settler type of equipment has the advantages of high individual stage efficiencies (and therefore reliable scale-up is not difficult), low headroom require ments, and flexibility with regard to flow rates and flow-sheet changes. Mechanically, they are more complicated than some of the tower types of extraction equipment. A variety of countercurrent extraction columns are in use. Packed towers have the

11-78

CHEMISTRY AND CHEMICAL ENGINEERING — DIP TUBES

HEAVY PHASE IN

(FOR

INTERFACE • LIGHT

[SEC.

11

MEASUREMENT)

PHASE OUT

PHASE DISENGAGING-*

-INTERFACE

SECTION

ERFORATED PLATES

PLATE

SECTION-^

-SUPPORT ROD WITH TUBULAR PLATE-TO-PLATE SPACERS


PHASE

DISENGAGING SECTION

)) PULSE TRANSMISSION

LIGHT PHASE IN

-

LINES

1! BELLOWS

HEAVY PHASE OUT

MOTOR VCAM

^SHAFT

Fig. 24. Schematic sketch of pulsed column. advantage of being inexpensive and reliable.

Because of their simplicity and ease of operation, packed towers are widely used in nuclear engineering solvent-extraction Their efficiency, however, is operations. not as good as that of some of the mixersettler contactors or some of the differen tial contactors in which agitation is employed. , , ^ . A mechanically pulsed, perforated^ P'ate column,"-5* called a pulsed column. I /^^^"-""^tftNv has proved to be a very efficient contact ing device and is now being successfully used with some solvent-extraction proc esses (see Fig. 24). It is essentially s sieve-plate column with holes in the plates so small that flow of liquid will not occur unless a regular pulsation is applied to the column contents. The pulse is applied FREQUENCY (CYCLES/ MINI AT ANY GIVEN PULSE AMPLITUDE by imposing a hydraulic-pressure cycle Fig. from a piston or bellows pump on one of Three types of pulsed-column operation. the feed streams. It has been found that the capacity and efficiency can be corre lated by the product of amplitude and frequency. Three distinct types of phasediapersion behavior in pulsed columns are shown in Fig. 25.

gi

Art.

BASIC

5]

OF SEPARATIONS

METHODS

11-79

The chief advantages of pulsed columns over simple packed columns are the higher efficiency, resulting in lower column heights; higher throughputs; and more repro The power consumption is relatively large, ducible extrapolation to larger sizes. The pulsing of packed towers is however, and mechanical complexity is increased. also employed, efficiences being improved by factors of 2 to 3 thereby. Performance Data. 5.36 Because of the simplicity of packed Engineering columns, they were the first type of contactor examined in the study of solventIt very early became extraction processes for use in processing radioactive solutions. obvious that HETS values of 3 ft or less were possible at reasonable throughputs, Critical design factors required were the using methyl isobutyl ketone as solvent. Most of relationships among column diameter, capacity, and extraction efficiency. the process development work was done on 1- and 3-in.-diameter columns. Con siderable study of scale-up factors has also been carried out. The variation of HETS with flow rate in 2- and 5-in.-diameter columns is shown in Fig. 26 for a solventextraction process employing methyl isobutyl ketone as solvent and aluminum nitrate The effect of flow rate upon column efficiency for various types of as salting agent. ORGANIC SOLVENT: METHYL AQUEOUS

PHASER

M

ISOBUTYL KETONE (HEXONE)

AIINO^

PLUS

SOME URANYL

T

3.0 2 -INCH

of

(y-INCH


UJ

j

COLUMN x -INCH

RINGS)

33-INCH

-

a.

INCH (4 RINGS1

e

£ Fia.

-

COLUMN *£ INCH

0

400

INCH

5 -INCH COLUMN (5 -INCH > £ -INCH RINGS)

COLUMN

(5 -INCH 1 £ -INCH RINGS) FLOODING POINT INCH RINGS

^FORj

0

NITRATE

FLOODING FOR

RINGSl

I

800 1200 1600 FLOW, GAL OF BOTH PHASES/IHRKFT2)

26. Effect of flow rate on efficiency Raschig rings.

POINT

j -INCH 2000

for uranium extraction

2400

columns

packed

with

operations is shown in Fig. 27. The general conclusion of the scale-up studies was that Raschig rings were suitable for all types of process columns provided that the ratio of column diameter to packing size was within the approximate limits of 5 to 20. Later, work with tributyl phosphate as solvent gave values of HETS considerably In an effort to overcome more than 3 ft for some operations, particularly stripping. the excessive column heights that would have been required, work was started on the The performance characteristics of a 3-in. pulsed columns described previously. pulsed, perforated plate column for extraction of uranyl nitrate by tributyl phosphate Some of the efficiency data are given have been reported by Woodfield and Sege."

in Fig. The

28. choice

of a contactor for fuel-processing operations is based on efficiency or HTU), capacity, reliability, adaptability to remote operation, solution holdup volumes, flexibility (performance not greatly affected by flow-sheet changes), and headroom requirements. Since the solutions to be handled are intensely radio active, the equipment must be operated and maintained behind several feet of con The cost of the contactor itself is negligible, but the type of contactor crete shielding. selected will have a great effect on the over-all plant cost because it determines the amount of shielding required (headroom), the complexity of remote-control and remote-maintenance provisions, the throughput capacity of the plant, and the future flexibility of the plant. Operating data from Hanford have been made available in TID-7534, "Symposium on Fuel Reprocessing," Brussels, Belgium, May, 1957.

{HETS


11-80

[Sec.

11

General Description of Solvent-extraction Separations Processes Used. 6.37 After removal from a reactor, the fuel is processed to recover fertile and fissionable materials and to separate these from one another and accompanying fission products. The two requirements of a separations plant are high product recovery ( >99 per cent) Fission-product concentrations relative to both uranium and high product purity. and plutonium must be reduced by factors of 10' to 10*. The solvent-extraction step of a fuel-processing cycle follows dissolution of the cooled fuel (Art. 7.3) and adjustment of concentrations, salting strength, and oxida tion state to give a suitable feed solution. The basis of the separation of uranium and plutonium from each other and fission products lies in the preferential solubility of The specific these materials in various counterflowing organic and aqueous streams. processes for which solvent extraction is used vary chiefly in (1) materials being (5-INCH DIAMETER ORGANIC

COLUMNS PACKED WITH j-x-^-INCH RASCHIG RINGS) SOLVENT: METHYL ISOBUTYL KETONE

1

1


5 2

z

< =>

p

10

I

1

STRIPPING (ORGANIC FILM BASIS)

.

\

\

2.0

1

1

\

z

1

FLOODING POINT FOR STRIPPING

Ul UJ 3.0

<

I

J

,

EXTRACTING

/^(AQUEOUS BASIS)

FILM

1

FLOODING POINT FOR EXTRACTION

0 Fio.

27.

1

1 1 1 1 1200 1600 2000 2400 FLOW, GAL OF BOTH PHASES/(HR)(FT2)

400

1

800

1

2800

Effect of flow rate on efficiency of packed columns for various types of

operations.

separated, e.g., fission products, uranium, thorium, and plutonium; (2) dissolution (3) extracting solvent; (4) salting agent; (5) contacting method. A typical solvent-extraction scheme is shown in Fig. 29. In the first contactor, uranium and plutonium are extracted into an organic phase while the fission prod ucts largely remain behind in the aqueous phase. Additional separation of these elements from fission products is effected in the upper portion of the contactor, where The organic-product the organic extract is scrubbed with an aqueous scrub solution. extract is then fed to the middle of the second contactor, where it passes counterThe current to an aqueous solution containing a reducing agent (ferrous sulfamate). plutonium is reduced to the trivalent (solvent-insoluble) state and is stripped into the Any uranium that is extracted into the aqueous phase is scrubbed aqueous solution. Finally, the uranium is stripped into water from the back by an organic solution. solvent in a third contactor called the stripping unit. The aqueous solutions of uranium and plutonium may then be put separately through additional solvent-extraction cycles if additional purification is desired. Generally, fission products are removed by factors of 10s to 10' in the first cycle. scheme;

Art.

BASIC

5]

METHODS

COLUMN:

ORGANIC

SOLVENT:

OF SEPARATIONS

11-81

3-NCH DIAMETER, PLATES SPACED 2 INCHES APART. PLATE HOLE DIAMETER, 1/8 INCH, PERFORATED . AREA, 23 PER CENT 12.5 VOL PER CENT TBP N KEROSENETYPE HYDROCARBON (O.I M N BOTH URANYL NITRATE AND NITRIC ACID

FOR STRIPPNG STUDES) PHASE: EXTRACTION STUDES: 0.18 M URANYL NITRATE 33 M SODIUM SALTS 26 M NITRIC ACID

AQUEOUS

STRIPPING STUDIES:

0.01 M NITRIC

ACID

/

•STRPPNG (ORGANIC FILM BASIS)

15

1.0

EXTRACTION (AQUEOUS

FILM BASIS)-"^


PULSE AMPLITUDE 0.5

(IN) 1 1 1

1 1000 FLOW, GAL OF BOTH

0

Flo.

500

28. Effect of flow rate on efficiency PLUTONIUM

AQUEOUS SCRUB SOLUTION

AQ

ORGANIC

PRODUCT

SOLUTION

I 1500

FREQUENCY (CYCLES/MIN)

~60

70

I

2000 PHASES/(HR)(FT?)

2500

of pulsed perforated plate columns. REDUCING

ORGANIC RAFFINATE TO PURI FICATION AND RECYCLE

ORGANIC

_ U PRODUCT

U, Pu

FEED

U. PU. AND

FISSION

-A CONTACTOR

-B

-C CONTACTOR

CONTACTOR

PRODUCTS AS

NITRATES AO WASTE

FISSION

PRODUCTS

SOLVENT EXTRACTANT

_ORG

PLUTONIUM PRODUCT (AQ)

AQUEOUS PRODUCT

U

ORGANIC FEED

29. Typical flrst-cycle solvent-extraction scheme for separating uranium, plutonium, and fission products.

Fio.

Solution

adjustments

are required between cycles (concentration,

acidity, etc., and,

in the case of plutonium, an oxidation step to put it in the organic-extractable, tetravalent or hexavalent state). The additional cycles are similar to the first except that only two contacting units are necessary (the extraction-scrubbing unit and the strip This would also be the case if only one element (uranium or plutonium) ping unit). were present in the initial feed solution.


11-82

[SEC.

11


Two principal processes based on the foregoing flow sheet are called Redox and Purex." In the Redox process, hexone (methyl isobutyl ketone) is employed as the

organic solvent and aluminum nitrate as the salting agent; in the Purex process, tributyl phosphate diluted with kerosene is employed as solvent and nitric acid as salting agent. Redox Process. Acidic and acid-deficient flow sheets have been devised for the Redox process. Acid-deficient conditions are produced by neutralizing (generally with sodium hydroxide) a portion of the normal acidity resulting from the hydrolysis Maximum plutonium extractability into hexone of uranyl and aluminum nitrates. Sodium dichromate has is realized when plutonium is in the +6 state (plutonyl ion). In the second column, plutonium is reduced been found effective for this purpose. to the trivalent (solvent-insoluble) state. The The aluminum nitrate salting agent is introduced in the scrub solution. presence of aluminum nitrate increases the distribution of uranium and plutonium into the hexone. This is particularly important in the lower end of the extraction section where the salting strength in the aqueous stream provided by the uranyl nitrate itself has been largely depleted by the mass transfer of uranium to the solvent. Packed columns have given good performance with this process system and are used throughout. Purex Process. The tributyl phosphate solvent and kerosene diluent have high chemical stabilities, so that nitric acid alone can be employed as the salting agent. Plutonium is best extracted when in the tetravalent state, which is produced by treat ment with nitrite ion. As in the Redox process, a reductant is used to reduce plu Mixer-settlera or tonium to the trivalent state for stripping into an aqueous phase. pulse columns are generally used as the contactors for this process. Typical decontamination data for these two solvent-extraction processes are given in Table 22. It may be seen that high fission-product decontamination is achieved with either process. The solvent-extraction operations are sometimes preceded or Such followed by auxiliary steps to augment the purification from fission products. steps include scavenging with a precipitate, ion exchange, and volatilization. Table 22.

Typical Decontamination Effected in Solvent-extraction Separations Processes Type of process

Redox

Variable

Uranium: Beta DF Plutonium:

Purex

1st cycle

2d cycle

Over all

1st cycle

2d cycle

Over all

10"

10"

10" 10*<

10" 10"

10" 10"

10" 10"

10" 10"

10" 10"

10" 10"

10" 10"

10'-'

I0M

10" 10"

10"

I0»

The Thorex process67 was developed for separating and decon Thorex Process. taminating thorium, uranium 233, and protactinium 233 from neutron-irradiated As in the Purex process, tributyl phosphate diluted with a kerosene diluent thorium. is employed to extract uranium 233 and thorium from a thorium nitrate solution. In a second column, the thorium is Protactinium remains in the aqueous phase. stripped from the organic phase by a dilute nitric acid solution, and in a third, the uranium 233 is stripped into water. The uranium 233 is subsequently further decon taminated and concentrated by ion exchange. Solvent-extraction processes are also used on enriched uranium fuels. Since no plutonium is present in these fuels, the second column in Fig. 29 is not needed.

Art.

BASIC

5]

METHODS

OF SEPARATIONS

11-83

The design of chemical processing plants based on solvent-extraction discussed in detail in Art. 9. Fractional Distillation

6.4

is,

Fractional distillation is another separative method used in fuel-processing opera If two substances tions. Specific applications are discussed beginning in Art. 5.44. possess at any temperature a marked difference in vapor pressures, it is possible to The basic make this fact the basis of a separation between them by using distillation. requirement for separation of components by distillation is that the composition of the vapor be different from that of the liquid from which it is formed. If all or part of the vapor is condensed, there will have been an enrichment of the more volatile component in the overhead and of the less volatile component in the undistilled liquid. If successive distillations are made, the enrich ment may be increased until the desired It is not theo degree of purity is obtained. retically possible to obtain absolutely pure components by this process, but in practice very high purities are often economically possible. Equilibria. 6.41 One of Vapor-liquid the basic data needed in a distillation problem is the liquid-vapor equilibria of the components over the composition range of interest. This discussion will be confined to binary systems for illustrative purposes. relationships equilibrium Quantitative are often complex and frequently must be determined experimentally. However, two generalizations can be made that hold in certain ideal cases and are approached, particularly over narrow ranges, in many others. Raoull's law states that for two miscible liquids the partial pressure of a component at a given temperature will equal the vapor pressure of that component in the pure state at the same temperature times the mole fraction of the component in the liquid phase. That = PaXa (28) Pa Pb = PbXb

B

is

is

a

is

it

is

is

is

A

B A

B

A

where pa, Pb = partial pressures of components and Pa, Pb = vapor pressures of pure and and in the liquid xa, xb = mole fractions of These relationships are shown by the dotted lines in Fig. 30. Raoult's law tends to hold when the components have similar molecular sizes and mix without molecular association or chemical combination. the exception and Consequently the rule deviations both positive and negative are common. In any case the actual partialpressure curve for either component generally tangent to the Raoult's law line for the component that often stated that Raoult's law applies nearly pure, so that to the solvent. A somewhat analogous statement Henry's law, which states that the partial dilute component pressure of proportional to its mole fraction in the liquid, or pA =

K

Kxa

Pb = Kxb

(29)

an experimentally determined constant. 30 show the nonideal behavior of system such as carbon disulfide-acetone. The areas in which Raoult's and Henry's laws hold are indicated. Vapor-liquid equilibria are presented in any of several ways. Figure 30 indicates

The solid lines in Fig.

a

where

is


methods is


11-84

[SEC.

11

the partial pressures of both components and the total pressure of the mixture as a function of composition at a single temperature. Figure 31 illustrates a normalboiling-point diagram wherein the equilibrium of liquid and vapor is shown as a function of composition at a single pressure (generally 1 atm). Figure 32 is illustrative of the representation of the equilibrium constant (K = y/x) as a function of tempera ture and pressure. This method is most commonly used in presenting hydrocarbon data. The most common representation is the equilibrium diagram illustrated in Finally Fig. 34 illustrates an enthalpy-concentration chart wherein the Fig. 33. heat content of liquid and vapor is given as a function of composition


Fio.

31. Normal boiling-point diagram stant pressure).

0

Fio.

32.

Equilibrium constants

of temperature

as function

and pressure.

IjO 0 05 MOLE FRACTION MORE VOLATILE COMPONENT IN LIQUID OR VAPOR

10

0.5 MOLE FRACTION MORE VOLATILE IN LIQUID, x COMPONENT 33.

Fio.

(con-

Fio.

Equilibrium diagram.

34.

Enthalpy-concentration chart.

The volatility of a substance is defined as its partial pressure in the vapor in librium with the liquid divided by its mole fraction in that liquid; that is, Fa

=

— XA

equi

(30)

When used in regard to a pure substance, the term has a value equal to its vapor pressure. The relative volatility a of two substances is the ratio of the two individual volatilities: a = VA = HZ* Vbxjl Vb and since

M =lA ya

a =

Vb

y-^ VBXA

pi) (32)

(33)

Art.

BASIC

5]

METHODS

OF SEPARATIONS

11-85

where y is the mole fraction of the component in the vapor. Relative volatility is a measure of the ease of separation of components by distillation. There are several ways in which separations by distillation may be effected. They fall into two general classes: in the first none of the condensed overhead vapor is allowed to return to the still ; in the second some of the condensate is returned to the still in such a way that the rising vapors and returning liquid are brought into intimate contact one or more times. The first method is usually called simple distillation, and the latter method is called rectification or fractional distillation. There are several important methods in the category 6.42 Simple Distillation. of simple distillation. Equilibrium Distillation. Equilibrium distillation involves vaporizing a definite fraction of a batch of liquid, maintaining liquid and vapor in contact with one another The vapor is then removed and condensed. The com until the end of the process. positions of liquid and vapor at the conclusion of the equilibrium distillation are therefore represented by a point on the equilibrium curve. In a binary mixture of A and B, consider F moles of feed of composition xr. At the conclusion of the distillation, V moles of vapor are taken overhead. A material balance on the more volatile component A gives Fx, = Vy + (F 7)i (34)


-

This equation contains two unknowns, x and y. The equilibrium diagram provides the other needed relationship between x and y. By combining the two, it is possible to determine the final liquid and vapor compositions as a function of the per cent vaporized. Differential distillation is a method wherein the vapor Differential Distillation. from a batch of liquid is continuously removed as fast as it forms, the volume of liquid The compositions of liquid and vapor will in the still decreasing continuously. This change continuously; consequently, the analysis must be done differentially. was first done by Lord Rayleigh, and the mathematical solutions for this case are known by his name. A differential material balance over a short time period will give where

L

-ydL —

d(Lx)

moles of liquid

dL —

or Integrating Eq.

L

.„..

dx

= y

-

(35)

x

(36)

(36) between the initial and final conditions,

In

£-/--*Jit L*

y — x

(37)

If the equilibrium relationship between y and x is known mathematically, the value y = f(x) may be put into Eq. (37) and an analytical solution may result. Since the relationship between y and x is generally known only graphically, Eq. (37) is usually solved graphically by determining the area under the curve l/(y — x) versus x between the desired limits. The vapors produced by either of the foregoing methods may be further enriched either by redistillation or by partial condensation. Steam distillation is a method sometimes employed, particu Steam Distillation. larly on organic materials whose boiling points are sufficiently high that decomposition may occur if a direct atmospheric distillation is made. Steam is therefore introduced into the still in excess, and advantage is taken of the fact that when liquids are Thus, the introduced steam reduces immiscible each exerts its full vapor pressure. the partial pressure of the high-boiling component necessary for distillation. When the gas laws apply, the weight ratio of the two components in the vapor is equal to the product of the molal ratio and the molecular weight ratio:

11-86

CHEMISTRY AND CHEMICAL ENGINEERING Wa

Ma

= Va_

Ww

Mw

yw

_

[Sec.

11

Ma

Pa_

pw Mw

and for a binary mixture of total pressure r, Wa Ww

_

r

MA

-Papa Mw

(39

As long as there are two liquid layers in the still, x — px must equal the vapor of the water and the temperature then becomes fixed at a value such that This results in a Pa plus the vapor pressure of water equals the total pressure. definite lowering of the temperature at which component A may be taken overhead. 6.43 Rectification, or fractional distillation, is a process wherein Rectification. vapors rising from a still are successively contacted with condensate of part of the An equilibrium curve will show that liquid formed by vapor previously distilled. partial condensation of a vapor must be richer in the more volatile component than Therefore, this condensate cannot the liquid from which it was originally distilled. be in equilibrium with the vapor, and when they are mixed some interaction must take place. The result is that some of the more volatile component in the liquid is Thus vaporized and some of the less volatile component in the vapor is condensed. an enrichment of the vapor takes place. It can be arranged to make a series of these contacts passing liquid and vapor countercurrent to one another. At each stage further enrichment takes place until the desired degree of separation has been obtained. Rectification is carried out in packed towers, bubble-cap columns, or sieve-plate The function of each of these is to provide intimate contact between liquid columns. and vapor and to allow countercurrent flow of the two streams. A great deal has been written about methods of calculation for rectifying columns. The subject is covered in detail in the general references to this article. The per formance of any actual column is best understood and analyzed in terms of an ideal column made up of theoretical plates. A theoretical plate is defined as one in which the The vapor rising from it and the liquid passing down from it are in equilibrium. degree of ideality for any actual plate must be determined experimentally. Two common methods of calculation are those of McCabe-Thiele and of PonckonSavarit.** The former is the more frequently used and will be described herein. The McCabe-Thiele method is a relatively simple graphical approach. Consider the still and rectifying column, shown schematically in Fig. 35, operating upon s system whose equilibrium relationship is indicated in Fig. 36. The first step in the calculation is to make heat and material balances. Consider first a single plate, the nth. The over-all material balance is


pressure

+ Vn-t = Ln + K„ and for the more volatile component is Ln+lXn+l + V„_,y„_,

-

LnXn + V»J/«

(40)

(41)

where the subscripts refer to the plate from which the stream is coming. A heat balance (using the temperature of the liquid on the plate as a datum) around this plate would indicate the following items: a = Latent heat in vapor from plate n — 1 6 = Sensible heat in vapor from plate n — 1 c = Sensible heat in liquid from plate n + 1 (below datum plane) d = Heat of mixing e = Latent heat in vapor leaving plate n = Heat losses

/

The heat balance is now a

+b

/

-

c

+d

-

e

+f

(42)

Ordinarily, the effects of 6, c, d, and are combined and disregarded. In so doing, the assumption is made that the vapor traveling up the column carries a constant quantity

Art.

BASIC METHODS

5]

11-87

OF SEPARATIONS

of

heat. This assumption is then combined with Trouton's rule, which states empiri that for large numbers of chemically similar liquids the molal latent heat divided by the absolute temperature at the boiling point is constant. This being so, V„_i may be taken equal to V» and, similarly, LB+i = L„ and the subscripts on V and L may be dropped.

cally

CONDENSER c; COOLING

00

-D„„

n + l

n+2

Vn,

n

1X"

Lotl.»n+l

V„ yn-i

n-l COLUMN

m+l +

Lm I,"mtl

STILL

HEAT

w '«w

=mole froclion of more volatile component liquid

QQtQ0

■mole fraction of more volatile component vopor

* heat added (or removed) to still (or from condenser)

LR" moles reflux returned

in

y

in

x

:

Legend

to column per unit time

=

L,L

F ■

D* moles overhead distillate withdrawn per unit time

V,V* moles vapor passing up

moles feed per unit time

column above and below feed point

W* moles of bottoms per unit time

diagram

of distillation column.

are

F

The over-all balances

35. Schematic

=

W

+

Fia.

moles liquid passing down column above and below feed point

D

Fxr = Wxw + Dxd Fhr +Qb =Qd + DhD +

(43) (44)

Whw

•» enthalpy of liquid where and the remaining quantities are defined in Fig. 35. Material balances may also be written around various sections. denser and the nth plate there obtained

(45)

is

h


FEED-

Around the con

ENGINEERING


11-88

V

Vy.-i Combining Eqs.

[SEC.

L+D

=

Li,

=

(46) and (47),

L +D

(46)

+ DxD

x, +

11

D

L+D

(47)

ID

(48)


If it is assumed that the plates in the column are perfect, the vapor and liquid Therefore, the point (x„,yn) will lie on the leaving any plate will be in equilibrium. Equation (48) represents the operating line. It has the slope equilibrium curve. L/(L + D), and since the vapor from the top plate has the same composition as

0

Fig.

"•

"F MOLE FRACTION OF MORE VOLATILE COMPONENT IN LIQUID, x

36. Equilibrium

diagram illustrating McCabe-Thiele

method

*0

l

of calculation.

yr = xd, the line must pass through the point (xd,Vt), which is point B in Fig. 36. The number of theoretical plates in the portion of the column above the feed intro

Xd, i.e.,

duction point (called the rectifying section) may be determined by stepping off the plates between the equilibrium line and the operating line, illustrated bv BGHJD in

Fig. 36. At the

feed plate a new stream is added, so a new material balance and operating line are required. Material balances around the bottom of the column give

V

Vy„-\

=L

-W

= Lxm

—

(49)

Wxa

(50

where V and L refer to vapor and liquid below the feed point. Combining Eqs. (49) and (50), ym-i =

L

W

Wx„ W

(51)

Art.

BASIC

5]

OF SEPARATIONS

METHODS

11-89

Equation (51) represents the operating line for the portion of the column below the It has the slope L/(L — W), and since the feed plate (called the stripping section). vapor leaving the still is in equilibrium with xw, the operating line must pass through point (y = xw, x = xw), which is the point E in Fig. 36. The lower operating line ED is therefore denned, and the step-wise process may be continued as illustrated by

DKLMNP.

The total number of steps (~4.7 in the example) represents the number of theoretical plates required for the desired separation of a feed of composition xr The actual number into overhead of composition xd and bottoms of composition xw. of plates is obtained by dividing the number of theoretical plates by the plate effi ciency, which must be determined experimentally. The line FC, formed by connecting the point F (i = xf, y = xf) with the point C (the intersection of both operating lines), represents the heat condition of the feed. It may also be determined by writing heat and material balances around the feed Doing so results in the equation

plate.

y~^L-x-^g-1 q

where

q is

the total heat needed to convert

by the molal latent


1

1

(52)

mole of feed into saturated vapor divided

heat.

The minimum number of plates required for the separation is obtained when all the Under these conditions overhead condensate is returned to the column (total reflux). It is represented, therefore, by the 45° line the slope of the operating line is unity. ABFEO. The minimum number of plates is obtained by stepping off between the equilibrium line and the 45° line. The minimum reflux that may be used is obtained from the slope of the operating line obtained by connecting the point B with the intersection of the feed line FC and the equilibrium line (i.e., the line BCt). When the assumption of constant molal latent heat is not permissible, it is some times possible to choose a fictitious molecular weight for one of the components such that Trouton's rule will hold. The equilibrium data are then recalculated on the fictitious basis, and this method then applied. Examples of the use of distillation in the 5.44 Applications to Atomic Energy. field of atomic energy are (1) the recovery and decontamination of solvent from solvent-extraction processes, (2) the recovery and decontamination of nitric acid from aqueous waste streams of solvent-extraction processes, and (3) the processing of irradiated uranium by fractional distillation of UF6 from a fluoride mixture. Much of the irradiated uranium processed in this country today Solvent Recovery. is handled by one of several solvent-extraction processes (see Art. 5.3). The used solvent has generally suffered some degradation from irradiation and contains radio activity which should be removed before reusing the solvent. It is often possible to remove much of the activity and some of the degradation products by simply washing the solvent with water, weak caustic, or soda ash. For more complete cleanup, particularly from degradation products, distillation is generally employed. While simple distillation has been used, steam distillation is the most commonly employed method. Rectification is seldom required. Virtually all separations processes employed to date com Nitric Acid Recovery. mence with the dissolution of uranium in nitric acid. Some of the solvent-extraction processes also use nitric acid as a salting agent. This acid, which amounts to a signifi cant cost item, is recoverable from the aqueous waste streams and from the gaseous products of the denitration of uranyl nitrate, in which step U02(NOi)2 is converted to uranium trioxide, U03. In the latter case the oxides of nitrogen are absorbed in a scrubber in the manner conventionally used to manufacture nitric acid. To recover the acid from the waste streams, distillation is employed. A typical flow sheet is shown in Fig. 37. Therefore, operating line for minimum reflux may intersect the equilibrium line only once. the case of systems with irregular equilibrium curves, it is often neceBsary to use an operating line tangent to the equilibrium line rather than the one defined here.

in

f The


11-90

[SEC.

11

The com Separations Processes Based on the Volatility of Uranium Hexafluoride. pound uranium hexafluoride, UF6, is easily volatilized, and work is in progress directed toward the development of processes that exploit this fact. None has yet reached the plant stage, but some pilot-plant work has been done. Uranium hexafluoride can be prepared by treating uranium with fluorine or, more cheaply, with hydrogen fluoride to produce UF4 and then fluorinating this with These are solid-gas reactions which are difficult to control. fluorine to give UFe. Other fluorination methods have been devised using interhalogen compounds such as bromine trifluoride, BrFj, or chlorine trifluoride, C1F3. Some of the properties of TJF« and other halogen compounds are given in Table 23. -WATER

I

RECTIFYING COLUMN

FEED

'I

M HN03

SALTS

FISSION


PRODUCTS

STILL »8 M HNOjl

..RECOVERED

I

ACID

_ FISSION

PRODUCTS AND SALTS TO WASTE

Fig.

37. Schematic

flow sheet for nitric acid recovery

from aqueous

wastes.

A schematic flow sheet of a possible process is shown in Fig. 38. Decanned uranium slugs are dissolved in boiling BrFj (~120 to 130°C) and the products of the dissolution, UF6 and Bra plus some BrF8, are taken overhead through a packed column and dephlegmator. The overhead is passed through a converter, where Fj is used to reconvert the Br2 to BrF3. The gas is then passed to one of a series of condensercollectors maintained below 64°C where the UF« condenses as a solid and BrFj drainout as a liquid and is returned to the dissolver column. Table 23.

Some Properties of UF8 and Other Halogen Compounds UFi

Molecular weight. . . . Boiling point, °C. . . . Melting point, °C . . .

352.07 56. 5« 64. 8 64. 8t

Density, K/cm': Liquid at °C 3.63/64 Solid at °C 4.85/64 Critical temperature, °C 217-249} Critical pressure, atra 44-63} * Sublimation point, t 1.135 mm. X Estimated.

F.

-

38 00 187.92

-217.96

BrF, 136.92 125.72

8.77

ClFi 92. 46 11.75

-76.32

BrFt

Br,

174.92 41.28

159.83

-61.3

58.78

-7. J

HF 20 01 19 5«

-«J.«7

1. 108/- 188 2. 848/8. £ 1. 85/11. 75 2. 40/40 0 957.'» 3.00/59 1. 2/— 218 3. 09/— 61 3.40/-7.3 l.637/-« 3.73/8.8

- 129 55

339}

69:

150}

213} «7}

311 102 ± 1

230.2

-

Art.

BASIC

5]

METHODS

OF SEPARATIONS

11-91

Inert gases and low-boiling fission products (for example, TeF«) are taken off the When the collector is full, flow is switched to another unit, the top of the collector. temperature raised, and the UF8 plus some BrF3 is melted out. The first 10 per cent The remaining 90 per cent is taken to a purification still is recycled to the dissolver. If additional product purity is required, this last where the UF» is taken overhead. step may be repeated. This process suffers from the fact that a volatile plutonium fluoride is not obtained and so the plutonium must be removed from the dissolver residue with an aqueous This adversely affects the economics of wash and recovered by solvent extraction. the process. In addition the intercycling of aqueous and interhalogen phases is timeconsuming and potentially hazardous. Therefore, the process is most likely to become of interest for highly enriched uranium fuels where the plutonium problem is absent. Unfortunately, many enriched fuels are prepared with alloying agents that are not amenable to low-temperature fluorination procedures including interhalogen dissolu tion. Some work is being done in which alloyed fuels are dissolved in fused salt mixtures at elevated temperatures (500 to 700°C) and the uranium converted to UF« Br2,UF6,BrFs,


SOME FISSION PRODUCTS

SOME BfFj, FISSION PRODUCTS

-fission products Plutonium

BrF, RECYCLE BrFj FISSION PRODUCT BLEED TO WASTE

Fio.

38. Schematic

with HF plus Fs, BrFj, or BrFs. tion for further purification. Much additional information 7534, "Symposium

flow sheet for possible volatility process.

The overhead vapor can then be treated by distilla

on these processes has been made available in on Fuel Reprocessing," Brussels, Belgium, May, 1957. 6.5

TID-

Ion Exchange

The property of certain solids of adsorbing ions from solution and at the same time This property has long been recog releasing a different ion is called ion exchange. nized in soils and various silicates. More recently organic polymers such as the sulfonic acid resins and the polyamine-type resins have been synthesized. These have much higher capacities than any naturally occurring materials known. Ion-exchange resins have been widely used in the production of deionized or softened water and in the processing of sugar. They have been used in the nuclear-energy industry to concentrate and purify solutions of heavy-element products, to isolate rare earth and other elements, to recover and separate specific fission products, to treat some radioactive waste streams, and to purify water for reactor cooling and for process uses. 6.61 The theory of ion exchange is not yet completely understood or Theory. even agreed upon by all workers. A complete treatment is beyond the scope of this work, and the reader is referred to the text by Kunin and Myers."


11-92


[SEC.

11

Three explanations proposed are (1) the crystal-lattice exchange theory, (2) the double-layer theory, and (3) the Donnan membrane theory. In the crystal-lattice concept, the constituents of the exchanger are considered to be present in the lattice structure as ions. Each ion is surrounded by a fixed number of ions of opposite charge. It follows that the attractive forces on ions near the sur face are less than those in the interior. If the exchanger is placed in a polar medium such as water, these forces may be diminished to such an extent that the exchange of an ion in the exchanger with one in solution becomes possible. In this view the event To account for is somewhat analogous to the mixing of dissimilar ions in solution. the high exchange capacity of many substances, it is also necessary to assume that they are sufficiently porous to allow the ambient solution to diffuse into the interior of the solid. The double-layer theory is based on the classical work of Helmholtz on the electrokinetic behavior of colloids. It assumes that the exchange particle is surrounded by an inner fixed layer of electrical charges of one sign and this in turn is surrounded by an outer diffuse layer of charges of the opposite sign whose concentration changecontinuously until it reaches the equilibrium concentration of the ambient liquid. These layers are due to the adsorption of ions different from those in the inner portion If the concentration of ions in the ambient liquid is changed, the of the particle. concentration of ions in the diffuse layer will have to change until a new equilibrium is established. The third explanation is an extension of the Donnan membrane theory. This assumes an unequal distribution of ions on either side of a membrane one side of which contains an electrolyte one of whose ions is not able to diffuse through the membrane. It may be easily shown mathematically that those ions which can diffuse through the membrane will redistribute themselves so that the concentration ratios, or more Tht properly the activity ratios, will be the same on both sides of the membrane. To extend this to ion exchange it is assumed has been shown to hold for membranes. that the colloidal micelle to which the exchangeable ion is attached is a nondiffusible ion, and the interface between solid and liquid is taken to be the membrane. All three theories are similar in that the exchange must satisfy the law of electroThe difference lies in the explanation of the position and origin of the neutrality. These sites are ionic groupings capable of forming an electrostatif exchange sites. The laws governing this heterogeneous exchange bond with an ion of opposite charge. are analogous to those governing solutions of electrolytes. Until a more rigorous and useful quantitative relationship can be established, the following set of empirical rules can be utilized effectively in formulating certain approximations: (aqueous) and ordinary temperatures, the extent of 1. At low concentrations exchange increases with increasing valency of the exchanging ion (Na+ < Ca~* < A1+3

< Th+4).

At low concentrations (aqueous), ordinary temperatures, and constant

valence, increases with increasing atomic number of the exchanging ion (Li < Na < K < Rb < Cs; Mg < Ca < Sr < Ba). 3. At high concentrations, the differences in the exchange "potentials" of ions of different valence (Na+ versus Ca+1) diminish and, in some cases, the ion of lower valence has the higher exchange potential. 2.

the extent of exchange

the 4. At high temperatures, in nonaqueous media, or at high concentrations, exchange potentials of the ions of similar valence do not increase with increasing atomic number but are very similar or even decrease. 5. The relative exchange potentials of various ions may be approximated from their activity coefficients — the higher the activity coefficient, the greater the exchange

potentials.

6. The exchange potential of the hydrogen and hydroxyl ions varies considerably with the nature of the functional group and depends upon the strength of the acid or base formed between the functional group and either the hydroxyl or hydrogen ion. The stronger the acid or base, the lower the exchange potential.

Art.

BASIC METHODS

5]

OF SEPARATIONS

11-93

5.62 Types of Resins. Ionic-exchange resins are high-molecular-weight polymers Anion containing particular ionic groupings as an integral part of the structure. exchangers contain amine groups with an equivalent amount of an anion such as chloride or hydroxyl ion. Cation-exchange resins contain phenolic, sulfonic, car boxylic, or phosphonic acid groups with an equivalent amount of a cation such as sodium or hydrogen ion. The polymeric structure is sufficiently cross linked to A partial list of commercially available resins is given render it virtually insoluble.

in Table

24.

Table 24.

Characteristics of Some Commercially Ion-exchange Materials

Available

Total capacity Name

Type

Manufacturer

Milli-

Milli-

equivalents/

equivalents/

g

ml

Cation Exchangers Amberlite IR-100 Amberlite IR-105 Don-ex

30


(Nalcite MX) Duolite C-3

...

Rohm and Haas Rohm and Haas

Dow Chemical Chemical Process American Cyanamid Wofatit P I. G. Farben Wofatit K I. G. Farben Wofatit KS I. G. Farben Permutit Permutit Amberlite IR-120.. . . Rohm and Haas Dowcx 50 (Nalcite HCR) .... Dow Chemical Research Products Amberlite IRC- 50. . . . Rohm and Haas Chemical Process Duolite CS-100 Permutit Permutit 216 I. G. Farben Wofatit C

Permutit Permutit Permutit

Phenolic methylene Phenolic methylene

sulfonic sulfonic

Phenolic methylene Phenolic methylene Phenolic methylene Phenolic methylene Phenolic methylene Phenolic methylene Sulfonated coal Phenolic methylene Nuclear sulfonic

sulfonic sulfonic sulfonic sulfonic sulfonic sulfonic

4.00 3.25 2.70

sulfonic

2.70 4.20

Nuclear sulfonic Carboxylic Carboxylic Carboxylic Carboxylic Carboxylic Aluminum silicate Aluminum silicate Aluminum silicate Aluminum silicate Aluminum silicate Aluminum silicate Silicic acid

1.75

2.70

1.35

2.50 2.45 1.62

4.25 4.95 10.0

3.85 5.30 7.00

0.8-1.2

0.65 1.00 1.35 1 00 0 81

0.53 1.00

0.90 0.60 0.89 2. 15

2.20 1.80

4.20 1.11 1.70

2.50

0.06-0.10

0. 18-0.2

1.0-3.0

0.01-0.04

Anion Exchangers Rohm and Haas Amberlite IR4B Rohm and Haas Amberlite IR-45 Amberlite IRA-410. . Rohm and Haas Amberlite IRA-400. . Rohm and Haas Permutit Chemical Process Chemical Process Duolite A- 3 American Cyanamid I. G. Farben Wofatit M Dowex 2 (Nalcite SAR)

Dow Chemical Dow Chemical

Source: Reprinted from R. Kunin and R. Inc.. New York, 1950.

Weak base Weak base Strong base Strong base Weak base Weak base Weak base Intermediate base Weak base Amphoteric Strong base Strong base

J.

10.0

6.0 2.5 2.3 9.3 7.0 6.8 7.4

2.50 2.0 1.0 1.00 1.5 1.20 1. 10 1.50 1.20

0.01

2.3 2.4

0.9 1.0

Myers, "Ion Exchange Resins,' John Wiley & Sons,


11-94

STREAM

ON

REGENERATION

EXHAUSTED

cr

h*

No +2

_L

Ca

CATION RESIN

CATION RESIN

R-H+-

R-No+

R-No+

R-Ca

CATION RESIN

rWI

+2

R-Cat2r R-H'

R-Co+2

7T

No+'Cf

+2 Co'

so; Fia.

so/2

39. Schematic operation of cation exchanger.

No+ Co+2


[SEC.

cr

ON STREAM

so;2

CATION RESIN R-H+— R-No+ R-Co+2

H*

ANION RESIN R'-OH"—

R'-cr r'-so;2

cr so;2 H-0 REGENERATION

EXHAUSTED

CATION RESIN R-No* R-Co+z

ANION RESIN

r'-ci", R'-SO,"2

HT.CI"

CATION RESIN

40. Schematic

OH"

ANION RESIN

R-Nd

R'-CI"

R-Co<

R'-SO;2

H R-H+

Na+ CI" Co+2

Fia.

No'

R'-OH"

No+

cr

}-

so«-* operation of dual-bed deionization.

11

Art.

BASIC

5]

11-95

OP SEPARATIONS

METHODS

6.53 Methods of Use. If it is desired to remove only cations from a solution, it is This operation will necessary to contact the solution with a cation resin only. The resin is regenerated by passing acid remove the cation and lower the pH. through it, followed by a water rinse. This is shown schematically in Fig. 39. Anions alone may be removed by the use of an anion exchanger. The effluent in

this case becomes more basic. ON

STREAM

No+ Co*2

CI~-2

EXHAUSTED

SO,2

MIXED

MIXED

RESIN

RESIN

R-H+

R-Co«

r'-oh;

R-Na* R'-CI"


Rd-No* R'-CI" r-co*2 r'-scSt

H2°

r'-so;2

REGENERATION No*

I OH"

+ +2 No, Co .

Fig.

41. Schematic

operation

of mixed-bed

CI.

-98

SO«

exchanger.

It

is possible to obtain complete deionization by passing the solution through both The con is accom plished by passing acid through the cation bed and base through the anion bed and

an anion and a cation resin bed. This may be done using either bed first. ventional scheme for deionizing water is shown in Fig. 40. Regeneration rinsing.

Complete deionization may also be accomplished by passing the solution through a bed of mixed anion and cation resins. Since the capacity of anion resins is about half that of cation resins, it is customary to employ 2 parts anion to 1 part cation resin. This operation is shown schematically in Fig. 41. To regenerate a mixed bed it is necessary first to classify the bed by backwashing with water, then to introduce acid into the cation portion and base into the anion portion of the bed and to follow this by a water rinse. Finally, the bed is remixed by blowing air through it. Any of these three methods is usually accomplished with the selected resin con tained in cylindrical beds. The bed should be not less than 30 in. deep, and manu Regenerfacturers generally recommend throughput rates no greater than 2 gpm/ft*. ant solutions generally employed are 6 N HC1 and 4 per cent NaOH.


11-96


[SEC.

11

The use of ion exchange in this field 5.54 Application to Separations Processes. An important use, however, has been as an adjunct to has been somewhat limited. Resins have also been used to isolate specific fission other separations processes. products from mixed fission-product activity, and they also have had several applica tions in the disposal of radioactive wastes. Their use in processing highly radioactive materials has been limited by their susceptibility to radiation damage and in waste processing by the relatively low concentration factor that they provide. One of the most important applications of ion-exchange methods has been as an Ion exchange has auxiliary step with other basic types of separations processes. proved to be a very effective method for concentrating solutions of plutonium, enriched Um, and U2*3. Comparative studies which have been conducted on various ways to concentrate intermediate or final product streams have resulted in the con First, clusion that ion exchange is superior to evaporation or precipitation schemes. it is a more convenient method than evaporation for accomplishing large volume Second, it is less likely to introduce impurities, such as corrosion reduction factors. products (in the case of evaporation methods), and added chemical reagents (in the Third, it frequently achieves significant additional case of precipitation methods). In some instances, the amount of purification is enough to enable purification. elimination of a complete cycle of solvent extraction. Concentration factors in excess of 100-fold have been demonstrated with simple ion-exchange operations on plutonium and other heavy element-containing solutions. At the same time, 2- to 10-fold (or more) decontamination from zirconium and niobium Some ruthenium removal is also effected. Organic exchangers is often obtained. have been most often used for this purpose. Silica gel, however, has been very successfully used to clean up concentrated uranium solutions from solvent-extraction The losses of valuable material have been negligible in applying ion processes. exchange to concentration or added purification steps. Cohn,60 Tompkins,*1 and Boyd"-'4 and their coworkers have done considerable work on the adsorption of mixed fission products on ion-exchange columns followed The mixed activity was generally by the selective elution of specific fission products. adsorbed from 0.1 to 0.2 M hydrochloric acid solutions onto a phenolic methylene sulfonic acid cation resin. Then, with the use of successive elutriants of oxalic, citric, or tartaric acid whose pH was adjusted with ammonia or hydrochloric acid, it was Purities as high as 98 per cent possible to obtain fractions of a specific fission product. for a particular fission product were obtained in a single elutriant fraction with a volume-reduction factor of 50 to 100. This process has not been used on full process levels of activity but has been used to supply specific fission products for distribution for research purposes. Because of the simplicity of ion-exchange processing and the possibility of devising a low-cost process, there has always been considerable interest in ion exchange as a Useful applications have turned out to be rela possible method of waste disposal. tively limited, however. This is due, in part, to sensitivity of resins to high levels of radiation but more generally to the fact that most waste streams contain a high concentration of inactive salts which quickly exhaust the resin. Thus, frequent Con regeneration is required and the concentration factor possible is quite limited. sequently, ion exchange is seldom used on a waste stream containing more than Table 25. Performance of Ion-exchange Resins (Basis: Waste containing 300 ppm total solids and mixed fission-product Type of resin

Decontamination factor

Capacity,

~30 4-5

6.000 > 150.000 400

10*

* Resin exhausted

gal/ft>

with regard to hardness ions still removes some activity.

activity)

Concentration factor

150 > 3.000

I

Art.

5]

BASIC

METHODS

OP SEPARATIONS

11-97

1,000 ppm total solids. Table 25 indicates the decontamination factor, capacity, and concentration factor that may be obtained from various types of exchange resins when operating on a waste containing mixed fission product and 300 ppm total solids. A very effective use of ion-exchange resins is as a second step following a saltremoval step such as evaporation. In such a use the capacity of the resin is very large and additional decontamination factors of 10s are often obtainable.

Pyrometallurgical

Processes

Problems of reprocessing fuel discharged from nuclear power reactors may be aggravated by the high fuel burnup and power density envisioned for these reactors. The use of conventional processes, such as solvent extraction, may require extended, and therefore costly, cooling periods of the fuel prior to processing in order to avoid deleterious radiation effects on the solvent. Other disadvantages of conventional processes are (1) large process volumes, (2) large waste volumes, (3) necessity of complete metal-salts-metal cycle, and (4) failure to recover expensive fuel diluents. Research and development work on pyrometallurgical processes was undertaken in an effort to find a suitable way of processing short-cooled, highly irradiated fuels. Ideally, pyrometallurgical processes are those in which the fuel is retained in the metallic state throughout. However, also included in the category of pyrometal lurgical processes are those in which the metallic state may be lost momentarily, such as in passage through an electrolytic molten salt bath, and those in which the metal may be converted to a salt in one step and back to metal in a succeeding step. Because pyrometallurgical processes employ high temperatures, they are also often called high-temperature or pyrochemical processes. Since pyrometallurgical schemes avoid bulk chemical conversion during purification, the usual need for converting uranium or plutonium salts back to metal is avoided. The purified metal, as obtained, is ready for fabrication into new fuel elements. Variations in fuel alloy compositions that sometimes cause difficulties in aqueous processing systems may be of much less concern in pyrometallurgical processes. Finally, the fission products are isolated in a very compact form and, therefore, are more likely to be suitable for utilization or disposal. Processing requirements are different for reactor core and blanket materials. For blanket material, it is considered necessary only to isolate the fissionable material bred therein or to upgrade its concentration to the point where it may be used for reconstituting the core fuel material. The core material must be reprocessed (1) to eliminate irradiation or thermal damage, (2) to restore the fissionable isotope con centration, and (3) to eliminate fission products. The required purification from fission products is considered to be only that neces sary for suitable reduction of fission-product poisoning and for restoration of satis factory physical and metallurgical properties of the fuel. The latter point arises if the presence of fission products adversely affects such properties as radiation stability or various desirable metallurgical properties such as castability and ductility. From a nuclear standpoint, removal of fission products by only a small factor (perhaps 2 to 10) is necessary, and this factor may also be adequate from a metal lurgical standpoint. Development of remote fuel-element refabrication methods therefore, a necessary adjunct for the successful application of a pyrometallurgical It should be noted that, for the high burnups contemplated for power process. reactors, the accumulation of uranium or thorium isotopes (such as uranium 237) or other transuranic isotopes will make some degree of remote fabrication necessary, even if fission products were completely removed. If the reactor were fueled with plu tonium, remote refabrication may be required from the standpoint of the plutonium

is,


6.6

hazard.

Because of the high temperatures employed, volatilization

of some fission products regardless of the basic process step. The vapor pressures of some elements and compounds of interest are given in Table 26. Fission-product elements that would be expected to be volatilized are xenon, krypton, rubidium, cesium, strontium, barium, iodine, tin, cadmium, and antimony.

will occur

11-98


[SEC.

11

The type of 1.

processes under consideration fall into two categories: Those chemical in nature:

Slagging Electrorefining Fused salt extraction Adaption of de Boer iodide process 2. Those physical

in nature:

Liquid metal extraction Volatilization Zone melting Slagging. Slagging may be defined as a process in which impurities in metal are preferentially converted to relatively stable compounds such as oxides or carbides, Those metal impurities which concentrate which are accumulated in a slag layer. 6.61

Vapor Pressure of Some Elements and Compounds at 1600°K

Table 26. Se.

Rb

Kr. Xe

Vapor pressure, mm Hg

Subatance

J. J.

As

Cd Cs Zn

Te


UI. 8r

Sb

Ba

UBn

Sources:

»I0»

3 X I0« 2 X 10' 3.0 X 10* 1.6 X I0« I. I X I0« 1.9 X 10' 3.3 X I0« 2.7 X 10' I I X 10' 4. 2 X 10' 3.0 X I0> < 10

Other elements occurring in fission L. Brewer. "High Temperature Decontamination and Separation Process." UCRL-3H,

1945.

" H. M. Feder, Pyrometallurgical Processing ence on the Peaceful Uses of Atomic Energy,

Table 27.

of Nuclear Metals,"

IX,

Free Energy (Gibbs) Oxide >i La:Oi

X X

CetO.

Nd.Oi

SrO

HFuiOi

X

UOi

X

PuOt BaO NbO

HZrOt

X X

MoO>

TeO. Cs.O

X RuOi

Paper P/544. International United Nations, New York, 1956.

of Formation

- Afo,

of Oxides at 1600

Confer

K

kcal/

g atom oxygen 109.7 107.0 105.7 102.0 99 0

98.9 97.0 95.3 95.0 66 38 4. 7

3.6

-2.7

in the slag layer form compounds of greater stability than those of the metal being Those impurities whose compounds are less stable than those of the metal purified. The source of oxygen and car being purified will remain behind in the metal phase. bon may be that previously present in the metal, the atmosphere, or compounds added in solid or liquid form which are reduced by some of the impurities but not by the The slagging separation process depends on the general physical metal being purified. phenomenon that the mutual solubility of metals and their oxides or carbides is low. There are a few exceptions to this rule; for instance, zirconium oxide is soluble in zirconium. Some pertinent thermodynamic data are given in Tables 27 and 28. Practical application of these data requires some knowledge of the actual species When graphite is Knowledge of the mode of formation is also important. formed. immersed in molten uranium, it would be expected that zirconium carbide would be

Art.


5]

11-99

Molybde preferentially formed and removed provided its solubility is not too large. num and ruthenium would not be removed in such a process. alloy occurs by Considerable purification of uranium or of a uranium-plutonium simply holding the material molten for a period of time in the presence of a limited supply of oxygen (such as under an inert atmosphere in a refractory oxide crucible). In this case, the ceramic oxide crucible may function as a solid slagging agent. Some of the fission products will be converted to oxides and will separate in a slag layer. Plutonium remains in the metal phase, its Some of the others will be volatilized. recovery being essentially the same as the uranium recovery. Physical separation of the bulk metal from the slag can be effected, for example, by The slag layer remains behind in the crucible. bottom pouring the bulk metal. Typical fission-product removals obtained experimentally by the combination of volatilization and oxidative slagging in a ceramic oxide crucible are shown in Table 29. Most of the fission elements are either removed well or essentially not at all. The elements not removed are those on the first peak of the familiar fission-product dis tribution curve (see Fig. 1). They constitute about 35 per cent of the total fission elements initially present, but the deliberate addition of some of these elements to Carbide improve the physical characteristics of the metal has been considered. slagging has proved beneficial in removing zirconium and niobium. Product recoveries are reasonably high (being about 95 per cent). Table

28.

Free Energy (Gibbs) of Formation of Carbides — AF°, kcal/mole 43. 1

Carbide


ZrC

UCi UC

-

<33 3 <0

Mo,C

Carbides of Ru, Rh, Pd 29.

K

33.8 ~35

PuC

Table

at 1500

Fission-product Removals Effected in Oxide-slagging

Operation

(4 hr at 1200°C) Greater than 90 % removed

0-10% removed

Rare earths

Niobium Molybdenum Technetium* Ruthenium Rhodium Palladium Zirconium

Yttrium*

Cerium Tellurium Strontium Xenon* Krypton* Iodine*

* No experimental data, but indicated result seems probable.

of Uranium. An attractive method for fission-product Electrorefining 6.62 In this process the metallic state of removal from fuel materials is electrorefining. the fuel is momentarily lost as uranium passes into solution at the anode and is The electrolyte consists of various fused halide salt mix deposited at the cathode. tures (chlorides and fluorides of sodium, calcium, barium, and uranium). Purification from the noble elements as well as those more active than uranium is theoretically possible in this process. Those elements much more noble than uranium remain behind as an anodic sludge. Those elements much more reactive than uranium tend to concentrate in the salt bath. The rare gases are vented off at the Periodic replacement of the salt bath would be required to prevent excessive anode. build-up of fission products in the bath. Plutonium has an oxidation potential very close to that of uranium, and it may be possible, by proper adjustment of conditions, to concentrate it in the salt bath or make it follow uranium. Metals that form low-melting alloys with uranium have been successfully used as At appropriate temperatures, this permits continuous removal of a cathodes."-67


11-100

CHEMISTRY

AND

CHEMICAL

ENGINEERING

[SEC. 11

Nickel cathodes have been successfully employed. molten product alloy. Chro mium, manganese, and iron are other possibilities. Subsequent removal of the alloying agent may be required. On small-scale runs, the material recovered at the cathode has represented 98 per cent of the material leaving the anode. Fission-product decontamination factors obtained have been about 600 for the noble elements (ruthenium and molybdenum) and about 100 for the active elements (zirconium, rare earths, and strontium), using fission products at trace concentration levels and synthetically prepared uranium alloys containing 1 weight per cent concentrations of ruthenium, molybdenum, zirconium, and lanthanum. 6.63 Fused Salt Extraction. In halide systems, plutonium is more electropositive than uranium. Consequently, through proper choice of metal halides in a fused salt mixture, plutonium can be made to extract from a molten uranium phase into a molten salt phase. Although the uranium largely remains in the metallic state, the plutonium is converted into a halide salt and must be subsequently converted to the metal. The use of salt mixtures of calcium, magnesium, lithium, and uranium fluorides has For instance, 90 per cent of the plutonium is been demonstrated to be effective. extracted from molten uranium by molten uranium tetrafluoride (equal weights of By successive extractions, essentially quantitative removal of the each phase). plutonium may be effected. Several of the active fission products (the rare earths, strontium, cesium) are also This is being exploited in extracted into the salt phase along with the plutonium. development of processes for purification of spent liquid metal reactor fuel (such as A large fraction of the fission products that constitute the uranium in bismuth). major poisons is removed in this way. Another high-temperature processing 6.64 Formation of Volatile Compounds. method may be the use of a scheme similar to the van Arkel-de Boer process which is used for the purification of zirconium.68 In this method, a volatile uranium iodide is formed at one end of a piece of evacuated equipment and decomposed on a hot The decomposition surface, liberating the metal in another part of the apparatus. temperature is a function of the total pressure; the lower the pressure, the lower the Uranium tetraiodide is decomposed in the temperature decomposition temperature. Many of the fission products range 1100 to 1400 C at pressures of a few microns. also form volatile iodides, but separations can be effected by utilizing the difference in vapor pressures and the difference in decomposition temperatures. However, except for zirconium and niobium, all the other fission-product iodides are less volatile It is than uranium iodide or are decomposed at temperatures less than 600°C. possible to condense the uranium iodide out of a mixture of uranium, zirconium, and niobium iodide and subsequently revaporize and decompose the uranium iodide. In laboratory runs, uranium has been decontaminated from both gross /S and y activities The behavior of plutonium in this process has not been by factors of the order of 100. investigated. Several metals such as silver, magnesium, cerium, 6.66 Molten-metal Extraction. and lanthanum are essentially immiscible with molten uranium, so that the possibility exists of using these metals to extract fission products or plutonium from uranium. The use of silver and magnesium as plutonium extractants has been investigated to Plutonium distributes between molten uranium and magnesium or some extent. silver with a distribution coefficient of about 2 (weight basis) in favor of the magnesium Molten-metal extraction is therefore an attractive possible method for or silver. concentrating the plutonium bred in a natural or depleted uranium blanket. The data of Table 30 show that the fraction of plutonium extracted by magnesium is concentration over a 300-fold range. In essentially independent of plutonium extraction with magnesium, account must be taken of the high vapor pressure of magnesium at the melting point of uranium. Distillation of the metal extractant away from the extracted plutonium is the In this case, the relatively high volatility of preferred method for its removal. Separation of plutonium from silver by distillation is magnesium is advantageous. more difficult.

Art.

OF SEPARATIONS

METHODS

BASIC

5]

11-101

Data for the silver extraction of irradiated uranium show the silver to have poor selectivity insofar as separation from fission products is concerned (see Table 31). The data show that the more reactive fission-product elements and plutonium are extracted by silver to about the same extent. These more reactive fission products are also well extracted by magnesium. Preliminary data show that ruthenium, a noble fission product, is only slightly extracted by either metal extractant. This fact can be the basis for a plutonium-ruthenium separation. Table 30.

Plutonium

Extraction in the Molten System

Initial plutonium

% plutonium in magnesium phase

1.0

60 60 71 67

0.3

0.9 0.7

23 55 97

Table 31.

1.0

Extraction of Irradiated Uranium with Silver (Equal weights of silver and uranium) % extracted by silver 96 94 89 86 73 65


Element Cesium

Strontium Lanthanum Cerium Zirconium Plutonium

Table 32.

Vacuum Volatilization Temperature,

°C

1680 1680

6.66

Volatilization.

Uranium-Magnesium

Weight ratio.

U/Mg

cone, ppm

:

of Plutonium

Surface

area,

em'/g

0.7 0.4

from Molten Uranium

min

% Pu volatilized

40 140

90.0 98.6

Time.

As already mentioned, purification of reactor fuel from many

The separation of fission products occurs through the mechanism of volatilization. plutonium from uranium by volatilization is also a possibility. M'70 Plutonium is more volatile than uranium, and its vapor pressure becomes high enough at temperatures of the order 1600 to 2000°C so that separation of plutonium

from

uranium by volatilization is possible. High vacuum is required, since even at the vapor pressure of plutonium is only of the order of 10 mm of mercury. The percentage removal of plutonium is a function of the temperature, time, and the surface area per unit weight of uranium. Experimental results have been found to agree with theoretical predictions. Results of small-scale tracer plutonium volatilization experiments are shown in Table 32. 71 principle, this is one of the simplest of schemes for isolating plutonium. How ever, at the temperatures involved, the materials problem is a very formidable one. Although vaporization of plutonium has been demonstrated, the collection of plu tonium has not. Extrapolation of the data obtained to full scale indicates that the evaporating surface must be relatively large (several feet in diameter). The design of the evaporator and condenser and the maintenance of high vacuum at the very high temperature involved are difficult engineering problems. 5.67 Purification by Zone Melting. Zone melting is a method whereby impurities in a metal bar can be concentrated in one or both ends of the bar by repeatedly passing a small molten zone through the bar in one direction only.72 Those solutes which lower the melting point distribute preferentially into the liquid phase and are swept

20O0°C

In


11-102

[SEC.

11

along with the zone, while those which raise the melting point distribute preferentially into the solid phase and move counter to the zone. The distribution coefficient k for any constituent is defined as the ratio of concen tration of the constituent in the solid to its concentration in the liquid at equilibrium. The distribution coefficient and the ratio of molten zone length to total bar length are important factors that determine the ultimate separation attainable and the change Theoretical in solute concentrations along the bar per pass of the molten zone. procedures have been developed for calculating concentration profiles.73-7' Distribution coefficients are nearer unity than the theoretical value obtained from a phase diagram and are called effective distribution coefficients. These are sensitive to the solidification rate.76 Effective distribution coefficients are shown in Fig. 42

i

1

1

1

1

1

1

r

^-MOLYBDENUM

o

0.9

o o

o.e

jZ

0.7

u. u. UJ

PALLADIUM

RUTHENIUM

m


0.6

o 0.5 0.4

0.3

-

0.2

— 8

RATE IN I0"3

10

J

L

12

16

18

INCHES/MINUTES

Fio. 42. Effect of solidification rate on effective distribution coefficient for various solutes

in

uranium. as a function of solidification rate for some elements of interest in uranium. The results are not very encouraging, since the effective distribution coefficients are close to unity unless solidification rates are extremely slow (less than Hso lin in./hr). Much additional information has been made available in TID-7534, "Symposium on Fuel Reprocessing," Brussels, Belgium, May, 1957.

BIBLIOGRAPHY L„ and W. L. McCabe: "Elements of Chemical Engineering," 2d ed., McGrawHill Book Company, Inc., New York, 1936. Brewer, L.: "High Temperature Decontamination and Separation Process," UCRL-314. Badger, W.

May 6, 1949. Brewer, L., et al.: The Thermodynamic Properties and Equilibria at High Temperatures of the Compounds of Plutonium, Paper 6.40 in "The Transuranium Elements," National Nuclear Energy Series, Div. IV, Vol. 14B, McGraw-Hill Book Company, Inc., New

York, 1949. Brown, G. G., and Associates: 1950. Dodge, B.

"Unit Operations," John Wiley

& Sons, Inc., New York.

F.: "Chemical Engineering Thermodynamics," McGraw-Hill Book Company. Inc., New York, 1944.

Art.

filtration, centrifugation,

6]

evaporation

11-103

Feder, H. M.: " Pyrometallurgical Processing of Nuclear Materials," Paper P/544, Inter national Conference on the Peaceful Uses of Atomic Energy, IX, United Nations, New York, 1956. G., and J. W. Kennedy: "Introduction to Radiochemiatry," John Wiley & Friedlander, Sons, Inc., New York, 1949. Glassner, A.: " A Survey of the Free Energies of Formation of the Fluorides, Chlorides, and Oxides of the Elements to 2500 K," ANL-5107, August, 1953. Hahn, O.: "Applied Radiochemistry," Cornell University Press, Ithaca, N. Y., 1936. Harty, W. M.: Nuclear Engineering — Part II, Chem. Eng. Progr. Symposium Ser., 50(13): 115-121 (1954).

Hyman, H. H., R. C. Vogel, and J. J. Katz: "Decontamination of Irradiated Reactor Fuels by Fractional Distillation Processes Using Uranium Hexafluoride," Paper P/546, International Conference on the Peaceful Uses of Atomic Energy, IX, United Nations, New York, 1956. Lawroski, S.: "Survey of Separations Processes," Paper P/823, International Conference on Peaceful Uses of Atomic Energy, IX, United Nations, New York, 1956. McLain, S.: "Reactor Engineering Lectures," ANL-5311, part III, Lectures 20, 30, 31 (prepared by R. S. Voris), pp. 188-207, 1954. Morello, Z. S., and N. Poffenberger: Commercial Extraction Equipment, Ind. Eng. Chem., 42:1021

(1950).

Motta, E. E.: "High Temperature Fuel Processing Methods," Paper P/542, International Conference on the Peaceful Uses of Atomic Energy, IX, United Nations, New York,


1956.

Perry. J. H. (ed.): "Chemical Engineers' Handbook," 3d ed., pp. 714-753, McGraw-Hill Book Company, Inc., New York, 1950. Quill, L. L. (ed.): "The Chemistry and Metallurgy of Miscellaneous Materials: Thermo dynamics," National Nuclear Energy Series, Div. IV, vol. 19B, McGraw-Hill Book Company, Inc., New York, 1950. Robinson, C. S., and E. R. Gilliland: "Elements of Fractional Distillation," 4th ed., McGraw-Hill Book Company, Inc., New York, 1950. Sherwood, T. K., and R. L. Pigford: "Absorption and Extraction," 2d ed., McGraw-Hill Book Company, Inc., New York, 1952. Stephenson, R.: "Introduction to Nuclear Engineering," pp. 323-343, McGraw-Hill Book Company, Inc., New York, 1954. Voight, A. F.: "The Purification of Uranium Reactor Fuel by Liquid-Metal Extraction," Paper P/545, International Conference on the Peaceful Uses of Atomic Energy, IX, United Nations, New York, 1956. Walker, W. H., W. K. Lewis, W. H. McAdams, and E. R. Gilliland: "Principles of Chemical Engineering," 3d ed., McGraw-Hill Book Company, Inc., New York, 1937. Wahl, A. C, and N. A. Bonner: "Radioactivity Applied to Chemistry," pp. 104-124, John Wiley & Sons, Inc., New York, 1949.

6

FILTRATION, CENTRIFUGATION, AND EVAPORATION

In Art.

5 several basic methods of separations and process flow sheets are presented. the framework of these basic processes, the unit operations nitration, cen trifugation, and evaporation are frequently used.

Within

6.1

Most

Filtration

of the applications of filtration used to date have been directed toward In solvent-extraction processes, for example, the process streams. presence of even small amounts of solids in process streams will cause the formation of scum, "fish eggs," or "crud" at the liquid-liquid interfaces. These scums are highly undesirable, as they cause maloperation of the extraction equipment, adsorb radioactivity, and make equipment decontamination difficult. Moreover, accumu lated bits of the scum can become entrained in process streams, greatly reducing the decontamination factor obtainable. It has been found in most cases that careful filtration of all influent streams will eliminate much of this trouble. The nitration of nonactive streams presents no particular problem. However, the nitration of a dissolver solution containing many thousand curies of activity is more difficult. It

clarification of

11-104


[SEC.

11

Pro has been done with sintered stainless-steel filter disks or with centrifugation. vision must be made for backwashing the filter remotely and combining the backwash with radioactive waste streams. It is also necessary to provide for the remote removal of the filter unit and its disposal in a shielded transfer cask to solid storage. Standard-practice use of nitration equipment is also encountered in other auxiliary operations of nuclear-energy installations. Examples are the use of gravity filters in the pretreatment of water and the use of deep-bed sand niters in sewage disposal. No problems peculiar to nuclear energy are encountered here, of course. Some lightly contaminated sludges in waste streams are handled on precoated rotary vacuum filters. 6.2

Centrifugation


Centrifugation is the separation by centrifugal force of a liquid and a solid or of two immiscible liquids. The supporting wall of the rotating member may be either porous or solid. If the wall is porous, the process is analogous to nitration through a filter medium in which the gravity, pressure, or vacuum (which would be used in If the supporting simple filtration) is magnified and replaced by centrifugal force. wall is solid, the resulting process is essentially one of settling in which gravity is magnified and replaced by the centrifugal force. The magnitude of the centrifugal force is proportional to the diameter of the basket and to the square of the rate of revolution. A convenient means of estimating the effect of centrifugal force is to compare it with the force of gravity. This may be determined by use of Eq. (53) :

ff

=

DN1

6*64

(53)

where G = number of times centrifugal force is greater than gravity D = diameter of basket, ft N — rpm In a practical centrifuge, sedimentation occurs at right angles to the throughput flow of liquid. In order that the separation be effective, sedimentation rates must be high enough so that the degree of separation desired may be accomplished in the time the fluid is in the force field. This may be accomplished by balancing throughput rates with the effective force. The higher the force developed, the higher may be the throughput rate and conversely. There is a practical limit to how high a force may be developed, since the higher the force the greater must be the strength of the bowl to hold it together. The largest application of centrifugation has been in separating out the precipitates formed in the bismuth phosphate process used at Hanford (see Art. 5.2). The pre cipitates formed in the early stages of the process are separated using large cen trifugals of the overhead-suspension, solid-bowl type. All parts of the centrifuges that may come in contact with the process solution are made of stainless steel. The units are equipped with magnetic brakes and plows both of which are used to cut the precipitate loose from the walls. Prior to cake removal some of the excess liquor in the bowl is removed with skimmers. Operating experience with these units in a remotely serviced canyon has been quite satisfactory. In later stages of the process where solution and cake volume are reduced, smaller The metathesis reaction solid-bowl centrifugals are used with considerable success. (see Art. 5.2), whereby a precipitate of LaF» is converted to La(OH), with KOH, is carried out in the centrifuge bowl using it as a reaction vessel. 6.3

Evaporation

The unit operation evaporation is often encountered in separations processes. It is used to reduce volumes of product streams between cycles in some of the presently used processes and has wide application in the reduction of waste volumes. The theory of evaporation is relatively simple, and practical operating considers

Art.

6]

filtration, centrifugation, evaporation

11-105

tions are generally more important in the selection of equipment than theoretical The rate of heat transfer (and therefore the boilup rate and capacity) calculations. of an evaporator is obtained from the same equations used for all other cases of heat transfer: q

-

UA&T

(54)

rate of heat transfer, Btu/hr heating surface, ft* AT = mean temperature difference between stream and boiling liquid, °F U = over-all heat-transfer coefficient, Btu/(hr)(ft')(°F) The over-all coefficient U is the resultant of a number of resistances in series. These might be the steam film, the wall of the metal tube, a scale factor, and the boiling film. In general, few data are available on specific film resistances, and even those on This lack of data is largely due to over-all coefficients are scanty and vary widely. the fact that the scale factor contribution is generally very low at startup but increases rapidly as the unit is used. Eventually the effect will become so great that the evaporator must be shut down and cleaned. Since an evaporator operating on radioactive materials cannot be dismantled with the ease of more conventional units, minimizing scale formation is of paramount It is highly desirable to select equipment that importance in evaporator selection. will admit of chemical rather than mechanical descaling. For this reason nearly all Washing with nitric evaporators used so far have been made of stainless steel. acid and/or Versene will usually restore heat-transfer coefficients. In ordinary evaporator practice, limiting entrainment carry-over to 1 per cent is often adequate. In handling radioactive solutions, much less carry-over is desired. Decontamination factors of 10» to 10* have been reported for various types of evapo where

q = —


A

rators

(see

Table 33).

Table 33.

Comparison of Evaporators

Capacity of equipment, gph

Type

of equipment.

400

Forced-feedflash

Site

Typical

feed (average solids), %

Over-all decontamina tion factor Volume reduction factor Slurry (average solids). %

285

Pot-type, removable heating coils

Used on Radioactive 600

Vapor com pression

Wastes

150

Vertical tube, natural circulation

100

1st effect

forced feed 2d effect — vertical tube,

natural circulation Mound

Knolls

Oak Ridge

Brookhaven

Argonne

0 3

1.0

0 8

0 2

3 0

I0«-I0'

10'* 10-15

lO'-IO1

10«-10» 100

~I0«

400 70

70

100 50

20

—

vertical tube,

30 60

Heat- transfer coefficient

Btu/(hr)(ft")(°F)....

Remarks

500-700

600

Scaling minimised

Trouble with scaling and foaming

Higher DF due to Fiberglas filtra tion of vapor

Chemical descaling of tubes satis

factory

The volume reduction possible is of great importance in determining the cost and It should be pointed out, however, that this factor is more value of the operation. Steam economy, which dependent on feed concentration than on equipment type. is of relatively less installations, in commercial many is of the greatest importance This is due to the fact importance in separations-process applications of evaporation.

11-106


ENGINEERING

[SEC.

11

that equipment, building, storage, and labor costs are high and the power cost is a relatively small percentage of the total. Particularly in connection with development work on the handling of radioactive These wastes, a number of evaporator types have been tested at various AEC sites. are summarized in Table 33. An attempt was Cost estimates have been made at each of the sites mentioned. made to cross-compare costs upon a weighted capacity basis. Within the limits of estimating error all the costs are of the same order of magnitude, being 6 to 8 cents per gallon of waste evaporated at 400 gph. This cost is exclusive of storage of evaporator bottoms but includes all operating costs including overhead and depre ciation of building and equipment.

BIBLIOGRAPHY Badger, W. L., and W. L. McCabe: "Elements of Chemical Engineering," 2d ed., McGrawHill Book Company, Inc., New York, 1936. Brown, G. G„ and Associates: "Unit Operations," John Wiley & Sons, Inc., New York, 1950.


Dickey, G. D., and E. M. Bryden: "Theory and Practice of Filtration," Reinhold Publish ing Corporation, New York, 1946. Perry, J. H. (ed.): "Chemical Engineers' Handbook," 3d ed., McGraw-Hill Book Com pany, Inc., New York, 1950. Walker, W. H., W. K. Lewis, W. H. McAdams, and E. R. Gilliland: " Principles of Chem ical Engineering," 3d ed., McGraw-Hill Book Company, Inc., New York, 1937. 7

AUXILIARY OPERATIONS 7.1

Ore Concentration

By M. Levensonf Several major steps are necessary in the preparation of a uranium metal. Ores of economic concentration must be found and mined, the uranium must be separated from the gangue and concentrated, the concentrate must be purified and converted into a compound suitable for reduction to metal, and finally, this reduction must be carried out. Chemical engineering plays an important role in the ore concentration and purification, which will be discussed in this article, and in the conversion of salts to metals, which will be discussed in Art. 7.2. At the present time some 27 different ores (Table 34) are mined in this country alone. Uranium is also recovered from imported ores and as a by-product from In other parts of the world, shales, phosphoric acid and fertilizer manufacture. sands, lignites, clays, and even some ground waters must be added to the list of uranium sources. Because of these widely differing sources, the ore-concentration process selected must be adapted to a particular type of uranium-bearing material. Two types of ore leaching processes and numerous concentration schemes are in use today. The flow sheet shown in Fig. 43 indicates the steps 7.11 Carbonate Leach. involved in a carbonate leaching process for handling carnotite. This flow sheet is typical of those used in treating ores containing vanadium as well as uranium. Car bonate leaching is a highly selective process by which uranium is dissolved in sodium carbonate solution. If an ore can be handled by this process, it is preferred because of low chemical costs, simplified equipment and material-of-construction problems, and the high degree of separation that can be realized. The ores are ground and then roasted with NaCl to solubilize the vanadium. The hot calcine is quenched with 5 per cent NasC03 — NaHCOj solution. The bicar bonate is necessary to neutralize the hydroxyl ion formsd during dissolution so that the uranium is not immediately reprecipitated. The solution containing 55 to 75 per cent of the uranium and vanadium is removed from the sludge and filtered. The sludge then goes to other cookers where it is eon-

t Argonne


Art.

auxiliary operations

7] Table 34.

United States Domestic Uranium Ore Processing Type of ore

Pegmatite, betafite, brannerite, samarskite Pitchblende, no risible sulfides Pitchblende with other metals Pitchblende with sulfides and metals Uranophane, CaO-2UOr2SiO«-6rhO , Uraniferous fluorite (requires destruction of fluorite) Autunite, CaO-2UOiPiOi-nHsO (n 8-12) Schroeckringerite, 3CaCOrNaiSO«UOrl0IhO Roscoelite Carbonaceous shale (60-100 ppm UiOtO Carnotite, low V, low lime, KtOÛOrVtCVnHiO (n 1-3) Carnotite, low V, intermediate lime, CaO-2UOrViO»-nHiO (n Carnotite, low V, high lime Carnotite, high V, high lime Carnotite, high V, intermediate lime Asphaltic carnotite, low V, low lime Asphaltic carnotite, low V, intermediate liine Asphaltic carnotite, high V, low lime Asphaltic carnotite, high V, intermediate lime Torbernite shale, CuOÛOiPiOinHiO (n = 8-12) Torbernite sandstone Uraninite, low V, low lime Uraninite, low V, intermediate lime Uraninite, low V, high lime Uraninite, high V, low lime Uraninite, high V, intermediate lime Uraniferous lignite (50 ppm to 0. 1 % UiOt)


-

Low V < 0.75 % ViOi High V > 0.75 % ViOi Low lime < 6.0% CaCOi Intermediate lime » 6 to 18% CaCOi High lime > I8f„ CaCOi Eng. Mining

J.,

1M(9):

Acid

Carbonate

leach

leach

CX BX BX BX A C

-

Source:

11-107

A A A

RB

-

A 4-10)

A A A

RA RA RA RA

AX AX A A A A

RB

BN B.N

BN

AX AX AX AX AX AX RAN RAN RAN RAN AN AN AN AN AN

A ■* uranium easily solubilized B = requires moderate conditions for uranium dissolution C ■»refraotory — vigorous treatment needed R =• requires roasting to remove carbon N = oxidant (e.g., MnOt, NaClOt) needed X — oxidant beneficial

106 (September,

1954).

After several successive teachings, the tacted with fresh, hot carbonate solution. The carbonate sludge is essentially free of uranium and vanadium and is discarded. solution containing small amounts of these metals is used as the quench solution for the next batch. This type of operation — the complete solubilizing or, preferably, the selective solubilizing of the metals from the ore— is the most difficult and most impor Once the uranium is in solution, tant part of the uranium concentration process. One of the many methods are available for further concentration and purification. most common techniques used with carbonate solutions is partial neutralization to Sometimes these separate the uranium and the vanadium by selective precipitation. precipitates are redissolved, oxidized, and reprecipitated to obtain further purification from such impurities as iron. This particular part of the process varies with the com A new technique finding favor for this step makes use position of the ore processed. of hydrogen gas at 65°C to reduce the uranium in the carbonate complex, causing it to A catalyst such as finely divided nickel must be used. precipitate as black UO». Advantages of this process are that the solution may be reused and the chemical consumption is lowered. A third scheme, employed primarily on solutions with low concentrations of extraneous metal ions, is the use of sodium or ammonium hydroxide to precipitate uranium as the diuranate. In all these process variations an oxidant must be present in order that the carbonate In carefully controlled experiments under inert solution can extract uranium. atmosphere where no oxidants were used and all solutions were oxygen-free, no uranium could be extracted, even from pitchblende. However, properly used.


11-108

[SEC.

11

carbonate leaching is an excellent method of recovering essentially all the uranium from carbonate-bearing ores. 7.12 Acid Leach. The second fundamental leaching process is the acid leach. This is used primarily on silicate- and phosphate-bearing ores (Figs. 44 and 45), since the acid consumption for carbonate-bearing ores would be high —about 1 lb of sulfuric acid per pound of lime, dolomite, or magnesite. The acid leach is very effective, so that, when used with an oxidant such as sodium chlorate or manganese dioxide, it dissolves 95 to 98 per cent of the uranium present in the ore. In processing ores that The amount of are salt roasted, hydrochloric acid is used instead of sulfuric acid. HC1 liberated during the roasting operation is more than sufficient to leach the calcine, The unit so that only partial recovery of HC1 from the furnace off-gases is necessary. CRUSHING AND SCREENING

NoCI •

ROASTING


Not CO] NoHCOj

•HCI GAS

QUENCHING

Hg0

NozCOj]

NoHCOj^-H

HjO

J

LEACHING

PPT HtSO*-

-SAND TO WASTE

AND WASHING

AND

URANYL VANADATE

FILTRATION

FURNACE LEACH

FILTRATE (SODIUM

PPT

HiS04-

AND FILTRATION!

Fio.

— No2C03 POWDER — SAWDUST |—- UOt

VANADATE)

-VtO,

43. Carbonate-leach

process.

operations involved are much the same as those used in carbonate leaching. The ore is crushed and leached, the liquor clarified, and the sludge washed and discarded. The uranium is then precipitated or concentrated by ion exchange. Because of the corrosive nature of the leaching agents, the material-of-construction problems are greater than those encountered in the carbonate process. Many other schemes are used to concentrate uranium from dilute sources, but most of them are used for specific raw materials and are not generally applicable. One of the most interesting of these is the use of coal to adsorb traces of uranium from oflwell waters and uranium-plant waste waters. The coal is then leached with HQ. In some cases it is first used for fuel and then the ashes are leached. 7.13 Ore Purification. Regardless of which process is used to extract and con centrate the uranium from the ore, further purification is required before the uranium is reactor-grade material. Ore concentration upgrades the uranium from the 10 to 100 ppm present in shale or coal or from the few tenths per cent present in carnotites to a product containing 50 to 75 per cent uranium. The subsequent purification

Akt.

AUXILIARY OPERATIONS

7]

11-109

to yield highly pure uranium is usually accomplished by solvent extraction. In some cases preliminary precipitations are used to recover radium or other valuable A typical process is shown schematically in Fig. 46. by-products. The ore concentrates are usually dissolved in nitric acid. A salting agent such as The salted aqueous solution is then con calcium nitrate is added to the solution. In some cases tacted with diethyl ether, which selectively extracts the uranium. tributyl phosphate is used as the extractant. Since tributyl phosphate is quite viscous, it is normally diluted with kerosene-type hydrocarbons. ORE

CRUSH AND SCREEN

NaCI

ROASTER

I

HCI (RECYCLE TO ACID LEACH)

\ WATER LEACH


WATER

HCI (FROM ROASTER)'

ACID LEACH

IRON

REDUCTION

NHj

PPT AND FILTER

NaCIOj

FeS0«

DISCARD

TO WASTE

FILTRATE

OXIDATION

\ J

NOjCOj

PPT AND FILTER

FROM VANADATE. RECYCLE TO SALT ROAST

PPT AND FILTER

ALUMINA-SILICA CAKE

DESTROY

H2S04

CARBONATE

I

PPT AND FILTER

NaOH

PRODUCT

Fig.

SOLIDS

SOLUTION

DISSOLVE

H2S04

Na2COj

VANADIUM

44.

FILTRATE

DESTROY

Na2U207

Uranium recovery from silicate ore by acid-leach process.

The uranium may be directly precipitated from the solvent, or the solvent may be distilled off. However, the usual practice is to strip the uranyl nitrate from the The uranium in the aqueous solution thus organic phase with a water solution. obtained may be precipitated by the addition of hydrogen peroxide or ammonium hydroxide, but the usual procedure is merely to evaporate the water, leaving a hydrated uranyl nitrate. 7.2

Conversion of Salts to Metalf

"

The product of ore-purification steps (see Art. 7.1) or of the spent fuel separations plants (see Art. 5) is usually an aqueous solution of uranyl nitrate, UO^NOjK The t The authors wish to acknowledge the assistance of M. Levenson, Argonne National Laboratory, in the preparation of this article.

Chemical Engineering Division,

AND CHEMICAL

CHEMISTRY

11-110

ENGINEERING

[SEC.

11

PHOSPHATE ORE 10% PjOs 100 PPM Ufig

LIMESTONE

"

.

CALCINING KILN

'

DRY

SCRUBBER

^jF2

1

H2SO,l

p*

HN03

ACID DIGESTION

1

FILTER

-WASTE

CAK

1

URANIUM PRECIPITATION

POTASH AMMONIA


1

FILTRATE.

FILTER

BLE NDER

1 URANIU U CAKE

Fiq.

DRYER AND

GRINDER

1

NITROPHOSPHATE FERTILIZER

45. Uranium recovery

from phosphate

ore by acid-leach process.

steps used in converting this product into either uranium metal (for fuel-element fabrication) or uranium hexafluoride (for introduction into a gaseous diffusion plant) are shown in Fig. 47. The reactions occurring in each of these steps are as follows: 1. The thermal decomposition of uranyl nitrate to form uranium trioxide UO,(NO„)2-6H20

AH 2. Reduction

->

Btu/mole

of uranium trioxide to uranium dioxide with hydrogen

UO, + H2AH 3.

UO, + 2NO, + H02 + 6H,0

~450°C = +262,000

=

-» U02

~600°C

+ HjO

-43,000 Btu/mole

Hydrofluorination of the uranium dioxide with anhydrous hydrogen fluoride UO, + 4HF-» UF, + 2HsO ~600°C AH

=

-84,000 Btu/mole

The UF» (known as green salt) may then be converted either to uranium fluoride or to metal.

hexa

Art.

11-111


7]

NITRIC ACID 1

ORE

CONCENTRATE -

RECYCLED

DISSOLVER

50-70%U

SLUDGE

CLARIFIER

Bo(NOs)2 No2S04

-

WASH LIQUOR

WASHER

RADIUM BEARING SLUDGE

PRECIPITATOR (OPTIONAL STEP)

DILUTE ACID

Ca(N03)2 WASTE SALT SOLUTION

EXTRACTOR

TRI BUTYL PHOSPHATE*!

" SOLVENT

STRIPPER

HfO

-HjO

EVAPORATOR


TO

RECYCLE

HYDRATED

URANYL NITRATE

Flo. URANYL NITRATE HEXAHYDRATE

46. Ore purification.

CALCINER

OXIDES ' H20 AND OF NITROGEN

UOj-KUjOe)

• HjO

REDUCER

UOt

HF ■

HYDROFLUORINATOR

•HjO

FLUORINE

FLUORINATOR

UF6

Mg-

REDUCTION

BOMB

•MgF2

REMELT FURNACE

U

Fio.

47.

Conversion of UNH to UF, and UF,.

" WASTE CAKE

11-112


[SEC.

11

UF'+Fl^5ÛF' AH

UF. + AH

-

-106,000

2Mg

Btu/mole U + 2MgF,

= -237,000

Btu/mole

7.3

Dissolution t

In any aqueous processing scheme, it is first necessary to dissolve the fuel element. Since the reactor designer attempts to develop as corrosion-resistant a fuel element as In general, the fuel elements are uranium, possible, dissolution is not always easy. They may be with each other or with other elements. of these or alloys plutonium,

0ISSOLVER SOLUTION

Fio.

48. Schematic

dissolver facility.

of,

clad in corrosion-resistant material (aluminum, titanium, zirconium, stainless steek which must be removed either before or during the dissolution of the base metal. 7.31 The removal of the cladding can be accomplished by mechanical Decladding. means or chemical dissolution. If the cladding can be cleanly separated from the fad How alloy mechanically, the chemical and waste-disposal problems will be reduced. ever, if the reactor has operated at high temperatures, if conditions of fabrication have caused the cladding and the fuel alloy to diffuse into each other, or if the cladding is Under these conditions one of very thin, mechanical separation may be impossible. three things must be done: (1) The cladding must first be dissolved and disposed and then the fuel must be dissolved; (2) the cladding must be ruptured, and the fuel alloy dissolved out, leaving the cladding; or (3) the cladding and fuel alloy may be The authors wish to acknowledge the assistance of M. Levenaon, Chemical Engineering Divisioa Argonne National Laboratory, in the preparation of this article.

t


STACK

Art.

7]


11-113

dissolved together. Three reasons for preferring (1) or (2) over (3) are that including the canning or cladding alloy in the dissolved fuel solution reduces plant capacity, increases waste storage costs, and may complicate the chemistry of the recovery


process.

Many of the fuel elements used to date have been clad in aluminum. If it is desired to remove the aluminum cladding, leaving the element intact, a satisfactory dissolving procedure consists of using a caustic solution as the dissolving medium. Following the decladding, the dissolver is emptied and rinsed prior to dissolution of the fuel elements. Aluminum cladding may be dissolved at 80°C in 10 per cent HNOa using a little mercuric nitrate as catalyst. This method is often used when the fuel element is an alloy of aluminum to dissolve both the cladding and element at the same time. Since this reaction is exothermic as well as catalytic, it is particularly sensitive to tempera ture and great care must be exercised in bringing the solution up to temperature. It is virtually impossible to avoid a surge of off-gases at the start of the reaction. Hydrofluoric acid is generally used for zirconium cladding, and sulfuric acid for those of stainless steel. In any event the solution chosen and the materials of con struction of the dissolver and of the waste storage facilities must be compatible. 7.32 Fuel Dissolution. The details of a dissolving procedure vary, of course, with the fuel material. Most of the fuels handled to date have been soluble in nitric acid, The fuel-dissolution step so that the separations plants are designed on this basis. will therefore be illustrated on the assumption that the fuel is soluble in nitric acid. On the laboratory scale, the dissolution of uranium in nitric acid is very simple. A piece of uranium placed in a flask containing nitric acid readily dissolves when heated gently, with the evolution of oxides of nitrogen. The reaction is easily con trolled, and the fumes are small enough in volume that they may be discharged into the hood. The following reactions are possible: U + 4HNO, — U02(NO,)2 + 2NO + 2HsO U + 8HNO* — U02(N03)s + 6NO, + 4H,0

It has been found that both reactions take place, approximately as

and the net result is expressed

U + 5.5HNO,-> U02(NO,)s + 1.25NO + 2.25N02 + 2.75H20 Because the dissolution of uranium in nitric acid involves the evolution of copious amounts of gas, plant scale dissolvers must be designed to handle potential foaming The gases cannot be discharged to the atmosphere without treatment (see problems. Art. 7.4). The dissolver system normally used is shown schematically in Fig. 48. 7.4

Handling

Off-gases

At several points in the separations processes, off-gases are produced that require These may be radioactive gases such as I2, Kr, or Xe; entrained special handling. particles of radioactive solution; or simply noxious but nonradioactive materials such as the oxides of nitrogen released during metal dissolution (Art. 7.3) or the denitration of UNH (Art. 7.2). Nearly all chemical processing is done in concrete cells or canyons that are them It is general practice to discharge ventilating air through stacks selves ventilated. that are at least 200 ft tall. Even so, additional treatment is often necessary for particular off-gas streams. 7.41 Dissolution Step. As indicated in Art. 7.3, copious quantities of the oxides of nitrogen are given off during the dissolution step. Not only are these noxious in themselves, but they also tend to carry with them entrained droplets of dissolver solution. In operation of separations plants this off-gas is often mixed directly with the cell ventilating air and discharged out the 200-ft stacks. The N02 fumes are clearly visible as they emerge, and the plume is often distinguishable for a mile or Hence, dissolving operations may be restricted to periods of more downwind.


11-114

CHEMISTRY

AND


[SEC.

11

Effective methods of off-gas cleanup have been favorable meteorological conditions. One of these consists in the passage of the dissolver off-gas diluted with developed. air countercurrent to a 4 per cent NaOH solution in a packed tower. In this way Some dismost of the NOj and a substantial fraction of the iodine are absorbed. Another effective method used to solvers are now equipped with caustic scrubbers. remove the iodine consists in passing the gases over a heated bed of silver adsorbed on ceramic packing. Another approach to the dissolver off-gas problem has been to carry out the dissolu In this scheme oxygen is tion in a closed system —so-called "fumeless" dissolving. introduced into the off-gas stream to oxidize all NO to NOj, and the gases are then conducted to a countercurrent absorber where the NOt is taken up in dilute HNOi to produce concentrated HNOj. 7.42 The Denization Step. The product from all current separations-process This must be dehy plants is a decontaminated aqueous solution of uranyl nitrate. drated and denitrated (see Art. 7.2). In the denitration step, the evolved oxides of nitrogen are recovered as nitric acid by conventional absorption techniques. 7.43 Ventilating Air. As mentioned earlier, cell ventilating air is generally mixed with process gases and discharged through tall stacks. It has been found that the stacks of plants which have operated for some time may discharge gases containing These particles have been traced to particles of solids that are highly radioactive. An effective solution to this problem has been flaked scale from the inside of ducts. to filter the mixed gases through large sand filters. Filters that have the approximate dimensions of 85 by 85 ft by 10 ft thick will handle about 35,000 cfm. They are made up of graded size material running from 3-in. gravel at the bottom to 50-mesh sand They remove approximately 99.7 per cent of Flow is generally upward. at the top. the activity. 8

SPECIAL PROBLEMS 8.1

Instrumentation

By L. F. Coleman f The use of instruments in reactor control, radiation detection, and analysis is discussed elsewhere in this handbook. In this article the use of industrial instrumen tation in separations processing is considered. Dependability is of utmost concern. Dependability, as used here, means almost an absence of maintenance sometimes under conditions of abuse. Maintenance problems are discussed in Art. 9.3. 8.11 A detailed discussion of process instrumentation General Considerations. is beyond the scope of this handbook. For a more complete discussion of specific instruments and principles, refer to the bibliography of this article. Most of the instruments in a radiochemical plant are installed in nonradioactive areas, and these are handled exactly as in any comparative plant. Those instruments installed in radioactive cells, however, constitute a special problem, and it is with this special case that this article is concerned. Success in applying instruments in radiochemical processing depends a great deal on adhering to the normal "rules" pertinent to all industrial instrumentation and several additional ones peculiar to this special application. Several of these are here listed : 1. Never install an instrument in a hot cell unless it is absolutely necessary. 2. Minimize the piping of instrument lines between radioactive and nonradioactive areas. 3. Standardize on instrument 4. Eliminate gadgeteering.

models.

5. Assure uninterrupted service by multiple installation. 6. Consider radiation damage to all instrument components. 7. Consider decontamination as a major process operation.

t Argonne National Laboratory.

Art.

SPECIAL PROBLEMS

11-115

Air lines and other service piping leading from the control room to radioactive equipment furnish a possible path for the backflow of radioactive liquid or Efforts should be made to avoid instrument installation where low-pressure air gas. systems are separated from high-pressure radioactive fluids by merely a thin dia Where possible, electrical transmission18 " between the two areas is highly phragm. preferred, but this is often limited by the breakdown of normal electrical insulators80 in intense radioactive fields and the lack of a good electrical process valve. Instrument standardization is important in any industry on the basis of simple It does not mean that stocking, familiar maintenance, and the ease of procurement. all instrumentation should be furnished by a single vender, but it does mean that all This is items of a specific type of element should be of the same size and design. particularly important in radiochemical processing because of the speed with which repairs must be accomplished. Gadgeteering is an essential part of research operations but is dangerous in largeToo often the successful operation or replacement of a gadget scale operations. depends on the availability of services or advice from a single individual or group. Where the operation of an instrument in a radioactive cell is essential to a process, continuity of operation should be assured by installation of several instruments. PROCESS in the case of remote mainte PROCESS A B Often, VESSEL VESSEL nance, it is sufficient to install extra piping 1 r or wiring. i -fOi O— j Figure 49 shows a system recommended where a simple open-shut valve is necesC D sary. With this system, the extra valves Fio. 49. Multiple-valve system, b, c, and d permit normal operation when An improved version would incor valve a fails in either the open or shut position. porate completely separate piping for the a-b and c-d runs. A great number of existing radiochemical processing plants work at a sufficiently low activity level so that no appreciable radiation damage occurs in the normal life The trend, however, is toward higher activity levels, and this time of an instrument. factor deserves serious consideration. Metals suffer relatively inconsequential damage in y fields, but plastic or rubber gaskets, lubricating greases, and electrical insulation may be extensively harmed. The need for decontamination can hardly be avoided, and, therefore, each instru ment should be installed to withstand this operation. Decontamination procedures should not be modified to permit the use of a specific instrument; rather, the instru ment should be originally chosen to permit the use of all contemplated decontamina tion methods. 8.12 Temperature Measurement. Temperature measurement in radiochemical Most temperature processing is one of the most important instrument applications. measurements can best be accomplished by use of thermocouples. Since in most radiochemical processes appreciable runs of extension leads are required, the use of millivoltmeter-type instruments is not advisable, and most instruments used are automatic balancing potentiometers. Various modifications of the Brown Eleetronik, Bristol Dynamaster, and Leeds and Northrup Speedomax have proved satis Electronic detection of unbalance is advisable for minimum maintenance. factory. The replacement of thermocouple extension lead wiring or any other wiring between the radioactive process area and the control room can be quite difficult. Therefore, spares should be installed and the wires should be thoroughly protected against Polyvinyl insulation provides adequate protection in many cases. For corrosion. extreme conditions, it is now possible to obtain ceramic- or Teflon-coated wire. Vaportight fittings are necessary where conduit is used. All hardware should be of a mate rial that will stand the corrosive vapors expected and any decontamination procedures contemplated. An example of such conduit material is the polythene-coated Dekoron conduit made by the Samuel Moore Company; it will not, however, withstand abrasive Conduit runs between the hot cell and the control decontamination procedures. room must be thoroughly sealed or air flushed. process


8]

11-116


[SEC.

11

The use of normal thermocouple wells in conjunction with conduit creates a hazard. A leak in a well installed in a pressure vessel may force liquid up the conduit and perhaps to the control room. This may be prevented by any of the following methods: Use of a double well. However, this introduces additional temperature lag. Connection to the well by short lengths of lead run without conduit. This may present a corrosion problem. 3. Installation of the thermocouple external to the process vessel or piping. Also introduces additional lag and error. While thermocouples serve for most temperature measurements, some must be Resistance thermometers are used for extreme accomplished by other means. accuracy and where narrow temperature spans are encountered. The high-precision laboratory models are usually too delicate to withstand the type of handling necessary. Extreme care must be taken to limit corrosion of the lead wires. For extremely high Total radiation pyrometers adapt temperatures, radiation pyrometry is used. Optical pyrometers can be used in conjunc reasonably well to hot cell applications. tion with periscopes.81 Pyrometers utilizing photocells are generally useless in s Radiation pyrometry in intense radiation areas is made difficult high-radiation field. by the darkening of glass; in extreme cases, appreciable darkening occurs in a few minutes. The problem can probably be alleviated through the use of metal mirrors or shielded periscope lens systems. Pressure Measurement. 8.13 Pressure measurement presents a greater problem than temperature measurement. There is no general solution, and each case must Indirect connections between the hot cell instrumentation be solved independently. and the control room are preferred to direct fluid lines. Most pressure measurements are made using pneumatic transmitters with metal bellows or diaphragm primary elements." Generally, bellows primary elements are more sensitive measuring elements but are not quite so dependable as diaphragm In either case, the metal must be thin and of a material suitable primary elements. This enhances the possibility of failure under for the necessary forming operation. corrosive conditions. Where pneumatic transmission or similar instrumentation is necessary, various protective measures can be taken. The extent of protection will depend upon the Several safety measures arc activity level and the particular instrument design. illustrated in Fig. 50. possible, pressure transmitters or other instrument connections 1. Wherever Rrlyimr should be introduced into a vapor space rather than below the liquid level. upon a trapped air space alone as a protective measure, however, is not recommended. 2. All transmitter cases are vented, and these vents serve as an air purge, lessening These vents should not be the attack by the corrosive vapors from the cell area. enlarged. Rather a low-pressure rupture disk should be installed in the case as protection in the event the vents should plug. of an enlarged section, or "fat spot," in the 3. In some cases, the introduction This should be at a high level and of transmitter output line is recommended. appreciable volume compared with the trapped volume of the line above it and the Normally, this type of chamber cannot be used in conjunction with receiver gauge. automatic control of flow. 4. A shut off valve in the output line is recommended, and it should be as near the The safest location is inside the hot cell, but, maintenance shielding wall as possible. problems might be great enough to warrant its installation outside. 5. As much as possible of the vertical sections of the output line should be inside the hot cell, and all horizontal lines should slope toward the process. 6. In cases of extreme hazard and where a large number of transmitters are used, the All filtering and installation of an air-supply tank in the hot cell may be advisable. The tank should have an reducing stations should be located outside the shielding. All transmitters should be connected appropriate liquid-level alarm and a drain. to the single tank. The most used method of pressure measurement involves pneumatic transmission Here an instrument converts a pressure measurement to a proportional air pressure 1.


2.

Art.

SPECIAL

8]

PROBLEMS

11-117

that may be measured by a conventional "receiver" gauge in the control room (see Many of the major instrument companies make good pneumatic trans Fig. 51). mitters for pressure or differential-pressure measurement. Examples of these are the Taylor Model 226R pressure transmitter and the Foxboro d/p cell, a differentialpressure transmitter. RECEIVER GAUGE

SHUT-OFF VALVE (4)

PRESSURE

TRANSMITTERhpup-IW AIR SUPPLY

In

LINE

f8

J\

AIR COMPRESSOR

AIR SUPPLY TANK (6)

(I)

SPACE

CONTROL

PROCESS VESSEL

ROOM

LEVEL ALARM

HOT

CELL

Fig.

50. Safety

DRAIN

measures for pneumatic pressure transmission.

corrosion rates are so high that the thin walls necessary for sensi a time to be practical. For this the Minneapolis-Honeywell application, Company developed their differential con verter, which utilizes a diaphragm consist ing of a Teflon coating over glass cloth. All the pressure transmitters mentioned so far are null-type instruments insofar as the transmitter section concerned. However, in each case the measuring ele ment must remain in a deflected position during measurement. The Booth-Cromer HOT CELL a true null pressure instrument. gauge The principle of this gauge shown in Fig. 52. The actual unit, described in the pat Fio. 51. Pressure measurement with pneu ent, utilizes a completely automatic sens matic transmission. ing and balancing unit. The balancing pressure, equal to the measured pressure, can be read on any standard gauge or manometer. 8.14 Flow Metering. The measurement of the rate of flow of radioactive liquids has received intensive study. Volumetric metering can usually be accomplished by weigh tanks utilizing force cells, by calibrated tanks with level measurement, or by an integration device in a rate-of-flow meter. The normal advantages of the orifice meter are somewhat offset by the features some processes,

is

is

f

is

tive diaphragms or bellows would last too short

t


VAPOR

DISC (2)

S. Cromer, et at.. U.S. Patent 2,842,905.


11-118

[SEC.

11


inherent in radiochemical processing. Orifice removal for cleanout or substitution All differential head meters suffer may be difficult under radioactive conditions. common shortcomings, the worst being the necessity for a differential-pressure transmitter in the hot cell. Where a differential head-measuring element seems

Fio warranted,

requiring

venturi-type

an

encased

52. Booth-Cromer pressure gage.

Remote elements are superior to orifices or flow nozzles. maintenance is equally complicated, but the selfcleaning and low-pressure-drop features warrant Many of the advantages of the extra initial cost. the true venturi can be realized at lower cost with "flow tubes" available from various sources. Area-type meters, especially the rotameter, have proved very useful in radiochemical processing. Recent improvements in the rotameter have made It has it a versatile and dependable instrument. the disadvantage of cutting off at low flows, but this is common to most alternate met hods. Differ ential head instruments cover large flow ranges only if several different range transmitters are installed. A very practical form of the rotameter for radio chemical flows is the all-metal unit with an induc Models of this tive sensing element (Fig. 53). type are available from the Brooks Rotameter Co., Co., and the Schutte and the Fischer-Porter Koerting Co. They are adaptable to either pneu All-welded con matic or electrical transmission. struction is recommended; annular compression A viscosity corrective gaskets should not be used. The design must float design should be used. provide for flushing of the armature chamber the complexity by even though this increases sensing coil.

Art.

SPECIAL PROBLEMS

8]

11-119

Appreciable development work has been done at Oak Ridge National Laboratory Similar meters for large to adapt the Thomas meter8* for use with small liquid flows. This type of meter functions by measuring flows have been available commercially. the temperature rise of a moving fluid when adding heat at a known rate. It is often more convenient to maintain a fixed temperature rise and measure the heating power required. The principle of operation is shown in Fig. 54. The unbalance resistor is adjusted so that the galvanometer reads zero when there is a 2° difference between the thermo When a flow change occurs, the rheostat is adjusted until a zero reading couples. on the galvanometer indicates this temperature difference. The wattmeter reading In practice this method has not been successful for is then an indication of the flow. It has led, however, to a very useful gas-flow meter. small flows of corrosive liquids. POWER

RHEOSTAT

^


|WATTMETER

FLOW

THERMOCOUPLE

^

A_

THERMOCOUPLE HEATER

UNBALANCE RESISTOR GALVANOMETER

Fio.

54. Thomas flow meter.

The Thomas type of meter depends on a temperature rise inversely proportional to volumetric flow rate when heat is furnished the fluid at a constant rate. This modi fied instrument depends on a temperature rise directly proportional to flow rate when heat at a constant rate is furnished to a portion of the outside of the flow tube. Tem peratures in the latter case are measured on the outside of the flow tube. The range may be changed by varying the heat-input rate. Development work on this instru ment is not yet complete, but it has proved extremely useful in measuring gas-flow rates of fluorine, hydrogen fluoride, and particularly corrosive interhalogens. A very important flow meter is the electromagnetic flow meter. Although this type of instrument has been used since 1930, mostly in the measurement of the flow of blood, only since 1950 have its industrial possibilities been realized. Today, it is the most important instrument used for the metering of molten metals such as NaK and sodium. It is also used for the metering of more common liquids, but with less The principle of this meter is shown in Fig. 55. The flow meter is in essence success. a d-c generator. The liquid, a conductor, moves in a magnetic field inducing a voltage at right angles to the field and the direction of movement ; electrodes are used to measure this voltage. The voltage generated depends only on the electrode space,


11-120


[Sec.

11

field strength, and linear velocity of the fluid, and it is independent of the nature of the The flow tube must be nonconductive relative to the liquid and of low liquid. magnetic permeability. Details of electromagnetic flow meters for liquid metals are readily available in open literature,84 and various sized models are commercially available. ELECTRODES 8.15 Liquid-level and Interface Meas measurement is urements. Liquid-level the determination of the location of a gasliquid interface, while interface measure ment normally refers to a liquid-liquid interface. The measurement problem is of varying complexity, depending on whether the desire is to know (1) that the interface is above or below a fixed point, (2) that the interface is between a specific pair of several fixed points, or (3) the exact posi tion of the interface at all times. Only simple on-off or floating control is possible MAGNETIC FIELD with cases 1 and 2, but case 3 may be used Fig. 55. Electromagnetic flow meter. with proportional-band control. Various means of measurement are pos sible,! all depending upon a difference in some physical property between the phases. The properties of major interest are density and electrical resistance. TORQUE

VAPOR

TORQUE

FLOAT

2

TUBE

H

iTRANSMTTES!

-FLOAT

TRANSMITTER

LIQUID

TUBE

LIQUID

—3

TYPE B

TYPE A

Fia.

5G.

Buoyancy-type liquid-level instruments.

Buoyancy-type instruments have few applications in radiochemical processing The two types shown in Fig. 56 have some application. Use of "torque" or "flexure" tubes permits all-welded installations. Either type may be used also for interface measurement. Type a works over a very limited range and is generally used only for simple high-low indication, while type b can give continuous level indication overs long range. Type 6 can also be used, when completely submerged, to measure specify gravity of a single liquid or to measure interface level when the specific gravities of the two liquids are known. The more useful forms of hydrostatic-head-type instruments are shown in Fig. 57. Types o and c are bubbling tube instruments and are used where air purges are per Types 6 and d are missible from both a process and radioactive-hazard standpoint. sealed-system counterparts of types a and c. Types a and 6 can be used only on vented systems, while c and d can be used on closed systems. Types c and d can also be used for specific gravity or interface measurement if both reference pipes are The measuring element is usually submerged. Type 6 deserves a special precaution. a thin metal diaphragm or bellows and hence very susceptible to corrosion failure. For this reason, this element should not be located at the liquid-vapor interface, < exceptional corrosion effects often occur. tSee Ref. 44. pp. 1289-1290.

Art.

SPECIAL PROBLEMS

8]

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58 shows five types of electrical liquid-level instruments. All types have advantage of all-electrical transmission and eliminate thin sealing diaphragms. Types o and 6 adapt only to high-low applications; the others give continuous level Figure

the

indications. Type a is normally used for overflow alarms.

It can, in some cases, be used to locate liquid-liquid interface if the electrical conductivities of the two liquids are suffi The main disadvantage of this type is that it may short out owing ciently different. to the formation of liquid films or salt crusts across the insulator. This electrode sys tem sparks slightly in operation and may be a fire hazard.


a

U

AIR

AIR

^DIFFERENTIAL

PRESSURE [TRANSMITTER

VAPOR

l-LIQUID TYPE C

VAPOR DIFFERENTIAL PRESSURE TRANSMITTER

H TYPE 0

Fig. 57. Hydrostatic-head liquid-level instruments.

Type 6, which is relatively new, utilizes a thermistor, separately or self-heated, and depends on the difference in cooling rates when immersed in liquid and gas phases. The instrumentation is more complicated than in the case of the simple electrode, but the sparking is eliminated. The effect upon thermistors of intense radiation has not yet been determined. The normal type of thermistor is limited to 150°C maximum temperature. The high-resistance-wire level indicator shown in Fig. 58 (type c) has not proved very successful in any radiochemical process to date. It may, however, prove useful in some liquid metal processing if the liquid metal used does not wet the fine wire appre ciably. This method can be used only with liquids of high electrical conductivity. The principle of inductance pickup is utilized quite successfully with remote rotameters; however, its application to level measurement is generally less successful. Ideally, the instrument should not be on a side arm as shown in d but in a guide tube inside the vessel utilizing an encased sensing coil. All materials except the armature must be nonmagnetic. A free-moving float such as this is very prone to bind, causing maintenance problems.

11-122


[SeC.

11

The capacitance probe method (Fig. 58, type c) of level or interface measurement is of the most promising methods. The instrument measures the capacitance between an insulated rod and a grounded tube surrounding it. The capacitance is a function of the dielectric constant of the material surrounding the rod. Since the dielectric constant varies between liquids and is very different for gases, this principle may be used to measure liquid or interface level. Conductive or nonconductive Long leads to the capacitance bridge introduce appreciable liquids can be used. error; hence the electronic circuitry must usually be installed in the hot cell. The one

HIGH RESISTANCE

LIQUID

nTYPE B

TYPE A


WIRE

INDUCTIVE BRIDGE .-INDUCTIVE

\l

TYPE C

JCARAOTANCE BRIDGE

REFERENCE GROUND

TUBE-

COIL

I^HNSULATED

»r FLOAT WITH SOFT IRON ARMATURE

rod

fc— LIQUID

TYPE D

—~

TYPE E

Fiq.

58.

Electrical liquid-level instruments.

need of insulation is a disadvantage. Capacitance probe systems are available the Foxboro Co. and the Fielden Instrument Co. 8.2

from

Corrosion

By W. B. Seefeldtf Much has been written about the tremendous annual cost of corrosion. The AEC, in its chemical-processing operations, has not been immune to the ravages of cor rosion, and large amounts of effort have been expended on programs to reduce i; Speller86 defines corrosion as the destruction of a metal by chemical or electrochemi cal reaction with its environment. In a very broad sense it is necessary to include erosion as a subheading, although erosion is fundamentally comminution by mechani Erosion, for example, can disturb or remove protective films, thus allow cal means. ing corrosion to proceed at a more rapid rate, and in any corrosion problem this must be taken into account. 8.21 Mechanism. Corrosion is a complex phenomenon that may take one oi several forms. corrosion involves the transfer of electrons from Fundamentally, metal atoms to some constituent in the solution. The entire corrosion reaction can be divided into anodic and cathodic portions that occur simultaneously at discrete points on a metal surface. The anodic reaction occurs at the point where the metal is oxidized and electrons are released. These pass through the conducting metal to s Here the electrons cotnbiof second location where the cathodic reaction occurs. t

Argonne


Art.

special problems

8]

11-123

either with some positive ions, usually hydrogen, or with oxygen and water to form hydroxyl ions. Anode reaction:

M -»


Cathode reactions:

M+«

+

2e~

2e- + 2H+ -» 2e- + HOs + H,0 —

H, 20H-

The reactions shown here are typical of those occurring at local anodes and cathodes for a divalent metal. If the locations where these reactions occur remain stationary, the conclusion must be reached that the metal would be attacked preferentially at Evi specific points, and this is the pitting kind of attack which is so undesirable. dently, where corrosive attack is uniform, these anodic and cathodic areas must be shifting about randomly. This explanation of corrosion is generally known as the electrochemical theory of corrosion. The products of the two reactions in turn can have secondary reactions of great consequence, either with each other or with other These secondary reactions and the point components present in the environment. where they occur frequently reduce the attack to an acceptable level, although freeenergy calculations indicate a very high tendency to corrode. For example, iron placed in pure water will undergo initial reactions at its anodes This will occur in both oxygenated water or water nearly to form ferrous hydroxide. The secondary reactions with additional oxygen to form ferric depleted in oxygen. In oxygenated water, the hydroxide occur in each case, but the location is different. reaction occurs adjacent to the metal surface, the insoluble product on the metal Such protection is not achieved in water nearly depleted in stifling further attack. oxygen, as the secondary reaction occurs at some distance from the metal. Generally the rate of attack of iron in water increases with oxygen content until a rather high oxygen concentration is reached; beyond this the corrosion decreases to nearly zero In this case and most others, the anodic reaction and as saturation is approached." its secondary reactions are controlling, but there are instances where the cathodic reaction controls. Specific anodic and cathodic areas can be induced in metals by various means. The tendency for metals to corrode is indicated in a general way by the position of the metal in the electromotive series. If metals are chosen whose positions differ mark edly and are connected electrically, a current flow will be generated that will produce an anodic reaction on one metal, a cathodic at the other. For example, iron and copper in sodium chloride solution produce anodic iron and cathodic copper. This is an example of the case where the corrosion products of the two areas react in the solution to form a nonprotecting precipitate. The more important forms of attack encountered are: 1.

2.

Direct uniform attack

Pitting

3. Galvanic 4. Concentration 5. Intergranular

cell (crevice)

6. Stress

Direct uniform attack may also be termed grain-face corrosion as differentiated from intergranular corrosion. Pitting occurs when anodic areas do not shift randomly. The pitting behavior of stainless steel in the presence of chlorides is well known. Galvanic attack can occur when dissimilar metals are in contact. Dissimilar metals have, as indicated by the galvanic series, t different potentials relative to the These are measurable potentials. When the metals are elec same environment. trically connected, the resulting current flow oxidizes one of the two metals in direct proportion to the electron flow. The current generated is limited by the polariza tion of the metals or in electrical terms by the internal resistance. It is generally

t The galvanic aeries differs from the electromotive aeries in that the metals are arranged in order of their relative potentials in a common environment rather than in solutions that are 1 AT in their respec tive tons. The galvanic series may change somewhat with environment.

11-124

CHEMISTRY

AND


[Sec.

11

assumed that galvanic attack will be greater the farther apart the two metals are in the electromotive series. This is not true when one of the metals polarizes rapidly. Concentration cell corrosion is that which occurs as a result of environmental There are two fundamental types: metal ion cells and oxygen cells. inhomogeneities. The metal ion cell is promoted by differences in tendency for anodic reactions to pro ceed; the oxygen cell is promoted by differences in cathodic reactions. An example of a metal ion cell is given in Fig. 59. The copper electrode in the right compartment will have a more negative potential than the other electrode and hence will undergo the anodic reaction, electrons flowing in the direction indicated.

1N CuSO,

\POROUS MEM8RANES7 -COPPER STRIPS-1 METAL ION CELL


Fig.

59. Illustration

<-IRON

STRIPS-

OXYGEN

CELL

of concentration-cell corrosion.

On the right in Fig. 59 is an example of the oxygen cell. The solution in each com partment is the same, but one is aerated and the other is air-free. As the presence of oxygen promotes the cathodic reaction, the left electrode would be expected to be cathodic and the right electrode would LOW OXYGEN do the corroding. Figure 60 illustrates CONCENTRATON-7 how each of these types might occur in a , HIGH OXYGEN -/— lapped riveted joint. CONCENTRATIONThe stagnant liquid in the crevice (in the top view) prevents fresh liquid con OXYGEN CELL taining higher oxygen concentration from HIGH METAL ION entering and hence sustains the reaction. CONCENTRATIONIn the lower view, a metal ion concentra LOW METAL ON .- C\ | tion gradient may be initiated by general C0NCENfrRATION^riVe \ corrosion of the metal by the stagnant solution in the crevice. Metal corroded METAL ON CELL at the crevice entrance is continuously Fio. 60. Concentration cells in lapped washed away by the movement of solu riveted joints. tion; the concentration gradient is thus Since the two cell types cause electron flow in opposite directions, they maintained. Likely locations for the occurrence of these are fortunately not mutually accelerating. types are crevices, and, in designing, these must be minimized or eliminated. Intergranular corrosion, as the name suggests, is attack that occurs along metal This is a particu grain boundaries as compared with the grain-face type of attack. larly vicious kind of attack, since it physically removes individual grains whose integ rity and structure are sound. This leads to very rapid rates of attack and early failure. The corrosion type occurs readily in stainless steels if proper heat-treatment pro cedures are not followed. Stainless steels that have been heated in the range of 800 to 1500°F suffer inter granular attack, while those heat-treated in the range of 1900 to 2100°F followed by rapid quenching are generally free of it. The chromium-depletion theory is a possible According to this theory, normally soluble carbon precipitates along explanation. grain boundaries when metals are heated in the sensitive range (800 to 1500°F). This carbon reacts with adjacent chromium, thus leaving a chromium-deficient area subject to preferential attack by the corrosive medium. The manifestation of Stress cracking is the last of the corrosion forms considered.

f£gH

—

Art.

8]

special problems

11-125

this type is a physical cracking of the metal in an area where the metal is stressed, It is not evident in annealed and unstressed material. either locally or generally. Some of the earliest examples of this type of attack involved brass cartridge cases, which made ammunition worthless during wet seasons, and riveted steel locomotive boilers that suffered "caustic" embrittlement and occasional failure by explosion. Fortunately, stress corrosion occurs much less frequently than other forms, but often failures are serious and unexpected. 8.22 Metal Factors. Other forms of attack such as corrosion fatigue, fretting, cavitation erosion, dezincification, knife-line attack, underfilm corrosion, and others are important in certain areas and are covered in the literature."'" Factors influencing all the foregoing forms of corrosion may be connected either Some of the more important metal with the metal itself or with its environment. factors are: Effective electrode potential of a metal in solution of hydrogen 3. Chemical and physical homogeneity of the metal surface 4. Inherent ability to form an insoluble protective film 1.


2. Overvoltage

The effective electrode potentials and their relationship to galvanic corrosion have The potentials indicated in the galvanic series do not already been mentioned. necessarily imply that the relative positions of the metals are fixed. Temperatures and concentrations have effects on potentials that can cause reversals in these posi By way of example, zinc is anodic to iron in fresh water at most conditions, tions. These changes but above 150°F the iron becomes anodic and corrodes preferentially. Polarization will not change the relative are separate from any polarization effects. positions of two metals but will determine the degree of acceleration of attack in the anodic metal. If hydrogen always plated out at the cathode at its reversible equilibrium potential, many metals would corrode at much greater than known rates. Fortunately, an additional increment of potential is required to plate out hydrogen, and this incre Low overvoltages tend to increase rates of ment is termed hydrogen overvoltage. attack. In some instances, materials can be added to the environment that will increase overvoltage and thus reduce the corrosive effect. If a metal is not homogeneous upon insertion in a solution, there will be local dif It is possible that this can ferences of potential at different points on the surface. result in pitting or that these effects can eventually be cancellable by shifting of the Causes of nonhomogeneity can areas. These may or may not affect the final results. be differences of metal composition or phase composition, incompletely removed mill scale, presence of stresses, and surface texture. The last accounts for the ini tially low rates of attack usually obtained on polished surfaces. These are often not permanent effects, and an initially polished surface may, after prolonged exposure, be attacked at precisely the same rate as a specimen not so prepared. The inherent ability to form an insoluble protective film is probably the most impor tant metal factor. Lead is resistant to sulfuric acid, silver to hydrochloric acid, and magnesium to hydrofluoric acid because the reaction product in each case is insoluble and forms an adhering protective film. This is not so with iron in these materials; attack is uninhibited because the reaction products are soluble. These are examples of thick protective films. Another example is the thin, adherent, and often invisible Stainless film that contributes to the passivity of such materials as stainless steels. Stainless steels in the steels that are not in the passive state are said to be active. active state are not far below carbon steel in the galvanic series, but in the passive Reducing conditions readily destroy state they are among the most noble materials. It is generally accepted that these the protective film and the associated passivity. films on stainless steels, chromium, Hastelloys, and Monels are formed by either reaction with or absorption of some constituent in the environment, usually oxygen. Where passivation is operative, it is important that a metal can be repassified in local areas which may have suffered mechanical abuse. Materials called passivaiors can be added to the environment in small quantities "A passivator," according to to assist metals in acquiring a passive condition.


11-126


[SEC.

11

Uhlig, f "is an inhibitor which appreciably changes the potential of a metal to a i For stainless steel, Fe+1, Cu+l, and NOj~ act as passivators. cathodic value." Sodium nitrite is used as an oil-pipeline passivator and in this service is said to make iron pipe several tenths of a volt more noble. Inhibitors generally do not change the potential of the metal but nevertheless do reduce the rate of corrosive attack. In this instance, inhibitors act by reducing the tendency for either anodic or cathodic reactions to proceed. Most often, anodic inhibitors act by forming a sparingly soluble product on the anodic areas. These protective films act as an additional ohmic resistance through which electrons and ions must pass, thus inhibiting the rate at which the anodic reaction can proceed. Generally, anodic inhibitors are more efficient than the cathodic. Typical of mate rials used as anodic inhibitors are phosphates, silicates, and chromates. The last, used in conjunction with iron and cooling water, react with soluble ferrous salts, throwing down a mixture of hydrated ferric and chromic oxides. In some instances, materials added as inhibitors accelerate the rate of attack in local areas because the decrease in anodic areas is more rapid than the decrease in rate of attack. When anodic inhibitors are added to an environment, certain areas are more susceptible to this accelerated attack than others, and mostly these are areas where fluids are apt to stagnate such as in corners and crevices, under sludge deposits, and the like. Cathodic inhibitors are, perhaps, not so efficient as the anodic but are considerably safer from the point of view of accelerating local attack. One of the better examples of cathodic inhibition is the addition of calcium bicarbonate to hard natural waters in steel pipes. Initial reactions produce an increase of pH at cathodic areas, which pre cipitates calcium carbonate as a protective coating. 8.23 Environmental Factors. Important environmental factors influencing cor rosion are: Concentration and pH Temperature 3. Aeration 4. Velocity 5. Additives 6. Time 7. Interface 1.

2.

Most of the factors listed here are perhaps obvious, but in the evaluation of a prac tical system they must be taken into account. In order to emphasize more strongly the importance of these, examples will be of systems that are sensitive to each. given | I The rate of corrosion of zirconium in sulfuric acid at 60°C is extremely low (<1 | I mil /year) at concentrations up to 70 per cent / acid, is still moderate (3.7 mils/year) at 75 per / cent, but from 75 to 80 per cent there is a / 50-fold increase in rate. Aluminum in nitric acid shows a reverse concentration sensitivity. The rate of attack is unacceptably high (80 mils/year) at 50 per cent, at 80 per cent is i reduced to 10 mils /year, and at 90 per cent is 300 200 so 100 i aimost nU. TEMPERATURE,? Figure 61 illustrates the effect of temperaFig. 61. Effect of temperature on corture on the system of 304 stainless steel in 68 rosion of SS-304 in 68 per cent HNOi. per cent nitric acid. The effect of aeration is illustrated by the results of exposure of Hastelloy B to a solution of 6 Af HC1 — 1 M HtSO, at room When the solution is aerated with air, the corrosion rate is 4 mils/year; temperature. when aerated with helium, the rate of attack is only 0.6 mil/year. As can be seen in Fig. 62, increasing velocity can be helpful or damaging, <

/

t Bee

Ref . 87, p. xxix.

Art.

8]

11-127

SPECIAL PROBLEMS

on the environment. In this case corrosion of the stainless steel is decreased while that of aluminum is increased. Type Additives or corrosion products can have accelerating or inhibiting effects. 18-8 stainless steels are very sensitive to corrosion product build-up in boiling concen This trated nitric acid. Chromium in such solutions causes intergranular attack. effect is illustrated in Fig. 63. On the other hand, a markedly beneficial additive, classifiable as a passivator, is cupric ion at concentrations above 0.03 per cent in phos Its effect is illustrated in Fig. 64 phoric acid solutions. Frequently, rates of attack are very high initially and decrease for some definite period until an equilibrium rate is achieved. This is characteristic of systems wherein

VELOCITY, ft/sec


Fio.

62. Effect of velocity on corrosion

VELOCITY,

of SS-347

ft/sec

and aluminum by fuming nitric acid.

PER CENT COPPER ADDED

Fio.

Fio.

212, 1949.)

1953.)

63. Effect of chromium content on cor rosion of 18-8 stainless steel by boiling 65 per cent nitric acid. (From W. B. Delong, " Testing Multiple Specimens of Stainless Steels in a Modified Nitric Acid Test Appara tus," ASTM Special Tech. Publ., no. 93, p.

64. Effect of cupric ion on corrosion of 18-8 stainless steel by 85 per cent phos phoric acid at 270°F. (from F . H. Beck " Corrosion by Aqueous and M. G. Fontana, Temperatures and Solutions at Elevated Pressures," 9, no. 8, p. 292, Corrosion,

metals depend on oxide films for protection; during film build-up, rate of base metal loss is high. A system in which the metal does not build up a thick film but nonethe less exhibits these same characteristics is Hastelloy B in contact with 6 M HC1 — The 1 M HsSO* plus simulated corrosion product such as 0.1 per cent ferric ion. depth of attack is more than 1 mil during the first day, then settles down to an accepta

ble

2 to 2.5 mils /year for the remaining exposure. Some materials are susceptible to localized corrosion at well defined liquid levels. By partially immersing a material, it is sometimes possible to induce anodic and If the anodic reaction is not inhibited in some way, preferential cathodic areas. attack will occur below the interface. It is possible to reduce attack by excluding air and providing an inert blanket over the solution. This is sometimes done in the nickel or Monel and hydrofluoric acid system, wherein interface corrosion can be a

problem.


11-128

[SEC.

11

8.24 Corrosion Experience in Separations Processing. In this article there will be considered some of the observed operating experience noted with aqueous processes A few observations are also of the solvent-extraction type discussed in Art. 5.3. included on the corrosion to be expected in interhalogen systems and in the processing

of molten uranium. The solvent-extraction 1.

process

consists of the following separate operations

:

Dissolution

2. Feed make-up 3. Solvent extraction 4. Solvent cleaning and storage 5. Waste processing and storage

Table 35 lists recommended materials of construction for various steps in a solventextraction process. Dissolution is usually accomplished with 70 per cent nitric acid and at a tempera ture corresponding to the boiling point.- A search in the standard corrosion references will reveal three basic materials that may be satisfactory: noble metals including tantalum, high-silicon irons, and austenitic stainless steels. Table 35.


Process

Recommended Materials of Construction for Nitrate Solvent-extraction Systems Material

step

!. Dissolution.

Limitations and comments

Tantalum SS 309. 310 SS 304L, 316L, 347 SS 304, 316

Zirconium 2. Feed adjustment.

SS 304L, 316L, 347 (if

3. Solvent extraction.

SS 304. 316 (if heattreated) SS 347. 304L

not heat-treated)

SS 316. 316L 4. Aqueous waste storage ■ 5. Solvent storage: a. Nonhalogenated solvents

solvent ex

Same

as

Same

as solvent

traction

High cost Sensitivity to fluoride ions Cannot be used for caustic decanning Sensitivity to accumulated dissolved corrosion product Same as SS 309 and 310; rates of attack may be as high as 30 mils /year in clean acid Same as above; these need to be properly heattreated to prevent intergranular attack No fabrication experience for chemical process ing None None

If

all weldments are heat- treatable, SS 304 would be satisfactory. These might be sub ject to pitting when a halogenated solvent is

used SS 316 and 316L are more resistant to pitting and might be satisfactory with a halogenated solvent where 347 and 304 may not be If a halogenated solvent is used, removal of solvent may be required before storage

ex

traction

b. Halogenated solvents con taining residual moisture. Hastelloy 6. Waste-volume reduction. . . , Tantalum

C

High cost Sensitivity to fluoride ion

If

flow sheet can be designed that will prevent the accumulation of corrosion products in high concentrations of boiling acid, the may be widened choices to include those listed under dissolution

Tantalum has virtually no limitations for this system except fluoride content and Five parts per million of fluoride ion in solution can accelerate attack to high pH. a dangerous point, while high pH precludes the use of the basic decanning solutions If the cost can be justified, tantalum can be used. frequently used.


Art.

8]

special problems

11-129

The high-silicon irons are suitable chemically but have physical restrictions. The material is hard and brittle and is subject to thermal shock. Dissolvings are usually done batch-wise, so that alternate heating and cooling are necessary. The resulting thermal-shock hazard may be too great in view of the high levels of activity handled and the value of material processed. The 18-8 and 25-12 stainless steels are satisfactory from the physical point of view In clean acid solutions the chemical resistance is satis but have other limitations. factory but not phenomenal, being of the order of 20 to 30 mils/year for the 18-8 -stainless steels. Furthermore and most important is the corrosion sensitivity of this When system to chromium in solution which is obtained when stainless steel corrodes. chromium concentrations exceed about 0.003 per cent, intergranular attack proceeds at an increasing rate as additional soluble chromium is produced. The higher alloyed stainless steels, such as types 309 or 310, have greater resistance to these acid solutions than the standard 18-8 alloys such as 304, 316, or 347. One factor that is unique in a dissolver is the presence of high levels of f) and y radiation. These high levels of radiation can conceivably have two effects: (1) to change the structure of the metal and in this way effect changes in the corrosion resist ance or (2) to change the environment by causing solution decomposition. Cor rosion studies involving the radiation variable on this system have been few. How ever, much has been done to determine radiation damage in reactors, but isolation of neutron damage from P-y radiation effects has not often been effected. From experi ence that has been obtained in operating dissolvers, it appears that there are no con The effect of radiation on corrosion of stainless steels sequential radiation effects. 309 Cb, 316, and 347 that had been exposed to 10' r of 7 radiation and then tested at room temperature in 20 to 50 per cent nitric acid for 35 days was determined. No increase in rates was observed. The feed make-up preparation offers no additional problems. Temperatures are low, and acid concentrations are moderate. With a reduction in both, rates of attack on stainless steel are reduced, as is the sensitivity to chromium ion. However, con ditions are not sufficiently less severe to consider other materials unless no heating or cooling is done, in which case the high-silicon irons can be considered. In the solvent-extraction operation, the material used must resist both phases plus In the nitric acid systems employed, the stainless steels are any reaction products. indicated as the optimum choice. With halogenated solvents, tests should be made to determine if hydrolysis reactions provide a detrimental quantity of chloride ions. If the chloride concentration becomes excessive, a material resistant to both chlorides and nitrates or, at least, one resistant to pitting would have to be chosen. Among the stainless steels, type 316 is more resistant to pitting than others and may be satisfac tory where other stainless steels are not. To effect certain separations, parts of the extraction equipment may be subjected to conditions more reducing in nature than others. Corrosion-wise, such equipment would probably be the most critical in the extraction process, as reducing conditions tend to remove the passivity of stainless steels. One of the more severe problems that can occur in contacting equipment is the acceleration of local attack in stagnant areas such as might occur in submerged flange This would be particularly true if porous types of gasket material were connections. used. Over a period of time, such areas can become oxygen deficient and anodic attack is possible. Gasket materials like Polythene or Teflon are more satisfactory from this point of view. Storage of waste material presents a long-range corrosion problem. If the wastes are not reduced to dryness, aqueous storage must be used. Since the wastes are highly radioactive, the storage should be virtually foolproof. The presence of the fission products requires additional consideration. The steady-state balance of heat production and heat loss results in some elevated temperature, perhaps even boiling. The temperature has an effect on the rate of attack as does the concentration of cor The presence of solvent rosion products that accumulate through long-range storage. may be helpful or objectionable; its stability must be considered as regards the effect of hydrolysis or decomposition products.


11-130


[SEC.

11

In the volatility separations processes, the solvent medium used is one of several These materials are strong fluorinahalogen compounds such as BrFj, BrFs, or C1F3. tors and react with spent fuel elements to form uranium and fission-product fluorides. Subsequent purification is effected by fractional distillation (see Art. 5.4). Nickel, nickel-base alloys, and copper appear to be very satisfactory for the indi vidual solutions and solution mixtures with or without the presence of uranium. The metals form thick, hard, impervious films that are not readily damaged by mechanical abuse. Measured rates of penetration on these systems have not exceeded 1 mil /year. Effects of variables such as corrosion products, velocity, etc., are not known. It is possible that nonvolatile residues may require an aqueous acid treatment for To prevent rapid attack of the nickel by acids such as nitric, sulfamic acid removal. can be added to inhibit corrosion. To promote fluorinating reactions that would otherwise not be possible or feasible, such as the dissolution of zirconium alloys, higher temperatures are being utilized. One method uses a molten salt bath through which fluorine, hydrogen fluoride, or an interCorrosion of nickel and nickel-base materials under halogen compound is bubbled. these conditions is severe. Several pyrometallurgical processing schemes involving molten metal systems have been proposed (see Art. 5.6). Generally, purified ceramic materials such as aluminum oxide, thorium oxide, zirconium oxide, and magnesium oxide resist the chemical attack of molten uranium. Other factors to be considered are contamination of the melt by the container material, ability of material to withstand steady-state temperature Ceramic materials are generally poor gradients, and sudden temperature changes. in the last two respects. The use of metal containers for molten metals poses other types of corrosion prob lems. Attack can be by a number of methods: intergranular diffusion, diffusion to form intermetallic compounds, dissolution, or mass transfer. The last results from differential solubilities of the metal at two different existing system temperatures. Metal dissolves at the hot location and precipitates at the cold. A metal possessing some usefulness for molten uranium is tantalum. The required processing temperature may be reduced by the addition of a second metal that forms a eutectic with the uranium. This can help the corrosion problem if the second element has no deleterious corrosion effect. 8.25 In view of some of the principles Design Criteria for Corrosion Resistance. discussed, it is evident that there must be an optimum way in which the equipment can be designed and fabricated to minimize or eliminate certain types of corrosion. Many corrosion failures that have occurred do not necessarily stem from a poor choice of material for the environment under question but rather from a design that allowed local attack. Some examples of poor and good design in stainless-steel sys tems will be discussed briefly. Examples of proper and improper design of joints are shown in Fig. 65. The prime objective in these welding techniques is the elimination of crevices and Fillet welds are preferable. Care must be taken in the selection of stagnant areas. weld rod so that the deposited weld metal is not substantially different from the base metal. Welds should preferably be ground in order to reduce further the likelihood of crevices or pits. Thorough inspections should be made, and, where necessary, second weld passes made. If the material is stainless, steel, the condition of the metal relative to intergranular corrosion must also be taken into account. During the welding operation, a tempera ture gradient exists between the cold metal and the point where molten material is being deposited and fused. Somewhere in that gradient adjacent to the weld lone the temperature is in the sensitizing region of 800 and 1500°F in which carbides pre Only heat treatment cipitate, making the area susceptible to intergranular corrosion. will solubilize the damaging carbides. If the equipment being fabricated is of such size or shape that heat treatment is not feasible, so-called stabilized and low-carbon These materials require no heat treat stainless steels should be used for fabrication. ment after welding. In flanged connections, nonfibrous gasket materials should be used if possible.

Art.

SPECIAL PROBLEMS

8]

11-131

Fibrous materials tend to have a wick action that draws corrosive solution into its These small volumes of stagnant solution can cause accelerated local attack pores. and eventual failure. More satisfactory materials are Koroseal, Neoprene, Polythene, and Teflon. 8.26 Corrosion Testing. Tests that can be conducted are of three basic types: laboratory screening tests, laboratory evaluation tests, and field tests. In screening tests it is desirable to test a large number of materials to determine those obviously suitable and unsuitable. Sufficient care, however, must be taken to prevent the possibility of excluding good materials. The environment should be carefully con trolled, and its conditions known as well as possible. Short cuts may be made in the

POOR POOR

GOOD POOR

GOOD GOOD

1

V

^

poor

GOOD

N

DrNr\D

1 r*

POOR

[_^

rA

LAP JOINTS

s\

n

POOR

GOOD

=ii

|_k=GOOD

GOOD

FLANGED

JOINTS TEE CONNECTIONS

Fig.

65.

Examples of proper and improper joint design.

specimen preparation, in measuring areas, and in the length of exposure before Unlike materials should not be placed in the same test, since the corrosion evaluation.

product

of one material may accelerate or inhibit the attack on another material. are screened, more carefully controlled tests are indicated in order to determine the limitations of those materials. This kind of test requires a more precise control of the environment and material. Such evaluations may indicate process changes that are minor as regards the process but major relative to corrosion. Field tests are the final test and imply the contacting of corrosion specimens with the process stream under precisely the same conditions as the equipment proper. applicable to the specific conditions under question only Information so obtained and none other. Application to a similar system can be done with limited confidence, aa minor changes in variables may make the material unusable in a second

Once materials

is


BUTT JOINTS

11-132


Data obtained in a circulatory evaluation.

or accumulating 8.3

system particularly

[SEC. need

11

careful

Critical Mass

The fact that Pu"9 and U"* are capable of self-sustained fission reactions if more than a critical mass of isotope is present must be considered in the design of chemical The amount of a given isotope required for criticality varies with processing plants.

30

20 RADIUS, CM

RADIUS, CM


Fig. 66. Critical mass of of radius of a sphere.

Pu"'

as a function

30

Fig. 67. Critical mass of Pu"' as of radius of an infinite cylinder.

a fum tion

the concentration of the isotope and of other chemical elements present, with the geometry of the system, and with the nature of surrounding material. There are given in Table 36 most of the basic data required for criticality considera tions in the design of an aqueous proc These are essing plant for plutonium. also plotted in Figs. 66, 67, and 68 for spherical, cylindrical, and slab geome A comparison of tries, respectively. critical masses for water solutions of U"6, Us", and Pu"9 is shown in Fig. 69. It can be seen that equipment can be designed that is infinitely safe, i.e., cannot This go critical under any conditions. requires, in general, the use of tanks in which the dimensions are seriously re The alternative to infinitely stricted. ?5

iLo

U. UJ 1.9

on

SO**

I

as: UMIN G NO CHAM 3E IN

HYC ROGE N DEr•JSITY ROC M TE VIPER STURf

1

-

V

5

10

20

1 BARE SY rTEM

"^REFLECT •EDSYSTEM l 1 60 70 40 50

30

500 1000 2000 1500 H ATOMS PER FISSIONABLE ATOM

THICKNESS, CM

Fia.

68.

Critical mass of Pu53' as of an infinite slab.

n

of thickness

function

Fio.

69.

Comparison of critical masses io: and Pus" in water for spherirs)

V", V", systems.

safe design is rigorous inventory batch control to ensure that at no time is more than a critical mass of material permitted to be in any one location. This restriction on

processing batch size frequently

dictates a multiplicity of equipment

in

Art.

special design problems

9] Table 36.

11-133

Calculated Criticality Limits for Plutonium

Solutions

Calculated limits Quantity

System

restricted Water-reflected

Any slab-shaped vessel, any thickness Any vessel, any mass (of plutonium) Cylindrical vessel, any mass Slab-shaped

vessel, any mass

Concentration Mass Mass/unit length Mass /unit area Volume

Cross section Diameter Diameter Thickness

6. 6 g /liter 464 g 10.0 g/cm 0. 39 g/cm' 4. 5 liters 98 cm' 20.5 om 12. 5 cm 4 0 cm

Bare

6. 6 g /liter 1.244 g 22. 4 g/cm 0. 24 g/cm' 20.3 liters 201 cm' 33. 8 cm ■ 25. 6 cm 17. 6 cm

Source: H. A. C. McKay and C. M. Nicholls, "The Criticality Safety of Chemical Plant Producing Fissile Materials," Paper 454, Conf. 8, International Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955.

A method by which criticality limits may be some meet throughput requirements. is to incorporate neutron poisons, such as cadmium or boron, into the equipment or the process solutions. The consequences of exceeding criticality are dependent upon the rate at which The incident might vary from a simmering solution to a gupercriticality is reached. steam explosion. The most serious aspects would be the creation of neutrons and of large amounts of fission-product activity that might well be forcibly scattered through out the cell. Natural uranium in water solution cannot be made to go critical but can in heavywater solution.


what increased

BIBLIOGRAPHY Alien,

R. E.: "Corrosion,

A Bibliography of Unclassified

Report Literature," TID-3048,

1953.

Kckman, D. P.: "Principles of Industrial Process Control," John Wiley & Sons, Inc., New York, 1945. Eckman, D. P.: "Industrial Instrumentation," John Wiley & Sons, Inc., New York, 1950. Evans, U. R.: "Metallic Corrosion and Passivity," Longmans, Green & Co., Inc., New York, 1948. Harty, W. M.: "The Design Philosophy of Remote Operations and Maintenance of Sepa ration Facilities," Nuclear Engineering, part III, Chem. Eng. Prog. Sym. Ser., 50 (13): 115-121

(1954). and R. Worthington: "Corrosion

McKay, R. J.,

Resistance of Metals and Alloys," Reinhold Publishing Corp., New York, 1936. Nelson, G. A. (ed.): "Corrosion Data Survey," Shell Development Company, San Fran cisco, Calif., 1950. Perry, J. H. (ed.): "Chemical Engineers' Handbook," pp. 1265-1337 and sec. 21, 3rd ed., McGraw-Hill Book Company, Inc., New York, 1950. Rabald, E.: "Corrosion Guide," Elsevier Publishing Company, Amsterdam, 1951. Speller, F. N.: "Corrosion, Causes and Prevention," McGraw-Hill Book Company, Inc., New York, 1951. Uhlig, H. H. (ed.): "Corrosion Handbook," John Wiley & Sons, Inc., New York, 1948. 9

SPECIAL DESIGN PROBLEMS RAISED BY RADIOACTIVITYt

The existence of radioactivity in the materials to be handled in any separations process does not in any way change the validity of the ordinary unit operations in chemical engineering. It does, however, raise several special problems in the design and operation of a plant. These problems are brought about by the high degree of t The authors wish to acknowledge the assistance of J. H. Schraidt, Chemical Engineering Division, Argonne National Laboratory, in the preparation of this article.

11-134


11

[SEC.

toxicity of some radioactive materials and by the biological damage that human beings suffer by exposure to radiation. Special design problems discussed in this article are ventilation, shielding, main tenance, and materials handling. 9.1

Ventilation

Of the various ways in which radioactivity may be introduced into the body, breath ing it into the lungs is one of the most hazardous. The proper ventilation of process equipment and surrounding area, therefore, is of great importance. Areas containing radioactive material should always be maintained at a pressure negative to the areas in which operating personnel are to be permitted. A negative pressure equivalent to about 0.025 in. of water is desirable. Sufficient air flow should be provided so that a linear velocity of at least 135 fpm will be maintained at all

» SUPPLY >

OFFICE >AT PRESSURE P,

AIR FLO*

EXHAUST AR FLOW

RELATIVE PRESSURE

j_

p2

'at pressure

AREA AT PRESSURE

CELL AT PRESSURE

P4

•OPERATION

1

1

1

IREACTION VESSEL AT PRESSURE

3PRE-FILTER

\

ISCRUBB^R

HIGH EFFICIENCY

Flo.

70. Desirable

FILTER-

ventilation flow pattern.

it is

is

is

times through any opening to the working areas. Moreover, such openings should be It desirable to arrange air-flow patterns so that the kept to an absolute minimum. clean air moves progressively through more and more contaminated areas and exits This procedure has the advan through the ducts from the most contaminated area. important in air-conditioned areas, and reduces the tage of air conservation, which size of treating facilities. Figure 70 shows an example of a schematic layout for good ventilating flow pattern. Since there often a long run of duct work from the cell to the disposal stack, usually advantageous to install a roughing filter or prefilter at the exit of the cell to help keep the duct work and final treating facility from becoming highly contaminated and to reduce the dust load on any subsequent filtering equipment. Prefilters are Where very high degrees of contamination are possible, generally made of Fiberglas. sel The prefilter provision must be made for remote handling of these prefilters. dom capable of providing sufficient cleanup to allow direct discharge of the ventilating air under all conditions of operation. Additional cleanup takes the form either (H of additional filtration through highly retentive paper media, such as the AEC filter (see Table 42), or through a deep bed sand filter (see Art. 7.4) or (2) of atmospheric or more) stack. dilution by discharge through It seldom tall (generally 200 is

ft

a

is

is


^

C0RR|D0R

P

U.

CLEAN AIR

pr*pt>»Wps

1

1 1

"

DIRECT SUPPLIED

t _j_

I

Art.

SPECIAL DESIGN

9]

PROBLEMS

11-135

necessary to provide for remote removal of the secondary filter, particularly if a prefilter is used. Sometimes in the design of a particularly complicated structure or in the case of an especially hazardous process step, it is advisable to provide separate ventilation for equipment even when it is located inside a ventilated cell. This approach may be


advantageous when the probability of general cell contamination from that particular piece of equipment is very high or when the gases evolved can cause deterioration of the ducts and/or filters. Examples of the latter are dissolver off -gases that contain oxides of nitrogen (see Art. 7.3), or gases containing interhalogen compounds (see Art. 5.4). In both cases, it would be necessary to provide a scrubber to remove these chemical contaminants before the gases could be discharged into the general cell ventilation. In a process evolving quantities of highly volatile or gaseous radioactive fission products, special provisions such as condensers, cold traps, or absorbers should be included. After the treatments indicated above, it is generally safe to discharge the gases to the atmosphere. Used filters are treated as solid waste and generally disposed of to a

Fig.

71. Dry box for handling a emitters.

landfill to be

Filters containing highly toxic materials (plutonium) (see Art. 10). sealed in a plastic bag before removal (see Art. 9.42). 9.2

The

subject

may have

Shielding89

of shielding is considered in detail in Sec. 7-3 of this handbook.

However, some comments are in order in connection with the design of chemical The term shielding generally refers to massive material interposed between plants. It also prop operator and sources of radioactivity to absorb penetrating radiation.

erly

includes the use of a continuous membrane to contain highly toxic materials. a emitters are present without the simultaneous presence of penetrating radia tion, membrane protection may suffice. It is then possible to make direct manual manipulations through glove ports. Equipment used in such cases is generally called a dry box. An example of a simple dry box for bench-scale manipulations is shown

Where

in Fig. of

71.

Penetrating radiation, on the other hand, requires the use of considerable quantities An approximation of radiation to be expected at a distance massive shielding.

through

air from a point source is given in Eq. (55).


11-136

[SEC.

11

/

= intensity of radiation, r/hr C = curies of activity at point source E = energy of radiation, Mev X =■distance from point, ft If, as is generally the case, the calculated value of in the working area is greater than the allowable design tolerance (generally 1 to 10 mr/hr) shielding must be used. Table 37 gives for a variety of materials the thickness in feet needed to reduce the radiation intensity by a factor of ten. Figure 72 shows half-thickness values for various materials as a function of y Note that the shielding effec energy. is roughly proportional to the tiveness density of the material chosen and that the higher the energy of the incident radiation, the greater is the half thickness. From these values it is possible to deduce the thickness required for any degree of The formula to be used is attenuation.

where

/

3.33 log

j


where n is the number of half-thicknesiet. Io is the original intensity of the y radia

1.0 1.5 2.0 2.5 GAMMA ENERGY, MEV

tion, and / the desired intensity. Shielding values are generally expressed in half-thickness values (HTV) or tenlhthickness values (TTV). The equiva lence of half-thickness and tenth-thick ness values is given by Eq. (57).

HTV

Fio. 72. Half-thickness values for various materials as a function of y energy.

= Q.ZOTTV

(57

Fact ors influencing the choice of specific shielding material are cost, space avail able, allowable floor loading, construction methods available, and compatibility with Table

37.

Approximate

Tenth-thickness Values of Various Gamma Radiation Material

Normal concrete Barytes concrete Magnetite concrete Steel Lead

Aluminum Air Water

Density, lb/ft1

ISO

-200 200 185 485 710 170

0.07 62.4 156

Glass: (Ce) Rolledt (Ce) Cast

(Pb) Cast

t Thickness to

170 200 385

Materials for 1-Mev

Tenth thickness. in.

t

6.5

3.9 3.9 4.3

2.0 1.15

5.6

2.900 ft 13.5 6 3

5.8 4.5

2.2

reduce radiation intensity by a factor of 10. X Specially chosen base glass formulation to which t to 2 per cent cerium is added ing under intense 7 irradiation.

to decrease

Art.

9]

SPECIAL DESIGN

PROBLEMS

11-137

The most popular shielding materials are concrete, highother parts of equipment. density concrete f, magnetite, lead, steel, and water. If both highly toxic materials and penetrating radiation are present, the problem is Both the membrane barrier and massive shielding must be very complicated. This complicates the materials-handling problem, as will be discussed in present.

Art.

9.4. 9.3

Maintenance


Ordinary concepts of maintenance are not applicable for equipment processing highly radioactive materials. Therefore, equipment must be designed so that a very minimum of maintenance is required for long periods of time, and when it is required, access and manipulation must be easy. There are two basic maintenance methods used in the industry, remote and direct. Both have their place and advocates. As yet, sufficient comparative experience has not been accumulated to permit clear-cut economic choice between the two methods. 9.31 Remote Maintenance.90 In plants utilizing remote maintenance, equipment The is located in cells such that all of it can be serviced by a single traveling crane. crane is operated from a shielded cab with vision provided by optical instruments or Each piece of equipment is carefully designed so that it can be removed by television. This requires that all connections may be broken remotely by the use by the crane. of special connectors.80 Interconnects must be made with special balance lugs and lifting hooks so that when they are disconnected and lifted by the crane, balance is maintained. Defective equipment is transported to decontamination cells, where it may be cleaned up suffi ciently to permit transport to a burial ground in shielded transfer cars. Sometimes Entry of personnel sufficient decontamination can be provided that repair is possible. into the operating cells once operations have begun is assumed to be impossible, although in extreme cases decontamination in situ and personnel entry may be

necessary. Provision

for remote maintenance is very expensive in original design and con

struction. It significantly reduces maintenance downtime, however. This system is in use at both the Hanford and Savannah River plants, where the operating experi ence with it has been good. 9.32 Direct Maintenance. An alternate to remote maintenance is the directly maintained plant. This system is generally used in pilot, plants and is also used at the In such a design, the equipment should be made as Idaho Chemical-processing Plant. simple and rugged as practical. Components should be isolated from one another by shielding, and decontaminatability be emphasized. Alternate process routes should be provided, and critical items installed in duplicate. The plant design must include sufficient capacity to allow periodic shutdowns for decontamination and maintenance. Provision must be made for decontaminating" equipment inside and out and for cleaning the cells themselves. Spray heads so located that decontaminating solutions can be directed at critical spots are highly desirable. The use of stainless-steel liners on walls and floors of the cells simplifies the cleanup operation. Easy access to process equipment by judicious location of ladders and platforms and by sufficient space However, quick discon allowance around process components is very important. nects and elaborate design of equipment to permit remote removal are not necessary. Before maintenance can be started, the equipment is emptied and flushed. A variety of decontaminating solutions is then put through the equipment. A list of effective agents is given in Table 38. From 1 to 4 weeks are generally required to get For small jobs, the cell may be cleaned to the point where a cell ready for entry. personnel have a few minutes' working time, and a multiplicity of workmen is then used for the repair. For larger jobs, decontamination is continued until at least an hour's working time is available. This method is difficult, but truly amazing jobs can be done. Perhaps the most

t A concrete of density 180 lb/ft* or greater obtained by the use of heavy aggregate barytes. steel punchings).

(magnetite,

11-138

CHEMISTRY AND CHEMICAL Table 38.

Decontaminating

ENGINEERING

[SEC.

11

Agents

Water Steam Soap or detergent Versene Organic solvents

solutions

Citric acid solution 10 % Oxalic acid solution 10% Sodium citrate Tartaric aoid Sodium hydroxide 10% and tartaric acid 2.5% Sodium hydroxide — 5. 10. 20% solutions Potassium permanganate solutions Nitric acid — -various concentrations Cleaning solution Hydrochloric acid solutions Nitric and hydrochloric acid mixtures Hydrofluoric acid solutions Sandblasting Grinding

remarkable job of this sort was accomplished by the Canadians in dismantling the NRX reactor. During a cleanup, large volumes of waste solution are produced. The This method cost of handling the solutions must be charged to the maintenance job. results in a lower initial cost and a higher operating cost than the remotely main tained plant.


9.4

Materials Handling

One of the greatest effects the presence of radioactivity has upon the design of Introducing and chemical-processing facilities lies in the field of materials handling. removing materials and sampling are difficult problems. Taking samples of a radioactive process stream is always a 9.41 Sampling. Whenever the activity is brought through the shielding potential source of trouble. The problem wall or barrier, there is a chance of spillage and release of radioactivity. of sampling has the following facets: 1. Obtaining a representative sample without cross contamination. 2. Keeping the amount of active material outside the cell proper to a minimum. 3. Providing for cleanup in case of a spill. These fall generally into two Many types of samplers have been devised and used. In the former case, a small classes: the repeated flush and the circulating types. amount of solution is drawn up into a sampler, generally by vacuum, and returned through the same or a separate line a number of times before the final sample is taken. Examples of this type are shown in Fig. 73. In the circulating type, a stream of solution is drawn from the process vessel Air or steam jets are through a small sample cup and returned directly to the vessel. generally used to move the liquid. When the circulation is stopped, all the solution An example of this except that in the sample cup drains back to the process tank. type is shown in Fig. 74. In either case, the sample point is located in a semiremote area with restricted access The general surroundings, particularly the and some local shielding is provided. floor area, are best covered with sheet stainless steel to facilitate cleanup of the spills that are bound to occur. Shielded carriers to transport the sample to the analytical facilities usually must be provided. It is necessary to introduce both 9.42 Introduction and Removal of Materials. All items removed are active and inactive materials into chemical-processing cells. considered contaminated, and some items are highly radioactive. The handling of spent fuel elements constitutes one of the more important materialsIf the chemical-processing plant is adjacent to the reactor, the handling problems. storage canal, in which the fuel elements are kept after discharge from the reactor, may run into the first processing cell, and rather simple crane arrangements can be devised to introduce the elements into the first process vessel. If the plants are separate, the fuel must be transported between them. This has

"

Art.

SPECIAL DESIGN

9]

11-139

PROBLEMS


been done in shielded railroad care that run directly into both plants or in a variety of An example of one of these casks to handle transfer casks by motor transport. cylindrical fuel elements is shown in Fig. 75. To introduce fuel into a process vessel from a transfer cask, the cask is positioned over a slug chute and the fuel elements dropped in one at a time through the bottom of the carrier. Inactive solutions are prepared and charged conventionally. Inactive solids may be charged through standard solids feeders, although some method of preventing the blowback of radioactive gases must sometimes be provided. The removal of material is more difficult. Liquid waste streams must be removed through shielded lines. In plants where buried tanks are used for storage of waste, the piping is also buried; often the piping is placed in concrete trenches so that leaks may be detected and repaired. Transfers should be made by means of gravity or

SAMPLE

VIAL

Fio.

73.

Flush-type samples.

Fio.

74. Schematic

circulating-type sampler.

If liquid wastes must be shipped any pressure systems. An example of one is shown in Fig. 76. shielded pots are required. Shipment of radioactive solutions is particularly hazardous and is to be avoided if It is preferable to reduce wastes to solids before transporting them any possible. distance. Large items such as contaminated equipment present a considerable problem. Unless the unit can be at least partially decontaminated, it is usually necessary to pro vide a shielded transfer vessel, which will afford some measure of protection during

vacuum rather than by distance,

special

transfer. The products uranium and plutonium of current operating plants

no longer require However, particular precautions must be taken in the handling massive shielding. of plutonium and U"3 because of the high radiochemical toxicity hazard. Sometimes the product is plated with an inert metal or canned in a metal can. Alternatively it may be transferred into a polythene or polyvinyl bag, which is then heat sealed and cut away from the lock. Wastes are treated in this latter manner. 9.43 Hot Laboratory Facilities.93-94 In conjunction with either research or plant operations, it is almost always necessary to design shielded working areas in which a

11-140

CHEMISTRY

AND


[Sec.

11

'FILL 8 EMPTY LINE VENT LINE HOLD

DOWN BOLTS

GASKETS

STAINLESS STEEL PRIMARY CONTAINER STAINLESS STEEL SECONDARY

CONTAINER

LEAD FILL STEEL SHELL

Fio.

75. Active liquid transfer pot.


, REMOVABLE TOP SHIELD

STEEL SHELL LEAD FILL SLUG

HOLDER

SLUGS DUMPING

GATE

PARTIALLY

WITHDRAWN

Fio.

76. Slug transfer cask — bottom discharge.

variety of operations may be carried out on samples of active materials. The facili ties — referred to as caves — are as varied as the number of designers. Most of them have in common: 1.

Shielding

2. Viewing windows 3. Manipulators 4. Ventilation

Much general information on this subject See Sec. 7-4. 10

can be obtained from the references cited.

WASTE DISPOSAL

and accumulation of fission products in reactor fuel have already The 2.2) as has the decay of these products with time (Art. 2.3). existence of large quantities of long-lived fission products ordains that, no matter what separations process is chosen, troublesome waste streams will result. The unique feature of nuclear energy wastes is that no process known will destroy the radioactivity of the waste. It is only possible to delineate the problem by assessing the environmental insult resulting from discharge of radioactivity, or to reduce the The formation

been discussed

(Art.

Art.

11-141

WASTE DISPOSAL

10]

magnitude of the problem by concentrating the activity into an economically storable volume. Tolerances

10.1

In an attempt to determine the amounts of radioactive material that may safely be discharged to the environment, much work has been done by biologists, physicians, roentgenologists, public-health officials, chemists, and radiological physicists in esti mating the tolerance of the human body for many radioisotopes and calculating per The factors that deter missible concentrations of these isotopes in air and water. mine the hazards of the various radioisotopes are discussed in some detail in Sec. 7-2 of the handbook. Table

Maximum Permissible Amounts of Radioisotopes in the Human Body and Maximum Permissible Concentrations in Air and Water

39.

Microcuriea in total body

Element

Ra»M + }$ dr

0. 1

0.2 0.009 0.04


Pu"» (sol) Pu"» (insol)

0 008 10 63

P" Ca" I»i

0.3

CoM

Sr" 8r« +

Y'i

3 2 1 13 90 50 5 4 90 5 5 120

Y*>

Nb'»

Mo".

Tc"

Ru1M +

Cs'"

Rh'"

+ Ba""» Ba'« + La1"

Ce"» + Pr1M

Pm'«

Microcuriea /ml of water

4 7 1.5

Microcuriea /ml of air

X 10-» X 10' X 10-'

2 X IO-« 5 X 10' 3 X 10' 2 X 10-' 7 X 10' 8 X 10'

0.2 4 X 10« 14

3

X 10' 0. 1

1.5 X 10-' 2 X 10' 4 X 10" 1

8 1.7 1. 7 2 2 1 3 3

X 10"" X lO"11 X 10-" X 10" X lO"" X 10-'

X

I0"»

X

10-'° 10"'

X 10-' io-» 2 X 10-' 2 4 4 2 3 3 2 6 7 2

X X X X X

10-' 10"'

10-' 10"'

X 10-'

X 10-■ X 10-»

X

10"'

Source: National Bureau of Standards, "Handbook 52," Maximum Permissible Amounts of Radio isotopes in the Human Body and Maximum Permissible Concentrations in Air and Water, 1953.

Table

40. Provisional Levels of Permissible Concentration of Radioactive Contaminants for Use Beyond the Control Area (October, 1951) Medium in which contained

Air.... Water.

a emitter, pc/ml 5 X 10" 10"'

Souhce: National Bureau of Standards, "Handbook 52," Maximum Permissible Amounts of Radio isotopes in the Human Body and Maximum Permissible Concentrations in Air and Water, 1953.

The best present estimates of tolerances for radioisotopes fixed in the body and for discharge into either water or air have been accumulated by a National Bureau of Standards Advisory Committee called the Subcommittee on Permissible Internal Doses. These recommendations have boon published in a National Bureau of Standards publication, "Handbook 52," entitled Maximum Permissible Amounts of Radioisotopes in the Human Body and Maximum Permissible Concentrations in Air These recommendations are given in Sec. 7-2 of this handbook. In and Water. Table 39 certain isotopes of particular importance in waste processing are given.

11-142


[SEC.

11

In

the event that nothing is known concerning the isotopic constituents of a waste, to assume that the most hazardous materials are present. The cur rently recommended levels of permissible concentrations of unknown mixtures of radioisotopes in air and water are set forth in Table 40.

it

is necessary

Philosophy

is

in

is

is,

is

is

it

is,

When confronted with the problem of handling a radioactive waste, it is possible either to dilute and disperse it or to concentrate and contain it. Dilution may be accomplished by mixing with sufficient air or water so that the tolerances given in Tables 39 and 40 are not exceeded. Many living organisms are able to reconcentrate activity. This effect can be mini mized by isotopic dilution of the waste. Isotopic dilution is particularly attractive when large quantities of an appropriate inactive isotope are available. For example, C14 is most conveniently disposed of by mixing it with coal in a large power plant. Isotopic dilution is applicable only for small quantities. Environmental dilution is also generally not practical for any large quantity of waste. The only alternatives available are to contain the waste completely under the control of the producer or to store it in such a manner that it is believed that the escape rate to the environment is so low that no important insult will occur or that the escape will be to an area where no human contact is possible. From the standpoint of cost it is always desirable to use the dispersal technique if at all possible. One important factor that limits the usability of this technique is that many of the lower living organisms reconcentrate certain radioactive species to such a point that they are dangerously inserted into an ecological cycle. Many factors must be considered in determining whether or not dispersal can be The location of the waste in relation to large urban populations and public practiced. In this connection of course, particularly important. drinking water supplies public relations rather than radiological physics considerations are often the control The quantities from limited research operations can often be dispersed. ling factor. The much larger quantities from production operations practically never can. In the final analysis, impossible to have any waste concentration process so Therefore, although the efficient that 100 per cent cleanup can be accomplished. not goal might be to discharge materials containing no activity, in practice this therefore, necessary to select some realistic disposal level that can be possible. It dependent obtained at a reasonable cost. This level will vary from site to site and on many factors. It impossible to discuss every conceivable situation. Hence, the next two articles the discussion will be confined to (1) research institutions and (2! production operations. 10.3

Research Laboratory Wastes

/

100

Monitored laboratory drain wastes Laboratory cooling water (unmonitored) radioaotive

35 0.

1

wastes Source: Argonne National Laboratory, calendar Based on total employment. Processed

Research Laboratory

Oallont {day) (employee)

Type o/ watte Sanitary sewage

year

1953.

t

Expected Waste Volumes from

5

Table 41.

a

a

if

is

a

In most research laboratories the quantities of waste that must be handled are In general, only few grams of o-emitting materials and perhaps somewhat limited. 100,000 curies of f}-y activity will be turned over to a waste control group in the course of a year. The biggest complicating factors are the wide variety of radioactive species It seldom, and the wide variety of materials with which these species become mixed. ever, possible to devise a single process that will handle all the problems presented. To give an idea of the quantities of waste volumes expected from a research laboratory. the results of year's operation of one of the national laboratories of the AEC, an institution located near a large urban center and employing 3,000 people, are given in Table 41.

t


10.2

Art.

11-143

WASTE DISPOSAL

10]

Once the laboratory has decided upon the discharge levels that it will attempt to meet and the general processes that will be used, the desired degree of segregation of It is invariably true that the laboratory waste at the source must be determined. It is worker would prefer to discharge all wastes into a single container or sewer. possible to set up a system in which this can be done, but it will result in the frequent necessity of processing relatively large volumes of wastes containing activity only a little above the discharge level. On the other hand, if segregation is practiced at the source, a system can be developed in which a relatively small quantity of high-level However, the large bulk of the waste will be so lightly con waste must be processed.

is

is

is

it

is

is

it

A

is

a

is

it

it

is

is

treatment. Specifications and Characteristics of AEC Filter

8X8

Depth, in.

iH

8J, 20 X 20 24 X 24 12 24 X 24 paper made from specially treated chemical

m

Filter

Thickness, in Ream weight, lb Resistance Percentage penetration on smoke tester generating

10.33

ratory

130 4. in. water at linear air velocity of 28 fpm 0.3m diameter particles with air at 28 lin 0. % max 1

ft/min

0.035-0.045 3

Sites, in.

.

Capacities at -in water pressure drop, cfm 50 450 500 800 wood pulp and blue Bolivian asbestos (~15%): 1

Table 42.

Liquids.

The problems encountered in handling liquid wastes from a labo

is

If

is

is

The greatest difficulty encountered in are generally the most complex of all. the extremely wide variety research institution processing this type of waste in and hazard of the liquids that can be met. The complexity very definitely depend segregation ent upon the decisions made in regard to segregation at the source. a


it,

taminated that it may be discharged directly. In any event, solids, liquids, and gases will all become contaminated, and it has been the general practice in all operating sites to treat these three separately, whether or not further segregation within these groups is practiced. 10.31 Solids. In any research laboratory a wide variety of solid material becomes contaminated. It is the general practice to place these solids in small containers within the working laboratories and to collect them in some routine manner and trans port them to either a burial ground or a central handling station. Some of the most popular transporting containers for small items are cardboard ice-cream cartons, For larger and garbage cans, drums, and other commercially obtainable containers. more heavily contaminated items, special provisions have to be made. At more remote sites these solid wastes are generally buried in trenches. In more populated areas it is often necessary to store them in a recoverable condition until sufficient quantity is collected to make transshipment to a final storage area economically feasi ble. It is possible to reduce the volume of material that must be handled by baling It also possible to segregate practiced at several sites (see Art. 10.5). and this The solid wastes at the source into combustible and noncombustible categories. combustible fraction, which often ranges from 50 to 80 per cent of the total volume, Incin may be incinerated, and this process has been tried out in at least four places. a relatively expensive process that eration discussed in some detail in Art. 10.5; cannot be justified economically unless the cost of storage quite high. 10.32 Gases. In research institution most of the gaseous waste problems result from ventilation of hoods, caves, and working areas in which radioactive work In almost all cases several areas, more or less widely separated, are being done. simpler to process the ventilation involved in this kind of work. In most cases highly air at the point of discharge than to conduct to a central plant or stack. retentive filter, variously called the CWS or the AEC filter, has been developed and widely used within the AEC projects. Specifications and characteristics of this filter general practice to dis are given in Table 42. These filters are so efficient that charge the filtered ventilation air through relatively short stacks at the point of

practiced, collection methods for highly active wastes in relatively small volumes must

11-144


[SEC.

11

be provided and processing methods for handling these high-level wastes must also If segregation is not practiced, provision must be made for processing be at hand. The processes most generally used very large volumes of relatively low-level wastes. All three for handling liquid wastes are ion exchange, Sacculation, and evaporation. of these are discussed in more detail in Art. 10.5. Higher level wastes may be either evaporated or immobilized in solids in some form or other. This latter process is also discussed in Art. 10.5. It is the general practice to mix all effluents below tolerance with the effluent from the sanitary sewerage system and to discharge these to the best available outfall. It has been the general experience of the larger AEC research institutions that the total yearly discharge of radioactivity can be held to about 1 curie without undue expense.


10.4

Production Wastes

Problems experienced at production sites are considerably different from those In general, the magnitude of the problem is encountered in research institutions. much greater both in quantity of waste to be handled and in level of radioactivity. On the other hand the compositions of these wastes, particularly in the case of liquid wastes, can be determined much more accurately and are much more nearly constant Therefore, it is more likely to be possible to devise a than are laboratory wastes. routine process that will handle a large fraction of the wastes. 10.41 Solids. The distinction between laboratory and production wastes is the least apparent in the case of solids. In both cases a wide variety of solid materials There is the additional problem that large bulky pieces of become contaminated. It is generally necessary to process equipment become very highly contaminated. provide within the production plants means of remotely decontaminating production Almost all production sites are located in remote areas equipment before discard. and upon large areas of government-controlled land. This has made it possible to In fact, some of the production sites also dispose of contaminated solids by trenching. serve as storage areas for the solid wastes produced by the smaller and research sites. Gases. The treatment of gaseous wastes at production sites is also handled 10.42 However, in conjunction with production plants ii has at the point of production. been the general custom to build large discharge stacks, at least 200 ft high, and to discharge the ventilating air from production canyons to the atmosphere either with When the associated laboratories are located or without previous treatment. sufficiently near the production stack, the discharge of ventilating air from such facilities is often sent to the same stack. Some pretreatment of the off -gases from This may take the form of wet chemical process vessels is nearly always practiced. scrubbing or of a specific process to remove a particularly hazardous contaminant. An example of the latter is the removal of radioiodine upon heated silver nitrate. After such pretreatment the process gases are vented into the general cell- ventilation streams. It has been found necessary in some instances to give some treatment to An effective the general ventilating air other than discharge through tall stacks. means which has been used is to pass the ventilating air through sand niters several fee; thick before discharge to the stack. In the case of processes that liberate gaseou? fission products (rather than filterable aerosols), it has often been necessary to coordi nate production operations with meteorological conditions. As in the case of research laboratories, the largest single problem Liquids. 10.43 From production of production wastes has always been the handling of liquids. plants thousands of gallons are obtained daily of very highly contaminated liquid wastes. At first no attempt was made to process these in any way other than to neutralize them. The neutralized wastes were then stored in huge (approximately This system worked very well million-gallon capacity) underground storage tanks. during the war, so well that it has not yet been completely abandoned and, in fact, appears to be an integral part of our waste-disposal system for a number of years to This system of storage of radioactive liquids has two disadvantages: (1) The come. accumulation of highly radioactive liquid represents a potential hazard should the Although some storage tanks have been tanks be ruptured, and (2) it is expensive.

Art.

10]

WASTE DISPOSAL

11-145

built for

less than 50 cents/gal, the general average at all sites has run in the order of a dollar or more per gallon of waste stored. Since the volumes produced each year reach millions of gallons, there is a considerable incentive to devise other methods. The volume of wastes produced can be reduced in any one of several ways, such as (1) by selecting processes that initially produce smaller waste volumes, (2) by the use of one of several possible concentrating techniques on the waste produced. A technique now in use at some sites is to cascade the waste supernatants through several of these large tanks in series, thus allowing time for decay, and finally to discharge the last supernatant to underground cribs wherein t he liquid is allowed to drain away into the It is possible to do this only under highly favorable geological conditions. ground.


10.5

Applicable Processes

In the foregoing discussions of the general problems of waste disposal, a number of These will processes useful in reducing the volume of wastes have been mentioned. be discussed in some detail in this article. 10.51 Evaporation. Evaporation is the most widely applicable technique for A wide variety of wastes can reducing the volume of liquid wastes (see also Art. 6.3). be handled, and a high decontamination factor can be obtained. Several types of In Table 33 (see Art. 6.3) evaporators are used, and all are satisfactory within limits. a list is given of the various evaporator types that have been used, their operating No attempt has been made in this table to characteristics, and performance data. give a value for the cost of operation. In almost all cases at relatively low capacity, The con the cost to evaporate a gallon of waste has been approximately 10 cents. centration factor possible in evaporation varies inversely with the solids content of the feed, but a broad range of feed solids content can be handled. Decontamination factors range from 104 to 108 for a single effect. 10.52 Flocculation.95'96 Variations of water-treatment practice have been studied exhaustively in an attempt to develop a generally usable waste-treatment process. appears that such a system may have value in treating large volumes of lightly contaminated wastes, but, as a method for handling high-level wastes, flocculation is Although a large number of flocculating agents have been tested, not satisfactory. over-all decontamination factors of only about 10 have been obtained for mixed If it is possible to tailor a process for a single radioactive species, fission products. much better results can be obtained. Advantages of the process are rather low cost, the ability to handle a wide range of solids content in the feed, and the production of a waste floe volume that is relatively independent of feed solids content. 10.53 Ion Exchange.97 98 Ion-exchange resins are useful for waste processing within prescribed limits. The capacity of the resins and the concentration factors obtainable vary inversely with the solids content of the feed. The method is of most interest for use on very low solids content feeds. Ion exchange is not practical on wastes containing much over 1,000 ppm total solids. If a mixed bed resin is used upon such a feed to obtain high decontamination factors, the capacity of the resin is only about 150 gal of waste processed per cubic foot of resin used. Furthermore, the regenerant volume will be nearly 15 per cent of the waste processed, so that under such circumstances it would be possible to obtain a volume concentration of only

It

about 7. Cation resins used alone give decontamination factors for mixed fission products of about 50; mixed ion-exchange beds will give decontamination factors of about 105. The cost of such a process is low if the solids content of the feed is low enough. One of the most useful applications of ion exchange is as a second stage after evapora tion. The overhead from a single evaporation will contain a very low solids content; therefore,

the ion-exchange column can be expected

give very high additional decontamination factors.

to last almost indefinitely

and

10.54 Incineration.95 About 50 to 80 per cent of the solid waste collected in a One possible research institution handling radioactivity is ordinarily combustible. means of handling this fraction of the waste is to incinerate it. The process of incineration has been tried out on an experimental scale. It is a feasible but expensive


11-146


[SEC.

11

It is possible to obtain a volume reduction factor of 20, to convert the waste process. to an easily storable solid that is in its final chemical and physical form, and to clean However, the cost is high, approxi up the resultant gases by factors of about 107. mately two dollars per cubic foot, and cannot be justified except in special cases or where the unit cost of storage is very high. 10.55 Another approach to the reduction of the volume of Bolid waste is Baling. This has the advantage that both combustible and noncombustible wastes baling. A volume can be handled and it is not necessary to segregate the solids at the source. reduction factor of about 3 is possible. Although the operation is messy and unpleas The cost of baling runs between 10 and 20 cents ant, the process is gaining favor. per cubic foot of collected waste. A number of methods have been tried on the pilot10.56 Solidification.100101 plant scale for immobilizing liquid wastes in solids. Wastes have been concentrated until they solidify on cooling, and the solids have then been stored in drums. Con centrated wastes have had portland cement added so that they set to a solid. Wastes are thus immobilized at a volume increase of about 2. It has also been suggested that the waste be absorbed on clays or glass- or ceramic-forming substances and then vitrified. Recently considerable success has been had in the use of a fluidized bed of inert The radioactive waste is solids as a drier and calciner of aqueous waste streams sprayed through a nozzle into the side of a heated, fluid-bed reactor filled with an inert oxide. Water and nitrogen oxides are driven off, and solids in the waste stream are calcined to oxides. The product formed is fluidizable and thus readily handled. It is not subject to further decomposition by heat or radiation. The activity, though reasonably immobile, is not completely unleachable, and there must be some integrity in the packaging technique. The methods being practiced for the disposal of radioactive waste produced today are sufficiently adequate so that no serious insult to the environment is resulting. However, should the present trend toward the application of nuclear energy to the production of electric energy on a large scale follow its probable and expected course, waste-disposal problems will become even more important. BIBLIOGRAPHY Ayres, J. A.: Ind. Eng. Chem., 43: 1526 (1961). Browder, F. N.: Ind. Eng. Chem., 43: 1502 (1951). Goodgal, S., E. F. Gloyna, and D. E. Carritt: /. Am. Water Works Assoc., 46: 66 (1953). Hayner, J. H.: Ind. Eng. Chem., 44: 472 (1952). McCullough, G. E.: Ind. Eng. Chem., 43: 1505 (1951). National Bureau of Standards: "Handbook 52," 1953. Rodger, W. A., and P. Fineman: Nucleonics, 9(6): 50 (1951). Straub, C., R. J. Morton, and O. R. Placak: J. Am. Water Works Assoc., 43: 773 (1951).

REFERENCES Dillon, I. G., and L. Burris: ANL-5334, 1954. Katz, J. J., and E. Rabinowitch: "The Chemistry of Uranium," National Nuclear Energy Series, Div. VIII, vol. 5, McGraw-Hill Book Company, Inc., New York, 1951. 3. Peppard, D. F., M. H. Studier, M. V. Gergel, G. W. Mason, J. C. Sullivan, and J. ¥. Mech: Report AECD-2890, June 2, 1950; J. Am. Chem. Soc, 73 : 2529, 3278 (1951). 4. Seaborg, G.: Chem. Eng. News, S3: 2190 (1945). 5a. Seaborg, G.: Nucleonics, 5(5): 16 (1949). b. Seaborg, G., el al.: "The Transuranium Elements," National Nuclear Energy Series. Div. IV, vol. 14B, McGraw-Hill Book Company, Inc., New York, 1949. 6. Debierne, A.: Compt. rend., 1S9: 593 (1899). 1.

2.

7. Debierne, A.: Compt. rend., 130: 906 (1900). 8. Mendeleeff, D.: Ann., suppl. 8, 2: 133, 196 (1871). 6: 324 (1918). 9. Hahn, O., and L. Meitner: Naturwissenschaftcn, 10. Hahn, O., and L. Meitner: Physik. G., 19: 208 (1918). 11. Hahn, O., and L. Meitner: Ber., 54: 69 (1921). 12. Soddy, F., and J. Cranston: Proc. Roy. Soc. (London), A94 : 384 (1918).

REFERENCES 13. 14. 15. 16.

17. 18. 19. 20.

11-147

Soddy, F., and J. Cranston: Nature, 100: 498 (1918). McMillan, E., and P. Abelson: Phut. Rev.. 57: 1185 (1940). Wahl, A., and G. Seaborg: Phyt. Ret., 73: 940 (1948). Seaborg, G., and J. Hamilton: Science, 102: 559 (1945). Seaborg, G.: Chem. Eng. Newt, 25: 358 (1947). Thompson, S., A. Ghiorso, and G. Seaborg: Phyt. Rev., 77: 838 (1950). Thompson, S., A. Ghiorso, and G. Seaborg: Chem. Eng. Newt, 28: 326 (1950). Thompson, 8., K. Street, Jr., A Ghiorso, and G. Seaborg: Chem. Eng. Newt, 28: 1030

(1950). 21. Thompson, S., K. Street, Jr., A. Ghiorso, and G. Seaborg: Phyt. Rev., 78: 298 (1950). 22. Hopkins, B. S.: "Chapters in the Chemistry of Less Familiar Elements," Stipes Publishing Co., Champaign, 111., 1939. 23. Coryell, C. D., and N. Sugarman: "Radiochemical Studies of the Fission Products," part National Nuclear Energy Series, Div. IV, vol. 9, McGraw-Hill Book Com pany, Inc., New York, 1951. 24. Simpson, J. A.: Rev. Sci. Inttr., 18: 884 (1947). 25. Jaffey, A. H., T. P. Kohman, and J. A. Crawford: Report CC-1802 (M-262), March, 1944. 26. Franz, N.: Phytik. Z., 63: 370 (1930). 27. Rutherford, E., F. A. B. Ward, and C. E. Wynn-Williams: Proc. Roy. Soc., 129: 211 (1930). 28. Wynn-Williams, C. E.: Proc. Roy. Soc, 132 : 295 (1931) ; 136 : 312 (1932). 29. Geiger, H., and W. Mailer: Phytik. Z., 29: 839 (1929). 30. Borkowski, C. J.: Anal. Chem., 21: 348 (1949). 31. van den Akker, J. A.: Rev. Sci. Inttr., 1: 672 (1930). 32. Strong, J.: "Procedures in Experimental Physics," chap. 7, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1945. 33. Korff, S. A.: "Electron and Nuclear Counters," D. Van Nostrand Company, Inc.. Princeton, N.J., 1946. 34. Hoag, J.: "Electron and Nuclear Physics," chap. 18, D. Van Nostrand Company, Inc., Princeton, N.J., 1946. 35. Hume, D. N.: Anal. Chem., 21 : 322 (1949). 36. Pollard, E., and W. L. Davidson: "Applied Nuclear Physics," John Wiley 4 Sons, Inc., New York, 1942. 37. Hahn, O. : "Applied Radiochemistry," Cornell University Press, Ithaca, N.Y., 1936. 38. Crowthers, J. A.: "Ions, Electrons, and Ionizing Radiations," Edward Arnold & Co., London, 1924. 39. Stearns, J. C, et al.: Report CP-2090, August, 1944. 40. Boyd, G.: Anal. Chem., 21 : 335 (1949). 41. Clark, H., and R. Overman: MDDC-1329, 1947. 42. Fajans, K.: "Radio Elements and Isotopes; Chemical Forces and Optical Properties of Substances," p. 92, McGraw-Hill Book Company, Inc., New York, 1931. 43. Wahl, A. C, and N. A. Bonner: "Radioactivity Applied to Chemistry," pp. 104-124, John Wiley & Sons, Inc., New York, 1949. 44. Perry, J. H. (ed.): "Chemical Engineers' Handbook," 3d ed., p. 554, McGraw-Hill Book Company, Inc., New York, 1950. 45. Morello, V. S., and N. Poffenberger: Ind. Eng. Chem., 42: 1021 (1950). 46. Davis, Jr., M. W., T. E. Hicks, and T. Vermuelen: Chem. Eng. Progr., 60(4): 188 (1954). 47. Standard Oil Development Co.: SOD-21, July 29, 1949. 48. Coplan, B. V., J. K. Davidson, and E. L. Zebroski: Chem. Eng. Progr., 60(8): 403


III,

49. 50. 51. 52. 53. 54. 55. 56. 57.

(1954). Scheibel, E. G.: Trant. Am. Intl. Chem. Engrt., 44: 681, 771 (1948). Fenske, M. R., and R. B. Long: Chem. Eng. Progr., 61(4): 194 (1955). Cohen, R. M., and G. H. Beyer: Chem. Eng. Progr., 48(6): 279-286 (1953). Sege, G., and F. W. Woodfield: Chem. Eng. Progr., 50(8) : 396-402 (1954). Sege, G., and F. W. Woodfield: Nuclear Engineering— Part III, Chem. Eng. Progr. Sympotium Ser., 50(13): 14 (1954). Stephenson, R.: Chem. Eng. Progr., 49(7): 340 (1953). Thornton, J. D.: Nuclear Engineering— Part III, Chem. Eng. Progr. Sympotium Ser., 50(13): 39 (1954). Van Dijck, W. J. D.: U.S. Patent 2,011,186, Aug. 13, 1935. Culler, F. L.: "Reprocessing of Reactor Fuel and Blanket Materials by Solvent Extraction," Paper R5.1-1030, International Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955.

11-148


[SEC.

11

58. Brown, G. G., and Associates: "Unit Operations," p. 326, John Wiley & Sons, Inc., New York, 1950. 69. Kunin, R., and R. J. Myers: "Ion Exchange Resins," John Wiley & Sons, Inc., New York, 1950. 60. Cohn, W. E., G. W. Parker, and E. R. Tompkins: Nucleonics, 3:22 (1948). Am. Chem. Soc., 71: 2504 (1949). 61. Tompkins, E. R., D. H. Harris, and J. X. Khym: 62. Boyd, G. E., A. W. Adamson, and L. S. Myers, Jr. : Am. Chem. Soc, 69: 2840 (1947). 63. Boyd, G. E., L. S. Myers, Jr., and A. W. Adamson: Am. Chem. Soc., 69: 2854 (1947). 64. Boyd, G. E., J. Schubert, and A. W. Adamson: Am. Chem. Soc, 69: 2818 (1948). " Electrorefining as a Method for Removing 65. Niedrach, L. W„ and A. C. Glamm: Fission Products from Uranium Fuels," Paper 200, Engineers Joint Council Nuclear Congress, Cleveland, Ohio, Dec. 12-16, 1955, and Ind. Eng. Chem., 48: 977-981 (1956). 66. Niedrach, L. W., and A. C. Glamm: "Electrorefining of Uranium — A New Approach," KAPL-1154, Aug. 13, 1954. 67. Niedrach, L. W., and A. C. Glamm: Uranium Purification by Electrorefining, Electrochem. Soc, 103: 521-528 (1956). 68. van Arkel, A. E., and J. H. de Boer: Z. anorg. u. allgem. Chem., 148 : 345-360 (1925). of Vacuum Distillation of Metal Mixtures," LRL-88, 69. Vivian, J. E.: "Principles March, 1954. 70. Motta, E. E.: "High Temperature Fuel Processing Methods," Paper 542/Conf. 9, International Conference on the Peaceful Uses of Atomic Energy, United Nations. New York, 1956. 71. Lawroski, S.: "Survey of Separation Process," Paper 823/Conf. 9, International Conference on the Peaceful Uses of Atomic Energy, United Nations, New York, 1956. 72. Pfann, W. G.: Trans. Am. Inst. Mining Met. Engrs., 194: 747-752 (1952). Metals, 5: 1531 (November, 1953). 73. Lord, N. W.: Metals, 6: 1953 (September, 1954). 74. Reuss, H.: Metals, 7: 1017 (September. 1955). 75. Burris, Jr., L., C. H. Stockman, and I. G. Dillon: 76. Wagner, C: /. Metals, 6(2): 154 (February, 1954). 77. Arnold, D. S., C. E. Poison, and E. S. Noe: "Production of Uranium Metal," Preprint 113, Nuclear Engineering and Science Congress, Cleveland, Ohio, Dec. 12-16, 1955. 78. Roper, C. G.: Electrical Transmitting and Transducing of Process Measurements. Instr. & Automation, 25: 338-341 (March, 1952). 87: 1962-1964 79. Gilliland, E. R.: Electronic Control Systems, Instr. & Automation, (December, 1954). 80. Mannal, C: Testing Electric Insulation for Use in Gamma Ray Fields, Nucleonics, 12:49-53 (June, 1954). 81. Monk, G. S., and W. H. McCorkle: "Optical Instrumentation," National Nuclear Energy Series, Div. IV, vol. 8, McGraw-Hill Book Company, Inc., New York, 1954. 82. Wildhack, W. A., and V. H. Goerke: "The Limiting Useful Deflections of Corrugated Metal Diaphragms," NACA Tech. Note 876, 1942. " Measurement Techniques in Mechanical Engineering," pp. 232-235, 83. Sweeney, R. J.: John Wiley & Sons, Inc., New York, 1953 84. Lyon, R. N. (ed.): "Liquid Metals Handbook," NAVEXOSP-733, June, 1952, 85. Speller, F. N.: "Corrosion, Causes and Prevention," p. 658, McGraw-Hill Book Company, Inc., New York, 1948. 86. Evans, U. R.: "Metallic Corrosion and Passivity," p. 544, Longmans, Green * Co.,

J.

J. J. J.


J.

J.

J.

J.

Inc., New York, 1948. H. S. (ed.): "Corrosion Handbook," pp. 69-70 (corrosion fatigue, fretting, cavitation erosion, corrosion by stray currents), pp. 1103-1110 (illustrations), John Wiley & Sons, Inc., New York, 1948. 88. Fontana, Mars G.: Ind. Eng. Chem., 40(3): 75a (1948) (concentration cell), 48(6): 63a (1950) (fretting corrosion), 48(1 1) : 99a (1950) and 43(8) : 87a (1951) (knife-line attack). 44(2): 63a (1952) (ring- worm attack). 89. Moteff, John: "Miscellaneous Data for Shielding Calculations," Report APEX-176. General Electric Company, Dec. 1, 1954. 90. Harty, W. M.: Nuclear Engineering — Part II, Chem. Eng. Progr. Symposium Ser„ 87. Uhlig,

50(13): 115-121 (1954). 91. National Bureau of Standards: "Handbook 48, 1951." 92. Landry, J. W.: "Sampling Devices for High Level Radiochemical Plants." Preprint 1955. 158, Nuclear Engineering and Science Congress, Cleveland, Ohio, December, 93. TID-5280, suppl. 1, Fourth Annual Symposium on Hot Laboratories and Equipment, Washington, Sept. 29 and 30, 1955. Material, AEC Program, Laboratory Equipment, 94. TID-5276, Selected Reference compiled by Brookhaven National Laboratory for the Geneva Conference, 1955.

REFERENCES

11-149

J. J.

Goodgal, S., E. F. Gloyna, and D. E. Carritt: Am. Water Workt Assoc., 46: 66 (1953). 96. Straub, C. P., R. J. Morton, and 0. R. Placak: Am. Water Works Assoc., 43: 775 (1951). 97. Ayres, J. A.: Ind. Eng. Chem., 43: 1526 (1951). 98. Nachod, F. C, and Jack Schubert: "Ion Exchange Technology," Academic Press, 95.

99.


100. 101.

Inc., New York, 1956. Rodger, W. A., and D. C. Hampson: J. Air Pol. Cont. Assoc., 6(1): 41-43 (May, 1956). Hatch, L. P.: Am. Scientist, 41: 410 (1953). Jonke, A. A., E. J. Petkus, J. W. Loeding, and S. Lawroski: "The Use of Fluidized Beds for the Continuous Drying and Calcination of Dissolved Salts," presented at Second Annual Meeting, ANS, Chicago, June 6-8, 1956.


SECTION

12

NUCLEAR-POWER-PLANT SELECTION BY

ALFRED AMOROSI, B.S.M.E., B.S.Ch., Technical Director, Atomic Power Develop ment Associates, Inc. (on loan from Detroit Edison Company, formerly Associate

Director, Reactor Engineering

Division,


L. E. LINK, B.S., Associate Chemical WALTER H. ZINN, Ph.D., President,

Argonne National

Engineer,

Argonne

Laboratory).

National

Laboratory.

General Nuclear Engineering Corporation, formerly Director, Argonne National Laboratory. FRANK B. DANIELS, B.S., Assistant Manager, Research and Development, Nuclear Products-Erco, Division of ACF Industries, Inc., formerly Assistant to Director of Mechanical Engineering, Industries Group, Allis-Chalmers Manufacturing Company. C. ALLEN, Director of Mechanical Chalmers Manufacturing Company.

ROBERT

Engineering,

Industries Group, Allis-

CONTENTS 12-1

SELECTION OF REACTORS BY ALFRED

AMOROSI

PAQE 12-2 1 Thermodynamics 2 Classification of Reactors 12-6 3 Factors That Influence the Design and-* Selection of Reactors 12-8 4 Principal Features of Reactor Types. . . 12-22 12-56 5 Coolants 12-59 6 Fuel Elements 7 Moderators 12-68 12-70 S Control Rods 12-76 References

ECONOMICS OF NUCLEAR POWER BY L. E. LINK AND WALTER H. ZINN 13-2

I

Fuel Consumption, Power, and Neutron

Flux

12-78

2 3 4 5 6 7 8

PASS

Neutron and Fuel Balance 12-80 World Fuel Resources 12-84 Conventional Plants — Power Cost 12-91 Cost of Nuclear Power Plants 12-95 Cost of Nuclear Fuels 12-100 Cost of Special Materials 12-106 Competitive Status of Nuclear Power. 12-108 References

12-112

POWER-GENERATING EQUIPMENT BY FRANK B. DANIELS AND

12-3

ROBERT C. ALLEN

1 Power Cycles Equipment and Aux 2 Power-generating iliaries 3 Operation and Maintenance 4 Shielding and Containment References

12-1

12-00 12-00 12-00 12-00 12-00

12-1

SELECTION OF REACTORS* BV

Alfred Amorosi 1


1.1

THERMODYNAMICS

Direct Utilization

of Fission Energy

In the fission process, 85 per cent of the energy appears as kinetic energy of the fission fragments and the remaining 15 per cent appears as radioactive decay energy, instantaneous -y-ray energy, and neutron kinetic energy. These particles lose their energy by collision with fuel, moderator, and structuralIn this process, atoms are ionized, generating electrons and photons. material atoms. The energy appears in the following forms: 1. The largest part of the energy is generated as kinetic energy. 2. A fair amount appears in the form of electrons. 3. A fair amount appears as y rays. It is natural to raise the question, "Is there some easier or more efficient method of converting atomic energy to mechanical energy than is available with the conventional heat source?" 1.11 Kinetic Energy to Mechanical Direct Conversion of Fission -fragment The fission fragments dissipate their energy by inelastic collision with other Energy. atoms. These fragments are usually stopped in about 10-1 cm of uranium or slightly As a possible mechanism to utilize the larger distances in other solid materials. kinetic energy of the fission fragments directly, consider the following: It can be hypothesized that the fissioning atoms could be arranged as an almost monatomic In this manner, the fission frag layer on a single face of a series of parallel plates. How ments going in one direction from the plate will cause a reaction in the plate. ever, this effect is counteracted by the fragments being caught from the adjacent Only the top plate can be made to have a true reaction. Very little energy plate. For example, assume a reactor that is a 2-ft cube can be obtained by this method. with a critical mass of 5 kg of Um arranged in flat plates with a 10~*-cm-thick layer The total number of plates would then be of uranium on the bottom of each plate. With only one plate active and only in one direction, the efficiency could not 7,300. PXCOPfi

K X

1/7,300

X

10' =

10-' of

1

per cent

Some orbital electrons are freed 1.12 Direct Utilization of Orbital Electrons. in the slowing-down collisions of the fission fragments and other particles. These electrons primarily come from the outer electron shell and are produced by ionization However, these electrons are randomly directed and tend to energies of 5 to 34 ev. As a result, there is no net current of electrons in replace other displaced electrons. Therefore, at this time, there does not seem to be any practical any one direction. method of producing electricity directly from the free electrons. Primary y rays are born from fission, and secondary ■> 1.13 Photoelectric Effect. These y rays have energies ranging rays are produced from fission-product decay. * E. Garelis and J. T. Kelly, Atomic Power Development Associates. Inc.. provided valuable assist Much of the information presented ance in preparing this subsection. experiences at Argonne National Laboratory.

12-2

is the result of the author's

selection of reactors

Sec. 12-1]

12-3

from a few kev to Mev and can penetrate several inches of material before they are If silicon transistors could be developed for y rays to the extent that they stopped. are developed for ordinary light, then direct conversion of the photon energy to elec tricity would be possible. The uranium could be arranged so that some of the y rays could be absorbed by the transistor and thereby produce electricity. Transistors are not developed to the degree required for this application. 1.2

It

The Reactor

as a Heat Source1*

appears from the foregoing that the reactor must be treated as a heat source.


The reactor heat source is different from a chemical heat source in that no oxygen is required and the heat does not have to be removed from gaseous combustion products with poor heat-transfer properties. The fission heat source can be distributed in solidfuel elements, in liquid, or in gas. One drawback of the reactor as a heat source is that it requires a minimum of 50 to 100 tons of shielding and, therefore, is not attrac tive where low weight is important. The most important gas and vapor cycles are reviewed, and their application to The diagrams for these cycles and their characteristics atomic power evaluated. are shown in Table 1. 1.21 The Carnot Cycle. The pressure-volume (JP-v) and temperature-entropy (T-S) diagrams for a Carnot cycle operating on a gas are shown in Table 1. The cycle consists of an isentropic compression ab, a reversible isothermal expansion 6c at temperature 7\, an isentropic expansion cd, and a reversible isothermal compression da at temperature Tt. The efficiency of a Carnot cycle is (Ti — Ti)/T\. Working between the same limits of temperature, no engine can have a higher efficiency than the Carnot engine, and the efficiency of all reversible engines equals the efficiency of the Carnot engine. The Carnot-cycle efficiency is, therefore, the maximum theoretically attainable for any heat engine, including a reactor power plant. Any real engine built to simulate the processes of the gas Carnot cycle would not be practical because of the difficulty in transferring large amounts of heat isothermally. Any inefficiencies in these processes would result in a highly inefficient This is a fundamental weakness of the gas cycle, not of the heat source, machine. and therefore no atomic reactor is being planned to run on the Carnot cycle. 1.22 The Rankine Vapor Cycle. The diagram for the ideal Rankine vapor cycle is shown in Table 1. Modifications of this cycle using reheat and feed-water heating are the conventional central-station steam plants. Because of limitations imposed by materials and the properties of the working fluid, the temperatures used in this cycle are lower than those used in the Diesel and Otto cycles. The processes in the Rankine cycle are effective for handling large power, and therefore the Rankine cycle is used for central-station power, even though the efficiency of the plant is lower than for internal-combustion-engine cycles. This cycle seems to be best suited to atomic power because: 1. At the present time, the maximum practicable operating temperature for a reac tor corresponds to or is lower than the temperature used in the conventional Rankine vapor cycle. 2. The problem of containment of radioactivity is better solved by this cycle than by other cycles. 3. Reactors are most economical in large sizes, as are Rankine engines. The thermal efficiency of the reactor plant is slightly higher than the conventional boiler-fired plant for the same conditions because of the negligible "stack" loss in the atomic plant. The Brayton Gas Cycle (Gas-turbine Cycle). The ideal Brayton cycle, or 1.23 gas-turbine cycle, consists of an isentropic compression, constant-pressure expansion (heat addition), isentropic expansion, and constant-pressure contraction (heat rejec tion). The high ratio of work required for compression to that obtained in expansion makes it important to manufacture the machine with: * Superscript numbers

refer to References


12-4 1. 2.

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

High compressor and turbine efficiencies High operating temperature (top temperature usually limited by materials to

1500°F) 3.


4.

Small compression ratio In some cases, intercooling during compression and regeneration

In a conventional machine, constant-pressure expansion is achieved by burning th>fuel in the working fluid. The compression work is usually equal to about half the Furthermore, turbine work, so that the net output is about half the turbine output. the operating temperature of the turbine is high. As a result, Brayton-cycle machines are used now only in small power plants. Because the air can be used as a working fluid and because of its light weight, the gas-turbine engine is of interest for aircraft The advantage of light weight is minimized in nuclear reactors because propulsion. of the heavy shielding necessary. A closed gas-turbine cycle can be considered with the uranium in the working fluid There are serious problems in the development of such a cycle. (e.g., as UF0). 1.24 The Otto Cycle. The Otto or the closed variant of the gasoline-engine cycle consists of isentropic compression, constant-volume heat addition, isentropic expan sion, and constant-volume heat rejection. This cycle could possibly be applied to atomic energy, using either a gaseous fuel or a solid fuel. Consider the case of using gaseous UFS. The adiabatic compression takes place until the gas reaches a critical At criticality, mass; the reactivity increases rapidly as the piston compresses the gas. the neutron population and heat generation increase very rapidly, giving a constantThe resulting high-pressure gas now expands adiabativolume increase in pressure. cally and hence drives the piston outward and returns the reactor to subcritieal With UF« as the working fluid, assume that 100 kg is required for criticality ; then such a device would have to be about 3 ft in diameter and capable of operating at 1,000 psi and would probably be designed for a maximum cyclic temperature of 2000'F. If moderating material were included in the walls of the cylinder, the diameter could A variation of this process could be applied to the use of a solid fuel. be much less. In this case, the uranium could be supplied in the form of a fine mesh located in the Water might be used as the working fluid and moderator top of the cylinder. Evaporation of the water returns the reactor to subcritieal and drives the piston outward. The push-pull reactor may be adaptable to this cycle. A concept of this type of plant has been advanced by M. E. Battat and M. Tsingin of Los Alamo Scientific Laboratory. Two cylinders connected to each other are partially filled with water containing uranium in solution. Oscillation of the water from one cylinder to the other makes them alternately critical. Very little work has been done on these types of engines. 1.26 Other Cycles. In principle, a variation of the Otto cycle, called the Stirling In the Stirling cycle heat is not generated in the cylinder but cycle, can be used. The Stirling cycle is outside and is transferred by a working fluid to the cylinder. limited in application to sizes below 10 hp, where it is not practical for atomic power. A variation of the Rankine vapor cycle, called the Hydroduct, is conceptually vaporized in the reactor possible. Water could be adiabatically compressed, constant pressure, and adiabatically expanded, using the steam jet for thrust. For electrical- or mechanical-power production, it appears that the reactor should be treated as a heat source, and for central-station application the Rankine cycle For special application, other cycles should be considered. Th* should be used. decay thermal energy in spent-fuel elements is not a large source of energy and, there fore, is not treated herein. 1.3

Coupling with the Working Fluid

The properties of the reactor coolant may or may not conform to the requirements of the heat-engine working fluid. Most nuclear plants do not have the same working fluid for the reactor as for the heat engine. The cycles under active development are discussed below.

s

a

V

P V C,(Tt

T.)

C.(r.

Cp(Tt

-

T.

T.

-

-

T.

T,

Ti)

Tt)

+

-

t.)

-

At

-

(nj/v.)

(7vro

-

©"

{Ti/T.)

s

♦

o

d

P

1

-

= -

1

■

+ +

t

where r. -

C1)7" V.' (»<■/»•)

(Tj/T.)

where r. = («/»•)

—

=

where

-

h, - hi h, - h, = pump work = hi - h.

where

-

Efficiency

,

cycle

an

-

Ti)

Qi

In Da

Ci -

ha ~ hi — hb

R(7\

-

Network

n

1

V

e

•et hi

In -

input Qi

F F

Otto

P ». -

RT,

Heat

Cycles

i,

F

Brayton cycle, complete expansion

diagrams

□

temp.-entropy

Heat-engine

r

vapor cycle

cycle

and

Ideal

i

Rankine

Carnot

A

Preas.-volume,

1.

Cycle

Table


q

s S

d

"

C

V


12-6

NUCLEAR-POWER-PLANT

SELECTION

[SEC. 12

A direct cycle can be used in which steam is generated in the reactor and goes directly This cycle requires a minimum of equipment and eliminates to the turbine (Fig. 1). the degradation of energy in the transfer It can be used either for solidprocess. or for homogeneous fuel heterogeneous reactors. The direct cycle has two main disadvantages: namely, the boiling proc ess does not permit the high power den sities that are attainable with liquid cooling, and radioactive steam is carried into the turbine and condenser. These problems are discussed in Art. 4.5. The indirect cycle (Fig. 1), in which heat is transferred by a primary coolant from the reactor to the boiler, is not so simple as the direct cycle. It gives greater power density and eliminates radioactivity in the steam to the turbine. A variation of this system is a boiling reactor where the steam generated in the in an evaporatorreactor is condensed condenser and the turbine steam is genFio. 1. Plant cycles, coupling with working erated there. This is discussed in more fluid. detail in Art. 4.4. A number of plants are being designed with an intermediate link that transfers the heat from the reactor coolant to the steam boiler. Where a boiler-tube failure could cause serious release of radioactivity from the primary sys tem, an intermediate link is used. 2

CLASSIFICATION OF REACTORS

Reactors are usually classified by (1) application, (2) nuclear characteristics, (3) coolant, and (4) mobility of fuel. Each of the foregoing classifications is treated below in detail. 2.1

Application

Application classifications include stationary power, mobile power, plutonium pro For stationary power, fuel economy is important, since the duction, and research. nuclear fuels must compete with fossil fuels. Stationary power reactors are designed with a high conversion ratio (over 0.7). Mobile power reactors are designed for Research reactors are designed to provide high power density and light weight. neutrons and produce 7-ray sources for research purposes. Materials-testing reac tors are classified as research reactors but are usually run at higher power than research reactors to provide a very high neutron flux. 2.2

Nuclear Characteristics

The following nuclear characteristics are generally used in identifying reactor typ»-' A thermal reactor is one in which the neutrons are Neutron Spectrum. predominantly at thermal energy. The average neutron energy is about 0.03 ev. A fast reactor is one in which the neutrons are predominantly at an average energy 2.21

above 0.1 Mev. It must be operated with an enrichment above 10 per cent, with U»" or with Pu"" fuel. 2.22 As a reactor classification, a homoftneHomogeneous and Heterogeneous. ous reactor is generally defined as one which has coolant, moderator, and fuel mixed, while a heterogeneous reactor has the coolant separate from the fuel. 2.23 Enrichment. Natural uranium and slightly enriched reactors usually cootain enough fertile material for the conversion ratio to be above 0.7. These reactors

power

Research

Small

Stationary

power

Application

Thermal

Thermal

Thermal

Fast

Neutron spectrum

Heterogeneous Homogeneous Heterogeneous

Heterogeneous

Heterogeneous

Homogeneous

Heterogeneous

Enrichment

Natural or slightly enriched Highly enriched Highly enriched Highly enriched H.O H,0 D.0

Graphite

HiO BeO HiO Be

DiO Boiling

DiO D,0 Highly

H.0 Boiling

HiO HiO

Enriched in varying degrees

H.0 HiO DjO

Air

H,0 Na Boiling Salt

HtO

DiO

HjO

Gas Organic

BeO Organic

Natural or slightly enriched

Bismuth Na Salt

Graphite

enriched

DiO

HtO

Gas HiO Liquid metal Liquid metal

DiO Boiling

DiO DiO Graphite

H,0 Boiling

Na Na Hg Pb (Bi)

Coolant

of fuel

coolant coolant

With With

SoUd With moderator Solid

Solid

Solid SoUd Solid With coolant

coolant coolant

coolant coolant coolant

With With

SoUd SoUd

With With With

common

water

graphite

HtO

DiO boiling

HiO boiling

Swimming pool, MTR, Water boiler CP-5 or heavy water

Air-cooled

Boiling,

Pressurized

Homogeneous, Homogeneous,

Homogeneous, Homogeneous,

ORR

DiO

HiO

fuel, graphite

Gas-cooled BeO Organic moderated

Liquid-metal

Gas-cooled graphite Wuter-eonUd Kraphitr Na-cooled graphite

Pressurized DiO Boiling DiO

Solid SoUd Solid Solid SoUd Separate carrier

Pressurised HiO Boiling H.O

fast Na-cooled. Mobile fuel, fast

Typical

Solid SoUd

Solid Separate carrier Solid Solid

Mobility

of Reactors

H.0 H.0

None

Moderator

Classification

Enriched in varying degrees

Natural or slightly enriched

10-40%

characteristics

Homogeneous vs. heterogeneous

Nuclear

Table


2 moderator

name

12-8

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

are capable of generating fissionable material to replace the material burned. They are, therefore, more economical fuel-wise than highly enriched unblanketed thermal reactors. They are generally used where fuel costs are important, such as in a cen tral station for power generation. Fully enriched unblanketed reactors do not have to provide neutrons for absorption in fertile material and, therefore, can tolerate greater neutron leakage and extraneous neutron absorption. Many research reactors are fueled with enriched material to provide high neutron flux at moderate power, and, because their power level is low, fuel economy is relatively unimportant. 2.24 The only elements that are commonly considered as moderators Moderator. are carbon, H or D in the form of H20 or D20 or as organic compounds, and beryllium Thermal reactors are, therefore, classified by these moderators. The fact or BeO. that these elements have different slowing-down, absorption, and physical charac teristics markedly affects the size and nature of the reactors. 2.26 Fuels and Fertile Materials. The fuels used in reactors are U,M, UlM, and Pu"9; U"8 is not usually considered as a fuel but does fission to some extent at high neutron energies and, therefore, contributes to the fission process. Although reactors are sometimes classified by either their fuel or fertile material, no attempt has been made to show this in Table 2 on the classification of reactors.


2.3

Coolant

The reactor coolant is an important factor in the design of the reactor. The prop erties of the coolant that must be taken into consideration are given in Art. 5. 2.4

Mobility

of Fuel

The mobility of the fuel greatly influences the reactor design. The degree of mobility of the fuel is as follows: The fuel does not move when the reactor is in operation. 2.41 Stationary Fuel. The fuel may be either a solid or liquid encased in tubes. The fuel is added to and removed from the reactor in batch quantities. 2.42 Fuel Movement by Separate Carrier. The fuel in this case is moved con tinuously through the reactor in form of a solution, a suspension, or a paste. The fuel is distinct from the coolant; that is, they are not mixed. The fuel may move through the reactor at a rate of inches per day. There are no reactors of this type in operation This scheme is being actively developed, but feasibility has not been demonstrated Fuel Movement with Coolant. The fuel moves with the coolant through the 2.43 reactor. 3

FACTORS

THAT INFLUENCE THE DESIGN AND SELECTION OF REACTORS

Establishing any one design feature of a reactor will usually influence all the remain Reactor engineering is a relatively new field, and some of the design ing factors. factors are not too well understood. Consequently, a firm picture of the interrela tion of the design factors cannot be established. In selecting a reactor type for » specified purpose, it is not always possible to say in advance which type is the best suited and most economical to build and operate. As the understanding of the effects of the design factors on the economics of reactors develops, the problem of selecting The signifi the best reactor for a particular purpose will undoubtedly be clarified. cance of the various factors as related to the design and selection of reactors in the light of present knowledge is developed in this article. 3.1

Nuclear Considerations

The nuclear reactor, in some respects, is similar to a chemical source. To main tain a constant heat output from a chemical source, the rate of burning must be

Sec. 12-1]

SELECTION OF REACTORS Table 3.

12-9

Neutron Balance Sheet for Typical Reactors Thermal reactors Fast reactor, U«» fuel. U"» blanket

Research air-cooled graphite,

Na-cooled graphite,

slightly

fully

fuel

fuel

fuel

2. 5

2.5 0.06 2.56

2.5

natural U


Neutrons produced per fission of U21* Neutrons produced by fast fission of TJ*18. Total neutrons available Neutrons to sustain chain reaction Neutrons absorbed in Um to form UM*. . . Neutrons absorbed in U"' to produce Pu. Neutrons absorbed in moderator Neutrons absorbed in coolant Neutrons absorbed in structural materials Neutrons leaving core Neutrons leaving blanket Neutrons for equilibrium poison Neutrons for Xe override Neutrons for control

Table 4.

2.5 0.35

0.06 2.56

2.85 1.0

Small power,

0.2 1.43

2.5

1.0

1.0

0.2 0.6

0.2

0. I 0 18

0.01 0. I

|0.4

0. 15

0.50 0.2

0.04

0.04

0 05

0.

0.04

0.05

0.2

(1.2) 0. I

0 01

0 04

Characteristics of Fissionable

II

Materials

1

Fuel

Handling

Radioactivity

Occurrence

Thermal*

Fast

V"

Natural uranium

2. 1

2.2

Tjm

Made from thorium

2 3

Pu"'

Made from U'»

2.0

* Thermal refers to the number of reactor. Fast refers to a fast reactor,

precautions

None

2.5

Weak >, weak a from associated U>" Strong y and a

Remote handling,

2.6

Strong a

Handle in dry box

fission

neutrons

produced

per neutron

captured

in a thermal

constant, and similarly for a nuclear reactor the rate of fission must be constant for a constant heat output. For any given reactor power this means that the neutron population must remain fixed. The total number of neutrons produced must be

equal to those which are absorbed and number of neutrons produced per fission

and

which leak out of the reactor.

The total

is dependent on the fuel and is between 2.5

3.0. One neutron is required to maintain the chain reaction. In central-station reactors, neutrons are absorbed in fertile material in order to minimize fuel and reprocessing cost. An attempt is made to keep the conversion ratio somewhere between 0.7 and 1.0 for thermal reactors and above 1.0 for fast reactors. The new fuel can be produced in the core or in the blanket. In these reactors, the designer also tries to keep the leakage and the absorption in the moderator, coolant, and struc ture to less than 0.4. Table 3 gives typical neutron balance sheets for several types reactors, indicating some of the absorptions that must be taken into account in the

power

of

design. 3.11

Fuels and Fertile Materials. The nuclear fuels and fertile materials play part in determining the choice of a reactor. The number of fission neutrons produced per neutron captured by the fuel is an important characteristic of a fuel and to a large extent determines the type of reactor to be used with a specified fuel. number, defined as n, and other characteristics of fuels that influence the choice » reactor are summarized in Table 4.

a large

This of

12-10

SELECTION

NUCLEAR-POWER-PLANT

[SEC.

12

The different characteristics of fuels make them adaptable to different reactor Where corrosion is a serious problem and fabrication is difficult, a fuel must types. Therefore, for hetero be used that permits easy access for inspection and handling. geneous water-cooled reactors, the preferred fuel is U2" because no special precau tions are required in handling it. For thermal reactors, tj for U2" is higher than for U2", and on this basis U"! might make thermal breeder reactors possible. The price of U2" in natural uranium is much less than that of U2"; however, the value of U2" should be comparable to the In reactors where highly enriched U2" is added, UMI value of highly enriched U2". For the sodium-cooled graphite reactors and or Pu can be considered as a substitute. for homogeneous reactors where handling of radioactive U2" may present no problem, enrichment or sweetening by the addition of U2" looks attractive. For plutonium, ri is so much higher in a fast reactor that its use is almost entirely relegated to those reactors. The above characteristics are summarized in Table 5. Table Fuel


Natural or slightly enriohed uranium U"», highly enriched

Preferred Fuels for Reactor Types

5.

Type of reactor

Mobility of fuel

Application

Thermal heterogeneous

Solid fuel

Stationary' power and

Thermal — usually

Usually solid fuel

Small power and

Mobile fuel or fuel in coolant Stationary or mobile

Stationary power

heterogeneous Thermal-homogeneous or heterogeneous

Fast

research research

Stationary power

fuel

There are only two naturally occurring fertile materials — U238 and Th2". Natural uranium or slightly enriched reactors have the fertile material mixed with the fission able material. Therefore, in these thermai reactors the fertile material is converted to fissionable material in the core of the re actor and is at least partly used in situ. The plutonium recovered from these re actors may be used as a feed for fast reactors. Thorium can find extensive use in the reflector or blanket region of all centralstation reactors to capture neutrons that would normally be lost by leakage. It will be used extensively in the blanket of homogeneous reactors because of the par ticular suitability of UM* for the homo The source of U24' will geneous reactor. be, therefore, by production in reactor blankets. Because of the high conversioci ratio of fast reactors, the excess neutrot? can be used to make Um from thorium. 7 a ~™4 6 5 9 3.12 To minimize the e-nLumping. o (inches) richment needed in thermal reactors, the Fio. 2. Effect of lumping on ifc. and confuel is usually lumped; i.e., a lattice ctf version ratio. rods (lumps) is arranged in the moderator. Figure 2 shows the effect The lumping increases the resonance escape probability. of lumping on the multiplication factor and the conversion ratio of sodium-cooJed graphite reactors. Curves for DjO reactors are similar. In establishing a design for a reactor, an effective neutron multiplication factor k,ff of 1.05 or higher is usually chosen for stationary power reactors. For mobile

SELECTION OF REACTORS

Sec. 12-1]

12-11

For a graphite-moderated power and research reactors, it can be higher than 1.10. reactor, the effect of lumping on conversion ratios for enrichments of 1.0 and 1.5 per cent is shown in Figs. 3 and 4, respectively. In H20-moderated reactors, because of the strong slowing-down power and rather high absorption cross section of H20, the H20-to-uranium ratio desired is between 2:1 and 4:1. In these reactors the distance between lumps is the water-coolant passage. In general, to minimize enrichment, thick fuel elements with a large H20 to fuelThis, of course, is inconsistent with the desire to achieve element ratio are desired. The enrichment used in boiling H20 reactors is usually found high power density. to be lower than in pressurized H20 reactors because, in part, the heat flux in boiling is limited and, therefore, the fuel plate thickness can be greater. In a D20 or graphite reactor, the slowing-down power is poor; therefore the modera tor-to-fuel ratio is usually greater than 20: 1. Because of the larger ratio of modera tor to fuel, more leeway is permitted in the arrangement and size of the lumps and, more important, in the freedom of subdividing the fuel within the lumps and not

K„, =I05


0.9

i

z o 0.8 l/l cr UJ >

z

o o II 10 a (INCHES) FOR 1% FUEL ENRICHMENT No COOLED GRAPHITE

12

0.7

\K"",(

'

6

15%

Via.

3. Effect of lumping on conversion (1.0 per cent enrichment).

ratio

0500

0.625

r

t

0.750

0=0 375"

0.6

REACTOR

\

n

No

7 8 a( INCHES) FOR

?

10

FUEL ENRICHMENT GRAPHITE REACTOR

COOLED

Eio. 4. Effect of lumping on conversion ratio (1.5 per cent enrichment).

destroying the lump identity as far as the neutron absorption reactors, lumping to minimize enrichment is consistent with

is concerned. For these fine division of the fuel

to

achieve good heat-transfer conditions. Table 6 indicates the types of 3.13 Structural Material and Other Absorbers. material and their approximate fraction composition that can be tolerated in centralstation power reactors. Every reactor must be considered individually. The effect of poison build-up and conversion of U"6 to plutonium fuel on the reac tivity of a natural uranium reactor is shown in Fig. 5. 3.14 Neutron Leakage. The fraction of neutron leakage from reactors is influ enced by the slowing-down and diffusion properties of the moderator and by reactor Table 6.

Per Cent

of Materials % tolerated

Tolerated by volume

Materials

Stainless

steel

Fast

Thermal

30 30 30 30

2 20 20 2

12-12


Fio.

/

5.

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

Variation of reactivity vs. irradiation time for natural uranium reactor. produced) /(neutrons absorbed): n = (neutrons

1. Based on power, 19.27 mw/ton: = (No. thermal neutrons absorbed in

U,,s)/(No. thermal neutrons absorbed). 2. « and m 5 per cent 2.. p are assumed constant. 3. 2«mI"i+*t™,ar,+™'>'"1+BM 4.
<

3 02

0

20

40

60

80

100

120

14.0

160

CORE DIAMETER (ft)

Fio.

6. Representative

neutron

leakage in thermal reactors

(unreflected).

Sec. 12-1]

SELECTION

OF REACTORS

12-13

size. Figures 6 and 7 show the effect of leakage and core composition on the reactor size for reflected and unreflected thermal reactors. For instance, these curves show that, if the neutron leakage from graphite and DjO reactors is to be kept small, these reactors must be large. Since the fraction of leakage is increased by the presence of voids, even H2()-moderated reactors must be large for small leakage if they operate CORE VOLUME

REACTOR

Zr

03

02

03 20

.015

4 Be MOD

10

05

5 H20,PRESSURIZED,UFUEL

20

15

1 (3RAPHITE

MOD

2 020 MOD 3 H20, BOILING,

25

COMPOSITION

U

\

U02

FUEL

10

OTHER

No

VOIDS

03

03

C=0.89

12

D20 =0835

10

30 .05

40

Be=070

65

l\

\i\

20

H2C

o

:ist

AKEr> AS! "HE S UMC FTH EPI THER MAL AND THER MAL LEAK AGE PRC >BAB LITIS S

LE* KAGf


5

,2

^N

5^

0

20

40

60

60

100

12.0

14.0

16 0

18.0

CORE DIAMETER (ft)

FlO.

7.

Representative neutron leakage in thermal reactors (reflected)

boiling and with UO. fuel. It is pointed out that the leakage neutrons are not In the case of a core with a blanket, most leakage neutrons necessarily always lost. are captured in fertile material. 3.2

Heat Removal

The design of the reactor must satisfy both nuclear and heat-transfer requirements.

For

example, the fuel-element thickness is governed by the heat flux and the allow The coolant passages through the fuel elements are determined by the coolant flow and by the amount of heat that has to be removed. These factors have to be reconciled with the arrangement of the fuel as dictated by the nuclear considerations. 3.21 Heat Transmission through the Fuel Element. The temperature drop through the fuel element is limited for mechanical or metallurgical reasons, and a maximum A 7* of 400°F is usually taken for metallic elements. If an oxide element is used, the allowable temperature drop must be high to attain a reasonable power density. It is, therefore, accepted that high temperature drops will be taken. The bond between the clad and the oxide must be designed to provide a minimum thermal resistance to

able temperature drop in the fuel elements.

12-14

SELECTION

NUCLEAR-POWER-PLANT

[Sec.

12

heat transfer from the fuel to the clad. The allowable temperature drop through a high-density oxide (about 95 per cent) is as high as 2000°F, and cracking of the ele ments is accepted. The cracks usually form in a radial direction and apparently do not adversely affect heat transfer. 3.22 Heat Transfer to the Coolant. Information is meager on the maximum heat flux that can be used. The design heat flux is usually chosen as a factor of 2 to The heat fluxes usually accepted as safe in reactor 3 below burnout heat flux.2 design are as follows: B/u/(*r)(/(>) 1. 0 X 10* 0 . 5 X 108 0. 2 X 10'

Liquid-metal transfer Liquid-water transfer Boiling-water transfer

The effect of these limits of heat transfer on power density as a function of the ratio of coolant to fuel volume is shown in Fig. 8 for metallic elements and in Fig. 9 for uranium oxide elements. For metallic fuel elements, the heat flux ql A is usually _\0/A=l.0»l06

1400

f =01"

typical design curve for (a) ^ fast reactor.

a

'

design curve for a (d typical pressurized water reactor /k— © TYPICAL DESIGN CURVE FOR A vy

\at =l30°f

i\

1200


notes:

"

t

l-

'

I

600

V

BOILING

\

WATER REACTOR

0/A=HEAT FLUX (BTU/SO FT HR.) t = FUEL PLATE THICKNESS(IN) flT =TEMPERATURE DIFFERENCE IN FUEL ELEMENT

3/A =05» I06 -U.I

iT =(>5°F

500

®

400

Q/A =05> I06 =02 oT =I30e F

300

200

(e I

^

—

0/A =05 «I08 t=0 4" aT =? fin °F

-

100

®

0 'A =0 ?»lf >6~ 1 t

0.5

Fio.

&e

.01

2.0 2.5 1.0 1.5 COOLANT TO FUEL VOLUME RATIO

8. The effect of hfint-trnnsfer considerations on power

=52 *F

30

35

density for metal

elements.

the limiting factor rather than the A7' limit of 400°F. For uranium oxide fuel ele ments, the heat transfer through the fuel is limiting as indicated in Fig. 9. 3.23 Heat Transport by the Coolant. The heat removed or transported by the coolant is equal to the product of the flow rate W, heat capacity cp, and the tempera ture rise of the coolant A7\ This is simply Wcv AT, or in volumetric units VAc ST, where V is the velocity, A the area of flow, and C the volumetric heat capacity of the coolant. As the foregoing discussion indicates, the ability of the coolant to remove heat is proportional to the temperature rise. In liquid-cooled reactors, it appears rather

Sec. 12-1]

SELECTION

12-15

OF REACTORS

difficult to eliminate thermal-stress fatigue problems in the vessel and piping system primarily caused by inadvertent reactor scram. Therefore, a rise of some 300°F is usually taken as a top limit. With experience and design ingenuity, it is expected that this limit will be raised. For pressurized water-cooled reactors, the rise is usually small and will probably be limited to 100°F, from 450°F inlet to 550°F outlet In this case, the top temperature is limited by the maximum desired temperature.


design pressure of the primary system and the lower (450°F) by the minimum accept It is interesting to note that coolant flow rate is about the same able steam pressure. for liquid-metal-cooled reactors (except lithium) as for a water-cooled reactor of the

2

34

56789 10

15

COOLANT TO FUEL VOLUME RATIO

Fio. UOi

9. The effect of heat-transfer considerations on power density for 95 per cent density fuel elements.

same rating because the difference in volumetric heat capacity* makes up for the dif ference in acceptable rise. The velocity of the coolant is usually taken as not greater than 30 fps. This is usually the limit because of pressure drop and pumping-power

limitations. The power density

is given by

VAC AT

LA/k

_ kVCAT L

where L is the reactor length and k the fraction of total area available for coolant flow. From the standpoint of heat transport, the above equation indicates that the power density is inversely proportional to reactor length; however, the total To reduce power is independent of length for given values of the other parameters. pressure drop, some reactors are made short in the flow direction, such as gas-cooled graphite power reactors. Because of this, some reactors are made with split, flow, This is done even at some sacrifice of critical mass. such as the Brookhaven reactor. « See Table 20.

NUCLEAU-POWER-PLANT

12-16

SELECTION

[SEC.

12

One of the greatest disadvantages 3.24 Operating Pressure and Temperatures. of pressurized water-cooled reactors is the high operating pressure that must be used The operating pres for only modest generated steam pressures and temperatures. sure is usually equal to the saturation pressure corresponding to the hot channel The operating coolant outlet temperature plus the pressure drop in the system. pressure for reactors using low-vapor-pressure coolants, such as liquid metals, is never more than 100 or 200 psi. HEAT GENERATION

HEAT GENERATION

HEAT = 1.26 AVE. HEAT

MAX

SURFACE

TEMP,

COOLANT

5

DISTANCE

£

ALONG FLOW DIRECTION


AND TEMPERATURE (o) HEAT GENERATION DISTRIBUTION FOR A THERMAL RE ACTOR WITH A GOOD REFLECTOR

DISTANCE

ALONG

TEMP.

FLOW

DIRECTION

HEAT GENERATION AND TEMPERATURE DIFFERENCE

(b) COSINE

.HEAT GENERATION HEAT GENERATION

SURFACE TEMP.

<

5 t-

COOLANT TEMP. DISTANCE

ALONG

FLOW

DIRECTION

(c) CONSTANT HEAT GENERATION AND TEMPERATURE DIFFERENCE 6 IS THE AVERAGE

Fio.

10.

TEMPERATURE

£

DISTANCE

ALONG

FLOW

DIRECTION

(d) HEAT GENERATION FOR A CONSTANT SURFACE TEMPERATURE

BETWEEN COOLANT AND SURFACE

Temperature distribution in a reactor for various heat-generation

pattcrns-

In order to attain high cycle efficiency, the coolant outlet temperature is made as The factors that usually limit the maximum operating tempera high as practical. tures are: 1. Vapor pressure of the coolant. 2. Thermal decomposition of coolant. Organic coolants are normally limited to a maximum of 700°F. 3. Structural-material The maximum comfortable operating tem limitations. perature for piping that is inaccessible and must last 20 years is around 10O0°F. Some designers will venture to 1200°F or higher. 4. Corrosion of fuel elements. 3.25 Nonuniform Heat Generation and Hot-spot Factors. For reactors with uniform fuel distribution, the average power density is smaller than the :


Sec. 12-1]

12-17

temperature drop through the fuel. It follows from reactor physics that, in reactors where the mean free path for events is large, the effect of localized concentration of particular atoms does not cause Fast reactors and DjO- and graphite-moderated reactors are of localized effects. this type; therefore, in these reactors, the exact location of control rods is not critical. The hot-spot factor in these reactors is small, since no localized flux disturbance exists. Whenever a strong moderator, such as H20, is used in a reactor, rapid thermalization of neutrons is likely to occur. Extreme care must be exercised in the design of H20 reactors, since a hot-spot factor as large as 2 can very easily develop. In homogeneous reactors, hot-spot factors have no significance because the heat is generated directly in the coolant. 3.3

Mobility

of Fuel

As mentioned in Art. 2.4, the fuel can be arranged in the reactor so that it (1) is stationary, (2) moves by separate carrier, (3) moves with the coolant. Rather dif ferent reactor characteristics are expected from these different fuel arrangements. Reactors in group 1 have been well developed, in group 3 partially developed, and in group 2 are in very preliminary development stage. Stationary Fuel.

3.31

Solid or fixed fuel has the advantages of staying in the

reactor and confining its fission products to the reactor. In the case of natural The fuel uranium or slightly enriched reactors, a fuel charge may last 2 to 10 years. is removed and then reprocessed or discarded. In many respects, this type of

plant

is the easiest to maintain; the fuel.

however, it requires mechanical handling devices to

remove

3.32 Fuel Moved by Separate Carrier. For this arrangement the fuel is contained tubes and is moved into and out of the reactor as a paste, solution, or suspension. eliminates the mechanical handling equipment required for stationary fuel. Rcestablishment of the fuel in the carrier may be much easier than refabrication of solid therefore, an attractive arrangement where frequent reprocessing elements. It radioactive. There a temperature difference between required or where the fuel of course, the fuel and the coolant as in a stationary fuel-element core. The carrier a necessary diluent that absorbs neutrons. In most cases, the desire to achieve high concentration of uranium in the carrier has resulted in an attempt to make the fuel into a paste or putty. Controlled pumping of these pastes remains to be demon strated on practical basis. The systems being studied are uranium and uranium oxide in sodium, uranium in bismuth, uranium compounds in lead and bismuth mix tures, and plutonium solutions.

in

is,

is

is

is

is,

It

a


The maximum power density is usually limited by the maximum fuel power density. temperature, the temperature drop through the fuel, or the fuel surface temperature. It is desirable to distribute the fuel, coolant, or moderator, so as to approach the In reactors where the maximum limiting power density at all points in the core. temperature of the fuel is controlling, it is desirable to produce the same maximum fuel temperature in each fuel element by the distribution of the fuel or coolant. In Some reactors are designed with orifice distribution practice, this is not easy to do. of flow to various channels, so that the temperature rise in each channel is more or For reactors where the heat flux is limiting, consideration is given to less the same. flattening the flux by a combination of factors such as location of control rods, choice of reflector, and a central region reflector. As the reactor art develops, there will be more of an attempt to make the average power level correspond to the limiting Many theoretical studies of the proper way to do this have been made, power level. but each application will dictate its own best method. Typical temperature-dis tribution curves for various idealized heat-generation patterns are shown in Fig. 10. In addition to the hot-spot factors caused by neutron flux distribution, hot spots may be caused by maldistribution of flow, nonuniform fuel distribution, variation in clad thickness, etc. Hot-spot effects usually require a correction in the calculation of maximum temperature by a factor higher than 1.5, applied to both coolant rise and

12-18

NUCLEAIt-POWEIt-PLANT

SELECTION

[SEC.

12

With the fuel in the coolant, there is 3.33 Fuel That Is Located in the Coolant. The reactor structure is usually quite simple, no temperature drop to the coolant. The system inven and fuel removal and return are easier than for stationary fuel. The radioactivity in the tory is usually quite high because of external holdup. The preferred fuel external sj'stem makes access for inspection and repair difficult. for these reactors is U"'. In Holland' consideration is being given to movement of dry particles of UOi. The particles are at about normal packed density of 50 per cent in the fuel tubes. They are moved through these tubes, which are vertical, by gravity and vibration of the outlet port on the bottom of each tube. The heat is transported by the particles themselves. Preliminary tests using sand have demonstrated the principle of operation. 3.4

Corrosion of Fuel Elements

The chemical reactivity of metallic uranium fuel elements and water presents one Unsuccessful attempts have been of the most serious problems for aqueous systems. made to find alloys of uranium that would provide reasonable corrosion resistance Alloying of in aqueous systems without seriously affecting the neutron economy. For the fuel elements also adds to the fuel fabrication and reprocessing problems. further discussion of this problem, see Art. 6, Fuel Elements. Alkali metals, lead, and organic coolants do not present any particular fuel corro sion problem.


3.6

Reprocessing

The fuel is reprocessed for one or more of the following reasons: (1) to repair irradia tion damage, (2) to restore reactivity caused by net loss of fissionable atoms, (3) to The degree to remove fission products because of their high neutron absorption. which these items affect reprocessing for various reactors is listed in Table 7. Table 7.

Factors That Affect Need for Reprocessing Thermal

Fast

Soli d fuel Solid fuel

1. Repair

irradiation

damage

or

Mobile fuel

Highly

Low

enriched

enrichment

Yea Minor

Minor Yes Minor

Minor Yes Yes

Yea

20 2 and 3

30 2

50-150 2 and 3

30 2 and 3

remove

2. Reenrich core 3. Remove poison effort of fission products Maximum tolerable burnup of fissionable material (atoms burned/fuel atoms), %. .

Yes Yes Minor 10

Mobile fuel or

Yf»

The steps involved in reprocessing depend to some extent on the method used. stops involved are (1) decladding or physical separation: (2) dissolution; (3) decontamination, or removal of fission products; (4) separation o: U-Pu or U-Th; and (5) metal reduction or conversion. The cost per kilowatthour of reprocessing is primarily affected by burnup. the cost In aqueous reprocessing, the cost is being almost inversely proportional to burnup. The amount and kind affected by the amount and type of alloying material used. of acid used in aqueous reprocessing are greatly influenced by the alloy composition Furthermore, storing the acid wastes is expensive and for some fuel elements car; amount to a charge of 2 to 3 mills/kwhr.

For aqueous reprocessing, the


Sec. 12-1]

12-19

The following general statements can be made in regard to the effect of reprocessing on the selection of reactors : 1. The fission-product buildup in fast reactors does not seriously affect neutron Irradiation damage in fast reactors is economy, whereas it does in a thermal reactor. the real limit to the permissible burnup, whereas in thermal reactors when corrosion is not a problem, the fission-product build-up and poor conversion ratio limit the fuel element performance. 2. Because the enrichment in a fast reactor is about 10 times that in a thermal reactor, reprocessing must be about 10 times as frequent per unit of fuel in the core. Reprocessing requirements are simpler, since partial removal of the fission products is sufficient. Pyrometallurgical reprocessing by simple melting and slagging seems feasible but remains to be demonstrated. 3. For natural uranium reactors and for reactors using up to about 1.5 per cent The fuel elements enrichment, the core will probably not be reprocessed for reuse. can be discarded without greatly affecting the fuel costs. 4. Reprocessing is costly but is necessary for solid-fuel-element thermal reactors The problem is worse for metallic elements that with enrichment above 1.5 per cent. are highly alloyed with corrosion-resistant metals, such as may be used in watercooled reactors, than for unalloyed elements because they are more difficult to dissolve. 5. The reprocessing steps are not necessarily reduced more for homogeneous or


liquid-fuel reactors than for solid-fuel types. For example, in the D20 homogeneous reactor the step of dissolving fuel does not exist; but the problem of removing the DjO from the uranyl salt, in order to minimize DtO losses, does exist. 3.6

Safety

One of the potential hazards of large nuclear reactors is the tremendous amount of If the fission products that build up in a radioactive material that they contain. reactor were accidentally released, a public hazard could result. The surest way to make a reactor safe from a public-liability standpoint is to remove the fission products continuously as they are produced. It is theoretically possible to do this in some homogeneous or mobile-fuel reactors. It is not always easy to do in a practical Furthermore, extreme care must be exercised in removing fission products manner. continuously to avoid the public nuisance of continuous release of radioactivity. 3.61 Containment. Most power reactors are built with a limited isolation area and provision for containment in a gastight building. For stationary power reactors, the isolation area is usually about 500 to 1,000 acres. Some of the considerations in the design of a building to contain the fission products in case of a nuclear incident are discussed below. 1. The building is designed to contain a complete release of the fission products. The release can be caused by a nuclear event or loss of coolant to the core. 2. In most cases, the energy release of a rapid nuclear transient does not control the building design, since the expected energy release is much less violent than the equivalent of 100 lb of TNT (300 kwhr of heat). If the nuclear event (or other possible accidents) can conceivably initiate a violent chemical reaction, this will control the building design. For a simple example, consider the air-cooled graphite reactor. A nuclear incident could cause meltdown of the uranium fuel, raising the temperature of the graphite to a point where the "air cooling" would ignite the graphite and the uranium. In a water-moderated reactor, a runaway accident might melt the core and a rapid reaction of the molten core materials and the water could ensue. In this case, some of the possible reactions that would have to be considered are the exothermic reaction of water with Al, Zr, or U with the formation of hydrogen and steam. A sodium-cooled reactor introduces the possibility of the sodium-air

reaction.

3. The

the

stored energy in the coolant must also be taken into account in establishing

building

3-62

accident

design.

Aside from the public-safety problem, even an Temperature Coefficient. Certain types of reactors have contained by the building is to be avoided.

12-20

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

I is

a

is

e

a

is

is

is

is

|3

/S,

k is

is

a

a

if

it is,

a

is,

of the reactor, thereby reducing the amount of thermalization of neutrons. Another factor is that tho cross section of U"6 drops faster with increased temperature than the absorbers, usually present in reactors. In the case of water-moderated reactors, the heat generated by the fuel elements can vaporize the water in the neighborhood of the elements, thereby causing the reactor to shut itself down before the fuel elements are seriously damaged. The Doppler effect can also contribute to an instantaneous negative coefficient to reactors provided the elements contain a large amount of U"8 as compared with U,3S. This nega tive reactivity change is brought about by the broadening of the resonance absorption The Doppler effect can, how because of the increased temperature of the uranium. ever, contribute to a positive coefficient rather than a negative one in fast reactors if the ratio of Us,a to Um is not high enough. This effect is rather small and does not play a major part in reactor safety. One of the advantages of homogeneous reactors is the large coefficient of expansion of the liquids as compared with the solids; this permits the fuel to expand out of the reactor and produce a large negative coefficient. It should be pointed out that what is really desired is a high negative temperature coefficient that comes into action one that does not depend upon the transfer of heat over any immediately, that appreciable distance. The process of transient heat transfer may require several seconds, and during this time interval the reactor may increase to such a power level as to cause nuclear incident. 2. Nonautocatalylic Reactors. Some reactors, on loss of their coolant, are autoWith sufficient care and some sacrifice catalytic; that they increase in reactivity. of performance, appears possible to design both graphite- and DsO-moderated reac To avoid this tors so that they can be cooled with HaO and still be nonautocatalytic. problem, North American Aviation, Inc., has designed the sodium graphite reactor so that the sodium coolant virtually cannot be drained from the otherwise autocatalyUc reactor. 3.63 Void Coefficient. It would be highly desirable to design the reactor so that all the control rods were removed the reactor would heat up, expand, and reach a stable power level short of melting. Some designs of the boiling-water reactor have this characteristic. Most of the boiling-water and pressurized-water reactors can W designed on this basis, but only at the sacrifice of operating temperature and power The surge tank in the pressurized water reactor can be located to permit density. The inertia effects of the expelled water in the reactor boiling as reactivity control. resulting from power surge can usually be minimized by locating the surge tank properly in reference to the reactor. 3.64 The time interval in which the neutron Large Neutron Generation Time. defined as the period of reactor. flux in reactor increases by factor of It can effective neutron-generation be shown that the period equal to l/(k — 1), where In the fission process, over 99 per the neutron-multiplication factor. time and cent of the neutrons are prompt and the remaining neutrons, the delayed fraction are emitted from the fission products over a much longer time. For U***, the equal to about 0.7 per cent and the effective generation time delay fraction primarily due to the delayed neutrons, since about 0.1 sec. This time of 0.1 sec less than 10~s sec. the generation time for the prompt neutrons For safety, a reactor should be operated so that short periods are avoided. In fact, in reactor with small reactivity changes to be accommodated, the safest way to to limit the maximum reactivity that can be imposed to less than that equal operate is

a


inherent safety features that tend to minimize the possibility of a runaway incident, some of which are discussed below. 1. Negative If the reactivity of a reactor Temperature Coefficient of Reactivity. decreases with increasing temperature, the reactor is said to have a negative tempera ture coefficient. This tendency for a reactor to resist large, sudden increases in power is almost a necessity and is achieved in most power reactors. Several factors can contribute to a negative coefficient. An increase in temperature can expand the reactor, thereby increasing the leakage and hence decreasing the reactivity. Also, if the moderator is a liquid, the increased temperature can expand the moderator out

Sec. 12-1]

SELECTION

12-21

OF REACTOKS

is

k

+

8.

8,

is

a

k

(1

(fc

l)/k, this Since the reactivity is defined as to the delayed-neutron fraction. for a reactor with Um fuel should be less than means that 0). When equal to 1.0064, then chain reaction can be sustained by the prompt neutrons alone. It important to prevent imposing large reactivity changes rapidly and to design so The that the reactor can rapidly and automatically create a large negative reactivity. delayed-neutron fractions and the average neutron-generation time for various fission the effective Using the values given in Table able materials are given in Table delay fractions and the effective generation times for typical reactors are given in —

9.

Delayed-neutron

0.

Table 8.

Times for

Fractions and Average Generation Fissionable Materials

delayed-neutron fraction. %t

I,

Table

average generation time sect

Material Thermal

Th"»» IT""

0. 19

0.61 1.6

0.2

0.21

0.045 0

0.048 0.082

Properties estimated. Thermal and fission spectrum refer to the energy of the neutrons

077

0.13 0.029

0.032

used to produce

fission.

Effective Delayed-neutron Fractions, Per Cent, and Average Generation Times, Sec, for Typical Reactors* Fast reactors

Fuel

Homogcueous

0.06

0.083 035

I

0.051

0

0.29 0.67 0.25

01

0.02

0. 13 20

0

0.34

0

0.055 0.085 0.039

76

1

0.38

0

I

0

Heterogeneous

0

Tjni TJl» Pu»»

Thermal reactors

0

9.

0

*

t

spectrum

1.9

Pu"»

Table

Fission

Thermal

0.26

0.26 0.64

U»" U»t

007

I

is

0 is

I is

* Where the effective neutron generation the effective delayed neutron fraction in per cent and These data are subject to revision particularly for the fast rcactur where very little time in seconds. The homugencous boiling reactor more nearly represented credit was taken for the fissions in J1". by the heterogeneous thermal data.

2

is

The large delayed fraction associated with the fast fission of U23" increases the delayed fraction for the plutonium-operated fast reactor. For homogeneous reactors, the effective generation time usually less than half that for fixed-type elements. The majority of the delayed neutrons in homogeneous reactors have half-lives that are long in comparison with the reactor core residence time; therefore, most of the neu trons are formed in the outside circulating stream and hence do not contribute to the However, homogeneous reactors are relatively safe, since the core generation time. fuel can be expanded rapidly out of the core. If large amount of reactivity, of the order of per cent, were introduced into any conceivable that the inertia force could readily rupture the pressure reactor, vessel before the fluid could expand out of the reactor and shut down the excursion. is

it

a


Fission spectrum

12-22 Table 10.

NUCLEAR-POWER-PLANT Compaiison

SELECTION

of Some Factors That Influence

[Sec.

12

Safety of Typical Fast and

Thermal Power Reactors Typical reactor using

Fast

Total Afc, to he handled,

%

fuel

Homo

Thermal

geneous

io-<

10"'

10 '

Effective generation life for small changes in -U , ace Reactivity changes, Afc, %: Cold to hot

UIJi

0.7 0.09

0.7 0.09

0.2 0.02

0.5 0.2 0.2 0.0 0.0 0.9

5.0 0-2

5.0 0.0 0.2 0.3

0 5 3 0 7 0 16.0

0 5 6 0

Some of the factors that influence the safety of thermal and fast power reactors arc shown in Table 10.


4

PRINCIPAL FEATURES OF REACTOR TYPES 4.1

Central-station

Power

Under classification of reactors, it can be seen that a great variety of reactors is conceptually possible. Table 11 lists the types of reactors now being considered for The number of reactors listed is rather large, but by no means central-station power. The degree of development of even those on which considerable work all-inclusive. has been done is limited, so that the features of any type cannot be accurately given. With the great difference in degree of development of the various reactors, the com parison is at best a rough estimate. 4.11 General Characteristics of Sodium-cooled Fast Reactors. The chain reac tion in fast reactors is sustained by fission events initiated by neutrons whose energies range from several Mev all the way down to about 50 kev. In general, the neutronenergy spectrum peaks in the vicinity of 0.2 to 0.3 Mev, and the median energy of fission-inducing neutrons is usually only slightly lower. Because the fission cross section of U"6 is only a few times greater than the absorption cross section of U"* in this energy region, enriched fuel must be used. No moderator is used in this type of reactor, the neutron energy being reduced primarily by inelastic scattering in diluent materials and in the coolant. Since the volume of the core is almost entirely made up of uranium and coolant, fast-reactor cores are characterized by small size. Since neutron losses in a fast reactor are rather insensitive to the quantity of struc tural material and coolant used, large latitude can be taken in the amount of these materials used. Table 12 shows the neutron balance for the Enrico Fermi Atomic Power l'lant. Even with 50 per cent sodium coolant volume, the neutron absorption Although there are approximately 20 per cent in the sodium is less than 0.1 per cent. structural iron and zirconium, their absorption is only about 1 per cent of the total The amounts of these materials are limited more by their effect neutrons produced. on neutron-energy level in the reactor and on safety as influenced by Doppler-effect considerations than by neutron absorption. Keact vity changes are caused principally by changes in neutron leakage accom Since fission-product panying core expansion and changes in density of the coolant. cross sections are small in the neutron-energy range of fast reactors, reactivity losses due to poisoning are considered negligible. The magnitude of the total temperature coefficient of reactivity in fast power reac

type

Homogeneous4: DtO

Liquid-metal

fuel*

fuel, graphite

Na-cooled Na-cooled, mobile DaO pressurized HaO pressurized DiO boiling, direct HiO boiling, direct

Heterogeneous: Graphite:

reactors:

reactors:

Thermal

Fast

Reactor

750

750

350 1.000 500 500 1.000 750 600 500

600

rating, Mw

Thermal

Table

11.

37.500

3.000 1.500 2.300 900

233 910 1.500

2.000-

Specific power, kw/kg U">

Comparison

60.0

60.0

Nat. 1.4 Nat. 1.2

Nat. 1.2 1.5

15.0

Enrichment, %

of Various

20

325 510 260 550

1.500 1.100 340

300*

Critical mass, kg U«»

Kinds

50 40 40 50

250 100 25

50»

Tons

-station

0.9

0.8

0.9 0.8 0.8 0.7 0.9 0.7 0.85 0.75

1.5

Conversion ratio

of Central


reprocessing

problems

Reactors

Corrosion

Corrosion

None Fuel-element corrosion None Fuel movement Fuel-element corrosion Fuel-element corrosion Fuel-element corrosion Fuel-element corrosion

Cheap

Unresolved

Power

Pressure

problems

vessel, repair, and safety

None Na handling Na handling Containment DsO Pressure vessel Containment DtO Stability

Na handling reaction Na-HiO Pu fabrication

Development

type

35

10 12 12

10 12 13 6X5 13 9X7

20 24 X 20

size, ht,

X X X

X

. . 700

24

3X3

Reactor diam

of Various

X

9

750

350 1.000 500 500 1.000 750 600 500

600

Thermal rating, Mw

Comparison

X

5

5 X

•Pu. Includes blanket. Fuel separate from coolant. not intended to discount the HiO homogeneous reactor. Author's estimate of installed cost of complete "third round" CFS gas flow at inlet conditions. 'Steam flow in pounds par hour.

modera

11.

X

Fast reactors: Solid fuel Thermal reactors; Heterogeneous: Graphite: Gas-cooled H20-cooled Na-cooled mobile fuel* Na-cooled. DiO pressurized HjO pressurized DjO boiling, direct. HjO boiling, direct fuel, graphite liquid-metal tor Homogeneous^: DiO

Reactor

Table

plant

turbine

500

700

400 450 600 550 450 450

600

tetnp, °F

Coolant inlet

generator.

550

1.000

800 550 900 850 550 550 486 486

900

temp. °F

Coolant outlet

of Central-station

including

2.500

100

150 1.500 100 100 1.300 1.300 600 600

100

Operating pressure, P8>g

Kinds


ft

*

e rf

is

It

'

'

1000 600 1100 1000 600 600 550 550

1100

temp, °F

Max surface

Power

120.000

45.000

450.000/ 80.000 40.000 40.000 80.000 60,000 2.5X 10" 2X I0««

50.000

Coolant flow, gpm

Reactors.

400

1.500

400 300 1,000 600 300 300 600 600

1.000

Steam pressure, PSlg

500

900

550 500 800 750 500 500 486 486

800

Steam temp, °F

(Continued)

200

240

37 26

240 220 220 200 220 210 180 170

220

34

26 25 34 33 25 25 25 25

Relative cost,* $/elec. kw Cycle efficiency,

Sec. 12-1]

SELECTION OF REACTORS Table

12-25

Neutron Balance in Clean Fast Reactor

12.

(Basis:

One fission in core Um)

Neutron production

TJm U"1 U1"

capture fission capture capture Stainless-steel Zirconium capture.

Boron capture Total

Neutron absorption

Core

Blanket

Total

Core

Blanket

Total

2.4800

0.0739

2.5539

0.2148

0. 1524

0.3672

1.0000 0.2106 0.0866 0.3443 0.0209 0.0037 0.0021 0.0250

0.0298 0.0073 0.0615 1. 1023 0.0112

1.0298 0.2179 0. 1481 1.4466 0.0321 0.0037 0.0037 0.0250

1.6932

1.2137

2.6948

0.2263

2.9211

0.0016

2.9069

Neutron leakage: Core to blanket, 1.0016; blanket to shield, 0.0142.


Table 13.

Distribution

of Fissions

Material

Core

Blanket

Reactor

25 28

0.849 0.074 0.923

0.025 0.052 0.077

0.874 0. 126 1.000

Total

tors

is usually of the order of — 10~s(Afc/A:)/°C. The largest component of reactivity is usually due to expansion of sodium (because Na decreases the energy level of the This component is several times larger than that obtained from linear neutrons). expansion of fuel. The prompt-neutron lifetime is extremely short; it is of the order of 10-7 sec. However, as shown in Table 9, the average generation time is similar to that in heterogeneous thermal reactors. For U"6 fuel, the generation time is about 0.09 sec, and for Pu"9 fuel, about 0.04 sec. The delayed-neutron fraction is also similar in fast and in heterogeneous thermal reactors, as shown in Table 9.

In

the following respects,

to thermal reactors:

the fast reactor possesses safety characteristics superior

It

is required to accommodate very little reactivity change. can be made to have a fair negative (instantaneous) Doppler coefficient. 3. There is negligible delay in the metal coefficient of reactivity, since the heat is generated in the fuel element. 4. The coolant coefficient is closely coupled to the reactivity change. 5. There is no chemical reaction that can be set off in the core as the result of a 1.

2.

It

nuclear excursion. The breeding ratio in fast reactors

is larger than in thermal reactors, not only because the larger fast fission effect (Table 13) but also because of the higher number of neutrons produced per neutron absorbed in the fissionable material as shown in Tabic The conversion ratio for U"s fuel is expected to be about 1.2 and for Pu fuel 14.

of

about 1.5. The importance of maintaining high neutron energy in the core of the reactor is illustrated in Figure 11, which shows the variation of or* with neutron energy. In the interest of producing high-quality plutonium, it is not so important to maintain a high neutron energy in the blanket, since the ratio absorption cross section of Pu*"

to U"s

does not drop except

at low energy, as shown in Fig.

12.

Owing to the small size and high power density of a fast reactor, the fuel must be finely subdivided in order to limit the internal temperature difference. As a result, * Alpha is the ratio of nonfission captures to fissions in the fuel isotope.

12-26 Table 14.

NUCLEAR-POWER-PLANT

SELECTION

[SEC.

12

Comparison of Nuclear Parameters for Fast and Thermal Reactors V

Fast 0.2 Mev

Thermal 0.025 ev

0.2 Mev

2.46

2.46 2.54 2.88

2.20 2.49 2.64

u»« U«" Pu»"

a

1

2.54

2.88

Fast

Thermal 0.025 ev

Fast

0.2 Mev

2.08 2.31 2.03

0.

II

0.02 0.09

Thermal 0.025 ev

0. 184 0. 1

0.42

coolant passages are quite small, around in. in diameter. Growth and deposition Plugging in the core due to mass transport does may tend to plug these passages. not seem to be a problem, since dissolved metal tends to deposit preferentially on cold Care must be exercised to prevent particle surfaces rather than on hot surfaces. accumulation in the small passages. —V

i o

0 8 ■

I


< 0 6 <

04

t

A (\

Pu"9 4

u«s"

0 2

—( I0"2

10"'

10°

I02

I03

10*

I05

10*

ENERGY, ev

Fig.

11.

Variation of a with energy.

In some ways, the small reactor size appears to be a help instead of a With a reasonable coolant velocity of 30 fps and a temperature rise of 300°F,

problem. 550 Mw

of heat, a desirable power output, is obtained from a reactor whose diameter and length are both 3 ft. A greater length 4.0 would not help the heat-transport prob lem, and this small size greatly simplifies ,7Te 7%ss 3.2 design and unloading the structural 2.4 problem. The use of sodium coolant permits 1.6 operation at high temperature without 0.8 high pressure and provides an opportunity for a relatively high steam cycle efficiency 0 Many structural materials are corro I0"z 10 10° lO2 I03 10* I05 10* Low-carbon sion resistant in sodium. ENERGY, ev steel, iron, stainless chromium iron, Fro. 12. Variation of plutonium quality nickel, and Inconel are all corrosion re with neutron spectrum. sistant at the temperature of interest. As a result, it is expected that the ! provided the oxygen content is kept low system equipment will not be very expensive. In many ways, the most serious problem in a fast reactor is the fuel cycle cost In a thermal reactor, perhaps only 1 to 2 per cent of the fuel element is fissionable material. If 1 per cent burnup of the total atoms can be achieved, a large frac tion of the initial fuel is burned before reprocessing. In the case of a fast rcart«r

Sec. 12-1]


12-27


about 25 per cent of the fuel element is fissionable material. If 2 per cent burnup of the total atoms is achieved, only 8 per cent of the fuel is burned before reprocessing. Thus, a fuel atom is on the average reprocessed 12 times before it is burned, and high inventory and loss charges may be associated with reprocessing. There are, however, some alleviating circumstances. Fission products can be allowed to build up to high levels without seriously affecting neutron economy, so that high burnup and crude The requirements here, too, are less stringent, since reprocessing can be tolerated. the corrosion resistance of uranium in sodium is such that a somewhat less perfect clad than is necessary in water-cooled reactors may be used.

These factors contribute to the possibility of using simple methods of reprocessing and refabrication. It is thought likely that pyrometallurgical reprocessing consisting of a simple melting operation with oxide slagging by the crucible will eventually be It is expected that this process will remove 80 per cent of the fission products. used. The recovered fuel will be sent to a "hot" fabrication facility for the manufacture of new fuel elements. Eventually, it is expected these reactors will be operated with mobile fuel, thereby eliminating the hot fabrication facility. The reactor has a low internal conversion ratio somewhere between 0.4 and 0.8, To avoid handling large reactivity changes, por depending on the fuel and design. tions of the core should be changed at frequent intervals, i.e., about once a month, even though the average core life is about 6 months. 4.12 The Enrico Fermi Sodium-cooled Fast Reactor. The Enrico Fermi Atomic Power Plant is described briefly in this article in order to illustrate some of the design features and problems of fast reactors. This reactor is designed to operate on U"s fuel, not plutonium, because of safety, availability of materials, and lack of plutonium tech It will produce about 300 MW of heat with a corresponding gross electrical nology. output of 100 Mw. The core, which is about 31 in. in diameter by 30.5 in. high, is surrounded by a blanket of depleted uranium 2 ft thick in the radial direction and 1 % ft thick on top and bottom. A perspective view of the reactor is shown in Fig. 13. Core Design. The reactor core, shown diagrammatically in Fig. 14, is an assembly of partially enriched uranium-alloy pins. Plutonium is produced in the reactor core and in a surrounding breeder blanket of depleted uranium rods. Insertion of The core consists of boron-containing poison rods provides shim and safety control. These square fuel-element subassemblies arranged to approximate a right cylinder. subassemblies contain the fuel elements and axial blanket elements. Two types of fuel elements are being considered: pin type (Fig. 15) and flat-plate type. The radial blanket subassemblies, shown in Fig. 16, are the same size as the core subassemblies; Both the core and the however, they contain cylindrical rods of depleted uranium. blanket are cooled by sodium which is pumped into the bottom of the reactor vessel , flows upward through these sections, and flows out near the top of the reactor vessel. By using up-flow through the core, decay heat can be removed by natural circulation. Control. Shim control is provided by two boron control rods located at the center of the reactor. Little reactivity must be accommodated by this control rod, so the loss in neutrons is very small. The decision to move poison instead of fuel was made because the heat-dissipation requirements are much less and because much more reactivity can be controlled per unit of volume. The poison control rod is lighter, permitting use of a relatively cheap drive. The safety or shutdown rods also contain l>oron poison. Although these rods contain approximately 6 per cent Ak, they do not absorb neutrons in normal operation because, under operating conditions, they are located outside the core at the upper edge of the top blanket. They are scrammed by gravity with an initial spring assist. Mechanical Handling. Unloading the core and blanket is accomplished by an elemenHiandling mechanism, shown in Fig. 13, that consists of a rotating shield plug and an offset handling crane mounted in the plug. A holddown plate is suspended below the plug to maintain radial alignment of the core subassemblies and to hold This them against the forces caused by the flow of coolant through the elements. plate also acts as a guide for the control element drives. The sodium level is maintained nearly 11 ft above the core. This is done to provide natural circulation cooling of the core during unloading or during a pump

NUCLEAR-POWER-PLANT

12-28

SELECTION

[Sec.

12

failure and to keep the fuel subassemblies under sodium during unloading in order to dissipate decay heat. Heat is removed from the reactor core Liquid-metal and Steam-power Systems. The heat is transferred to an intermediate and blanket by circulating molten sodium. sodium system and is then transferred to water and steam in a once-through-type steam generator, a schematic arrangement of which is shown in Fig. 17. Steam is utilized in a conventional steam turbine which is directly connected to an electric The intermediate circuit is used to prevent a sodium-water accident generator. MACHINERY

COMPARTMENT CONTROL

OFFSET HANDLING MECHANISM

ROD ACTUATORS

SCRAM MECHANISMS

HOLD-DOWN

EXIT ELEVATOR-

HOLD-DOWN MECHANISM MOUNTING PLATE

SHIELD ROTATING ,PLUG ASSEMBLY

iVWi"


MECHANISM

DECAY

COOLANT OUTLET DRIVE SHAFT EXTENSION

CONTROL

■COOLANT

INLET

HOLD- DOWN DRIVE SHAFT HOLD-DOWN PLATE

TRANSFER ROTOR CONTAINER

CORE

RADIAL BLANKET COOLANT INLET ESSEL THERMAL

SHIELD

SUPPORT COLUMNS STRUCTURE SUPPORT COLUMNS STRUCTURE

SUPPORT

Fio.

13.

PLATES

Perspective view of reactor.

A plan view of the plant is shown in Fig. 18. The from releasing radioactivity. only items of equipment in the primary system that may require removal for inspec tion and repair are the pumps and the tube bundles of the heat exchangers. These are made as sump-type units with a free surface above the active portions and can be removed from the system without cutting the piping and without draining the system. No isolating valves are used in the primary or secondary systems, since they are con sidered more vulnerable than the piping, heat-exchanger housing, and pump housing The valves also would unnecessarily increase that they would be called on to isolate. the pressure drop, which should be kept to a minimum, since the coolant flows by A siphon break is gravity from the reactor tank to the heat exchanger and pump.


Sec. 12-1]


12-29

provided on the piping at the reactor so that a rupture in the piping will not drain Surrounding the reactor vessel is a so-called primary shield tank the reactor tank. which provides containment for sodium in the event of a reactor vessel leak. /"cbwwtment " The large sodium pool above the reactor, containing approximately 15,000 gal of sodium, contributes the largest part of the 67-sec cycle time for the sodium flow through the primary system. The pool minimizes thermal shocks to the piping system which tend to occur during reactor scram. Thermal shock to system parts has -CORE also been minimized by the following REACTOR design features: (1) All the high-pressure VESSEL(100 psi) parts of the reactor vessel are at the cool inlet temperature; (2) the rise through the reactor is limited to 250°F; (3) the top pool and piping are COOLANT IN COOLANT IN fabricated of thin walls, with the piping in. thick and the pool tank wall being 1 in. thick; and (4) the need for scram is Fio. 14. Reactor arrangement. minimized. The thermal shield, composed of metal plates, is positioned against Shielding. the inner wall of the reactor vessel as shown in Fig. 19. Its purpose is to reduce the fast-neutron flux reaching the reactor vessel in order to minimize the damage that results from neutron embrittlement and neutron- and y-induced thermal stress. It

FUEL SUBASSEMBLY-OVERALL LENGTH 98%"

Fio.

15.

Pin-type fuel subassembly.

also acts as a reflector, slightly decreasing the blanket neutron leakage and decreasing the neutron and y heating in the borated graphite. Compartment shielding is used to provide flexibility in the design and location of the system components. The radial

shielding,

which is located close to the reactor, is primarily

neutron shielding to

reduce activation of the secondary coolant system and to eliminate the need for cooling the concrete compartments of the shield. This primary neutron shield is made of borated graphite and is located in the primary neutron shield tank so that it can be confined and cooled by forced gas circulation. The secondary neutron shield is the structural concrete of the building and docs not require cooling.

12-30

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12


The space between the concrete shield and the building is used as the heat-exchanger and pump-machinery compartment. The building concrete around this compartment The level of sodium activity is about provides the required biological shielding. 0.1 curie /cm3 and is mainly generated in the sodium pool above the core.

BLANKET SUBASSEMBLY

RADIAL

Fig.

16.

Radial-blanket subassembly. TURBINE- GENERATOR

REACTOR

INTERMEDIATE HEAT EXCHANGER

CONDENSER

LEGEND EZZ3

nak

H8

NA

ES3 E53 I

Flo.

17.

STEAM WATER

I INERT GAS

Diagram of nuclear power plant.

There seems to be no net positive temperature coefficient of reac Safety. Uranium expansion alone seems to be fast enough to handle any conceivable The operating temperature has been chosen low enough fast reactivity transient. so that even if all the control rods are removed the reactor would shut off short of melting. A serious cold sodium slug accident is avoided by the low operating The reactor is temperature and by the fact that the system cannot be isolated. Reactor

tivity.

Sec. 12-1]

SELECTION -EQUIPMENT

OF REACTORS

12-31

DECAY PIT

DISASSEMBLY AND SHIPPNG raOUTY

REACTOR INTERMEDIATE HEATER EXCHANGER SODIUM PUMP-

-NAKPUMP

SOOUM STORAGE TANK:

STEAM GENERATOR


Via.

18.

FEEDWATER HEATERS

Plan view of plant.

OVERFLOW SECTION 4" INSULATION OVERFLOW PIPE

DOUBLE BARRIER EXPANSION JOINT OUTLET

Reactor veasel, elevation.

PIPE

12-32

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12


unloaded at operating temperature, so that little reactivity is required in the shim The maximum reactivity of the two control rods is less than that equal to control. Therefore, if the control rods are accidentally removed the delayed-neutron fraction. from the core when the reactor is critical, it would not get into a prompt period. The maximum speed of the control rods gives 0.007 per cent Ak per second. The fuel sub assemblies are also inserted at this rate. Containment. An airtight steel cylindrical reactor building, shown in Fig. 20, encloses the reactor, the fuel-handling mechanisms, the intermediate heat exchangers, and the sodium pumps and piping. It is approximately 72 ft in diameter and has a wall thickness of about 1 in. The purpose of this building is to contain the products of any reactor accident that results in release of fission products and radioactive sodium. General Design Considerations. Flexibility has been designed into the system. All mechanisms are removable. The rotating plug is removable in layers. The control I -GASTIGHT BUILDING Z - REACTOR TANK 3 OFFSET HANDLING CRANE A - TRANSFER ROTOR 5 -FUEL DECAY PIT 6 - FUEL TRANSFER PASSAGE 7 - FUEL ELEMENT DISASSEMBLY ROOM 8 - SHIPPING ROOM 9 - INTERMEDIATE HEAT EXCHANGER 0 - Na PUMP 11 - STEAM GENERATOR 12 -NAK EXPANSION TANK 3 - NAK PUMP 4 - NAK SYSTEM

Fig.

20.

Perspective

GRADE ELEVO0-0'

view of reactor building.

The core subassembly support plates can drive, rods, and thimbles are removable. be removed. The fuel and blanket subassemblies are the same size, so that with minor modifications they are interchangeable; therefore, the core size can be adjusted if necessary during initial start-up to achieve criticality. Furthermore, after a num ber of years of successful operation at design power, the core size can be increased to increase the power output. The design of the reactor has been established, and a full-scale mechanical test of the reactor and one primary loop are being constructed to prove the mechanic*: and hydraulic operation of the reactor. The components, which will not contain fissionable material, will be completed by early 1959. The reactor plant is expected to go into operation late in I960. Table 15 gives design data for the plant. Many design features of the reactor remain to be demonstrated, but there appears to be no problem for which a solution is not foreseeable. Liquid-fuel Fast Reactors. The fuel of primary interest in fast reactors is 4.13 The degree of contamination of Pu"9 with I'u110 is of little consequence, plutoniuin. since the fission cross section for Puwo is 0.6 that of Pu5" at high energy and it has i good »j. Since plutonium melts at considerably lower temperature than uranium, it is conceivable that liquid plutonium fuels can be developed which melt below WW'F.



12-33

Design Data for the Enrico Fermi Plant

Plant capacity:

Grose electric capacity, Mw Net electric capacity, Mw Turbine-generator rating, kw Reactor specifications: Power (heat), kw Core diameter, ft Core length, ft Critical mass UM», kg Fuel enrichment, % Core sodium flow rate, lb/ hr velocity, fps Sodium Neutron flux, neutrons/fern1) (sec) Liquid metals and steam systems: Net thermal efficiency, % Primary sodium temperature, °F: Leaving reactor Entering reactor Sodium flow, lb/hr Intermediate link Na temperatures,

Entering boiler Leaving boiler Na flow, lb/hr


Steam pressure, psig Steam temperature, °F Feed-water temperature, Steam flow, lb/hr

°F

100 90 100,000 300.000 2^6 2)4 450 ~25 11.22 X 10" 27.6 0.5 X I011 30.0

°F:

800 550 13.2 X 10' 750 500 13.2 X 10' 600 730 400 1 X 10"

Studies of these fuel systems are being conducted at Ames Atomic Laboratory.

The

Los Alamos Scientific Laboratory is working on a small molten plutonium reactor that is described in Summary of Major Power Reactor Experiments by W. K. Davis:* A promising reactor concept on which work has begun involves the use of molten plu High temperature would be possible with this system and a fast breeder tonium alloy. Many problems must be solved before reactor using such a fuel is potentially feasible. this reactor concept becomes a reality and the date for operation of a reactor experiment Preliminary work is can only be considered quite tentative, perhaps as early as 1959. being done at Los Alamos. The project is known as the Los Alamos Molten Plutonium It is designed for 1 megawatt heat and will use sodium Reactor Experiment (LAMPRE). as the coolant. 4.14 Graphite-moderated Reactors. The first reactors to be built were graphite moderated. Graphite was chosen because it had a sufficiently low thermal neutron absorption cross section to operate successfully with natural uranium and because it was available. Graphite has a relatively small slowing-down power, which gives rise to a larger leakage during thermalization as compared with other moderators. Therefore, to minimize leakage these reactors must be large (see Figs. 6 and 7). To permit operation with natural or slightly enriched uranium, the fuel is lumped in order to reduce resonance absorption in U2*8. The size of the lump and the lattice spacing can be varied. In the past the practice was to use about a 1.5-in.-diameter fuel cylinder as the lump and to use about an 8- to 10-in. lattice or pitch between these cylindrical lumps. With lattice arrangement of fuel, it is a simple matter to arrange the fuel in tubes and to provide individual cooling for each tube. This eliminates the problem of designing the reactor shell for the pressure of the coolant. For gas- and water-cooled graphite reactors, the tubes are arranged horizontally to facilitate access to the coolant headers and fuel. The neutron lifetime of these reactors is long, around 10~3 sec. The long neutron lifetime and the small effectiveness of individual control rods in such large reactors make a rapid nuclear excursion improbable. If they are cooled with an inert gas or licjuid, they are also relatively free from hazard of any rapid chemical reaction that might be produced from a nuclear event. Although the water-cooled reactor is subject to the possibility of a water-fuel reaction, the nuclear event takes place so

*

From Industrial Applications of Atomic Power: A fact sheet based on remarks of W. K. Davis before Association for the Advancement of Science, Atlanta, Ga.. Dec. 20. 1955.

A merican

NUCLEAR-POWEIt-PLANT

12-34

SELECTION

[Sec.

12

Extensive chemical slowly that the possibility of a rapid chemical reaction is slight. reaction of Al, Zr, or U with water has been shown to take place only when the metals Because become molten, and in some cases a high rate of heat transfer is required. of these facts, the reactor building need only be made gastight to prevent spread of radioactivity in case of the unlikely and mild nuclear accident. The most likely nuclear accident is the slow melt-down of the fuel caused by decay heat on complete failure of coolant flow. In a reactor of this size it is difficult to provide natural circulation for emergency cooling. This type of accident for a large graphite reactor must be guarded against by providing a gravity or auxiliary power supply for the coolant. Gas-cooled Graphite-moderated Reactors. The main reason gas-cooled graphitemoderated reactors are attractive is because they require no special materials for their construction other than natural uranium. Some countries have no supply of hafnium-free zirconium and no enriched fuel and have no desire to make DjO. In this case, there is little to choose but a gas-cooled graphite-moderated reactor. Helium Furthermore, to get even a modest power density, it is and C02 are suitable gases. If specialized fabrication tech necessary to operate at 5 to 10 atm of gas pressure. nologies are not available, the uranium can be canned in magnesium or aluminum. If aluminum is used, the operating temperature must be limited to avoid formation of Niobium could also be con low-melting-point eutectic of aluminum and uranium. sidered as the canning material if made thin enough.4 Table 16. Design of C alder Hall Reactor* level, heat, Mw Neutron flux average, neutrons/(cm*)(sec) Moderator ami reflector Arrangement Feed Fuel type Average metal temperature, °F Lattice spacing, in

182 ~1013

inlet temperature, °F exit temperature, °F inlet to boilers, °F pressure, psia Gross power, electric, Mw Auxiliary power, electric, Mw Turbine steam pressure, psia Turbine steam temperature, °F Turbine exhaust pressure, in. llg

284 637 637 115 42


Power

Coolant Coolant Coolant Coolant Coolant

Nuclear Data * The World's Reactors

No.

797 8

COi

7.5 200 590 28. 25

^ ^ ^

6 — Calder Hall, Nuclear

A representative reactor of this type

Graphite Interlocking blocks and tilea Natural uranium Extended surface

Engineering,

vol.

j-^ 1. no. 9, Dec..

1956.

is the Calder Works Nuclear Power Station by the United Kingdom Atomic Energy Authority and placed in service in October, 1956. The design and operating characteristics are shown in Table 16. The outlet gas temperature is 637°F. The reactor core is 21 ft high and 31 ft in diameter. It uses C()2 as the coolant at an operating pressure of 100 psig and has a heat output of 182 Mw with an electrical output of 42 Mw. Since the fuel channels are vertical, the unloading method is carried out by a discharge machine which lifts the fuel ele ments from the core and stores them in a shielded magazine. The magazine, when The full, is then lowered to a coffin and carried away for storage and processing. channels which have been emptied are then recharged by a charge machine which is much lighter because of the reduced shielding required. The more advanced British reactors have been designed for much higher output and for charging and discharginc under load. Some details of the Brookhaven open-cycle air-cooled graphite reactor are given in Sec. 13-5. A study of a typical ga.s-cooled graphite reactor was reported to the United States Atomic Energy Commission on Nuclear Power Reactor Technology. If some activity in the coolant were accepted, it is conceivable that an uncanned ura erected

Sec. 12-1]


12 35

The element could be coated with BeO, which nium oxide element could bo used. This design could be operated at temperatures would act as a partial fission barrier. If air were used as a coolant, it would high enough to be used with a gas turbine. have to be separated from the graphite and the top temperature would be dictated by the material and the design of the tubes. Reactors. Light-water-cooled graphite-moderated Waler-cooled Graphite-moderaled reactors have the same general features as the gas-cooled graphite reactors. These reactors use slightly enriched uranium because of the neutron absorption in the struc


tural material required for the tubes and in the alloy material required to reduce the By use of a coolant that is separated from the modera corrosion of uranium in water. The difficult tor, the coolant tubes can become the pressure-containing member. pressure vessel design and fabrication problems present in pressurized water reactors With water cooling, large graphite reactors can be designed for very are avoided. At the present time it appears that the largest practical size is high power ratings. This size is limited by central-station operation, not by reactor 1,000 Mw of heat. However, there are many design features that are adversely affected performance.

by going above this rating. As mentioned in Art. 3.2, the temperature rise is limited to about 100°F, and with an inlet temperature of 450°F this gives an outlet temperature of 550°F, which requires about a 1,400-psi operating pressure for the primary system. Because of the poor corrosion resistance and poor dimensional stability of uranium The fuel elements are discussed in more metal, it is desirable to use oxide elements. detail in the fuel-element article. It is important in the design to limit the size of the water annulus, particularly on This is done to minimize neutron capture by the H2() so that the outside of the tube. For the the reactor will not become autocatalytic on loss of coolant from the tubes. desired lattice arrangement, header design is a problem. The neutron flux in reactors of this type is usually around 3 X 10", so that there is a fair amount of equilibrium poisoning. Enough control is usually incorporated to provide about 1 hr xenon override. Reactors.* The use of liquid sodium as a reactor Sodium-cooled Graphite-moderaled coolant permits the attainment of high operating temperatures without requiring pressurization of the reactor. Sodium under atmospheric pressure boils at 1620"F. High coolant temperatures, in turn, permit the use of modern steam conditions with Sodium also has a moderately low net plant efficiencies greater than 30 per cent. neutron capture cross section —approximately half that of H2(). In addition, there are no corrosion incompatibilities between sodium and either uranium or Because of the high temperatures involved, graphite is generally considered thorium. as the moderator material in thermal reactors using sodium coolant. It would appear that the sodium-cooled graphite reactors might resemble the water-cooled graphite reactors with steel and zirconium substituted for aluminum. The flow rate for the same power is about the same because the reduced heat capacity of sodium is compensated by its greater temperature rise. However, the problems in handling sodium are so different from handling HjO that the reactors must be physically quite different. Containment features, reloading problems, and the autocatalytic nature of the reactor upon loss of sodium from the tubes make it unde sirable to use the horizontnl-tube-type construction characteristic of gas- and waterTherefore, the reactor is made with the coolant channels cooled graphite reactors.

vortical.

A

small sodium graphite experimental reactor, the SRE, has been constructed by American Aviation's Atomics International Division. This reactor is of the and tank" type as shown in Fig. 21. Considerable precautions have been taken to ensure that the coolant cannot be lost from the core. A large upper pool is provided containing sufficient sodium to fill the void spaces around the reactor should the core tank and outer tank develop leaks. In this manner it is virtually impossible for the reactor to lose all its coolant. Two independent cooling circuits are provided

North "pool

* Edited Inc.

anu contributed

to by S. Siegel. Atomics International Division. North American Aviation,

12-36

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

In many respects, the features of this type to ensure that one will always function. of reactor are similar to the fast reactor described in Art. 4.1. Sodium-cooled reactors generally use an intermediate, or secondary, cooling circuit to prevent escape of radioactivity in case of a sodium-water reaction in the steam generator (see Fig. 22). If the same lumping is used in the sodium graphite reactor as is used in waterA concept of this cooled graphite reactors, the reactor will be physically very large. type of reactor has been studied by the Monsanto Chemical Company.5 Such a reactor could operate on slightly enriched fuel with perhaps less than 1 per cent U*" in the uranium. The difficulties of constructing, loading, and unloading such a large High power reactor are so formidable that no real effort has been made to build one. may be achieved in a much smaller reactor by using a cluster of rods instead of a single rod element on approximately the same lattice distance. CONTROL ELEMENT DRIVE

SEAL ROLLER

ROTATABLE SHIELD RING SHIELD BELLOWS SODIUM LEVEL DIAPHRAGM

INNER LINER


MAIN SODIUM INLET LINE BIOLOGICAL SHIELD CORE TANK

THERMAL SHIELD OUTER TANK THERMAL INSULATION BEARING PLATE GRID PLATE CONTROL ELEMENT FUEL ELEMENT MODERATOR ELEMENT GUARD PIPE AUXILIARY SODIUM INLET LINE

Fig.

21.

Reactor core, elevation.

The fuel element is a group, or The Atomics International SGR is such a design. With this higher uranium-to-graphite cluster, of 19 rods. ratio, the enrichment required is greater than for the water-cooled or gas-cooled graphite reactors, generally being somewhere around 2 per cent. Because of the corrosion resistance of uranium in sodium, the life of the fuel element does not depend upon the integrity of the clad as in the case of the water-cooled graphite-moderated reactor. In the sodium-cooled reactor the life of the elements is From Fig. 5 it can be seen that for a primarily controlled by the loss of reactivity. higher conversion ratio the decrease in reactivity is much smaller; therefore, in order to attain a longer fuel-element life a system that gives a higher conversion ratio is to be preferred. The Th-U2" cycle gives a conversion ratio close to unity as compared with a U"'-enriched fuel, which might have a conversion ratio of around 0.8. There fore, a longer fuel-element life and a better fuel economy are expected to result I the use of a thorium cycle. Remote handling is necessary with uranium 233 1 The fuel element and its fabrication are sufficiently i of its 7 and a activity. so that this fuel element appears to be no more difficult to make than a nonradioactive The Atomics International SRE (Sodium Reactor one for a water-cooled reactor. Experiment), which is designed for 60-Mw heat output, has been successfully operated The performance of the reactor itself has been quite satisfactory, and experience with

12-37


Sec. 12-1]

The usual minor difficulties were experienced sodium technology has been very good. with such things as adjustment of the control system and with heat-exchanger per formance. Because of improved irradiation stability of uranium oxide compared to uranium, consideration is being given to operation with oxide fuel elements. Because of the possible awkwardness of loading and unloading the sodium-cooled Putty fuels could be con reactor, there is some incentive to consider mobile fuels. In this case, the volume fraction sidered where the uranium and thorium are mixed. of fuel and fertile material in the tube would be about 60 per cent of the theoretical Even this high concentration leaves considerable sodium carrier to act as density. A fully enriched U233 slurry could also be considered an added parasitic absorber. with thorium located in the blanket. This reactor would have greater parasitic losses, but the fuel might be easier to handle because the core slurry could be made quite dilute. PRIMARY ,HEAT

SUPERHEATERS 825° F

STEAM TO TURBINE

8I5.9T

ELIMINATORS 520. 4"F


EVAPORATORS 300° F

ÂTER

f-ARGON FILL AND VENT LINE LEGEND PRIMARY SODIUM LOOP SECONDARY SODIUM LOOP STEAM AND FEEDWATER SYSTEM ARGON LINES

SURGETANK PRIMARY-HEAT EXCH

PRIMARY

LOOP

PUMPS

Fia.

SECONDARY LOOP PUMPS

22. Reactor flow diagram.

In the pressurized Pressurized Water Reactors. Pressurized H20 Reactors. The reactor size is primarily reactor, the coolant also acts as the moderator. dictated by heat-transfer requirements. The average heat flux g/A is around 300,000 Btu/(hr) (ft*) with a maximum of about 600,000. This heat flux corresponds to the The resonance absorption of use of a metallic fuel rod with about a %-in. diameter. neutrons with this small-size rod is greater than with a large rod and hence requires more enrichment; however, it is economically warranted because the increased power To provide sufficient neutron moderation, the density reduces inventory charges. H20-uranium volume ratio is kept slightly above 2:1, even though it is not neces sary from a heat-transport standpoint. There is enough coolant flow area to per mit the use of a two-pass flow; however, the improvement in performance may not be great enough to warrant the added complexity of a two-pass design. A reactor generating 750 Mw of heat requires about a 6-ft-diameter core and, therefore, needs about an 8-ft-diameter pressure vessel. The operating pressure and temperatures are the same as for the water-cooled graphite reactor discussed in Art. 4.22. Very little water is required as a reflector for this type of a reactor. To reduce the neutron and y heating of the pressure vessel walls and to limit irradiation 4.16

H»0

12-38

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

damage, some thermal shielding is needed. For a design pressure of 1,500 psi, 6-in. vessel walls are required. The technology of these reactors is well developed. The use of oxide fuel eliminates the corrosion problem associated with the use of metallic elements. The design problems are reasonably understood, but another generation of these reactors will have to be undertaken before sizable gains are made on capital cost. The FUEL ELEMENT closure to provide for loading and unload HANDLING MECHANISM

FUEL SU8- ASSEMBLY

-CORE


CONTROL

-DISCHARGE

Fig.

23.

POT

Lurge-throw-manipulator

ing scheme.

DRIVE

unload

because of the high is awkward Two schemes are commonly pressures. used, either separately or together. Id one scheme a modified MTR method is used in which the fuel clement is handled by an offset crane and dropped into a detachable pot as shown in Fig. 23. The other scheme is to remove the entire top of the pressure enclosure. The core liftis expected to be about 2 years, so that unloading will not be very frequent and can be tolerated. a major shutdown Scheduled unloading can be limited to one every half year. The main problem is from unscheduled unloadings due to fuel-element failures through corrosion. For any unloading scheme, it will be somewhat difficult to determine which Most reactors have fuel element failed. elements that are designed to last several

ing

weeks after a rupture without introducing too much activity into the coolant. The preferred design is to use uranium oxide fuel as discussed in Art. 6. The enrichment in these reactors is high, compared with other thermal stationary power reactors, for the following reasons:

The use of cladding to prevent corrosion The large absorption cross section of water 3. The use of thin plates or small rods required for high heat transfer, which increases the resonance absorption 1.

2.

The enrichment usually runs between 1.5 and 3.0 per cent, depending primarily on the Some designs are based degree of risk taken on alloying for corrosion resistance. This is pri on using natural uranium and enriching by using fully enriched plates. marily done when the achievable burnup of the partially enriched plates is low. Because of the strong moderating property of H2G, the control rods must be placed on about 1-ft centers. To prevent these rods from interfering with unloading, they are usually operated from the bottom. This location and the operating pressure make it difficult to design for easy access for inspection and repair. Pressurized D20 Reactors. The pressurized DjO reactor is made large so that it can be operated with natural uranium. The core size for a large central-station-iyp.' The power reactor using natural uranium fuel is usually above 12 ft in diameter. uranium is lumped in a similar manner to that used in graphite reactors. Such a reactor is capable of producing 1,500 Mw of heat with a metallic fuel and 1,000 Mw of heat with U02 fuel elements. The chemical properties of D.O are similar to H;tl. The thermodynamic properties of 1)2C) are similar to H50 on a molal basis. The pressure vessel may be designed for the full operating pressure or the pressurr may be taken by the tubes around the fuel. The reactor is usually made vertical in order to contain the D20 easily, particu larly during unloading. It can be unloaded by an offset device as shown in Fig. 24 In this scheme one hole is provided in the top head for each seven subassemblies in


Sec. 12-1]

12-39

By this arrangement, the ligament efficiency around the a

1

a

is

if

\

The reactor type which we propose shows very consider able promise of being a type which ultimately can achieve the low cost of electricity necessary for nuclear fuel to com pete with conventional fuels. In the following paragraphs, some of the characteristics of the reactor are listed and our reasons are given for believing that these characteristics are achieved a long step toward our goal will have been

taken. 1. The coolant gas, either COj or helium, at a pres The exit temperature of the gas sure of 500 psi. 1050°F. is

is

Fro. 24. Small-throw-ma nipulator unloading scheme.

is

is

is

is

1

2. Superheated steam at a temperature of 1000°F and a pressure of .200 psi is generated, hence generating equipment of standard design used. 3. The fuel material UO2 or other form of ceramic uranium. The high exit temperature of the gas makes the use of uranium metal impractical. 4. The high pressure of the gas prohibits the use of an enveloping pressure vessel. Instead, the gas confined by pressure tubes which pass through the reactor. 5. The reactor proposed is a prototype which, in size, large enough so that scale-up to a reactor of 200 Mw or more will not require the solution of new basic or development

is

is

is

is is

problems. 6. The prototype proposed will be fueled with uranium enriched to 1.15 per cent, but a full-scale reactor would operate satisfactorily on natural uranium. The reason that the prototype must be enriched the large neutron leakage into the shield. The large leakage of neutrons into the shield due to the fact that the core diameter of the reactor only 10 feet. 7. Heavy water the moderator for the reactor and has been chosen because more spare reactivity available with this moderator than any other.

The

to be in excess of 10 Mw of heat per ton of fuel, comparable to what is obtained with liquid-cooled reactors, and the direct result of the use of high pressure and high temperatures for the gas. Raising the specific power of a gas-cooled reactor expected to largely remove the o-apital cost disadvantage which up to this time gas-cooled reactors have been supposed

It

is

specific power of the reactor

is very high for gas-cooling.

is

13.

which

is

is

to incur.

Boiling-water Reactors. The following dis Direct-cycle Boiling Reactors* of boiling reactors using solid-fuel elements limited to the direct-cycle of tho information in this article See Ref. taken from ANI. studies. 6,

is

4.16

cussion * M ost

is


A

is

A

1

is

It

a

is a

5

is

is

is

is

A

is,

It

high.

1

holes in the top 6-ft-diameter tank for operat therefore, possible to construct ing pressures of 1,500 psi and not encounter too much difficulty with the mechanical design of the head. unloaded more frequently than an D20 reactor HjO reactor because the specific power greater. Because of the high D20-to-uranium ratio of about 20:1, the total uranium used relatively small. The flux higher, around X 1013 for 1,000 Mw, and there To great extent, problem in overriding xenon. the popularity of this reactor will depend on the avail ability and the assigned inventory charges chosen for expected that the fuel elements in this the DjO. reactor will have a burnup of per cent of the total core atoms and then will not be reused. variation of the pressurized D20 reactor the DsOmoderated gas-cooled reactor. Such a reactor was pro posed by the General Nuclear Engineering Corporation to the East Central Nuclear Group and the Florida West Coast Nuclear Group, and in turn proposed to the AEC as a joint project of the electric company groups. brief description of the reactor quoted from statement made by W. H. Zinn to the Subcommittee on VA Legislation of the Joint Committee on Atomic Energy on March 14, 1958: the core.

head is

12-40

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

The problem of radioactive carry-over in direct-cycle boiling reactors does plants. not appear to be very severe. Some of the problems introduced by radioactive carry over, such as maintenance difficulties on the turbine and condenser and the design of seals to prevent radioactive leakage, are all minor as compared with the cost of an evaporative condenser. A simplified drawing of a boiling-water reactor is shown in Fig. 25. An advantage of the boiling reactor is that it is basically simple. It operates at moderate pressures as compared with the pressurized water reactor, and consequently the primary vessel can be made using relatively thin-walled vessel and pipe material. SHIELDING

PLUG

1- STEAM

OUTLET

STEAM COLLECTION RING

CONTROL

ROO

SHROUD

FUEL ELEMENT GRAPPLING FIXTURE

FEEOWATER DISTRIBUTION RING

-FEEOWATER INLET FUEL ELEMENT

CORE SUPPORT

PLATE

Flo.

25.

=*

Boiling-water reactor.

1.1

TO ION EXCHANGE AND FILTER SYSTEM

CONTROL DRIVE MECHANISM

(Courtesy of Argonne National Laboratory.)

The fuel temperatures

1

is

2

is

is

it

a

is

»

is

A

are only slightly higher than the steam temperature as com pared with an additional 50 to 100°F required in the pressurized water reactor. boiling reactor inherently safe because the steam void volume increases on transient power increase. Most of the disadvantages of a boiling reactor can be avoided by proper design. Instability in boiling avoidable as demonstrated by the operation of the Borax reactor.' The exit voids, i.e., the vapor volume of the leaving steam-water mixture from the reactor core, limit the power density of the boiling reactor. The reactor cannot be operated with the high exit voids attained in It coal-fired boiler. desired to operate with an exit void volume of not greater than GO per cent, correspond ing approximately to a water-to-steam mass ratio of 20: at 600 psi. At this time suggested that 45 per cent be used in H20 plants and 60 per cent in DsO plants. Operating with this ratio permits the generation of additional voids in order to pre vent nuclear excursions. Another limit usually imposed that the \k equal to the steam voids not greater than Load control to the turbine cannot be per cent.

is


THERMAL SHIELD

Sec. 12-1]


12-41

handled by adjusting the turbine throttle, because a reduction in the steam flow will an increase in pressure on (ne reactor which, in turn, will collapse the steam bubbles and increase the reactivity. Load change must be accomplished by steam The bypass steam is used to heat the feed water, so bypass control or control rods. there is little loss in efficiency. The low power density contributes to high inventory fuel charges, but it also permits the use of thicker fuel plates to minimize resonance absorption and, hence, cuts down enrich The low operating pressure permits ment. the use of larger reactor tank sizes, and as a result a reactor output of about 600 Mw can be achieved with U02 as a fuel. cause


Typical power densities obtainable with natural circulation are shown in Fig. 26. I )esign information is given in a report by Lottes and Flinn.8 The power density can be increased by placing risers or chimneys on the core or by increasing the circulation These by means of mechanical pumping. pumps are designed to handle large flows with a small pressure drop. Mechanical pumping can easily double the power den sity, using less than 1 per cent of the total For instance, with a plant power output. 40:1 recirculation ratio, a 10-ft developed head desired from the pump, and 50 per cent

pump and motor efficiency, the pump power required is (40 X 110)/ (0.5 X 778) = 1 One pound of steam gives Btu/lb steam. about 200 Btu of electrical energy. The The flow rates are required pump output for this case is 0.5 per cent of plant output. only about as large as required for the primary system of a pressurized water reactor. Boiling reactors can be designed against almost any conceivable accident, even The against a possible core melt-down, provided the reactor vessel does not rupture. accidental removal of all control rods can be carried out safely provided the reactor has a sufficiently low steam void fraction. In this case the void fraction in the core will increase with increased power and the reactor will stabilize itself, provided the relief valves prevent an excessive pressure buildup. In case of a lattice reactor like DjO, the lattice must be arranged to prevent burnout. In case of a power transient the free surface of the boiling water in the core provides space for expansion. This space also acts as an entrainment separator to minimize radioactive carry-over. The evaporation process in the boiling reactor is a very effective means of decon tamination. If the liquid entrainment in the vapor is limited, a decontamination However, factor of 10-6 can be achieved in the process of concentrating liquid wastes. for the operating conditions in a boiling reactor it is estimated that a decontamination factor of 10-4 can be expected. This means that if the water had an impurity of 1 ppm, then the steam would have an impurity of 10-4 ppm. There are essentially two major sources of impurities that find their way into the water system of the reactor. One is from the corrosion of the system itself. The system is made of conventional materials except for the reactor tank, which is lined with an 18-8 stain less steel. The second major source of contamination results from the possible failure of the fuel elements. In this case the impurities are generated in the reactor tank. Water contamination is controlled to about 1 ppm by an ion-exchange system. The flow to the ion-exchange system is from the reactor liquid space and is equal to about The amount of impurities carried over to the 1 per cent of the total steam flow. external system is relatively small because of the low decontamination factor of about 10""4. Estimated carry-over performance for a 1,000-Mw plant is given in Table 17. Gaseous fission products, which are given off on a Blug rupture, find their way into the external system along with the steam. They are removed from the

12-42

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

system through the condenser air vent and can be trapped on a refrigerated graphite absorber. The net decomposition rate of the water into hydrogen and oxygen gas is very much greater than in a pressurized water reactor. In the latter case the gaseous products In a boiling reactor the decomposition formed strongly promote a back reaction. product hydrogen and oxygen gases immediately leave the reactor with the steam, The net decomposition that takes place so that very little back reaction takes place. is quite high* and can give 1 ppm 02 in the steam if steps such as the introduction of H2 or hydrazine are not taken to prevent it. This oxygen in the steam may increase corrosion in the turbine. The design features of typical HsO and DjO boiling reactors are listed in Table 18. Table 17.

Estimated Carry-over Values for a 1,000-Mw Heat Boiling Water Reactor Plant* Corrosion Kate of External System Pounds per month Material Iron 53 Copper 43 Radioactivity at the Turbino with Casing Removed Decontamination

Miliiroentoens at I in. 600 100 10-' 10 from a l-kg Break of a Natural Uranium Slug Burned to 1 Per Cent at the Turbine with the Casing Removed factor I0"> I0"<


Activity

Milliroentgcns at 1 in. Decontamination factor

io-> I0"« I0~» * ANL-5208. Quarterly Report, p. 147.

Initial

17.000 1.700 170

After 1year

1.000 100 .0

(Classified.)

Because of the strong moderating properties of HjO, a small HsO Boiling Reactor. moderator-to-fuel ratio is required; that is, the ratio of water space to uranium usually lies in the range from 2 up to about 4. These reactors tend to be autocatalytic if This may also limit the they are large and have a high water-to-uranium ratio. maximum size of the reactor. Because the power density is somewhat limited with boiling, this type of reactor has a low specific power. The specific power is of the order of 5 Mw per ton of uranium as compared with about 25 for a pressurized water reactor. A typical design of an HjO boiling reactor is given in Table 18. The reactor may be about 7 ft high with a 9-ft diameter, this size being a compromise of several con siderations that affect an H20 design. Such a reactor is capable of producing a heat With output of about 250 Mw using natural circulation and with 0.5-in.-thick plates. forced circulation a reactor of this size can produce 400 Mw if the plate thickness is reduced to 0.25 in. A further decrease in the plate thickness will correspondingly increase the heat output. Provision must be made for adequate separation of the steam and water at the top of the reactor. The recirculation water velocity must be limited at the top of the core so that its radial component is only 1 fps; otherwise, it will interfere with the steam flow at the top of the outer elements. The head of water above the core is usually maintained at least equal to half the diameter of the core; if not, special separator baffles are required. The reflector, i.e., water surrounding the core, also acts as a down-comer for the recirculating water. Since little head is available for



Design Features of Typical

HjO

12-43

and D«0 Boiling Reactors D,0

HiO Natural

Thickness of uranium fuel-clement Clad thickness, in

corn, in

600 16 12 13.5 55

1.000 16 12 13.5 92

25 20 1.1 50

25 32 1.1 50

0 12

0 20

5.0

8.0

2: 1 45 40: 1 12.6 0.7 X 10'

2: 1 45 40: 1 20 1.6 X 10"

0.46 0.02

0.22

0.015


None* 1.75 1.45

i.3

Perturbations due to control rods

Prompt neutron • No danger

lifetime

Forced

400 12 7 9 64

0. 50 1.0

3.3

Heat flux, average Btu/(hr) (ft') Heat flux, maximum Btu/(hr)(ft')

Natural

240 12 7 9 40

Power density, kw/liter of coolant Core volume devoted to fuel assembly side plates, clearances, inactive ends of fuel elements, % . . .

Exit steam voids ratio Water/steam

Forced

160.000 530.000 100 310 1.6

0.4 X I0<

of burnout; assume that some channels

Nat. U

Nat. U

40 120 15 14:1 60 20: 1 24 2.4 X I0« 0. 22 0.015

40 120 25 14: 1 60 20: 1 40 4.0 X 10* 0. 12 0.015 0. 15 0.375

0.25 0.50

0.25 0.62

1.75 1.45 1.3

1.78 1.27 1.0

3.3 160.000 530.000 100 310 1.6 0.4 X 10<

2.3

1.78 1.27 1.0

2.3

150.000 400.000

150.000 400.000

1.0 6.0 X IO-«

1.0 6.0 X 10 '

will run above 40 per cent exit steam voids.

circulation, the velocity in the down-comer is limited to about 4 fps and the reflector diameter is usually about 1.3 times the core diameter. With the essentially evenly spaced plates in an H20 reactor, unloading is somewhat

of

It is usually solved in the same manner a problem because of the lack of space. pressurized H20 design (Fig. 23), although a D20 design (Fig. 24) can also

as for a he used.

Corrosion of the uranium

is a problem, but not so severe as for a pressurized water As in the case of pressurized

reactor because of the lower operating temperature. water reactors, consideration is being given to the use I Jresden plant uses uranium oxide fuel elements.

of oxide fuel elements.

The

In a great many respects, the boiling D20 water reactor is Reactors. example of an inherently well-balanced system. The optimum design power for a reactor operating at 600 psig is about 150 Mw of electrical output, i.e., 600 Mw of heat output. The power density attainable with boiling is not very great, but for the large core size, dictated by the use of D20 and natural uranium, 600 Mw of heivt can be attained without difficulty. This power can also be obtained by using for a fuel. Therefore, nuclear requirements, heat transfer, and fuel-element ilesign are mutually compatible. The design of the pressure vessel for 600 psig is no problem compared with 1,500 >3ig for a pressurized D20 reactor. The unloading scheme contemplated is the same shown for the pressurized D20. The steam space used above the core is utilized provide for flooding of the elements during core unloading. By use of displacement links, the extra D20 required during unloading can be minimized. The power density D2() Boiling

a good

fJOi is o

12-44

NUCLEAR-POWER-PLANT

SELECTION

[Sec. 12

in terms of kilowatt per liter of total DjO is little greater for a boiling D20 reactor than for a pressurized DsO unit, since very little holdup of 1J20 exists in the external sys tem. The large volume spaces are all low-pressure, low-density steam. The equipment required for containment of D20 is little different from that required to contain the radioactive leakage from the H:0 boiling reactor. The turbine steam is prevented from leaking by using a dry air buffer. Drying equipment is required This equipment is shown in Sec. 12-3. To avoid unnecessary loss of D20, bleed-off from the valves and from the turbine casing is re turned to the system. The valve bleed-off is contained by a bellows, and the loss from the turbine by a sheet-metal container. L T ^n a ^J<~* r*8,0*01, the age and the diffusion H I I II I I I I I length are large enough so that serious localia?d n U H M hot spots do not occur. The reactivity change 2 from normal operation to full steaming in the tubes of a D20 reactor is not great enough to handle removal of all the control rods if a standard "lumping" is used. Some method must be used to make sure that steaming will cause enough loss of void to shut the reactor Fig. 27. Three-dimensional accom down. One method of doing this is to use a modation of lumping. combination of radial and axial lumping, a* shown in Fig. 27. In this design, about 60 per cent of the area is devoted to flow passage. Unloading can be accomplished by using one unloading station for every five subassemblies. The control rods can be arranged between subassemblies. 4.17 Aqueous Homogeneous Reactors. In the aqueous homogeneous reactor the fuel is intimately mixed with the water, which acts as both the coolant and moderator. The fuel is usually a sulfate or nitrate of uranium or a suspension or slurry of an insolu ble oxide. The reactor is essentially a large pot that is an enlargement in the circu lating system. There are no fuel elements, and consequently no superfluous struc tural materials are required. Since the heat is generated directly in the coolant, nc At exists between the fuel and the coolant. The circulating fluid is pumped through the reactor to a heat exchanger where the steam is generated and back to the reactor. Since the fuel is intimately mixed with the moderator, lumping cannot be employed to reduce resonance escape and the fuel must be enriched, even if D20 is used as s coolant moderator. The fuel is reprocessed continually by bleeding off a small amount of the circulating system and reprocessing this stream. The aqueous homogeneous reactor lends itself to one of the simplest systems of removing fuel for reprocessing. Procedures for remote maintenance of the radioactive primary system must be carefully considered during plant layout. Either homogeneous reactors can be made with a single region in which the breeding material is mixed with the fuel material, or they can be made with two regions, the The tw<>core primarily containing fuel and the blanket primarily fertile material. region reactor seems to be preferred if the fuel is in solution, and the single-region reactor if the fuel is in suspension. Both systems will be very briefly described. The main reason the two-region reactor is Two-region Homogeneous Reactors. preferred for a solution-type homogeneous reactor is that corrosion is much less with the reduced uranium salt concentration that is possible, since highly enriched uranium

II

i

IiI


J_

is used.

The large pot through which the circulating fluid flows is surrounded by an out*i that contains a reflector consisting of a thorium slurry or solution in DjO. This reflector also acts as a blanket to capture the leakage neutrons for breeding. The outer concentric vessel is the pressure vessel, thereby allowing the thickness of Because of the the inner vessel to be minimized and economizing on neutron loss. vessel


Sec. 12-1]


12-45

absorption of neutrons in the blanket, the main or outer pressure vessel is subjected to less irradiation damage and less neutron heating than the inner vessel. It appears that the maximum temperature at which the uranyl salt solutions can be operated is about 550°F because of the corrosion and chemical-stability problems The corrosion increases with the concen associated with the uranyl salt solutions. tration of the uranyl salt solution, and for this reason a DsO-moderated reactor is preferred to an H20-moderated reactor, since the concentration of uranium for a given enrichment is much lower. The lower concentration markedly reduces the A fully corrosion problem and minimizes extraneous neutron absorption in H20. enriched fuel is preferred to a low enrichment for the same reason. There is no par ticular loss in neutron economy because the neutrons are absorbed by leakage into the blanket. The core coolant solution must be reprocessed to remove fission prod ucts and can be refueled as part of the same operation. The ThM2-U133 cycle is preferred to the U238-U23S cycle because of its higher con The Th232-U233 cycle using D20 can be designed for a conversion ratio version ratio. greater than unity. Once the concept of using enriched fuel in the core is accepted, it is preferable to go to the thorium cycle. For a fluid-fuel reactor such as the aqueous homogeneous, the problem of handling y and a radioactive U"3 is not serious. The core diameter of a fully enriched D20 reactor is usually limited to about 5 ft Such a reactor would have in order to provide the necessary leakage to the blanket. a blanket diameter of about 8 ft and would produce an output of around 1,000 Mw of heat. A total pressure of about 1,500 psi is maintained on the system by heating D20 in a pressurizing chamber attached to the main circulating stream and the blanket near This pressurizing chamber acts as a surge tank the top of the sphere of the reactor. similar to the surge tank in a pressurized water reactor. The D2() (or H20) modera tor-coolant is decomposed by the fission recoil to produce a stoichiometric yield of deuterium (or hydrogen) and oxygen. To minimize decomposition, a catalyst is The quantity and material must be chosen to minimize the added to the water. Even with a catalyst, gases are generated at the extraneous absorption of neutrons. rate of thousands of cubic feet per minute (STP). These D2 (or H2) and 02 gases arc recombined and returned to the system. In addition, 20 liters per day of intensely These gases are continuously removed from the radioactive fission gas is produced. fuel solution. Since some of the delayed neutrons have mean lives much longer than the coolant residence time in the reactor, the effective delayed-neutron fraction is out to about 0.2 per cent (see Table 9). These delayed neutrons, as well as the fission products, contribute to an extremely active external circuit. Considerable shielding is required, The system must be and access for inspection and repair is extremely difficult. designed to prevent leakage. The integrity of the system required to contain radio activity makes the problem of containment of D20 trivial by comparison. The loss of delayed neutrons is not very serious from a control standpoint, since the temperature coefficient of reactivity is strongly negative owing to the expansion of the fuel moderator coolant system. The reactor is operated without control rods. Some measure of nuclear control is exerted by the level of the reflector. With a 550°F outlet temperature from the reactor, a 70°F rise and a 40°F thermal difference in the boiler, the boiler can deliver 500-psi steam at about 500°F. This docs not give a high cycle efficiency; however, with the low fuel cost this is not important. In this system, it is possible to achieve a low fuel-cycle cost and, therefore, justify an inefficient turbine, thereby reducing the capital investment charges on the reactor Because of the low fuel-cycle and low capital costs possible for this generating plant. reactor, it may readily become one of the outstanding thermal reactors. The problem of corrosion is one of the unresolved problems of the aqueous homo The fuel moderator solution is strongly oxidizing and also very geneous reactor. strongly acid. Aside from the noble metals, only a few metals seem suitable for the Titanium appears to be satisfactory. In addition to the homogeneous reactor. problem of corrosion of the core liner, there seems to be a major problem with "fret


12-46

NUCLEAR-POWER-PLANT

SELECTION

[Sec. 12

ting" and crevice corrosion of the tubes in the boiler if the fuel solution is on the out side of the tubes. Corrosion is aggravated by radiolytic decomposition. Single-region Homogeneous Reactors. In a slurry-type homogeneous reactor the water is maintained free of impurities except for the suspended fuel-bearing particles. Corrosion, therefore, may not be a serious problem. A slurry can be made with 5 per cent by volume of suspension. It seems desirable to investigate use of fuel slurries of slight enrichment. One concern about slurry reactors is the problem of remov ing the fission products from the system. The Dutch10 believe that fission products can be successfully removed from the suspended particles if these particles are opti mized in size at about 10 u and if the gas that is reabsorbed on the particles is removed. They also believe that a charge must be maintained on the particles to keep them from agglomerating. They have demonstrated this in laboratory work. The Dutch expect to be able to remove 90 per cent of the fission products continu ously from the reactor by purification of the water. This, therefore, reduces the health hazard in case of a nuclear incident. The other problem in slurry homogeneous reactors is the prevention of agglomera tion under irradiation and the resulting settling of particles. Boiling Aqueous Homogeneous Reat;tors. One variation of the homogeneous reac tor that holds promise is the boiling homogeneous reactor. This reactor is essen tially a container of a boiling uranium solution. A reactor of this type is attractive because effective decontamination takes place in the boiling process so that corrosion is not a serious problem in the evaporative condenser. In the solid-fuel-type boiling reactor, power densities up to about 50 kw per liter of core volume can be obtained. If a boiling homogeneous reactor were operated at this power density, the boiling would be very ebullient and unstable. Therefore, in the boiling homogeneous reactors, power densities may be much lower than in a heterogeneous boiling reactor. With the introduction of baffles, it may prove possi ble to operate a homogeneous reactor with forced-circulation boiling and hence at higher power densities. Alternately, internal tubes placed in the reactor could be used to increase the natural circulation by a chimney effect and hence increase the power density of the reactor. Vortex tubes, tubes with an internal spiral ribbon on the inside, might be used for this purpose. In this type of reactor the neutrons stay in the reactor, so that the delayed-neutron fraction is comparable to that in a solid-fuel type. However, gaseous fission products can be removed in order to eliminate the xenon-control problem. An indirect cycle using an evaporative condenser could be used with this reactor to The evaporative condenser in this case would prevent radioactive carry-over. be subjected to less corrosive conditions than the steam generator in the conventional homogeneous design because of the decontamination factor attainable in the evapora tion

process. 4.18 Liquid-metal

The major advantages of inti Homogeneous Reactors.* mately mixing the fuel with the coolant are: 1. High temperature and high cycle efficiency are possible without requiring highpressure equipment. 2. There is no fuel-element fabrication, and there is no problem of transferring the heat from the uranium to the coolant. 3. The fuel can easily be transferred from the reactor to the processing facility by means of pipes and pumps. Large savings are possible in eliminating fuel-elementhandling equipment and its associated shielding. 4. There is a possibility that fission products may be separated from the solution without removing the uranium. Lead and bismuth appear to be the only liquid metals that have a significant ura nium solubility at normal operating temperatures, say 250 to 600°C (500 to 1100°F). The thermal neutron absorption cross section of lead is too high to consider it as a carrier for uranium at the low concentrations attainable at these temperatures. The solubility of uranium in bismuth is shown in Fig. 28. If one demands that the fis sionable material be in true solution at reasonable operating temperatures, only * Contributed by R. B. Nicholson, APDA, a member of study team on LMFR.

Sec. 12-1]

SELECTION

12-47

OF REACTORS

There are, of course, several bismuth can meet the neutron economy requirements. It is transmuted to polonium, which is a strong objectionable properties of bismuth. Bismuth is rather a emitter and is one of the most difficult materials to handle safely. The melting point, 273°C, expensive and is not being produced in large quantities. is not so low as is desired. The characteristic of expansion upon freezing is not a corrosive metal, and as yet there is a Bismuth is a particularly desirable one. minimum of data indicating that a satis |i5,000 1.5 ( factory container material is available. 1 1 1 | 1 1 Nevertheless, there is no other liquid J metal in which uranium is sufficiently soluble. D u An alternate that is being studied is oUl the use of slurries rather than true solu Q- 1.0 0,000 f The compound UBi could pos tions. tX sibly be held in suspension in bismuth o with sufficient stability for reactor use. Ul Some work has been carried out on ura 2 nium oxide in sodium or sodium-potas twork has not This gone sium slurries. 5.0001=)0.5 ID far enough to be evaluated at the present ID 3 time. It may well be that, if the slurry 3 . . o in concept is accepted, some other liquid o ■ metal may enter the scene. ■ ■ The liquid-metal-fuel reactor discussed 0 here is one in which the fuel is a solution 250 300 350 400 450 500 550 600 TEMPERATURE. C of enriched uranium in molten bismuth Fig. 28. Solubility of uranium in molten and the blanket is a slurry composed of It is a bismuth. thorium bismuthidc in bismuth. thermal reactor moderated with graphite. It is homogeneous in neutron behavior, although it is structurally heterogeneous. The conceptual design made by the Babcock & Wilcox Company under contract with Brookhaven National Laboratory will be used as a basis. This study was made in 1955 and is one of the many liquid-metal-fuel reactors being studied by Brookhaven Figures 29 and 30 were taken from Report B & W-2. National Laboratory. The core is a Figure 29 shows a vertical section of the reactor vessel and internals. graphite structure that is drilled to form passages for the flow of liquid bismuthThis core structure is suspended in place by a thin chromiumuranium alloy. The blanket molybdenum steel baffle tube that separates the core and blanket fluids. is also a graphite structure through which a slurry of thorium bismuthide in liquid bismuth circulates for breeding and cooling. The flow is up through both the core and blanket, and a fountain effect is used in the core degasser to effect the removal of the gaseous fission products. The heat is removed from the circulating reactor fluids in intermediate heat ex A lead-bismuth alloy was chosen for the intermediate coolant, although changers. other metals such as sodium or sodium-potassium could have been used. There are four parallel intermediate loops of this type for removal of core heat and two smaller loops for removal of blanket heat. The purpose of the intermediate loop, as in several other reactors under consideration, is to prevent activation and contamination of the This intermediate coolant is pumped through natural circulation steam system. boilers and superheaters, producing steam at about 975°F and 1,850 psi at the turbine


/

/

throttle. The design philosophy required that major com Figure 30 shows the plant layout. ponents be isolated from one another with concrete shielding so that maintenance of Whether any one of the loops may be possible while the others remain in operation. or not this type of maintenance is practical has not been established. The neutron economy in a reactor of this type is fairly good. If it is fueled with TJ"', then conversion ratios of about 0.9 are possible; with improved designs, con The use of Us" as a fuel version ratios greater than unity can probably be attained.


12-48 NUCLEAR-POWER-PLANT SELECTION [SEC. 12

Sec. 12-1]

SELECTION OF REACTOKS

12-49

will result in about, a 20 per cent lower conversion ratio. The total fuel required in the circulating system is about 300 to 400 kg of U233 or U235. Other LMFH designs arc possible that would require less fissionable material. The large negative temperature coefficient, long neutron lifetime, and the rapid expansion characteristics of the fuel solution are features that tend to give inherent

.VI

CIRCULATING

WATER

||lNLET )

if// I

V"">

f>ISCHARGE

500 FT

REACTOR AND


TURBO-GENERATOR ROOM CONTROL STEAM-GENERATOR

SUPERHEATER REHEATER —

HEAT CORE

EXCHANGERS AND

' HEAT V> CORE

EXCHANGERS

BLANKET

O

CM

CONTROL

ROOM CORE

Fid.

\-

60-0-

30. Plant layout for a liquid-metal-fuel reactor.

safety and ease of control. The circulation of the fuel causes a major fraction of the delayed neutrons to be released outside the reactive region and sacrifices some of this inherent safety. The high concentration of fission products in solution would make a rupture in the system quite serious if it were to occur. Chemical reprocessing would be accomplished by continuously drawing off streams

12-50

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

of the liquid-fuel solution and blanket slurry and pumping them to the on-site chemi Figure 31 shows a simplified flow sheet of one of the pro cal reprocessing plant. Decontamination would be accomplished by posed chemical reprocessing schemes. extracting many of the salt-soluble fission products into a molten salt such as BiCli In the next step, under reaction conditions that do not permit uranium extraction. the uranium would be removed into a salt stream, leaving the salt-insoluble metals These fission products would then be removed from the bismuth in the bismuth. by oxide slagging and zinc extraction techniques.

FP-FISSION

PRODUCTS FPS -SALT-SOLUBLE FISSION PRODUCTS FPN-SALT-INSOLUBLE FISSION PRODUCTS FISSION PRODUCTS VFP-VOLATILE SPARGING

DEGASSER

GAS

CARBON ADSORPTION

VFP

BED

VFP -» TO WASTE DISPOSAL

FUEL SOLUTION

LMFR U-BI FP

U-BI FPN

FPS FUSED SALT| IN SALT EXTRACTION COLUMN


U-BI

SALT

U-BI FPN

J

WASTE DISPOSAL 'AND SALT RECOVERY TO

Bl

SALT

Si BISMUTH

U-BI EXTRACTION COLUMN

U

IN SALT

n

FUSED SALT EXTRACTION COLUMN

SALT

Bl

FPN OXIDANT

OXIDE SLAGGING

REACTIVE

FPN

OXIDE SLAG Bl NOBLE

ZINC

FPN ZINC

EXTRACTION

ZINC NOBLE FPN

.TO WASTE DISPOSAL

TO WASTE DISPOSAL AND ZINC

RECOVERY

Bl

Fig.

31.

LMFR

processing flow diagram.

An attractive feature of LMFR reprocessing is the possibility of integrating a small chemical plant with the reactor plant. Since fuel burnup is not limited by radiation damage, the amount of fuel withdrawn for chemical reprocessing can be very small for even a large reactor. This results in a plant that contains small equipment, requires little space, and is easily shielded. It is hoped that these features can mskf inexpensive chemical reprocessing possible. There are many who have acclaimed the LMFR as being a reactor that would pro The proof of this claim must await the compila duce extremely economical power. tion of a considerable amount of development effort and perhaps even the construc tion and operation of a prototype. There are three major problems to be solved in the LMFR program: 1. It must be demonstrated that a satisfactory container material exists for the fuel solution.

Sec. 12-1]


12-51

2. It must be demonstrated that the proposed salt-extraction chemical reprocessing techniques will operate as predicted theoretically and do so economically. 3. The stability of this very dilute uranium solution must be demonstrated under reactor conditions. There are also a number of less formidable obstacles and difficulties such as encoun A few of these are: tered in all reactor development work. 1.

Availability and cost of bismuth

of suitable pumps and valves for operation in bismuth, leadbismuth, and thorium-bismuthide slurry and fused salt solutions 3. Demonstration of reactor safety 4. Development of an economical and effective method of recovering fissionable material from the blanket slurry and reconstituting the slurry 5. Demonstration of the removal of the fission-product gases 2. Development

4.2

Small Power Reactors


A description of small power reactors Energy.* Table 19. Heterogeneous General: Name: STR or SWl reactor. Purpose: Mobile power. Neutron energy: Thermal. Status: In operation for over 1 year. Materials: Fuel; U-235. Fuel elements: Sandwich plates, U, Zr. Fuel element jacket: Zr. Moderator: HsO, high pressure. Reflector: HiO, high pressure. Primary coolant: HsO, high pressure.

is quoted from the Panel Report on Atomic

Pressurized Water Reactor

Fuel:

t ■'>. Maximum temperature in fuel clement, °F: <> Sheath temperature, °F: 551. Average cycle time: 000 hours. Limitations: Reactor temperature limited by pressure required to prevent boiling; high-pressure vessels, piping, and heat exchangers required; steam generated at relatively low pressure; high coolant pumping cost; expensive fuel clement. Advantages: Reactor has relatively few hazards, has been proven in service, expensive moderator not uted.

Table 20. Heterogeneous Pressurized Water Reactor General: Name: MTR type power producer. Purpose: Power production. Neutron energy: Thermal. Status: Preliminary design and feasibility study. Materials:

Fuel:

Fuel: UOj. Fuel elements: Plates, flat, rectangular. Fuel element jacket: Stainless steel 347 type, UOj, stainless steel. Moderator: HsO. Reflector: HiO. Primary coolant: HiO, high pressure.

Maximum temperature in fuel element, °F: 575. Sheath temperature, °F: 525. Average cycle time: 9 months. Limitations: Reactor temperature limited by pressure required to prevent boiling; high pressure vessels, piping, and heat exchangers required; steam generated at relatively low pressure; high coolant pump ing cost; relatively inexpensive fuel element not yet proven. Advantages: Expensive moderator not required, relatively few hazards. * Report of the Panel on the Impact of the Peaceful Uses of Atomic Energy to the 2, pp. 228-235, 1956.

Atomic Energy, vol.

Joint Committee

on

NUCLEAR-POWEK-PLANT

12-52

Table 21.

SELECTION

Boiling-water-cooled

[Sec.

12

Reactor

General:

Name: Boiling reactor for heat and power production. Purpose: Power and heat for remote locations. Neutron energy: Thermal. Status: Preliminary design and feasibility study. Materials: Fuel: Zr, U slightly enriched. Fuel elements: Plates, flat, rectangular. jacket: Zr. Fuel-element Moderator: II»0. Reflector: HtO. Primary coolant: HtO, boiling. Fuel: Sheath temperature, °F: Above steam temperature of 486°F. Average

cycle time:

12 months.

Limitations: Turbine and auxiliaries are radioactive, lower heat flux in core than with pressurized water reactor, high rate of water circulation required, possible effects of motion on the o juration of the reactor, expensive fuel element, not yet proven. Advantages: Heat exchangers and primary coolant circulating pumps not required, reactor is inherently availability of higher steam pressures than with pressurized water reactors. self-regulating,

Table 22.

Heterogeneous Supercritical Water Reactor


General :

Name: Supercritical water reactor (8C-WR). Purpose: Mobile power. Neutron energy: Intermediate. Status: Design and development. Materials: Fuel: UOi dispersed in stainless steel matrix. Fuel elements: Plates, 1 inch square box with parallel plates and sine wave fillers. Fuel-element jacket: Type 347 stainless steel 0.007 inch thick. Moderator: HiO. Reflector: IIiO. Primary coolant: H2O, high pressure (about 5.000 psi). Fuel: Maximum temperature in fuel element, °F: 1,300. Sheath temperature, °F: Not determined. Average

cycle time:

144 hours.

Limitations: Materials for high pressure and temperature service a problem, retaining mechanical seal*. etc., at high pressure and temj>erature. Reactor not proven. Advantages: High-pressure steam with resultant good cycle efficiency, small size of reactor, no phase change in reactor.

Table 23.

Homogeneous Aqueous Reactor

General:

Name: Homogeneous reactor experiment (HRE). Purpose: Experimental. Neutron energy: Thermal. Status: Operation. Materials: Fuel: UOtSOi. Fuel elements: HiO solution. Fuel-element jacket: Stainless steel tank. Moderator: HtO. Reflector: DiO. Primary coolant: Circulating fuel solution, secondary coolant boiling HsO. Fuel: Maximum temperature in fuel element, °F: 482 out, 407 into reactor. Average cycle time: Time in core, 7 seconds. Limitations: Corrosiveness of uranyl sulfate solutions, evolution of radioactive gas, dissociation of wmt*-* highly radioactive solution in heat exchangers. DiO, high-cost item. Advantages: High burnup a possibility, continuous fission product removal a possibility, mechanicAli/ simple,

reactor is stable,

refueling

Table 24. General:

Name: SIR or SGI. Purpose: Mobile power. Neutron energy: Intermediate. Status: Design and construction.

would

be simple.

Sodium-cooled

Intermediate

Reactor

Sec. 12-1]


12-53


Materials: Fuel: UO:. Fuel elements: Stainless steel tubes. Fuel-element jacket: Stainless steel tube. Moderator: Be. Reflector: Be. Primary coolant: Na. Fuel: Maximum temperature in fuel element, °F: 1700 ± 300. °F: 900 maximum. Sheath temperature, Average cycle time: 900 hours. Limitations: Coolant becomes radioactive, heat exchnneer requires heavy shield. Be moderator is expen sive, Na reacts explosively with water, fuel clement removal is a problem. Advantages: High reactor temperature and temperature difference possible, low pressure in primary loop, somewhat higher thermal efficiency.

Table 26. Liquid-metal-fuel Reactor General: Name: Brookhaven liquid metal fuel reactor (LMFR). Purpose: Power and fissionable material production. Neutron energy: Thermal. Status; Research and development (BNL). Materials: Fuel: U-233, alternate fuel U-235. Fuel elements: U-233 in bismuth. Moderator: Be or graphite. Primary coolant: Bi-U fuel, externally cooled by Na in heat exchanger. Fuel: Maximum temperature in fuel element, °F: 950. Average cycle time: Continuous fuel processing. Limitations: Requires highly enriched fuel, preheating for startup, formation of radioactive polonium, low solubility of uranium in bismuth, unproven design. Advantages: High temperatures at low pressure, continuous fuel processing and high burnup possible, possibility of breeding, no fuel element fabrication, small heat exchange surfaces. Table 26. Gas-cooled Reactor General : Name: ANP Purpose: Test stand operation. Neutron energy: Thermal. Status: Detail design. Materials: Fuel: UOi. Fuel elements: Plates, sandwich. Fuel-element jacket: Stainless steel. Moderator: HiO. Primary coolant: Air. Fuel: Maximum temperature in fuel element, °F: 1800. Limitations: Low heat transfer coefficients, induced radioactivity in cooling air requiring operation of " Unproven hot" turbines, high pumping costs for circulating air. Successful operation not proven. design.

Advantages: High operating temperatures,

relatively low pressures, light weight.

Table 27. Organic -liquid-moderated-and-cooled Reactor General: Name: Shipboard nuclear powerplant. Purpose: Shipboard power. Neutron energy: Thermal. Status: Preliminary design. Materials: Fuel enrichment: U. Fuel elements: Rectangular plates. Fuel-element jacket: Aluminum. Moderator: Terphenyl. Reflector: Terphenyl. Primary coolant: Terphenyl. Fuel: Sheath temperature. °F: 800. Average cycle time: 90 days. Limitations: Heat transfer coefficient low, polymer formed in coolant, requires terphenyl purification system. Possible high terphenyl losses, unproven design. Advantages: Negligible corrosion, low pressure at high temperature.

12-54

NUCLEAR-POWER-PLANT 4.3

SELECTION

[Sec.

12

Research Reactors

There are three types of research reactors. The first is used for basic science studies and activation studies. This type includes the training reactors that universi A second type is for materials testing. The primary purpose of ties are installing. this type is to study the effect of integrated neutron flux on the behavior of structural materials and fuel materials. These tests are conducted on small samples, and a record of flux and temperature is required. In most cases temperature must be con In this type trolled, but this can be done by providing the desired conditions locally. of reactor it is desirable to have a high thermal flux and to be able to load and unload easily. If a high fast flux is desired, it can be produced by the use of converter The third type is for engineering tests. In this type it is desired to test a prototype fuel element, the coolant, or a prototype mechanism. Test holes or channels are desired that are several inches in diameter. The test channel should receive a good sup ply of neutrons at constant flux over several feet of its length. It is highly desirable that the test channel be operated at its own temperature and pressure and that :i failure in this test channel will not seriously affect the reactor. Ease of instrumenta tion is important. It is acceptable to take a lower flux in this type of reactor than in a materials-test reactor. Research reactors are made either with highly enriched uranium fuel or with natural uranium fuel. The natural uranium type has a large critical mass of U,ls, and hence has a much lower neutron flux for a given power rating. Because it is also To provide physically large, it does provide large working areas around the reactor. a moderately high flux, the NRX and BNL reactors are run at high power. Fully enriched research reactors are attractive because they give a higher neutron The ANL CP-5 reactor uses enriched fuel and D:0 flux for a given power rating. moderator. As a result, it has a small critical mass which provides a high flux and is also moderately large so that ample working area is provided around the reactor. All research reactors operated at low power are relatively safe because they are designed to handle very little excess reactivity and have a strong negative temperature coefficient. The water-boiler-type (SUPO) reactor is extremely safe from any nuclear event because the fission products are continuously removed. Even if a nuclear incident were to take place, there would be very little radioactivity to be released The water boiler is really a misnomer; it gets its name from the fact that during: operation there are a slow decomposition of the water and an evolution of hydrogen and oxygen which causes a bubbling similar to that of boiling. It has great The Oak Ridge swimming-pool type of reactor is a plate-type unit. inherent safety and does not require the continuous attention to chemical cleanup that the water-boiler type requires. It is less likely to get into minor contamination difficulties but does face the extremely remote possibility of an aluminum-water reaction. The swimming pool provides flexibility in that the reactor can be moved within the pool. It also provides visual observation of the operation of the reactor. These reactors can be built like the MTR in a closed tank or in an open tank or swimming pool. Fully enriched reactors operated at low power do not burn up enough uranium to However, » incur high costs in reprocessing and refabrication of fuel elements. reactor like the MTU running at 30,0(X) kw requires a new fuel loading every 2 weekto a month and, as a result, does incur large costs for fuel reprocessing and refabrication. The MTR reactor provides a high thermal flux and a somewhat higher fast therefore, useful for testing structural materials under It flux within the core. neutron bombardment. The types of research reactors and their characteristics are given in Table 28. The detailed design of these reactors given in the article on kinds of reactors. is

is,


slugs.

gj U H.O HiO

Al Al

U

2

10" 1.5 XI0"

30.000 175.000

100

I0'«

X

4.0 kg 12.0 kg

HiO

, Al

Over Over

16 28 24 in. high 32 in. diam, 36 in. high

18

Over

Over

Over

X

-ft diam

ft high

650

Cost, millions of dollars*

750 14.700

90 % 90 %

17 17.2

0.25

28 3.6 90%

~I2.0

0.5

44 0.8

90 %

3.0

25

46

1540

10

460

1.3

90 %

Natural

24 in. high

15

Sphere

X

+ ++

coat of building.

+

t

3.6 kg

35

10"

1

HiO

ft diam,

2

0.8 kg uo.so

2.000

X I0»

X 1

4

6.5 tons DsO

22 ft high

65

16.5

Specific power, kw/kg

4

Including

reactor

1 2

Superpower water boiler BSF Oak Ridge swimming-pool MTR Materials testing ETR

U Al

X

1.3 kg

17

X

reactor

Type

Uranium

17

Natural

230

U

research

C X 30.000

10.5

Natural

Critical mass, kg V"''

U

CP-5 Argonne 8UPO

5 10"

core

8.75 ft diam, Midi

5-f

Enrichment

U

Enriched

2

730 tons

7 1

metal

C

00 tons

30.000

X 10"

—

Reactors

Core size

.

tons DiO

3,800

5 10"

X

Power level, kw

of Research

8 1

reactor

metal

U

10 tons

620 tons

Flux

Kinds of Various Natural Uranium Type

Moderator

Comparison

t

graphite-moderated

metal

U

35 tons

Fuel

28.

ft

•pBNL BNL

X-IO natural ura O.R. grapl lite-moderated, nium reactor NRX reactor Chalk River DiO-moderated

Table


5

X

X

X

UU

*

12-56

NUCLEAR-POWER-PLANT 5 6.1

SELECTION

[Sec.

12

COOLANTS Desired Properties

The choice of coolant for a nuclear reactor is governed by a number of factors. of the factors to be considered and their desired characteristics are outlined below : 1. Heat Transfer and Transport Properties. The coolant should have a good heattransfer coefficient and a good heat capacity on a volume basis. 2. Nuclear Properties. For thermal reactors, a low absorption cross section is necessary for good neutron economy, and good moderating properties are desirable, 3. Vapor Pressure. The coolant should have a low vapor pressure at the operat ing temperature in order to minimize the piping and container thickness required. 4. Corrosion and Chemical Activity. There should be a minimum of corrosion or chemical reaction between the coolant and the material contacted by the coolant, i.e., fuel, structure, air, etc. 5. Itadioactivity. Coolants that become activated and form hard 7 emitters with long half-lives should be avoided. Activated coolants that are a and 0 emitters can easily be shielded but must be contained. 6. Decomposition. Coolants should be stable under irradiation and at the operating temperatures of the reactor. 7. Melting Point. The melting point of coolants that are solid at room tempera ture should be easily achievable by compartment heating. The choice of a coolant will usually depend on a compromise, since no known coolant embodies all the desired characteristics.

5.2

Types of Coolants

Coolants can be characterized into five broad classes: (1) water, (2) liquid metals, To date, only the first three (4) inorganic fluids, and (5) organic fluids. Each of the foregoing classes is dis types have been used extensively as coolants. A comparison of the various coolants, cussed below in reference to a typical coolant. including their physical properties, advantages and disadvantages, and major feasi bility problem, is given in Table 29. 6.21 Water Coolants. Water is a fairly good heat-transfer agent , is easy to handle, provides some lubrication, offers no serious corrosion problems that are not sur mountable by proper choice of materials, and is readily available and economical. For these reasons, it has found rather wide application as a reactor coolant. Although the (n,p) reaction with O16 forms N17, a very hard 7 emitter, its half-life of 7.5 sec is short enough so that there is no serious activation problem. There is also a minor activation of O18 with an 18-sec half-life of no particular consequence. The water must be continuously purified with mixed-bed ion exchangers or other means to keep down the activity of corrosion products. It is relatively easy to main tain a purity of 1 ppm in the water and minimize transport activity. The most serious drawback with water is its high vapor pressure; even the modest coolant tem perature of 550°F requires a primary system pressure of 1,500 psi. Heavy water, DjO, has essentially the same physical and chemical properties as Its nuclear properties are markedly different. The absorption cross section H20. is !"<;oo of that for water, and the slowing-down power {£, is of that for water. The slowing-down power of H2() greater than necessary or even desirable for some applications (see discussion on lumping, Art. 3.5, and hot-spot factors, Art. 3.6); con much more attractive from the nuclear standpoint. sequently, DjO At present the low operating temperature for both II.O and DsO systems limits the plant thermal efficiency to about 25 to 30 per cent. 5.22 Liquid-metal Liquid metals are attractive because they give Coolants. high heat-transfer rates and do not require pressurizing to operate at high tempera ture. The alkali metals are of the most interest. Sodium most attractive because is

}>&

(3) gases,

is

is


Some


Sec. 12-1]


12-57

its melting point is relatively low, it is not corrosive to most common structural mate rials up to 1200°F provided low oxygen concentration is maintained, and it has fair Sodium has a thermal absorption cross section of 0.5 barn. nuclear properties. Sodium, on the absorption of a neutron, forms radioactive Na24, a 2.7-Mev y emitter It may also form Na22 on an (n,2n) reaction with a 2.6-year with a 15-hr half-life. The Na22 activity has not proved to be a serious problem in reactors that half-life. Considerable external shielding is required to shield have been operated to date. against the Na24 activity and a waiting time of approximately 2 weeks is required for Sodium reacts violently with water, and hence care must be exer access and repair. The poor lubricating properties of cised in the design of the sodium-water boilers. sodium and its reaction with air further restrict the mechanical design of these reactors. The high boiling point and good heat-transfer properties of sodium seem to more than offset its undesirable properties and give it rather widespread interest. Lithium 7, because it has a higher heat capacity than sodium and is easy to shield, It has a thermal absorption cross section has attracted attention for power plants. Isotope separation must be used to obtain Li', but perhaps complete of 0.033 barn. separation is not essential, since the residual Li* may bo used as a burnable poison to The cost of such separation justifies its use only for small specialsome extent. purpose reactors. Lithium is considerably harder to contain than sodium or sodiumIt appears that its corrosion might be kept to tolerable limits by reduc potassium. ing both the O2 and N2 concentration. Gallium has attracted some attention as a coolant because as a metal it gives an exceptionally high heat-transfer coefficient and melts at 86°F. It has a thermal absorption cross section of 2.7 barns. At elevated temperatures, gallium is one of At temperatures up to the most corrosive liquid metals that has ever been tested. 800°C, tungsten is the only metal that is completely unaffected; next is beryllium, being substantially unaffected at temperatures up to 500°C. Stainless steel appears to be effective up to 200°C. The corrosivencss coupled with its high absorption cross section are major objections to gallium as a coolant. 6.23 Gas Coolants. Gases are inherently poor heat-transfer agents because of their low density as compared with liquids. This difficulty can be overcome to some degree by operating the gas system under pressure; however, this introduces other The number of gas atoms per unit volume is sufficiently small (when problems. compared with condensed materials) so that gases do not particularly affect the neutron economy of a reactor. A comparison of the relative ability of various gases to remove heat is given in Table 30. The data presented in this table take into account both heat transfer to the gas and heat absorption by the gas. The main advantage of gas coolants lies in their application to high-temperature reactors, since For gas-turbine application, some gases are very inert at elevated temperatures. other properties may readily overshadow heat-transfer considerations in determining the choice of gas coolant. Most of the salts are ruled out because of their 5.24 Other Inorganic Coolants. high melting point, high cross section, irradiation instability, or extreme corrosivencss. The halides should be stable under irradiation and should not be too difficult to con tain. They have, of course, a negligible vapor pressure at operating temperature. Sodium hydroxide has been considered as a coolant because it is a possible moderator due to its hydrogen content and also because of its temperature stability. It is probably stable under irradiation; however, it has a high cross section and high melting point and is somewhat corrosive at elevated temperatures. Yellow phosphorus has a thermal absorption cross section of 0.15 barn and melts at 1 10°F, but it has a transition to solid red phosphorus within the operating range of The solid phase can be avoided by special handling, but the acceptability interest. phosphorus for a reactor coolant system remains to be demonstrated. Diphenyl and other organic liquids arc attractive coolants 5.25 Organic Coolants. 3. nd have found considerable application in the heat -transfer field because they are
of

12-58

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

Table 29. Reactor coolant.

Reactor type Molecular weight , . Boiling point, °F. . Melting point, °F, . Maximum operat ing temperature, °F Operating presHeat-transfer coeff. (20 fps),

HiO

DiO

Thermal ■e

20 212 32

600 2.000 8000

Btu/(hr)(ft>)


TO

Specific heat 60 . Btu/(ft')(°F). Thermal absorp tion crosssection 0 II 85 X 10 • Z« at 600°F Slowing-down 1.5 0 18 power £2. Pipe-shielding thickness, Fe. . . . 13in. 13in. Corrosion Chemical activity .

Corrosion-resist ant structural material

300sc ic , Zr, Uf, Ti, StetliUs

Major advantages

Available, experi ence, moderator

He gas

Na

NaK 50% by weight

Pb-Bi 45-50% by weight

Thermal

Thermal or fast 23 1621 208

Thermal or fast

Thermal or fast

4

-452 -458 2000

1500

Comparison Hi

200 6 675

1508 52

3040 257

1500

1500

1200

0-100

100-200

-40

2.000 1000(at 150fps I500T, 2.000psig)

0-100 25.000

0-100 15.000

I2.000(?)

0.9

18.7

14.2

22

25.7

2X 10'

0.0064

0.013

0 0014

9 6

0.0003

0.005

0 003

0.002

0.003

None None

20 in. Corrosion active with air + HiO 300series,Ni alloys, W, Mo, low-C Bteel Nonrcactive with U, low vapor pres sure, high cycleefficiency

20 in. Corrosion active with air + HiO Same as Na

Chemical activity, radioactive

Chemical activity. radioactive

Unloading (?)

Unloading (?)

Any

Inert, high cycle efficiency

Same as Na

None Corrosion

12

I

Ye. (?)

Cr-Mo steel

Fe. 300sent*

High cycle efficiency

Low melting point, inert to! air and EM) 1 1 1

Major disadvan tages

Feasibility problems

Poor cycle effi ciency, high pressin

Minor

Pressure, temperature, pump power

Corrosion, melting point, pump power, polonium 1'orrosion, polonium activity, unloading

Expeme, pumping power, toik

1

1

The possibility may exist of temperature without dissociation or decomposition. decomposition products being deposited out on the fuel-element plates and inter fering with the heat transfer. Diphenyl is shown in Table 29 because it is the cheapest of the relatively stable fluids of this group. Decomposition of the coolant is primarily by neutron interac tion. About 2 per cent of the neutron energy is dissipated in the coolant. If it is assumed that one-tenth of the neutron energy dissipated in the coolant breaks bonds, then the make-up is approximately J-oo lb of diphenyl per 12,000 Btu of total heat generated. This is the equivalent of assuming that such a plant needs a diphenyl "fuel" make-up of one-fiftieth of the fuel oil flow of a conventional oil-fired plant. The make-up requirements for a ship could be quite small. The make-up could be used as shielding and displaced by water. The decomposition products can be easily removed by distillation and could be burned. Experiments have proved that these products in many cases are just as good coolants as the starting mate


Sec. 12-1]

12-59

of Reactor Coolants NaOH

Nal + NaCl

UO2SO4in D2O solution

U02 in DiO slurry

U in Bi solution

Homogeneous fused salt*

Thermal

Thermal

Thermal

Thermal

Thermal

Thermal

Thermal

154.2 490 158

40 2530 604

700

1500

1500

500

650

0-100

0-100

2000

2500

Diphenyl

(Cdhh

250 2000

110 1004

1500


25

1500

49

0 009

0 007

~0.04

1.0

0.67

None None

20 in. Corrosion

Many

Low-carbon Ni, graphite, AhO,

~5 ft concrete ~5 ft concrete! -5 ft concrete -~5ft concrete Stability, Corrosion corrosion 300 series

Titanium

Same as HjO

Cr-Mo steel

Simple core

Simple core

Simple core, low vapor pressure

Low vapor pressure, high cycle efficiency, stability

Erosion, ag glomeration

Corrosion, distribution of uranium

High melting point

Reasonable vapor pres sure, lubri cant, nonreactive to U, noncorroeive, non radioactive Decomposi tion

Moderator, good temp, coefficient, high cycle efficiency

Low vapor pressure, inert to air

Melting point

Melting point, Corrosion, system dissociation activity

Decomposi tion, depo sition

Corrosion,

Corrosion

Corrosion

Reprocessing, Containment, agglomera distribution tion of uranium

Corrosion, remote maintenance

* Molecular per cent: NaF, 53.5; ZtF*. 40; UF*. 6.5.

rial. In the case tially terphenyl,

of diphenyl starting material, the decomposition product is par The decomposition which is a better coolant than diphenyl. products can be allowed to build up to something like 15 per cent without affecting the behavior of the coolant. This makes the cost of distillation cleanup relatively cheap — probably under 0.3 mill/kwhr. The make-up diphenyl, even at Sl/gal, will not be more than 1 mill/kwhr. 6

FUEL ELEMENTS

The fission recoil distance is of the order of 10"4 in., so that the heat is generated essentially at the source of fission. This heat must be transferred through the fuel element and coolant interface without excessive temperature drop. The degree of

SELECTION

NUCLEAR-POWER-PLANT

12-60

Mole

Gas

Specific-heat

weight

Hydrogen, Hi

2 4 10 10 10 16 18 20 28 29 30 32 44 124 200

Hypothetical gas (monatomic) Hypothetical gas (diatomic) Hypothetical gas (triatomic) Methane. CH«

Ethene,

12

Specific Power of a Reactor for Various Gases*

Table 30.

Steam,

[SEC.

HiO

CjH.

Mercury vapor, Hg

ratio, 7

1.40 1 67 1.67 1.40 1.33 1.31 1.33 1.40 1.40 1.40 1.23 1.40 1.33 1.17 1.67

Ratio of thermal output compared with air

3.8 1.7 1. 1 1.7

2.2 1.8 1.5 1.2 1.0 1.0 1. 8 1.0

0.8 1.2

0.24


* This table shows the ratio of reactor thermal output obtainable with various gases compared *itL air for a fixed size of reactor with the same operating pressure, temperature, and per cent blower output based on room-temperature values of y.

subdivision of the fuel is primarily dictated by the problem of transmitting the heat through the fuel element and coolant film to the coolant. In reactors that use highly enriched fuel, the fuel is diluted with structural material Usually these in order to provide sufficient structural strength to the fuel elements. structural materials also provide corrosion resistance to the fuel. In central-station reactors, the fuel is only partially enriched, so that the diluent is fertile material. The use of uranium or thorium in metallic form as the fertile material introduces a serious corrosion problem in water-cooled reactors. A clad is provided on fuel elements to prevent fission products from escaping into the coolant and is also used in water-cooled reactors to protect the fuel from cor rosion at least partially. 6.1

Fuel Materials

o. Unalloyed Uranium. Orthorhombic lattice (a phase) to Metal Systems, tetragonal lattice (0 phase) to 1405°F, body-centered cubic (-y phase) to melting at 2010°F. Uranium corrodes in water, a phase suffers bad anisotropic k = 16.* The operating temperature is usually limited to irradiation growth, below the a-f) transformation (1225°F) because cycling through it increases growth. Similar to unalloyed uranium. Some alloys b. Alpha-phase Alloys of Uranium. reduce corrosion in water. Made by addition of 9 to 12 weight per cent molybdenum, if c. Gamma Uranium. mctastable. Its corrosion resistance is much better than a-uranium alloys. Thi* Its maximum operating tem phase is also attainable by other alloy compositions. perature is limited by irradiation growth to about 1300°F. d. Thorium. Face-centered cubic lattice, no transformations, and melts at 3074°F It corrodes badly in water. There are presently no known alloys that make it cor rosion resistant in water. Zr with up to about 5 atom e. Metals with Uranium as a Minor Constituent. per cent U is reasonably easy to work and has somewhat the same properties as Zr: Al with up to about 5 atom per cent U is reasonably easy to work and has somewhat the same properties as Al. Uranium Oxides, High Density (above 90 per cent). The melting point is 6.12 5000°F; k = 2; isotropic growth under irradiation. It is brittle and will fracture Most fission gases are retained in lattice. It has high corrosion under high AT. resistance in HaO. conductivity, Btu/(ft')(hr)(°F/ft). ** 6.11 1225°F,

-

Sec. 12-1]

12-61


Material. The dis a. Matrix Structural Dispersions, Heterogeneous 6.13 persed materials are UOj, UC, UN. The matrix materials are Fe, Al, Zr. The dis These materials are, persed fraction is kept low, so matrix properties still dominate. Conductivity is a function therefore, limited to under 50 per cent dispersed material. of the conductivity of both constituents and packing fraction. UOj (U'" or U"3) in depleted U, Th, and their alloys. 6. Matrix Fertile Material. These pockets will grow under irradiation and strain the matrix material. 6.2

Corrosion (and Fabrication)

For slightly Uranium and thorium corrode badly in high-temperature water. enriched reactors, it is very difficult to find a low-cross-section alloy that will make The effect of some structural materials on reac these materials corrosion resistant. tivity is given in Table 31. Table 31.

Effect of Alloy on Enrichment

Weight % of alloying material

and Conversion Ratio*t

% enrichment required

Conversion ratio

0.91

0.82 0.77 0.74 0.8 0.7 0.85

0.99


Silicon.

1.05 0. 96 1.5

3.8

0.94

• Prepared by G. A. Freund, Argonne National Laboratory. t Baaed on HiO-cooled and -moderated reactor with an HsO:U volume ratio of 2.5 and a fixed waterIt The fuel plate thickness was varied to maintain the 2.5 HtO:U ratio. thickness of 0.5 in. passage of 1. 10 and on 10 per cent volume of zirconium as clad material. is also based on a

For water-cooled reactors, the following approaches to the corrosion problem are being pursued: The problem of achieving corrosion resistance by alloying High Alloying. 6.21 is quoted from a paper concerning the

PWR

reactor.

*

It is highly unlikely that cladding can be sufficiently perfect to never expose the uranium Also, whether or not radiation damage can be prevented by mechani to the coolant water. The other possible solution is to add something to the uranium cal restraint is not known. to improve its characteristics, which can be done by making a uranium alloy, such as uranium-molybdenum, uranium-silicon, or uranium-niobium, or by utilizing the uranium in the form of uranium-oxide. All alloying elements that improve the radiation-damage resistance or the corrosionresistance properties of uranium alloys tend to give poor neutron economy, inasmuch as the alloying elements have relatively high cross sections for capture of neutrons. Even if neutron economy is ignored, none of the alloys arc suitable from both a corrosion-resistant viewpoint. This means that either the life is seriously limited by and radiation-damage The radiation may cause a rupture of the radiation damage or by corrosion, or both. cladding and subsequent failure of one element due to corrosion, or the corrosion may be In either event, after several days of exposure to the hot initiated by a cladding defect. water, all of the uranium alloy and the contained fission products in the affected fuel element might be released to tho coolant stream. 6.22

Use of UOs.

U02 is practically insoluble in high-temperature water.

Table

22 indicates that it can be used in place of unalloyed uranium with only a slight effect on the nuclear properties of the reactor. A major problem is that of attaching a clad to prevent fission recoil particles from entering the coolant. An adherent coating as yet has not been developed. A nonadherent clad is therefore used. The features of this type element used for the Shippingport PWR reactor are described below :f * Proceedings of a meeting of the Atomic Industrial Forum, Inc., Washington, D.C., September, Ibid.

t

1955.

12-62

NUCLEAU-POWEll-PLANT

SELECTION

[SEC. 12

Uranium oxide is an excellent material for the natural uranium fuel elements because is completely inert in high-temperature water and is also satisfactory from a radiationdamage viewpoint for a relatively high burn-up. The uranium oxide can be readily cold compacted into pellet form and then sintered to increase the density to in excess of 90 per cent of theoretical density. By mass production techniques, the dimensions can be ground to extreme accuracy and the uranium oxide loaded into Zircaloy tubing. The center temperatures of the uranium-oxide fuel elements may be as high as 2200 degrees F, but this appears to cause no difficulty because it is still approximately 3000 degrees F below the melting temperature of uranium oxide. The Zircaloy-2 tubing is ideal from a performance viewpoint. Its inherently high cost is a disadvantage; however, this disadvantage is more than compensated for by the decreased cross section for capture of neutrons of this material.

it

6.3

Shapes

Many factors influence the choice of the shape or geometry of a fuel element. Primarily, these are power density, bond integrity, geometric stability, subassembly fabrication, and fabrication cost. The characteristics of the various shapes are dis cussed to indicate how they accommodate these factors. 6.31 Rounds (Slugs, Rods, and Pins). Rods and pins are used

primarily

where

).{

bonding between the clad (a tube) and the fuel is difficult and where the tube is The tube has good structural and expansion properties. required as a pressure vessel. A round shape is used to maintain a tight fit between the fuel and the clad in highdensity oxide fuels and for unbonded or poorly bonded metal fuels. For reduceddensity oxide fuels, the tube is used as a pressure vessel to contain the gaseous fission products. Figures 32 to 36 show various configurations of round fuel elements. When the heat throughput is small, large rods or slugs can be used. In large sizes (3>2-'n- diameter and over) a round shape is cheap to fabricate. Tubular Elements. 6.32 These elements also maintain their geometry well. They are more difficult to fabricate than rods. In some cases, they are easier to arrange into a subassembly than rods. Fewer tubes are required than rods for the In water-cooled graphite reactors, hollow slugs, with the major same performance. cooling on the inside, can be used to circumvent the increase in reactivity on loss of They arc intermediate in all characteristics between rods and plates. Fig water. ure 37 shows a tubular fuel element. 6.33 Plates or Flats. These are used primarily where a large surface area is For equal heat -transfer performance, the thick required (for high power density). Since plates can be made ness of a plate should be about half the diameter of a pin. in any width, fewer are required than pins. For instance, in the PRDC fast reactor, one hundred forty-four J^-in.-diameter pins are required per subassembly compared In some cases, the cost of fabricat with ten 6-in. plates for the same performance. They may be easier to braze into an integral sub ing plates less than pins. assembly. They should not be used where only poor bond can be obtained between the clad and the fuel. They are used successfully whore sodium used as the bond ing agent, where zirconium bonded to uranium, or where the "meat" has matrix of the same material as the clad. Such an element a UOj fuel in an iron matrix bonded to iron. Twisted ribbons (Fig. 43) are a form of plates. They are used to simplify the assembly of fuel elements into subassemblies. Their primary application is

a

is

is

a

is


6.23 Other Methods of Protection. Subdivide the metallic ura Subdivision. nium, and surround it in a corrosion-resistant matrix. Cladding. Rely totally on clad, but use special methods of inspecting the clad integrity. (The British are considering insertion of a small capsule of high-pressure helium under the cap of the closure and rupture capsule. A helium leak tost is then used to determine the clad integrity.) Lead Seal. On a limited Use the punctureproof-tire principle with Pb as the seal. has demonstrated that this method will work. scale, Argonne National Laboratory Noncorrosive Coolants. Organic coolants sodium and helium do not corrode ura Fabrication of fuel Only a fission barrier must be provided. nium and thorium. elements for these coolants is relatively easy.

12-63


Sec. 12-1]

FUEL ASSEMBLY

FUEL ELEMENT TYPE:

Slug _^

072

^-WELD

5—CAP

Al TUBE

8.70

L_ 1.440"— ^ FUEL:

Natural

CLAD:


CORE

086" COOLANT ANNULUS

uranium

2S Aluminum

CONFIGURATION

Process lubes set in graphite-block lattice spacing

COOLANT.

H20

MODERATOR:

Grophite

REACTOR APPLICATION:

Plutonium

Via.

production (Honford-type 32.

reactor)

FUEL ASSEMBLY

Slug

AL CAN

-l.08"O.D.

I 0.0. U-SLUGX8'

FUEL:

Natural

CLAD:

2S Aluminum

CORE

with sguare

Hanford-type reactor.

FUEL ELEMENT TYPE:

stack

CONFIGURATION:

Ouatrefoil

D20

MODERATOR:

D20 Plutonium

Fig.

is to reduce the cost arc, therefore, being

OUATREFOIL

SLUG

uranium

COOLANT:


LONG

process tubes placed on hexagonal lattice

producer -Sovannoh

33. Savannah

River (CP-6)

River reactor.

of making a subassembly where brazing cannot be used. They considered for zirconium-clad fuel elements. Figures 38 to 44 .show various configurations of plate or flat fuel elements. 6.34 Wire Mesh and Compact Surfaces. These surfaces are used where the heattransfer and hydrodynamic properties of the coolant do not suit the simpler shapes For instance, wire mesh (Fig. 45) is considered for small power plants. mentioned. In one case, the core size was such that the velocity of the coolant was 10 fps. This velocity was not high enough to attain the desired heat-transfer coefficient with a

12-64

SELECTION

NUCLEAR-POWER-PLANT

[SEC.

12

conventional surface. A wire-mesh design increased the pressure drop but also increased the heat-transfer coefficient correspondingly. In small gas-cooled power reactors, the shape desired for high heat-transfer per formance and minimum pressure drop is a pancake thin in the flow direction. This is not compatible with nuclear considerat ions. The heat-transfer surfaces are, there fore, arranged for multisplit flow with an equivalent length of only several inches. The velocity through the element itself is very small, and the surface can be very finely subdivided as a compact surface. Such an element is shown in Fig. 46. FUEL ELEMENT


TYPE-

2wt-%

FUEL: CLAD: CORE

FUEL ASSEMBLY

Rod

uranium

Aluminum CONFIGURATION:

COOLANT:

D20

MODERATOR:

D20

REACTOR APPLICATION: REFERENCE:

-90%

U

-

■

aluminum

alloy 0850 in. diameter

jacket 0.059 in. thick 116 2% U-AI rods + 2 special

rods containing 29 in. of 6% U-AI alloy grouped on 5^8"in. centers in 6ft diameter x 7ft 9 in effective height

Argonne

heavy water reoctor (CP-3'1

ANL-WHZ-250 Fiq. 34. Argonne heavy-water

reactor.

6.36 The formation of a core by making many Integral Castings and Extrusions. thousands of pins or plates is not intriguing from a standpoint of cost. The casting Such units (Fig. 47) have been of a complete subassembly as a unit is attractive. cast, but consistent fabrication of these intricate shapes requires careful control oi mold temperature and pouring temperature and a uniformity of fuel-material com They have not been reproducibly made in large quantities to the tolerarjc position. If the casting is made directly around the clad as a mold, good bonding i> required. difficult to achieve. If casting is made in a separate mold and then the clad is insert?*! as tubes, a liquid bond such as sodium must be used and it is difficult to determine il" integrity of the bond in such a complex shape. Integral extrusions can be made by casting into oversize channels and then extrud Some success has been achieved with this approach, but it must 1"" ing to final size. demonstrated on a production basis.

Sec. 12-1]

SELECTION

12-65

OF REACTORS

NINETEEN ROD FUEL CLUSTER

SECC-C AREA=5054in2

FLOW

FLOW

SEC. D-D

AREA=5.868in2

FLOW

ST. STEEL

495"0D.X. OIO" WALL

ORIFICE PLATE

AREA=4.287in.2

-I20in-

FUEL JACKET LOWER

GUIDE

COOLANT FLOW

-UPPER

-FUEL


CORE

HANGER

010" NAK BOND 455* FUEL DIA. 4ln. 0 2 1,1,1,1,1,1,1,1,1 SCALE

GUIDE

TUBE

215 31^"lubes eoch containing pitch spacing lOin. triangular

CONFIGURATION:

COOLANT:

Na

MODERATOR:

Graphite


North American

SEC. A-A

a 19 rod cluster of fuel elements.

concept of o sodium- cooled graphite

Tubes on a

250mw heat reactor

Fig. 35. Sodium-bonded rod-type elements.

FUEL ASSEMBLY

FUEL ELEMENT TYPE:

Pin

k It

diameter

0.010

pin

0 005" clad thickness FUEL:

Enriched

CLAD:

Zr if metallurgical bond 18-8, or Nb if No bond

alloyed

U

METHOD OF FABRICATION:

CORE

144 pins per subassembly arranged within hex or square con

CONFIGURATION: Approximately

COOLANT:

No

MODERATOR:

None


50-100 assemblies

EBR-TX ond study of PRDC

Fig.

36.

EBR-II

and

PRPC

reactors.

per core- full length

NUCLEAR-POWER-PLANT

12-66

FUEL ELEMENT TYPE

SELECTION

[Sec.

12

FUEL ASSEMBLY

Tube

COOLANT

l-'/2" OD, l" ID

FUEL:

Uranium olloy

CLAD:

Zirconium, OOI5" thick

BOND:

Metallurgical

TLADDING Cluster of seven orronged in a 6in. diameter tube or without tube. Supported at ends only.



CORE

Clusters of seven tubes orranged in 6in. diameter clusters spaced on lOin. pitch to next. Tube length-lift.

CONFIGURATION:

COOLANT:

D20

MODERATOR:

D20


Boiling D20 central station power

Fia.

37. Heavy-water boiling reactor, Argonnc concept. FUEL ASSEMBLY

FUEL ELEMENT TYPE:

Plate

Spot weld side plates

FUEL:

U235 in Al or Zr

CLAD:

Al or Zr

METHOD OF FABRICATION: Cast in Zr can, roll to sizeCORE

CONFIGURATION:

COOLANT:

H20

MODERATOR:

H20

EBWR

approximately

Fig. 6.4

38.

100 subassemblies

EBWR

reactor.

Radiation Effects

In the fission process, two atoms are produced per fission, approximately 30 per cent of these atoms are gaseous, and some 10,000 atom displacements are produceJ per fission. The mechanism of damage by these phenomena is discussed in Sec. MM. and the following discussion serves merely to point out some of the fuel systems whioh have been considered and used.

Sec. 12-1]


12-67


This results in nonuniform Unalloyed alpha (a) uranium is highly anisotropic. A similar growth, below about 850°F, is encountered on growth on thermal cycling. In This growth phenomenon restricts allowable burnup of a uranium. irradiation. special cases the burnup limit has been increased somewhat by the use of structural Increased burnup has been achieved by minor additions of alloying mate restraints. In this case, growth is minimized by virtue rials such as molybdenum or zirconium. Burnups* up to of a two-phase structure and random orientation of the a phase.

Via. 39. EBWR fuel assembly. However, in been achieved with alloys of this type. operating temperature exists above which excessive swelling occurs. This temperature limit has been increased from about 850 to 1000°F by the use of minor alloy additions and has been further increased by the use of 7-stabilized

0.5 per cent and higher have

all

cases,

a maximum

alloys.

By alloying uranium with -^-stabilizing elements such as molybdenum or niobium, the cubic y phase may become stable under irradiation conditions. For a 10 weight per cent Mo alloy and at temperatures below about 1200°F burnups up to 3 per cent

How can be achieved with density decrease of 2 to 3 per cent per 1 per cent burnup. ever, at higher irradiation temperatures, a lesser amount of burnup, perhaps as low as 0.2 to 0.5 per cent, can result in density decreases in excess of 10 per cent per 1 per cent of burnup, accompanied by considerable swelling of the fuel. Another method to raise the burnup limit is to alloy the uranium with aluminum or zirconium, thus forming a compound or alloy of relatively low uranium content. Although many specimens have been irradiated, both the lack of control of material * As

used herein, burnup means atoms

fissioned

divided by total atoms in the fuel alloy.

12-G8

NUCLEAH-POWEH-PLANT

SELECTION

[SEC.

12

composition and the difficulty of accurately measuring the temperature during irradia tion produce a large uncertainty in interpreting the results. Ceramics, such as uranium oxide, show much promise in overcoming the short comings of metallic fuels. Apparently, the fission products can fit into the larger atomic lattice sites of the ceramic, giving the ceramic the ability to accommodate a Also, the fission gases appear to be able to diffuse larger number of fission products. through the oxide lattice and thereby either escape from the structure entirely or seek Cracking of the oxide has been observed under high voids within the structure. temperature gradient conditions but over-all growth appears very low. The burmip FUEL ELEMENT TYPE:

FUEL ASSEMBLY

Plote

LENGTH;

Standord, 24% in over-oil

Outside,28%

over-oil

Active, 25% in over-oil

2.845 in before curving


WIDTH:

5.504"R. OUTSIDE FUEL PLATE STANDARD FUEL PLATE

*Comb is at ends only

20wt-%uronium-90%

FUEL:

10 to

CLAD:

Aluminum, 0.020 in. thick

CORE

CONFIGURATION:

Subassemblies

U235 oluminum

grouped

olloy, 0.021 in. thick

into slob; octive core approximately

by 27 in long by 23% in. high; beryllium

COOLANT:

H20

MODERATOR:

H20


Moteriols

REFERENCE:

ORNL-963

9m wide

reflector

testing reactor (ARCO)

Fio.

40. Materials-testing reactor.

limit of this fuel system

has not been determined, but appears to be higher than that of the metallic systetns. One of the major problems associated with the oxide i= containment of the fission gases. 7

MODERATORS

Table 32 lists some of the more pertinent nuclear properties of potential moderators. measure of the ability of a material to slow down neutrons is the slowing-down power i.e, the product of the scattering cross section 2, and the mean log energy loss per collision {. The efficiency of a substance in slowing dom neutrons without excessive absorption is given by the moderating ratio, the ratio o: the slowing-down power to the absorption cross section. For small reactors, modera tors with a high value for the slowing-down power are preferred. For large reactors, moderators with a high value for moderating ratio are preferred.

The most direct


Sec. 12-1]

12-69

The amount of moderator required in a reactor varies greatly with the moderator and with the degree of leakage that is acceptable. The solution-type H20-moderated The H20-to-U volume ratio in reactors are only 1-ft-diameter spheres. research slightly enriched H20 reactors is usually kept below 4: 1. In reactors that use modera tors of poor slowing-down power, such as graphite or D20 reactors, the moderator-touranium volume ratio required for natural or slightly enriched reactors can be higher FUEL ELEMENT TYPE

FUEL

ASSEMBLY

Plate

—

-2.940 -

]k.l20" r-J54' I

-243"

(-

17/,6—

GAP

5'/2"R.


FURNACE

FUEL:

H1

Approximately

BRAZE

10 to 20wt-% enriched

uranium (-90% U255) -oluminum

olloy,

0.020 in. thick

CLAD:

2S oluminum, 0.020 in. each side

LENGTH.

Inoctive ond plotes(2

eoch), 24%"in.

Active plotesdOeoch), CORE

—

6

D

23%"in

□ D Provisions for 17 subossem *rj rj c) D blies os shown, reactor stort-up will be mode using the 10 to 20% O □ D D enriched uronium-oluminum sub.-, .-. '-' assemblies; estimated 12 required-, two subassemblies odded later for burn-up compensation, three additional subassemblies available for initial loading, subsequent reloodings to be made with less enrichment per subassembly to utilize oil locations; subassemblies contained in 6ft. diameter tank

CONFIGURATION

a

"

COOLANT:

D20

MODERATOR:

D20


Arqonne reseorch reactor

REFERENCE:

Communication

Fio.

than 20:1. The shown in Figs. 6

41.

(CP-5)

from J.M.West

Argonnc research reactor (CP-5).

influence of moderator on the size of slightly enriched reactors is and 7. In addition to having good moderating properties, a moderator should be compatible with other requirements. It should be chemically inert to its environment, should not suffer undue irradiation damage, and should have the mechanical and thermal prop erties required for its operating conditions, as well as be reasonably available and not too expensive. A tabulation of some of these properties for solid moderators is given in Table 33. The properties of liquid moderators are discussed in Art. 5. A big advantage of liquid moderators is that they can also be used as the coolant and, as a result, greatly simplify reactor design.

NUCLEAR-POWER-PLANT

12-70

Table 32.

Element or compound

A,

Nuclear Properties

[SEC. 12

of Moderators* Slow-

'V, (10" atoms)/ (cm')t

epithcrmal, barits

1 85

0.124

6.1

1 60 2 8

0.080 0.067

4 8 9.8

2.5 1.00

0 027 0 033

Den sity, mole g/cm'

SELECTION

0.025 ev, barns

r..

2«. cm-1

cm"'

0.009

0 0011

0.76

0 0045 0.009

22.4 49.0

0 2 0 66

i

Moderat in*down ing ratio power <».)£ ».)£

CommenU

150

Good moderator

0 0OO37 0 19 0.158 0 063 0 66 0.173 0 11 0.0006

170 180

Good moderator

0 005 0 022

20 69

Strong moderator

Beryllium, Be 9.01 Carbon (graphite), C 12 0 25 0 Bcryllia, BeO Boron carbide, 56 0 B».C Light water, HtO. 18 0 Heavy water (99.75% D,0).. 20.0 Sodium hydroxide, NaOH •WO

1.10

0 033

10.5

0 0026 85 X IO-« 0 035 0 51

2.1

0.32

27 1

0.82

Polystyrene (CH). 17.0

1.07

0.0 J8J

25 0t 0 335«1 0.013

0 026

0 206 0 16

0.6 0.172 0 14 1.64 0 93 1 50

0.87

0.77

0 18

21.000

0.67

26

0 95 0.842 0 80

62

Good moderator Fair moderator: neutron lo&i high Hadution dam age a problem

• In part from 0. E. Kvans, Materials, Sueitonici, 11 (6) (1953). t Number of atoms or molecules per cubic centimeter. Number of (CH) units per cubic centimeter. Per (CH) unit.

J Generated for wjivans (University of Florida) on 2015-09-23 02:57 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

f

Table 33.

Melting point, °F Thermal conductivity k,

Btu/(ft)(hr)(°F/ft)

Modulus of elasticity E Coefficient of thermal expan sion per °F, 6 Resistance to thermal stresst

*0

- *)P/Eb

Tensile strength P. psi Ductility Cost, fabricated, $/lb

Properties of Solid Moderators Bo

BeO

B».C

6600*

2340

4900

4410

60 at IO00°F 1.2 X 10'at

60 at 800°F 40 X 10«

I00°K 5 X 10'at

10.5 X 10« at 800° F

IOOO°F

18 at 800' I 38 X I0« up to

I800°F 5.3 X 10«at

X

10*

0.22 X 10-«

1000°F

1.800 3.300 at !000°F

2.900 27.000 at 800°F

700 10.500 at I800°F

Low

None

1

100-200

25

Low

16

66

16.000 40.000!

None Price unavailable (may not be higher than Be)

* Sublimation temperature. t The resistance to thermal stress is really the relative allowable power density in the moderator. It is a measure of the permissible thickness of the moderator blocks. It is based on Hooke's law, and represents Poisson's ratio. X Bending strength.

7.1

Graphite

Graph Because of its low slowing-down power, graphite reactors are usually large. ite permits use of natural uranium because of its low absorption cross section. The maintained strength of graphite at elevated temperatures, its high melting point, and excellent resistance to rupture by thermally induced stresses make it attractive for high-temperature reactors. Graphite is susceptible to radiation damage at low temperatures and to oxidation or corrosion by some fluids at high temperature. High purity is required for reactor quality, and the refinement process is somewhat expensive. Graphite in finished form for a large reactor can cost over a million dollars. The properties of graphite vary widely with its purity and density. Representa tive high-density values are given in Table 33. Pile-grade graphite is relatively soft

12-71


Sec. 12-1]

FUEL ASSEMBLY

FUEL ELEMENT TYPE.

Plote

COOLANT RIBS

CLADDING

FUEL-

FUEL:

Notural or enriched U

CLAD:

Zr Plates ore rolled or pressure bonded 10 clod.

CORE CONFIGURATION;

COOLANT:

H20 or No

MODERATOR:

H20 or none

REACTOR

Boiling reactor such as EBWR or fost reactor. For coses where clod motenol cannot be easily brazed to side plotes.


APPLICATION:

Flo.

42. Flat fuel plate concept.

FUEL

FUEL ELEMENT TYPE:

ASSEMBLY

Plote

FUEL PLATE -

coolant

ESSSaSSS^S^^

3^

•

STAGNANT No BOND -

0050"

2l/2"x2'/2" squore subassembly 30" long. Stainless steel rodiator made by brazing. Plates are slipped into slots and No bonded. 2 mil Na bond

FUEL:

Alloy with 25% enriched uranium

CLAD:

18-8

METHOD OF FABRICATION: CORE

Rolling

or casting

CONFIGURATION: Ninety-two assemblies moking up a core approximately 30 in. diameter x 30in. long. Design is based on rodiooctive fabrication

COOLANT:

No

MODERATOR:

None

REACTOR APPLICATION :

Fast reator

REFERENCE.

Section 4.1

Fio.

43.

APDA

of plates

advanced

design concept.

NUCLEAR-POWEH-PLANT

12-72

SELECTION

[Sec.

12

The reaction of graphite with various gases, aqueous and can be readily machined. solutions, and solids is shown in Table 34. Sodium will attack graphite by two mechanisms: (1) carbon loss by mass transport The presence of attack is indicated by a weight loss and and (2) thermal shock. FUEL ASSEMBLY

FUEL ELEMENT TYPE:

Twisted ribbon

FUEL:

Uronium or uranium zirconium

CLAD:

Zirconium



CORE

Ends ore fastened together and paralled so thot oil ribbons are similarly oriented and hence spoce each other.

Oval roll and draw to final shope

CONFIGURATION:

Arranged in 5in. diameter tubes II ft long for D20 reactor

COOLANT:

D20


Central station D20

Fig. 44. Pressurized

heavy-water reactor, Argonne concept.

FUEL ELEMENT TYPE:

FUEL

ASSEMBLY

Wire

Wires brazed together

FUEL:

U02 in stainless

CLAD:

Stainless

CORE

steel or

U-Zr

alloy

steel or Zr

CONFIGURATION:

COOLANT:

C02

MODERATOR:

BeO


Small

Fig.

power plant or maritime

45. Gas-cooled

ship propulsion

small power-reactor

concept.

physical damage such as fractures, cracks, and spalled areas. Solubility probably accounts for only a small part of any weight loss. A theory has been suggested, based upon actual test work, that sodium reacts with carbon to form the carbide but the carbide that has been detected is more probably calcium carbide. This carbide could form from the calcium present in the sodium more readily than sodium carbide. Carbon loss apparently occurs only in the presence of a material that is susceptible

Sec. 12-1]

SELECTION OF REACTORS FUEL ELEMENT

TYPE:

12-73

FUEL ASSEMBLY

Smoll cooling passages, large headers COOLANT

Washers as above or screens or porous tube

PATH

Washers as obove, screens, or porous tube

FUEL: Metal or oxide CLAD:

If ceramic fuel, glazed clad

CORE CONFIGURATION:

Split flow design

COOLANT:

Gos

MODERATOR:

Be, BeO


Gas cooled reactor Gas turbine possible


Fig. 46, Gas-cooled

reactor concept. FUEL ASSEMBLY

FUEL ELEMENT TYPE:

Integrally cost

V ID

FUEL:

25 w/o Cr-U olloy with 25% enriched U-235

CLAD:

18-8 stainless

CORE CONFIGURATION:

TUBES, '/3J" THICK WALLS

16 subassemblies 2 f t long

Core opproximotely 2ft diameter, 2ft long, 16 assemblies 6in. across flats of hexagonol moke -up core

COOLANT:

No

MODERATOR:

None

REACTOR APPLICATION

Study of fast reoctor by APDA

FlQ. 47. Fast reactor concept. Sodium acts as the mode of transportation for the carbon between graphite and the susceptible material. Unlike liquid tin, bismuth, and aluminum, liquid sodium has the ability to wet graphite. This apparently will occur at temperatures above 300° F and results in the graphite absorbing the sodium by capillary action. The effect of density would be to control or limit the amount of sodium absorbed. Graphite is not totally impervi-

to carburissation.

the

NUCLEAH-POWER-PLANT

12-74

Table 34.

SELECTION

[SEC. 12

Reaction of Graphite with Corrosive Agents*

Agent

Temperature, °F

Oases: Oxygen

Hydrogen Nitrogen Water vapor Carbon dioxide.

Aqueous solutions: Dilute acid or alkuli Strong acid (HNOi and HiSO.). KOH (50 % solution) Solids: Alkali hydroxides (inert utiuos. ). Metals (most) Metal oxides

Effect or product

840t and higher 1650-1830 Up to 5430 1300 and lower I 300 and higher I 300 and lower

Produces carbon oxides Acetylene (traces)? No effect Negligible reaction Increasingly severe attack. CO Negligible reaction

Less than boiling 570 and higher 660 and higher

No attack Carbon oxides Attacks graphite

At fusion

No attack Metal carbide Metal carbide and carbon oxides

2730 and higher 2730 and higher

* U.S. Atomic Energy Commission, " Reactor Handbook," McGraw-Hill Book Company, Inc., 1955. t Attack at 660 F is reported; the weight loss is not measurable, but a slight, permanent increase in length is noted. Later information indicates accelerated attacks under neutron flux.


ous to sodium regardless of density. This wetting of graphite by sodium and the sodium absorption is the basis for damage due to conditions of thermal shock. 7.2

Beryllium

Beryllium is a relatively good solid moderator, both from the standpoint of slowingFurthermore, it has a down power and from the standpoint of moderating ratio. Because of these qualities, it is used as the modera very high thermal conductivity. tor in the Sea Wolf submarine reactor and as a reflector in the MTR and similar reactors. Pure Be has fair corrosion resistance to water up to 500°F. It is corrosion resistant in sodium to 1000°F and to air attack to 1 100°F. It has a noticeable vapor pressure at 1400°F and, therefore, is not considered for use even in inert-gas systems much above 1200°F. The following properties have limited its widespread use. It is expensive to make and fabricate, costing around $100 to $'200/lb in fabricated form. It is extremely toxic and must be handled so as to prevent inhalation or ingestion. It has poor

ductility.

7.3

Beryllium Oxide

BeO is similar in nuclear properties to Be. It has a very high conductivity for a ceramic and good resistance to thermal shock. Being a ceramic, it has very good It can be used in the presence of air, sodium, and Cd high-temperature properties. without difficulty. It is volatile in water vapor above 1800°F. When dense, it resists attack by sodium or sodium-potassium at a temperature of 1000°F. Because a considerably smaller reactor can be made by the use of a BeO moderator in place of graphite and because BeO is corrosion resistant to sodium, it is being considered as the moderator for sodium-cooled thermal reactors. It can be impregnated with uranium and used as a fuel-element material. Low A BeO coat can be applied to cut down on density improves its resistance to shock. fission-product release to the system. BeO presents the same toxicity problems as Be but is much less expensive to fabricate. 7.4

Boron Carbide

BiC is listed both as a control material and as a moderator. The isotope B" is a very strong neutron absorber, while B" has fair moderating properties. B"4C is discussed in place of Bu because it is cheaper to produce and about as effective as

Sec. 12-1]

SELECTION

OF REACTORS

12-75

# y/rs?ssA

y/ss/ssM. FLATS

- 4 TO 8 CROSSES

None

'16

18-8

732"Niorl8-8

Ag

Up to 650

Ni

Up to 650

None

Up to 650

2 «/o Boron in 18-8

None

Up to 650

48. Typical control rods for IT?0-moderatcd reactors. CLUSTER OF PINS

RODS

CORE

low power

F

Up to 550

Metallurgical

Hofnium

Fig.

F

Up to 450

Metallurgical

F

'/i8" Aluminum

'/e" Ferroboron

MATERIAL

Boron

steel

"25"*

Hoynes

CLAD MATERIAL

None

CYLINDERS

U6

Cd,Cd Ag

B„C

Boral

HfB

B4C

Zr, 18-8

Ni

WAFERS

B«C

18-8

Aluminum

18-8 BONDING

Metollurgical

None

with Zr clad,

Metallurgical

others

APPLICATION

Graphite

ond

D20 moderated

»35

or No

No

No

All types,

All types,

except H20

except H20

Fast

»/o cobolt

Fio.

49. Typical rod-type control rods.

The primary

is

that the field of solid moderators

is

reason for listing BU4C as a moderator to illustrate This material has no by no means exhausted. moderator. immediate application as It B«C has attained industrial importanoe because of its extreme hardness. used in molded shapes to resist wear, in precision gages, in sandblast nozzles, and in

is

a.

moderator.

Li

METHOD OF

a


Aluminum

APPLICATION

F

V8" Boral

METHOD OF BONDING

F

/i6 Cadmium

V'Cd

CLAD MATERIAL

F

MATERIAL

/e"

CORE

12-76

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

Molded pieces are extremely resistant to chemical powder form as an abrasive. They resist any combination of acids or alkalies under ordinary conditions attack. but are attacked slowly by concentrated nitric acid or other comparable oxidizing Although U,C is attacked by agents when heated under pressure in a closed vessel. fused caustic salts, its resistance to liquid metals, such as sodium, is very good. It In neutral or reducing gas reacts with metals in the iron group at around 1800°F. It can be heated in carbon atmospheres, it is stable up to quite high temperatures. monoxide, argon, or nitrogen almost up to its melting point, 4410°F. It is slowly etched by II2 at temperatures around 2200°F. Exclusive of isotopie separation, it costs between $5 and $250/Ib, depending on The lower density material has a greater resistance to shock and purity and density. is much cheaper. The cost of isotope separation is not expected to be particularly expensive. 7.6

Other Moderators

Such materials as wood, paraffin, oils, and metal hydrides can be considered as Li7, although it has a low atomic weight, is not moderators for special applications. any better than sodium as a moderator because of its low scattering cross section.


8

CONTROL RODS

The common ways of controlling a reactor are by moving poison, moving fuel, or The choice of method of control is usually made after a prelimi changing leakage. nary design of the reactor has been established. leakage or reflector control is gen erally used when it is desired to control without putting mechanisms in the reactor itself. Movement of fuel is used when other methods seriously affect neutron econ omy. The method generally used is moving poison. This method does not have the neutron economy of moving fuel, but in most reactors the amount of control that must be in the reactor under normal operating conditions is small and the penalty is not so great as it woidd seem at first glance. Moving poison can give as strong control For instance, the micro per unit volume as fuel movement, even in a fast reactor. scopic cross section of B10, because of high neutron-energy resonance absorption, is comparable to that of U255. Cooling requirements for the rod for moving control are much less than for moving fuel. 8.1

Control -rod Shapes

There are two shapes of control rods in general use: rods or cylinders and flat plates In crosses. The reactors, flat plates or crosses arc used almost entirely. moderating property of II20 is so strong that control rods must be located within 1 ft of each other. To keep from using too much space, these control rods are flat, usually about J4 in. thick and about 4 in. wide. The spacing of the control rods and volume occupied in other reactors is not critical, and the shape generally used is lubes or cylinders. The various control rod shapes being considered are shown in Figs. 48 and 49. or

REFERENCES 1. 2. 3. 4.

Keenan, J. A.: " Thermodynamics," John Wiley A Sons, Inc., New York, 1941. Jens, W. A., and I'. A. Lottes: " Analysis of Heat Transfer, Burnout , Pressure Drop and Density Data for High Pressure Water," ANL-4C27, May, 1951. deBruyn, H., et al: "The Design of a Small Scale Prototype of a Homogeneous Pow*f Reactor Fueled With a Uranium Oxide Suspension," Geneva Paper P-936. Mcintosh, A. B.: Metallurgy of Nuclear Power Production, Engineer, Nov. 25. 195.V p. 762.

5. 6.

Reports to the United States Atomic Energy Commission on Nuclear Power Re»<-t-or Technology, Government Printing Office, Washington, D.C., May, 1953. Power Breeder Reactor Progress Reports, Boiling Experimental Reactor Reports.

Sec. 12-1] 7. 8. 9.

10.


11.


12-77

Reactor Engineering Division Quarterly Report, September, 1953-November, 1953, ANL-5208 (Classified). Dietrich, J. R., H. V. Lichtenberger, and W. H. Zinn: "Design and Operating Experi ence of a Prototype Boiling Water Power Reactor," Geneva Paper P-851. Lottes, P. A., and W. S. Flinn: "Method of Analysis of Natural Circulation Boiling Systems," Argonne National Laboratory, March, 1956. Zinn, W. H., H. V. Lichtenberger, M. Novik, G. K. Whitham, C J. B. Zitek, J. G. Feldes, and R. O. Haroldsen: Operational Experience with the Borax Power Plant, Nuclear Science and Engineering, vol. 1, no. 5, October, 1956, p. 420. deBruyn, H., el al: "Power Reactors Fueled with 'Dry' Suspensions of Uranium Oxide," Geneva Paper P-938. McCullough, C Rogers: "Summary Report on Design and Development of Hightemperature Gas-cooled Power Pile." Sept. 15, 1947, MON-N-383.

12-2

ECONOMICS

OF NUCLEAR POWER BY

L. E. Link and Walter H. Zinn 1

FUEL CONSUMPTION, POWER, AND NEUTRON FLUX

Before the various factors under the above subject matter are discussed, it is well the meaning of some of the frequently used terms as they are used in this section. Other terms are defined as a part of the discussion. 1. Fissionable material — nuclides having high thermal-energy fission cross sections, It is recognized that there are other nuclides which have U233, U235, Pu259, Pu"1. appreciable fission cross sections at energies above thermal. 2. Fertile material — nuclides that, after capturing one neutron, become a fissionable nuclide or will decay to a fissionable nuclide, e.g.,

+

Th»J Um

3.

U"3

Pa'33

^

n ->

tjim + „ Np2" U"9 Pu239 — Fuel fissionable material or homogeneous mixtures of fissionable and fertile

material. 1.1

Fuel Consumption and Power

calculated in the simplest form and in one set of con

P

The reactor power output sistent units as follows:

Power, Flux, and Fuel Loading is

1.2

V

P

= where = = N = = 07 =

.

«

is

a

it

1

is

A

is

5

is

The rate of fuel consumption based on an energy release of 200 Mev per fission. Approximately 95 per cent of this energy results from primary fission, and per The total fission energy recovered cent from secondary sources. equivalent to common term for the destruction rate of fis 3.22 X 10~" watt-sec per fission. sionable material Since grams per megawatt-day (Mwd) of heat produced. 2.68 X 1021 atoms fission per megawatt-day, the amount of material fissioned varies from 1.04 to 1.07 g/Mwd, considering the range of fissionable nuclides, U1" to Pu'". The Fissionable atoms are destroyed not only by fission but by parasitic capture. ratio of capture to fission a varies with isotopes and with energy. At thermal energies Thus, the initial ranges from 0.10 for U233 to 0.18 for U255 to 0.42 for Pu239.* consumption rate of fissionable material varies from 1.14 to 1.52 g/Mwd in reactors Where fast fission effect having no fast-fission contribution from fertile material. less than the 1.14 to 1.52 exists, the consumption rate of fissionable material g/Mwd by the factor €.

*


U234

^>

+ n ->

Th232

*->

to define

= Av(N
reactor power output, Mw conversion factor, 3.22 X 10~17 Mw-sec per fission mean thermal-neutron flux, neutrons/(cm2)(sec) number fissionable atoms in fuel, atoms/cm3 of fuel microscopic fission cross section, cm2/nucleus volume of fuel, cm3

Superscript numbers refer to References


12-78

(1)

Sec. 12-2] In

OF NUCLEAR

ECONOMICS

POWER

12-79

terms of mass of fissionable material in a reactor, Eq. (1) is

P

=

BvC/W)

(2)

where B = conversion factor, 8.18 X 10' Mw-sec/kg of fissionable material W = weight of fissionable material in reactor, kg The numerical value given for B is an average for the fissionable nuclide range U"3 to Pu»>.

Equations (1) and (2) apply to a reactor fuel consisting of a single fissionable species. a type that will result in the ultimate growth of a second fissionable species, the power calculation must account for both as follows:

In reactors having fertile material of

P

=

A(WV>/,F,

+

(3)

is

it

P

A

loaded with uniformly enriched* fuel. (4) applies where a reactor a reactor having fully enriched fuel elements (spikes) is

Equation

fourth classification exists in

is

distributed among fuels having low enrichment, most probably natural uranium. This would require an additional term in Eq. (4). the heat generation per unit mass of fissionable or 1.21 Specific power (P.) The usual units of specific power are: fissionable plus fertile material. Megawatts per kilogram of fissionable material. 2. Megawatts per ton (2,000 lb) or metric ton (1,000 kg) of fuel. This numerically equal to kilowatts per kilogram or 3. Watts per gram of fuel. megawatts per metric ton of fuel.

P.

_

= p

"*

P

is

1.

m

(5)

specific power, watts/g of fuel = kw/kg - Mw /metric ton of fuel conversion factor, 3.22 X 10~" watt-sec /fission = fuel density, of fuel/cm3 of fuel m — mass of fuel in reactor, metric tons The other units are same as in Eq. (1). 1.22 Power density (Pd) the heat generation per unit volume of reactor core. The usual units of power density are kilowatts per liter or watts per cubic centimeter, which units are numerically equal. =

g

--

where

Pj

= = = =

Vc =

=

AvNa/C

=

V e

Pi

£

is

p

A

where P,

PCA


1

2

if,

where the subscripts 1 and 2 designate the different fissionable isotopes. Generally speaking, both Ni and
(6)

power density, kw /liter conversion factor, 3.22 X 10~M kw-sec /fission volumetric fraction of core occupied by fuel reactor power, kw core volume, liters

* By common usage in the United States, "enrichment" has come to mean the fraction or percentage of Um in a mixture of Um and U319. In some other countries, enrichment rnejins the factors of U,r' concentration above that in natural uranium; i.e.. 0.712 per cent U*« = 1.0 enrichment.

NUCLEAR-POWER-PLANT

12-80

2

SELECTION

[SEC. 12

NEUTRON AND FUEL BALANCE

The operation of a nuclear reactor necessarily implies the consumption of fissionable material. Concurrent with the consumption, it is possible to generate fissionable material from fertile material that is built into the reactor for such a purpose. Conversion and Breeding


2.1

If the neutron captured by fertile material originates from a fissionable nuclide different from the form into which the fertile material will change, the process is termed a conversion process; e.g., neutrons from U2" fissions are captured in U2" or If the neutrons originate from Th232, decaying to Pum and U2", respectively. fissioning Pu23' or U233 and by capture and decay reactions form new Pu23' or U"3, the process is termed a breeding process. Breeding is not always used in accordance with the preceding definition. The other meaning sometimes given to breeding is that of producing more fissionable material than is consumed. The gain in fissionable material is discussed in a later portion of this article. Depending upon the purpose for which the reactor is designed, the fissionable iso topes produced may be removed from the reactor to be used as fuel at some future time or they may be retained in the producing reactor to replace fuel being consumed. Reactor design should be such that neutrons are used in the most efficient manner in the generation of fissionable material. The efficiency influences the cost of the converted material or of the power generated. Thus, it is of interest to observe the history of neutrons in a critical, chain-reacting system operating under a steady-state condition. 2.2

The Neutron Balance

The following neutron-balance equation applies for any steady-state critical reactor regardless of classification according to neutron energy or fuel distribution: Neutrons lost by leakage from core per sec + neutrons absorbed by all materials in core per sec = neutrons produced in core per sec (7)

If the neutrons lost by leakage from the core are considered to be those which never return to the core, the above equation applies to both bare and reflected reactors. This equation exhibits no detailed information about the factors that contribute to its terms. Such detail is unique for each reactor design and depends upon such things as the materials used, the distribution of the materials, geometry, and the neutronenergy spectrum. If the core materials are homogenized, the neutron balance for a thermal reactor may be depicted graphically, showing in considerable detail the possible experiences of the reactor neutrons and the respective probabilities of the occurrence of these events (see Fig. 1). Table 1 gives a neutron cycle showing the possible range of events within the cycle. Table

1.

Neutron Balance

— Large,

Low-enrichment

Virgin fast neutrons from thermal fissions Fast neutrons lost by fast absorption Fast neutrons gained by fast fission Fast neutrons lost in leakage Resonance capture in fertile material Thermal neutrons lost in leakage Thermal neutrons captured in structural materials, moderator, reflector, Thermal neutrons absorbed by fission product Thermal-neutron capture in fertile material Thermal neutrons captured parasitically in fissionable material Thermal-neutron fission captures 2.3

Thermal

coolant,

Reactors

control. .

+ 100. 0 — 0 to 3 + 0 to 8 — 2 to 10 — 15 to 25 — I to5 — 5 to 15 — 2 to 5 — 10 to 20 — 6 to 8 — 40

Neutron Economy

2.31 Available Neutrons. In designing a critical breeding or converting reactor, the choice of core and reflector materials, the distribution of these materials, and the

Sec. 12-2]

12-81

OF NUCLEAR POWER

ECONOMICS

Critical

condition

ke-B2T

, l + L2B2" Thermol neutrons obsorbed in reactor

2n2 s—~ I + Lz Bz Thermal neutrons leaking from reactor

l-f

te'

Capture of thermol neutrons in "poisons" moderator, control rods,| fertile and fissionable motenal

kp,pr = ke-^ Neutrons thermalized in reactor

Thermal neutrons obsorbed in fuel

kP((l-Pr)

f(l-a)

Neutrons leaking out between resononce and : hernial energies

'Nonproductive capture of thermal neutrons by fuel

V.pPf

=

of

kP(

Neutrons which escape resonance copture

i)Ml-p)P,

Thermal neutrons captured in fuel leading to fissions

Neutrons captured in resonance region

Thermal


¥«Pf

a»f-i)f

Neutrons which enter resonance region

ijMi-P,) Neutrons ieoking out before resononce region

Fast neutrons produced by thermal fission

,f« Total number of fast neutrons produced by fission

Flo.

fission

3

Fost fission effect

Neutron cycle in critical thermal reactor. M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Noetrand Company, Inc.. Princeton, N.J., 1952. Source:

GUaatone,

1.

S., and

geometry of the reactor must be properly synthesized to meet the neutron balance indicated above. In order to achieve the maximum efficiency it is necessary to design so that a maximum number of neutrons is available for breeding or converting. Neutrons available for breeding or converting Neutrons absorbed in fissionable material neutrons produced by fissionable material neutrons absorbed in fissionable material neutrons absorbed by moderator, structure, fission-product poisons, control rods, and fissionable material neutrons absorbed in fissionable material neutrons lost from core by leakage neutrons absorbed in fissionable material

The last

(8)

term of the equation may be written Neutrons lost from core by leakage

neutrons produced by fissionable material

Neutrons produced by fissionable material

neutrons absorbed in fissionable material

(9)

NUCLEAR-POWER-PLANT

12-82

SELECTION

where L is the probability of neutron leakage from the core. hand side of the equation may be represented by „

but L = Hence,

1 —

P

where

P

_I

- hn - ,(1 - L) -

is the probability

[SEC.

12

Therefore, the right-

(10)

-f

that a neutron will not leak from the core.

Neutrons available for converting Neutrons absorbed in fissionable material

_ p_

1

/

It is obvious that the design should provide for a maximum /. If all the fertile mate If the rial is contained in the core, the design should provide for a maximum P value. reactor includes a blanket of fertile material, P must be optimized in order to maxi It is implied that the blanket is designed to yield a maximum mize the conversion. blanket catching efficiency that is defined as Neutrons captured by fertile material in blanket


Neutrons that leak into the blanket As fuel is irradiated, the amounts of the original 2.32 Reactivity Changes. This change is generally fissionable and fertile materials in the fuel will change. accompanied by reactor reactivity changes due to variations with time of the effective values of v, a, f, e. As a result, reactors fueled with a fissionable-fertile material mixture may lose reactivity even though, at the start of a fuel cycle, the production rate of new fissionable material from the fertile material may approach or equal the The reactivity loss can be at a rate destruction rate of existing fissionable material. that results in short irradiation periods of the fuel. Reactivity may also increase at the start of a fuel cycle. An initial gain does not necessarily mean long-term irradiations, for the gain may be small and the reactivity may subsequently decrease at a very rapid rate. When an initial reactivity gain is appreciable, a fuel-management problem results. To avoid a reactor-control problem, certain fuel may be removed after a short irradia Reactivity may also tion period and later reloaded when reactivity has decreased. be maintained by shifting fuels from one zone of a reactor core to another or by dis charging some zones in a reactor after short periods of time and others after longer periods. The management of fuel is a complex problem. However, fuel management is It is particularly important necessary if fuels are to be used in an efficient manner.* in achieving long irradiation cycles at a minimum concentration of fissionable mate rial in an initial fuel loading. Neutron losses by leakage and parasitic absorption in a reactor represent a loss of fissionable material in that such neutrons are not available for capture in fertile Equally detrimental to the fuel cycle are the physical losses to waste material. The actual physical losses of fertile and fissionable streams external to the reactor. material in the separations operations of irradiated fuels from a reactor or in the fabrication of fuel for a reactor should be less than 1.0 per cent. 2.33 Breeding or conversion ratio C is the ratio of the fissionable atoms formed from fertile material to the fissionable atoms destroyed by fission and capture. The words breeding and conversion have the same connotation as previously stated. Ia broad terms 1 I ij C (121

- - -

where I is neutrons lost by leakage and parasitic capture in all but fissionable niateriiis per neutron absorbed in fissionable material. For a thermal reactor having U2,s and U"s in the fuel, the

would be C<" =

£S7fia + vv"**a

- p)«-**

initial conversion

ratio

(13)

Sec. 12-2]

OF NUCLEAR

ECONOMICS

POWER

12-83

where ff„u"',
-

G = C

1

G = i)

or

when C >

- -I 2

1

(14) (15)

At thermal energies, and vFa"' = 2.03, 1 so that conver vv'" sion gains do not appear probable in thermal reactors. However, a breeding gain at thermal energies appears to have good possibilities using the U233-thorium cycle. From the thermal value of iju'3J it is observed that the term I may be relatively high and still leave the value of G to be a positive one. In the resonance energy region a increases over its value in the thermal region. As the neutron energy increases beyond the resonance region, a decreases until it is about 0.1 for all fissionable nuclides in the Mev region. Thus, in fast reactors the possibilities of both breeding and conversion gains are good. The possibility of breeding gain in a fast reactor is especially good for the Pu239-U23S cycle if the neutron spectrum can be kept high so that the average a is low. The neutron yield per fission Assume that v does not change with the v is about 2.9 for Pu"a at thermal energies. energy spectrum of the fissioning neutron; then if an a for Pu23S of 0.2 can be realized,


r,v"' = 2.31,

= 2.08,

the possibilities for an appreciable breeding gain are quite good. In a fast reactor if fertile material is present in the core, the fast effect t becomes appreciable especially for the nuclide U238. In reactors having a breeding or conversion gain, theoretically, the fuels can be utilized until all fertile and fissionable materials are destroyed. Where breeding or conversion ratio is less than unity, the ultimate utilization of fuel is limited by the ability of the reactor to convert fertile to fissionable material. This ultimate conver sion is the sum of the series rC + rC2 + rC* + ■ ■ ■ (16) which reaches the limit 1

-C

(17)

where r is the fraction of fissionable material in the fuel. The ultimate utilization of fuel becomes the sum of the fraction of fissionable mate rial in a fuel plus the amount of fertile material converted to fissionable material

fEq.

(17)], or

r +

For

1

-C

=

-

—!— 1 c

(18)

example, in a reactor operating on natural uranium and having a conversion ratio

of 0.8, 0.00712 1

-

0.8

0.0356

■epresents the ultimate fraction of the fuel that can be fissioned. The above example illustrates why reactors having a breeding or conversion gain are The interest is not merely from a viewpoint of possible economic advan>f interest. ages in the fuel cycle but also because the resources of fissionable and fertile materials ire markedly extended. With reactors producing more fissionable material than is lestroyed, the reason for stopping short of 100 per cent utilization of fissionable and ertile materials will be the build-up of nuclides such as U235. Such a nuclide is vasteful of neutrons to convert it into a fissionable form, yet this nuclide cannot be conomically separated from the more useful uranium isotopes U235 and U238. Howver, with a breeding or a conversion gain, over 50 per cent utilization of fuels should

realized. 2.36 Doubling Time.

>e

erm

With reactors having

doubling time is evolved.

Frequently

a gain in fissionable materials, the this has been defined to mean the length

NUCLEAR-POWER-PLANT

12-84

SELECTION

[SEC.

12

of surplus fissionable material equal to the quantity required in the initial fuel charge. In a more practical sense, the quantity of fissionable material considered in a calculation of doubling time must also include the quantity held up in the portion of the fuel cycle that is external to the reactor. This would cover fuels held in decay storage, processing, fabrication, ship Doubling time in days then ping, and reenrichment if the last two steps are required. of time required for a reactor to create a quantity

becomes

Fissionable material in total fuel cycle, g O

X

fission

rate, g/Mwd,

X

(1

+ a)

X

reactor power, Mw

It should be noted that high specific power and fast recycling are important factors the rate of increase of the stock of fissionable material. 3

WORLD

in

FUEL RESOURCES

The nuclear power industry is just beginning to establish itself. Some small-scale However, no large nuclear-power-plant or pilot plants are or have been operating. Conse experience is available from which cost and operating data can be extracted. quently, it is particularly important at this stage to present information on the total It is necessary to examine the current and potential competition energy picture. that nuclear energy faces from both resources and economics viewpoints. The world fuel resources are divided into three broad classifications: Generated for wjivans (University of Florida) on 2015-09-23 02:57 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

1.

Fossil fuels

2.

Plant and animal products

3.

Nuclear fuels

The other energy

sources

used or potentially 3.1

usable are discussed in Art. 3.8.

Fossil Fuels

Fossil fuels cover those fuels which are decomposition products from plant and/or animal life. This classification of fuels covers coals, peat, petroleum, natural gas. All these fuels, except peat, have been in the formation oil shales, and tar sands. Peat is a young, somewhat coal-like fuel whose forma process for millions of years. tion and rate of change are relatively fast. 3.11 Coal. The coals include, as broad groupings, anthracite coals, bituminous These fuels are, at present, the most important in the power indus coals, and lignites. Of the world Coals are widely, but unevenly, scattered throughout the world. try. reserves of coal, 95 per cent is believed to be in the Northern Hemisphere.* 3.12 Peat is unique among the fossil fuels in that its continual formation in surface deposits gives it the potential of an agricultural product that can be harvested. Although peat is an important domestic fuel in certain portions of the world, it has not been a significant contributor to the world's energy. Present indications are tha: it will not be a large energy source. Petroleum and Natural Gas. These fuels are closely associated, since a 3.13 Petroleum large portion of natural gas is recovered from petroleum-producing wells. The bulk of the known reserves are located in the North is found on all continents. American continent, the northern part of South America, the Middle East section in Asia, and in territories under the control of the U.S.S.R. The rate of consumption of these fuels is increasing very rapidly. Oil shales are a mixture of rock and bitumen. This material has not been, but is The mixture is mined and crushed, and the potentially, an important fuel source. Oil shales are of crushed rock heated in a closed vessel to distill off an oil product. interest not only because of the large quantity available but also because of the ura nium content of some of the shales.4 Tar Sands. The tar sands differ from oil shales in that the organic phase is a liquid The organics in tar sand are soluble in ordinary solvents, while dispersed with sand.

KCONOMICS

Sec. 12-2]

OF NUCLEAR

POWER

12-85

Tar sands are usually sticky and difficult to the solid bitumen in oil shales is not. No significant quantity of oil is being recovered from this source. The handle. material is of interest because of the large quantity of oil contained in known deposits.4" Plant and animal products have been and still are important contributors to 3.14 the world's energy supply. Fuel wood is the most important item in this classifica tion. Fuels of this classification are largely used as domestic fuels and make only a small contribution to the energy used in the power industry. In forested countries short of fossil fuels (e.g., Brazil) wood is a very important fuel. Animal wastes are an important domestic fuel in India. Plant and animal fuels have a record of decreas ing importance as energy sources because of their large bulk per unit of heat and of their higher monetary value for other uses, such as pulp, wood products, or fertilizer. 3.2

Nuclear Fuels

The U*" isotope contained in uranium is the only naturally occurring fissionable found in any appreciable quantity. The second isotope found in natural uranium, U,,s, is classed as a fertile material, since by the capture of one neutron and Thorium is also a naturally subsequent decay to Pu2" a fissionable nuclide is formed. By a similar neutron capture-decay process as in U"8, occurring fertile material. the Th"* goes to the fissionable nuclide U"s. Neither thorium nor uranium is a The earth's crust is calculated to contain about 4 ppm of uranium scarce element. and 12 ppm of thorium. This places the abundance of uranium above that of gold and silver. In relation to some of the more common metals, lead is estimated to be more abundant than uranium by a factor of 6, copper by a factor of 30, and iron by a factor of about 12,000.


material

3.3

World Input o' Energy

Putnam6 estimates that in the period 1860 to 1946 the annual world input of energy 0.017 to 0.093 Q.* The input of energy in the same period in the United States is estimated to have increased from 0.004 to 0.032 Q. The total cumulative energy input in the period a.d. 1 to 1850 has been estimated by Putnam to be about 9 Q, having reached a rate of 1 Q per century by the end of the The rate has continued to increase, and by 1950 the rate of energy input was period. 10 Q per century and the cumulative total from a.d. 1 had reached 13 Q. The projections' for energy demand within the period 1950-2050 are large. Popu lation growth plus increasing energy needs per person give requirements that will raise the cumulative total of input of energy into the range 80 to 500 Q by the year 2050. The world fossil-fuel resources (Table 2) total 158 Q.

from all fuels increased from approximately

3.4

United States Fossil Fuels Consumption

The

rate of fossil-fuels consumption in the United States is given in Fig. 2. The are plotted in a cumulative fashion. The vertical distance between any pair of curves represents the energy release for a particular type of fuel. The quantity of hydroelectric power is plotted with the fossil fuels, since it represents a saving of fossil fuel. The hydroelectric power is plotted on the basis of its fuel equivalent in a central station plant. The projected consumption beyond 1954 is taken from Paley

curves

Commission reports.'

The

projected increase in hydroelectric power is not great, since there is a relatively to be developed in the United States. The projected increase in petroleum consumption, though a vigorous one, is per;entage-wise not so great as in the past owing to an approach to saturation of the aassenger-car market. The projected consumption of natural gas is approximately at the same growth Gas has been a very cheap fuel for industries located in the large ■ate as petroleum. It has been piped in large quantity into the urban areas of the Southern gas fields. * 1 Q equals 10" Btu or is equivalent to i X 10" tons of coal.

small number of good sites still

NUCLEAR-POWER-PLANT

12-86

SELECTION

[Sec.

12

North and has been a low-cost domestic fuel. Heavy demand has been decreasing the cost advantage of this fuel. However, the convenience of gas as a domestic fuel is expected to keep the total demand increasing, even though its rising cost may cause industrial consumers to use other fuels. Figure 2 illustrates the position coal has had in the United States energy picture. In the period 1930-1950 oil and gas consumption increased steadily with only minor dips along the way. Consumption has increased by a factor of 2 for oil and by a fac tor of 3 for gas during the 20-year period. The rate of coal consumption has shown major setbacks in this 20-year period and by 1950 had recovered to a rate only 0.07

0.06


0.05

//

~ 0.04

/

/

- 0.03 O

'//"/ for, oal £ \

0.02

I'.',

'",

'//"

"/' '/'',

V/'/,-

f

s

ss

s

[

t *

"/// '/'//, ■;'/'/* '/
'//"}, *$'•!/<

2 1910

//

/

S

r'\i" 's/'A'sA'ji r"T',

V", /'///Y/s //,Y', /// ■///.;

/s

k

s

/

/

/s

m

/,//,

/

t

/

/

V*

1

o

0.01

/

/

/

1

f4

7—

1920

1930

1940

1950

1

I960

Year Fia.

2. Energy consumption

in the United States.

slightly higher than its rate for 1930. Factors such as strikes, depression, and the convenience of fluid fuels have caused the differences in growth. Coal represents the largest potential fossil-fuel energy source in the United States. As gas and oil reserves decline and as lower cost coal production and shipping methods are perfected, the energy derived from coal is expected to assume again the dominant position among the fossil fuels. In the case of all fossil fuels, discovery and production costs are increasing. Drilling and mining depths are increasing; more inaccessible locations, such as tidelands, are being explored. Advances are Thinner, lower grade coal scams are being mined. being made in the techniques and equipment for discovery and production that are

economics of nuclear power

Sec. 12-2]

helping to offset the difficulties brought on by depletion. expected that fossil-fuel costs will increase steadily. 3.6

12-87

However, it is generally

World Fossil Fuels Consumption

5

3.6

World Reserves of Fossil Fuels

a

2.

of

The world reserves The coal reserves are large fossil fuels are listed in Table but are concentrated in the United States (35 per cent), U.S.S.R. (23 per cent), and Paley Commission report6", about half of China (19 per cent). As pointed out in the listed United States reserves are inferred rather than proved. Though coal deposits have been worked and studied for many decades, much uncertainty still

2

a

a

it

1,

is

1,

is

is

1,

1,

is

2

is

if

exists as to the recoverable quantities. The estimated recoverable reserves range from 507 to 10 per cent' of the total. Most certainly new mining and recovery tech an appreciable quantity of the coal reserves to be niques must be developed recovered. The petroleum reserves listed in Table are estimated recoverable reserves. The 1955, estimate of proved recoverable world reserves as of January 153 X 109 bbls compared with the proved recoverable reserve of 98.5 X 109 bbl on January However, the average depth of all 1951. •* The rate of discovery still very great. wells drilled steadily increasing and the oil yield per well drilled has been decreasing. 1955, The proved recoverable United States reserve as of January 30.3 X 109," compared with 29.5 X 10s bbl as of January 1951. ** Actually, the United States has for some years imported more petroleum than has exported petroleum and petroleum products. The oil shales and tar sands are large potential source of liquid fuels. The bulk of these deposits are found in the United States and Canada. The oil shales are mined, crushed, and destructively distilled to recover crude oil. Ordinary solvents will not dissolve out the solid organic material contained in the rock. The quantity of The tech crude oil listed under shales in Table represents a recoverable quantity. At the present time only economic conditions niques for recovery have been solved. The big problem with oil shales prevent wide-scale recoveries of crudes from shale. will be the disposal of the large quantities of spent rock. The United States reserves involve about tons of spent rock to be disposed of per barrel (42 gal) of oil recovered. The tar sands are a mixture of a liquid or semiliquid and mineral particles. Oil can be extracted with common organic solvents or by flotation with hot water. The tar sands are sticky and consequently difficult to handle. The handling techniques for 2


Q.

3

A

5

3

2

is,

The world fossil-fuels consumption is not so well known as is that in the United States. Available production data and known conditions in other portions of the world lead to an appreciation of the possible magnitude of the growth pattern. Putnam's data5 shows that in 1950 the United States, with about 17 per cent of the world's population, consumed about 40 per cent of the world's output of energy. As living standards increase in other portions of the world, the energy requirements are expected to increase at a greater rate than the expected growth rate in the United A predicted trend is given in the crude-oil production and use figures listed States. The projected United States crude-oil usage in in a Paley Commission report.6* The projected crude-oil usage of the remain 1975 is a factor of 2.1 times that of 1950. der of the free world in 1975 is a factor of 3.7 times that of 1950. The rate of increase in petroleum consumption, though higher than the expected rate of increase of the nevertheless, total energy requirements, indicative of possible world consumption The projected increased energy requirements should living standards be improved. for the United States given in Fig. represent a rate of increase of about per cent Some reports indicate a per cent rate possible on a world per year compounded. wide basis. per cent rate represents a doubling every 25 years; a per cent rate, As a base line for projection, the free world's energy production in every 14 years. 1950 from coal and petroleum was of the order of 0.06

12-88

SELECTION

NUCLEAR-POWER-PLANT Table 2.

[SEC. 12

World Fossil-fuel Reserves Crude oilt Natural gast

Coal* Petroleum

Quan tity, short

value,

tons

Q

X 10-'

2.01 1 99

50.3 2 5

no

107

2.7

1.323 1.11}

33. 1 27.9

98

2^4

x North America: United States Canada Remainder (in cluding Alaska). Asia: U.S.S.R. (both Europe and


Asia)

China Middle East (in cluding Egypt) . . Remainder Europe (excluding U.S.S.R.) Germany United Kingdom . . Remainder Africa (excluding Egypt) Union of South

io-»

371 190 161

226 2 59 5

Remainder Australia South America Brazil

Heat

Quan tity, bbl

40*

Heat value, Q

0.6 0.2

150

0.9

155 54

0 9

13

0. 1

8

0. 05

8

0.05

Oil shale

Quan tity, bbl x io->

365

Heat

Quan tity,

3

Heat value.

value,

Q

X 10'

Q

x io-»

Q

2. 1

100 500

0.6 2.9

400

0 4

0.3

ft'

50

0.3

150

0 8

800

4.6

0 04

0.02

4 7

4.0

5.6 0.05 1.5 0. 1

300

5.767

Quan

tity, bbl

value,

35

9.3

Venezuela

Remainder World total

Heat

Tar sands

144.2

80

0. 5

618

3.6

668

1.7

3.8

1.8001

1.8

NOT«: The following conversion factors were used: 25 X 10* Btu /short ton of coal. 5.8 X 10" Btu/bbl (42 gal) of crude oil. I01 Btu/ft1 of natural gas. * "Bituminous Coal Annual — 1953," Bituminous Coal Institute, Washington, 1954. t E. Ayres and C. A. Soarlott. "Energy Sources—The Wealth of the World." McGraw-Hill Book Company, Inc., New York, 1952. t Includes all of North America excluding the United States. If P. C. Putnam, "Energy in the Future," D. Van Nostrand Company. Inc.. Princeton, N.J.. 1951.

the tar sands have not been solved to the degree that reliable predictions can be made as to recoverable quantities of oil. At the present time natural gas is a prime fuel resource only in the United States. The proved United States reserves as of December 31, 1954, were 211.7 X 10" ft*.' Ayres'" lists the ultimate reserves in the range of 200 to 400 X 10" ft* based on a The higher figure is shown in Table 2, since the proved reserve 1948 estimate. already exceeds the 1948 low estimate. 3.7

Reserves of Fissionable

and Fertile Nuclear Fuels

The U"5 isotope occurring to the extent of one part in 140 of uranium is the only The other significant isotope, Uu', is s naturally occurring fissionable material. fertile material, as is thorium. By The known reserves of uranium and thorium ores are not well established.

Sec. 12-2]


12-89

figures were released by the three major uranium-producing countries, the United States, Canada, and South Africa, indicating inferred ore reserves of about Good estimates of ultimate 1.5 X 10* tons containing about 825,000 tons of Us08. recoverable reserves are not available since major exploration for uranium is rela tively recent and little intensive search for thorium has existed. Areas of occurrence of major uranium ore supplies are known to be widely scattered The richest deposit is the Shinkalobwe mine in the Belgian throughout the world. At least three major ore areas are being worked in Canada, Great Bear Lake, Congo. and the Blind River-Algoma areas. The Colorado Beaver Lodge-Athabaska, plateau area in the United States has had a great discovery boom in recent years. Economically recoverable ores have also been located in Wyoming, South Dakota, The Witwatersrand and Orange Free Washington, California, and Pennsylvania. State gold fields in Africa are already producing significant quantities of uranium as a Australia has the Rum Jungle field in the north and the Radium Hill by-product. France, Ger Madagascar is reported to have important ores. field in the south. many, Portugal, Spain, and Israel of the free-world nations are producing or planning production from ores located within their continental borders. The Joachimsthal mine in Czechoslovakia has been producing uranium for many years. In addition, phosphate rock and shales are low-grade ores available in large quantity in many por By-product uranium is being recovered in the United States from tions of the world. phosphoric acid production processes and in Sweden from oil-shale processing. Thorium is available in significant quantities from the beach sands in India and Idaho has at least one active thorium mine, and by-product thorium is being Brazil. recovered from mining operations in some of the southeastern states. The magnitude of the uranium reserves must be judged by recent releases. J. E. Johnson, Manager of the AEC Raw Materials Division, announced in confer ences at San Francisco46 and Geneva, Switzerland,10 that the producing nations of the West had between 1 million and 2 million tons of uranium in high-grade ores based This uranium could be produced on present development and geological evidence. for an average cost of about $10/lb of U>08 in a high-grade concentrate. In the twenty-third semiannual report to Congress," the Atomic Energy Commission reports an inferred reserve of 76 X 10* tons of uranium ore averaging 0.27 per cent U308. The United States production rate of uranium from ore was 10,000 tons of U|08 in 1957 and a mill capacity of 20,000 tons per year should be reached in mid-1959. In addition, the United States alone has phosphate and shale deposits with a uranium content of between 5 million and 7 million tons. The cost of high-grade U808 from these sources may be between $30 and $50/lb. Known deposits of phosphate rock and shale in other parts of the world equal or exceed those of the United States in grade and tonnage. Johnson also reports that large tonnages of uranium exist which could be recovered The quantity of at intermediate costs of between $10 and $30 per pound of Ua08. uranium in these deposits may be as large as several millions of tons. The conclusion from the Johnson releases and from others at the International Conference on the Peaceful Uses of Atomic Energy at Geneva, Switzerland, in August, 1955, is that the known deposits of uranium and those inferred from the known deposits recoverable for $50 or less per pound of U808 already exceed the 25 X 10«-ton reserve estimate released by the AEC a few years ago. In April, 1955, J. E. Johnson44 stated that the low-priced uranium ($10/lb of U808) will be available until the annual demand exceeds 30,000 tons. This represents a If, because of enriching and large quantity of uranium on a power-reactor basis. holdup in various manufacturing and cooling stages, the average burnup of the uranium was only one-fourth of 1 per cent, the quantity of useful power at 25 per cent cycle efficiency would approximate the annual United States electrical production in


1957,

1953.

The thorium reserves are probably not well established, since the financial incen tives for discoveries of ore have not been furnished. Thorium reserves of 10s tons However, it is quite probable that much more thorium will be have been quoted.6 available at reasonably low cost.

12-90

NUCLEAR-POWER-PLANT 3.8

SELECTION

Other Sources of Energy

The sources of energy other than those covered previously 1. 2.

Wind

4.

Temperature differences Earth heat Waves and tide

6.

are as follows:

Water power Solar energy

3. 5.

[SEC. 12


For exam Obviously these classifications are not clearly distinct from one another. They are kept sepa ple, water power is a result of solar energy evaporating water. rate only for discussion purposes. 3.81 Water power is the only one of these energy classifications that, at present, yields significant quantities of power. In countries such as Norway and Sweden, the bulk of the power generated is by hydroelectric energy, and Norway possesses sufficient undeveloped hydroelectric sites that they can provide the bulk of the addi tional power demands for many years.11 In the United States Northwest, hydroelec tric power is the largest source of electricity. This is the only large geographical section of our country in which hydroelectric power is the dominant source of elec trical power, and even in the Northwest the best locations for hydroelectric power have been developed. 3.82 Solar energy strikes the earth at a rate of about 3.4 Q (1016 kwhr) daily. Energy benefits are already derived in the forms of the photosynthesis process neces sary to sustain plant life and of water evaporation, which eventually returns as rain. While solar energy is interesting because of its large quantity and "free" characteris tic, economic large-scale utilization has, as yet, not been made. Photosynthesis, using algae,6 shows promise of providing relatively high yield per acre and of being free of the seasonal growth aspect that exists with the agricultural crops. While such a process shows promise as a low-cost food source, the present stage of development indicates that it will be an expensive fuel source. Solar energy can be used directly as a heat source. Considerable work has been carried out on solar furnaces. Trombe1' has developed a 75-kw solar furnace. Appre ciable development efforts have been made on space heating by means of direct These have had some commercial importance in the Southern portions collection. Solar cookers of the United States, where climatic conditions are most favorable. suitable for home use have been developed. The use of such low-cost devices is especially important to tropic countries, such as India, that have poor fuel resources. One of the important recent advancements is the solar battery developed by Bell According to Chapin et al. the efficiency of these units has reached as Laboratories. high as 11 per cent.13 This represents a major improvement in the efficiency of con verting solar energy to electrical energy. At the present time the developments in solar energy indicate that specialty uses are However, much work remains to be done on the storage of solar energy practical. for use during nights or periods of unfavorable climatic conditions. Although the solar energy can be considered free, the means of converting it to electrical energy, as yet, are so costly that it is not being actively considered as a substitute heat source by electrical utilities. 3.83 Wind power has been utilized as an energy source for centuries. Windpowered pumps have been used in the Netherlands to keep out the sea water from diked coastal lands. Small wind-powered generators or pumps have been used on farms. Locations having steady, high-velocity winds are not sufficiently numerous to provide a significant percentage of world energy requirements. Tides, Tides, Temperature Differences, and Heat from Interior of Earth. 3.84 temperature differences between surface and deeper waters or between air and water, and the heat from the interior of the earth have been considered as energy sources. Where very high tides exist, serious consideration has been given to power projects.


Sec. 12-2]

12-91

No large-scale projects have been realized, though one at the Severn in southern England seems most likely. <0 The tidal project at Passamaquoddy Bay in Maine has Experiments have been a much-discussed power development in the United States. been made on power generation at ocean areas where appreciable temperature dif ference exists between the surface water and deeper waters near by. The French have experimented with this method using the Claude process. The good locations for utilization of this process are in tropic locations far from the energy-demanding Similar power considerations have been studied at sites where air tempera areas. tures are low and significantly higher ground temperatures can be readily obtained. Earth heat has been commercially utilized in Iceland and Italy, where it emerges at the earth's surface in the form of steam. However, the deep drilling of large holes necessary to capture earth heat is too costly for economically usable power. 4

CONVENTIONAL PLANTS— POWER COST


4.1

Definitions

In discussions involving the electrical generating industry, it is well to define certain terms that are unique to the industry. Generating Unit — A generator with its prime mover and energy source. Station — A unit or group of units at one location. System — The total integrated generating facilities of a utility. Capability — The maximum load that a unit, station, or system can carry under speci fied conditions for a limited time interval. This definition, while technically correct, has become somewhat obsolete. Common usage is tending to make capability synony mous with capacity. Capacity or Rated Capacity — The maximum load that a unit, station, or system can carry under specified conditions for steady, long-term operations. Net Capacity — The resultant power output from a unit, station, or system after subtracting the quantity of power necessary to operate the generating apparatus from the rated capacity figure. Plant or Capacity Factor — Ratio of the average generated load for a period to the total rated capacity of the unit, station, or system under consideration. Load Factor — Ratio of the average generated load for a period to the peak load for the same period. The peak load may be instantaneous or an average for periods of an hour or less. Utilization Factor — Ratio of the peak load for a period to the rated capacity of the Peak-load definition is the same as under Ix>ad Factor. unit, station, or system. Heat Rate — Amount of fuel burned in terms of Btu per kilowatt hour of electricity generated. 4.2

Development of the Power Industry

The progress of the electrical industry is best illustrated by the power costs, their history, and trends. Figures 3 through G,,9~" not only illustrate the competition that nuclear power is facing but also may illustrate a prospective trend for nuclear power itself. This figure, which Figure 3 is the U.S. Bureau of Labor Cost of Living Index. measures the price changes in commonly purchased goods and services, increased by In this same period of time, the average a factor of 1.33 from 1920 through 1953. unit revenue from the sale of residential power in 1953 was 37 per cent of the 1920 This reduction resulted because of both increased thermal efficiency figure (Fig. 4). (Fig. 5) and increased demand per unit of population (Fig. 6). Figure 5 shows the average station heat rate per net kilowatthour of electricity In terms of coal used, the average utility coal generated from 1920 through 1954. consumption in 1920 was 3.0 lb per net kilowatthour generated, and in 1954 it was A few of the most efficient plants were running at a rate of about 0.7 lb of coal 1.0 lb. The latest advances in boiler and turbine (13,100 Btu/lb) per net kilowatthour. designs are being incorporated in the Philo plant of the Ohio Power Company and in

12-92

NUCLEAR-POWER-PLANT

1920

1930

1940

SELECTION

1950

[Sec.

I960

Year

Fig.

3.

U.S. Bureau of Labor Statistics cost-of-living index.

6

1920

1930

1940

1950


Year

I960

Fio. 4. Average resale power cost

1920

1930

Fig.

1940

1950

I960

Year 5. Average

station

heat rates.

4000

3000 2000 1000

1920

1930

Fig.

1940

1950

Year 6. Average

electrical

consumption

I960

12

Sec. 12-2]

ECONOMICS

OF NUCLEAR POWER

12-93

of the Philadelphia Electric Company system. These are supercritical double-reheat units and are expected to produce power at a station heat rate of about 8500 Btu (0.65 lb of coal) of heat input per net kilowatthour of elec tricity generated. In addition to improving thermal efficiency, the steam plants have improved eco nomics by use of larger units that result in lower unit labor and maintenance costs. Cheaper types of building construction have been utilized, and even outdoor-type construction is commonly used today. a plant

pressure,

4.3

Cost of Electric Power


The power production costs are generally divided into two categories: (1) fixed A third factor of minor economic importance is the costs and (2) operating costs. administrative and general expenses, or overhead. The first factor includes the cost of money (interest on bonds, dividends on stocks), depreciation, ad valorem, and income taxes and insurance. The second factor covers the regularly occurring out-ofpocket expenses for supervisory, operating, and maintenance labor; fuel; materials; supplies; and miscellaneous items. 4.31 In an average situation, the privately Capitalization and Fixed Charges. owned utilities have their capitalization made up of 50 per cent stock and 50 per cent long-term debt, which is essentially bonds. The interest on bonds plus the expected dividend on stocks will amount to a fixed charge in the range of 4.5 to 6 per cent of The taxes are on an average about 60 per cent Federal the initial plant investment. income tax and 40 per cent other taxes. The other taxes are largely ad valorem taxes, although in some sections state income taxes are applied. The fixed charge to cover taxes is in the range of 4 to 6 per cent of the initial plant investment. Deprecia tion is in the range 1.5 to 3.0 per cent, assuming 5 per cent sinking-fund amortiza tion over a 20- to 30-year period. The insurance and miscellaneous items represent a small fraction of a per cent. The total annual fixed charges in privately owned utili ties is in the range 10 to 15 per cent of the plant investment. Table 3.

Average Steam-electric Unit Investment Costs* (Dollars per kilowatt of capability)

Plant capacity, Mw

Southern

<10

...

10-25 25-50 50-100 > 100 * Figures taken from Federal

locations

Northern locations

184 169 142 135

Power Commission

227 206 162 182

Average

252 207 181 151 155

data.

The average unit plant investment costs in steam elec 1946 to 1953 are given in Table 3. The table illustrates the influence of generating-unit size on cost. The average costs are essen tially constant at about $150 per kilowatt of capability for plant sizes above 50,000 kw. While the average figure has some value, the range of investment costs for these large generating stations is of more interest to a new industry such as the nuclear-power industry. In Southern locations some modern plants have been built with invest These plants can utilize lowment costs as low as $100 per kilowatt of capability. cost, fully outdoor type of construction, and the use of gas as fuel eliminates coal- and ash-handling systems. The highest cost, large, modern plants have unit investment These plants are generally located in the costs in the range of $200 to $225/kw. The plants are mainly coal-fired plants. The large metropolitan areas of the North. influences resulting in high costs are the equipment necessary for highest thermal 4.32

Investment Costs.

tric plants put into operation from

12-94

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

efficiency to combat high fuel cost, adverse climatic conditions, use of solid fuel, and the effects that densely populated areas can have on industries. Table 4 illustrates a breakdown of costs of steam-electric generating units. The columns high, low, and average are data from Electric World's Tenth Steam Station Cost Survey.14 These are figures from 28 plants having unit sizes over 75 Mw and put into operation in the period 1949 to 1956. The costs in Table 4 are listed per kilowatt of capability. These costs might be as much as 15 per cent higher if given on a net capacity basis. Table 4.

Cost Breakdown of Steam-electric (Dollars per kilowatt of capability) High

Item*

Structures (311)

Miscellaneous

plant equipment

(316)

Totalt


* "Uniform

System

15.0 84. 1 83.7 53. 1 23. 1 5.5 20. 1 238.7

of Accounts Prescribed

Unit Costs

Low

0.2 6.2 38.4 32.0

5.8 0.3 4.8 101 . 7

Average

2.0 29.5 58.0 40.4 10.8 2. 1

9.2 152.0

for Public Utilities and Licenses." Federal

Power Com

mission.

t Totals for high and low are the totals listed by J. J. Kearney in Tenth Steam Station Cost Survey. They are not the sum of the high and low components listed in Elcc. World, 148 (15) (Oct. 7. 1957). the columns above. 4.33 Operating Costs. The operating costs are usually divided into three sub headings: (1) operating labor, supplies, etc.; (2) maintenance labor and materials; and In large, modern plants the total manpower requirements are about (3) fuel costs. 0.5 man/Mw. The operating costs for subheading (1) are 0.3 to 0.4 mill/kwhr. The costs for subheading (2) are 0.2 to 0.3 mill/kwhr. Comparable costs in older or smaller plants would total 1.0 to 1.3 mill/kwhr for subheadings (1) and (2). Fuel costs are quite a variable, since this cost is dependent on the type of fuel, the location of the consumer relative to the fuel-production point, and the generating-unit efficiency. At certain locations in the large gas-producing regions of Louisiana and Texas, fuel gas is available at less than SO.10/106 Btu. Large, coal-fired plants, if located adjacent to areas in which large quantities of strip-mine coal are available, However, S0.20/106 Btu is generally considered can have fuel costs of the same size." This cost would cover a good fuel value for coal, even in the most favorable areas. In contrast, large plants requiring long rail hauls of coal or fuel handling charges. oil will pay $0.35 to $0.40/10" Btu for their fuel. Small plants in out-of-the-way locations may have fuel costs as high as Sl/108 Btu. The efficiency of any particular power-production unit also influences the effect of fuel cost on the power-production cost. Using the best current net thermal effi ciency of 37 per cent (station heat rate 9200 Btu/kw), fuel costing SO. 10 to $0.40/10* Btu would contribute between 0.9 and 3.7 mills/kwfir to the production cost of elec In older plants having a 25 per cent net thermal efficiency, the fuel influence tricity. for the same fuel-price range is 1.3 to 5.2 mills /kwhr. Roberts15 reports that for The administrative and general expenses vary widely. estimating purposes these costs may be taken at 15 to 25 per cent of the total annual production costs, exclusive of fuels. Another factor having a marked influence on power-production Plant Factor. The electrical load of a system varies widely on a daily and costs is the plant factor. A daily load curve given by Gaffert17 shows a variation by a factor of yearly basis. 2.2 between the high- and the low-load extremes in a large privately owned utility Similar curves for some municipally owned utilities show daily load variation by s

Sec. 12-2]

ECONOMICS

POWER

OF NUCLEAR

12-95

factor of 4. Utilities must provide generating capacity not only for the daily and peak loads but also to cover the planned and some of the emergency shut Since the generating capacity of a utility is appreciably above its average The average plant factor on a nation-wide basis is load, much equipment is idle. about 60 per cent. The most efficient units in a utility system are kept in operation as much as is practical, and those having lower efficiencies operated to take most of These efficient units or stations are termed base load units or the load fluctuations. An 80 per cent average is stations and have a plant factor of 70 per cent or higher. The least efficient stations may operate generally used in calculations for new units. with plant factors of 40 per cent or lower. Any change in plant factor gives an inversely proportional change in all powerproduction costs except fuel. Even fuel costs are affected to some degree by load Fuel is most efficiently used during steady operation at the rated capacity variation. of a boiler. Lower capacity or intermittent operations decrease efficiency. 4.34 Total Cost of Power. Table 5 lists broad parameters that influence the The power-production costs in modern, large, conventional steam-electric plants. high and low limits are not the extremes that may be found but rather limits that occur with reasonable frequency. The assumed heat rate is shown in a manner indicating in part, for higher thermal efficiency. Table lists that higher investment cost data the power-production costs based on Table 5

6

is,

seasonal downs.

Modern, Large, Conventional Steam-electric

5.

Table

Fuel cost, cents /million Btu Annual fixed charges,

%

Table 6.

High

Average

10 10.000 100 10 80

* Low and high heat rates arc not applied do not represent highest efficiency; high-cost

25 9.500 150 13 80

to low and high costs, respectively, plants do represent high efficiency.

40 9.000 200 15 80

since low-cost plant*)

Modern, Large, Conventional Steam-electric-plantPower-production Costs (Mills per kilowatthour) High

0. 0. 1.0

Administrative and general expenses* Total production costs

1.4 0.

2.8 0.

2.9

6.0

4.3 0.2 9.2

6

* Applied as 20 per cent of all operating

costs, excluding

4

5

0.

1

Fuel

1

0.6

0. 2.

.

0 3. 1 4 3 4

Average

2 2

(estimated)

Low

1

Operating costs: Operating labor, supplies

3.6 4.7

fuel.

COST OF NUCLEAR POWER PLANTS

In

is

a

it

has become standard practice to rate calculations involving nuclear reactors, This causes confureactor heat output in kilowatts (kw) or megawatts (Mw). reactor power system. To avoid also a part of .- ion when electrical generation fii-s confusion, the units kw or Mw will mean electrical power. The units used for x*»sactor heat output will be kw-heat or Mw-heat.

the

t


Low

Plant Data

12-96

NUCLEAR-POWER-PLANT

SELECTION

[Sec. 12

Published Costs of Nuclear Power Plants

5.1

For nuclear power plant units there is insufficient commercial construction and operating experience to determine costs. Most of the presently operating large reactors are primarily producers of fissionable materials only. The characteristics of these reactors are such that they are not a basis for estimating power reactor costs. The firms developing the nuclear power reactor industry are in the complex situation of evaluating and estimating untried designs in a period of rising costs. Conse The United States program involves work mainly quently, cost estimates are rising. in the areas of water-cooled and -moderated thermal reactors. Work is being carried out in organic-moderated reactors, sodium-cooled graphite-moderated, aqueous, and liquid-metal homogeneous reactors, and sodium-cooled fast reactors. The United Kingdom has committed itself to a major program of gas-cooled graphite-moderated The initial reactors in this development program are successfully power reactors. operating. The revised total cost estimates of some of the large power reactors under develop ment are tabulated in Table 7. These represent the basis for appraising the possible cost range for nuclear power plants. The range of costs is $327 to $689/kw. The figures reflect a large quantity of development costs and are expected to be reduced to a major degree after experience is gained. It appears quite reasonable to expect some second-round costs to reach $300/kw.


Table

Nuclear-power-plant Cost Estimates

7.

Electric

Reactor

PWR, oil-fired PWR'

power,

superheater"

Na-faste

HiO boiling, coal-fired

superheater''

Na graphite* HjO boiling' Organic/ Graphite-moderated gas-cooled*.

..

Mw

275 134 100 22 75 66 12.5 90

Total cost, millions of dollars

Unit cost, doUars/kw

90 57.5

""'Forum Memo, April. 1958, Atomic Industrial Forum. « Consolidated Edison Indian Point Reactor (includes $10,000,000 research not include core). b Yankee Atomic Energy (does not include core). * Power Reactor Development Co., Enrico Fermi Plant. * Elk River Boiling Reactor (includes research and development and core).

327 429 511 518 668 327 689 630

and development;

does

' Nebraska Sodium Reactor. of Piqua, Ohio. * Forum Memo, June, 1957, Atomic Industrial Forum, Northern StateB Boiling Reactor. » Herron, D. P., A. Puishes: "PWR and Calder Hall— How Do They Compare," Nudconict, vol. 15. no. 6: Cost of reactor built in United States; Comparable cost in United Kingdom — $425/kw.

/ City

5.2

Components of Nuclear Power Plant Costs

Some of the first nuclear power Land and Land Rights. 6.21 Account 310. plants may acquire fairly large plots of ground in order to eliminate a portion of the However, this personal liability that might result from faulty reactor operation. danger can also be eliminated by the "containment" principle, whereby the reactor is housed in a building capable of withstanding internal pressures of the order of Consequently, the substitution of a nuclear heat tens of pounds per square inch. source for a fossil-fuel heat source need bring about no significant change in the cost For evaluation or rough estimating purposes, $1 to $2 per of land and land rights. kilowatt of capability should be adequate. and Improvements. 311. Structures The items under this 6.22 Account account comprise the total building costs, including building services (e.g., drains,

Sec. 12-2]

OF NUCLEAR POWER

12-97

potable water supply) but excluding foundations for major equip reactors, boilers, turbogenerators, condensers). Also included in this account are roads, railroads, and yard improvements and services. In an average situation the conventional power plant having generating units of 50 Mw or higher will have an investment of about $30 per kilowatt of capability in this account. The substitution of a nuclear reactor for a conventional steam generator can appreciably affect the costs in this account. Most power-reactor proposals are making provisions for a reactor building that is capable of containing the pressure resulting from a sudden release of steam from the power-reactor unit. Such buildings are spherical or cylindri cal welded-steel structures capable of withstanding 10 to 30 psi internal pressure. For example, the structure housing the submarine test reactor (SIR) at West Milton, N.Y., is a sphere 225 ft in diameter. The boiling reactor (EBWR) at Argonne National Laboratory is in a vertical cylindrical building about 80 ft in diameter and 120 ft tall. A design by the Nuclear Power Group of Chicago uses a 200-ft-diameter spherical In these first stages of nuclearbuilding for a 180-Mw boiling reactor power unit. reactor development, the cost of the bare building is not the only cost factor which is The large steel buildings must higher than that used in conventional power plants. An internal have thermal insulation to prevent condensation on the inside walls. building lining may be required to protect the steel walls from fragments resulting from rupture of high-pressure equipment. A "shadow shield" (internal or external) may be required as biological protection in case the building becomes a radiation source following an incident. The cost of items in the 311 account for nuclear-reactor power units is probably in the range of $40 to $80 per kilowatt of capability for the first stages of nuclear-powerAs reactor technology advances; as improvements are made in plant development. mechanisms, instrumentation, and associated equipment; and as confidence develops with actual power-reactor operating experience, the total capital costs entered in account 311 should approach those of conventional power plants. 5.23 Account 312. Boiler or Nuclear Reactor Plant Equipment. This includes the cost of all items used directly or solely for the production of steam or other vapors In a coal-fired plant, this covers the to be used primarily for generating electricity. cost of the boilers with the fuel-, water-, and ash-handling equipment and the steam In a nuclear plant this account system for carrying the steam to the turbogenerators. number should include costs of the reactor with all its associated parts — moderator, reflector, shielding, controls, instrumentation, fuel-handling systems, and primary coolant systems. The cost of equipment used in any secondary coolant and boiler The cost of the initial charge of fuel is one reactor systems would also be included. item that should be kept as a separate account (Account 317), since it requires a different type of accounting. Essentially all equipment in the 312 account will be completely depreciated. In addition, the equipment that will become radioactive must carry a fixed charge reflecting the estimated cost of safe disposal, at the end of its lifetime, for a period as long as harmful radiation exists. The costs of items in account 312 are the ones that cause the greatest difference between the nuclear power units and the conventional power units. In conventional power units, the average 312 account costs are $60 per kilowatt of capability. The estimates for the comparable items in nuclear power units are in the range of $80 to sprinkler

ment


ECONOMICS

systems,

(e.g.,

$ 160 kw. 5.24 Account 314. Turbogenerator Units. This includes the cost of the turbo generators, the turbine condensers, and the accessory equipment associated with these

major

equipment items. Included also is the water-supply system for the condensers. classification of equipment is essentially the same for conventional or nuclear power units. However, there are certain deviations from the modern conventional jower plant practice that can and will affect costs in a nuclear plant. Most reactor ,ypes under consideration have operating conditions such that relatively low-pressure iaturated steam is generated by the reactor primary coolant. Most water-cooled eactors being considered generate steam at a pressure of 600 psi or less. The use of ow-pressure steam limits the maximum power output of a turbine. Thus, multiple

This


12-98

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

turbines may be required, and this affects cost. As an illustration, Gaffert's data:T indicate that the cost of a 100-Mvv turbogenerator unit of a tandem compound double flow type would cost about 20 per cent less than two 50-Mw units of the same type. In the case of turbines supplied with saturated rather than superheated steam, condi tions of excessive moisture are sometimes encountered in the low-pressure end of the turbine. Special blading capable of producing moisture separation, greater use of blading of special erosion-resistant metals, or external moisture-separation devices Another solution is to installed between stages of a turbine may be necessary. The Consolidated Edison reactor proposal19 uses an oil-fired superheat the steam. superheater. The liquid-metal-cooled reactors under study and development gener ally have steam conditions in the 700 to 1,200 psi and 700 to 900°F range. While these conditions are lower than modern conventional steam plants, the unit cost of turbo generators does not change much from these 700 to 1,200 psi units to units operating above 2,000 psi in modern conventional plants. The turbine condensers are in some cases more costly in a nuclear power system. More condenser surface is required for generating units having low cycle efficiency. In a boiling reactor system where the reactor-generated steam is led directly to a turbine, added precautions must be taken to keep to a minimum the cross contamina tion between the high-purity reactor cooling water and the relatively impure con Should heavy water be the primary coolant in a boiling reactor denser cooling water. system, the condenser must be even more elaborate in order to avoid excessive loss or isotopic dilution of the costly heavy water. Studies have indicated that it is eco nomically more sound to keep condenser construction and materials quite conven tional, to use ion-exchange resins to correct ionic contamination, and to use instrumentation to avoid major isotopic dilution. The cost of providing condensers constructed of highly corrosion-resistant materials such as the 300 series stainless steels is economically prohibitive. In the average conventional power plant, the costs are $45 per kilowatt of capability for the account 314 items. In early stages of the nuclear power unit development*, the cost is in the range of $45 to $80 per kw for account 314 items. This cost should certainly approach that of the conventional plants in future reactor development. 5.25 Account 815. Accessory Electrical Equipment. This covers the cost of auxiliary generating apparatus, conversion equipment, generator connections, and All items in this aceoun: main and auxiliary electric circuits at a unit or a station. are independent of the differences in the heat sources used for steam generation Consequently, the average conventional power unit cost of $10 per kilowatt of capa bility for account 315 items also applies to a nuclear power unit cost for similar item*. Account 316. 5.26 Miscellaneous Power-plant Equipment This includes all types of equipment in and about a generating unit that are not properly included solely in one of the previous accounts. Cranes, fire-fighting apparatus, compressed air, cleaning systems, locomotives, and ventilating systems illustrate the type of equipment covered in this account. Although differences may exist between the type and size or capacity of some of this equipment in conventional and nuclear power The average cost of $3 per kilowatt plants, the differences should not be significant. for 316 account items in a conventional plant is applicable to nuclear plants. 5.27 Account 317. Initial Fuel Charge. The Federal Power Commission has no The new account is instituted account 317 in their Uniform System of Accounts. since conventional power plants have no comparable item and since the annual fixed charges on fuel may be applied at a different rate from that applied on the other capital items. The quantity of fuel involved in an initial charge stays essentially constant through At present, the law does not permit private industry to out the lifetime of a reactor. own the fuel. Thus, in place of an annual fixed amortizing charge based on the value of the fabricated fuel, the AEC rental charge of 4 per cent annually must be appbed to the value of fuel in the form in which the AEC supplies it. In addition, it is neces sary to account for the decrease in fuel value due to loss of fissionable material and accumulation of fission products during the fuel lifetime. The fuel "as received" from the AEC will probably not be in its final, usable form. For example, uranium

Sec. 12-2]

ECONOMICS

OF NUCLEAR

POWER

12-99

To put this metal into a form may be obtained from the AEC as metal billets. acceptable for a solid-fuel reactor may require a large capital outlay for alloying, The fabrication charges for the initial forming, cladding, and assembly operations. fuel might then be amortized over the reactor lifetime. Should private ownership of fuel be allowed, each reactor unit will require a sepa rate examination of its accounting for the initial fuel charge. The initial fuel charge could conceivably vary from a fully depreciating one, with disposal and long-term "hot" storage costs, to an appreciating one. The former situation can exist for a The latter situa reactor using natural uranium and achieving long-term irradiation. tion might exist for a reactor using highly enriched fuels and having an appreciable conversion or breeding gain. 6.28 Account 343. Station Equipment This is the final account usually con This account covers the cost of equipment sidered in an estimate of power plants. DUU

500


o o

400

300

5>nn

0

50 Net

Fio.

100

150

electrical output -

mw

200

7. Costs for boiling-flashing power reactor.

required for transforming, conversion, and switching equipment that prepares the As with items in account 315, this type of equip generated power for transmission. ment is independent of differences in characteristics of the heat source. The average cost of $10/kw in conventional power plants is directly applicable to nuclear power plants. 6.3

Capacity of Nuclear Power Plants

The major nuclear power plant design efforts to date have been with the relatively large units of the order of 100 Mw or more. Most of the reactors in the AEC's Fiveyear Program are considered to be prototypes of the large generating units. However, Stacbler18 points out that in this group of five reactors, ranging from 2- to 60-Mw power and having estimated costs of $020 to $l,050/kw, the possible improved per formance leads to unit costs of 8375 to SGOO/kw. In addition, the Nuclear Power Group made a study20 of the influence of size on the unit cost of a boiling-flashing reactor system. This relationship is given in Fig. 7. The ultimate costs of commercial nuclear power units are difficult to determine until construction and operating experience is available on the AEC prototypes and the first-round commercial units. However, it is reasonable to assume that in a 10- to

12-100

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

15-year period the unit costs of nuclear power units of 100 Mw or more should approach The possible capital-cost the upper limit of costs for conventional coal-fired plants. picture for small power-output units (less than 10 Mw) is not so favorable for the nuclear systems. Conditions of criticality, neutron leakage, enrichment, and shielding are such that low-power reactors are not small reactors whereas low-power fossilfueled boilers are small boilers. Consequently, the low-power nuclear reactors do not appear to be competitive in the usual central-station application.

COST OF NUCLEAR

6

FUELS

The cost of nuclear fuels is not covered merely by the cost of the fissionable and In fact, the costs of fissionable and fertile material are quite proba fertile materials. bly a minor portion of the total fuel cycle costs. 6.1

Cost of Fuel Recycling

The flow diagram in Fig. 8 illustrates the factors that can enter into a heterogeneous reactor system where the fuel recycles through a solvent-extraction type of separation

I

Storage


Reactor

Ship

Flss. a Fert

Declod

Material

Alloy

Fabricate

Scrap

Metals

Fuel

Recovery

Dissolve

Gaseous Waste

1

Cladding Metal

To

Reduce Metal

Slag Recovery

Head End Treatment

Watte

Solvent Extraction

Solv. a Chem Recovery

Conver t

Tall

To

UF4

Treatment

Fig.

8.

End

Flow diagram for fuel recycling process.

The solvent-extraction system is illustrated, since this is the best of the plant. developed types of separations processes. Costs are not well established for the separations process suitable for the many varieties of fuels to be used in power-reactor systems. The AEC has published some information which is discussed in Art. 6.21 The factors that make up the fuel recycle costs are discussed, but the parameters ut too many and too broad to allow for much generalization. Using the flow diagram (Fig. 8) as the reference, the factors influencing the cost c: fuels are discussed starting with the irradiated fuel discharged from the reactor 6.11 Cooling and storage of fuels are a necessity. A few days must elapse befw most fuel can conveniently be handled in air without danger of fuel overheating as »

Sec. 12-2]

ECONOMICS

OF NUCLEAR

result of the decay of fission products.

POWER

12-101

Under conditions found in solvent-extraction

processes the cooling and storage period is weeks or even months to allow for decay The major cost of such nuclear species as Np2" (2.3 days) or Pa2'8 (27.4 days). The rate of effect from cooling and storage is build-up of the fuel-inventory value. irradiation degradation of organic solvents is also affected by the length of the cooling period.

The quantity of fuel held in decay storage is in the range of 10 to 50 per cent of the


annual reactor fuel requirements. 6.12 The influence of shipping costs is dependent on the particu Shipping Costs. Should the economics of a separations process allow lar separations process selected. the construction of a unit sized and built in conjunction with a reactor unit, shipping However, if the separations units are economical only in sizes requir costs are zero. The ing the fuel output from many reactors, shipping costs can be appreciable. shipping charges will be mainly from the weight of the heavily shielded shipping cas The currently developed solventkets, not from the weight of the irradiated fuel. Shipping extraction processes are not economical in sizes required by single reactors. charges are likely to be in the range $5 to $10/kg of fuel with a shipping distance of 2,000 miles. Decladding. The decladding operation can be a negligible 6.13 Reprocessing. cost factor provided the cladding has no metallurgical bond with the fuel. If no bond exists, it is possible to declad by mechanical means and have the cladding mate If a metallurgical bond exists rial remain as an easily stored, solid, active waste. between the cladding metal and the fuel, separation of the two is by chemical means. Some clad metals are difficult to dissolve, and, in any case, liquid active wastes result that are bulky and costly to store. In addition, storage of wastes in the liquid state is Only in a solid-state form can active wastes considered by many to be impermanent. be assumed to be permanently controlled in storage. Consequently, wastes must ultimately be reduced to a solid state for permanent disposal. The fuel-dissolving step is similar in nature to the decladding Fuel Dissolving. If the fuel is unalloyed, dissolving is not difficult and adds dissolution operation. little to the fuel-cycle costs. However, if the fuel has been alloyed for corrosionresistance and/or radiation-stability purposes, dissolution can be both difficult and slow. Head-end Treatment. The presence of alloying elements in the fuel is the most Head-end treatments are adjunct processes likely cause of a head-end-treatment step. necessary only when the main separation process is incapable of removing a particular element or of removing the quantity of a particular element that may be present. The cost of head-end treatments is felt not merely because it is an added processing step but also because it increases the quantity of active wastes. The separations process is the most critical step in the cost Separations Process. Solventpicture. Figure 8 illustrates this as being a solvent-extraction process. extraction processes are both proved and highly effective separation methods. In addition, they are flexible to a degree to achieve separation of chemically different fissionable elements, fertile elements, and fission products in very pure forms or to make only a relatively good separation of fertile-fissionable element mixture from the fission products. However, the solvent-extraction processes appear to be economical only when the plant capacity is high. From various sources11 chemical-processing plants having capacities in the range 60 to 350 metric tons per year might have invest Davis"" reports that the unit capital cost ment costs of $6,000,000 to $18,000,000. of the chemical plant varies inversely as the square root of the design throughput. A chemical separation plant set up to process fuels from a number of reactors can provide a favorable cost picture only if the irradiated fuels supplied by all the reactors are similar in composition, i.e., fertile and fissionable materials, cladding metal, and alloying additions. Without such fuel similarity, the economic benefits from highcapacity processing plants will be markedly decreased. 6.14 Pyrometallurgical Processes. Various high-temperature pyrometallurgical In very-long-term processes under development82 can improve the fuel-cost picture. irradiation, particularly where neutron flux is high, second-order reactions will

12-102

NUCLEAR-POWER-PLANT

produce 7-emitting thorium,

SELECTION

uranium, and plutonium

[SEC.

12

isotopes of moderate half-life


The presence of significant quantities of such iso (e.g., Th228, U232, Pu238, Pu"1). topes eliminates the possibility of "cold" refabrication procedures. If cold refabrication is not possible, separations need not have high-level decontamination factors. The treatment stage or stages for irradiated fuels then serve to ( 1 ) remove or reduce the concentration of fission products that impart deleterious effects to the physical properties of fuels, (2) repair possible damage caused by radiation (e.g., warpagc, swelling), and (3) adjust concentration of fissionable material in fuels. The high-temperature processes are not developed to a point where sound economics can be applied. They appear attractive, since the "hot " wastes products are very concentrated and, in at least some processes, the process steps of dissolving and later reducing uranium salts to metal or ceramic form are eliminated. 6.16 Tail-end Treatments. The tail-end treatments are "polishing" type of operations used only if high-purity end products are required or if certain highconcentration fission products or alloying agents have an excessive tendency to follow the recycling fuel stream. This step is not likely to be a significant cost factor in a power-reactor system. For most types of processing it will not exist. 6.16 Fuel Fabrication. The fuel-fabrication processes are in a state of develop ment similar to the separations processes. Fuel geometry and composition van Standardization of these items has not been evolved as yet. greatly. Consequently, consideration must be given to all features of fuel fabrication. Reactor designers quite naturally strive for high specific power (heat output per unit mass of fuel). This gives fuel elements having a high surface-to-mass ratio. I: also has a tendency to cause narrow coolant passages to maintain high coolant velocity while holding the flow quantity to reasonable levels. The narrow coolant passages, in turn, bring about rigid manufacturing tolerance on fuels. Reactor fuels are being used or developed in many forms. The solid-fuel forms are generally metals and oxides. Also being considered are metallic fuels dissolved or The oxides may be in dense ceramic forms or suspended suspended in liquid metals. in liquid metals or water. The salts may be dissolved in water or in other inorganit Fuels dissolved or suspended in fluids have little or no recycle step that can liquids. be termed fuel fabrication. A step may exist that involves preparation of the proper fuel salt or oxide for solution or suspension. However, the costs for homogeneousreactor fuel preparation are generally low, approximately Sl/kg. Solid reactor fuelhave been designed in any metallic or ceramic form that might be manufactured. The most common forms are rounds or flats. The metallic fuels have been tried in The cladding metals have been applied to fuel by alloyed and unalloyed condition. roll bonding, metallurgical bonding, or liquid bonding. Finally, fuel elements are generally assembled into clusters by fusing into rigid frames or by tying a number of elements together with spacers to hold the proper coolant channels. In some instances these clusters are assembled into an expendable coolant channel structure. No fuel form or composition has had any general acceptance. In considering the cost influence of fuels, it is necessary to study not merely the factors involved in fabricating fuels but also how these factors affect other portions of the fuel recycling. The materials used for fuel coolant channels may be reused provided the fuel cluster can be designed to allow easy disassembly and assembly by remote means. Fuel clad metal has less possibility of being reused after irradiation. The possibility exists only for fuels not having metallurgical bond with the clad. Metals used to alloy with the fuel will certainly go with the waste stream in a solventSome of the metals that might conceivably be used as alloying extraction process. agents are very expensive (see article on special materials) and can represent an appreciable operating cost. 6.17 Reenrichment. Another cost factor that may enter the fuel picture is the reenrichment cost for reactors using low-enrichment U236 fuels and having conversion ratios less than unity. The lowest cost fuel for such a reactor would be found by reenriching the depleted uranium in a gaseous diffusion plant. However, diffusion plants may not accept highly irradiated uranium, which would certainly either contain U238 or would have a higher ratio of U234 to U235 than is found in natural uranium.

Sec. 12-2] Reenrichment or plutonium.

OF NUCLEAR

ECONOMICS

POWEK

12-103

must, in such a case, be made by blending in highly enriched uranium These are more costly forms of fissionable material. 6.2

Cost of Fuel

As an aid in estimating the cost of recycling fuel, certain unit costs are available or have been approximated for some fuel situations. In 1955 the Atomic Energy Com 6.21 Cost of Natural and Enriched Uranium. The base price is $40/kg of uranium mission released the price of natural uranium. On November 17, 1956, AEC released the declassified schedule metal in billet form. Figure 9 of base charges for enriched uranium, as UF6, f.o.b. Oak Ridge, Tennessee. is a plot of these prices with the unit cost in terms of dollars per gram of contained 18

16

14


CM12

/

/

s //

>

s

,-""

."""

1 MO

z

o o

z

1 1 1 1 1 1 1 1 1 1

00072-0.90 00072-0.10

04 10 0.6 08 004 0.06 008 01 FRACTION OF U-235 IN URANIUM Fig. 9. Price range of enriched uranium. 0 0

02 002

U"s in the UF6. Costs for reducing UF6 to metal have not been established. These depend on factors such as order size, value of material, and criticality consideration. For estimating purposes the reduction cost may be in the range of 8'2/kg of uranium metal to $100 /kg of uranium metal in going from natural to 90 per cent enriched metal. The base uranium charges released by the AEC follow closely the results which may be obtained using the ideal cascade equation:

r-±[<*

yd

- i)

\x

- y)

-y)(l yd

2y)~

where V = value of contained U236, dollars per gram of U535 A = cost of a unit of separative work, dollars per gram of U5" = atom fraction of Um in product y = atom fraction of U"6 in tailings

i

NUCLEAR-POWER-PLANT

12-104

SELECTION

[Sec.

12

For purposes of the plot in Fig. 9, it was assumed that a negligible cost difference exists between uranium in the metallic state and in the hexafluoride state used in a diffusion plant. The loss of fissionable material is termed burnup, which covers losses due both to In fuels having both fissionable and fertile materials, new fission and to capture. The ratio of the integrated fissionable material will form from the fertile material. rate of formation of fissionable atoms over any period of time to the integrated rate of burnup over the same period of time is defined as the cumulative conversion ratio. The influence on power costs due to loss or gain in fissionable material can be deter mined from Fig. 10. Reactors having cumulative conversion ratios less than 1 must Those having cumulative conversion ratios purchase make-up fissionable material. Figure 10 gives results in terms of kwhr-heat. greater than 1 have a surplus for sale. It is necessary to divide this by cycle efficiency to determine the influence on electri cal cost.


10

|

1

1

1

1

1

8/gm Fia.

10.

1

1

1

1

1

1

1

1

Fissionable Material

Cost or credit for fissionable material.

The above treatment is a simplification. Changes in reactivity are governing in the quantity of fissionable material needed for make-up or available for sale. How ever, cost calculations based on the cumulative conversion rate should be acceptable, 6.22 Cost of Thorium. In January, 1956, the AEC announced a price of $43 kg for thorium metal. Thorium oxide may be purchased from commercial sources for about $25 /kg. 6.23 Cost of Produced Fissionable Materials. The AEC has announced that

guaranteed prices for plutonium metal will vary from $45 to $30/g with plutonium having PuS4° contents varying from 0 to 8.6 per cent, respectively. This price schedule lasts to June 30, 1962. For 1 year thereafter, all plutonium will be purchased at the flat rate of $30/g regardless of quality. No price is established beyond June 30, 1963. However, earlier releases had given an ultimate price of 812/g. The AEC has a guaranteed fair price of S15/g of 100 per cent U"' as uranyl nitrate. At lower purities it is assayed along with U236 and the price falls until uranium con taining 1 per cent U"5 plus Um has a value of $0.07 /g of total uranium." 6.24 Costs of Fabricating and Separating Fuels. The costs of both fabricating and separating are not commercially established. This results from a combination of lack of production experience in fabrication and lack of volume to interest private capital in separation.

1

Sec. 12-2]

ECONOMICS

OF NUCLEAR

12-105

POWER

Cost of Separation. Ohlgren, Lewis, and Weech"6 report estimated total separation costs of $13 to $37 per kilogram of natural uranium plus plutonium. This estimate is based on solvent-extraction techniques. Jackson and Sadowski,lc give a cost estimate for a solvent-extraction plant having more than twice the capacity of that The cost estimate for both plants is the same. given by Ohlgren et al. These process ing plants have capacities that represent the fuel output from reactors developing approximately 5,000 to 12,000 Mw-heat, assuming that the fuel was irradiated until 10,000 Mwd of heat had been liberated per metric ton of fuel. In March, 1957, the AEC gave an indication of fuel processing costs." A plant handling 1 ton/day of natural or slightly enriched uranium would have a $15,300 "standard daily processing charge." The plant would produce purified nitrate salts of uranium and plutonium. In the spring of 1958 the AEC announced a price of


20

|

1

1

1

20 11.

1

40

t/kg Fig.

1

Fissionable

1

1

60 plus

.

80 Fertile Material

i—T—i

100

Fabrication and /or process cost.

$5.60 /kg of uranium (<5 per cent U"5) for converting from nitrate to hexafluoride Also given was a price of $1.50/g of plutonium for converting from nitrate to metal. Cost of Fabrication. The various steps in fabricating any particular fuel element will have to be estimated step by step to determine the fabrication costs, since fuel assemblies vary widely in compasition and shape. Present indications are that suc cessful fuels should be developed which will have a fabrication cost not in excess of Costs of $25/kg are not unrea $100 per kilogram of fertile plus fissionable material. sonable to expect. These costs should be applicable for fuels of the slightly enriched range (< 3 per cent U"6). Cost of Separation plus Fabrication. Figure 1 1 gives the influence of unit fabrication and separation charges up to $100/kg of fertile plus fissionable material. As in Fig. That the unit costs for 10, the ordinate scale is in terms of mills per kwhr-heat. fabrication and separations may each be as high as $100 /kg is not in itself alarming. This places the emphasis on long-term irradiation of the order of 10,000 Mwd/metric ton or higher. Should recycle costs of fuel for a power reactor be 5200 /kg and the reactor unit have a net cycle efficiency of 0.25, the recycle cost at 10,000 Mwd/ metric Assuming also that a cumulaton would be 3.2 mills per kilowatthour of electricity.

form.

12-106

NUCLEAR-POWER-PLANT

SELECTION

[Sec. 12


tive conversion ratio of 0.8 is available and that the make-up fissionable material is replaced at a cost of $30/g, the burnup cost is 0.8 mills/kwhr, giving a total fuel cost This is not far from the fuel costs in modern steam-electric plants. of 4.0 mills/kwhr. For special-purpose thermal reactors having highly enriched, highly subdivided fuels, the fabrication costs may be a factor of 10 higher than the top range of Fig. 11. The final major factor in the fuel-cost picture is the Inventory Charges. 6.26 The first element to consider in inventory charges is the influence of inventory. Fuel-inventory annual charges are usually in a range of 4 to 15 percentage charge. The 4 per cent figure represents a charge that might per cent of the initial fuel value. be made if the government retains ownership of the fuel and makes only a rental A 15 per cent figure represents a private-utility ownership and full deprecia charge. tion of the fuel over a period of the order of 20 years. The second element in calcu It is lating inventory charges is determining the quantity of fuel held in inventory. necessary to count as inventory not only the fuel held in the reactor but also the fuel tied up in all stages of recycle. It is possible that cladding and alloying materials may also add a significant amount to inventory value. Figure 12 gives the inventory charge in terms of kwhr-heat as in Figs. 10 and 11. The inventory charge applied is 10 per cent. Different percentage charges are directly proportional.

10

15

S/gm

Fig.

12.

Fissionable

20 Material

Inventory charge.

In a single fuel recycle, physical losses of fertile and fissionable material from any process will most likely be less than 0.5 per cent. These losses are into waste streams. The fuel-fabrication and separation steps may have temporary losses in the form of Such materials will be recovered. scrap and slags containing valuable materials. It is necessary to analyze this situation carefully, since the costs of recovery from scrap and slags may add a significant amount to the fuel recycle costs. In reactor systems using heavy water, an additional cost factor is introduced both in inventory value, and, owing to losses, in operating costs. Although DsO is in no sense a fuel, it is generally considered in the economic analysis of the fuel cycle. The reason for this is that D20 has a marked influence on fuel in that it tends to improve conversion ratio and hold down enrichment. D20 as an inventory item is treated as a nondepreciating asset. Until operational data on D20 power-reactor units are available, a reasonable DjO loss rate is unknown. The annual loss rate most frequently assumed for estimating is 5 per cent of the system volume. The quantity of D20 required should be in a range of 1 ton per 2 to 10 Mw-heat. The presence of DjO in a reactor system will also add a cost for insurance against accidental loss. The magnitude of this cost has not been appraised. 7

COST OF SPECIAL MATERIALS

The special materials for reactor usage are divided into the classifications of (1) moderators and coolants, (2) internal structural or alloying metals, and (3) control materials. The special aspects arc taken to include materials that are rare, unique to nuclear reactors, or common materials that are costly in the form or purity required

Sec. 12-2]


The following notes on these materials contain by nuclear reactors. prices and some information of availability : 7.1

12-107 approximate

Moderators and Coolants


7.11 Graphite. ~81.10 to $l/kg ($0.50 to $0.75/lb) for tonnage lots of reactor grade, unmachined stock having a density of about 1.6 g/cm'. Supply can probably be made to meet demand without excessively increasing the price. 7.12 Heavy Water (D,0). ~$62/kg ($28/lb). Price released by the AEC; specifications presumed to be the 99.7 per cent grade of D20 listed by Selak and Finke." Probable current production rate over 100 tons /year. 7.13 Price depends on the Beryllium. ~$100 to $400/kg ($50 to $200/lb). Beryllium is a relatively scarce material, and the metallic form is form desired. difficult to produce and work into finished shapes. 7.14 Beryllium Oxide. ~$20 to $300/kg ($10 to $150/lb). The range covers powders to complex fabricated shapes. — $0.40/kg ($0.20/lb). Sodium. 7.16 Production rated in tens of thousands of tons annually. ~86/kg ($2.50/lb). 7.16 Potassium. Production probably several thousands of tons annually. 7.17 ~$4/kg ($2/lb) for eutectic alloy having Sodium-Potassium Alloy (NaK). 77 per cent potassium. 7.18 Bismuth. ~$5/kg ($2.25/lb) in tonnage lots. Total world production has been of the order of 1,000 tons /year. 7.2

Internal Structural or Alloying Materials

Zirconium (Low Hafnium Content). 7.21 $11 to $16/kg ($5 to $7.50/lb) for Zirconium sponge; ~$33/kg ($15/lb) for ingot; up to $1 10/kg ($50 /lb) for thin sheet. Planned production rates are a few million pounds per year. supplies are adequate. ~$65/kg ($30/lb) for pure ductile metal. Because of slightly 7.22 Vanadium. high thermal neutron absorption cross section, vanadium is not suitable for use in a thermal reactor core except as a minor alloy constituent. Supplies of this material should be adequate. 7.23 Molybdenum. ~$15 to $30/kg ($7 to $14/lb) for rod or sheets in small quantities. Usage in the core of thermal reactors is restricted because of fairly high Molybdenum has potentialities of highthermal neutron absorption cross section. temperature applications that may warrant its use as an alloying or cladding material. Supplies should be adequate. ~$175/kg (S80/lb). 7.24 Niobium. Niobium has potentialities as an alloying Niobium agent, although its thermal neutron absorption cross section is fairly high. Supplies at times have been inade is not only an expensive metal but a scarce one. quate to meet the demands of the steel industry. 7.3

Control Materials

7.31 Boron. ~$10 to $16/kg ($4 to $7 /lb) as boron carbide; $33/kg (S15/lb) as Supplies of the material are adequate. Wrought stain amorphous natural boron. less steel containing up to 2 per cent boron can bo obtained on special order. These The prices on these are being used for control and neutron shielding purposes. Acceptable stainless-steel boron castings have materials will run up to about $4/lb. The B10 isotope is available at prices been made containing up to 5 per cent boron. ranging from $3.10 to $8.60 /g of metal in enrichments ranging from 30 to 95 per cent B10. 7.32 Hafnium. ~$65/kg ($30/lb) of crystal bar. Hafnium is found associated The hafnium content of zirconium ores is 0.5 to 2.0 per cent of the with zirconium. At present all hafnium is recovered as a by-product from the zirconium content. Costs should be reduced as demand increases. purification of zirconium.

NUCLEAR-POWER-PLANT

12-108

SELECTION

[Sec. 12

This class of materials contains some elements having very 7.33 Rare Earths. Costs will run from about S4/kg for high thermal neutron absorption cross sections. a mixture of oxides having about 2 to 3 per cent of the high absorbers, samarium and The gadolinium oxides, to about $9,000/kg for a fairhy high purity gadolinium oxide. Prices are likely to supply of these materials should be adequate for control usages. remain high largely because of difficulty of separation and lack of tonnage demand.

COMPETITIVE

8

STATUS OF NUCLEAR POWER

is,

At the present stage of nuclear power, too little information is available to allow firm statements as to its probable economic status. The predictions are made based on the rapid progress made in the past and the promise that rapid progress will con The greatest uncertainty lies in the economics of the fuel cycles. Prime tinue. concern not with the basic price of natural uranium and thorium, but in the possible length of irradiation from reactivity and irradiation-damage standpoints, in the costs of preparing fuels for reactor use, and in the costs of separating the residual fissionable and fertile portions from the fission products. Near -future and More-distant-future Predictions

if

is

7

is

The PW'R System24 has been completed as the first large-scale nuclear power It has been estimated that the power production cost will be reactor in this country. about 52 mills/kwhr for the first core loading and will fall to about 15 mills /kwhr for In this period an expected increase in the electrical output the third core loading." from 60 to 90 Mw will aid in lowering this cost. The gas-cooled, graphite-moderated power-reactor system under development in the United Kingdom86 expected to have a power-production cost of about mills/kwhr from the second stage of develop ment of this type of reactor." The 7-mill figure arrived at by United Kingdom calculated by the accounting methods and probably represents 10 to 12 mills accounting system used for the PWR figures. Data — Nuclear Power Plants " Near future"

Concentration of fissionable material in fuel. c.'r Specific power. Mw. heat /kg fissionable materiul in cycle Irradiation level, Mwd, heat /metric ton fertile plus fissionable material. Cost of fuel, dollars /g of fissionable material* Fuel fabrication and separation costs, dollars /kg of fissionable plus fer tile material Minimum electrical capacity, Mw/generating unit Cycle thermal efficiency, % Plant factor, % Capital costs, dollars/Mw Annual fixed charges: Depreciating itemB, % Nondepreciating items, % is

*

assumed that regardless of concentration of fissionable It purest form (see Fig. 9). Assumed ultimate cost of fissionable material in pure form.

"

More distant future"

1-3

1-3

0.5-3.0

3.0-4

0

Predicted

5.000 I5»

20.000

150-300 100 25 80 250-400

100 100 35 80 200

I0t

15 12

15 12

material in fuel its cost

is

Table 8.

that of the

t

Nuclear power has its best possibility of competing in the central-station field when Table gives the basic data for the generating units are large (greater than 100 Mw). estimating this relative competitive status with time. The "near future" implies early, large nuclear reactors that are built after design, construction, and operating This experience has been gained on the first round of power reactors. of the order of 10 years. The "more distant future" implies not only more of the aforementioned experience but also time to bring in advanced-type power reactors having favorable is

8


8.1

Sec. 12-2]

ECONOMICS

fuel characteristics years.

OF NUCLEAR


12-109

This is probably of the order of

and high cycle efficiencies. 8.2

POWER

25

Fuel Influence

The quantity of fuel in inventory appears to fall in the range resulting in specific This broad range powers of 0.5 to 3.0 Mw-heat per kilogram of fissionable material. of specific power does not represent a similarly broad range for inventory charge because the low specific power is representative of low unit cost fuel such as natural uranium and the higher specific power, of the more highly enriched fuels. Conse quently, the inventory charge per kilowatthour of electricity is expected to vary by a factor of only about 2.5. While the inventory charges for fuels or fuels plus DiO are expected to be a significant addition to power-production costs in the first generation of power reactors, the importance of this factor should markedly decrease with later reactors. Higher specific power, shorter processing and fabricating periods, and longer irradiations should combine to bring this about. In the early operating stages of power reactors, it is not unreasonable to expect fuel irradiation levels of 5,000 Mwd-heat per metric ton of fissionable plus fertile material. This should advance to 20,000 Mwd-heat per metric ton in the more dis tant future. In early reactors, the cumulative conversion ratios are expected to be in the range 0.7 to 0.9 using the U"' cycle and perhaps as high as 1.1 if the U2" cycle is used. When the fast reactor emerges as a developed power-reactor type, cumulative conversion ratios as high as 1.6 will result in a significant quantity of salable surplus fissionable material from the reactor. Therefore, with time, the make-up fuel for some systems will change from a debiting factor on power-production costs to a credit ing factor. Operating and Maintenance

8.3

Costs

The operating and maintenance costs for the first power-reactor units will probably be a factor of two to three times the similar type costs for modern conventional elec As operating experience is gained and additional mechanical and cycle tric plants. efficiency improvements are made, the operating and maintenance costs should be of the same order as conventional power plants. Table 9 gives the estimated power-production costs based on Table 8 data. These costs include no element of research and development, nor is cost escalation considered. Table

9.

Power Reactor Electrical Production (Mills

"

Near-future" reactors

Fixed charge: Capital Fuel inventory Fuel fabrication and separation Fissionable material make-up. . Operating and maintenance. . . . Total

8.4

Costs

per kilowatthour)

Geographical-location

5.3-8.5 1.0-2.5 1.2—1.7

0.0-0.4

1.0-1.5 8. 5-14.6

"

Morc-distantfuture" reactors

4.3 0.5 0.2

-0.3

0.5 5 2

Influence

Nuclear power has, in its early stages of development, the potentiality of providing power at a cost competitive with conventional steam power in those areas of the The New England country having high power demand and fairly high fuel costs. area is probably the best example of a high-fuel-cost, high-power-demand area. With

growth

of power demands turoughout the United States, the non-fuel-containing areas Northeast, Southwest, and a portion of the Rocky Mountains will be favorable regions for nuclear-power units. The competitively unfavorable areas are in the

of the

12-110

NUCLEAR- POWER-PLANT

SELECTION

[SEC.

12

central portion of the United States, where natural gas and coal are still abundant, cheap fuels. The longer range future should find nuclear power in a competitive position in most The depletion of natural-gas reserves plus the high demand sections of the country. for this fuel for domestic heating and chemical uses are expected to remove it from its Petroleum or petroleum present low-cost classification even in the producing areas. Coal products are not a low-cost energy source now and should not be in the future. is still an abundant fuel and is cheap, provided high transportation costs are not In locations where power plants are near thick-vein, reasonably high-grade incurred. underground mines or near good strip mines, it is unlikely that nuclear power can economically compete for many years. These areas cover the Eastern and Middle Western coal fields, some portions of the Rocky Mountain states, and the large lig nite field located mainly in North Dakota.


8.5

Cost Reduction

A highly competitive position for nuclear power plants can be achieved by a com The most important of these factors are those involved in the bination of factors. fuel cycle. 8.51 Processes are needed that cost less for fabri Fabricating and Separating. cating and separating irradiated fuels in plants of relatively low capacity. Such work is under way, and some high-temperature processing techniques appear promising. The homogeneous reactors have no element of fabrication as do the heterogeneous reactors, and compact separations processes will probably develop that will become a part of homogeneous reactor design. 8.52 Equally important as a means of reducing costs will be Fuel Irradiation. This will be accomplished for solid-fuel reactors by develop long-term irradiations. ment of fuels having low irradiation damage characteristics. Such fuels are within A second step to achieve long-term irradiation is fuel management. sight today. This essentially consists of moving fuel assemblies within a reactor from time to time so that the average irradiation level of all fuels within a reactor core is about half of the discharge level for fuels. Homogeneous reactors do not have a fuel-management aspect in the same sense that heterogeneous reactors do. These reactor types can charge and discharge fuels continuously, and the rate is fixed by considering optimum of insoluble material in the fission-product build-up or preventing precipitation reactor. 8.53 Investment. Another major factor in cost improvement will be the reduc tion of construction costs. Nuclear reactors are in the early stages of development. There is no reason to expect that nuclear reactors should differ from the investment trends of other industries. The experience gained in design and construction of nuclear power systems and the technical improvements that will be made regularly will reduce the unit investment costs. Nuclear systems contain large quantities of Possibility of radioactivity, which is a new hazardous feature for industry to face. personal and property liability due to reactor malfunction has resulted in safety fea tures that have added significantly to the unit investment costs. As reactor hazards are better understood, as detection devices are developed, and as reactor operating experience is gained, the costs attributable to safety should decrease. 8.54 Table 9 lists the operating and maintenance Operating and Maintenance. Modern con charge on power production costs in the range 1.0 to 1.5 mills/kwhr. The higher costs for firstventional power plants have a charge of 0.5 mill/kwhr. generation power reactors will, in a large part, be chargeable to factors such as highlevel safeguard provisions and greater maintenance expense on these first units. It is expected that the operating and maintenance charge for nuclear power plants will be The cost reductions that will the same as for a comparable conventional power plant. result are reflected in the low power-production costs listed in Table 9. 8.55 Insurance. Insurance companies have established rates and procedures for coverage for injury to employees working on reactors and associated facilities and for damage to equipment. The premium on these types of insurance should have a

Sec. 12-2]


12-111

The insurance covering liability to persons and negligible effect on power costs. A group of stock companies are property outside a utility is handled in two ways. prepared to issue policies up to 350,000,000 and a group of mutual companies coverage The premiums on these policies are fairly to 810,000,000 for any one installation. appreciable and may exert an influence on power costs of a few tenths of a mill per kwhr. In 1957 an Insurance Indemnity Bill was passed providing a maximum The fee on this policy covers a Government indemnity of $500,000,000 per incident. charge for paper and administrative work only.


8.6

Special Reactor Application

The main emphasis thus far has been on large power reactors for central-station There are three broad classifications for applications for nuclear power applications. The first classification is where other than the central-station power-only variety. the possible low bulk value of nuclear fuels is of prime importance, the second where by-products of the reactor are of prime economic importance, and the third where the heat is used for purposes other than power generation. In sections where it is difficult and costly to 8.61 Special Power Reactors. transport fossil fuels, nuclear power reactors using highly enriched fuels can be of For example, a reactor using fuel having 10 weight per cent fissionable importance. material and operating to 50 per cent burnup of the fissionable material will require just 5 kg of fuel elements per year per megawatt of reactor heat. The fuel cost would The capital cost for a nuclear-reactor power unit using amount to 8 to 20 mills/kwhr. highly enriched fuel can range from $375 to $450 per kw.18 The total power-produc tion cost would be 15 to 50 mills/kwhr. Maritime uses are also possibilities for nuclear-power applications. 8.62 Cargo and oil-tanker ships have shaft horsepower requirements in the range from 5,000 to 25,000. These would take power-reactor units producing 4 to 20 Mw of electrical It is not likely that the capital costs of "near-future" power reactors, even power. at the 20-Mw electrical output, can equal conventional oil-fired boiler power units. Although the nuclear fuel costs may not exceed the fuel-oil costs, the capital cost of the system will exceed conventional boiler costs. Economic advantages for the nuclear power ships are possible dead-weight savings and cargo-volume increase resulting from the use of a nuclear reactor unit rather than a conventional boiler with its fuel oil. The best possibilities for nuclear power reactors in maritime service will be for the largest, fastest, long-haul service ships having a high load factor (fraction of total The class of ships best meeting these specifications are the oil tankers time at sea). The large, dry-cargo ship such as the hauling crudes on an international scale. However, the dry-cargo C-4 Mariner" is a possibility for nuclear-power application. ships do not have high-load-factor records as do oil tankers and thus cannot so readily Large passenger lines may tolerate the higher investment cost of a nuclear reactor. be good possibilities for nuclear power especially if the ships are designed for military use in times of emergency. 8.63 One of the By-product power reactors are another special classification. This reactor is frequently studied varieties is the so-called "dual-purpose" reactor. merely one in which fuel is irradiated to a degree set by a maximum concentration of some undesirable isotope in the fissionable material, e.g., concentration of U"4 in tjjjj or Pu*40 in Pu238. The tolerable radiation level for such specification fissionable material may be so low that a power utility could not afford the frequent shutdowns or the fabrication and separation charges from such a high fuel feed rate. The formed Should the demand for fissionable materials must, therefore, be premium-priced. such premium-grade fuels be high, the unit price can be correspondingly high, and utilities may find dual-purpose operation an economically attractive feature. Another profitable reactor by-product is the utilization of the radiation from the Much work is in progress on using the radiation for sterilizing foods fission process. At present, the sources of radiation and drugs or for promoting chemical reactions. have largely been from discharged fuel or specially irradiated materials such as

NUCLEAR-POWER-PLANT

12-112

SELECTION

[SEC.

12

However, direct reactor radiations could be utilized and thus provide an cobalt. income. Fission products recovered from irradiated fuels may become a source of income. It is quite conceivable that future developments may make the radiation and or the potential high-temperature characteristics of nuclear reactors sufficiently impor tant that certain chemical industries will use them to promote reactions.

REFERENCES la. Hughes, D. J., and J. A. Harvey: "Neutron Cross Sections," BNL-325, Brookhaven National Laboratory, Upton, N.Y., U.S. Government Printing Office, Washington D.C., July 1, 1955. b. Hughes, D. J., and R. B. Schwartz: "Neutron Cross Sections," Supplement Number Printing Office, Washington, D.C., January 1, 1957. 1, BNL-325, U.S. Government 2. Lewis, W. B.: "Some Economic Aspects of Nuclear Fuel Cycles," Paper 4. Interna tional Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. 3. "Bituminous Coal Annual — 1953," Bituminous Coal Institute, Washington, 1954. " 4a. Ayres, E., and C. A. Scarlott: Energy Sources — The Wealth of the World," McGrawHill Book Company, Inc., New York, 1952. Eng. Mining 6. Johnson, J. E.: Free World Uranium Resources — -Where Do We Stand? J., 156(5) (May, 1955). 5. Putnam, P. C: "Energy in the Future," D. Van Nostrand Company, Inc., Princeton


N.J.,

1953.

6a. The President's Material Policy Commission: "Resources for Freedom," vol. 2, The Outlook for Key Commodities, Government Printing Office, Washingon, D.C., 1952. 6. The President's Material Policy Commission: "Resources for Freedom," vol. 3, The Outlook for Energy Sources, Government Printing Office, Washington. D.C.. 1952. Yearbook," vol. 2, Fuels, 1952, Bureau of Mines, Department of the 7. "Minerals 8. 9. 10. 11. 12. 13. 14.

Interior, 1954. World Oil. Feb. 15, 1955. "Proved Reserves of Crude Oil, Natural Gas Liquids, and Natural Gas," vol.9, American Gas Association and American Petroleum Institute, Dec. 31. 1954. Johnson, J. E. : " Nuclear Fuel for the World Power Program," Paper 8/P/470. Inter national Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. Vogt, F.: "Energy Consumption in Norway," Paper 877, International Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. Daniels, F., and J. A. Duffie (eds.): "Solar Energy Research," University of Wisconsn: Press, Madison, Wis., 1955. Chapin, D. M., C. S. Fuller, and G. L. Pearson: The Bell Solar Battery, Bell LaM Record, 33 (7) (July, 1955). Kearney, J. J. (ed.): Tenth Steam Station Cost Survey, Elec. World. 148 (15) (On. 7, 1957).

"Progress in Peaceful Uses of Atomic Energy," July-December, 1957. U.S. Atonur Energy Commission, U.S. Government Printing Office, Washington, D.C.. Jan., I95& 16. Johnson, A. F.: Coal as a Source of Power for Production of Aluminum. Mining E*i. 7 (4) (April, 1955). 17. Gaflert, G. A.: "Steam Power Stations," McGraw-Hill Book Company, Inc.. Xe» York, 1952. 18. Staebler, Civilian Reactor Pro V. M.: "Objectives and Summary of U.S.A.E.C. gram," Paper 816, International Conference on the Peaceful Uses of Atomic Energy Geneva, August, 1955. 19. Power, 99 (5) (May, 1955). 20. "Boiling and Flashing Reactor Central Station Power Plant Design Study." NPG-l'.l Nuclear Power Group, Chicago, February, 1955. 21a. Davis, W. K. : "Capital Investment Required for Nuclear Energy," Paper 477 International Conference on the Peaceful Uses of Atomic Energy, Geneva, Auru^ 15.

1955. b. Ohlgren,

H. A., J. G. Lewis, and M. Weech: Processing Reactor Fuels. Xvdeonin (March, 1955). Jackson, H. K., and G. S. Sadowski: Design and Operation for Direct-maintenisoFuel Separation, Nucleonics, 13 (8) (August, 1955). Motta, E. E.: "High Temperature Fuel Processing," Paper 542, International Con ference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. 13 (3)

c. 22.

Sec. 12-2]


12-113

J. Finke: Heavy Water — A Review

P. J., and Large Scale Production, " 24. Simpson, J. W., et al.: Plant at Shipping Port, Peaceful Uses of Atomic

of Processes and Plants foi Chem. Eng. Progr., 50 (5) (May, 1954). Description of the Pressurized Water Reactor (PWR) Power Pennsylvania," Paper 815, International Conference on the Energy, Geneva, August, 1955. Nucleonics, 14 (1): 14 (1956). Hinton, Sir Christopher: "The Graphite-moderated Gas-cooled Pile and Its Place in Power Production," Paper 406, International Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. Jukes, J. A.: "The Cost of Power and the Value of Plutonium from Early Nuclear Power Stations," Paper 390, International Conference on the Peaceful Uses of Atomic Energy, Geneva, August, 1955. Witzke, R. L., and S. A. Haverstick: Nuclear Power Plants for Ship-propulsion Application, Elec. Eng., 74 (2) (February, 1955). Monthly Labor Review, Bureau of Labor Statistics (February, 1953). "Electric Utility Industry in the United States," Statistical Bulletin for the Year 1953, Edison Electrical Institute. Advance Release of Data for Statistical Bulletin 22, Edison Electric Institute, 1954. Nucleonics, 16 (2): 27 (1958). Sizing up Uncertainties in Nuclear Fuel Costs, Nucleonics, 16 (1): 19 (1958).

23. Selak,

25. 26.

27.

28. 29. 30. 31. 32. 33.


BIBLIOGRAPHY Glasstone, Samuel: "Sourcebook on Atomic Energy," D. Van Nostrand Company, Inc., Princeton, N.J., 1950. Glasstone, S., and M. C. Edlund: "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc., Princeton, N.J., 1952. Glasstone, Samuel: "Principles of Nuclear Reactor Engineering," D. Van Nostrand Com pany, Inc., Princeton, N.J., 1955. Stephenson, Richard: "Introduction to Nuclear Engineering," McGraw-Hill Book Com pany, Inc., New York, 1954.

12-3

POWER-GENERATING EQUIPMENT BY

Frank B. Daniels and Robert C. Allen 1

1.1

POWER CYCLES

Power Cycles for Nuclear Power Plants

Power cycles used in stationary nuclear power plants are of the same basic types as The choice of a cycle will depend used in power plants burning fossil fuels. upon the reactor coolant used, type of reactor, capacity, efficiency level desired, and other factors. Gas power cycles are an attractive and practical means of power generation with a nuclear heat source when the reactor coolant is a gas, such as helium, that does not If the gas does not con become radioactive in its passage through the reactor core. tain impurities that may become radioactive, it can be used directly in the power However, the use of any cycle, eliminating the need for a secondary heat exchanger. gas other than air will dictate the use of a closed cycle, requiring a precooler which will partially offset the advantages of eliminating the secondary heat exchanger. An open gas cycle requires the use of atmospheric air, which may present a health problem because of radioactive air being discharged from the turbine to the atmos phere. This might not be considered sufficiently serious to preclude the use of an open cycle for aircraft propulsion, but it probably would impose a stricter limitation For further discussion of the application of gas cycles for a stationary power plant. to nuclear power plants, refer to See. 13-4. The most common power-generation cycle today is one in which the cycle fluid It can be a simple, reheat, or regenerative cycle. passes through the vapor phase. In the simple, or Rankine, cycle, the fluid is heated, evaporated, and sometimes super heated in a boiler; expanded in a turbine or engine; condensed; and pumped back to It is the least complicated but also the least efficient of the three cycles. the boiler. In the reheat cycle, the cycle fluid is withdrawn from the turbine or engine after partial expansion, reheated to a higher temperature, and returned to the turbine or It is more efficient than the simple cycle, engine to complete the expansion process. but the necessary complication is believed not to be justified for a nuclear power plant In the regenerative cycle the condensate is partially heated in at the present time. An appreciable gain in one or more stages by fluid extracted from the turbine. efficiency is realized by this cycle at a relatively low increase in cost, but the justifica tion for its use will depend upon the value of the fuel saved and upon the effect of feedwater temperature on reactor operation. Of the various liquids that have been suggested for use in the power cycle, ordinary water appears to be the best because of its availability, its well-known and suitable thermal properties, ease of handling, and experience of operating personnel with its use.

it

is

is,

In indirect systems, that with two separate circuits used for reactor cooling and of standard construction except as power generation, the power equipment may steam conditions. In direct systems, in which a be influenced by unconventional used for both reactor cooling and power generation, some additional single fluid minor modification of standard equipment may be required to prevent escape of radioactive steam to the station atmosphere. It may also be necessary to take some extra precaution in the treatment of the reactor feedwater, such as the use of ion 12-114 is


those

power-generating equipment

Sec. 12-3]

12-1

15

exchangers and full-flow or bypass filters, in order to remove the products of corrosion and erosion and supply the reactor with extremely pure water. Heavy water may be desirable as the fluid in direct systems of large power ratings because of neutron economy in the reactor. It is also a satisfactory fluid for power generation because it is chemically the same as H20 and has nearly the same thermo dynamic properties on a molal basis. For properties of D20, see Tables 25 to 31 of Sec. 9-1. However, because of its high cost and the detrimental effect of even a slight amount of H20 in a D20 system, all points of possible leakage must be provided with special seals to prevent the loss of D2() and also to prevent contamination of the D20 These modifications of by inleakage of atmospheric moisture or other form of H20. conventional equipment designs are described later. 1.2 Examples

Typical Nuclear Power Plants

of current trends in design of nuclear power plants are shown in Figs.

1 to 3.


Figure 1 shows a diagram of the power plant of the Duquesne Light Company at Shippingport, Pa. The system is the indirect type, using H20 at 2,000 psi as the The secondary circuit consists of a conventional nonrcheat, primary coolant.

Fia. Pa.

1. Pressurized-water

nuclear power plant of Duquesne November, 1955.)

Light Company, Shippingport,

(From Westinghouee Engineer,

regenerative steam-turbine cycle operating with initial steam conditions of 600 psi The plant is nominally rated at 60 Mw with the expectation saturated. that it will ultimately operate at 100 Mw or more. Figure 2 shows the arrangement of the experimental boiling-water reactor plant in operation at the Argonne National Laboratory, Lemont, 111. The system is the direct-cycle type, using H20 as the cycle fluid. The power equipment is designed with special seals in case it is desired later to operate with DjO. The plant is rated at 5 Mw with initial steam conditions of 600 psi, dry and saturated, and with an exhaust pressure of 2% in. Hg abs. To provide the electrical output of 5 Mw, the reactor was designed for a heat output of 20 Mw but has actually been operated at over three This times this output, the surplus heat being absorbed in an auxiliary condenser. output has been obtained with natural circulation in the reactor; still higher output is expected when present plans for conversion to forced circulation are completed. Another application of a boiling-water reactor is the Dresden station of the Commontvealth Edison Company, near Chicago. It is a dual-pressure, direct-cycle plant, with

dry and

12-116

NUCLEAR-POWER-PLANT

SELECTION

[Sec. 12

steam generated partly in the reactor and partly in a secondary steam generator sup This arrangement is expected to increase plied with saturated water from the reactor. The rating of the plant is 180 Mw with the reactor output and improve the control. steam conditions of 1,000 and 500 psia, both dry and saturated, at the steam drum and the secondary steam generator, respectively. Figure 3 shows the general layout of the Atomic Power Development Associates, Inc. fast-breeder nuclear power plant being built in Michigan to deliver power to the The system is indirect with three circuits, the Detroit Edison power network.


CONTROL

ROOM

STEAM TURBINE AND ELECTRIC GENERATOR

STORAGE HOLES FOR RADIOACTIVE FUEL ELEMENTS

CONDENSER REDUCES STEAM TO WATEn FOR RETURN TO STEAM GENERATOR

Fio. teay

2. The experimental Allia-Cholmera.)

boiling-water reactor at Argonne National Laboratory.

(Cour

primary and the intermediate coolants being sodium. The third circuit uses HjO in a steam-turbine cycle. The plant output will be 100 Mw (turbine 150 Mw) with steam conditions of 600 psig, 730°F, and a condenser pressure of 1.5 in. Hg abs. Each of these installations includes a containment shell designed to withstand the maximum pressure resulting from the maximum credible accident. 1.3

Thermodynamics

The two most important fluid properties used 1.31 Thermodynamic Properties. computations, in addition to pressure and temperature, are the in thermodynamic The specific volume also must be known if flow areas are to enthalpy and entropy. be calculated.

Enthalpy of a substance at a given condition or state point is the energy possessed by unit weight of the substance by virtue of its pressure, temperature, and state of In the case of steam, the aggregation, measured above an arbitrary reference point. The unit of enthalpy in the reference point is taken as saturated water at 32°F. The Btu, or British thermal unit, is equal to English system is Btu per pound. 778.26 ft-lb and is approximately equal to ^go of the heat required to increase the temperature of water at atmospheric pressure from 32 to 212°F.


12-117

12-118

NUCLEAR-POWER-PLANT

Enthalpy

SELECTION

[Sec.

12

is defined by the equation

where h = enthalpy, Btu/lb of substance u = internal energy (in engineering thermodynamics, this is the sum of the potential and kinetic energies of the molecules), Btu/lb p = fluid pressure, psf v = specific volume, ft3/lb = conversion factor, 778.26 ft-lb/Btu

J

Enthalpy is a point function; its value at a given condition is independent of the In the conventional simple steam cyclt path traversed from the reference point. energy is supplied to the fluid in two forms. First, mechanical work is done on the liquid to compress it to the desired pressure, and then heat is added to arrive at the desired temperature and state. In this case, assuming saturated water at 32°F as the initial condition and no losses from the system, the enthalpy of the fluid at the given condition may be considered as the sum of the heat added and the heat equiva lent of the work of compression. Entropy of a substance at a given state point is defined by the equation

where s = entropy, Btu/(lb)(°Rankine) dQ = heat added in a reversible or frictionless process, T = temperature, "Rankine ( = °F + 459.7) = summation of increments from an arbitrary

Btu/lb

reference

point to given state

point

where hm = = = = =

il

- q)h,

- - q)hlt (1

fc„ =•

h,

h. -qlh

or from

+

is,

In the case of steam, the reference point corresponds to saturated water at 32°F, at which the entropy is taken as zero. The definition indicates entropy to be a function of heat transfer, a path function However, entropy itself is a point function, or true property, and may be used as a coordinate in plotting the other properties of a substance. A frictionless process in which no heat is transferred to or from the substance is » In an isentropic or reversible adiabatic process and occurs at constant entropy. constant entropy expansion the difference in enthalpies at the two end points is there fore the energy available for doing work in an ideal engine. 1.32 Steam Tables. The thermal properties of ordinary water and steam have been determined accurately over a wide range of conditions by extensive research, and a set of internationally accepted values has been adopted. Condensed tabulations of these values for the saturated and superheated states are given in Tables 1 and 2. The properties for intermediate values of pressure and temperature may be obtained by straight-line interpolation with sufficient accuracy for ordinary steam-cycle calcu for mixtures of water and vapor, may lations. Properties in the wet region, that For example be calculated from the proportions of water and vapor in the mixture. the enthalpy of a mixture at a given pressure and quality, or vapor /mixture weight ratio, may be computed from the equation

h„ q

enthalpy of mixture quality of mixture enthalpy of saturated vapor enthalpy of saturated liquid enthalpy change by evaporation h/t The entropy and specific volume of the mixture may be computed in like manner hf


.- Jof«T

POWER-GENERATING

Sec. 12-3]

12-119

EQUIPMENT

The available energy, or isentropic enthalpy drop between a given initial pressure pi and temperature ti and final pressure p2, also can be computed from the steam tables if desired. The initial enthalpy hi and entropy si can be read or interpolated from the tables corresponding to the given values of pi and ti. If si is greater than the entropy se* of saturated vapor at pressure p2, the isentropic end point is in the super heated region and the corresponding enthalpy h* can be obtained by interpolation If «i is less than s„2, the from the table for the values of pressure p2 and entropy si. end point is in the wet region and the enthalpy h^ can be calculated by the equation hi = h„2 — (s„2 — «i) (459.7 + k), where hg« and <2 are the enthalpy and temperature of saturated vapor at pressure p>. The available energy, or heat drop, is then hi — hi. Example: Given pi From Table

= 450 psia, d = 750°F,

=

1.5 in.

Hg abs.

2

hi

= 1387.3

s, =

From Table

p;

1.6481

1

h„, = 1101.7 = 2.0041 8„2

fj

Since «i is less than fta =

1101.7

-

<= 91.7


-

0.3561 X 551.4 = 905.3 905.3 = 482.0 Btu/lb

Available energy = 1387.3

A revision and extension of the present steam tables to higher values of pressures and temperatures is the subject of a recently established research program. For properties of heavy water, see Tables 25 to 31 of Sec. 9-1. 1.33 Enthalpy-Entropy (H-S) Diagram. The thermodynamic properties of a substance can be represented graphically in several ways. The most convenient for steam-cycle calculations is the Mollier diagram, in which entropy is plotted as the abscissa and enthalpy as the ordinate. One disadvantage is that the lines of constant specific volume are very nearly parallel to the constant-pressure lines over a large portion of the diagram, making reading difficult, and they are usually omitted. A portion of the H-S diagram for steam is shown in Fig. 4. An important advantage in using these coordinates is that the available energy between specified initial condi tions and final pressure is represented by the vertical distance between the two pres For example, using sures on the initial entropy line and is therefore easily determined. the conditions assumed previously and reading from the chart, = 450, U = 750 p2 = 1.5 in. Hg abs = 0.736 psia

Pl

S, = 1.648

Si

=

1.048

A, = 1387 905

h, =

Available energy =

482

Btu/lb

1.34 The Carnot Cycle. This is an ideal heat cycle consisting of four processes: Heat is supplied to the working fluid at a constant temperature 7\; the fluid is expanded adiabatically without friction (at constant entropy) from temperature Ti to a lower temperature T2; heat is extracted from the fluid at temperature r2; the fluid is compressed adiabatically without friction from temperature Tt to temperature T\. It is shown graphically on a temperature-entropy (T-S) diagram in Fig. 5 and on the H-S diagram in Fig. 6. The efficiency of the cycle is (7'i — Tt)/Ti. Although this cycle is not directly applied in actual engines, it represents the highest efficiency that can be attained by any heat engine and is therefore of value in judging the performance of other cycles. Actual steam power plants operate basically on the 1.36 The Rankine Cycle. Eankine cycle. In this cycle, heat is supplied in the steam generator at constant pressure pi, the steam is expanded adiabatically in the engine, unavailable heat is rejected in the condenser at a constant lower pressure p2 by complete condensation of the steam to saturated liquid, and the liquid is compressed adiabatically in a pump to the initial pressure pi. The cycle is shown on the T-S diagram in Fig. 7 and on the H-S diagram in Fig. 8. The efficiency is the ratio of useful heat to heat supplied and

12-120

NUCLEAR-POWER-PLANT Table Specific

Absolute

Hg Hg Hg Hg Hg Hg 1.0

2.0 3.0 4.0

Tables*

Enthalpy

volume

Internal energy

Entropy

°F

abs abs abs abs abs abs

Liquid

79.03 0.01608 91.72 0.01611 101. 14 108.71 115.06 125.43

0.01614 0.01616 0.01618 0.01622

101.74 126.08 141.48 152.97

0.01614 0.01623 0.01630 0.01636

162.24 170.06 176.85 182.86 188.28

Vapor

652.3 444.9 339.2 274.9 231.6 176.7

333.6 173.73 118.71

Vapor

Liquid

47.05 59.71 69.10 76.65 82.99

1049.2 1042.0 1036.6 1032.3 1028.6 93.34 1022.7

1096.3 1101.7 1105.7 1108.9 6 1116. 0

0.0914 1.9473

69.70 1036.3 93.99 1022.2

1106.0 1116. 2 1122.6 1127.3

0. 1326 0. 1749 0. 2008 0.2198

1131. 1 1134.2 1136.9 1139.3 1141.4

0.2347 1.6094 0.2472 1. 5820

Liquid

Evap.

llll.

Evap.

0.1147 1.8894 0. 1316 1.8481 0. 1449 1.8160 0.1560 1. 7896 0.1738 1.7476

Vapor

2.0387 2.0041 1. 9797 1.9609 1.9456 1.9214

Evap.

990 0

981.4 974.9 969.8 965.5 958.5

1. 9782 1. 9200 1. 8863 1. 8625

974.6

933.0

0.2581

1.5586 0.2674 1.5383 0.2759 1.5203

1.8441 1.8292 1.8167 1.8057 1. 7962

1.8456 1.7451 1.6855 1.6427

957 9

947.3

90.63

109.37 120.86

1013.2 1006.4

0.01640 0.01645 0.01649 0.01653 0.01656

73.52 61.98 53.64 47.34 42.40

130.13 137.96 144.76 150.79 156.22

1001. 0

38.42 35.14

979.3 976.6 974.2 971.9

1143.3 1145.0 1146.6 1148. 1 1149.5

0.2835 1.5041 0. 2903 1. 4897 0.2967 1.4763 0.3027 1.4638 0.3083 1.4522

1. 7876 1. 7800 1. 7730 1. 7665 1. 7605

911.1

28.04

161.17 165.73 169.96 173.91 177.61

982.1

12 13 14

193.21 0.01659 197.75 0.01662 201.96 0.01665 205.88 0.01667 209.56 0.01670

14.696

212.00

26.80

180.07

970.3

1150.4

0.3120 1.4446

1. 7566

897

969.7 967.6

1150.8 0.3135 1.4415 1152.0 0.3184 1.4313 1153. 1 0.3231 1.4218 1154.2 0.3275 1.4128 1155.3 0.3317 1. 4043

1. 7549 1. 7497 1. 7449 1.7403 1. 7360

896.7

5.0 6.0 7.0 8 0

9.0 10

II Generated for wjivans (University of Florida) on 2015-09-23 02:58 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Saturated-steam

[Sec. 12

Temp,

pressure. psi

1 in. 1. 5 in. 2. 0 in. 2. 5 in. 3.0 in. 4. 0 in.

1.

SELECTION

0.01672

32.40

30.06

996.2 992.1

988.5 985.2

939 3

927

5

922.7 918.4 914.6 907.8 904 8 901 9 899 3 5

15 16 17 18 19

213.03 216.32 219.44 222.41 225. 24

0.01672 0.01674 0.01677 0.01679 0.01681

26.29 24.75 23.39 22.17 21.08

181.11 184.42 187.56 190.56 193.42

20 22 24 26 28

227.96 233 07 237.82 242.25 246.41

0.01683 0.01687 0.01691 0.01694 0.01698

20.089 18.375 16.938 15.715 14.663

196.16 201.33 206. 14 210.62 214.83

960. 1

1156.3 1158. 1 1159.8 1161. 3 1162.7

0.3356 0.3431

1.3962 1.3811 0.3500 1.3672 0.3564 1.3544 0.3623 1.3425

1.7319 1.7242 1.7172 1.7108 1.7048

885.8 882 0 878 5 875 2 872.1

30 35 40 45 50

250.33 0.01701 259.28 0.01708 267.25 0.01715 274.44 0.01721 281.01 0.01727

13.746 11. 898 10.498 9.401 8.515

218.82 227.91 236.03 243.36 250.09

945.3

1164. 1 1167. 1 1169.7 1172.0 1174.1

0.3680 0.3807 0.3919 0.4019

1.6993 1.6870 1. 6763 1.6669 1.6585

869. 1 862.3 856.1 850.5 845.4

55 60 65 70 75

287.07 292.71 297.97 302.92 307.60

0.01732 0.01738 0.01743 0.01748 0.01753

7.787

256.30 262.09 267. 50 272.61 277.43

919.6

1175.9 1177.6 1179. 1 1180.6 1181. 9

0.4193 1.2316

1.6509 1.6438 1.6374 1.6315 1.6259

840.6 836.0 831.8 827.8 824 0

80 85 90 95

312.03 316.25 320.27 324. 12

0.01757 0.01761 0.01766 0.01770

282.02 286.39 290. 56 294.56

901. 1

1183. 1 1184.2 1185.3 1186.2

0.4531 0.4587 0.4641

1. 1. 1. 0.4692 1.

1676 1571 1471 1376

1. 6207 1.6158 1.6112 1. 6068

820.3 816.8 813.4 810.2

298. 40 305.66 312.44 318.81 324.82

888.8 883.2 877.9 872.9

1187.2 1188.9 1190.4 1191.7 1193.0

0. 4740 0.4832 0.4916 0.4995

1 1286 1. 1117 1.0962 1 0817 0.5069 1.0682

1. 6026 1.5948 1.5878 1 5812 1.5751

807 1 S01 . 795 6 790.2 785 2

100 110 120 130 140

327.81 0.01774 334.77 0.01782 341.25 0.01789 347.32 0 01796 353.02 0 01802

* Source: Keenan, Sons, Inc., New York,

J.

7.175

6.655 6.206 5.816

5.472 5. 168

4.896 4.652 4.432 4.049 3.728 3.455

3.220

965 5

963.6 961.9 956.8 953.7 950.7 947.9 939.2 933.7 928.6 924.0 915.5 911.6

907.9 904.5

897.8 894.7 891.7

868.2

0.4110

1.3313 1. 3063 1.2844 1. 2650 1.2474

1.2168 0.4342 1. 2032 0.4409 1. 1906 0.4472 1. 1787 0.4270

894.3 892.0

889.9 887.8

H., and Keyes, F. G.: "Thermodynamic Properties of Steam," John WiW A

1947.

power-generating

Sec. 12-3]

Table Absolute


pressure, psi

1.

Saturated-steam

Liquid

Vapor

3.015 2.834

Tables.

Liquid

Evap.

759.0

372. 12 379.76 386.98

828 5

1.5453 1.5372 1.5298 1.5229 1.5164

393.84 400. 39

809.0 803.0

1202.8 1203.4 1203.7 1204. 1 1204.3

0.5879 0.5952 0.6022 0.6090 0.6153

1.5104 1. 5046 1.4992 1.4941 1.4891

724.3

1204.5 1204.6

0.6214 0 8630 0.6356 0.8378

1.4844 1.4734

1204.4 1203.9 1203.2 1202 3 1201.2

0.6487 0.6608 0.6720 0.6826

0 6925 0.7371

1.4634 1 4542 1.4454 1.4374 1. 4296

659. 1

1200.0 1198.6 1197. 1 1195.4 1193.7

0.7019 0.7205 0.7108 0.7045 0.7194 0.6892 0.7275 0.6744 0 7355 0.6604

1.4223 1.4153 1. 4086 1. 4020 1.3958

649.4 630.4 611.7 593.2 574.7

1191. 8 1187.8 1183 4 1178.6 1173.4

0.7430 0 6467 0 7575 0.7711 0.7840 0.7963

0 6205 0 5956 0.5719 0.5491

1 3897 1.3780 1 3667 1.3559 1.3454

556.3

1167.9 1162. 1 1155. 9 1149.4 1142.4

0.8082 0.8196 0.8306 0.8412 0.8516

0 5269

1135. 1 1119. 2 1101. 1 1080.2 1054.8

0.8619 0.4230 0. 8820 0.3826 0.9023 0.3411 0.9232 0.2973 0.9459 0.2487

1 2849 1. 2646 1.2434 1.2205 1 1946

1020.3

0.9731 0.1885 1.0320 0.0532 1. 0580 0

1. 1615 1.0852 1 0580

417.33 423.29 428.97 434.40 439.60

0.01890 0.01899 0.01908 0.01917 0.01925

1.5433 1.4485 1.3645 1. 2895 1.2222

406.66 412.67 418.45

791.4 785.8

400 450

444.59

0.0193 456.28 0.0195

1. 1613 1.0320

424.0 437.2

780.5

500 550 600 650 700

467.01 476.93 486.21 494.90 503. 10

0.9278 0.8422 0.7698

449.4 460.8 471.6 481.8 491.5

755.0

750 800 850 900 950

510.85 0.0207 518.23 0.0209 525.26 0.0211

500.8 509.7

699.2 688.9 678.8 668.8

0.0201

0.0204 0.0205

531.98 0.0212 538.42 0.0214

0.7085

0.6554 0.6094 0.5687 0.5328 0. 5006

518.3 526 6

0.4718

534.6

0.4456

542.4 557 4 571 7 585.4 598.7

1000 1100 1200 1300 1400

544.61 556.31 567.22 577.46 587. 10

0.0216 0.0220 0.0223

1500 1600 1700 1800 1900

596.23 604.90 613. 15 621.03

0.0235

2000 2200 2400 2600 2800

635.82 0.0257 649.46 0.0268 662. 12 0.0280 673.94 0.0295 684.99 0.0315

0. 1878 0. 1625 0. 1407 0.1213 0.1035

3000 3200

695.36 0.0346 705.11 0.0444 705.40 0.0503

0.0858 0.0580 0.0503

3206.2

0.0227 0.0231

0.0239 0.0243

0.0247 628.58 0.0252

0.4001 0.3619 0.3293

0.3012 0.2765 0.2548 0.2354 0.2179 0.2021

780.5

0.5435 1.0018 0.5537 0.9835 0.5631 0.9667 0.5719 0.9510 0.5801 0.9363

300 320 340 360 380

0.0197

1. 5694 1.5640 1.5590 1.5542 1.5497

1198.4 1199.6 1200.6 1201.5 1202 3

0.01870 0.01880

0.0200

1.0556 1.0436 1.0324 1.0217 1 0116

Evap.

843.0

404.42 411.05

2.288

Vapor

355.36 364.02

2.087 1.9183 1.7748 1.65 II

0.01839

389.86 0.01850 397.37 0.01860

Evap.

0.5138 0.5204 0.5266 0.5325

381.79

2.532 2.404

Liquid

1194. 1 1195. 1 1196.0 1196.9 1197.6

200 220 240 260 280

2.675

Vapor

energy

863.6 859.2 854.9 850.8 846.8

358.42 0.01809 0.01815 0.01822 373.06 0.01827 377.51 0.01833

Internal

Entropy

330.51 335.93 341.09 346.03 350.79

ISO 160 170 180 190

363.53 368.41

12-121

(Continued)

Enthalpy

Specific volume Temp, •F

equipment

611.6 624. 1

636.3 648.3

835.6 821.8 815.3

797. 1

767.4 743. 1 731 6 720.5 709.7

538.0 519.6 501. 1

660.1

482.4

671.7 694.8 718.4

463.4

743.0 770. 1

337.2 284.7

802.5

872.4

217.8 62.0

902 7

0

424.4 382.7

934.4 902.7

0.5381

0.9225 0.9094 0.8970 0.8851

0.8738

0.8147 0.7935

0.7734 0.7548

0.5053 0.4843

0.4637 0.4433

1 3351 1 3249 1.3149 1 3049 1 2949

775.8 771.4 767.1 763.0

751 3 744.1 737.3 730 7

718.3

712.4 706 8

701.3 695.9 683 2

NUCLEAR- POWER- PLANT SELECTION

12-122

Table 2.

Superheated-steam

12

Tables*

Temperature of steam, °F

Pressure. psia

(Saturation temp, °F)

500

550

600

650

700

750

800

900

32.35

1000

25 (240 07)

V 22.74 h 1286.2 B 1.8668

23.95 1310. 1 1.8911

25. 16 1334.1 1.9144

26.36 1358.3 1.9367

27.56 1382.6 1 9581

28.76 1407.3 1.9785

29.96 1431.9 1 9988

1481 9 2 0371

34 74 1532 S 2 0732

50

V 11.309 h 1283.9 s 1.7887

11.922 1308.2 1.8133

12.532 1332.5 1.8368

13. 139 1356.9 1.8593

13.744 1381 4 1. 8809

14.348 1406. 1 1.9017

14.950 1430.9 1.9219

16. 152 1481. 1 1.9602

17 352 1532.1 1 9964

7.496 V h 1281.5 a 1.7421

1306.2 1.7672

8.731 1355.4 1.8137

1380.1 1 8354

9.543 1404.9 1.8564

1429.9 1.8766

10.752 1480 3 1,9151

11.555 1531 5 1.9514

5.589 V h 1279. 1 s 1 7085

8.052

1304.2 1.7340

4.445 V h 1276.7 1 682 s

1303.2 1.7079

1327.4 1.7323

3.681 V h 1274. 1 a 1.6599

3.899 1300. 1 1.6862

1325.7 1.7109

2.726 V h 1268.9 s 1. 6240

2.895 1295.8 1.6513

(281.01) 75

(307.60) 100 81)

(327

125

(343.53) 150


[SEC.

(358.42) 200

(381.79)

1.5949

300

V 1.7675

h 1257.6 8

1.5701

T 1.4923 h 1251 5

8.323 1330.8 1.7910

6.218

6.527

9.138

9.947

1329. 1 1.7581

1354.0 1.7810

6.835 1378.9 1 8029

7. 141 1403.8 1.8240

7.446 1428.9 1.8443

1479 5 1.8829

8.656 1530 6 1.9193

4.956

5.205 1352.5 1.7554

5.454 1377.6 1.7775

5.700 1402.7 1.7987

5 945 1427.9 1.8191

6 433 1478.7 1.8579

6 918 1530 1 1 8944

4.323 1351. 1 1.7343

5.352

1376.3 1.7566

1401.5 1.7779

5.758 1529 4 1 8740

1322. 1 1.6767

3.221 1348.0 1. 7006

3.380 1373.6 1.7232

1399.2 1.7448

1424.8 1.7655

1291.4 1.6234

2.427 1318.5 1.6495

2.559 1344.9 1.6739

1371.0 1. 6969

2.816 1396.9 1.7188

1422.7 1.7397

1474.5 1 7793

1.8891 1286 8 1 5998

2.005 1314 7 1.6268

2. 118 1341.8 1.6517

1368. 3 1.6751

2.336 1394 6 1.6972

2 442 1420.6 1.7184

1472.8 1.7582

1.7036 1310.9 1.6070

1.8021 1338.5 1.6325

1.8980 1365.5 1.6563

1.9916 1392. 1 1.6788

2 084 1418.5 1.7002

1471. 1 1.7403

2.445 1523 8 1 7777

2.292

h 1263.4

(417.33)

5.906

4.702

2.151

250

(400.95)

7.912

4.113

3.060

4.532

2.688

2.227

4.738

3.537

4.944 1426.9 1.7984

3.693

2.942

1477.8 1 8374

4.002 1476.2 1.8048

3.192

2.652

2.266

4.309 1528 0 1.8415 3.439 1526 6 1.8162 2.859 1525 2 1 7954

a

1.5481

1.6020 1282. 1 1.5792

h s

1.2851 1245. 1 1.5281

1.3843 1277.2 1.5607

1.4770 1306.9 1.5894

1.5654 1335.2 1.6155

1.6508 1362.7 1. 6398

1.7342 1389.7 1. 6626

1.8161 1416.4 1.6842

1.9767 1469.4 1.7247

2 134 1522 4 1 7623

450

V 1. 1231

(456.28)

h 1238.4

1.2154 1272.0 1.5437

1.3005 1302.8 1.5735

1.3810 1331.9 i . 6003

1.4584 1359.9 1.6250

1 5337 1387.3 1.6481

1.6074 1414.3 1.6699

1.7516 1467.7 1.7108

1 8928 1521 0 1 7486

1.0798 1266.7 1.5279

1. 1591 1298.6 1.5588

1.2333 1328.4 1.5863

1. 3044 1357.0 1.6115

1.3732 1384.8 1.6350

1.4405 1412. 1 1.6571

1 5715 1466 0 1.6982

1 6996 1519.6 1 7363

1.0431 1294.3 1.5451

1.1124 1324.9 1.5734

1. 1783 1354.0 1.5991

1.2419 1382.3 1.6228

1.3038 1409.9 1.6452

1.4241 1464.3 1.6868

1.5414 1518 2 1 7250

0.9463

1. 01 IS 1321.3 1.5613

1.0732 1351. 1 1.5875

1. 1324 1379.7 1.6117

1. 1899 1407.7 1.6343

1.3013 1462.5 1. 6762

1 4096 1516 7 1 7147

350

(431.72) 400

(444.59)

a

1.5095

500

y 0.9927

(467.01)

h 1231.3 a 1.4919

550

V 0.8852 h 1223.7 a 1.4751

0.9686

0.7947

0.8753 1255.5 1. 4990

(476.94) 600 (486

21)

* Source:

Sons,

1261.2 1.5131

h 1215.7 a 1.4586

Kecnan, Inc. New York,

J.

1289 9 1 5323

H., and Keyes. F. G.: "Thermodynamic

1947.

Properties of Stcarn.'

John Wiley A


Sec. 12-3]

Table

Superheated-steam

Tables.

(Continued)


Pressure, psia (Saturation temp, °F)

500

550

600

650

700

750

800

900

1000

....

0.7277 1243. 2 1 4722

0. 7934 1280. 6 1.5084

0.8525 1313.9 1.5391

0.9077 1345.0 1.5665

0.9601 1374 5 1. 5914

1.0108 1403 2 1.6147

1. 1082 1459.0 1.6573

1. 2024 1513.9 1.6963

h a

0.6154 1229.8 1.4467

0.6779 1270.7 1.4863

0 7328 1306. 1 1 5190

0.7833 1338 6 1.5476

0.8308 1369.2 1. 5734

0.8763 1398.6 1. 5972

0 9633 1455.4 1.6407

1.0470 1511.0 1.6801

(531.98)

h a

0. 5264 1215 0 1.4216

0. 5873 1260. 1 1 4653

0.6393 1298.0 1.5002

0.6863 1332 1 1.5303

0. 7300 1363.7 1. 5570

0.7716 1393.9 1. 5814

0.8506 1451.8 1.6257

0.9262 1508 1 1.6656

1000 (544.61)

v h a

0 4533 1198.3 1 3961

0 5140 1248 8 1.4450

0. 5640 1289.5 1.4825

0.6084 1325.3 1.5141

0.6492 1358. 1 1.5418

1.6878 1389.2 1. 5670

0 7604 1448. 2 16121

0.8294 1505. 1 1.6525

1200 (567. 22)

v h a

0. 4016 1223. 5 1 4052

0.4498 1271.0 1.4491

0.4909 1311.0 1. 4843

0. 5277 1346.4 1.5142

0.5617 1379.3 1.5409

0. 6250 1440.7 1.5879

0.6843 1499.2 1.6293

h ..

0.3174 1193.0 1 3639

0. 3668 1250.6 1 4171

0. 4062 1295.5 1.4567

0.4403 1334 0 1.4893

0.4714 1369. 1 1. 5177

0. 5281 1433. 1 1.5666

0. 5805 1493.2 1. 6093

V !) . . a

0.3027 1227.3 1.3850

0.3417 1278.7 1.4303

0. 3743 1320.9 1. 4660

0.4034 1358.4 1.4964

0.4553 1425.3 1.5476

0. 5027 1487.0 1. 5914

v h a

0.2506 1200.3 1.3515

0. 2907 1260. 3 1. 4044

0 3225 1307.0 1.4438

0.3502 1347. 2 1.4765

0. 3986 1417.4 1.5301

0.4421 1480.8 1. 5752

v .. h 8

0. 2058 1167.0 1.3139

0 2489 1240.0 1.3783

0 2806 1292.0 1.4223

0.3074 1335. 5 1.4576

0.3532 1409. 2 1.5139

0.3935 1474.5 1. 5603

0 1686 1176. 8 1 3073

0. 2032 1249.7 1. 3689

0. 2294 1303. 6 1.4127

0. 2710 1387. 8 1.4772

0. 3061 1458. 4 1 5273

0.0984 1060 7 1 1966

0. 1485 1197.8 1 3128

0. 1760 1267. 2 1. 3690

0.2159 1365 0 1.4439

0.2476 1441.8 1 4984

0. 1295 1171. 7 1 2869

0 1583 1250. 5 1. 3508

0. 1981 1335 2 1.4309

1 2288 1434 7 1. 4874

700 (503. 10)

800 (518.23)

h

900


2.

12-123

1400 (587. 10)

1600

(604.90) 1800

(621.03) 2000

(635.82) 2500

(668 13)

h

3O0O

(695.36) 3206.2 (705.40)

V b

h

0. 0287 763.8 0.9347

0.0638 1010.6 1 1420

0. 1052 1174.8 1.2757

0. 1462 1314 4 1. 3827

0 1743 1406.8 1. 4482

h

0.0268 746. 4 0 9152

0 0337 856. 1 1.0076

0.0593 1047. 1 1. 1622

0. 1036 1256.5 1 3231

0 1303 1369.5 1. 4034

v h

0.0262 741.3 0 9090

0 0313 836.5 0. 9892

0.0463 985.0 1 1093

0.0880 1224. 1 1 2930

0. 1143 1349.3 1.3821

4000

5000

5500

.


12-124

NUCLEAR-POWER-PLANT

m

Fig.

4.

ENTROPY Enthalpy-entropy diagram for steam.

SELECTION

16

Source:

[Sec.

12

1.7

Keenan,

J.

H. and F. G. Keyes.

is usually taken as (hx — hi) /(hi — h/i), where hi is the enthalpy of the steam to the engine, ht is the enthalpy after adiabatic expansion, and h/2 is the enthalpy of the saturated liquid at pressure pi. Rankine-cycle efficiencies, or theoretical thermal efficiencies, for a range of initial conditions and exhaust pressures are shown in Figs.

inclusive. Theoretical Steam Rates. Steam rate is the weight of steam required to produce a unit of work or, stated differently, is the rate of steam flow required to produce a unit of power. It is usually expressed in pounds per horsepower-hour 9

to

13,

1.86


-ass [e-si

ONixvHaxao-HaAiOd

SZT-ZT

XNaivdinba

AdoaiN3 .»

anumiA'pouiJaqx sapjadojj jo ,,'uiB3jg uqof Aa[i^ pue 'suog '-aui

M8]>i "H-io^

aouig jBuuaqt poo praiuBqaaui (jq-dq/qi) jo spunod jad jnoqi}BA\o|p( -(jqA\)j/q[) saiSjaua bjb Xj[Bn^nui 'a|qijjaAuoa aqj uiBajs ajuj jo no aui8ua ubb aq paindraoa uiojj b uoisjoauoo jopBj pun aq$ ^Baq pasn jad pnnod jo :pmu

uiBajg 'aiBJ jq-dq/qj

=

UIBBlg 'ajBI JUM3j/qi

=

Sf>9Z

jq-dq/n»a

iBaq 'pasn

qj/tiiji

NUCLEAR-POWER-PLANT

12-126

SELECTION

[Sec. 12


Steam rates for the Rankine cycle, or theoretical steam rates, for various initial steam conditions and vacua are given in Table 3. Gases. Several gases have been proposed as the power-cycle fluid for nuclear 1.37 plants for both the direct and indirect systems. Those most commonly considered are air, nitrogen, helium, and carbon dioxide.

S

S

Fio.

5.

Camot-cycle

temperature-entropy

diagram.

Fig. 6. diagram.

Carnot-oycle

Fio.

Rankine-cycle

enthalpy-entropy

S

S

7. Rankine-cycle temperature-entropy diagram.

Fio.

8.

enthalpy-entropy

diagram.

The values of the thermodynamic properties of air have been well established and in tabular and" graphical form. For air at low pressures, Ref. 1 is

are available

recommended as a convenient and quite accurate means for determining the proper * is An enthalpy-entropy diagram for air* ties and isentropic enthalpy changes. of the in values properties, disparity some is there shown in Fig. 14. Although enthalpy changes read from the chart are generally within 1 per cent of those calcu lated from the tables. This is within the accuracy required for preliminary work. * Superscript numbers

refer to References


.

Rankine-cycle theoretical thermal with in. Hg ahs. exhaust pres-

Fig.

200 400 600 800 1000 1200 1400 INITIAL PRESSURE, PSIG

10.

liankine-cycle theoretical thermal with 1.5 in. Hg abs. exhaust pres-

efficiency

HG. ABS. EXH. PRESS

■■ij=im 11. Rankine-cycle theoretical thermal in. Hg abs. exhaust pres-

ficiency with ire.

2

ig.

200 400 600 800 1000 1200 1400 INITIAL PRESSURE, PSIG

200 400 600 800 1000 1200 1400 INITIAL PRESSURE. PSIG Fig. 12. Rankine-cycle theoretical thermal efficiency with 3.5 in. Hg abs. exhaust pres sure. 0

14

0


effi. :iency

200 400 600 800 1000 1200 1400 INITIAL PRESSURE, PSIG 1

Fic

!).

0

12-127

0


Sec. 12-3]

NUCLEAR-POWER-PLANT

12-128

Table 3.

pres sure, psig

in. Hg abs

50

100

1511

200

250

inn


350

100

450

500

550

1,00

650

700

750

BOO

850

Mil)

1.000

1.200

12


haust pres sure,

25

[SEC.

Steam Rates, lb/kwhr*

Theoretical

Ex

Ini tial

SELECTION

31)11 41)0

,1)0

600

650

700

750

BOO

850

900

10 22 12 07

11.55

9.75

9.41 10.89

9.08

8.76

10.47

10 06

8.91 10.15

8.62 9.79

8.34 9.44

8.06

8.73 9.87

8.46 9.54

8.20 9.22

7.94

7.69

9.81

8.61

9 57 8 97 10 79 III 10

8.44 9.46

8. 18 9. 16

7.92 8.86

7.68 8.56

7.44

6.987

8 28

7 74

9 32 10 45

8 75 9.80

8.22

7.97 8.89

7.73 8.60

7.49 8.32

7 24 8 05

6 827 7 54

5 3 5

9 07 10 13

8.59 9.58

8.06

7.81 8 67

7 58

8 97

8.40

7.35 8.87

7 12 7. 13

Oil" 6 701 1.2 7 37

5 3 5

8.47 9.41

7.94 8.80

7.69 8.52

7.46 8.24

7.23

7.98

7.02 7.73

6 601

9 88

5 3 5

8.38 9.29

7 84 8 67

7.59 8.38

7 36 8 II

7. 14

6.924

9.68

7.86

7 61

5 3 5

8 56 9 54

7.51

6.849

8 56

8.28

7.28

7.06

9.20

8.01

7 75

5 3 5

9.35

8 27 8 83

7 69 8.47

7.44

5 3 5

8 41 9 23

8.21 9 01

7 64 8 40

5 3 5

9, 13

8.32

8.22

5 3 5

9.02

5 3 5

1.03 3. 15

13 75 16. 73

5 3 5

12 22 14.46

II 32 10.80 10. 10 13 59 12 67 11.77

II

5 3 5

10.82 12.47

10 43 9 81 9 20 12 01 11.26 III 31

5 3 5

10. 10 II 49

9.88

9.30

11.24

10.54

5 3 5

9 64 0 88

5 3 3

8 72

8 32

8 47

9. 18

7.76

7 60

32

1000

9.84

10.62 12 60

13.51 12.71 11.87 16 42 15.38 14 26

1.5 3 5

950

9.11

403 02

7.24 717 6 518 37 7. 14

325 915

7.51

6 447 7 05

257 829

7.21

6 995

8. 19

7.92

7.67

6.784 7.43

6.3876. 200 6 016 6 970 6 75(, 6 547

7.39

7 16

7.85

6.938 7.50

6.729 7.36

6 334 6 904

7. II 7 79

6 888 7. S3

6 680

7.29

47" 6 288 On 6 845

104 5.92S 638 6.433

8.

II

7.34 8.05

149 5.960 692 6 487

9.02!

8.34

8.22

7.57

7 30

7.07

6 846

9 02

8 29

8.00

7.73

7.48

6.636 7.24

437 6 247 0 1 6 793

065 5 68s 586 6.383

8 18 8 97

7 54 8 26

7.27

7.03 7.68

6.813

6.602

7 43

7. 19

401 6 211 963 6 746

030 5.855 540 6 342

5 3 5

8 09 8 86

7.52 8 22

7.92

7.00 7.64

6.778 7.39

6.568 7.15

6 179 6 705

998 5.824 500 b W3

5 3 5

8.07 8 83

7.50

7.22 7 88

6.975 7.60

6.750 7.35

6.539

8.21

340 6. 150 884 6.668

"7u 5.79* 404 6 26/

5 3 5

8 03 8 78

7.48

7.21

7.86

6.954 7.57

6.725

8. 18

7.32

6.512 7.08

312 6. 122 84916.636

"44 5 772 431 6.237

5 3 5

8 01 8 73

7 48 8. 17

7. 18

6.934 7. 55

6.702 7.28

6.490 7.05

289|6. 101 5 820 6.606 6

5 3 5

7 97 8. 79

7.49 8. 16

7.27

6.905 7.50

6.667 7.24

6.451

7.91

5 3 5

7.88 I

7.55 8.21

1.450 i

5 5

• Source:

8 22

8 57

7 746

7.844J

8.5II1 Keenan,

8.402

J.

7.95 7.25

7.83

7

II

6.993

7. 16

6.87

7.79

6.624 7.18

6.398

7.45

7.248 7.848

7. 105 7.456

6.611 7 137

6.366

H., and F. G. Keyes: "Theoretical

Steam

6 921

6 871

5 747 6 20t>

6.060.5 .879 6.553 350 190 5 999 688 6 474

70? I5t,

til

5.645 2o« 6 076

5.948 5 762 5 590 6 406'6 198 6 005

Rate Tables." ASME,

1938.


-^s fe-?.i

£-

"OIJ 'f\

Z-

onixvh3N30-h3avo
I-

0

I

2

Xdcuiua-.tcliBmug nnu8np

E

joj -jib

t>

iNaiwinba

«

wojjj)

9

•/•

\i

•D)mjvjp}j

6SI-ST

i

syx svq •autqunx

12-130

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

"Gas Tables"1 includes similar data for nitrogen, carbon dioxide, and other gases. Properties of helium, with a temperature-entropy diagram, are given in Ref. 3. 1.4

Steam Cycles

1.41 Steam Conditions. In central-station practice, the trend has been and still is to higher steam pressures and temperatures in order to obtain the benefit of the better thermal efficiencies possible. In modern plants burning conventional fossil fuels, the average pressure and temperature levels are around 2,000 psig and 1000T, with a number of plants in service operating at and others being built for a temperature of 1050°F. A few are designed for 2,400 psig and 1 100°F, and two will operate initially at 4,500 to 5,000 psig and 1 150°F. It is planned to operate at least one of the latter eventually at 1200°F. In the interest of standardizing steam-turbine generator units and other powerplant equipment, the American Society of Mechanical Engineers and the American

IELECTRICAU. GENERATOR

AIR EJECTOR (IF USED)


STEAM GENERATOR!

Fig.

15. Simple

steam cycle.

Institute of Electrical Engineers, in cooperation with the power industry, consulting turbine-generator rating*

engineers, and manufacturers, have selected the following and steam conditions as "Preferred Standards": Max rating, kw 12.650 16.500 22.000 33.000 44,000 44.000 66.000 66.000 100.000 100.000 150.000

Steam conditions, psig at °F 600 (u. 825 850 (d) 900 900 850 850 (ft 900 850 (a) 900 1.250 @ 950 850 (S, 900 1,250 Oil 950 1.450 ® 1000 1.450
Standardized units may not be compatible with a specific reactor design: however, they should be considered as the ultimate goal in order to obtain the benefit of lower manufacturing costs and shorter delivery time. The first nuclear power plants to go into service are designed to generate only satu rated steam by nuclear energy. As reactor technology progresses, steam condition* will advance into the superheat field. In the meantime, separately fired super heaters using fossil fuels are being specified for some nuclear plants. Although this


Sec. 12-3]

12-131

requires additional equipment, the added cost is more than offset by the improved cycle efficiency and reduction in other equipment costs. 1.42 The Simple Cycle. The simple cycle, consisting essentially of a steam gen erator, a turbine-generator set, a condenser, and a feed pump, as shown in Pig. 15, involves a minimum of equipment and, although of relatively low efficiency, might well be used for small plants in remote areas where extreme simplicity and reliability It has also been considered in some quarters for are of paramount importance.

1 1 1 1

nTTi-U-U-q:

p«ir. THROTTLE PRESS!

SATURATED THROTTLE STEAM — EXHAUST PRESSURE"!

10

20

30

40

50

60

70


RATING

Fig.

16.

0

Fig.

Approximate turbine-generator

20

40

60

80

80

90

100 110 120 130 140 150

MEGAWATTS efficiencies for saturated

100

120

140

steam.

160

RATING MEGAWATTS 17. Approximate turbine-generator efficiencies for 825°F steam.

medium-size and large plants on the assumption that because of the low cost of nuclear fuel, the improvement in cycle efficiency obtainable by more complicated cycles does not warrant the cost of additional equipment. However, the effect of efficiency on size and cost of the basic equipment must not be ignored. As the effi

ciency is improved by tions, the size and cost The performance of

the use of more efficient components and better steam condi of at least some of the main equipment will decrease. an actual turbine or engine is usually judged by the ratio of heat used by the turbine to the isentropic heat drop or heat used in an ideal heat engine working on the Rankine cycle. This ratio is called Rankine-cycle efficiency ratio,

12-132

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

With the turbine efficiency given, the thermal efficiency, or engine efficiency. efficiency, or ratio of net heat used to heat supplied, can then be computed as the product of the turbine efficiency and the Rankine-cycle efficiency. The present level of combined efficiencies of turbine-generator sets is shown in Figure 16 is based on the Bcldecos-Smith paper,4 for 100- Mw units, Figs. 16 and 17. calculated with no moisture separation, and on the performance of standard machines at the lower ratings. Figure 17 is based on the Elston-Knowlton paper' at high ratings and on the performance of standard machines at lower ratings. In using turbine

Table 4. Approximate Market Prices (May 15, 1968) of Commercial Straight Sets, 1.6 In. Hg Abs Exhaust Pressure Condensing Turbine-Generator (Prices

include Steam pressure, psig

standard

accessories exlusive of condenser, freight, and superintendence Prices are in dollars per kilowatt of turbine capability.)

With rated capability. Mw, of 5

20

60

200

Single case, 3.600 rpm 69.46

Single cose, 3.600 rpm 55.49

Single case. 1.800 rpm 60.56

Tandem comp., double flow, 1.800 rpm 59.25

Tandem comp.. double flow. 1.800 rpm 48.17

600



Tandem coinp., triple flow,


Tandem comp.. double flow, 1.800 rpm 43.42



Tandem comp., triple flow, 3.600 rpm



Tandem comp.. double flow, 1.800 rpm 58.39


Dry and Saturated

1.000

100

Steam at Throttle

3.600 rpm 53.00

52.76 I.4S0



150

Tandem comp.. triple flow, 3,600 rpm 52.76

825°F Initial Steam Tempi raturc Single case, 3,600 rpm 62.83


Tandem coinp., triple flow,

600



1.000

Single case. 3.600 rpm 74.06

Single case, 3,600 rpm 74.06

200

1.450


Tandem comp.. double flow, 1,800 rpm 44.19



Tandem comp.. single flow, 1.800 rpm 43.08


Tandem comp.,


Tandem comp..




Tandem comp.. single flow. 1.800 rpm 43.51

3.600 rpm 57.31

double flow, 3.600 rpm 41.63

single flow. 1.800 rpm 43.51

the Elston-Knowlton paper, a constant exhaust loss of 20 Btu/lb was assumedFor other corresponding to a last stage axial leaving velocity of about 850 fps. exhaust velocities, the efficiency may be corrected in accordance with Fig. 10 & Ref. 5 and the total isentropic drop available. Turbine and cycle performance are often given as heat rate. This is equal to total heat supplied to the turbine throttle less the total heat in the condensate returns'to the heat source divided by the power output, or

thy


of installation.

Hli

F(hy

-

h.)

Sec. 12-3] where

HR F


= heat rate,

12-133

Btu/kwhr

throttle steam flow, lb/hr hi = enthalpy of steam to throttle, Btu/lb hc = enthalpy of liquid returned to heat source, Btu/lb P = output, kw In the simple cycle, he is the enthalpy of the liquid leaving the condenser, and in the regenerative cycle it is the enthalpy of the liquid leaving the last feed heater and assumes that the flow leaving the last heater is equal to the turbine throttle flow. The heat rate is related to thermal efficiency by the equation: thermal efficiency = 3,413/heat rate. Over-all plant efficiency is obtained by multiplying the thermal efficiency of the turbine-generator set by a factor representing boiler losses, auxiliary power required, and miscellaneous station losses. In central stations, the average power required for —

Table

6. Approximate Market Prices (May 15, 1958) of Commercial Surface Condensers of Standard Construction Serving Straight Condensing Turbinegenerator Units, 1.6 In. Hg Abs. Exhaust Pressure and 70°F Cooling Water

(Prices include two half-capacity circulating pump units, two full-capacity condensate pump units, one twin two-stage steam jet air ejector, admiralty metal tubes, a set of spring supports, freiKht, and super intendence of installation. Prices are in dollars per kilowatt of turbine capability.) With rated capability, Mw, of


Turbine throttle pressure, psig

5

20

60

100

150

Dry and Saturated Steam at Throttle 200 600 1.000 1.450

15. 3 13. «

12.8 11.0 10.8 10.8

11.1

10.8

10.5

9.7 9.3 9.2

9.5

9.3 9.0 8.9

9. 1

9.0

825°F Initial Steam Temperature 200 600 1.000 1.450

auxiliaries

12.0 10.8

9.6 8.0 8.0 8.2

8.3 7.0 6.8 6.9

8.3

8. 2

7. 1

7.0 7.0 7.0

7.0 7.0

is about 6 per cent, varying from 4 to 8 per cent, with no definite correlation rating or steam conditions. In the power-generation section of direct cycle nuclear plants, auxiliary power will run somewhat lower because of the elimination of certain equipment such as forcedand induced-draft blowers, fuel-preparation and -handling equipment, etc., but in the case of indirect cycles, the total power may be higher because of primary coolant pump requirements. Miscellaneous losses include line pressure drops, temperature drops, leakages, etc., and may average around 1 per cent in a well-maintained plant. It is estimated that auxiliary power and miscellaneous losses in a nuclear plant will average 5 to 6 per cent plus the power required for primary circulating pumps, if any. Equipment costs will depend upon rating and steam conditions. Representative prices as of May 15, 1958, of the major equipment — turbine-generator sets, con densers and auxiliaries, and feed-pump units — are given in Tables 4 to 6. The prices These prices must be considered are given in dollars per kilowatt of capability. approximate, as the specification requirements for particular plants will vary greatly. Prices are also subject to changes resulting from market conditions, rise or fall in labor and material cost, etc. Prices for final budgeting purposes must be based on manufacturers' quotations. Approximate dimensions of generating units are shown in Fig. 18.

with

NUCLEAR-POWER-PLANT

12-134

SELECTION

[Sec. 12

Substantial improvement in plant efficiency can The Regenerative Cycle. In this at comparatively low cost by use of the regenerative cycle. cycle, shown in Fig. 19, one or more heat exchangers are used to heat the feed water, the heat being supplied by steam extracted from the turbine after partial expansion. In this way, the extracted steam does work in the high-pressure section of the turbine and all its latent heat, a large part of which is otherwise lost in the condenser, is used to heat the feed water. If desired, one of the feed heaters may be of the direct-contact or deaerating type. Central-station practice is divided in the use of deaerators, the trend in some plants being to exclude them in new designs and in others to include them after having eliminated them in older designs. Good deaeration can be obtained in the modern condenser if it is properly designed and operated, so that the deaerator is often con However, a deaerator is useful in that it will extract gases sidered unnecessary. entering the condensate after leaving the condenser, or gases due to maloperation of the condenser, and will also serve as a surge tank in cases where this is desirable. 1.43

be obtained

Table

6.

Approximate Market Prices (May 15, 1958) of Commercial Boiler Feed Pumps for Straight Condensing Turbine -generator Installations

(Prices are for one full-capacity pump and driver with standard materials, construction, and accessories. Standard accessories include manually operated bypass orifice, provisions for forced lubrication of driver bearings, and one set of special tools. Prices are in dollars per kilowatt of turbine capability.) With rated capability, Mw, of


Turbine throttle pressure, psig

5

20

60

100

ISO

Dry and Saturated Steam at Throttle 200 600 1.000 1.450

0.36

0.26

0.22

0.20

1. 12

1.30

0 76 1.28 1.88

0 67 1.14 1.68

2.13 3.03 825°F

200 600 1.000 1,450

0. 19

0.64 1.08 1.59

Initial Ste:im Temperature

0.36

0.26

0.22

1.12

1.08 1.90 2.80

0.61 1.06 1.59

0.20 0.52 0.88

0 19

1.36

1.22

0.46 0.79

Improvement in efficiency, increase in throttle flow, and reduction in flow to the condenser due to feed heating are shown in Figs. 20, 21, and 22. Economic justification of the regenerative cycle in a conventional plant depends upon the value of fuel saved by improved plant efficiency compared with the increased In a nuclear plant it will depend also upon the saving in fixed investment charges. The increase in investment will be due mainly charges on the fuel inventory required. to the additional cost of the heaters and somewhat more piping, but these costs will be partially offset by a reduced cost of the condenser and heat source. 1.44 Initial Steam Separator. Where space permits, the moisture content of steam leaving the drum of a steam generator can be held to very low values by internal moisture separators and steam scrubbers. However, it is very important that no slugs of water be permitted to enter the turbine when operating. To prevent this when superheating is not used, a steam separator should be installed in the line imme It diately ahead of the turbine stop valve in order to catch condensation in the line. is particularly desirable in the case of a direct cycle boiling-water reactor plant, where it may not be possible to provide sufficient space for adequate internal separation. 1.46 The Moisture Problem. As the steam expands in the turbine, doing work. its enthalpy is reduced, forming moisture when the saturation line is passed, which


Sec. 12-3]

12-135

the turbine efficiency adversely and also causes erosion of the last rows of blading and the casing. The amount of moisture at the various stages depends upon the initial steam conditions, the rating as it affects stage efficiency, and local pressure. The calculated moisture content of the exhaust steam of straight condensing tur bines as a function of rating and steam conditions is shown in Table 7. Actually, the moisture leaving the last row of blades is somewhat less than calculated, as some of the moisture thrown from the blades passes through interstage drain holes near the affects


SINGLE

APPROXIMATE 600 PSIG-825 RATED CAPABILITY MW 5

CASE, SINGLE FLOW TURBINE

DIMENSIONS, TURBINE-GENERATOR UNITS HG ABS EXHAUST PRESSURE

FAT THROTTLE. 15" TURBINE TYPE

SINGLE

CASE

SPEED RPM

OVER/ VLL DIMENS ONS LENGTH A

-6"

WIDTH C ■ 0'- 11"

HEIGHT B

6'-

3600

27'

16'

-6"

9'

10"

-

20

SINGLE CASE

3600

41' - 4"

60

TANDEM COMR TRIPLE FLOW

3600

64'-8"

20'

-H"

I3-4"

100

TANDEM COMR DOUBLE FLOW

1800

89'-9"

2 4'

-8"

l4'-7"

150

TANDEM COMR DOUBLE FLOW

1800

93'

- r

Fio. 18. Approximate dimensions of straight condensing 825°F at throttle, 1.5 in. Hg abs. exhaust pressure.

24-8"

14'

turbine-generator

9"

-7" units, 600 psig

bottom of the casing. In the larger machines, moisture catchers of various types are provided to trap some of the moisture, reducing the amount carried into the blades by the steam. One type of moisture catcher used in modern turbines is shown in Fig. 23. Some reports indicate that up to about 25 per cent of the moisture present has been removed

by these devices.

Recent developments may provide means for removing

even greater amounts. The greatest erosion occurs on the rotating blade inlet edges because of the high relative velocity between the water droplets and the blades. To reduce erosion, the

inlet

edges of the last few rows of blades in large turbines are usually protected by

12-136

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

ELECTRICAL

GENERATOR

DE AERATOR

STEAM

AIR EJECTOR (IF USED)

CONDENSER

GENERATOR

CONDENSATE PUMP

B°'LER

HIGH PRESSURE HEATER 19.

LOW

PRESSUREl HEATER

Flow diagram of regenerative

steam cycle.

NUMBER OF STAGES Flo. 20. Improvement in cycle efficiency due to feed heating. Dotted lines indie*" optimum temperature rise of feedwater from condenser hotwell to last heat«r discharge

inserts of a hard material, such as stellite, over the outer portions of their lenpb; where the moisture is concentrated and the relative velocity is highest. However experience has shown that for satisfactory blade life the calculated moisture eont«>; of the exhaust steam should not exceed 12 per cent for low-pressure blades operatini at tip speeds not exceeding 1,250 fps. For higher tip speeds the moisture rentes'1 should preferably be held below this figure. For example, at a tip speed of 1,550

If*


FlQ.

PUMP PUMP

POWER-GENERATING EQUIPMENT

Sec. 12-3]

12-137


30 29

I

Fig.

the

2

21.

34

9

67

8

9

10

NUMBER OF STAGES Increase in throttle flow due to feed heating.

moisture content of steam in the exhaust should be held, if possible,

cent.

Valve

desired. 1.46

bine for stage is

seats

are

usually hard-faced, and casings

below 8 per

can be protected by shields if

Intermediate Moisture Separator. When steam is extracted from the tur feed heating or other purposes, a relatively large part of the moisture at that carried out with the extracted steam which may, in small machines, reduce

12-138 Table 7. Turbines

SELECTION

NUCLEAR-POWER-PLANT

[Sec.

12

Calculated Moisture Content of the Exhaust Steam of Straight Condensing Operating with 1.6 in. Hg Abs. Exhaust Pressure, No Moisture Separation (Moisture values in per cent of total exhaust flow)

Initial pressure, psig

200

600

1.000

1.450

With rated capability, Mw, of Initial temp, °F

Dry and sat. 725 825 925 Dry and sat. 725 825 925 Dry and sat. 725 825 925 Dry and sat. 725 825 925

5

20

60

150

13.9 3 4

14.7 7 3 5. 2

16.0 10. 1

17.5 12 1

15 1 8 3 6 2 4. 2 17.9 13.4

7.6

9.7

15.4 8 6 6 3 4.5 18.3 13.8 11.6 9 6 20 1 16.4 13.9

5.8

3.0

7.4 18 9 14.4 11.7

9.2 20. 1 16.5 13. 2 10.2

III

9. 1 19.8 15.9 13.4 11.0 21 4 18. 0 15. 1 12.4

117 21 8 18.8 15.9 13.2

In the case of large machines operating with saturated or low superheat steam at th* throttle, it may be found necessary to dry out the steam after partial expansion at least once, possibly two or three times, in order to hold the exhaust moisture wit his This can be done by reheating or by moisture separation. Tbf acceptable limits. former is not considered a practical solution at present, since reheating in the reactor 1.17 1.16 would complicate the reactor design and 1.15 operation, and reheating by live steam is 1.14 inefficient. < 1.13 Provision for external separation can read 1.12 In a compound turbine, the ily be made. ys separator can be placed in the line between I. II 1.10 1.09 1.08 1.07 l
1.06 1.05

_ rvu

1.04

HG ABS

pprcc tor

1.03

J.OZ 1.01 )


the exhaust moisture to an acceptable level when operated with initially saturated steam.

400 600

800

INI TIAL PRESSURE,

Fig. 23. Moisture catcher used in standard large turbines.

10OO

PSIA

Fig.

24. Improvement in cycle efficiency due to use of moisture separators.

the high-pressure and low-pressure elements. In a single easing design, the turbiEcan be provided with a blank diaphragm at the appropriate stage or stages, all the steam passing out of the turbine ahead of the diaphragm, through a separator, and tfcet back into the turbine. The moisture separator may be of the baffle type in which separation is due to many

Sec. 12-3]

POWER-GENERATING

EQUIPMENT

12-139

reversals of the steam flow, the moisture clinging to and flowing down a series of baffles, or the centrifugal type in which the steam is forced into a spiral path at rela tively high velocity, throwing the moisture to the outer container, where it is collected and drained off. Both types are effective, reducing the moisture content to less than However, 1 per cent moisture should be an per cent in the effluent if desired. Pressure acceptable value and may permit a smaller and less expensive separator. drops through the separators are low, but when cycle performance is calculated, the total drop from the high-pressure to the low-pressure section of the turbine must be taken into account. This will average about 5 per cent of the high-pressure-section discharge pressure. Space requirements are rather large. For example, a 100-Mw turbine operating on 600 psi dry and saturated steam requires two separators 8-ft OD by 15 ft long, each handling 550,000 lb/hr and reducing the moisture content from 14 per cent at inlet to 1 per cent at discharge. The price of these separators, which are of the baffle type, is about $0.90 /kw total, and the weight about 32,000 lb. The improvement in cycle efficiency obtained by one and two stages of moisture separation at the optimum points is shown in Fig. 24. The curves are based on the Beldecos-Smith paper4 and on reducing the moisture content to 1 per cent in each case. 1.47 A separately fired superheater using a con Separately Fired Superheater. ventional fuel may be considered incongruous in a nuclear plant and may nullify some of its advantages. However, the great advantage of superheated steam in


\i

improving plant efficiency, using standard equipment, and reducing or eliminating the moisture problem should not be overlooked. It is believed that until superheated steam can be generated by nuclear energy, separately fired superheaters deserve careful consideration. In one nuclear power plant with an oil-fired superheater, the superheater accounts for about 40 per cent of the total plant output, resulting in a plant heat rate of 10,700 Btu/net kwhr, or an efficiency of about 32 per cent. 2

POWER-GENERATING EQUIPMENT AND AUXILIARIES 2.1


Power-generating equipment will be of more or less conventional design, modifica tions, if any, depending upon the type of system, the power-cycle fluid, operating The unusual problems involved are those conditions, and reactor characteristics. associated with radioactivity, containment, and control. In the indirect system the power-cycle fluid can be ordinary water, which should not become radioactive except in the case of a double casualty such as the simultaneous rupture of a fuel element and the occurrence of a leak in the heat exchanger that admits reactor coolant into the power-cycle fluid. With proper detection instruments, it should not be necessary to provide for a double casualty and conventional power However, reactor operating characteristics may equipment designs can be used. dictate the use of additional equipment such as a steam bypass line around the turbine to the condenser or an auxiliary cooling system to absorb excess heat generated in the reactor upon reduction in power demand and after shutdown. In the direct steam system, although the radioactivity of the steam is normally low, it must be contained because of the possible presence of minute quantities of radio active isotopes that would be extremely dangerous if inhaled or ingested. Contain ment is also necessary in case radioactive particles are carried over from the reactor upon failure of a fuel element. If the fluid is DjO, all leakage must also be recovered because of the high cost of D20, and contamination by H20 must be prevented. Inventory of D20 must be kept low, which may influence the selection of the number arid type of feedwater heat exchangers and method of liquid-level control. The radioactive isotopes present in the steam are N", which has a half-life of only 7.3 sec, and O", with a half-life of 29 sec. Test results of boiling-water-reactor experi ments indicate that the radioactivity of the steam, if free of impurities and water ca.rry-over, is sufficiently low to permit limited access to the power equipment when in operation for inspection and minor adjustments. No permanent shielding is required. For maintenance requiring long periods of time on equipment that can be

12-140

NUCLEAR-POWER-PLANT

SELECTION

[SEC.

12

isolated from the system, portable shielding can be provided for protection of per sonnel. The short half-life of the radioactive isotopes will permit continuous access within 5 or 10 min after shutdown. With the types of fuel elements that will be used in direct-cycle power reactors the possibility of serious contamination of the system due to fuel-element rupture is rather remote. However, until adequate operating experience with this type of reactor has been obtained, certain precautions should be taken in anticipation of such an event. These precautions should include avoidance of all pockets and crevices in which radioactive materials could lodge, good drainage of the system with remotely controlled drain valves, and means of injecting water or solvents into the various The flushing valves should also be remotely components of the system for flushing. A flushing operation will probably reduce the radioactivity to a suffi controlled. ciently low level to permit opening the equipment in the usual manner for further decontamination. 2.2

Seals

2.21 Shaft Seals. Standard shaft seals may be used without change for a direct cycle system designed for operation with ordinary steam because the problem is one of containment only. Labyrinth seals are provided with leakoff connections that lead to a small exhaust blower to create a pressure slightly below atmospheric in the leakoff


SEALING STEAM OR LEAKOFF

Fig.

25.

A turbine shaft seal for operation

with DiO.

The mixture of air and vapor spaces, thereby preventing leakage to the atmosphere. may be passed through a small surface condenser, and the condensate discharged to s retention tank before going to drain or back into the system. A gland exhauster and condenser are standard equipment for large steam turbine-generator units equipped with these seals.

A water gland, which depends upon an annulus of water maintained by centrifuital force to seal the atmospheric end of the gland assembly, is equally effective in pre venting steam leakage to the atmosphere and the inleakage of air. If standard labyrinth seals and glands were to be used for a heavy-water system, atmospheric moisture drawn in through the outer gland section by the exhauster The resulting loss of heavy system would contaminate the gland leakoff steam. Thi? water would represent a large operating expense and could not be tolerated. problem is readily solved by adding a section to the standard seal and supplying dry air or gas, at a pressure slightly higher than atmospheric, to the space between tisr Some of the dry air will escape to the atmosphere, sealing the out> r two outer seals. gland against inflow of atmospheric air, and the rest will flow inward to the leakoff The mixture of air and steart. space, preventing a leakage of heavy steam outward. is discharged to a recovery system (see Air-drying and D20-recovery System) wherr the steam is condensed, the air dried, and the heavy water and dry air returned to their respective systems. As in the case of the standard gland, a slight vacuum is

POWER-GENERATING

Sec. 12-3]

12-141

EQUIPMENT

in the leakoff space. A turbine shaft seal and a feed pump seal of this An all-labyrinth gland could be used on the pump typo are shown in Figs. 25 and 2G. also; the mechanical seal included in the pump gland shown in Fig. 26 is an alternate Water glands, if used, must be provided with heavy water. type undergoing testa. 2.22 Valve-stem Glands. Valve-stem leakage may be collected in the same manner as shaft leakage with little or no modification to the standard gland, and, if desired, provision can be made for sealing with dry air for operation with D20. Valve steins may also be sealed by a bellows fastened to the valve bonnet and to the stem, with a leak-off from the bellows to a collection system in order to prevent a For large valves with large travels, this might build-up of pressure in the bellows. mean an excessively long bellows. In this case, the length can be reduced by using double or triple telescoping bellows. However, a bellows is subject to fatigue stresses,

Fig.

20. A feed-pump

seal for operation

with DjO.

especially when used on a control valve, which may be moving constantly. studies must include full consideration of all factors affecting reliability.

Design

A

hermetically sealed valve, having no valve stem or gland, is shown in Fig. 27. valve in effect, built into the piping and controlled, through a pilot valve, by the system fluid taken from a higher pressure source. It will be seen that as the pilot valve admits fluid to one side of a stationary piston to move the valve, also releases the fluid on the other side of the piston. The discharged fluid can be col lected and returned to the system. This design modification of the Johnson valve used in hydraulic turbine systems. 2.23 Flange Seals. During steady-state operation, equipment joints remain tight, but during startup and shutdown periods and during large load changes, tem perature fluctuations may cause the joints of large machines to "breathe" slightly. If this possibility considered undesirable, leak-off grooves can be provided in the flange face or the entire joint can be provided with a light, nonrigid housing as shown in Fig. 28. The space around the housing can then be connected to collection sysit

is

is a

a

is

This

is,


maintained

12-142

NUCLEAR-POWER-PLANT


Fig.

27.

SELECTION

[Sec. 12

A hermetically sealed valve.

tem, or, in the case of a heavy-water system, it can be connected to the dry-air sealing and DjO-recovery system. 2.24 Condenser. Condenser-lube Seal. Condenser tubes rolled into tube sheets There is some question, how are normally considered to produce very tight joints. ever, regarding the degree of tightness required to prevent the contamination of heavy water by small quantities of the condenser cooling water. Until rolled joints have been proved in service to be sufficiently tight, it may be desirable to consider other means to ensure against contamination. Welding the tubes to the tube sheets, if properly done, will produce leakproof joints that will remain tight indefinitely. However, this requires the selection of weldable tube and tube-sheet materials, and it may be desirable to provide a welding lip around each tube-sheet hole to prevent distortion and cracking. On large condensers, this would entail a very large number of welds. Another method of preventing contam ination is by the use of double tube sheets as shown in Fig. 29. In this case, the TO AIR DRYING tubes are rolled into both tube sheets and a the space between the tube sheets is RECOVERY SYSTEM A leak in purged and filled with dry air. an inner tube-sheet joint will be indicated Fig. 28. Turbino casing a flange seal. by a drop in the air pressure, and a leak age of cooling water will be indicated by a water-detection device at the bottom of A continuous supply of dry air can be pro the chamber between the tube sheets. A condenser of this design is used in the vided in case of a leak in both tube sheets. boiling-water-reactor power plant at Argonne National Laboratory. In addition to providing insurance against contamination of the cycle fluid, this installation will give a good indication of the tightness of rolled tube joints.

Sec. 12-3]

POWEK-GENEKATING

EQUIPMENT

12-143


Other possible causes of condensate contamination in Condenser Tube Failures. the condenser are tube failures due to tube vibration, steam impingement, chemical action, and cooling-water erosion. Failures due to vibration are rare in modern con densers and are avoided by proper spacing of the tube-support plates and baffles. Erosion due to steam impingement and cooling water is progressive, and potential Erosion is minimized by steam failures can be detected by periodic inspection. baffles and tube shields on the upper rows, by the proper design of water boxes, and by holding cooling-water velocities to conservative values. Chemical action, which may cause tube cracking, is avoided by careful choice of tube materials for compatibility with boiler-water treatment and pH values. Clean liness and water treatment of the cooling water also are of prime importance. Further protection against failure by cracking may be obtained by the use of duplex tubes consisting of two concentric tubes in intimate contact at the interface. In these tubes, a crack in the outer surface will tend to stop at the interface, thereby preventing

Fio.

29.

Double tube sheets.

a complete failure of the tube. Duplex tubes may be of the same material inside and out or may be of different materials for greater compatibility with both steam and cooling water. Continuous contamination of the condensate, should a leak occur, may be avoided without shutting down the entire plant by compartmentation of the

steam and cooling-water sides of the condenser. With suitable leak-detection devices and multiple connections or valving, this will provide a means for isolating that sec tion of the condenser in which a leak occurs. 2.3

Air-drying and D20-recovery

System

The tightness of the system required to prevent loss of D2G and the quality of air required for gland sealing in order to limit contamination by H2C) to within acceptable limits will depend upon the rating of the plant, the size of the system, the cost of D20, and the cost of repurifying. At the present time, these costs are so high that the loss and contamination must be held to very low values, possibly of the order of only a pound per day. To achieve this, practically all moisture must be removed from the gland sealing air and a highly efficient recovery system must be provided to reclaim heavy-water leakage. The drying and recovery system used in the Argonne experimental plant is shown in Fig. 30. In this system, the air-vapor mixture from the glands and all other vapor that might otherwise escape are taken to a vent condenser where the free vapor is condensed and the condensate drained to a drain-collecting tank and returned, with other drains, to the system.

12-144

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

Saturated air leaving the vent condenser passes through a refrigeration unit where the specific humidity is reduced to about 40 grains per pound of dry air and then through a chemical drier that dries the air to a moisture content of less than 0.25 grain/lb. The dried air is returned to the air sealing system, and the recovered water back to the main system via the drain-collecting tank. The chemical drier used is a sealed, dual unit, using activated alumina as the adsorb ing medium, in which one unit is in service while the other is being reactivated.

DRY AIR TO OTHER SEALS


MAIN TURBINE SHAFT SEAL

CHEMICAL DRYER

AIR-VAPOR FROM SEALS VAPOR FROM AIR EJECTOR OTHER

DR/liN

D,0 FROM

REACTOR PUMPS

-

FROMAIR EJECTOR

CONDENSATE

REFRIGERATION UNIT

SUMP

MAKE-UP AIR —

Q Fio.

TO

30. Air-drying

-

SYSTEM DRAIN

and DiO-recovery system.

Other chemical driers are available that Cycling of the units is entirely automatic. provide continuous drying and reactivation. To provide makeup air to replace that lost through the outer seal sections, atmos pheric air is dried by means of a second refrigeration and chemical drying system This air also will have a moisture and discharged to the main air sealing system. content of less than 0.25 grain/lb. 2.4

Materials

The materials used in the construction of turbines for the indirect cycle may be those generally used for similar steam conditions in the central-station industry today. It may be desirable, however, to extend the use of erosion shields because of the higher moisture content of the steam in the turbine blade path when supplied with saturated steam at the throttle. For the direct cycle, consideration must be given to the extent of corrosion and ero sion of the various materials and to possible effects if the products should pass into t he reactor. It is particularly desirable to minimize the use of objectionable materials such as cobalt, which is an important constituent of the Stellitc used on low-pressure blade edges and high-pressure valve scats. Cobalt is also present as an impurity in

POWER-GENERATING

Sec. 12-3]

12145

EQUIPMENT

Full-flow filters and full- or part-flow deionthe nickel of austenitic stainless steels. izers are recommended in order to keep foreign materials out of the reactor and to hold the concentration of impurities in the reactor water at a low level. The following materials are used in modern large steam turbines: Cast chromium-molybdenum High-pressure casing: Cast carbon steel to 800°F. steel 800 to 1050°F. Forged chromiumHigh-pressure spindle: Forged carbon vanadium steel to 650°F. steel 650 to 10'25°F. molybdenum-vanadium Low-pressure casing: Plate steel or cast iron. steel forging. Low-pressure spindle: Nickel-molybdenum-vanadium Blading: 13 per cent chromium low-carbon steel. The alloy is modified with small additions of tungsten, molybdenum, nickel, and vanadium for the long, high tip speed blades. In addition to these materials, a leaded-bronze or a high-sulfur chromium steel Stellite or other hard-facing materials may be may be used for labyrinth shaft seals. used on inlet edges of some of the low-pressure blading. Table

8.

Estimated Rates of Wear of Material in a 150-Mw Turbine Operating with a Condensate pH Value Greater than 7 (Kates in pounds


Element

Carbon Chromium. . . Cobalt Copper Iron Manganese. . Molybdenum Nickel Phosphorus. . Silicon Sulfur Tungsten. . . Total

All

Cast iron

per year)

Cast and wrought

13 chrome

steel

steel

blading

2 1

0. 187

0.012

0.06

0.075

1.5

0.3 55.53 0.36 0.06 0.15

0.15 73.34 0.60 0.075 0.23 0.04

0. 18 1.2

0.06 60

0 263

0.04 75

Haynes No. 6

Stellite

0.05 0.90 1.72

10.26

0.06 0.06 0.05 0.004 0.05 0.004 12

0.09 0.09

Totals

2.349 2.535 1.72

0.45 139.22 1.02

0.195 0.52 0.224 1.513

(M5 3

these parts arc subject to corrosion and erosion in varying

0.104 0.15 150

degrees,

depending

upon their location in the turbine, peripheral velocity of the blading, steam conditions, and the pH value of the condensed steam. Estimates of the amounts of materials

that might be removed from the internal parts of a Table 8. These estimates are based on observations

150-Mw turbine are shown in of a number of central-station machines over a period of more than 15 years operating at pH values of condensed Changes in feedsteam above 7 and for a wide variety of designs and conditions. water treatment during the life of a unit will produce different rates of loss of materials. In some cases, erosion shields have later been fitted over areas subjected to the greatest rate of wear in a turbine casing. The data given should be regarded as general indications rather than precise The need indices of the amounts of materials lost over a given period of operation. for precise data on this subject was not acute until the nuclear heat source entered the field as an accepted prime-mover energy source. Additional data are needed before accurate predictions can be made of the rates at which materials are lost by corrosion or erosion of steam-turbine parts. Steam condensers usually include admiralty metal tubes, Muntz metal tube sheets, steel shells, and steel tube-support plates. Other tube materials in fairly common use are aluminum-brass, arsenical copper, and copper-nickel, such alloys being employed to satisfy the requirements of the chemical conditions existing in particular installa Naval brass, copper-nickel, and steel are sometimes used for tube sheets. tions.

12-146

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

Estimated rates of wear due to corrosion and erosion on the steam side of a larp condenser, based on observations over extended periods, are shown in Table 9. All these elements are not necessarily present in all condensers. Those present will depend upon the compositions of the materials used for the various components. The same materials may be used, if desired, in condensers serving nuclear power However, some additional security against contamination of heavy water plants. by tube corrosion in plants using this cycle fluid may be obtained by use of duplex In bot h eases the condenser cost w ould tubes or stainless-steel tubes and tube sheets. be increased appreciably owing to the high unit cost of material and the additional surface required because of lower heat-transfer coefficients. Aluminum tube? li. either aluminum or steel tube sheets also deserve serious consideration. The use of Table 9.

Estimated Rates of Wear of Material in a Surface Condenser Serving a Condensate pH Value Greater than 7

a

150-Mw Turbine Operating with Element

Autirnony

Pound* /year 0.1 0. I

Arsenic

Carbon Chromium Cobalt Copper

0.35 255 1J5

Iron

I.I


Manganese

Molybdenum Niokel Phosphorus Sulfur Silicon

0. I 0. 1 0 1

J.

Tin

Tungsten Zino

6

106

Total

502

aluminum tubes in condensers is relatively new, and extensive reports of operating experiences are not yet available; however, the results in a few test installations hav been quite encouraging. The choice of materials will depend largely upon analystof the circulating water and condensate. 2.6

Load Control

The type of load control required for a nuclear power plant will depend upon tbreactor characteristics and the method of reactor control. In steam power plants using conventional fuels, a reduction in load demand dote the turbine control valves, reducing steam flow and momentarily increasing the steati:supply pressure. These effects react on t he boiler combustion control to reduce the fuel to the burners. A similar system might be used to actuate the control rods of s reactor, but the pressure surge due to load change may not be compatible with tb operating characteristics of the reactor. In the experimental boiling-water-reactor power plant operating at Argor.r.f National Laboratory (Fig. 2), the reactor is maintained at substantially constant Reactor output will be controlled by manual adjustment of the contn
POWER-GENERATING

Sec. 12-3]

12-147

EQUIPMENT

proportion to maintain constant steam pressure. An increase in load, within the margin provided by the normal bypass flow, maintains pressure by opening the inlet After a load change is made in either direction, valves and closing the bypass valve. the bypass flow is brought back to the desired 5 per cent of total flow by manual '

TURBINE CONTROL VALVES

— GENERATOR signAi

SIGNAL

generator responsive to control

GENERATOR

RESPONSIVE TO STEAM PRESSURE

VALVE POSITION


DUAL REACTOR

TRANSDUCER

I

OIL

FiQ. 31. Turbine control system for constant reactor pressure.

-AVFio.

32. Turbine control system for rising reactor

pressure with load.

adjustment of the reactor control rods. As shown on the diagram, a second signal generator, responsive to steam pressure, is provided to operate the bypass valve

directly without going through the speed output due to causes other than changes

governor in case of change-* in reactor steam in electric load.

NUCLEAR-POWER-PLANT

12-148

SELECTION

[Sec.

12

For large power plants using this type of reactor, a similar control system could he used but adjustment of the reactor control rods would probably be automatic. A boiling-water reactor may possess an important self-regulating characteristic due A reduction in pressure tends u. to the effect on output of steam bubbles in the core. increase the volume of steam bubbles and, by thus removing moderator in the core, This direct variation of output with pressure is the same as the reduce the output. flow-pressure characteristic of a constant area passage but is opposite to the norms! load-pressure characteristic of a steam turbine due to action of the governor valves. However, this latter characteristic can be reversed by the use of suitable equipment. The usual Figure 32 shows a control designed to take advantage of this effect.


centrifugal governor is arranged to operate the turbine inlet valves through the ventional hydraulic system. A second control system, operated from the same ernor lever, functions to adjust the compression on a valve spring that in turn lates the pressure in the reactor itself. As indicated by the diagram, a loss of

con gov

re load

REACTOR

CONDENSATE PUMP

EACTOR PUMP

Fig.

33.

Dual-cycle control system.

operates simultaneously to close the turbine inlet valves by the required amount and at the same time to adjust the constant-pressure valve balancing-spring so that th pressure in the reactor and the consequent amount of evaporation in the core will he adjusted to the degree required to balance the reduced load. In order to compensate for extremely short-time load changes, the power system that operates the main turbine valves is arranged to change the compression on thWith a sudden loss of load the pressure counterbalancing spring of a dump valve. in the main steam line is immediately dropped by the functioning of this dump valvr A safety valve also is provided with a connection directly from the upper portion
Sec. 12-3]

POWER-GENEKATING

12-149

EQUIPMENT

Drains high-pressure and low-pressure steam is used in a mixed-pressure turbine. from the flash tank, which are at saturation temperature corresponding to the lower pressure, are pumped back into the reactor where they mix with and subcool the recirculating water. As the turbine load increases, the turbine governor adjusts the low-pressure inlet This drops the pressure in the flash valve to admit more steam from the flash tank. tank, causing more water to flow into the tank and thus providing an increased This additional subcooling of the recircu flow through the drain line for subcooling. lating water in the reactor decreases the steam voids in the core and increases the In a later design, a closed heat exchanger reactivity and steam output of the reactor. MOISTURE SEPARATOR

BOILING WATER REACTOR

STEAM

TURBINE GENERATOR


SURFACE CONDENSER

COMBINED CONDENSATE REACTOR FEED PUMP

AND

DUAL MIXED BED EXCHANGE UNITS

ION

"PARTIAL SYSTEM

FLOW

DEMORALIZING

"PUMP

Flo.

34. Flow diagram

of water treatment

system.

It functions in a similar manner and has the was substituted for the flash tank. advantage of requiring less pumping power. Boiling water reactors may also have a strong negative temperature coefficient In this case the reactor can which will at least partially offset the effect of pressure. be designed so that the output will closely follow load changes and none of the rather Automatic control rod adjust complicated control methods described is necessary. ment may then be made responsive directly to steam flow or load change. In indirect systems in which the reactor coolant does not change phase, the pressure The coefficient of reactivity is usually small and cannot be used for reactor control. temperature coefficient, however, can be made sufficiently large and in the right


12-150

NUCLEAR-POWER-PLANT

SELECTION

[Sec.

12

direction to provide a completely self-regulating characteristic without the aid of control-rod adjustment. In the usual arrangement, the average coolant temperature remains constant and the reactor inlet temperature will rise with a decrease in power This results in an increase in steam temperature and a corresponding output. increase in steam pressure as load demand drops off. The turbine is provided with the usual speed governor and must be designed for the no-load pressure. The system can also be designed for constant steam pressure, but this requires & variable average reactor coolant temperature and a much larger pressurizer in the In primary system to provide for additional change in volume of primary coolant. either case, the response of the reactor to load changes is rather sluggish and a steam bypass to the condenser should be provided to take care of large load changes and also to provide means of cooling the reactor when the turbine is shut down. It has been stated many times that a reactor should completely control its heat It is recom output and be independent of the turbine prime mover in this respect. mended that consideration be given to the interconnection of the controls because of the probability that the turbine will suffer instantaneous loss of load a number of times during its life. It would seem desirable that the earliest possible signal should be transmitted to the reactor controls, which, of course, would come from the electrical circuit of the turbine-generator unit, from its speed-control mechanism, or from the rate of steam flow. Another problem may be presented when turbines operating at low initial super heats or with dry and saturated steam are required to have large moisture separator! at an intermediate point in the expansion. If the quantity of steam contained in larpr moisture-separating tanks is great enough, the trapped energy may result in undesira ble overepeed. The same is true if the steam is ret urned at an intermediate stage to a source of heat for reducing the moisture content or for superheating. Present solutions in fuel-burning conventional plants for similar conditions employ intercepting valves or dump valves leading to the condenser, actuated by the speed governors. 2.6

Feedwater Treatment

The basic requirement for the water in a boiling-water reactor is that it will be as free as possible of all solids, both ionizable and suspended, and free of corrosive gases. For the initial filling of a system, feedwater that has been deionized and deaerated, such as would be found in a modern high-pressure central-station system, would be satisfactory. Since the water, being free of ionizable solids, has a pH value of approximately 7, it could be expected to be actively corrosive. This makes necessary the use of cor rosion-resistant material, such as stainless steel, for the reactor and for other equip ment in contact with the feedwater, such as pumps, heat exchangers, and piping. Present practice indicates the acceptance of conventional materials of constructioc for the steam piping, steam turbine, condenser, and condensate pump. It must be expected th:it some quantities of corrosion products will be formed in these compo nents. It is therefore necessary that the condensate be filtered to remove such solids prior to its return to the reactor. To ensure an acceptable control of the dissolved solids in the reactor to a level not exceeding 1 ppm, a bypass ion-exchange system, such as shown in Fig. 34, should be The flow is first cooled by heat exchangers, after which it is carried employed. through filters for the removal of suspended solids and then through mixed-bed ioa exchangers.

The effluent from the ion exchangers is again filtered, passed through the regenera tive heat exchanger, and returned to the reactor. Filtration is employed to prevent particles of the ion-exchange resin from entering the reactor. The ion-exchange resin becomes radioactive and is discarded when exhausted. A full-How filter should be installed in the feed line between the condenser hot wri. and the reactor. Future experience may also dictate the desirability of installing full-flow ion-exchange units in certain types of installation.

Sec. 12-3]

POWER-GENERATING

EQUIPMENT

12-151

A possible major source of difficulty in a system where water passes through a neutron flux is the breakdown into the component gases oxygen and hydrogen. Argonne National Laboratory estimates total gases by dissociation in EBVVR as 1 cfm at 20 Mw reactor output, which is about 62 cm'/liter of condensate. The corrosive effect of these gases on system materials has not yet been published, but at any rate This may introduce a serious corrosion problem, particularly they must be removed. in a turbine handling wet steam from a boiling-water reactor. A standard condenser ind ejector system will remove oxygen to a level of 0.03 cm'/liter in the condensate This can be reduced to 0.01 cm'/liter under good operating eaving the hot well. :onditions and by minor modifications of the condenser. A deaerator will deliver eedwater having an oxygen content not exceeding 0.005 cm'/liter. These values ire considered too high for modern boilers, and oxygen scavengers are ordinarily used 0 remove substantially all remaining oxygen in order to inhibit corrosion in both the >oiler and the turbine systems. However, upon recycling through the reactor, the


>xygen

and hydrogen content may again reach an undesirably

high level.

These

;ases can be reeombined catalytically. Another method of approach is described in the paper by Welinsky, Cohen, and iearaon.' The work of these investigators indicates that feeding excess hydrogen nto the steam is effective in recombining the oxygen to form water, resulting in a elativcly low oxygen content at the turbine inlet. The use of hydrogen also permits he reaction with any nitrogen which might be present to form ammonia that in turn ,-ould be removed by the cation resin in the ion exchanger. The use of hydrogen an also he expected to eliminate the possibility of the formation of nitric acid through lie reaction of nitrogen and oxygen and thus prevent the corrosive effect of this comound on the steam turbine and condenser system. The foregoing indicates the necessity of providing deaerated water for the initial

lling. A general statement and summary of the problem might well be that any solids or ases are undesirable in a reactor system and that these elements or compounds lould be prevented from entering with the feedwater. In like manner, compounds r gases that are formed within the system or that otherwise enter the feedwater cirlit should be removed and prevented from concentrating to objectionable levels by le methods described. 2.7

Instrumentation

and Remote Control

The type of instruments and the need for remote control of the power-generating juipment of a nuclear power plant will be dictated by the type of reactor and the level of radioactivity that may exist in the power-generating area of the the level is low and access to the equipment for reasonable lengths of time ;n normally be permitted, much of the auxiliary equipment need not be remotely «ntrolled, even in large installations. Provision should be made, however, to shut On the other >wn all equipment from a central control room in case of emergency. ind, when local controls are not accessible during startup or operation, because of 1 expected high level of radioactivity or the presence of bulky biological shielding, mote-operating features must be provided. For large steam turbines (50 Mw and above), the following instrumentation is ually provided: tensity

ant.

1.

2. 3. 4. 5. (i. 7. 8. 9.

If

Throttle pressure gauge — indicating Throttle temperature-indicating instrument Pressure gauge for first-nozzle inlet pressure Exhaust pressure gauge Exhaust temperature-indicating instrument End-position indicator for high-pressure turbine spindle End-travel indicator for expansion of high-pressure casing from anchor point Oil-pressure gauges at critical points in governor and bearing supply systems Oil-temperature indicators at critical points in bearing supply system

12-152

NUCLEAR-POWER-PLANT

SELECTION

[SEC.

12

means to shut down turbine in the event the main and supple 10. Automatic mentary means of oil supply fail to maintain a predetermined minimum oil pressure 11. Electric tachometer with indicating instrument 12. Turning gear for rotating spindles during cooling and heating periods 13. Valve-position indicator 14. Load-limit-position indicator 15. Speed recorder that takes its signals from the electric tachometer. Two com plete and independent means of recording speed are desirable vibration pickup at each bearing with vibration recorder in 16. Electromagnetic control room 17. Eccentricity indicator, with remote recorder, for observing eccentricity of shaft when starting after a shutdown 18. Remote recorder for end-position indicator on high-pressure turbine 19. Remote recorder for axial expansion of high-pressure turbine casing 20. Remote signal device to operate if oil pressure drops below predetermined value 21. Remote valve-position indicators in control room

The following extra instruments are usually purchased direct from an instrument manufacturer by the utility for convenience in operating and for maintaining records Recording gauges for throttle pressure and first-nozzle inlet pressure Recording temperature instrument for throttle temperature 3. Gauges for pressures at extraction points 1.


2.

All the foregoing turbine operating and supervisory instruments are either devel oped or capable of being modified so that the turbine can be started, operated, and When this is desired, means must be provided for stopped from a remote point. remotely controlling the turbine throttle and control valves and all drains from thf

turbine system with feedback indicators which will show in the control room that the desired valve movement has taken place. It may also be considered desirable to provide means, remotely controlled, for flushing the turbine and other power-cycle components with water or a weak acW solvent in case of contamination of the system by rupture of fuel elements. Standard instrumentation for the generator includes: 1. Pressure gauges, differential gauge, temperature indicator, and chart for deter mining hydrogen purity in system 2. Temperature-operated alarm switch, with high-temperature limit setting, respon sive to hydrogen temperature leaving coolers 3. Pressure switch with alarm for low hydrogen pressure 4. Liquid alarm responsive to the presence of oil or water in the hydrogen space of the generator 5. Pressure switches for starting auxiliary pumps if hydrogen-seal oil pressure drops below predetermined level. An alarm is sounded in this event 6. Thermocouple elements in armature coils 7. Thermocouple elements in cooling gas at inlet and outlet of hydrogen coolers 8. Thermocouple elements in cooling air at inlet and outlet of cooling system for exciter

Electrical include:

instruments usually purchased by the utility for switchboard

mounting

All meters for measuring outputs and voltages of exciter and main generator Recording instruments for temperatures at various points in main generator and exciter assemblies 3. All instruments used for starting, synchronizing and changing load and posvr factor 1.

2.

In some systems oil-flow indicators at the generator provide a sensitive means of determining if the oil-flows through the hydrogen seals are excessive. If desired, s to a give can be remote of condiarranged indication system closed-circuit television

Sec. 12-3]

power-generating

equipment

12-153

lions at these points. The condition of the brushes on the main generator slip rings and exciter commutator can easily be observed by television.


3

OPERATION AND MAINTENANCE

The operation of steam-turbine units for a reactor system should in general be no different from that for any conventional power plant except that remote control may be required. Some of the main considerations are: 1. Complete drainage of steam spaces in all components during shutdown and starting cycles. 2. Reliable means of ascertaining from a control center that circulating water flow is established and condensate pumps are running before turbine is started. 3. Operate turbine on turning gear during the cooling cycle when shutting down and prior to admitting steam when starting in order to equalize temperatures and elimi nate temporary thermal distortion that may have developed if spindle has been at rest during shutdown. 4. Manufacturer's instructions regarding time required for running on turning gear, rate at which the unit is brought up to speed, and rate of loading vary greatly. It is necessary that the instructions of each manufacturer be followed in relation to each particular size and type. One illustrative example of starting a 100-Mw, 3,600-rpm tandem turbine unit from a relatively cold condition follows: 1. Roll the unit on the turning gear for 2 hr. 2. Rolling should continue until any temporary thermal distortion or runout of the turbine shafts at the gland spaces does not exceed 0.0015 in. Gland steam is to be admitted during the latter part of this period, and condenser vacuum built up to the desired starting value. 3. When ready to start, open the throttle and /or inlet valves to run the turbine at a speed not exceeding 500 rpm with the turning gear disengaged. •J. Continue rolling on steam until the axial expansion of the high-pressure turbine casing assembly begins to increase. Manufacturers' instructions should be observed regarding the length of rolling time at slow speed. Then gradually accelerate the turbine to full speed and synchronize. 5. Apply 15 per cent of full load as soon as this can conveniently be done. Do not run large turbine units for long periods at no load or extremely light load. 6. After 15 per cent load has been established, further increases in load will be governed by rates of temperature change in the component parts and by allowable temperature differentials in critical areas. Such instructions can be taken only as a guide because of differences in steamturbine characteristics and in the recommendations of the builders. It is very important that the operators become accustomed to observing carefully the axial expansion of the high-pressure turbine elements from the anchor point. This expansion can be a most valuable index of the rate at which large turbine units <-an be loaded. The excellent vibration indicators available today make it possible frequently to observe and record the amplitudes of vibration of turbine elements. Operators must make routine records of vibration amplitudes and be trained to interpret changes due to load, pressure, and temperature. The foregoing comments have been directed toward the operation of the steam turbine. Of equal importance is the observance of corresponding instructions for the operation of the main generator and all the numerous power-plant auxiliaries. The operating procedures of any power plant must include careful schedules for the starting of auxiliaries and their control which will be appropriately timed with the requirements of the steam-turbine unit. It is self-evident that the operating schedule for all auxiliary apparatus, as well as for the main turbine generator, must be properly coordinated with a corresponding operating cycle of the reactor. Many new problems are introduced in the over-all operation of power machinery with a reactor heat source. The time rate of change of heat output may be quite

12-154

NUCLEAR-POWER-PLANT

SELECTION

[SEC.

12

different from the corresponding rate of change in a conventional plant. The neces sity for absorbing heat generated by isotope decay in the reactor at times when the reactor may be noncritical introduces other modifications in the operating procedures for a nuclear power plant. 4

SHIELDING AND CONTAINMENT 4.1

Shielding Power Equipment

3

1

a

4.2

Building Construction

is

It desirable that the reactor and all piping and vessels containing radioactive water be inside a pressuretight building to contain the steam that would leak from a failed joint or escape from a broken pipe for a sufficient length of time to permit radio active decay to an acceptable tolerance level. The building should be of a geometry and construction to withstand the maximum internal pressure that could be built up in case of the maximum credible accident. Building construction should be planned with adequate escape doors and hatches to permit personnel to leave any room in the building group without delay in the event of destructive occurrence. The site for nuclear power plant should be carefully selected on high and firm ground in order to avoid any possible floods from nearby lakes or streams that might make necessary the abandonment of the plant. survey must also be made of the seismological geology and history of the proposed location to ensure that the possi bility of earthquake disturbances will be extremely remote. Elsewhere in this handbook considerations relative to plant location with respect to general safeguard requirements are enumerated in greater detail. In general, power-plant services, machinery arrangement, handling equipment, etc., A

a

a


is

is

a

if

is,

Shielding engineering is covered in detail in Sec. 7-3. In the selection and arrange ment of the shielding around the prime mover, special consideration must be given the problems involved in handling rotating machinery. Steam turbines and generating units to be used for nuclear power plants are now developed to operate with an extremely high order of reliability. The same applies to station auxiliaries. It nevertheless, desirable to arrange shielding, required, in such way that operating personnel can have easy access to each bearing of the unit. The purpose of this to be able to check shaft eccentricity or bearing vibration in the event the remote-control instruments fail. It also desirable that there be ready access to the thrust-adjusting mechanism so that the end position of the rotor can easily be changed in the event of some unforeseen The safety of personnel attending to these matters in an installation emergency. where the turbine unit might become radioactive must be ensured by very careful monitoring. Consideration must be given to radiation effects on lubricating oil in selecting the position of the oil tank and the oil-supply and drain lines. Radiation effects on the insulation in the generator and its auxiliary equipment must be carefully considered, and the appropriate steps taken to prevent undue deterioration where such are necessary. Large steam turbines should be dismantled after year of service for the first inspec tion. After this initial inspection they can be expected to run years or more between overhaul periods provided recorded data do not show varying pressure and tem These overhaul operations include a complete perature readings for the same loads. dismantling of the turbine, requiring lifting the cover and removing all rotating ele ments to permit thorough inspection of all internal parts. Operating experiences with a specific unit may indicate the advisability of less extensive inspections oi easily accessible components at more frequent intervals. The placement of any shielding must be considered so that these scheduled inspections may take place without an undue amount of reconstruction work.

Sec. 12-3]

power-generating

equipment

12-155

little different from those currently employed for conventional stations pro vided the known fundamental differences are treated with good judgment by the plant designer. need be

REFERENCES 1. Keenan, J. H., and J. Kaye: "Gas Tables," John Wiley & Sons, Inc., New York, 1948. 2. Rettaliata, J. T.: The Gas Turbine, Allis-Chalmers Elec. Rev., December, 1941. 3. Akin, S. W.: The Thermodynamic Properties of Helium, Trans. ASMS, August, 1950. 4. Beldecos, N. A., and A. K. Smith: Comparative Performance of Turbine Units in Satu rated Steam Cycles, Combustion, July, 1954. 6. Elston, G. W., and P. H. Knowlton: Comparative Efficiencies of Central-station Reheat and Non-rcheat Steam-turbine-generator Units, Trans. AS ME, 1952. 6. Welinaky, I. H., P. Cohen, and J. M. Seamon: Chemistry of a Pressurized Water Nuclear Power Plant: A Review of Two Years Operating Experience, Proc. Am. Power Con/., 1956.

BIBLIOGRAPHY "Thermodynamics," John Wiley & Sons, Inc., New York, 1948. "Thermodynamics," McGraw-Hill Book Company, Inc., New York. 1948. Keenan, J. H., and F. G. Keyes: "Thermodynamic Properties of Steam," John Wiley & Sons, Inc.. New York, 1947. Salisbury, J. K.: "Steam Turbines and Their Cycles," John Wiley & Sons, Inc., New York, Keenan,

J. II.:

Lichty, L.


1950.

C:

Harrer, J. M., A. S. Jameson, and J. M. West: "The Engineering Design of a Prototype Boiling Water Reactor Power Plant," Columbia University Press, New York, 1955. Simpson, J. W., M. Shaw, R. B. Donworth, and W. J. Lyman: PWR Power Plant, Westinghouse Engr., November, 1955. Le Clair, T. G., and S. Untermyer: Commonwealth Edison's Nuclear Power Plant, Gen. Elec. Ret., November, 1955. Amorosi, A., A. P. Donnell, and H. A. Wagner: "Developmental Fast Neutron Breeder Reactor," Columbia University Press, New York, 1955. Powell, S. T.: "Water Conditioning for Industry," McGraw-Hill Book Company, Inc., New York, 1954. Allen, R. C: "Steam Turbine Condenser System for a Direct Cycle Boiling Reactor," unpublished, 1953.


SECTION

13

MECHANICAL DESIGN AND SELECTION OF REACTORS BY

UNTERMYER II, B.S., Consulting Engineer on Reactor Technology, Vallecitos Atomic Laboratory, Atomic Power Equipment Department, General Electric Company. B. BRIGGS, B.S.Ch.E., Director of Homogeneous Reactor Project, Oak Ridge National Laboratory.

SAMUEL

R

ARTHUR H. BARNES, Engineering Division,

A.B., A.M., Ph.D. (deceased), formerly Director, Argonne National Laboratory.

Reactor

C. LEVERETT, B.S., M.S.E., Sc.D., Manager, Development Laboratories, Aircraft Nuclear Propulsion Department, General Electric Company.


MILES

HERBERT

S.

ISBIN, Ph.D.,

ing, University

Associate Professor, Department of Chemical Engineer of Minnesota.

ALF KOLFLAT, M.E.,

Senior Partner, Sargent & Lundy,

GEORGE A. ANDERSON,

Engineers.

B.M.E., Manager, Reactor Engineering, Nuclear Prod-

Division of ACF duots-Erco, Argonne National Laboratory.

Inc.,

Industries,

formerly

Project

Engineer,

CONTENTS 13-1

1 2 3 4 5 0 7

WATER-COOLED REACTOR SYSTEMS BY SAMUEL UNTERMYER II

Water as a Reactor Coolant Pressure Vessel Pressurized-tube Water-cooled Reactors* Components of Water-cooled Reactor Systems Special Features and Applications of HjO-cnoled Reactors Boiling and Flashing Reactors Other Types of Hydrogenous Coolants.

13-2

1 2 3 4

13-23

References

PAGE 13-80 13-98 13-99

13-4

13-100 13-102

GAS-COOLED REACTOR SYSTEMS BY MILES C. LEVERETT

1 Gas-cooled Reactors Systems 2 Once-through 3 Recirculating Systems References

13-49 13-50 13-5H

13-5

13-C>2 13-73 13-74 13-75 13-79

H. BARNES

Alkali-metal Systems Heavy-metal Systems Homogeneous Metal Systems Design Considerations of Liquid-metalcooled Reactors References

13-35 13-41 13 -48

BY R. B. BRIGGS General Features and Reactor Types. . . Fuels Reactor Flow Sheets Components of Homogeneous Aqueous Reactor Systems 5 Reuctor Control tj Maintenance Properties of Homo 7 Thermodynamic geneous Reactor Fuels

SYSTEMS

BY ARTHUR

PAGE 13-3 13-12 13-18

HOMOGENEOUS AQUEOUS REACTOR SYSTEMS

1 2 3 4

LIQUID-METAL REACTOR

13-3

KINDS OF REACTORS BY HERBERT

1 Reactor Classification 2 Nuclear-reactor Catalogue 3 Reactor Descriptions References

* Article 3.1 contributed by A. B. Carson. 13-1

13-105 13-108 13-119 13-122

S.

ISBIN 13-12:1 13-123 13-141 13-157

13-2 13-6

MECHANICAL REACTOR BUILDING

DESIGN AND REACTORS

DESIGN

BY ALF KOLFLAT 1 Building

with

No Gaslight

Require

ment

13-160

2 Gas tight


Building with Nominal Pres sure Specifications 3 Gastight Enclosures with High Pressure Specifications

4 Press u re- resistant All-steel Containment

Housing

PAGE

13-102 13-165

[SEC.

5 Plant and Laboratory Layout 6 Building Details

13-7

13

PAOt 13-166 13-173 13-173

REACTOR CONSTRUCTION BY GEORGE A. ANDERSON

1 The Reactor 2 Shielding 3 Circulating System

13-177 13-181 13-1S4

WATER-COOLED

13-1

REACTOR SYSTEMS

BY Samuel Untermyer

1

1.1

WATER

AS A

II

REACTOR COOLANT

Pressure and Temperature Requirements


The same considerat ions that have resulted in the preeminence of water and steam as general-purpose heat-transfer mediums have also caused their widespread applica tion as a reactor coolant. The one major disadvantage of water results from its rela tively high vapor pressure, and boiling is sometimes allowed within reactors to extract Figure 1 summarizes power from water without excessive pressure requirements.

C a

, 1,000

01

0

REQUIRED

Flo.

1.

I

,

2,000 3j000 REACTOR PRESSURE, PSIA

Typical pressure requirements

of water-cooled

reactors.

pressure requirements of water-cooled reactors. It can be seen that useful thermal efficiencies are obtainable with pressures above 300 psi in boiling reactors or above 1,000 psi in nonboiling reactors. It is apparent that water-cooled reactors operating at very high pressures and temperatures can generate power with reasonable efficiency. For instance, a boiling reactor operating at 3,000 psia has an efficiency of about 36 per cent, while a nonboiling reactor at this pressure has an efficiency of about 29 per cent. These figures are typical values, such as might be obtained in large plants of these types.

typical

1.2

Methods of Containing Pressure

There are two ways in which the vapor pressure of the water can be contained, and the mechanical designs of water-cooled reactors may be subdivided into two classes. The entire chain-reacting core can be submerged in water contained within a pres sure vessel (Fig. 2). This pressure vessel must be large enough to contain the reactor jmd its reflector, and sealed openings must be provided for control rods, water sup plies, and various other equipment. The maximum-size tank-type reactor that can be built is usually fixed by the pressure vessel. As an alternative, the fuel and high-pressure coolant can be subdivided and con13-3

13-4

MECHANICAL

,- WATER

INLET

DESIGN AND REACTORS

HEAD

SEAL WELD CONTROL ROD MECHANISM

WATER OUTLET

REFLECTOR —TANK

Fio.

2.

Containment of vapor pressure of

water by

submerging

core

in

pressure


vessel.

Range of Sizes

The specific choice of fuel, coolant, and moderator, as well as the mechanical-design choice between pressurized tubes and a pressure vessel, will serve to place practi cal limits on the size of the reactor plant. Table 1 indicates normal size ranges of typical water-cooled reactors. 1.4

Allowable

Coolant Leakage

1.41 Methods for Reducing Leakage. Escape of reactor cooling water or radio-

WATER WALL

SLIDING SUPPORT FOR THERMAL EXPANSION ENLARGED VIEW CONNECTIONS TO WATER WALL

SHOWING

Fio.

13

tained within individual pressure tubes These tubes must be capable of (Fig. 3). operating in an intense neutron flux and must be able to withstand the internal Since they arc pressure of the coolant. located in the core of the nuclear reactor, neutron absorption must be minimized. Consequently, materials such as aluminum, zirconium, or extremely thin steel must be used for these tubes. 1.3

THERMAL SHIELD

[Sec

3. Use of individual pressure tubes to contain fuel and high-pressure coolant.

Sec. 13-1]

WATER-COOLED

REACTOK Table

Type of reactor

Heterogeneous: HiO, enriched, D»0, enriched,

tank type

tank type

H:0, partially enriched, tank type. . . . OzO, natural uranium, tank type DiO, natural uranium, tube type Graphite, partially enriched tube type Graphite, natural uranium, tube type.


Homogeneoua: HiO, enriched DiO, partially enriched

13-5

SYSTEMS

1

Range of core diameters, ft

>2-5 1- 8 2- 10

6-U

10 18 15-25 Over 20

Application and power range electric generation, kw

Research; 1,000 20.000 Research 20.000-200.000 Research; 10.000 I 10.000 Research; 50.000 400.000 Research; 100.000-400.000 Research; over 400.000 1.000-10,000 Research; Over 10.000

active steam may constitute a health hazard, while leakage or contamination of D20 Thus successful operation of a high-pressure may be a serious economic burden. water-cooled reactor places stringent requirements on the leaktightness of the primary There are two general methods for reducing leakage: cooling system. Such equipment includes sealed-rotor1. Zero-leakage equipment may be used. type pumps, bellows-seal valves, seal-welded containers, and magnetically actuated, sealed-type control rods. All piping must be welded, and the entire system can be checked for leakage with helium leak-detecting equipment. 2. Some leakage may be allowed, but this leakage must be collected by ventilation A slight negative pressure is maintained hoods over each possible point of leakage. in each ventilation hood so that leakage and air are collected. Where HjO is used as a coolant, leakage can be discharged up a stack, but when D20 is used, provisions must be made to collect leakage without introducing HiO contamination. 1.42 Escape of radioactive coolant may introduce radio Biological Tolerances. Normally, contam nation is limited to active contaminants into the atmosphere. Higher doses can be tolerated for 400 disintegrations per cubic meter per second. brief periods or if breathing masks are used, but it is generally inadvisable to exceed this radiation level within an atomic plant. There are three principal sources of radioactive 1.43 Sources of Contamination. contamination: In a typical water-cooled reactor the 7-sec oxy 1. Short-lived Oxygen Activities. gen activity might amount to 10' disintegrations per cubic centimeter per second. 2. Longer-lived Activities Resulting from Dissolved Impurities in Reactor Coolant. Since these impurities are not volatile, air contamination is limited to the material Thus, this activity will be entrained in water droplets carried along with the steam. reduced by the separation factor of water from steam on boiling by a factor of about 10,000 (Fig. 31). 3. Fission Products. These may be introduced as a result of a fuel-element failure in a heterogeneous reactor or leakage of fissionable solution from a homogeneous reactor. These activities include volatile fission products such as the noble gases and iodine. In most cases, homogeneous reactors include fission-gas-removal systems, so However, the gases formed that gases are removed within 2 hr of their formation. during this period still constitute a serious health hazard in event of leakage. In Table 2 it is assumed that the reactor 1.44 Magnitude of Allowable Leakage. is contained in a room 1,000 cubic meters in volume and that the atmosphere within this room is changed every 20 min. Coolant is assumed to contain 10' disintegrations per cubic centimeter per second of short-lived oxygen activity and 10s of long-lived The homogeneous reactor is assumed to operate at 20 kw of nonvolatile activity. heat per liter of solution, and 5 per cent of the fission products are assumed to be gaseous. DiO is assumed to cost $28/lb. A 200,000-electric-kilowatt reactor is This table provides an ordcr-of-magnitude appraisal assumed in these estimates.

13-6

MECHANICAL

DESIGN AND REACTORS

[SEC.

13

Table 2

Type of reactor

Homogeneous Heterogeneous Heterogeneous

Reason for limitation of leakage

, . . . Gaseous fission products . . . Short-lived oxygen . . . Lung- lived activity

DsO

from water impurities Contamination of D2O

DiO

Loss of DiO

Magnitude of allowable

Methods of limiting leakage

leakage

2 em'/year 6 cm* /in in 50 liter /min 2 liters HsO per day 20 liters DaO per day

Zero-leakage and collection at equipment points of possible failure Either aero-leakage equipment or collection Reasonable leakage rates

Dry air at points Special heat exchangers. of leakage. DtO pressure higher than HiO pressure Special heat exchangers Monitors in condenser cooling effluent Leakage recovery system Precautions against accidental loss of DrO due to error of personnel

of the limitations on coolant leakage. However, other factors must be considered ia the design of a specific atomic power plant.


1.6

Reactor Cooling-water

Supplies

1.51 Water Purity in Secondary Cooling Loops (Boiler Feed Water). Since there is always a remote chance that the boiler feed water might inadvertently enter the reactor, high-purity water is required to minimize scale formation in the shielded, highly radioactive boiler. Consequently, it is customary to maintain the highest Resistivity is generally main standards of purity for boiler water in reactor plants. tained at about 500,000 ohm-cm, while total solids are usually kept below 10 ppm. When mild-steel boilers are used, the oxygen content must be reduced or hydrogen may be added to reduce corrosion; with stainless steel boilers, chloride-ion concentration must be minimized. 1.52 Water Purity in Primary Cooling Loops. When the water passes through the reactor neutron field, it becomes radioactive. Radioactivity in water may be due to the short-lived products of neutron capture by oxygen or to activation of impurities dissolved in the water. Owing to the oxygen activities, water that has passed through a reactor remains intensely radioactive for about 5 min. Consequently, heavy shielding is needed around the cooling pipes leading from the reactor, while boilers or heat exchangers require thick shields. Figure 4 shows a closed cooling system such as is used in a pressurized water reactor. Either H20 or D»0 can be used as the primary coolant in the closed cooling system. The steam that is generated in the boiler is not radioactive; hence, such a system does not normally discharge any radioactivity. However, in a homogeneous reactor, where the primary coolant contains a fuel solution, delayed neutrons are transport oil from the reactor to the boiler, so that the steam system may become slightly radio active. Treatment of water for the primary cooling system must meet all the usual purity standards for boiler water, and, in addition, those impurities which might result Use of high-purity in residual radioactivity must be reduced to a bare minimum. water reduces the problems associated with maintenance of primary coolant pumps, piping, and heat exchangers. Such materials as manganese, cobalt, or sodium pro As a result of these stringent duce hard 7s, so that these cations must be removed. purity requirements, the quality of primary cooling water should be maintained by Either such a system may continuous operation of an ion-exchange and filter system. process a bypass stream from the main coolant circuit (Fig. 5), or else a continuous make-up stream of purified feed water can be added to the system while a continuous bleed is removed from the primary loop (Fig. 6). The first arrangement requires that the demineralizer handle radioactive water, and the impurities removed from the water

water-cooled reactor systems

Sec. 13-1]

13-7

are intensely radioactive. Moreover, since ion-exchange resins cannot operate except at low temperatures, water must be cooled before passing through the demineralizer. On the other hand, with the continuous make-up stream (Fig. 6), radioactivity is discharged in the bleed, so that a retention basin is necessary. Experience has indicated that the great bulk of the radioactive materials is removed during the passage of water through the ion exchanger. There are two general classes For materials that form part of the reactor core such as of radioactive contaminants.

STEAM

PRIMARY COOLANT

BOILER

REACTOR

TURBINE

PUMP CONDENSER


PUMP

PUMP

RIVER NON RADIOACTIVE

|

Fig.

4.

A closed cooling system for

use

in

a pressurized

water reactor.

PUMP

FILTER HEAT EXCH ION EXCHANGE

Flo.

5.

Purification by

use of bypass stream from main coolant

circuit.

the fuel cladding or tank walls, the radioactivity developed by corrosion products varies inversely with purificat ion rate. The same reasoning applies to fission products released from the fuel. On the other hand, in the case of isotopes dissolved in the coolant outside the reac tor and entering the reactor through the cooling stream, activity varies inversely with the square of the purification rate. An economic balance should be achieved in the design of the ion exchanger. Such units may vary in size from a capacity to purify the system in 1 week up to a capacity to clean up the entire system in 1 hr. In most cases, it is advantageous to discard used ionic resins without regeneration so that they can be used as a means of storing radioactive material in solid form. Since resins are available that absorb over a pound of ions per cubic foot before regen

MECHANICAL DESIGN

13-8

AND REACTORS

[Sec.

13

eration, this method of storing wastes may often be more economical than the storage of liquid wastes resulting from the regeneration of resin beds. Mixed-bed resins are to be preferred when regeneration is not used, since they produce the highest quality water. 1.63 Radioactive Turbine Contamination with Boiling Reactors. Calculations and available operating experience indicate that the radioactive deposits on the turbine prove to be only a minor nuisance. Consequently, special operating procedures are not required, since the radiation level during turbine overhaul does not exceed safe


RIVER

Fio.

6.

Purification by 600 PSI STEAM

use of continuous

make-up

stream of purified feed water.

STEAM

|

SEPARATION

/

/

STEAM DOME

\\

DRAIN1

FLASH TANK

REACTOR CORE

350 PSI STEAM

600 PSI WATER

350 PSI WATER

CONDENSER

CON

REACTOR PUMP

DENSATE PUMP

DEIONIZER

CP-

PURIFIER

PURIFIER

PUMP

HEAT EXCHANGER

Fio.

7.

A

dual-cycle

boiling-reactor

FILTER

1_

plant.

working tolerances. All radioactivity present during the repair period originates from impurities entrained in the minute droplets of water that are carried into the turbine with the steam. Figure 7 shows a dual-cycle boiling-reactor plant designed for the production of 200,000 kw of electricity. Water within the reactor boils to form steam, which is used to operate the turbine. Water from the condenser is pumped back into the reactor. Purity of water in the reactor vessel is maintained by continuous repurification. The effluent is essentially free from mechanical filtration, and a resin dcmineralizer. Steam separators are used to minimize the entrained water that enters impurities. the turbine with the steam. Experience (Fig. 33) indicates that steam containing less than one ten-thousandth part of water by weight can be delivered to the I


Sec. 13-1]

WATER-COOLED REACTOR

SYSTEMS

13-9

The bulk of the radioactivity in the steam passes through the turbine without being In the calculations, it is assumed that one-fourth deposited on the turbine surfaces. of the solid radioactive contaminants in the steam are deposited in the turbine. Corrosion tests on zirconium and on stainless steel have been used to estimate the rates used in the calculations. Experience indicates that a turbine of this size corrodes or erodes at the rate of about 50 lb /year. The calculations are based on this rate. The condenser corrosion assumption is taken as 0.1 mg/(meter2) (day). Corrosion at condenser temperatures is negligible, but there is erosion due to impingement of water droplets on condenser tubes. It is assumed that maintenance work will not be started on the turbine until one Since it is necessary to allow the steam system time to day after reactor shutdown. cool off in the thermal sense, this period of waiting docs not handicap the plant opera tor unduly. One day after shutdown, the principal activities result from species having greater than 8-hr half-lives. The water repurification system assumed can process ttie equivalent of the entire liquid content of the reactor vessel in about 2 hr. The calculation considers the case in which the reactor has operated until all activi In the case of cobalt, saturation is not reached for about 10 years. ties are saturated. Another possible source of radioactive contamination arises from leakage of cooling water into the condenser. This leakage introduces other contaminants such as The calculations include a leakage rate of 1 gph into the sodium into the reactor. condenser. In the special case of sea-water cooling, this leakage rate introduces serious chloride corrosion problems, so that it cannot be permitted to persist for reasons other than radioactivity. Fuel -element failures probably constitute the most serious source of turbine radio activity. These failures do not occur often, yet it is expected that there will always The jackets on these be a small number of defective fuel elements in each fuel charge. elements may fail, permitting the fission products in the fuel to become dissolved in the water at some rate. The case that is calculated assumes that 1 lb of irradiated fuel suddenly becomes dissolved in the reactor water. Again, it is assumed that the turbine must be repaired one day after this incident. For the purpose of these calculations, the turbine is considered to be a large cylinder about 60 ft long and 15 ft in diameter. It is assumed that the -/-emitting contami nants are deposited so that they will emit radiation outwardly without any inter The noble ference, even though actually the turbine serves to shield the radiation. fission-product gases pass through the turbine and are removed by t he condenser airremoval system. The results of these calculations are shown in Table 3. Possibility of Flushing out Turbine before Servicing. Experiments in several labora tories have indicated the feasibility of flushing out most of the turbine contaminants with special solvents. A dilute solution of periodic acid has generally been found to be the most effective solvent for this purpose. Decontamination factors of 10 to 100 are usually obtained by these procedures. It is important to use very 1.64 Water Purification with Mixed-bed Resins.* pure water as a coolant and moderator in a nuclear-reactor system. Mixed ion-exchange resin columns can yield water of more than 15 megohm-cm with a neutral pH. Experiments indicate successful use of mixed resin beds at unit flow rates up to 100 gpm/ft5 (beds 3 ft deep) at 3 to 192 ppm influent concentration. The normal industrial rate for deionization of water in separate resin beds is about These results have been obtained 5 gpm/ft2; higher flow rates result in ion leakage. with strong-base polystyrene resins such as IR-120, HCR- and IRA-410, and SAR (Dowex 2), respectively. The more thermally stable resins Dovvex and IRA-400 The capacity of the are more desirable for use in water at higher temperatures. cation beds should be one to two times the capacity of the anion resin beds. The resins may be purchased generated to 5 meq (milligram equivalent) per dry gram for the cation resin and 3 meq per dry gram for the anion resin. + By Michael SiveU of General Foods Co. Based right 1955 by the American Chemical Society.

on an article in Ind. Eng. Chrm..

May, 1955. copy

V . . . .

2 %

X 10 <

5

X

'.

2 X

I

Na

gph)

solids

Approx 50

0.9 Na" 14.8 1.000 Na«< 14.8

of radioactive

passing through

10*

gph)

turbine

90.000 (at

10 ppm Na

Normal cooling water

leak

products

90.000 (at

1.2%

Sea water

Condenser

Fission

factor in boiling, (3) one-fifth

2

2

hr, (2) 10* decontaminated

0.2

1.3

1 1.3

<

on the average every

10

17 Cu" 0.65

2 Co" 2.750

products

(instantly)

550 Rapid 454

Fission

Fuel-cladding rupture, fuel

Overhaul

1

6

rcpurified

10

0.2

78% Cu 20.000 100 20

60 % Co

shields

Con denser

during

2

X

5

reactor water Rased on adhere to blades.

1.3. 0.3

0.7

Cr

Bucket

components

in Turbine

1

2

0.

59. 1.2 Fe", Cr" 65. 39

27 Zr" 94

Fe,

Structure

60

98

Level

system corrosion

Turbine

Zr 900 sso 90 30 %

1.3

0.0017 Co" 2.750

Co

cladding

Fuel

Normal

of Radioactive

I

ingredient corrosion or leakage, g/day Active isotope Mean life, days Gamma energy, Mev per disintegra tion Calculated in turbine mr/hr during repair

0.07% 150 500 15 2.2

Reactor tank

Estimates

g

Active

Activated ingredients to Surface area, meters' Operating temperature, °F rate, nig/(meters')(day) O Corrosion Total corrosion or leakage, g/day.

Source of material

Table


3.

1

I)

(

Sec. 13-1]


13-11

In use, cation resins will pick up 1 gram-equivalent per liter and anion resins will pick up 0.5 gram equivalent per liter before the effluent quality of mixed resin beds falls below the magnitude of megohm-centimeter (specific resistance) quality. Mixed ion-exchange resin columns can yield water of more than 15 megohm-cm from water, and their ability to do so is valuable. However, mixed resin beds should be flanked by 100-mesh screens and be protected by a prefilter so that they are not A postfilter is desirable to prevent fine resin frag plugged with particulate matter. ments from entering the main water recirculatory system. A characteristic of resin beds used in conjunction with dcmincralization of reactor cooling waters is the con centration of radioactive ions at the top of the bed. When these "hot" resin beds no longer pass megohm-centimeter quality water, then the resin may be ashed and the ash may be stored for decay of radioactive elements.


Table 4.

Effect of Temperature for Pure Water

Temp, °C

pH

0 25 SO 75 too

7. 5

7.0 6. 7

6.4

Specific

resistance,

ohm-cm

75 X I0« 18 X 10" 5.6 X 10'

6. 1

The most important factor in measuring water pH and resistivity is that these properties be measured in the flowing stream. Recorders for pH and resistivity are preferred. Samples removed from the flow system for measurement in the laboratory do not give true readings, not only because of contamination of water by handling and glassware but also because of rapid carbon dioxide solution in the water from the atmosphere. Megohm-centimeter water is unbuffered and can give erratic pH read ings. Oxidation of organic matter in a reactor usually results in carbon dioxide. The resistivity of the reactor water should be recorded before and after it passes the Commercial industrial conductivity recorders have been found bypass resin column. reliable to 1 megohm-cm; however, above several megohm-centimeters with 0.01 cell constants, varying resistivities were obtained, depending upon rate of water flow past the electrodes. This is believed to be the result of leaching phenomena. Platinum metal electrodes are recommended for exact work where such deviations are to be eliminated. The resistivity of the inlet water to the resin bed indicates the quality of dissolved ionized solids in the bulk reactor water, while the resistivity of the resin-bed effluent indicates the absorptive ability of the resin bed. Table 4 shows the pure water (H20) resistivity and pH variance with temperature. High-purity heavy water (D20) has a specific The resistivity varies appreciably. resistivity about three times greater than H20 of the equivalent purity, and its dis The pH of pure DsO is 7.0. sociation constant at 26°C is 1.95 X 10"". If water resistivity is above 1 megohm-cm, the pH, being related to water resistivity, When the bulk water resistivity is so near neutral that its corrosion effects are slight. falls below 1 megohm-cm, the pH record of resin column influent indicates better than resistivity when to replace a fresh resin bed. It may be advisable to have monthly spectrochemical determinations of all metal elements in the water and chemical determination of particular anions. In addition, the curve of specific radioactivity and decay for the reactor water is useful. A typical bypass installation for mixed-resin purification is shown in Fig. 8. Since the radioactivity is concentrated by the ionic resins, monitors located near the deraineralizer normally give accurate and early indications of fuel-element failures or unusual corrosion. Distillation has also been used to clean up accumulated radioactivity. Evaporative stills are preferable for grossly contaminated water. After cleanup is completed, ionic resins maintain purity more conveniently and at lower cost.

MECHANICAL DESIGN

13-12

AND REACTORS

[SEC. 13

TO PUMP SUCTION FROM PUMP

PH RECORDER

Fig.

8. A

typical bypass installation for mixed-resin 1.6

Deuterized Ion-exchange

purification.

Resins*

is

}>i

To avoid isotopic dilution of DjO systems, H20 must be removed from ion-oxchange resins. This is accomplished by passing D2C) slowly through the resin bed, and upflow To is necessary to minimize mixing H20 (density 1.00) with DiO (density 1.10). eliminate gas voids, which may result in channeling and inefficient deuterization, the resin bed should be first filled with H20. Experiments indicate that a flow velocity The course of deuterization can be of cm/min may be used for deuterization. Before an expended followed by measuring the specific gravity of the effluent water. deuterized resin bed is discarded, the D20 is reclaimed. H20 is passed down through reclaimed as the effluent. the bed at cm/min, and isotopically diluted D20 Deuterization of resins requires a volume of heavy water equal to about one-fourthcolumn volume of the ion resin bed or results in half the resin-bed volume of D-O of Dedeuterization results in the same degree of dilution of 50 per cent isotopic purity. 2

D,0.

PRESSURE VESSEL

2.1

Size Limitations

is

a

it

is

it

ft

long can be Rail Shipment. Ordinarily 13-ft-diameter tanks up to 45 2.11 Tanks should be shipped with transported by rail on specially designed flat cars. stub nozzles attached, so that welding pipe connections after erection will not affer; In some locations these clearances are not available, while there the reactor vessel. In any specificare some routes that can accommodate 14-ft-diameter vessels. instance necessary to consult the railroads that will be concerned with the ship ment, as the detailed clearance limitations are frequently altered. Water Shipment. Barge shipment allows transportation of the largest pres 2.12 For instance, 20-ft-dianiet«r sure vessels that would ordinarily be used for reactors. vessels may be readily shipped by water. not possible to ship the pressure vessel to 2.13 When Annealing at the Site. the site owing to insufficient rail clearances, then the vessel can be fabricated at the This procedure has serious disadvantages and should be adopted only reactor site. last resort. The pressure vessel may be partially constructed at the factory, and as Annealing will be necessary after welding, the final welding can be done on the site. and a temporary annealing furnace must generally be constructed at the site for this After fabri purpose, although electric heating can provide annealing in some cases. cation, X-ray inspection of welds, and annealing, the tank must be pressure- and leak-tested. *


CONDUCTIVITY RECORDER

Bused on a paper by M. Sivetz and C. H. Scheibelhut.

Sec. 13-1]


13-13

SYSTEMS

From the foregoing, it should be clear that field fabrication of the reactor pressure Consequently, at is expensive and results may not be wholly satisfactory. the present time, reactor plants do not ordinarily use pressure vessels larger than can be shipped from the manufacturer's plant to the site. vessel

2.2

Structural Limitations

All reactor vessels must meet construction codes, though 2.21 Pressure Stresses. it is not clear at present whether the ASME Boiler Code or the ASME Unfired These codes specify the allowable strength of mate Pressure Vessel Code applies. allowance for rials at various temperatures and also indicate welding efficiencies, Final design of the reactor vessel should nozzles and other openings, and so forth. As a rough preliminary indication of follow the principles laid down in these codes. For the nominal tank-wall thickness, a working stress of 15,000 psi can be assumed. the walls of a cylinder, (internal pressure) (diameter) X working stress

The walls of a spherical section need be only half as thick as those of a cylinder of the same diameter operating under the same internal pressure. The walls of a reactor vessel are heated 2.22 Thermal Stresses, Thermal Shock. during operation by y rays, which originate from two sources: 1. 7s produced by fission in the reactor core 2. Capture y rays emitted in or near the tank walls owing to capture of thermal neutrons by the material in the walls or by other material near by Fission ys arc generally assumed to have an effective range corresponding to an About 5 per cent of the reactor heat is emitted as 7s. They are energy of 2 Mev.

emitted by a layer of the reactor core which is effectively two or three half thicknesses. They are attenuated in passing through the reflector and other material in the tank. These 78 produce more heat in the inner part of the tank wall. Of all the neutrons emitted in the reactor, the fraction (k — l)/k will escape. After attenuation through the reflector and thermal shield, some neutrons will impinge Capture of a upon the tank, and some of these neutrons will be of thermal energy. thermal neutron by iron is usually taken as liberating 5 Mev of hard 7s. These are captured in the tank walls and result in additional thermal stresses. Table 5 suggests the most likely sources of nuclear heat generation in pressure vessels of water-cooled reactors: Predominant Unreflected HiO reflector DjO, graphite,

heat eoure*

Both neutrons and gammas Gammas

heryllium, or lead reflector.

Neutrons

The stresses from continuous heat production in tank walls and any rapid change in water temperature may cause severe thermal shock. Such a situation requires indi vidual calculation, but it is apparent that thermal shock is most serious with thick

Thermal shock can be minimized by reducing the rate of change of the reactor period. rays or Irradiation with 2.23 Radiation Hardening and Neutron Barriers. Irradia fast neutrons from the reactor may affect the properties of the reactor vessel. analogous to cold work, so that the tion generally has an effect on metals that The transition temperature tensile strength increases while the ductility decreases. necessary to consider from ductile to brittle states may also be reduced. Thus, the material and the applicable design tensile strength and elongation in the light of the results of irradiation tests. In order to diminish the effect of irradiation on the tank wall, neutron barriers may Neutron barriers take be interposed between the reactor and the pressure vessel. the form of fast-neutron shield materials such as iron-water combinations and also pressure vessels.

7

is

it

is

reactor output, that

is,


2

Thickness

13-14

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

In cases where the thermal-neutron flux at normally provide sufficient 7 shielding. the tank is high, resulting in the production of capture 7 rays near the tank, it is possi ble to depress the thermal-neutron flux without producing 7 rays by the use of hydro materials, which absorb the neutrons without gen-, boron-, or lithium-containing emission of hard 7 rays. 2.3

Pressure Closures


2.31 Head Seal Weld. Since little leakage can be tolerated, special types of seal arrangements are frequently used on tank heads, control rods, and other critical HjO tank-type reactors require leaktight openings up to 10 ft in diameter points. When absolute leaktightness is required through the top of the reactor vessel. The load on the head where ventilation is not practical, seal welds should be used. is taken by bolts or other mechanical means, while leaktightness is secured by a final seal weld. This weld must be sacrificed to open the reactor for removing fuel, and a new weld must then be made. Figure 2 shows a typical seal weld for H20 reactors. Such welds should be used only where ventilation is impractical. 2.32 Several types of gasket seals are obtainable Special Gasket Arrangements. At pres that may be used when ventilation is available to remove minor leakage. sures up to 350 psi and for small openings more conventional gaskets of the Flexitallic type can be used, but for higher pressures all those closures rely on the principle that internal pressure on the head should be used to seat the gasket. Where collection of all leakage is impor Some of these closures are shown in Fig. 9. tant, a light secondary seal is provided and the leakage is collected between the two seals. 2.33

Multiple Openings. In the case of DsO reactors, a large number of smaller openings arc required in the upper pressure head to allow withdrawal of fuel. A typical large D20 reactor requires over one hundred 5-in. openings about 10 in. apart. Seals for such openings are described in Art. 4.11. 2.4

Tank-type

HsO Reactor Designs

These reactors generally use vertical coolant flow. They are characterized by their high solid content. Typically, the ratio (solids in core) /(total core volume) might range from 0.3 to 0.7. Consequently, the main design problems are those associated with packing a large amount of material into a small volume while leaving space for coolant flow. When the reactors are constructed in small sizes, the coolant pass through them may be too short, so that a double-pass design may be used (Fig. 10). The water is frequently used as a coolant, moderator, reflector, and thermal shield, The water may also and an annulus of water is provided around the reactor core. be used as a shield so that fuel assemblies can be exchanged and the reactor repaired during shutdown by a worker who is stationed above the reactor vessel and observes For such a shield 15 to 18 ft the work through a protective layer of water (Fig. 20). of water is required. Many Control rods in a power reactor must operate in high-temperature water. designs have been proposed, and some have achieved a record of successful operation. The fuel-unloading operation is made easier by location of controls below the reactor, while in many cases control-rod mechanisms are simplified by placing them above the The nuclear properties of the H20 coolant result in severe restriction^ reactor core. on control-rod design, since introduction of large water passages within the reactor causes local overheating of the adjacent fuel. Consequently, provisions should be made to fill spaces left by withdrawal of control rods, and rods must be designed 50 that they do not introduce water passages. Fuel may be removed through the head of the pressure vessel. This head need not be quite so large as the reactor core, since the design often permits moving fuel side ways during removal or pulling light fuel elements out at an angle. Fuel may be removed eit her as a unit in case of small reactors or in the form of rods or assemblies in the case of larger reactors. Where the entire core is removed as a unit, it is withdrawn through the top opening into a coffin, but when fuel is withdrawn in smaller sec-

WATER-COOLED REACTOR SYSTEMS

Sec. 13-1]

CHANNEL BARREL

HEAD

LTD-

—

13-15

SEGMENTAL SHEAR RING BLOCK

SET SCREW BEARING COLLAR

COMPRESSION RING GASKET RETAINER RING

OUTER GASKET


INNER GASKET

ELLIPTICAL HEAD LOWERED. TURNED TIPPED TO PERMIT REMOVAL

AND

WALL OF VESSEL SLIGHTLY ELLIPTICAL NEAR TOP —

ASSEMBLED

THE SILLERS

FLEX

BREDTSCHNEIDER

ELLIPTICAL

GASKET BEING

RING

HEAD

REMOVED

CLOSURE

Flo.

DESIGN

9. Several types of closures

PLUG BEING

GRISCOM-RUSSELL HEAD CLOSURE used.

REMOVED

13-16

MECHANICAL DESIGN

AND REACTORS

[SEC.

13

tions, it may be either removed through the top of the tank or lowered through a tube into a storage basin. 2.5

Tank-type

DÔ-cooled and -moderated Reactors


Those reactors normally use vertical coolant flow, directed upward through indi vidual cooling passages which surround each fuel assembly. The main limitation on

TWO PASS CORE

SINGLE BASS CORE

Fia.

Single- and (for small HjO-cooled reactors. 10.

reactors)

double-pass

coolant-flow designs for tank-type

Fig. 11a. Top view of tank-type DiO-cooled reactor illustrating design problem caused by the number of large openings required. As a consequence, i< this type of reactor is the large-size pressure vessel required. generally is not feasible to use a single pressuretight closure, and so an opening is pro vided for each fuel tube. A mechanical-design problem is presented by the multi Extren* plicity of large openings (Fig. 11) through the upper part of the vessel. thicknesses of metal are often required to resist the pressure stresses at these point*

Sec. 13-1]

WATER-COOLED

REACTOR

SYSTEMS

13-17

Fuel is removed through the top of the reactor, and provisions may be made for removing and reclaiming some of the D20 droplets from the fuel. Control rods are usually arranged vertically, with a rod replacing a fuel lattice position. Fuel may be arranged in a triangular or square lattice, but the triangular arrange ment is generally preferred for pressurized reactors, since it provides stronger liga-

A


HAND HOLES

„ UPPER ASSEMBLY SUPPORT GUIDE

UPPER SHIELD SUPPORT BEAM UPPER ASSEMBLY SUPPORT

ELEVATION

-THERMAL

SHIELD DzO EMERGENCY DRAIN"

Fig.

1 16.. 'Cross-sectional

*

S

/

WATER

INLET HEADER

view of reactor shown in Fig. 11a.

ments in the tank hcfad between fuel positions (Fig. 25). With triangular-lattice spacing, controls are oiften arranged in either "fused" or "unfused" hexagon pattern. The fused pattern provides one control for each two fuel channels, while the unfused pattern uses one contitol rod for each six fuel channels. In practical designs it is usual to assemble a cluster i of rods or plates into a fuel assembly. In this way, fewer but larger fuel channels are required. This design improves nuclear properties and requires fewer but larger openings in the tank. Typically, a large power reactor

MECHANICAL DESIGN

13-18

AND REACTORS

[Sec.

13

might require hundreds of fuel assemblies within a 10- to 18-ft -diameter vessel. Control-rod operating mechanisms are most conveniently located under the pressure However, vessel so that the top of the reactor is clear for the reloading operation. some compromises are often required because of mechanical-design considerations in connection with control-rod drives. In most cases, D20 is used as a reflector in this type of reactor. The high cost of this material necessitates a careful study of the optimum reflector thickness. 3

PRESSURIZED -TUBE WATER-COOLED REACTORS 3.1

Pressurized-tube

Graphite-moderated

Reactors


By A. B. Carson* The reactor generally consists of a rectangular block of Plant Description. 3.11 graphite containing a large number of uniformly spaced horizontal tubes of lownuclear-cross-section corrosion-resistant metal for the placement of canned cylindrical fuel elements and cooling water. The tube spacing, amount of fuel, and coolingwater volume can be varied over fairly wide ranges as needed for heat-transfer and structural requirements as long as the ratio of graphite to uranium remains fairly constant. Acceptable nuclear performance can be obtained with spacings from 5 to 10 in. or more with weight ratios of graphite to uranium of about 5:1. These tubes extend out through the thermal and biological shielding that surrounds the graphite block and are connected to an external cooling-water-supply system. Channels arc also provided to allow control rods containing neutron-absorbing material to pene trate the graphite block. Some of these are water-cooled and can be accurately positioned for fine adjustment of reactivity during operation, while others are equipped for rapid insertion and are used to ensure reactor shutdown when demanded by abnormal operating conditions. The water plant may be equipped to provide a continuous supply of cold water for single-pass cooling of the reactor. Alternatively, the cooling system may be of the closed-loop recirculating type. In this case, eqV'Pracnt f°r removing the heat from the coolant stream must be supplied, which may Result in higher capital cost than for the once-through type of system. However, higher water quality can be eco muc^ nomically maintained in a closed loop, making it adaptable to high-pressure, hightemperature operation and efficient power recovery. ^ Plant auxiliaries include gas-circulation and control equipment for providing a properly inert atmosphere for the moderator graphite, Viuipment for discharging and recharging fuel elements, and storage facilities for both few and spent fuel. 3.12 Capacity. Graphite moderation is applicable t? reactor sizes and power rat ings ranging from essentially zero-power test or research Installations to the relatively high-power Hanford production reactors. The minimum reactor size is a function of fuel enrichii*nt> nuclear quality of the in-pile materials, specific power of reactor operation, and tV manner in which the fur! is distributed through the moderator. As an illustratio ' °f s'ze> a "zero-power" critical assembly can be obtained with as little as 40 tons $ natural uranium distrib To accommodate the uted in a graphite cylinder about 10 ft in diameter and lengthreactivity-loss effects of cooling-tube materials, fuel-elem41' cladding, moderator voids, and fission-product build-up encountered in long pposure or high-power operation, either the active volume must be increased by asmuc'1 43 a factor °f 10 or the Um content of the fuel by a factor of 1.5 to 2. The maximum reactor size will generally be limited by the û*on'um Product'on tuê con" rate or power-generating capacity desired in a single unit, sinc("le tyPe struction and the structural capabilities of the moderator make P088'"'6 to build and operate water-cooled graphite reactors of very large size. 3.13 Design Problems. Moderator. Criteria for the KraPn'te >n(,'ude modefc°r *,8t'CSdensity car. density, purity, machinability, and irradiation-damage charact be adequately controlled during manufacture and, in fact deli*ate var'at'ons can

\

* General

Klectric Company, Hanford Works.

Sec. 13-1]

WATER-COOLED

REACTOR SYSTEMS

13-19

by the designer to obtain the most desirable nuclear characteristics in various Purity can also be closely controlled during manufacture, and parts of the reactor. the most desirable degree of purification for it is possible to evaluate economically Both raw materials and manufacturing methods must be different components. closely controlled in order to obtain predictable irradiation behavior. The moderator consists of a "stacked" assembly of individual graphite bars having dimensions that are some convenient multiple of the basic lattice spacing but with machining and handling. Dimensional over-all size small enough for convenient tolerances on individual bars are not stringent, since the necessary over-all stack dimensional control can be maintained by tailoring individual layers after "lay-up" Bar configuration is designed to provide process tube support, circu is under way. lating gas passages, control-rod openings, and means of keying the bars together for Specific provisions are also made, by choice of material over-all structural stability. and dimensions, for thermal expansion and long-term irradiation effects on structural properties and dimensions. Shield. The relatively large size of these reactors increases the incentive to use cheaper shielding materials and has led to extensive application of heavy aggregate concretes. Design of such shields calls for obtaining practical placement conditions Design of shield perforations for control rods, and limited operating temperatures. instrument channels, and the thousands of process channels is a major item, but by utilization of the shielding and scattering properties of the cooling water and the


be used

GRAPHITE

THERMAL

Fig.

12.

SHIELD

BIOLOGICAL SHIELD

An example of shield perforation assembly.

usually generous space available,

suitable designs can be obtained without resorting close dimensional tolerances or special materials. This allows shield erection to be accomplished by normal construction crews employing conventional practices. An example of a shield perforation assembly is shown in Fig. 12. Control-system requirements are not greatly different from those of other types of thermal reactors if lattice geometry is adjusted for optimum reactivity behavior. (The effect of coolant on reactivity is shown by Fig. 15.) Conventional rack-andpinion hydraulic- or electric-drive mechanisms can be applied to operational control systems with simple hose-reel arrangements for rod cooling-water supply and disposal. Fail-safe gravity systems are generally adequate for safety controls, although other acceleration methods may be incorporated where more rapid insertion is required to ensure that power excursions can be prevented under all operating conditions. Cooling-systems material limitations and water-treatment requirements are similar to ot her pressurized water applications. In most cases graphite-reactor process channel spacing permits the use of crossheaders to service rows of process tubes, with separate connectors to individual process tubes. Similar systems can be used for effluent and influent lines, permitting flow control as well as temperature and fission-product release monitoring on an individual tube basis. Care must be taken in the design of fittings, connectors, valves, and strainers to minimize pressure drop and flow turbulence in order to maintain good pumping efficiency. One of the major design requirements of this type of system is allowance for thermal expansion, especially between process tube and crossheader. Common practice has been to incorporate a flexible connector between these components so that each tube is free to expand independently. Cooling conditions requiring very high flow rates or the use of alternate lattice geometry (small lattice spacing or groups of small process tubes without intervening graphite)

to

13-20

MECHANICAL DESIGN AND REACTORS

[Sec.

13

may make the application of individual-tube cooling-water supply less attractive. Alternate designs featuring common water supply to groups of tubes allow some sim To reduce the pressure-drop and pipe-layout problems plification in piping layout. further, an integral water chamber design encompassing the entire process channel pattern may be used in place of the crossheader system. Process tubes extend through the water chamber by way of tube sleeves, which also serve to tie the chamber walls together and to provide an anchor for process-tube fittings. Process-tube thermal expansion is accommodated by permitting relative movement between inlet and The flexible connectors for individual tubes are replaced by outlet water chambers. A typical piping scheme is rigid fittings more suitable for high-pressure service. shown in Fig. 13 Since continuity of cooling-water supply is of primary importance to the safety of a reactor, cooling-system design provides for reserve and emergency pumping power, spare pumping units at each station, and pump decay characteristics such that if reac tor shutdown is initiated at the moment nonstandard conditions are detected, the CONTROL EMERGENCY

VALVE

OUTLET WATER

WATER

VENT

OUTLET NOZZLES (TYPICAL 4 SIDES)

THERMAL SHIELD COOLING

OUTLET Generated for wjivans (University of Florida) on 2015-09-23 02:58 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

LOOP

WALL

INLET

HEADER

LOOP

HEADER

EFFLUENT, WATER HEADERS

INLET NOZZLE

INLET NOZZLES (TYPICAL 4 SIDES)

THERMAL BARRIER INLET WATER WALL

CHECK

BACK

VALVES

FlO.

13.

FLUSHING LINE

Typical piping scheme.

cooling-water supply will always be adequate to prevent physical damage to the reac In the primary loop of recirculating coolant systems, tor or melting of fuel elements. The most economical practice for seal leakage must be eliminated or controlled. installations the size of graphite-reactor plants is the use of gas-seal pumps or injec tion-type mechanical seals rather than the zero-leakage canned-rotor designs. This practice also permits the use of flywheels to provide continuity of cooling-water supply in event of pump power failure until lower pressure emergency systems can be applied. To provide cooling during shutdown, primary system pump drives may be designed for low-speed operation with pump characteristics capable of supplying reduced flow, which may be as little as 5 per cent of normal. It is sometimes more desirable to provide a separate cooling system for shutdown use which may be incorporated with the emergency cooling system mentioned above. The reactor design must provide for efficient unloading of spent Loading Systems. fuel elements and reloading of fresh material. Because of the channelized nature of Eleva graphite reactors, these operations are performed on an individual tube basis. tors are provided on both inlet (charging) and outlet (discharging) faces of the reactor for this service. If operating conditions permit unloading to be done with the reactor at zero power, design problems are greatly reduced, since radiation levels and coolingwater requirements are much less severe and process channels can be opened directly to atmosphere so that no special equipment is required at the discharge end of the

WATEU-COOLED

Sec. 13-1]

REACTOR

13-21

SYSTEMS

Low-pressure charging machines are designed to handle over 1 ton/hr of uranium, and proper elevator layout will permit two or three such machines to be Provision must be made for control of alignment and charging used simultaneously. forces to ensure that the reloaded fuel column is undamaged and properly positioned. For reactors designed for irradiation service only, loading during shutdown is usually the most economical method of operation, since normal maintenance requirements of control systems, instrumentation, and pumping plants make periodic shutdown neces The same may be true for power reactors if resistance to damage of the fuel sary. However, element and long-term reactivity behavior permit long fuel-loading life. to ensure maximum operating efficiency for all conditions it may be possible to design In this ease both inlet and outlet for unloading and reloading during operation. operations have to be carried on remotely and at full cooling-water flow. Although the design problem is more difficult and attainable charging rates considerably lower, the potential improvement in operating efficiency provides strong incentive to develop the elaborate systems required. Process monitoring is required to maintain accounting of source and fissionable material, to permit maximum reactor operating efficiency, and to ensure reactor safety. Individual tube power is monitored by measurement of flow rate and tem perature rise of the cooling water. Accuracy and speed of response are important for control and safety, but the major factor in the design of such instrumentation is Since there may be thousands of sampling points and operating serviceability. continuity may be very important, average service-life requirements of sensing ele ments, amplifiers, switches, indicators, etc., are among the most severe encountered anywhere. Monitoring of groups of channels rather than individual tubes reduces the cost and complexity of the instrumentation greatly, with some loss in detailed information for safety and accountability but no significant effect on operating Other necessary features include alarm or control-actuating signals upon efficiency. detection of abnormal flow or temperature conditions, graphic display of power dis tribution either continuously or at 3- to 5-min intervals, and automatic digital readout of individual tube power for accounting purposes. These requirements are met by careful system design and selection of components from currently available equip Total reactor power level is monitored by measurement of coolant conditions ment. and of neutron flux level. The latter quantity is not precisely related to reactor therefore, power but provides a very sensitive indication of over-all reactivity and Safety circuits are designed for maxi used in both safety and control applications. mum reliability and inherent "fail-safe" characteristics. Positive indication of fail ure or malfunction and either backup components or automatic reactor shutdown are is,

provided.

Since fuel-element failure may result in contamination of the primary coolant sys

problems.

y, or neutron activity

is

it

desirable to provide sensitive fission-product tem with fission-product activity, detection in the effluent system in order to minimize operating difficulties and cleanup £),

can be monitored at the tube outlet or by side-

stream sampling for indications characteristic of fuel-element failure. Available detection and amplification equipment can be used to provide adequate serviceability

and

sensitivity. 3.2

Pressurized-tube

D20-moderated

Reactors

In the past, D20-moderated pressurized-tube Arrangement. of vertical-tube construction with either upward or downward water flow. Coolant may be either DjO or H20. If H20 used, nuclear considera tions usually require that its volume be minimized. D20-moderated gas-cooled reactor has also been constructed for research purposes in France. While all D2() no reason why a horizontal-tube reactors have been of the vertical-tube type, there .O-moderated reactor could not be built, although support of the horizontal fuel tubes might present some difficulties. The coolant can flow either upward or down ward in vertical-tube reactors. The advantages of upflow are: 1. Steam bubbles rise with coolant stream. 2. Exit monitoring simplified. 3.21

Fuel-tube

been

is

A

is

reactors have

O

is


tube.

13-22


[Sec

13

"

power distribution may be obtainable with 3. An advantageous "roof-topped poison controls. 4. Natural circulation can be used to provide shutdown cooling. On the other hand, downward coolant flow also has certain advantages: 1. Inlets to tubes are accessible, so that it is readily possible to adjust coolant flow according to heat generation in tubes. 2. Coolant is discharged at the lower end of tank, so that the most radioactive portion of coolant loop is removed from operating area. 3. Seals for water feed to fuel tubes are more readily accessible for inspection and repair. The volume of uranium metal fuel might vary from Design of Fuel Tubes. 3.22 4 to 10 per cent of the moderator volume, and when the volume of coolant, coolant tubes, and coolant cladding is added to these figures, the total volume within the fuel Naturally, this tubes might range from 10 to 20 per cent of the total core volume. introduces problems associated with closely spaced cooling tubes. The following table illustrates the situation for a square-lattice reactor in which the fuel coolant and material comprise 15 per cent of the total volume of the reactor core: associated •


Coolant tube dinmeter, in

I

2

3

4

2.3

4.6

6.8

9. 1

5 11.2

6

13.7

Since, because of heat transfer, fuel dimensions must be less than 1 in., it is clearly advisable to pack a number of fuel elements into a single cooling tube, thereby permit ting a larger lattice spacing as well as appreciable nuclear benefits. Such a "clumped lattice" may consist of a cluster of fuel rods (usually 3, 4, 6, 7, or 19). These rods may be supported within individual cooling tubes, while, on the other hand, a single

SINGLE FUEL ROD

Fig.

MULTIPLE ROD CLUSTER 14.

Fuel-tube designs.

Alter large cooling tube may contain a complete cluster of fuel tubes (Fig. 14). nately, fuel plates may be used and a number of plates can be assembled as a unit within a single coolant tube (Fig. 25). It can be seen that clumped lattices have several practical advantages, since the larger tubes are stiffer and more easily handled while the reduction in the number of fuel assemblies notably simplifies fuel reloading operations. 3.23 Either H20 or D20 may be used to cool D-O-moderH20-cooled Designs. ated reactors. However, when H20 is used, its volume must be limited, both because it reduces the conversion ratio of the reactor and because loss of a large amount of Figure 15 indicates design limi H20 may cause an excessive increase in reactivity. tations for H20-cooled D20 reactors. Reactors that lose reactivity upon loss of coolant fail safe, while those which gain reactivity present unusual safety problems. In order to minimize the amount of H20 coolant within the reactor, high coolant velocities are normally desirable in an H»0-cooled, DjO-moderated reactor. The*


Sec. 13-1]

SYSTEMS

13-23

allow removal of a relatively large amount of power from a small cooling cases, such a small coolant channel is used that the assembly of and coolant channel is replaced as a unit, installed within vertical fuel, cladding, tubes passing through the I)20 vessel, as shown in Fig. 19. In such a case, there would be gas space between the H.O coolant and the DjO container. This reduces the likelihood of mixing HiO and !)■.<) in the event of a leak, but it also serves to iso late the fuel from the moderator, so the thermal capacity of the moderator is not available to store the heat generated in the fuel during shutdown periods. high velocities

channel.

In many

1.10

>

I 00


O

090

0.4

0.6

LIGHT-WATER

Fiu. 4

15. Design

08

10

12

VOLUME/URANIUM

1.4

1.6

VOLUME

limitations for HjO-cooled D30 reactors.

COMPONENTS OF WATER-COOLED REACTOR SYSTEMS* 4.1

Soft

Seals

Organic (e.g., Neoprene) or fluorocarbon (e.g., Teflon) be used where temperatures and pressures are moderate and where radiation intensities are not excessive. Such seals may be used in the form of adjust able glands, O rings, or gaskets. Normally, leak Figure 16a shows an O-ring seal. age through the first two 0 rings would be caught by the third O ring and would be 4.11

materials

Packing.

can

These seals have a low leakage rate but usually cannot tolerate repeated

collected.

axial motion. Slow rotary motion can be permitted with these seals. Temperatures of organic seals should be kept below 212°F, while fluorocarbons may be used up to 400°F. Organic materials generally can be subjected to radiation exposures of 10" r without damage, but no more than 107 r can be tolerated by the common fluorocar bons. After such exposures, Teflon, for example, becomes hard, "chalky," and

"brittle." 4.12

Metallic

Steel Ring Labyrinth.

Figure

166

shows a type of ring labyrinth

seal that can be used for high-temperature applications or when motion must be transmitted through the seal as for control rods. The seals have been tested for extended periods in hot water for both rotational and axial motion. Since these seals

always leak returned to

slightly, provision must be made to drain off the leakage, which may be the system. To reduce leakage, the seal may be cooled, thereby greatly increasing the water viscosity. Clearances between parts of a 1-in. seal might initially range from 0.001 to 0.003 * For

pressure

vesaela, see Art. 3.


13-24

[Sec

13

in.; however, if wear occurs, these clearances will increase and leakage will become Compatible materials must be used to fabricate the rubbing elements of this greater. For instance, where stainless steel must be used, 400 series hardened type of seal. Chrome plate against stellite alloy may be used against 300 series austcnitic steel. has proved to result in long-life operation, since these materials are compatible and arc


not corroded by hot water. Bleed-off Systems. As previously mentioned, most seals leak slightly, and it 4.13 is generally desirable to provide at least two leakage barriers, draining off the seepage In the case of O rings, several rings may be that passes through the primary seal. used as a primary leakage barrier, followed by a bleed-off and a final secondary Similarly, with metallic seals, several rings may be used as a pri O-ring barrier. mary barrier, followed by a bleed-off and two or more secondary rings. When seal* are used to prevent leakage into a low-pressure region, then water may be adniittol under higher pressure to leak through the seal and avoid drawing in air.

0

- Rl NG

METALLIC (a)

RING SEAL (J/)

Fio.

16.

4.14 In the case of DjO reactors, any possible contamination Dry-gas Systems. with H30 must be avoided. Consequently, dry air or helium may be provided at points of possible air leakage into the system. 4.2

Pumps

4.21 Conventional Pumps, Special Seals. Standard centrifugal water pumps fitted with special seals can be used in many water-cooled reactors. Such pump* have been used with good results on DsO and on HjO. Generally, two seals are pro vided, with leakage being collected between the seals and returned to the low-pressure side of the system. It has generally not been thought desirable to use organic pack ings in these pumps, since the organic materials might introduce impurities into the cooling system. Graphite seals may be used in many cases, while the metallic-ring type is probably desirable for extreme conditions of service. Because of their lower cost and higher degree of development, standard pumps with special seals are more economical than special canned-rotor pumps and are used whenever possible. 4.22 Special Canned-rotor or Submerged Pumps. When no possibility of leak age can be tolerated, as in reactors operating in confined spaces or in the case of homogeneous reactors, it is necessary to provide special zero-leakage pumps. In such pumps, the motor, together with its pumps, is contained in a hermetically sealed vessel. There are no gland or other openings, so that it is possible to maintain

water-cooled

Sec. 13-1]

reactor systems

13-25

ROTOR

STATOR

SUBMERGED PUMP


THIN SEPARATING SLEEVE

GAS

SEAL PUMP

BEARING (3 SECTIONS)

CANNED

ROTOR

A- PRESSURE B- METERING

PUMP

HYDRAULIC

Fig.

17.

Throe pump designs and

a design

FEED FROM

PUMP

ORIFICE

BEARING

for a hydraulic bearing.

essentially zero leakage. There are three general classes of pumps of this type (Fig. 17): 1. Canned-rotor Pumps. Motor rotor and bearings operate under water. Bear ings are water-lubricated. There is a thin presstiretight shell between the rotor and stator of the induction motor.


13-26

MECHANICAL

DESIGN AND REACTORS

[SEC. 13

Motor Pumps. The motor, including the stator, operates under 2. Submerged The motor windings and bearings must be designed to operate under these water. conditions. These are vertical pumps in which the motor is located above 3. Gas-seal Pumps. The upper part of the tank, including the motor, the pump within a pressure tank. is filled with gas under pressure, while the lower part of the tank, including the pump, Provision must be made to maintain the water level in is submerged under water. this tank. Because of difficulties in operating motor windings under water, the second type of pump has not been widely used in high-temperature service, while the third type of Thus, canned-rotor-type pumps pump requires accurate level-regulating equipment. The bearings on have been used in most of the cases where zero leakage is required. these pumps must operate under water, so that oil or grease lubrication is not practi cal. One type of bearing that may be used employs a graphite sleeve or collar operat Provided unit loads of 50 psi ing against hard stainless material such as Stellite. are not exceeded, this type of bearing will usually give satisfactory service with water In order to support maximum unit loadings, dense forms of graphite lubrication. These bearings represent a critical impregnated with tar (Graphitar) are often used. point in the design of canned-rotor pumps, and detailed test work is generally required Another design feature of the canned-rotor to fix the exact design of this component. pump arises from the need for a thin, pressuretight container of high electric resistance Some clearance is needed outside this can for reasons of assembly, around the rotor. and inside the can clearance is required between the rotor and the can for mechanical clearance and for hydraulic reasons. Because of hydraulic drag, electric losses in the shell, and increased rotor clearance, the electrical efficiency of canned-rotor motors is More heat generally lower than that of comparable motors of conventional design. is developed in the stator windings, and special provisions are required to remove this Another type of pump bearing, also shown in Fig. 17, uses the hydraulic pres heat. The periphery of this sure of water from an auxiliary pump to support the journal. bearing is divided into three sections, and water from the auxiliary pump is admitted through an orifice to each section. 4.3

Pressurization

and Surge Tanks

Since the volume of the cooling water changes with tem 4.31 Expansion Tanks. perature, provision must be made to allow for the expansion without developing exces sive pressure. In a hermetically sealed system, this may be accomplished by estab This tank is partially lishing an expansion tank at the highest point in the system. filled with a gas, such as hydrogen, helium, or steam, the gas volume being sufficient Expansion tanks may also be used as deaerating tanks to to allow for expansion. In this application, the coolant trap dissolved gases accumulated within the coolant. may be sprayed into the tank, so that more complete deaeration takes place. The pressure within a water system can also be main Pressurizing Pumps. 4.32 tained by a small centrifugal-type pump. This pump must operate continuously to In order to provide some cooling flow through this pump, maintain the pressure. The pressurizing pump is often used to a bleed is generally used in such a system. Thus, this pump often functions circulate water through the ion-exchange resin bed. both as a pressurizing pump and as an ion-exchanger circulating pump. Steam Surge Tanks for Boiling Reactors. In a boiling reactor, it is desirable 4.33 to reduce the possibility of sudden pressure changes within the reactor vessel, since these changes can affect the nuclear behavior of the reactor. One way of reducing the magnitude of pressure transients within the reactor tank is to pass steam from the reactor through an auxiliary drum, so that pressure fluctuations are dampened through the combined effect of the flow resistance of the steam lines and the steam This drum is also a convenient location for centrifugal capacity within the drum. steam separators and provides a point at which additional water can be removed from the moist steam. The effective capacity of a steam surge drum to prevent pressure transients may be greatly increased by maintaining a level of saturated water within

Sec. 13-1]

water-cooled

reactor systems

13-27

In normal practice, steam drums may the steam drum (as in a steam accumulator). contain as much as half the reactor tank volume.


4.4

Valves

Some leakage is tolerable with heterogeneous water-cooled reactors if local ventila Conventional valves can be modified to provide for collecting tion is provided. In the case of D20 reactors, economic leakage between an inner and outer seal gland. factors associated with the cost of manufacturing and reclaiming DjO compel more attention to the possibility of small leaks, and bellows-type valves may be required. In the case of homogeneous reactors, the serious results of even the smallest leaks (see Art. 1.44) require use of the highest quality double-bellows valves throughout the primary system. 4.41 Conventional Valves with Double Seals. These valves utilize conventional A valve packing glands with soft packing, such as Teflon, or metallic packing rings. second packing gland is provided, and the leakage through the first gland is collected from the space between the glands and returned to the system. Valves of this type Serious failure of the primary gland can be detected are reliable and economical. by increased flow through the collection system, and the situation can be remedied before damage occurs. Leakage-detecting systems may use electrodes within the Under ordinary conditions, this space is essentially void, space between the glands. When a large leak occurs, and slight water leakage drains into the collecting sump. this space fills with water, which shorts the electrodes and sounds an alarm. 4.42 Bellows Valves. Metallic bellows can be substituted for conventional glands to provide zero-leakage valves. Because of the ever-present possibility of fatigue failure, bellows valves generally use a backup seal, of either a conventional or bellows type, and the pressure between these seals is measured in order to detect failure of Bellows valves are costly, and their use is the primary seal as quickly as possible. ordinarily confined to homogeneous reactors and other reactors where even slight leakage of the coolant cannot be tolerated. 4.5

Heat Exchangers or Boilers

4.61 Materials and Containment. In the absence of oxygen, the corrosion rate of carbon steel is about ten times greater than the rate of stainless steel, while in the Consequently, stainless-steel presence of oxygen, carbon steel corrodes much faster. tubes and headers have generally been used on the primary coolant side of heat There is a possibility that ordinary carbon steel may exchangers for water reactors. be found acceptable in the future if means can be discovered to eliminate oxygen from Stainless steel has notable disadvantages of poor thermal con the cooling water. ductivity, large coefficient of expansion, and high cost. Moreover, stainless steel Corrosion of stainless steel introduces may be subject to intergranular corrosion. traces of cobalt, associated with the nickel in the alloy. The cobalt on being exposed to neutrons becomes activated with a long half-life and produces hard y rays. Where operation is at low temperature, aluminum or copper tubes may be used in When aluminum is used, water must be extremely pure and the the heat exchangers. pH should be maintained between 5 and 7. Where the expense is warranted, nickel tube exchangers can be used. Nickel has a thermal conductivity comparable to that of carbon steel and is not subject to rapid corrosion. Where H20 is used as the primary coolant, conventional (rolled-in tube) heat In some instances, tubes have been welded or exchangers have proved satisfactory. Whenever complete brazed into sheets to provide greater assurance against leakage. freedom from tube-sheet leakage is required, however, a double-sheet type of con struction seems desirable. In some cases the tubes are rolled into each of the two tube sheets, while in other cases tubes are welded into the outer sheet and rolled into Leakage accumulating between the two sheets can then be detected, the inner sheet. Double-tube sheet exchangers and steps can be taken to repair the heat exchanger.

13-28


[Sec

13

DiO reactors, and they should always be used on homo reactors. Both straight-tube and Design and Performance of Heat Exchangers. Although the U-tube arrangement elimi U-tube heat exchangers have been used. nates thermal stresses due to tube expansion, this design does not lend itself to con Thus, it is common practice to venient repair or replacement of individual tubes. Single-pass straight-tube heat exchangers are space-con block off defective tubes. suming because of their great length, and an expansion joint may be required on the However, tube replacement is simplified in shell side to allow for thermal expansion. Large straight-tube heat exchangers have been built on this type of heat exchanger. Normally, flatcars, rail-mounted so that they can be readily removed for repairs. The primary hot water passes through the tubes and steam is raised in the shell. smallest temperature difference across the heat exchanger occurs at the place where In order to promote vigorous boiling the secondary coolant reaches the boiling point. and stimulate heat transfer on the shell side, a temperature differential of at least It is desirable to divide heat exchangers into preheater 30°F is required at this point. and boiler sections in order to take advantage of counterflow operation and increase have found acceptance for


geneous 4.52

Fig.

18. Designs

for a tube filler and a special head, to reduce inventory.

this temperature differential. In the design of the boiling section it is not possible to consider the shell-side coefficient as constant, because the rate of steam production has a marked effect on the coefficient. As used in reactors, over-all coefficients gen erally range from 300 to 1200 Btu/(ftJ)(°F)(hr). To obtain the higher figure, high water velocities are required through the tubes together with vigorous boiling in the shell. 4.53 In the case Special Heads and Tube Fillers to Reduce Coolant Inventory. of D2O or homogeneous reactors, the liquid in the primary system is so valuable that unusual efforts may be made to reduce the external inventory. This can be done in several ways: Tube diameter may be reduced; however, small tubes do not lend them selves readily to fabrication, so that groups of small tubes may be brazed into a sine!? Another design uses stand large tube, which is then assembled into the tube sheet. ard tubes filled with a solid, square "tube filler." This tube filler may be twisted to

promote turbulence (Fig. 18). Figure 18 shows a typical Special heads may also be used to reduce inventory. Heat exchangers may be designed having an inventory low-inventory head design. as low as 1 £ lb of coolant per square foot of heat-exchange surface. 4.6

Safety Devices

4.61 Pressure-relief System. The water within the primary reactor system is radioactive, so thought must be given to the location of vents for reactor safety valves. During the normal course of operation, relief valves are generally vented


Sec. 13-1]

13-29

condenser or stack so that the plant operators will not be exposed to radiation. Reactors are often contained in pressuretight enclosures with the idea of discharging all vents, including the pressure reliefs, within the enclosure in event of a major accident. This can be accomplished with quick-acting valves responsive to the radioactivity level in the enclosure. In the case of boiling reactors, it is often convenient to discharge the reactor relief However, sudden loss of vacuum may destroy the turbine valve into the condenser. by overheating the final stage blading; consequently, auxiliary discharges should be Steam-relief valves from small reactors may provided in case of loss of vacuum. discharge to large tanks of cold water. Either relief valves or rupture diaphragms can be used to protect reactors. Instal lations should meet applicable boiler codes. Rupture diaphragms are less precise in their functioning than are relief valves, but the use of a rupture diaphragm set at high pressure ensures safety in event that the relief valve completely fails to open. Besides relieving pressure, operation of the relief system in a water reactor may result in for mation of steam bubbles within the reactor core, which will affect the nuclear proper Water-moderated reactors are usually designed so that forma ties of the reactor. tion of bubbles reduces reactivity. Consequently, operation of the relief system can be made to shut off the reactor, acting, in effect, as an additional safety system. However, when a relief valve is used, this effect may be temporary, since the boiling will stop soon after the relief valve is reseated. Consequently, when relief valves are used, provision should be made to insert control rods when the relief valve operates. 4.62 1. Some heat is generated within the Shutdown Cooling Requirements. reactor core after the reactor has been shut down. This heat results from the decay of radioactive fission products. During the first minute after shutdown, additional heat is produced as the result of fissions caused by delayed neutrons. For instance, after 2 years of fuel irradiation, the fraction of operating power heat produced after shutdown is indicated in the following table:


to the

Time since shutdown. Shutdown power

...

1 sec 1

0.06

1 hr

6 hr

1 day

1 week

1 month

6 months

1 year

0 015

0.009

0. 006

0.004

0.002

0.0006

0.0004

In the case of large power reactors, provision must be made to dissipate this heat under all conceivable circumstances, since failure to remove shutdown heat might result in melting or even vaporizing the radioactive portions of the core. Shutdown heat must be dissipated in the absence of all normal power supplies, since interruption Provi of power supplies may occur during a period when the reactor is shut down. sions for removal of shutdown heat will vary, depending on the design of the reactor. Therefore distinction will be made between tank-type or vertical-tube reactors, where natural convection can generally remove this heat, and horizontal reactors, where forced circulation may be required. 2. Tank-type and Vertical-tube Reactors. Natural circulation can be used to pro vide shutdown coolant circulation within the primary loop. Boiling of the primary coolant will result in sufficient circulation to provide shutdown cooling in almost all Since the primary loop circulation can be provided without an auxiliary instances. power supply, it is desirable, if possible, to cool the primary coolant without requiring auxiliary pumping power. In the case of power reactors, which operate at elevated temperatures, cooling may be provided by boiling water at atmospheric pressure in a secondary system. The primary loop may pass through tubes that are submerged in this water or through a boiler supplied from an elevated reservoir. In the case of boiling reactors, cooling may be provided by condensing steam from the reactor through the evaporation of water within a small, atmospheric-pressure condenser

boiler. The procedures outlined above are appropriate for large, high-power reactors, but many small research reactors may not need special equipment . In many cases, the heat capacity of the shield may be sufficient to absorb shutdown heat generation without reaching too high a temperature, and in many instances normal heat losses from the reactor may be sufficient to dissipate losses.


13-30

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

3. Horizontal Reactors. Since natural circulation may not provide shutdown cool ing, positive provisions are required to ensure circulation in event of complete failure of normal power sources. Auxiliary pumping plants or water from elevated tanks may be used for this purpose. Enclosure Requirements. 4.63 In order to preclude any possibility of release of fission products in event of a major reaction incident, several water-cooled-reactor plant designs have included pressuretight enclosures for housing the more radioactive portions of the nuclear system. It would seem reasonable to design such contain ment for the pressures to be encountered under most severe conditions. In most power reactors, the amount of nuclear energy liberated during a reactor incident is found to be trivial as compared with the energy content of the hot water. In some cases, the chemical energy of the reaction of the fuel with the water may be greater than any other source of energy. Chemical reactions that could release considerable energy might conceivably occur between water and zirconium, aluminum, or ura nium metals. When there is no possibility of chemical reaction, then the container is often designed to withstand the pressure developed through rupture of the primary reactor system and resultant flashing of a portion of the water into steam. In event that a chemical reaction is possible, then more energy may be available, sometimes enough to vaporize all the water within the reactor system. When this has occurred, however, any chemical reaction will stop, since liquid water is generally Thus, the largest container required might be necessary to sustain these reactions. the one that contained the steam resulting from vaporizing all the contents of the reactor system. If the ratio (volume of container) /(volume of water in reactor) is denoted by Ry and the ratio (absolute pressure after incident) /(initial absolute pressure) by RF, then the size and pressure requirements for reactor containers can be calculated from the following approximate formula:

Rv

=

(v*~

\

Vw

M' (W1'" - 1/\

In

this formula, Vs denotes the specific volume of the steam after the incident, and The symbol IV value for saturated steam can be used in this calculation. The polytropic expansion denotes the specific volume of water before the incident. constant n for air may be taken as 1.3, so that 1/n is equal to 0.77. Typical values resulting from the solution of this equation are given in Table 6. the

Table Hv

Rr

210 380 510 770 1.150 3.600 4.500

6 5 4 3 2.3 1.7 1.3

6

Psig in container (for 15-psia initial preasurc)

105 60 45 30 20 10 S

Ab an example of this calculation, a reactor containing 100 ft3 of water can be housed either in a container of 1 15,000 ft* capable of standing 20 psig, or in a 51,000-ft* container which could stand 45 psig. 4.7

Piping and Materials of Construction

Continuous technical progress makes it difficult to specify exact materials for reac tor construction. However, at the present time, the following specifications would constitute good practice.

Sec. 13-1]

WATER-COOLED

KEACTOR

SYSTEMS

13-31

Nuclear Components. 4.71 Because of nuclear considerations, a relatively small Table 7 indicates the group of materials is used within the chain-reacting core. applications of the principal materials of construction for water-cooled reactors. Table Material

Zirconium

and alloys.

Aluminum

and alloys.

M:iKn<\sium

Functions

Fuel clad, structural parts Fuel clad, structural parts

Fuel clad, structural parts

Beryllium

Auxiliary moderator

Graphite

Moderator

Stainless steel

Fuel clad, structural parts, tank

7

Remarks

Up to 600°F, low neutron absorption,

good mechanical properties, expensive may be used up to 430°F Low neutron absorption, soft, weak, ductile, inex under special conditions, pensive Kxtremcly low neutron altsorption, but poor corrosion resistance limits use to 150°F, suscoptible to rapid corrosion under certain conditions Excellent nuclear properties, probably may be used to 350°F, expensive, brittle, hard to fabricate, toxic Excellent nuclear properties, easy to fabricate, low strength, high porosity, some radiation damage High neutron absorption limits use to thin sections within neutron held; used for supports and tanks

At the present time, it has proved Primary System, External to Reactor. to remove the corrosion products originating from stainless-steel piping and With special water treatment, it may be possible equipment operating at over 500°F. to use low-carbon steel in conjunction with a larger demineralizer and to maintain adequate water purity. At the present time, however, stainless materials are used in


4.72 feasible

high-temperature reactor water systems. In the steam system of a boiling reactor, corrosion rates are much lower, and so it seems practical to use carbon steel for the steam piping and turbine components of a boiling-water reactor. However, care must be taken to avoid acid conditions in the turbine, since droplets of low pH have been known to cause severe blade corrosion. When temperatures are lower, as in room-temperature research reactors, the range of materials is much greater. Aluminum is often a suitable material in such cases; however, when aluminum is used, a pH between 5 and 7 should be maintained in the It may be possible to use aluminum condenser tubes where good-quality coolant. cooling water is available. Standard condenser materials may also be used at low temperatures. 4.73 Since radioactivity is not normally present, Secondary Cooling System. conventional materials can generally be used in the secondary cooling system. In homogeneous reactors, however, delayed neutrons emitted from fuel within the heat exchanger may result in some radioactivity in the secondary system; hence it may sometimes be desirable to maintain high-purity water and use special materials in such systems. 4.74 Expansion Stresses in Piping. Any piping system that operates over a wide temperature range will require allowance for the thermal expansions which occur in Though water-cooled reactors the course of bringing the system up to temperature. operate at relatively low temperatures, there are several reasons why piping expansion must be considered. 1. Austenitic stainless steel frequently used in reactor piping expands about 40 per cent more than low-carbon steels. 2. The components of water reactors are frequently heavy and are anchored. This may cause excessive stress in interconnecting piping unless provision is made for expansion. Either expansion joints or expansion loops may be used to compensate for thermal In some cases, where the "pipe" is of very large diameter, expansion of the piping. expansion bellows are required. For instance, when it is desired to divide support of a research reactor between the top and bottom of the tank, then an expansion joint

13-32

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

in the form of a single bellows must be incorporated in the tank. This was done in the Argonne research reactor (CP-5). On the other hand, expansion of horizontal fuel tubes may be allowed for through the use of "gooseneck" connections at the ends of the tubes. Expansion of steam pipe can be provided through the use of expansion loops. Finally, in some cases, it may be possible to support the pumps, valves, or other equipment flexibly so that the equipment can move to provide for the expansion of For instance, where a "water-wall" construction is used to provide the piping. water in a horizontal tube reactor, the water walls may be movable so that expansion of the tubes moves the massive water-walls (Fig. 3).


4.8

Miscellaneous

Systems

Electric leak detectors are often used to pre 4.81 Leak-detecting Equipment. vent loss of DjO from research reactors. They may consist of a pair of wire screens A drop of DjO leaking from a fitting Bpaced close together but electrically insulated. makes electric contact between the screens, causing an alarm to indicate DjO leakage at that fitting. Beta-counting rate meters or air count chambers are used to detect steam leaks. Such equipment must be thoroughly shielded from the effects of 7 radiation from the coolant system. 4.82 Periscopic viewers are often the only means for locating defects Periscopes. within the core of a power reactor. Such devices are commercially available in Periscopes are most effec sizes that can be inserted through a typical fuel channel. Therefore it is recommended that tive when the subjects are strongly illuminated. Extreme care should be exercised special provision for intense lighting be provided. to avoid breaking lamp bulbs or otherwise introducing foreign material into the reactor. Magnified photographs can be taken through periscopes to bring out small defects more clearly. Canals and Underwater Fuel Handling. Water-filled canals provide one 4.83 of the most convenient means for handling and viewing intensely radioactive objects. Fuel can be handled with the assurance that the water in the canal will remove the heat from fission-product decay. A variety of fixtures and viewing devices are used in the operation of a canal. Glass-bottomed trays can be used to look through the water. Long rods tipped with various fixtures are employed in manipulating fuel at the bottom of a canal. Vitreous tile has proved to be a suitable material for the interior of a canal. Permanent illumination is provided so that the equipment at the It is most satisfactory to limit illumination bottom of the canal is strongly lighted. above the canal and provide strong lighting within the immediate working area at the bottom of the canal. As radioactivity decays, Fresh fuel elements should be shielded by 18 ft of water. Materials other than fuel may often be they may be handled through less water. handled through a few feet of water. Thus, it is often desirable to provide a variety of depths within the working area, since it is much easier to work through less water thickness. Experience has indicated that some machine work can be performed in a Metal canal and the resultant contamination may not prove a serious health hazard. More serious chips from machine work can be swept from the bottom of the canal. In general, therefore, under contamination may result from fission-product gases. water cutting operations should be carefully studied before work is started. 4.84 Special Gas Systems. Special gases, such as helium or carbon dioxide, are used to provide an inert atmosphere around the moderator of graphite reactorsIn this case, Special inert gas may be used above the liquid level of DjO reactors. the gas is normally circulated and the deuterium and oxygen are combined using s noble-metal catalyst. Helium is generally used as the blanket gas above the D-0 systems. Helium has the advantage that it neither activates nor reacts with the water. 4.86 Auxiliary Cooling Systems for Moderator, Reflector, and Shield. Approxi mately 6 per cent of the heat produced by the reactor is generated within the moders

Sec. 13-1]


13-33


When the moderator also serves as a coolant, this heat may be dissipated by the tor. main cooling system, but when the moderator is not used as a coolant, special means Of the 6 per cent energy produced within may be required to dissipate this energy. the moderator, about 2 per cent is due to neutron moderation while approximately 2 per cent is due to 7s produced in the fission process. These sources of energy cease when the reactor is shut down. The remaining 2 per cent energy is produced by Provisions may be required fission-product ys which decay slowly after shutdown. to remove heat from the moderator even though the reactor is shut down. Another means of cooling the graphite reflector is exemplified by the MTR (Fig. 21), where the graphite is arranged in the form of small round spheres piled in the reflector Air is blown through this bed of spheres to provide the necessary cooling. space. Where callandria are used with an HiO-cooled, D20-moderated reactor, the D20 moderator is cooled by H20 using a heat exchanger. Shields or pressure vessels may also require cooling. In tank-type reactors, most of the cooling is provided by the primary coolant but some auxiliary cooling may be needed. For instance, shields may be cooled by water coils wrapped around the hottest portions. Although this water develops very little activity, it may be unde sirable to return it to the sanitary drains. Therefore a closed system is sometimes preferred for this application. 4.86 Coolant and Reflector Dump and Disposal Systems. In event of a major leak, provisions should be made to dump the D20 moderator into a tank in order to When possible, the reactor should be scrammed to allow shutdown prevent its loss. heat to decay and the moderator should be dumped as soon as the shutdown heat can be removed in the absence of the moderator. An underground tank is usually pro vided as storage for the moderator. The moderator dump system can also be used in an emergency as a means for shutting down the reactor. Generally this can be done by lowering the water level to the top of the fuel elements so that cooling is still pro vided by the moderator. The moderator dump tank can serve as a storage tank. A pump is provided to return the moderator to the reactor. 4.87 1. Horizontal Prcssurized-lube Reactor. Loading and Unloading Systems. Horizontal tube-type reactors are usually loaded with short fuel slugs. When such reactors are unloaded, the slugs are pushed through the fuel tubes and fall out the far, or discharging, face of the reactor into a storage basin. This arrangement greatly simplifies charging and discharging operations, since the operator can work from the nonradioactive, or charging, face while fuel is pushed out the radioactive discharge face into the basin. However, residual radioactivity may accumulate at the charg ing face of a power reactor, where water is recirculated through the primary reac tor loop. Because of the facility with which fuel can be reloaded, horizontal reactors have found application in isotope-producing reactors. Materials of fuel may be com pletely unloaded from one channel, or, alternately, only the materials near the far end of the tube can be replaced. In the latter so-called "segmental " discharge scheme, materials receive approximately equal radiation during their residence in the reactor. The simplicity of the unloading process in horizontal-tube reactors has prompted the design of mechanical arrangements to permit reloading during the operation. Pre liminary studies indicate that future pressurized-tube reactors might be reloaded with out shutting down the reactor or reducing the operating pressures and temperatures. 2. Vertical Pressurized-tube Reactor. Most reactors of this type have used 1)20 moderation. When D20 is used also as the coolant, its loss during reloading opera tions must be reduced to a minimum. Then the fuel must be withdrawn out of the top of the reactor and transferred to an HjO canal. Shielding casks can be used where reloading time is not an important factor, but when rapid reloading is necessary, special remotely operated crane manipulators are used to transfer the fuel from the reactor vessel into the canal. Operators can view the process through glass shielding

windows.

Figure 19 shows the arrangement of the Chalk River reactor. To unload a fuel element, the element is disconnected above the reactor tank, and is raised into a cask, and transferred to a canal. Positive cooling is maintained during the transfer.

13-34

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

In the case of mobile tank-type reactors, simplicity in fuel 3. Tank-type Reactors. The fuel must be loaded with handling favors removal of the entire core as a unit. poison materials to prevent criticality, and the entire core can then be raised through the tank lid opening into a coffin. Cooling must be provided to remove fissionproduct heat during the unloading period. In order to maintain more uniform reactivity, central-station tank-type reactors must be unloaded in sections, portions of the core being replaced at regular intervals. Thus the fuel must be subdivided and is withdrawn in small assemblies or individual rods. Fuel may be loaded into coffins which are lowered into the space above the


COOLING WATER TRAVELS DOWN ALUMINUM TUBES IN WHICH ARE RODS OF

URANIUM HELIUM CADMIUM CONTROL ROD HEAVY WATER MATERIAL TO BE IRRADIATED PLACED IN ALUMINUM BALL AND INSERTED HERE SELF-SERVE UNIT PLACES BALLAT REQUIRED DISTANCE FROM TANK

COOLING WATER FROM OTTAWA RIVER CONCRETE SHIELD ALUMINUM TANK

BEAMS OF FAST NEUTRONS EMERGE FROM EXPERIMENTAL HOLES

IRRADIATION COMPLETE

BALL ROLLS OUT

HERE INTO"CASTLE"

COOLING

AIR

—1 COLD WATER

PUMP CIRCULATES HEAT-EXCHANGER

Fig.

19a. Arrangement

FROM OTTAWA

RIVER

HEAVY WATER THROUGH TO COOL IT

of Chalk River reactor.

reactor core, or an unloading chute may be provided to pass the fuel into a cans! below the reactor vessel. 4.88 1. Monitoring Instruments. Monitoring instru Measuring Instruments. ments are often used in large reactors to ensure more efficient operation and to detect The temperature rise and flow in a water-cooled reactor tube can incipient failures. be used to indicate power generation within the tube. Venturis or orifices in tbe tube inlet can be used to measure flow, while thermocouples at the inlet and outlet are used to measure the temperature rise. Control rods and fixed poison columns can then be adjusted to give the best operating conditions. This can be accomplished by making those changes which reduce the power from the "hot-test" tube, and by a succession of adjustments a considerable degree of radial power distribution "flatten ing" can be achieved. When several per cent of reactivity is available, it is often possible to double the permissible output of a large reactor through flattening. At the present time few measurements of axial flux within a large high-power reac tor have been made. Small ion chambers or fuel-element thermocouples can be used

Sec. 13-1]

WATER-COOLED

REACTOR

SYSTEMS

13-35

for this purpose, and, if axial measurements are available, then the axial power dis tribution can be adjusted by poisons to give maximum power output. This can be ac


by reducing the power output complished near the outlet side of the reactor and raising the output near the inlet. Monitors are also used to detect abnormal For instance, partial conditions in tubes. stoppage of a tube results in a higher inlet pressure and a higher outlet temperature. The radioactivity of the water from the reactor and from individual coolant channels The principal purpose is often monitored. of such monitors is to detect incipient fuelclement failures. Methods are used that minimize the sensitivity of the detector to normal water activity while retaining the sensitivity to fission-product radiation. 7 scintillation spectrometers are often used, as they can be set to detect only y radiation characteristics of particular fission products. A means of holding up the water sample is often used in order to allow the short-lived oxygen activities to decay before the sample reaches the monitor. 2. Power-level Instruments. The power of water-cooled reactors is normally meas ured by instruments that multiply the cool ant flow rate by the temperature rise In the case of boiling through the reactor. reactors, steam nozzles are used to measure power. 3. Water-level The Indicators. water level in reactor cooling systems can often be measured by direct means, using a gauge This usually gives reliable indica glass. tion, and in some cases gauge glasses can be read remotely with television equipment. Differential pressure cells are convenient to measure water level because they provide a direct signal at the control panel. How Fig. 19b. Close-up view of single tube in ever, such systems are rarely reliable under Chalk Hiver reactor. all conditions, so it is desirable to install duplicate water-level indicators in power reactors where gauge glasses cannot be

used. 6

SPECIAL FEATURES AND APPLICATIONS OF H.O-COOLED REACTORS 5.1

6.11

Research

Reactors.

Use

H-O-cooIed Reactors of

Hs<)

in research

reactors provides practical

advantages in handling radioactive apparatus and fuel, since work can be performed under large thicknesses of water. This is illustrated by the operation of the swim ming-pool reactors as shown in Fig. 20. This low-cost cooling system is an additional advantage. Therefore HjO cooling has been used in the MTR (Fig. 21) and the

NTR

(Fig. 22) reactors for research use.

research reactor located at Chalk River.

The

H20 was also used to cool the first Canadian

principal disadvantage of H20 reactors for research use is the small size of the

13-36

MECHANICAL DESIGN

AND REACTORS

[SEC.

13

active core. As compared with D20 reactors, they have lower thermal flux for the same power, and, because of their shorter neutron lifetime, they are less safe in event of a major incident. Small Power Reactors. 6.12 Small power reactors can use H2O with dilute en riched uranium elements. Stainless steel or zirconium fuel plates are preferred, but aluminum fuel elements can be used in water up to about 430°F. The Army package power reactor shown in Fig. 23 is an example of a pressurized H50 design using stain less steel U235 fuel plates to produce 2,200 kw of electricity. 6.13 Either partially enriched uranium or thorium-Li"' Large Power Reactors. combinations may be used in large HiO reactors. Theoretically, graphite-moderated


U

Fio.

20.

HANDLING

TOOL

HiO-cooled swimming-pool

reactor.

HjO-cooled reactors could operate with natural uranium, but actually some enrich ment is usually beneficial. As indicated in Art. 3.1, the graphite design is most advantageous in very large sizes, and such units have greatest appeal when such Urge blocks of power can be utilized. No H20-cooled D20-moderated power reactors are planned at present because mechanical problems arise in constructing the callandria, because of difficulties asso ciated with the coolant connectors and channels at high pressure, and because such reactors do not have material nuclear advantages over graphite-moderated piles. Tank-type pressurized water reactors are limited in capacity by the sizes in which Several reactors of this type are planned, heavy pressure vessels can be fabricated. and one is under construction. These reactors can be made to fail-6afe under all However, in large sizes, conditions, and very high power densities can be obtained. the pressures that are required necessitate extremely heavy-walled reactor vessels, and closures, piping, and pumps must be designed to operate under these conditions.

Sec. 13-1]

water-cooled reactor systems 5.2

D.O-cooled

13-37

Reactors


6.21 1. Research Reactors. Types of D20-cooled Reactors. Because of its low neutron absorption D»0 has notable advantages for use in research reactors. D20 reactors can provide large experimental volumes at high thermal flux, while the long

prompt-neutron lifetime of The high cost concerned.

these reactors is a great advantage in so far as safety is of DsO is a serious drawback, and because of its high special precautions must be taken to prevent its loss. These measures generally make 1)20 research reactors less flexible and more complicated than similar H20

cost

machines.

The

fuel elements in D20 reactors can be separated by several inches, so that it is to cool moderate-size research reactors by natural convection of DiO between fuel elements. Argonne's first Ds0 reactors (CP-3 and CP-3') were cooled in

possible

the

13-3S

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

this manner. The arrangement of these reactors is shown in Fig. 24. Rods aro in a tank of D20, and the water within the tank is cooled by circulation When higher power outputs are required, through an external heat exchanger. forced circulation of D-0 is provided through the fuel elements. The present Argonne research reactor (CP-5) is of this design, as shown in Figs. 5 to 9 of Sec. 13-5. 2. Large Reactors. Use of D20 affords major nuclear advantages, but these advan Large DjO reactors operating tages are offset by practical construction problems. suspended

.TYPICAL REFLECTOR 2 ALUMINUM CAN

SUB-ASSEMBLY BEAM HOLE

OVER

DIRECTLY REMOVABLE REFLECTOR ASSEMBLY


UPPER SHIM ROD GUIDES

IHORIZ. PNEUM

SPENT ASSEMBLY

DISCHARGE CHUTE (SCHEMATIC)

LOWER GUIDE GRID

Fio. on natural

21b.

Exploded view of

MTR

core.

uranium can be designed to produce more fissionable material than is consumed, and use of the U"'-thorium cycle on such reactors affords a possibility of appreciable breeding gains in a thermal reactor. Figure 25 shows the lattice arrange ment for such a design.' Tubes in such reactors might be up to 15 ft long, while pressure vessels up to 16 ft in diameter might be required. Normally, DjO is used as moderator, coolant, and reflector. D20 is pumped through the coolant tubes to provide circulation, while DsO between tubes provides moderation. Only the central portion of the tank is loaded with fuel, so that the DjO near the tank walls serves as » reflector.

Sec. 13-1]


REMOVABLE ALUMINUM CLAD GRAPHITE PLUG FUEL LOADING SLOT SAMPLE HOLE ANNULAR CORE CAN COARSE

5FT x5FTx5FT

6

13-39

FT CONCRETE WALL

CUBE OF GRAPHITE

4FTx4FTx4FT

LOGS-

THERMAL COLUMN

ROD

CONTROL MECH

FINE ROD CONTROL MECH SOURCE ROD MECH

EXPERIMENT

HOLES

SAFETY SHEET MECH (6-TOTAL) SAFETY

ROD MECH

CIRCULATING CONTROL


Fio.

22.

NTR

MECH

(4 -TOTAL)

SYSTEM MOUNTING PLATE

reactor.

A 20,0(M)-k\v electrical DXt-cooled power reactor while preliminary designs of reactors of this type have been prepared by Argonne and

is being designed in Canada,

h- 20"R-

Commonwealth Edison. Effect of Leakage on Economy. 5.22 A major practical problem in the design of large D«0 reactors results from the need to reduce DjO leakage to a very low value. If D20 costs $28/lb, then loss of only 50 lb/day from a 100,000-kw electric plant would result in a penalty of more than 0.28 mill/kwhr. Consequently, extreme pre cautions must be taken to reduce loss, and this adds to the plant cost. While there is experience in the handling of DiO in low-pressure reactors, there are indicat ions that the problems would be far more diffi cult in high-temperature reactors. Article 1.41 discusses some of the methods of reducing normal leakage to an unavoidable In the major reactor incident minimum. at the Chalk River reactor, the DjO was contaminated with H20, though less than o per cent of the DjO was lost. Such con tamination and loss are ever-present dan gers in UjO reactors and cooling systems. ]Por example, a major boiler tube failure could cause rapid loss of much valuable

u2o. In

the design of a D-O-reactor plant, are generally made to reclaim For instance, trays J_Z>2^ ' from major leaks. watertight compartments are located below the reactor tank and below the principal

provisions

or

STAINLESS STEELCLADDING Fig. 23. Small power reactor,

piping.

Plant drains may be col


MECHANICAL DESIGN

Fig.

»

-f-

J>— -o

]>"<'

+

a,

+

-f

0 + +

y— + ~*

V-

24. Arrangement

[Sec

AND REACTORS

of CP-3.

o

e

e

,

+

i

-« °

«l

"O +

V—•

+

o — -" /*- —

P «

+

+ o

-f

*b

"FUSEO HEX" PATTERN

"UNFUSED HEX" PATTERN

o FUEL RODS + CONTROL RODS

o FUEL RODS + CONTROL RODS

COOLANT TUBE - '/is" THICK 5" DIAMETER

A

\

*

+ '»

N BETWEEN TUBES-5"

SPACING

FUEL PLATES - '/e" THICK URANIUM CLAD WITH ZIRCONIUM

Fig. 25. Lattice arrangement

of large DsO-rooled reactor.

a

>


Sec. 13-1]

13-41

and tested before release to the external drainage system. Isotopic enrich The cost is ment of contaminated D20 is generally accomplished by distillation. D»0, and facilities may not always an appreciable part of the cost of manufacturing lected

be available.

There are three ways in which the Arrangement of Fuel in D20 Reactors. can be contained in the fuel tubes of a D20 reactor. As described in Art. 5.21, item 2, the entire reactor may be contained in a vessel and the tubes would Alternately, the moderator could then contain only a nominal pressure differential. operate at low pressure in a callandria, as exemplified by the Chalk River design Finally, the pressure differ (Fig. 19), except that D20 might be used as a coolant. For instance, the coolant ential might be divided between the tank and the tubes. in the tubes might operate at 600 psi while the moderate in the tank operated at 300 All these designs have been studied, and it is not clear which of these arrange psi. ments is to be preferred. The capital cost of the D20 may be a considerable 6.24 External Holdup of D20. At the present time, it appears that as much as 1 portion of the total plant cost. At $'28/lb, ton of D»0 is required per megawatt of electric-generating capacity. Naturally, advances in the design of heat this represents an investment of $56/kw. exchangers may reduce this figure, but at the present time external D20 holdup is In one of the principal factors to be considered in the design of a plant of this type. order to reduce this holdup, high velocities may be used in the heat exchangers and Flow velocities of over 30 fps have been suggested for such systems. piping system. 5.23


required pressure

6

BOILING AND FLASHING REACTORS 6.1

Special Features and Applications

Small, Enriched Boiling Reactors. 6.11 Experiments with small enriched naturalcirculation reactors have demonstrated that stability can be maintained with up to Under these conditions, substantial 3 per cent excess reactivity in steam bubbles. power can be removed from a small natural-circulation boiling reactor. Such reac tors are normally designed to have a negligible void coefficient at room temperature, so the effect of steam formation on reactivity is negative but as small as possible. Under these conditions, 20 per cent steam voids in the coolant reduce the reactivity Table 8 indicates the power output obtainable from a 5-ft-high from 1 to 3 per cent. boiling reactor with 20 per cent average steam voids in the coolant. Table Pressure, psia

15 100 300 600

Temperature,

°F

212 328 417 486

8

Power, kw tie.it/liter of coolant

10 12 22 35

Thus, it can be seen that a small natural-circulation reactor operating at 300 psia produces 25 kw per liter of coolant. Such a reactor could use aluminum fuel plates, as it has been found that aluminum is not attacked by suitably treated water at this temperature. A cubical reactor core 2J.£ ft on a side contains 440 liters, or about 200 liters of coolant. At 25 kw /liter, such a reactor produces 6,500 kw of heat or

about

1,000 kw of electricity.

The obvious simplicity of this type of plant makes it

extremely attractive for application in such sizes. The EBWR is a larger reactor of the same general design in operation at Argonne National Laboratory. As can be seen from Fig. 26, this reactor has a core about 4 ft in diameter contained in a 7-ft pressure Pro This reactor uses H2G cooling, and generates 5,000 kw of electricity. vessel.

visions

are made for testing both natural and forced circulation.


13-42

MECHANICAL

Fig.

DESIGN AND REACTORS

26. Small enriched

[Sec.

13

boiling reactor.

Flashing Reactor. By use of a "flash cycle" (Fig. 27) it is possible to dis with heat exchangers and yet generate steam outside the reactor. Unfor tunately, the pumping requirements of this cycle are considerable, especially when steam is generated at high pressures. Pumping power requirements for such reactors range from 5 to 15 per cent of the electric For this reason, the simple flash output. cycle has not been used in reactors. 6.13 Large H.O-moderated Boiling TURBINE Reactors. WATER Small reactors as described in Art. 6.11 can be constructed in larger sizes ) Icon denser up to at least 50,000 kw of electricity. The General Electric Company has de REACTOR veloped a modification of the boiling re actor that has notable advantages with WATER respect to performance and regulation over simple boiling reactors.* WATER WATER In dual-cycle boiling reactors, as shown PIJMP in Fig. 3, steam generation is confined to PUMP the upper portion on the reactor core. Fia. 27. Flash cycle. During full-load operation, water enters the lower part of the reactor at a temperature well below the boiling point. This water passes up along the fuel rods, anil boiling starts after the coolant has gone two-thirds of the distance through the reactor core. Consequently, steam generation is con6.12

pense

rLs

rhn

"S3--

* Since this section was conceived, principal changes; 1. 2. 3. 4.

detailed

design of the Dresden

plant has resulted

in the following

Increased steam pressure — 1,000 psi primary, 500 psi secondary Forced circulation, with steam separation in external drums Use of heat exchangers instead of flash chambers Location of control and turbine buildings separate from reactor enclosure

See J. R. Woolcott, V. A. Elliot, E. P. Peabody, G. M. Roy, J. L. Schanz, G. Sege: "A Design Descrip tion of the Dresden Nuclear Power Station," A.S.M.E. Annual Meeting, Nov. 26, 1966.

Sec. 13-1]

reactor systems

13 43

to the upper one-third of the core, so that steam has only a short distance to travel in order to escape. Either natural or forced circulation can be used with this design. When natural circulation is used within the reactor vessel, "chimneys" are provided above the core to promote water flow. Steam is produced within the reac tor under a pressure of 600 psia. Passing through the steam drums, it enters the At full load, about half the steam is generated high-pressure stage of the turbine. within the reactor core. A portion of the water that did not boil in passing through the reactor core is expanded to 350 psia through nozzles into the flash tank. Consequently, 8 per cent of this water flashes to steam. This steam is piped to the intermediate stage of the Since the heat content of the turbine to provide the other half of the steam supply. coolant entering the flash tanks is conserved, the temperature of the remaining water is reduced to 431°F, and this water, together with the feed from the condenser, is pumped into the reactor core to provide the subcooling already mentioned. Most of the turbine governor load adjustments are made by controlling the inter If the demand for power diminishes, then mediate-stage turbine admission valves. As a consequence, less water the governor reduces flow of steam to the flash tank. flows through the nozzles into this tank and the pumps, controlled to hold a constant Thus when the load slackens, level, return less subcooled water into the reactor vessel. the temperature of the water entering the reactor rises, so that more steam is pro duced in the reactor, reducing reactivity and causing the reactor output to fall in For example, when the reactor is operating at 25 per cent response to the loss in load. load, the intermediate-pressure turbine admission valve is closed and the pressure in the flash tank is the same as in the reactor, or 600 psia. Under these conditions, the only subcooling within the reactor is that provided by the condensate feed and boil ing starts about one-third of the way up the reactor core. The reactor is then operat ing as a simple boiling reactor, and the steam content of the core is identical with the steam content under full-load conditions when four times as much power was being Under full-load conditions, twice as much steam is generated in the produced. reactor, but this steam is formed nearer the top of the reactor and travels only onehalf as far before leaving the core. Consequently, the volume within the reactor When the reactor is operating at full occupied by steam is the same in each case. power, an additional and equal amount of steam is generated within the flash tank. Including the power derived from this steam, the reactor operating at full load on the dual cycle produces four times as much power as is produced by a simple boiling reactor under the same conditions. This illustrates the improvement in power density achieved through the use of the dual cycle. The output of large boiling reactors is usually limited so as not to exceed The use of the dual cycle, therefore, a tolerable steam content within the core. affords an effective means for improving the power density of boiling reactors within permissible limitations of reactor steam content. The previous discussion was not intended to restrict the use of the dual cycle to For temperatures, and components. any particular combination of pressures, instance, it is apparent that a comparatively small heat exchanger can be substituted for the flash tank. When pressures are increased, the use of a heat exchanger becomes increasingly attractive, since the fraction of steam that must be generated in the heat exchanger outside the reactor vessel becomes smaller while the pressure differential in the flash cycle, which would have to be overcome by pumps, becomes larger. The performance of boiling reactors is dependent on operating pressures and tem The operating conditions listed above were chosen to ensure successful peratures. operation of a dual-cycle reactor under the present conditions of technological devel However, it is expected that experience and research will provide a back opment. When ground of technology permitting the use of higher pressures in later plants. higher pressures and temperatures can be used, the performance of dual-cycle reac tors will benefit, as shown in Fig. 28. The thermal efficiency will improve, and the amount of steam to be generated outside the reactor will be reduced until, at a pres sure near the critical pressure, all the steam will be produced in the core. A 180,000-kw electric dual-cycle reactor is being designed by General Electric fined


water-cooled

MECHANICAL DESIGN

13-44

AND REACTORS

[SEC.

13


This is the largest of the reactors being for the Commonwealth Edison Company. built under the AEC's 1955 demonstration program. Figure 29 shows the arrangement of the reactor tank, pumps, and associated Figure 30 shows the arrangepiping. ment of the reactor components in a 60 power plant. 6.14 Large D;0-cooled Boiling Re 50 actors. Argonne Laboratory has investi of Di(>-cooled gated the feasibility 40 These re direct-cycle boiling reactors. 30 actors resemble the designs described in PORTION OF STEAM 1ENERATED 0U"~ Art. 5.2 except that the steam is generated 20 REACTOR within the reactor core. Forced circula tion is required for high-power ratings, 10 and much of the coolant volume is filled with steam. Aside from the obvious 500 1,000 1,500 2PO0 2,500 3P00"*\ difficulty of reducing leakage throughout CRITICAL REACTOR PRESSURE.PSIA PRESSURE the "heavy steam" power plant, there Fig. 28. Performance obtainable from dualis some question as to the heat-transfer cycle reactors. capabilities of the steam-water mixture. Several mechanical arrangements have been proposed. The pressure may either be contained within a pressure vessel or be divided, part of the pressure being contained by the coolant tubes.

Fio.

29. Components

of 180,000-kw

dual-cycle

power reactor.

If the heavy steam is used in a heat exchanger to produce light steam for use in the plant, then it will be much less difficult to reduce leakage of valuable 1)4). The Norwegian reactor is designed to boil I)j< ) within a pressure vessel. The power rating of 20,000 kw of heat is obtained with natural circulation within a 9-ft tank. Heavy steam rising from the tank enters a boiler where light steam is produced. DsO condensate returns from the boiler by gravity. With no pumps in the primary sys tem, this design represents the ultimate in simplicity. Unfortunately, the output obtainable from DjO reactors without pumping is severely limited, so this arrangement is not applicable in large i


Sec. 13-1]

13-45

SYSTEMS

The Argonne boiling reactor (EBWR) operates on H.O, but it is arranged to test feasibility of a low-leakage power system. Until the practicability of such a system is well established by test, it seems unlikely that direct-cycle, boiling D20 reactors will find large-scale application. the

Reactor — D20- or Graphite-moderated. D20so that less of the H2() coolant reduces reactivity. The conditions associated with this requirement are indicated in Fig. 16. If only a limited amount of moderation is provided by graphite or 1)2(), then the moderating effect of the H20 coolant can exceed the neutron absorption, so that loss of coolant causes an increase in reactivity. 6.16

Large H20-cooled

Boiling

or graphite-moderated reactors can be designed

180,000 KW- DUAL CYCLE

BOILING REACTOR PLANT PRELIMINARY ARRANGEMENT TO SHOW PRINCIPAL COMPONENTS) (DISTORTED

SHUTDOWN COOLING WATER TANK

SPHERICAL PRESSURE TIGHT CONTAINER


TURBINE-GENERATOR 190,000 KW

-AIR EJECTORS . H P HEATER

-L P HEATER

-

RECIRCULATING/ FEED PUMPS VENTILATION DUCTS EXHAUST TO STACK-

CIRCULATING WATER OUTLET

CONDENSER

SCALE C

WA^N^T

CP°UMDPESNSATE

FiQ. 30. Plant layout of 180,000-kw dual-cycle

power reactor.

Use of H20 as a coolant simplifies the design and operation of the power plant, steam is used and leakage requirements arc not as stringent as where "heavy" steam must be contained. With modifications, graphite reactors of the type described in Art. 3.1 can operate as boiling reactors. Figure 31 shows two arrangements that can be used. Although is possible to generate all the steam within the reactor, the power that can be removed in this design is limited. Reactors of this type have relatively long coolant channels, and if much power is removed, most of the coolant space will be occupied by steam. The dual-cycle design, also shown in Fig. 31, permits higher output from a boil ing reactor. Here, part of the steam is vaporized within the reactor, while the hot water from the reactor is passed through orifices into a "flash chamber," where a portion is vaporized to form steam. The remainder of the water will then be at reduced temperature, and as a result, the water that enters the reactor is below the boiling point. In a typical graphite-moderated reactor of this type, the water is

since "light"

it

MECHANICAL DESIGN' AND REACTORS

13-46

[Sec.

13

below the boiling point at the mid-point of the cooling channel, where the heat pro duction is most intense. Thus, by improving heat transfer the dual cycle makes possible higher power outputs from boiling graphite reactors. As alternates to the arrangements shown in Fig. 27, several stages of flash chambers may be used in series, and, if desired, heat exchangers can be used instead of flash chambers to remove heat from the nonboiling portion of the coolant. 6.2

Deentrainment

of Steam and Water Circulation

During the development of boiling reactors, it was found that 6.21 Steam Slip. Steam slip velocity, steam bubbles could rise through water at very high velocities. which is the difference between wat er and steam velocities, is strongly dependent on the saturation temperature or pressure. STEAM DRUM The effect of pressure or temperature on It can be steam slip is shown in Fig. 32. seen that the slip velocity varies from over 60 fps at atmospheric pressure to about 5 It is presumed t hat this fps at 600 psia.

60

g g £

SIMPLE BOILING

REACTOR (LOW POWER OUTPUT)

50|

3 -I

|I>

Z 1/1

,

61 9

(J
20|

10

100

Fig. 31. Two arrangements

showing how can be oper

reactors graphite-moderated ated as boiling reactors.

SATURATION PRESSURE, PSIA 15 50100200

CRITICAL CONDITIONS^

500

7X

500 600 200 300 400 SATURATION TEMPERATURE,*?

FlO. 32. The effect of pressure or temper Based on data taken ature on steam slip. from article by S. Untermyer. Xuclconia it (7): 43 (July. 1954). also P. A. Lottct. Proc. 1955 Conf. on Nuclear Eng., University of California, Los Angeles, California Co., Ltd., Berkeley, Calif. Y

DUAL CYCLE FORCED CIRCULATION BOILING REACTOR

curve may be extended, as shown by the dotted lines, so that the slip velocity becomes zero at the critical pressure of water, where steam and water have identical proper Other variables may also affect steam slip, but they have far less effect than ties. saturat ion pressure, and at the present time the significance of the secondary variables not clear. Nelson and Martinelli have derived a method for cal Water Circulation. 6.22 As applied to most culating the resistance to flow with two-phase circulation. will be found that the acceleration and entrance loss boiling reactors, however, ea» portions of this formula predominate, and in many instances the water velocity at the point heads velocity two to corresponds which velocity be calculated as that Using the curve of steam slip through highest. where the water velocity is

it

is


70


Sec. 13-1]

13-47

shown in Fig. 32, together with the frictional resistance of water equal to two velocity heads, it is possible to estimate the steam void distribution within a boiling reactor. 6.23 Deentrainment of Steam Bubbles outside the Reactor Core. It is necessary to limit velocities in the downcomers of the water-return system so that steam bub bles can rise faster than the water flows downward. Some margin should be provided to ensure that the smaller bubbles which 586° rise more slowly have an opportunity to / r*— escape.

Deentrainment of Water from Centrifugal separators are gen erally used to separate residual water droplets from steam. While various de signs are available for this purpose, data generally indicate that this separator is much more effective at lower pressures 6.24 Steam.

80°C

J

I

'

TEMPERATURE

—110 lb/in2 GAGE

V

T

I09

8

64 3

0 l-

«

s

8

1 5

4 3


36,400

AMPERES

122.8 VOLT

VOLTS

,

AMPERES

1

Hro%0SEC

VWWWVWWWY

u 2 io"

1 10

1 PRESSURE

J 8| 61 4

/ PRESSURE _ INDICATOR'

3| 21

40CF

Flo.

500°F

600°F

/ HGNITRON

700°F

33. Typical performances of steam sys tems operating at various temperatures.

Fio.

AND TIME

34. Test record showing formation of steam bubbles.

time delay in

and temperatures. Figure 33 shows typical performances of steam systems operating In the past, detection of water droplets has been so difficult that at various pressures. few data are available. Available data indicate that almost all the solid material

contained in steam is associated with minute water droplets. 6.25 Time Delay in Formation of Steam Bubbles. Time delays are extremely important in reactor self-regulation. Most thermal reactors have prompt-neutron lifetimes ranging from 10"° to 10"' sec. If a typical reactor has a lifetime of 10~J sec, then a sudden increase in reactivity to 1 per cent beyond prompt criticality causes the reactor power to increase by a factor of 2.7 in 0.01 sec. In order to quench such a reactivity excursion before the fuel plates melt, steam bubbles must appear within a few hundredths of a second. Therefore, experiments were performed at Arfconne National Laboratory to determine the time delay inherent in the boiling process. Water-filled metal tubes, ranging from 1 to 4 cm in diameter and from 10 to 30 cm in length, were heated suddenly by the momentary passage of a low-voltage alternat ing current. This current was furnished by the low-voltage windings of a transformerprimary current, and the surge was controlled by an ignitron. Sufficient power was available to heat the tube wall over 500°F within 0.01 sec. This condition corre sponds to a momentary heat flux of many kilowatts per square centimeter.

13-48

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

Tests were performed both with saturated water and with subcooled water. A typical test record is shown in Fig. 34. Steam formed within the tube started to Thus, it has become clear expel water within 0.003 sec after power was applied. that the boiling process itself is fast enough to control fluctuations in power due to These data have been substantiated by the behavior of an actual reactor boiling. in tests performed by the Argonne National Laboratory in Idaho. 7

OTHER TYPES OF HYDROGENOUS COOLANTS


Although water has the advantage of low cost and melting point and a background of well-developed technology, its high vapor pressure and corrosive character have stimulated the search for alternate hydrogenous coolants. Considerations of sta bility under prolonged neutron irradiation unfortunately rule out most organic materials. The most promising organic coolants are of the polyphenyl type. Because of their ring structure these compounds are more stable under neutron bombardment than are other types of hydrocarbons. These materials seem to be thermally stable Mixtures of 30 per cent biphenyl with 70 per cent up to temperatures of 700°F. orthoterphenyl and of 30 per cent metaterphenyl with 70 per cent orthoterphenyl have low melting temperatures, so that they can be used in a reactor without provisions for auxiliary heating. On the other hand, vapor pressures under operating conditions For instance, the following table gives the vapor pressure of are not excessive. biphenyl : Temperature, 200 300 400 500 600 700 800

°F

Pressure, psia 0 060 0 701 4.117 16.29 47.01 110. I 221.0

Since these materials do not appear to corrode ordinary metals, their use may per mit substitution of ordinary carbon steel for coolant system and aluminum cladding for the fuel. While little information is available, it might be expected that some type of "cleanup" system should be incorporated into reactors using these coolants so that the products of irradiation dissociation can be continuously removed from the coolant. Since these coolants do not contain materials that become radioactive under neu Because of the low tron bombardment, the need for secondary shielding is reduced. pressures required in the reactor system, the coolant can be expediently employed as part of the biological shield. Consequently, materials of this type may be useful in mobile marine applications where a small, light power plant is required.

13-2

HOMOGENEOUS

AQUEOUS REACTOR SYSTEMS BY R. B. Briggs

1

GENERAL FEATURES AND REACTOR TYPES

Aqueous homogeneous reactors are unique among the aqueous-moderated and reactors because they employ a fluid that is fuel, moderator, and coolant. H-0 or DaO is the moderator and coolant, depending on the application. Fissile and fertile fuels are dispersed in the water as solutions of soluble salts or slurries of insolu ble compounds. Fissioning in the fluid causes the temperature to rise or causes steam to be formed if the water is at the boiling point. Heat is extracted by circulating the fuel through heat exchangers, by withdrawing and condensing the steam, or, for a few low-power-density applications, by circulating a separate coolant through Hydrogen or deuterium and oxygen are produced tubes installed in the reactor core. by radiolytic decomposition of the water. The gases are recombined, the vapors are condensed, and the water is returned to the reactor.


-cooled

1.1

Advantages and Disadvantages

The attractive features of aqueous homogeneous reactors of a fluid fuel and moderator. They include:

are derived from the use

1. Fuel handling and processing are simplified, and continuous purification is possible. 2. The large negative temperature coefficient provides the reactors with a high Control by adjustment of fuel concentration can be sub degree of nuclear stability. stituted for control elements in the reactor core. 3. The primary heat-transfer function can be removed from the reactor core to heat exchangers whose design is independent of most nuclear high-performance considerations. 4. The reactor core is simplified and the neutron economy is improved by elimina tion of most of the structural materials from the core region. By proper selection of the fuel compounds parasitic absorption by the fuel can be made negligible. The use of an aqueous fluid fuel results in the following undesirable characteristics: 1. Power reactors contain large volumes of highly radioactive fluids at high tem There must be no leakage. perature and pressure. 2. All the equipment in contact with the fuel becomes highly radioactive and diffi cult to maintain. 3. Uranium, plutonium, and thorium compounds; water; and materials of con Operating conditions are struction form complex physical and chemical systems. I imited by the physical and chemical characteristics of these systems and by corrosion firul erosion effects. 4. The presence of stoichiometric mixtures of hydrogen or deuterium and oxygen constitutes a serious potential explosion hazard. 5. Owing to dispersion of fuel most of the benefit of Um or thorium fissions by fast rieutrons is lost. The major engineering problems of aqueous homogeneous reactors arise in design to overcome the undesirable characteristics. 13-49

ing

13-50

MECHANICAL


1.2

DESIGN

AND REACTORS

[SEC. 13

Types and Applications

Aqueous homogeneous reactors have been designed for producing power, fissionable materials, and neutrons for research. The power reactors include UMi burners for power only, and U"8-Pu converters and U"'-Th breeders for simultaneous production of power and fissionable materials. Some reactors have been designed for production of fissionable material alone. Reactor construction has been limited to research reactors of the "water-boiler" type,1-* which was developed at the Los Alamos Scien tific Laboratory, and to the homogeneous reactor experiment*'* and the homogeneous reactor test, ' two power reactor experiments at the Oak Ridge National Laboratory. Homogeneous reactors are classified as "one region" or "two region" according to the distribution of fissile and fertile materials in the reactor vessel. They are "cir culating fuel," "boiling," or "internally cooled" depending on the method of remov A hare reactor containing a homogeneous mixture of fissile and fertile ing heat. A typical two-region reactor con material in the moderator is a one-region reactor. sists of a core of predominately fissile material surrounded by a blanket of predomi In some two-region reactors a reflector is substituted for the nately fertile material. fertile blanket. If the core of a reflected reactor is large in terms of neutron-diffusion lengths and the reflector is of minor importance, the reactor might be classified as a one-region reactor. One-region reactors have the advantage of simplicity because there is only one fluid to handle. Two-region reactors are versatile and have good neutron economy in small sizes. The core and blanket regions may be kept separate so that they can contain different types of fluids, such as a uranyl sulfate solution in the core and a thorium oxide slurry in the blanket. To achieve good neutron economy, the tank that sepa rates the core from the blanket must absorb few neutrons. Circulating-fuel reactors are those in which the fluid fuel is circulated through the reactor vessel, where the heat is generated, and through a heat exchanger, external The cir to the critical volume, where the heat is transferred to a secondary coolant. culation may be accomplished by a pump, by a gas or vapor lift, or by thermal con In a boiling reactor, water is vaporized by the heat produced in the reactor, vection. During steady separated from the liquid, and condensed in external condensers. operation the condensate is returned to the boiling fuel. Internal cooling is accom plished by passing a coolant through tubes that are installed within the critical volume of the reactor where they are surrounded by the fluid fuel. Thermal convection, supplemented by local density differences resulting from the formation of gas or steam, causes circulation of the fuel about the tubes. Circulating-fuel reactors can be designed to operate at high power densities with For boiling reactors to operate at low volume fractions of decomposition gases. comparable power densities, it appears necessary to permit average vapor fractions Internal cooling is usually applied only to those of 25 per cent or more in the fluid. The reactors where the total power is low and the power density is unimportant. homogeneous reactor experiment and the homogeneous reactor test are examples of The "water-boiler" type of research reactor is internally circulating-fuel reactors. Boiling homogeneous reactors are still in the study stage. cooled. 2 2.1

FUELS


Fuels for aqueous homogeneous reactors include solutions and slurries of uranium, The large neutron absorption thorium, and plutonium compounds in H2() and D»0. cross section of hydrogen requires that the atomic ratio of hydrogen to U,3S or Th!" in the fuel be 5 or less to achieve high conversion ratios with H»t) as the moderator. It is difficult to obtain satisfactory fuels in such high concentrations, so the use of HjO may be restricted to research and power applications where the fuels are highly enriched and where there is reason to minimize the reactor volume. D«0 is used * Superscript numbers

refer to References


Sec. 13-2]

homogeneous aqueous reactor systems

13-51

when a high conversion ratio is important and when the fuel concentration must be If the volume of fuel in a reactor employing a highly kept low for corrosion reasons. enriched fuel is dictated by heat-removal considerations, the cost of the fuel may be less and the materials problem simplified if a fuel is used that contains a low concen tration of fissionable material in D2O as opposed to the higher concentration required with H«0. The number of uranium, thorium, and plutonium compounds that can be used in aqueous homogeneous reactor fuels is limited by considerations of compatibility with water and constructional materials, thermal and radiation stability, neutron-absorp tion cross sections of associated elements, and solubility or ability to form satisfactory slurries. Concentrations that are of most interest with DjO as the moderator range from 1 to 10 g/liter of uranium or plutonium in highly enriched fuels and 100 to 2,000 g/liter of uranium or thorium in fuels containing low concentrations of fissionable materials. With highly enriched fuels and H20 as the moderator, the interesting concentrations are from 10 to 50 g/liter. Organic compounds are usually unaccepta ble because of their poor thermal and radiation stability. 2.2

Uranium Solution Fuels


Uranyl sulfate, fluoride, phosphate, and nitrate are the uranium compounds that have been considered most often for use in solution fuels. Uranyl sulfate and nitrate solutions have been used in research reactors; uranyl sulfate solutions were the fuels for the homogeneous reactor experiment and the homogeneous reactor test. Uranyl phosphate solutions are proposed as the fuels for power reactor experiments at the

MECHANICAL DESIGN

13-52

AND REACTORS

[Sec.

13

Uranyl fluoride solutions have not been used Los Alamos Scientific Laboratory.1 because of lack of information on corrosion. Phase diagrams for the systems UO,S04-H,0, UOjFj-HjO, and UO.(NO,)rHiO, respectively, are presented in Figs. 1, 2, and 3.,,T The operating temperatures and concentrations are generally assumed to be limited to the unsaturated solution areas. Both uranyl sulfate and uranyl fluoride are stable to temperature and nuclear radia tions. Uranyl nitrate decomposes at elevated temperatures (the reaction is reversi ' If excess sulfuric ble), and elemental nitrogen is produced in fissioning solutions. acid is added, the two-liquid-phase temperature for the uranyl sulfate system can bo raised above 500°C as shown in Fig. 4. The addition of excess acid is effective in


400

30

40

50

60

WEIGHT PER CENT U02F2

Flo.

2. Phase diagram

for L'OiFt-HiO.

raising the precipitation temperature in the nitrate system and is presumed to rai* the two-liquid-phase temperature in the fluoride system. The parasitic absorption of neutrons in the anions of the fuel compounds is impor NIS must be used tant in regenerative reactors. Losses to fluorine are negligible. If high concentrations of uranyl sulfate i.'1 to minimize the nitrogen absorptions. used to obtain a large amount of resonance absorption in U"8, the captures in sulf'-T can be limited to < 0.1 neutron per total absorption in U"B. Although uranyl phosphate is insoluble in water, UOj dissolves readily in orthophosphoric acid. Data that relate the uranium solubility to the phosphate ronrer.tration and the temperature are shown in Fig. 5.6fl The solutions are stable to nuclear radiations, and compositions that are greater than 2 molar in phosphorv acid are thermally stable above 400°C. If U03 is dissolved in 95 per cent orthopho?phoric acid, solutions arc obtained that have vapor pressures near 700 psi at 450 t

Sec. 13-2]


■

13-53

Neutron losses to phosphorus appear to limit the application of the fuels in regenera tive systems. The uranyl dichromate and the sodium earbonate-uranyl carbonate systems have been studied also.8 The available data have not yet suggested suitable applications for these systems. 2.3

Uranium Slurry Fuels

Slurries are attractive alternates to solutions as reactor fuels because they can be neutral or alkaline and would be expected to provide fewer corrosion problems and


450,

20

30

70 40 50 60 PER CENT U02(NO3)2

WEIGHT

Fig.

fewer

3.

Solubility and thermal stability of U02(N0j)2 in II20.

To be satis problems of chemical stability than would highly acid solutions. fuels the slurries must be easily maintained as homogeneous suspensions with rnoderate agitation. They must be free from tendencies to flocculate or recrystallize Buch a manner that caking would occur and result in plugging of pipes and diffi in dispersing the particles after long settling periods. The pumping and heattransfer characteristics should be similar to those of water. Radiation should not adversely affect the slurry properties. An oxide of uranium seems to be the most attractive slurry fuel compound.* 0 f£exavalent uranium appears to be the stable oxidation state in water containing Only slight reduction occurs hydrogen and oxygen pressures of similar magnitude.1" r i the presence of a large excess of hydrogen. Uranium trioxide monohydrate has been found to be the stable form of the oxide elevated temperatures. It can exist in three allomorphic modifications — platelets. * See Rcf. 0, pp. 059-009.

factory

in

culty

i.t

MECHANICAL DESIGN

13-54

AND REACTORS

[Sec.

13

—depending


on its mode of preparation and treatment as shown rods, and bipyramids in Fig. 6.' The rods are bright yellow in color, normally 1 to 5 it long; the platelets are pale Bipyramids are also pale yellow and yellow, 6 to 50 it on edge and about 1 it thick. Rodlets, platelets, and bipyramids all several hundred microns along each edge. have an orthorhombic structure, but the crystal-lattice parameters differ slightly. Easily suspended slurries have been prepared of both the rods and platelets. Bipyramids, because of their size, require violent agitation to keep them in suspension. Slurries of rods have been pumped satisfactorily at temperatures below 200°C

WEIGHT

Fio.

4. Coexistence

PER CENT URANIUM

curves for two liquid phases in system UOiSO(-HsSO<-HrO.

Slurries of platelets, although easily pumped, have shown a tendency to form «nf' cakes on the pipe walls at temperatures above 200°C. In slurries that are circulated by pumping, the rods are abraded into ellipsoitii! Platelets are abraded, and the corner* shapes with about 5 it as the largest dimension. The hardness oi are rounded to give particles that are 1 it thick with 1- to 5-m sides. Platelets and rods, respectively, uranium trioxide crystals is 3.5 on the Mohs scale. have densities of 5.7 and 6.0. The settled rod and platelet slurries contain abou: 900 g of uranium per liter. The viscosities of the slurries decrease with increasing temperature in a manner similar to the variation in the viscosity of water. Below 100°C the viscosity of i slurry containing 250 g of U per liter is about 50 per cent greater than the viscosity o" The convective heat-transfer coefficient for t water at the same temperature. 250-g/liter slurry is only about 10 per cent less than that obtained for water under tr*

same conditions.


Sec. 13-2]


6

8

40

20

10

13-55

60 80 100

P04 MOLARITY

Fio.

5.

Solubility of UOi in HiPO, solution

U04-Hz0

200-300 '

iss-wcr00'

U02 (N0j)2-

»*>-

(Platelets)

Fio.

6.

Several

C hundred

ppm UOj*^.

I85-300°C

Preparation of UOrHjO

H20 •UO. H20

(Bipyramids)

The

reactors being studied by the reactor development group in the Netherlands dioxide slurries. It is reported" that small particles form stable slurries if pH is maintained from 7 upward. The density of UOi crystals is 10.7 g/cm3, a settled slurry of irregularly shaped particles contains about 3,000 g of uranium liter. It is proposed that hydrogen be used to operate a gas lift pump to circulate slurry. The hydrogen would be depended upon to prevent the oxidation of

employ

the and per

the

uranium.

Uranyl

carbonate, magnesium diuranate, and uranyl phosphate slurries have been

13-56

MECHANICAL

DESIGN

AND REACTORS

[Sec.

13

Uranyl carbonate decomposes at elevated temperatures to form the studied also. trioxide, although the decomposition can be suppressed by maintaining a high partial pressure of CG2 in the liquid. Magnesium diuranate forms soft cakes on the pipe walls when circulated at 250°C. Uranyl phosphate slurries have been circulated successfully at 250°C. The phosphate slurries are bulky, and the uranium concen tration may be limited to 200 or 300 g/liter. 2.4

Plutonium Fuels

Plutonium is a component of the fuel in power reactors that burn U"*. Compati ble uranium and plutonium compounds are required to achieve a satisfactory fuel. Too few data have been published to permit an evaluation of the plutonium fuels. 350

300


250

200

150

100

-100

30

40

50

60

70

WEIGHT PER CENT Th(N03)4

Fio.

7.

Solubility and thermal decomposition 2.6

of Th(NO«)4 in

HjO.

Thorium Fuels

Thorium-fuel compounds may be limited to the nitrate and phosphate for solution fuels, to the oxide and fluoride for slurry fuels, and to thorium hydroxide, which forms both slurries and colloidal solutions. N15 must be used to achieve good neutron economy with thorium nitrate. The nitrate decomposes at elevated temperatures as shown in Fig. 7.* The precipita• See Ref. 6, pp. 642-«54.


Sec. 13-2]


13-57

tion temperature can be raised by addition of nitric acid. Nitrogen is generated in the solutions by reactor radiations. Phosphoric acid solutions have been prepared containing 1,000 g of thorium per The neutron liter at 260°C with a ratio of 5 moles of phosphate to 1 mole of thorium. absorption cross section of the phosphorus is one-seventh that of the thorium, which places a severe penalty on the use of the compound in thermal breeder reactors. Thorium oxide slurries have been studied at concentrations from 100 to 1,200 g of There are no thorium per kilogram of water and at temperatures up to 300°C* Thorium oxide phase transformations of the solid phase at elevated temperatures. The properties of crystals have a density of 10 g/cm' and a Mohs hardness of 7.5. the slurries depend upon the method of preparation of the oxide, whether they are freshly prepared or have been pumped for long periods, and the impurities that are present. Thorium oxide, which has been found satisfactory for use in a high-temperature cir culating system, can be prepared by controlled thermal decomposition of thorium The final calcination temperature is in the range of oxalate or thorium formate. 650 to 800°C. The average size of the thorium oxide crystals is about 200 A, but the crystals are found in agglomerates ranging from 1 to 10 p in diameter. Thorium oxide slurries settle slowly even at elevated temperatures, and no diffi culty is encountered in redispersing the pure oxide after it has settled for several days. The slurries appear to behave as Bingham plastics after they have been pumped or A yield stress in the range of 0.1 to 2.5 psf must be overcome to agitated violently. initiate flow. Although the slurries are pumped easily, the accepted correlations for pressure-drop and heat-transfer coefficients developed for Newtonian fluids may not The magnitudes of the corrections are unknown, be valid without some modification. although there is some evidence that they are small for high rates of fluid flow. Thorium fluoride slurries that have been made were bulky. This property may limit the practical thorium concentration to 500 g per kilogram of water. Thorium Solutions hydroxide forms slurries or can be peptized to form colloidal solutions. have been made containing as much as 3,000 g of thorium per kilogram of water. 2.6

Corrosion

One of the most important problems of aqueous homogeneous reactors is that of The solution fuels are highly acidic and corrosive selecting materials of construction. to most materials. Some of the slurry fuels have a tendency to cause erosion of the materials. Whether a particular material is acceptably resistant to corrosion by uranyl sulfate depends on the temperature, the solution velocity or local turbulence conditions, the uranyl sulfate concentration, and the concentration of additives such as sulfuric At temperatures below 100°C there are many materials that can be used to acid. Included among the metals contain uranyl sulfate solutions at all concentrations. are the austenitic stainless steels represented by types 347, 321, and 304; the highchromium steels, types 443 and 446; the precious metals gold and platinum; and Inconel, tantalum, zirconium, titanium, and the Stellites. Graphite or Graphitar, Teflon, titania, alumina, zirconia, quartz, and Pyrex glass are among the satisfactory nonmetals. The number of acceptable metals decreases rapidly as the temperature is raised. Titanium has demonstrated superior corrosion resistance in uranyl sulfate solutions Good under essentially all temperature, velocity, and concentration conditions. quality zirconium and Zircaloy 2 appear to be as corrosion resistant as titanium. Gold and platinum are not corroded and are useful for special applications. Type 347 has been found to be the most corrosion resistant of the stainless steels that meet the structural requirements of reactors. Low corrosion rates are found with fluid velocities as high as 35 fps and temperatures at least as high as 572°F in uranyl sulfate solutions that contain less than 10 g of uranium per kilogram of water. At higher uranium concentrations it is necessary to reduce the flow velocity until, at « See Ref. 9 and Ref. 0. pp. 069-670.

13-58

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

concentrations of 300 g of uranium per kilogram of water, the maximum permissible Although other austenitic stainless steels and the highvelocity is below 10 fps. chrome steels are generally less corrosion resistant than type 347, their behavior is similar and they have some application in the same temperature and concentration


ranges.

Quartz, Teflon, and the synthetic gems —spinel, titania, and zirconia — are only slightly affected by uranyl sulfate solutions at temperatures as high as 250°C. Corrosion of materials by uranyl fluoride has received less attention because of the expectation that both the solution and the vapor phase would be very corrosive at Tests that have been run show that the corrosion of stainles.* high temperatures. steels and titanium by the solution phase is no more severe than corrosion in uranvi Zirconium and its alloys are incompatible with even dilute uranyl sulfate solutions. Gold and platinum are not corroded by uranyl fluoride. fluoride solutions. Phosphoric acid solutions of uranium are so corrosive that it is difficult to take Gold and platinum may be the only advantage of their superior thermal stability. materials that have the necessary corrosion resistance at high temperatures. Some tests have indicated that stainless steel is corroded at least as rapidly by uranyl nitrate solutions as it is by equally concentrated uranyl sulfate solution.* Little is known about the corrosiveness of uranyl chromate and sodium uranyl car bonate solutions. The uranium trioxide, uranyl phosphate, and thorium oxide slurries are essentially neutral when free from impurities and should be no more corrosive than distilled water. Slurries containing the uranium compounds have been circulated in stainlesfsteel systems at temperatures from 100 to 250°C with no evidence of erosion and very little corrosion attack. The velocity in the piping was usually 20 fps or less, but there were local velocities in excess of 50 fps. The slurries had little effect on Stellitt pump bearings at 100°C. Very little corrosion or erosion has been experienced with thorium oxide slurries as temperatures as high as 290°C and with velocities as high as 20 fps in straight pipes or in smooth bends. Some attack is found in pumps, where the velocity is high and the turbulence near the metal surfaces is great. The presence of slurry in pump bearings has resulted in excessive wear rates. Titanium has been found to be mutt more resistant to attack than stainless steel and is used for parts where the service is severe. 2.7

Radiation Decomposition

Fissioning in aqueous homogeneous reactor fuels results in decomposition of tte moderator and the production of hydrogen and hydrogen peroxide. The hydrogen peroxide may decompose to form water and oxygen, or it may react with uranyl ion to form uranyl peroxide. Uranyl peroxide is rather insoluble and unstable; it may precipitate as the monohydrate or decompose into uranium trioxide and oxygen gas Under practical reactor operating conditions important volumes of a stoichiometn mixture of hydrogen and oxygen arc produced in fissioning solutions. At low tempera ture and moderate power density, uranyl peroxide precipitates have been observed. The yields of hydrogen gas and hydrogen peroxide in solution fuels are essentially constant with temperature and vary only slightly for different uranyl salts. The yield G?h, in molecules of H. formed per 100 ev of fission-fragment energy absorbed by the solution is about 1.8 for dilute solutions and decreases to about 0.5 for solu tions containing ~ 800 g of uranium per liter.* This compares with a Gh, of 0.4 to 0.2 for the absorption of (3 and y radiations in the solutions. The rate of decomposition of hydrogen peroxide is directly proportional to concen The rate constant increases with increasing temperature and varies with tration. impurities in solution as shown in Fig. 8.' Uranyl peroxide will precipitate in » reactor or in solution taken from an operating reactor when the power density resum ing from the absorpt ion of radiations is

Sec. 13-2]


13-59


where C is the solubility of U04-H20, K is the rate constant for the decomposition of hydrogen peroxide at the temperature of the solution, and G is the peroxide yield for the radiation involved. A solubility frequently used for uranyl peroxide is 0.004 mole/liter in the absence of excess acid. It has been estimated that precipitation will occur in reactors that are operated at power densities as low as 0.2 kw/liter when the temperature is below 100°C. When samples are being taken from an operating reactor, the sample lines must be kept warm to avoid precipitation which might plug the line or affect the quality of the samples. The presence of small

quantities of dissolved iron and copper has been found to catalyze strongly the decomposition of hydrogen peroxide in uranyl sulfate solutions. Hexavalent chromium is reduced to the trivalent state in uranyl dichromate fuels. Elemental nitrogen is produced in nitrate fuels in nuclear reactors. Gn. values for fission-fragment radiations were found to be 0.002 and 0.06, respectively, for tho rium nitrate solutions containing 60 and 000 g of thorium per kilogram of water. A Gn, value of about 0.02 has been ob served for the Los Alamos Supo reactor. The amount of gas liberated in slurry fuels will depend on the particle size and, therefore, on the way in which the absorp tion of the energy of fission is distributed between the liquid and the solids. Gasgeneration rates for fuels where the particles are submicron size should approach those found with solutions. 3

3.1

REACTOR FLOW SHEETS Subdivisions

of General Flow Sheet

A complete flow sheet for an aqueous homogeneous the following systems: 1.

reactor can be subdivided into

Reactor and primary heat-removal system

2. Gas-handling and recombiner system 3. Fuel-handling and -storage system 4. Heat-disposal or power-generating system 5. Chemical-processing systems 6. Waste-disposal systems 7. Plant-services systems

Not all the systems would be included in every reactor installation; research reac tors which are used as neutron sources require a minimum of facilities. Only very large reactor installations would have complete chemical-processing and waste-disposal facilities. Systems 1, 2, and 3 and part of 6 are discussed below. The heat-disposal, power-generating, and plant-services facilities are similar to those which are required other types of reactors. Until methods for simple elimination of fission products from fluid fuels are demonstrated, it must be assumed that the fuel-purification proc Disposal facilities for most of the esses will be similar to those used for solid fuels. liquid and solid wastes will be associated with the fuel-processing plants; however, will usually be necessary to make provision for handling radioactive gaseous wastes the reactor site.

for

it

»t

MECHANICAL,

13-60

DESIGN

AND REACTORS

Reactor and Primary Heat-removal

3.2

[SEC.

13

System

flow sheet is illustrated by a typical The reactor and primary-heat-removal-system diagram for a two-region circulating-fuel power breeder presented in Fig. 9." The core system consists of the reactor core, pressurizer, gas separator, steam generator, and circulating pump. Several parallel circuits may be required to remove the heat Similar equipment is provided for the blanket system. The selec from a single core. tion of flow rates and temperatures is based on corrosion and physical-properties data and studies related to the economics of the reactor plant and the associated With a given maximum fuel temperature, higher steam steam-generating facilities. temperatures and improved power-plant efficiencies are achieved at the expense of The higher circulation rates, larger heat-transfer areas, and greater fuel inventory. usual circulation rates vary between 0.05 and 0.1 gpm per kilowatt of heat generated, STEAM TO TURBOGENERATOR 8 PLANT AUXILIARIES 435°F a 365 PSIA 22,000 LB/MIN GAS TO RECOMBINERS 30 LB/MIN D2 120 LB/MIN Oz

530 LB/MIN D20 10-100 GAL/MIN


ENTRAINED

LIQUID

5,000 LB/MIN GAS TO RECOMBINERS 3 2 LB/MIN D

I28 LB/MIN

0 LB/MIN DO | 10-50 GAL/MIN

128

ENTRAINED

LIQUID

FEED 8 DISCHARGE

FEEDS DISC^^CE

LINES

LINES

Via.

9.

Flow diagram for high-pressure

circulating system.

Excess and the fluid temperature rises by 65 to 130°F in passing through the reactor. pressure, above the vapor pressure of the circulating fluid at its maximum tempera ture, is provided to limit the volume of saturated decomposition gases in the reactor to less than 10 per cent of the total volume. The flow sheets for other reactors vary depending on the type and purpose of each A one-region circulating-fuel reactor has only the core circuits. No pump reactor. "Water-boiler" research reactors have is required for a thermal-convection circuit. the heat exchanger included as part of the core, the gas is separated at a free surface in the core, and the reactor pressure is governed by the pressure in the reeombiner system. Only vapor is circulated to the heat exchangers of a boiling reactor, and condensate is returned to the core at a rate of 4 to 10 lb per kilowatthour of heat gen erated. A separate pressurizer might be provided for controlling some types of Dol ing reactors. 3.3

Gas Handling and Reeombiner System

Depending on the operating conditions and the type of fuel, stoichiometric mixtures of hydrogen and oxygen are produced at rates as high as 12 std ft3/(min) (Mw) or tkf gas production can be suppressed completely by the addition of catalysts to the fuei


Sec. 13-2]

13-61

to reeombine the gases in situ. With the maximum rate of gas production, about 4 per cent of the energy liberated in the reactor is converted into heat by recombining the gases. This heat is usually wasted in research or experimental reactors; it would be recovered in a large power reactor possibly for superheating the steam from the main boilers. Flow diagrams for flame and catalytic recombiner systems for handling the gas from the core circuit of the reactor of Fig. 9 are shown in Figs. 10 and ll12. The flame-rccoinbiiicr system flow diagram is similar to the one that was used for the homogeneous reactor experiment. Gases from the reactor are reduced in pressure and diluted below the explosive limit, about 10 mole per cent of hydrogen, by steam in the evaporator. The diluent steam is removed in the precondenser; the gases are burned, cooled, and condensed; and the condensate flows to a storage tank, to the evaporator, or to the reactor. (The distribution of condensate between the various LET-DOWN

STREAM

30LB/VIN I20LB/MIN 530LB/MIN

D2 Oj Dj,0

GASEOUS FISSION PRODUCT TO STORAGE


10-100 GAL/MIN ENTRAINED LIQUID

Fio.

10.

Flow diagram for low-pressure

flamc-recombiner

system.

tanks is used in controlling the uranium concentration and the reactivity of the Gaseous fission products appear as noncondensables in the gas cooler and reactor.) condenser and are discharged from the system. The heat consumed in the evaporator in this system is greater than the heat pro duced in burning the hydrogen and oxygen. Flame arresters might be used to con trol flash-backs from the burners, thus reducing the diluent requirements. Catalytic recombiner systems are attractive because it appears that they can be Gases from the reactor are designed to operate safely at both high and low pressures. diluted below the explosive limit with steam or a mixture of steam and a gas such as oxygen or helium. Explosions and excessive temperatures are prevented by keeping the hydrogen content of the saturated gas below 3 and 6 mole per cent, respectively, for oxygen and helium diluents. Although a blower is shown for recirculating the diluent, these systems can be designed for thermal-convection recirculation. Handling and recombining the gases present major problems because of the hazards that exist when explosive mixtures are present. The hazards can be reduced by recombining the gases, using catalysts added to the fuel. A small amount of copper sulfate is an effective catalyst in uranyl sulfate solutions at temperatures above 450°F.

MECHANICAL

13-62

800-1100° F

DESIGN AND REACTORS J3AS

COOLER

[Sec

13

AND

.CONDENSER

BLOWER TO FISSION PRODUCT

j,

ADSORBERS

2000 LB/MIN

GAS 8 ENTRAINED LIQUID FROM

Oj

3000 LB/MIN DjO

SEPARATOR

30LB/MIN I20LB/MIN 530LB/MIN

ADSORBERS ►FROM FISSION PRODUCT

OR 110 LB/MIN 1230 LB/MIN

Cfe 0£ D20

He Cy3

100-1000 GAL/MIN OF ENTRAINED

LIQUID

572-F-2000

PSIA

572-F-2OO0 PSIA


LIQUID RETURN TO SUCTION OF CIRCULATING

PUMP

Fia.

11.

Flow diagram for high-pressure 3.4

catalytic-rerombincr

system.

Fuel Handling and Storage System

Facilities must be provided for storing the fuels when the reactor is being main The tained, for adding new fuel to the reactor, and for removing fuel for processing. equipment consists of critically safe dump tanks, evaporators for changing the fuel concentration, pumps for feeding fuel to the reactor, small recombiners for recombining the decomposition gases, and heat exchangers for removing the fission-product decay heat. The system operates at low pressure and may share equipment with a low-pressure recombiner system. 3.5

Gaseous Fission-product

Disposal System

Gaseous fission products that are discharged from the reactors must be disposed If they are undiluted, they can be bottled of without creating a biological hazard. Where large and stored in shielded areas until most of the activity has decayed. amounts of diluent are present as in a catalytic recombiner installation, the fissionproduct disposal system consists of cold traps or driers to recover D-O, adsorption beds containing activated carbon or other adsorbents to retain the fission products until all except the 10-year KrS6 have decayed, and a tall stack and blowers to dilute the krypton and dispose of the gases to the atmosphere. The adsorption beds must he It may eventually become necessary to provide cooled to maintain their capacity. additional facilities for complete retention of the Kr*6, which is released at a rate of 1 to 1.5 curies per megawatt-day of reactor heat. 4

COMPONENTS

OF

HOMOGENEOUS

AQUEOUS

REACTOR SYSTEMS

Much of the equipment for aqueous homogeneous reactors is similar to that used for other types of water-cooled reactors. Greater care in fabricating the equipment and inclusion of some special features are dictated by the value of the fuel and the Major departures from the designs or requirenecessity for complete containment.

Sec. 13-2J


13-63

merits of other types of reactors are found in the reactor-vessel configurations, gas separators, the recombiners, and the fuel-etorage equipment. 4.1

the

Reactor Vessels

Conceptions of two-region reactor vessels for circulating-fuel power reactors are shown in Figs. 12 and 13." The unique features of the designs are related to obtain ing a satisfactory flow distribution in the core, where most of the heat is generated. Structural requirements are similar to those of containers for other types of watermoderated reactors. The core tank and pressure shell, respectively, would be


FUEL OUT

FUEL IN

Fio.

12.

Two-region reactor with straight-through core.

expected to vary in diameter between 1 and 8 ft and 3 and 12 ft, depending on the application and power level. The two configurations differ in the attachment of core piping and in the flow dis tribution in the core. With the "straight-through" core the fuel flows in at the bottom of the core tank and out at the top. Diffuser screens provide a smooth expan sion of the flow. As the fluid passes through the core, the temperature rises and gas For bubbles are produced uniformly according to the heat-production distribution. many calculations the average temperature in the core and the average vapor fraction are assumed to be the arithmetic averages of the temperature and vapor fraction in the inlet and outlet streams. The number of screens or perforated plates required for a conical diffuser can be estimated from the equation

MECHANICAL

13-64

DESIGN AND REACTORS

[Sec.

13

where A0 = area of the inlet nozzle An = area at the point of tangency of cone and sphere k = screen constant which varies linearly between 1.2 and 2.0, respectively, for solidity ratios* of 0.4 and 0.5 n = number of screens The maximum wire diameter should be less than one-tenth the minimum distance between screens, the ratio of cross-sectional areas of adjacent screens should be kept CORE OUTLET

DISCHARGE LINE

CORE

INLET


BLANKET OUTLET

THERMAL SHIELD

BLANKET INLET

Fig.

13.

Two-region reactor with "concentric-inlet-and-outlet"

core.

The pressure drop through constant, and the solidity ratio should be less than 0.5. a set of screens is about 1.6 inlet velocity heads. In the concentric inlet and outlet core the fluid is introduced as a jet which passes through the center of the core, reverses direction, and flows out through an annulus around the inlet pipe. There is a very high degree of mixing in the vessel, so that the average temperature and vapor fraction in the region surrounding the jet approach The pressure drop through the temperature and vapor fraction in the exit stream. the vessel is about 1.5 inlet velocity heads. The concentric inlet and outlet design has an advantage in mechanical simplicity. Neither type of core appears to have an important advantage in nuclear or hydrodynamic design. Reactor vessels for one-region circulating-fuel power reactors are similar in con* Solidity ratio is the fractional area of the screen or perforated

plate that is solid.

Sec. 13-2]


13-65

figuration to the core tanks and similar in construction to the pressure shell for twoThe size of a one-region reactor is expected to be between 1 ft in region reactors. diameter for research or special-purpose reactors and 15 ft for large power reactors. Other designs for aqueous homogeneous reactor vessels are those used for the homogeneous reactor experiment, Fig. 14, and the Los Alamos water-boiler reactors, The fuel solution entered the core of the former through a tangential inlet Fig. 15. setting up a vortex in the vessel and causing the gas bubbles to be displaced to a cen-

PRESSURIZER

- SAFETY PLATE DRIVE

VENT-

.COOLING

JACKET

- GAS AND FUEL LET DOWN


-FUEL OUT

-FUEL IN

-REFLECTOR VESSEL

CONDENSATE

- REFLECTOR FLOW LINER

SAFETY PLATES

"ION

CHAMBER

THIMBLE

BRIDGE

TUBE

APPROX SCALE

CORE TANK DRAII

'

0

Via.

14.

HRE

I

IN FEET 2

i

core vessel.

tral was

void. The gas was withdrawn through a nozzle at the north pole, and the liquid discharged through an annulus around the nozzle. This type of flow is less attractive for large reactors because the pressure drop is large, the effectiveness of the core as a gas separator decreases with increasing size, and there is some difficulty in stabilizing the void to eliminate fluctuations in reactivity. The pressure drop for a reactor with one or more tangential equatorial inlets and polar outlets can be estimated from the equation13

^.(r"[®'-']+w®'}

MECHANICAL

13-66

DESIGN

AND REACTOKS

[Sec.

13

where AP = pressure drop across sphere, lb-force/ft* liquid density, lb-mass/ft' p V0 = total volumetric flow, cfs conversion factor = 32.2 (lb-mass) (ft) /(sec*) (lb-force) go ri = inlet pipe radius, ft r2 = outlet pipe radius, ft R = sphere radius, ft Ni = number of inlet pipes Nt = number of exit pipes

"

"

4.2

Gas Separator

An axial-flow pipeline device for separating gases from the fluid fuels is shown in The fluid is caused to rotate by the vanes at the inlet. Bubbles that are Fig. 16. entrained in the liquid are displaced to the central vortex and withdrawn through the Vanes are provided at the outlet of the separator to straighten the outlet nozzle. flow and to recover some of the energy of rotation. CONDENSER

COOLANT LINES

CIRCULATION LINES


GAS

REFLUX

CONDENSER "

COOLANT LINE

^

FILL a DRAIN LINE

Mat

>

./-VERTICAL THIMBLE

HORIZONTAL

THIMBLE

Fig. ROTATION

15.

Water-boiler core vessel.

VANES -

\

RECOVERY VANES ^RE

GAS WITHDRAWAL PIPE

Fio.

16.

Gas separator.

Sec. 13-2]


13-67

Stable flow and good gas separation arc obtained in the separators when the Kroude number is greater than 4.

Fr

=

\\'/Rg

(4)

= tangential velocity at It, fps = inside radius of pipe, ft g = gravitational acceleration The efficiency of the separator depends on the properties of the liquid and the vapor, the size distribution of the bubbles, the thickness of the annulus of rotating fluid, the axial and rotational velocities, and the length of the separator. The length required to obtain an efficiency greater than 0.9 can be estimated from the equation where

W R

l

=

-150(1

-

m

L -

[i

-

^^rr,]

(5)

length between hubs, ft inside radius of pipe, ft radius of hub /radius of pipe volumetric flow rate, cfs tangential velocity at R, fps a constant with dimensions ft'Vsce''* Experimental separators with 0 = 0.4 and with simply curved vanes having dis It was determined that charge angles near 45° have performed satisfactorily. ~ lAv', where v' is the superficial axial velocity through the separator and is equal va to V/irR*. Values for k vary between 1 and 1 .5 for most power-reactor fuels, and the calculated separator lengths are usually in the range of three to eight pipe diameters. where

R


= 0 = V = Wo = k =

4.3

Heat Exchangers

Shell and tube heat exchangers of all-welded construction are used for removing heat from the fuel. Fuel is ordinarily circulated through the tubes, and the coolant The inventory of fuel in the exchanger decreases but the is contained in the shell. fabrication cost increases with decreasing tube diameter. Over-all investment and operating costs for power-reactor installations are estimated to be at a minimum with

Fig.

17.

HRT 5-Mw

steam generator.

13-68

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

tubes that are about %-in. OD and with fluid velocities between 10 and 20 fps. Over-all heat transfer coefficients from 700 to 1200 Btu/(ft')(hr)(°F) are calculated for exchangers of optimum design. The heat exchanger of homogeneous reactor test is shown in Fig. 17. Circulating Pumps

4.4

Canned-rotor pumps are considered to be the most promising pumps for circulating the fluid fuels. Standard Westinghouse canned-rotor pumps with some modifica tions were used in the homogeneous reactor experiment and in the homogeneous reac tor test. Maintenance of large pumps can be simplified by making all the internal Most satisfactory parts removable through a flanged opening at the top of the pump. performance is obtained when a few gallons per hour of condensate is fed into the rotor cavity to keep the bearings flushed free of fuel solution or slurry. The pumps are constructed of stainless steel. Titanium is used for the impeller and for other parts where high velocities and turbulence are encountered. They are available for 2,000-psi service in capacities from 5 to 20,000 gpm with developed heads in the range of 10 to 300 ft of fluids. 4.6

Pressurizers


Circulating-fuel reactors are pressurized by vaporizing fuel or condensate or by supplying a constant gas pressure in vessels that arc attached to the main circulating

Fio.

18.

HRT

pressurizer.

The vapor volume in the pressurizer, the distance from the reactor to the free surface, and the size of connecting piping are specified to limit, to an acceptable value, the pressure rise that occurs in the reactor as fluid is ejected during a rapid increase in Provision must be made for measuring the liquid level and for venting or reactivity. loops.

recombining any hydrogen and oxygen that collect in the pressurizer in order to The pressurizer for the homogeneous prevent explosive mixtures from forming. reactor test is shown in Fig. 18. 4.6

Recombiners

Two types of flame recombiners have been considered for homogeneous reactor A drawing of the rccombiner used in the homogeneous reactor experiapplications.


Sec. 13-2]

13-69

tnont is shown in Fig. 19. It was water-cooled and had a single-burner nozzle and a spark plug igniter (which operated continuously). Satisfactory operation was obtained as long as the gas velocity through the nozzle openings was greater than the flame-propagation velocity. -SPARK

-BURNER

PLUG

NOZZLE

,_HEAD

Fdetail

-COOLING


Fio.

19.

WATER JACKET

HRE

flame recombincr.

A second type of flame recombiner is shown in Fig. 20. Gas entering the recombiner is diluted below the combustion limit with steam. As the steam is condensed, the gas is ignited by spark plugs, hot wires, or small catalyst beds and burns quietly in the condenser. This type of recombiner is attractive for boiling-reactor applications. Platinum and palladium catalysts deposited in concentrations of 0.03 to 0.3 weight per cent on }£-in. right cylinders of alumina have proved effective for recombining hydrogen and oxygen in noncombustible mixtures. 100 per cent re Essentially combination is obtained in packed beds of NON-CONDENSIBLES TO ABSORBERS 0.3 per cent platinized alumina when the space velocity of the gas, converted to standard conditions, is 400,000 hr~'* or THERMOCOUPLE

-THERMOCOUPLES

H202i STEAM

MIXTURE

Fio.

20. Volume combustion recombiner.

Fio.

21. Catalytic recombiner.

No important pressure effects have been observed. Flow rates that are greater a factor of 10 are acceptable in recirculating systems where lower efficiencies are permissible. The effectiveness of the catalyst is reduced by moisture and by adsorbed poisons. Provision must be made for heating the bed or preheating the gases to ensure that the

less.

by

* Space velocity = (flow rate of Ras) /(volume of packed bed).

13-70

MECHANICAL

DESIGN

AND

KEACTORS

[Sec.

13

catalyst is dry when operation is started. Heat liberated by the recombination will keep the pellets dry during normal operation. Iodine is a volatile fission product that can poison the catalyst. Poisoning by iodine can be prevented or the catalyst can be rejuvenated by operating the bed at temperatures above 700°C. A typical catalytic recombiner unit is shown in Fig. 21. The design of a particular unit will depend on the capacity required and the pressure drop permitted. Minimum pressure drop is important in thermal-convection systems. 4.7

Fuel-handling Equipment

Fuel-storage Tanks. Fuel-storage tanks must be designed to be critically safe and to withstand the pressure that results when the fuel from the reactor is dis charged rapidly into them. The latter requirement usually dictates that the tanks


4.71

FEED PUMP

TO

Fig.

22.

HRT

fuel-storage

tanks.

for high-temperature reactors be designed for pressures near 500 psi. The critically safe requirement is calculated simply from the critical equations for low-enrichment fuels but gives rise to some difficulties for highly enriched fuels. Because it is possi ble for the uranium to precipitate or to be concentrated in other ways, the tanks that contain highly enriched fuels usually must be horizontal and contain less than 0.5 kg of fissionable material per foot of length. Essentially no restrictions are placed on If vertical tanks are used, they must be about 6 in. in diameter the tank diameter. or heavily poisoned or both in sections where the uranium can concentrate. Means for inventorying the fuel and changing its concentration are usually provided Inventory can be obtained by measurement of liquid with the fuel-storage tanks. Evaporators of a variety of designs can level or by weighing the tanks and contents. The fuel-storage-tank installation for the homo be used for concentrating the fuel. geneous reactor test is shown in Fig. 22. 4.72 Fuel-feed Pumps. Aqueous homogeneous power reactors require equip ment for pumping radioactive fuels from the low-preasure storage tanks to the highpressure reactor system. Leakage cannot be tolerated in most cases. Diaphragmtype positive-displacement pumps have proved most satisfactory for this application. Pumps are available with capacities in the range of 0.1 to 2 gpm when pumping


Sec. 13-2]

against a pressure

A typical pump

of 2,000 psi.

head and installation

13-71 are shown in

Fig. 23. 4.8

Valves

requirements for the reactors are similar to those for most chemical systems around the stein to the surroundings can be tolerated. Bel lows-sealed stems are used on valves for operation at pressures as high as 2,000 psi. A typical valve construction is shown in Fig. 24. Valve

except that no leakage

4.9

Fission -product Adsorbers

Approximately 60 per cent of the fission products have xenon or krypton isotopes of the decay chains. Half of the isotopes (~70 cm1 STP/Mwd) are stable or have half-lives that permit them to be removed from a fluid fuel before they In most reactor designs the fission-product gases are accompanied by rela decay. tively large volumes of air, helium, or other diluent. Adsorption of the krypton and xenon on activated carbon or other adsorbents is a convenient method of storing the gases until most of the radioactivity decays and they can be discharged to the


as members

atmosphere. Design of the adsorption units is based on the approximate solution of equations for the depletion of radioactive isotopes from a gas stream by isothermal elements of an Assumptions used in adsorption bed and for the temperature distribution in the bed. solving the equations include: 1. Steady-state distributions of temperature and concentrations of radioactive iso topes exist in the gas and on the adsorbent. 2. Capacity of the adsorbent is unaffected by the retention of solid decay products. 3. Heat is produced at the point of emission of /3 particles, and in most instances the contribution of y photons to the heating can be neglected. Depletion of isotopes along a length of an adsorber bed at constant temperature

is given by ln

W>

= ~ L

Kg)

+

Kg-)

22ôb-J

Az

(6)

of radioactive isotope i in total gas stream leaving an increment of the bed with length z concentration of radioactive isotope i in total gas stream entering the increment linear distance through increment of packed bed, ft decay constant for isotope t, sec~ 1 density of the gas, lb/ft' fraction of voids in the bed density of charcoal in the bed, lb charcoal /ft' bed total mass flow rate of gas per unit cross-sectional area of bed, lb/(fts)(sec) equilibrium distribution ratio for the isotope

= concentration

where

= z = X, = p = =

/

pb = G =

Ki

=

(std cc of t adsorbed on solid) (g charcoal) (mm Hg partial pressure

P Mi

of i in gas)

= total pressure of gas in the bed, mm Hg = molecular weight of isotope i

Adsorption beds for homogeneous reactors are designed to delay the radioactive To accomplish this, ?ases until essentially all but the 10.3-year Kr" have decayed. mfficient bed volume must be provided to reduce the concentration of 5.3-day Xem, :he gas of next longest half-life, by a factor of about 100. Although the distribution •atio K decreases rapidly with increasing temperature (Fig. 25)," problems encoun-

ered in cooling emperatures of

olet sections

beds usually require that the adsorbent operate at 200°F and that the heat-transfer path be relatively short in the the short-lived isotopes are adsorbed and release their decay

the adsorption 100 to

where

13-72

MECHANICAL

DESIGN

AND

REACTORS PRESSURE I SYSTEM

TO HIGH

-DOUBLE-WALL RUBBER

PULSATOR


Flo.

CAST ALUMINUM COVER

23. Injection feed pump.

SPRING

STEEL SPRING

AIR IN FOR NORMALLY OPEN OPERATION -

CAST ALUMINUM YOKE

AIR

IN

CLOSED

LEAK

\

FOR NORMALLY OPERATION

ASBESTOS AND CARBON PACKING

DETECTOR-

FULTON

PACKLESS.

ASSEMBLY

SLYPHON BELLOWS

VALVE

STELLITE NO. 6 PLUG AND SEAT •jiummrft.

Fig.

24. Bellows-sealed

vulve.

[Sec

13

Sec. 13-2]


13-73

The volume of an isothermal adsorption bed can be estimated from Eq. (6) appropriate constants for Xe1" and the adsorbent, but a complete design requires a careful analysis of the heat production and removal and the temperature distribu tion in the bed, usually for short increments, to ensure adequate cooling and proper selection of values for K. Adsorption beds for the homogeneous reactor experiment and homogeneous reactor test were made of pipe packed with 8- to 14-mesh carbon The pipes varied in diameter pellets and immersed in a tank of water for cooling. from in. at the inlet to 6 in. at the outlet. energy.


and

6

REACTOR CONTROL

Although mechanically operated control elements may be necessary to achieve the precise regulation desired in some types of research reactors, the complete control of

power

reactors can be based on the large negative power coefficient of reactivity, con the fuel concentration and enrichment, and control of the power demand. Shim control to effect the reactivity changes required for normal starting, stopping, and changing of the temperature level of the reactor can be accomplished by adjusting the concentration or enrichment of the fuel in the core as shown for a typical reactor Fig. 26. Dependence is placed on the negative power coefficient of reactivity to satisfy the regulation and safety requirements. The fuel temperature and the volume of gas or vapor in the reactor increase with increasing power, resulting in a large negative coefficient. Values of dk,,//dT between —IX 10"4 and —5 X 10-' per degree centigrade have been calculated for

trol

in

of

13-74

MECHANICAL

DESIGN AND REACTORS

[Sec.

13


reactors of a wide range of sizes and configurations at temperatures from 20 to 300°C. The negative coefficient resulting from vapor production can be greater or less than the temperature coefficient, depending on the design of the reactor. Although the negative power coefficient is sufficiently large and certain to provide regulation and to limit foreseeable power excursions in all reactors that have been considered, the safety of the reactor depends on the coupling between the core and A rapid increase in the reactivity would be accompanied the pressurizing system. by rapid increases in power, temperature, and vapor volume in the core of the reactor, resulting in expansion of the fuel and ejection of part of it into the pressurizer. The pressurizer and its connection to the core must be designed so that the transfer of fuel can be accomplished without producing damaging pressures in the core. These requirements are satisfied by making the vapor volume of the pressurizer large, by making the connection to the core as short and as large in cross section as practical, and by limiting the reactivity rise rates to 0.5 and 2 or 3 per cent in t)A-,//, respectively, for source-power and full-power operation.

10 URANIUM

CONCENTRATION

100

I0OO

(gm U/kg D20)

Fio.

enrichment, and critical concentration of uranium 26. Curves relating temperature, for aqueous homogeneous bare reactor of 6-ft diameter.

The power level of a reactor adjusts automatically to meet the demands of the heatremoval system. A complete analysis of the nuclear, hydrodynamics, and powerremoval characteristic of the reactor is required to be certain that only damped oscillations can be initiated when changes are made in operating conditions. 6

MAINTENANCE

Equipment that contains the reactor fuel becomes highly radioactive by absorbing delayed neutrons that are emitted by the circulating fuel and by adsorbing fission products that are dissolved or suspended in the fuel. The induced activity is negligi ble in comparison with the fission products, which coat the surfaces and are contained in crevices. Although no quantitative data are available for estimating the radiation levels, it seems certain that all maintenance of radioactive equipment must be done with special handling devices. Replacement of equipment in piping systems requires numerous connections. Standard ring joint flanges modified for leak-detection connections to the ring groove (Fig. 27) were used in the homogeneous reactor experiment. Bolts were installed with long-handled wrenches, and the equipment was manipulated with long-handled tools. As the piping and equipment become larger, the problems of remote assembly

Sec. 13-2]


Fia.

27.

13-75

Ring-joint flange with leak detector.

of such joints become more difficult. Remotely operated cutting and welding machines now being developed for industrial operations may provide a better method of making connections in large piping systems.

THERMODYNAMIC PROPERTIES OF HOMOGENEOUS REACTOR FUELS


7

For most applicat ions the fuels for aqueous homogeneous reactors contain more than 90 mole per cent and 80 volume per cent of water as H20 or D20. The variations in heat capacity and thermal conductivity with fuel concentration are usually of minor importance. Prop Variations in density and viscosity can be of major importance. erties of uranyl sulfate solutions and of some slurry fuels are presented here. Proper ties of other fuels can often be approximated by comparison with these fuels on the basis of molar quantities of the constituents. Uranyl Sulfate Solutions

7.1

Density. equation*

7.11

the

The densities of uranyl sulfate solutions can be calculated from =

(78.66/1/) pKln = "(TT.0/17)1^

-

1.046

0945

+ P"'° + pu'°

where

p = density, g/cra' U = weight per cent of uranium in fuel Results agree within ±2 per cent in the temperature range from 25 to 300°C for solutions containing less than 50 weight per cent uranyl sulfate. 7.12 Thermal Conductivity. Solutions are usually assumed to have the same thermal conductivity as water. Corrections can be approximated using the formula

£

where

k = thermal conductivity

*

1

of solution thermal conductivity of water at same temperature V «• uranium content, weight per cent where a is estimated to be 0.3 X 10~2 for solutions containing less than 25 weight cent of uranium. ko

and per

- - aU

—

See Ref. 0, p. 875.

MECHANICAL DESIGN

13-76

AND REACTORS

[Sec

13

The relationship among concentration, temperature, and vis Viscosity. 7.13 cosity of uranyl sulfate solutions is presented in Fig. 28. * Curves obtained at tem peratures below 100°C have been extrapolated to 300°C. The viscosity of the 39.5 per cent U02S04 solution at 300°C could be 50 per cent greater than that obtained from the curve.

40 60 100 200 400 TEMPERATURE, °C

for HiO, DiO, and aqueous UOjSO<

28. Absolute viscosity vs. temperature

solutions.

7.14 Heat Capacity. Values for the heat capacities of solutions of uranyl sulfate in H20 and D20 at 25°C are presented in Table l.f The values are estimates based on a comparison with measured values for uranyl nitrate and for salts of other metal. Heat capacities at other temperatures can be estimated with the equation . , {Cp)T

Table

1.

.

.

= (Cp)ts

[cp(H2Q or D20)]r [cp(H20 or D20)]«

Heat Capacities of Uranyl Sulfate Solutions Estimated heat capacity at 25°C. cal/dtiCC)

UOjSOi, wt %

HiO

DiO solutions

0 10 20 30 40 50 60 70 80

1.005 0.916 0.830 0.745 0.658 0.568 0.474 0 370 0.252 0. 174

85.9 (UOjSO<-3D.O) 87.2 (UO«SO«-3HiO)

solutions

0.9983 0.905 0 809 0.714 0.619 0 523 0. 428 0 333 0.238 0. 170

7.2

Slurry Fuels

7.21 The density of a slurry can be calculated from the volume frac Density. tions and the densities of the constituents using the equation Pulurry * See Ref. 0. p. 579. pp. 580ff. f Sop Ref. fi,


Flo.

800

— Pit

+ V p(j>P

—

pj)

Sec. 13-2] where


13-77

pz = —

pf

Vf

density of H20 or D20 density of fuel solids = volume fraction of fuel solids

p*Wf pzWf + PJ-d

=

Wf

-

Wf)

= weight fraction of fuel solids

Densities of thorium and uranium oxides are given in Table 2. Table 2.

of Thorium and Uranium

Properties

*>, K/cm1

ThOt

UOi U.O. UOiHtO UOi-HiO

kr, cal/(sec)(cm)(°C)

(<»»,

cal/(it)(°C)

0.002* 0.002* 0.002* 0.002* 0.002*

10.0 10.9

7.3 6.3 5.7

(rods) (platelets)

Oxides at 26°C

0 06

0.06 0 07 0. II* 0. II*

* Estimated.


The thermal conductivity of a suspension can be 7.22 Thermal Conductivity. estimated using an equation similar to the Maxwell equation for electrical fields: , fcalur

where

kr-

2VF(k, \2kt + j + kF + VF(kt

-

-n

- krkF)

= thermal conductivity = thermal conductivity

of H20 or D20 of fuel solids The thermal conductivities of thorium and uranium oxides are included in Table 2. 7.23 Heat Capacity. The heat capacity of a slurry can be estimated with the equation fcj

kr

fcp).lurry = (cP)t

-

Wf[(,Cp),

-

(cv)f\

where (cp)j = heat capacity of HsO or D20 (cp)f = heat capacity of fuel solids Heat capacities of the fuel solids are listed in Table 2. Slurry fuels can be Newtonian or non-Newtonian fluids, depend 7.24 Viscosity. ing on the degree and type of flocculation, which, in turn, depends on the properties of the solids and the electrolyte atmosphere in the suspending medium. With a Newtonian slurry the viscosity is independent of the rate of shearing strain. Ein stein" has shown that the viscosity of a dilute suspension of rigid spherical particles is related to the volume fraction of solids and the viscosity of the suspending medium by the equation Slurry = Hid + 2.5JV) Einstein's equation has been extended16-23-* to higher concentrations by theoretical and empirical methods; the results are compared in Table 3. Data for a variety of slurries containing particles of unknown sizes and shapes but in which the deviations from Newtonian characteristics wore small were correlated with fair agreement by Orr and Dallavalle" using the empirical equation ^slurry

— [1

where

Vfb

- (Vf/Vfb)Y*

is the volume fraction of solids in a settled bed of slurry. a variable viscosity that is a function of the rate of shear" and the electrolyte atmosphere around the slurry particles" as well as the volume fraction solids in the slurry.

Non-Newtonian ThO« slurries have

* See Ref. 16, p. 300.

MECHANICAL DESIGN

13-78 Table 3.

AND REACTORS

[Sec.

13

Comparison of Theoretical and Empirical Values for the Constants in tie Equation Wurrjr = M(l + A,P> + A*Vrt + A,Vr,)* Author

Reference

Ai

A. Einstein

15

2.5

V. Vand E. Guth and R. Simha. H. de Bruyn N. Saito

16 17 18 19 20 21 22 23 16 23

1.5 2.5

A,

At

0

0

Comments Mutual hydrodynamic int«action for spheres in dilutt suspension

....

Oden

Routaric and Vuillaume. . . .

7.349 14. 1 2. 5 2 5 0 30 60 75 9-13 7.17 8F«

2.5 2.5 3.35

2.5 2.5 2.5 2.5 2. 5F

* This is a revised version of Table 1 in "Solids Dispersed Oct. I. 1956.

0 0 2.5 2.5 0 0 0 0 16. 2

40f

in Liquids,"

For

F F

-

rod-shaped particles: 2.1 for L/D 5 to = 32 for L/D 100

-

-

ORNL-CF-56-

10-35. by D. G

Thomas,


Thorium relation

oxide slurries may be considered as Bingham plastics,17

dVr - - ~j~ dr g,

ri

(T

defined by the

T„)

where the yield stress t„ and the coefficient of rigidity y are constants characteristic of the particular slurry. The relation between the yield stress and the slurry concen tration, for a variety of different slurries, can be expressed as

t,

- X,C'

K\ is a function of the particular slurry and varied from 1.4 X 10~10 to 1.2 The relationship between the 10"9 psi/(g Th/kg HjO)3 for the slurries studied." concentration and coefficient, of rigidity for thorium oxide slurries (having a range of

where

X

concentration from 200 to 1,400 g Th/kg H20) can be expressed v =

as

A'**/""

where Kt is a function of the particular slurry and varied from 0.8 to 1.6 centipoises for the slurries studied.28 7.26 The pressure drop Use of Slurry Properties in Engineering Calculations. due to friction for viscous flow of non-Newtonian fluids may be calculated from the dimensions of the system and the fluid properties using the relations given by .Mves.* The Reynolds number corresponding to the transition from laminar to turbulent flow Slurries flowing under fully may be determined by the method given by Hedstrom." turbulent conditions may be considered to have the viscosity of the suspending medium.*0

An analysis of the effect of the rheological properties of Bingham plastic slurries slurry heat-transfer coefficient indicated that data in the turbulent region could correlated by an equation of the form

on be

where n was taken to be % based on a momentum heat-transfer analogy and m was 0.2." Preliminary data indicate that n may be as high as 3, in which case m •» 0.

Sec. 13-2]


13-79

references King, L. D. P.: Design and Description of Water Boiler Reactors, Proc. 1st Intern. Conf. on Peaceful Uses of Atomic Energy, P/488, vol. 2, Paper 46, 1955. Reactor Experiment, Chem. Eng. 2. Beall, S. E., and C. E. Winters: The Homogeneous Progr., 5(5): 256 (1954). Reactor Experiment, Mech. Eng.. 77(7): 575 (1955). 3. Gall, W. R.: The Homogeneous Reactor Test, Proc. 1st Intern. 4. Beall, S. E., and J. A. Swartout: The Homogeneous Conf. on Peaceful Uses of Atomic Energy, P/498, vol. 3, Paper 24, 1955. 5. Froman, Darol, R. P. Hammond, and L. D. P. King: Los Alamos Power Reactor Experiments, Proc. Intern. Conf. on Peaceful Uses of Atomic Energy, P/500, vol. 2, Paper 25, 1955. 6. U.S. Atomic Energy Commission: "Reactor Handbook," vol. 2, Engineering, pp. 555657, AECD-3646, McGraw-Hill Book Company, Inc., New York, 1955. Reactor Chemical Problems, Proc. 1st Intern. 7. Secoy, C. H.: Survey of Homogeneous Conf. on Peaceful Uses of Atomic Energy, P/821, vol. 9, Paper 38, 1955. 8. Schreyer, J. M., and C. F. Baes: Solubility of Uranium VI Orthophosphates in Phos Am. Chem. Soc., 76: 354 (1954). phoric Acid Solutions, 9. Kitzes, A. S., and R. N. Lyon: Aqueous Uranium and Thorium Slurries, Proc. 1st Intern. Conf. on Peaceful Uses of Atomic Energy, P/811, vol. 9, Paper 35, 1955. 10. Gillies, D. M.: "Some Studies of the Reactions of Uranium Oxides with Hydrogen. Oxygen and Water (Columbia University)," MDDC-647, U.S. Atomic Energy Com mission, 1946. 11. Went, J. J., and H. de Bruyn: Liquid Fuel Reactor with Uranium Oxides, Nuclear Eng., 1.

J.


12. 13.

14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

2: 120 (1954). Briggs, R. B., and. J. A. Swartout: Aqueous Homogeneous Power Reactors, Proc. 1st Intern. Conf. on Peaceful Uses of Atomic Energy, P/496, vol. 3, Paper 20, 1955. Garber, H. J., and F. N. Peebles: "Pressure Drop for the Rotational Flow of Liquids through a Spherical Container," Department of Chemical Engineering, University of Tennessee, Knoxville. Oct. 20, 1952. Peters, K., and K. Weil: Z. pkysik. Chem., A-148: 1 (1930). Einstein, A.: Ann. Physik, 19: 289 (1906), 34: 591 (1911); Kolloid-Z., 27: 137 (1920). Vand, V.: J. Phys. & Colloid Chem., 52: 277 (1948). Guth, E., and R. Simha: Kolloid-Z., 74: 266 (1936). De Bruyn, H.: Proc. Intern. Congr. on Rheology (Amsterdam), 2: 95 (1949). Saito, N.: J. Phys. Soc. Japan, 5: 4 (1950). Gosting, L. F., and M. S. Morris: J. Am. Chem. Soc., 71: 2005 (1949). Oden, S.: Der Colloide Schwefel, Nova Acta Regiae Soc. Sci. Upsaliensis, 4: 3 (1913). Boutaric, A., and M. Vuillaume: J. Chem. Phys., 21: 247 (1924). Eirich, F., H. Margaretha, and M. Bunzl: Kolloid-Z., 76: 20 (1936). Orr, Jr., C, and J. M. Dallavalle: Heat-transfer Properties of Liquid-Solid Suspension, Chem. Eng. Progr. Symposium Ser. No. 9, 50: 29 (1954). Alves, G. E., D. F. Boucher, and R. I. Pigford: Pipe Line Design for Non-Newtonian Solutions and Suspensions, Chem. Eng. Progr., 48: 385-393 (1952). Lyon, R. N., el al.: HRP Quart. Progr. Re.pt., ORNL-1813, July 31, 1954, p. 125. Kitzes, A. S., and R. N. Lyon: "Aqueous Uranium and Thorium Slurries," Paper 933, Session R-4, Geneva Conference on the Peaceful Uses of Atomic Energy, August. 1955. Thomas, D. G.: "Solids Dispersed in Liquids," ORNL-CF-56-10-35, Oct. 1, 1950. Hedstrom, B. O. A.: Flow of Plastic Materials in Pipes, Ind. Eng. Chem., 44: 651-656

(1952). 30. Lawson, C. G.: 31. Lawson, C. G.:

HRP HRP

Quart. Progr. Rent., ORNL-214S. Quart. Progr. Rept., ORNL-2222,

Julv

31, 1956. pp. 55^56. 1956, pp. 59-61.

Oct. 31,

13-3

LIQUID-METAL REACTOR SYSTEMS BY


Arthur H. Barnes

(Deceased)

*

Liquid metals possess the following desirable properties as reactor coolants: lowvapor pressures, negligible corrosion of reactor fuels and structural materials, and Disadvantages stem from their chemical excellent heat-transfer characteristics. activity, neutron absorption, and induced radioactivity. Metals that are of prime interest are sodium, sodium-potassium alloy (NaK), bis muth, lead, and lead-bismuth alloy. Mercury has been used as a fast reactor coolant, Li' would be very attractive as a but it is no longer receiving serious attention. coolant for thermal reactors if its cost could be significantly reduced. Choice of a liquid-metal coolant for a particular reactor application is based mainly on nuclear properties and the engineering difficulties associated with melting tempera Table 1 presents the pertinent physical properties of tures and corrosion effects. metals and alloys of interest. 1

ALKALI-METAL

SYSTEMS

Extensive operating experience has been acquired with NaK in the Experimental Sodium has been employed widely in nonradioactive Breeder Reactor I (EBlt-I). Lithium, on the other hand, systems, and its use as a reactor coolant is increasing. In the articles that follow, therefore, the has been used only in small-scale equipment. emphasis is on sodium and NaK, although in a general way the information presented can be applied to lithium also. 1.1

Container Materials

Materials suitable for containing liquid metals are those which possess desirable high-temperature physical properties and at the same time are not susceptible to cor rosion. Liquid-metal attack on a container material may occur by any of the follow ing mechanisms: solution attack (in which the metal is carried off in solution), alloying (in which surface films are formed), intergranular penetration, and attack on the container by impurities in the liquid metal such as oxygen, nitrogen, or carbon. A large number of metals are suitable for use in the construction of sodium and NaK systems. Table 2 lists structural materials and indicates the safe maximum In general, ferrous alloys with low carbon temperatures for extended operation. Types 304, 304L, and content and high nickel and chromium content are preferred. 347 stainless steels have been the most widely used container and structural materials. Final choice is based on ease of fabrication, cost, and corrosion resistance within the operating temperature range. Corrosion by sodium and NaK is determined almost entirely by the temperature For extended operation in the range 400 to COOT, and the oxide (Na.O) content. Solubility oxide content should not be greater than 0.005 ± 0.002 weight per cent. of oxide in liquid sodium increases rapidly with temperature1 (by a factor of approxi* The publishers and editor-in-chief deeply regret the untimely death of Dr. Barnes. The editor-inchief moreover deplores the loss of a close friend and wishes to thank Mrs. Rosemary Barnes for so care fully editing her husband's contribution.

13-80

675

1400

1670

1317

357

760

257

354

-38.0

47

125

179

-38.9

3038

200 400 600 800 1000 200 400 600 800 1000

10.46 10. 19 9.91 9.64 9.36 0.507 0.490 0.474 457 0.441

g

1

160 200 240 300 320 225

60 120 160 220 200 300 400 500 600

0 0.022 0.023 0.024 0.026 0.027 0.09

0.023 0.026 0.028 0.030 0. 107 0. 101 0.096 0.090 085

0

II

18.7 25.0 31.4

98.4 103.2 114.2 127

45

113 118 123 128

94.6 107.2 116.4 125.7

5

167 250 400 700

330 500 700

039 0.037 0.036

128.9 145.3 153.5

41

50 100 200 300 150 250 350

230

0.0

200 300 400 500

-3.32

327 600 800 1000

300 600 750

"C

change

of solid volume

Volume

3

0.331 0.258 0. 191 0. 136

300 500 700

0.041 0.037 0.037

ôhm-cm

resistivity

Electrical

6

100 250 400 550 700

1.55 1.21 1.01

°c

conductivity

cal/(sec)(cm)(°C)

Thermal

Metals*

|

0.819 0.783 0.747 0.711 0.676

183 208 285

0.592 554 0.455

20 100 200

332 458 500 550 600

1.70 1.38 1.29 1.23 17

0

13.5 13.35 13. 12.88

441 551 844

12 1.70 185

°c

304 451 600

10"

1.66 1.28 966

1.

20 100 200 300

400 600 800 1000

10.51 10.27 10.04 9.81

9

3159

300 600 962

10 03 9.66 20

Poises

2. 1.

5

63.7

2403

1737

621

327.4

°C

g/cm>

of Liquid

Viscosity

Properties

0

Lead-bismuth alloy, Pb 44 w/o, Bi 55.5 w/o...

1477

520

271 2691

°F

°C

°F

°C

Density

Physical

X

Bismuth

Metal

point

Boiling

point

Melting

Table


%

6 3 2

0

1.

1

2 (ri'v.), U.S. Atomic

6 T-733

6

2d i-d.. NAVEXOS

Energy Commission

11

Handbook."

0.

and

Department

0

Mntala

of the Navy.

51.4 56. 62 68.6

240 292 417 498 0.080 0.081 079 0.078

300 400 500 600

1.67 1.38 18 05

1952.

87

"Liquid

409 523 648 704

6.83 76 6.67 64

0

0.

41 18

35. 38.0 41.0

100 200 300 400 500

0.062 0.063 065 066 0.0675 104 168 250 400 700

0.546 412 0.316 0.230 163

0

2270

100 250 400 550 700

0. 886 850 0.814 778 0.742

00

449

1518

825

41.0 44.0 47.0

100 400

0.058 0.064

104 168 250 400 700

0.468 0.359 279 0. 205 146

9.65 13. 18 16.70 18. 44

*iohm-cm

5

231.9

66.

19

100 250 400 550 700

0.847 0.811 0. 775 0.739 0.703

°C

resistivity

Electrical

100 200 300 400 500

cal/(sec)(cm)(°C)

conductivity

0. 206 0. 195 0. 181 170 160

°C

Thermal

(Continued)

104 168 250 400 700

10»

Metals.*

0. 686 0.504 381 0.269 0. 182

0

Na.

1443

784

100 250 400 550 700

928 0. 891 0.854 0.817 0.780

Poises

Viscosity

of Liquid

0

K, 56w/o

12

1621

883

°C

K/cm'

0

11

208

97.8

°F

°C

Density

Properties

0

44w/o

-

°F

°C

point

Boiling

Physical

X 0. 0.

alloy Sodium-potassium (N.K): 22w/oNa.. 78w/oK,

Metal

point

1.

Melting

Table


2.5

2.6 400 600 800 1000

2.9

100 150 200

100 150 200

2.5

100 200 300 350

°C

Volume change on fusion, of solid volume %

•

Sec. 13-3] Table 2.

liquid-metal reactor systems

13-83

Materials in General Use for Container and Structural Purposes* Max temp for long-term use, °Ct

Material

Resistance

of Materials to Sodium

Aluminum (2S, 3S) Armco iron

and

200 900 900 600 200 900 900 400 300 900 600 > 1000 > 1000 600

Cobalt

Niohrome V (80 Ni-20 Cr) Monel (30 Cu)

Nickel

Max temp for temporary use, °Ct

NaK 450 > 1000

>I000

900 550

600 400

800

>I000 450 450 >1000 600

600 800 900


>I000 600 500 600 200 150 200

AhO. BeO

MxO

650 450

of Materials to Lithium

Resistance

Austenitic stainless steel

Nickel and nickel-hase

600

500 1000 800 900 1000

alloys 350 1000

Tantalum

* L.. R. Kelrnan, W. D. Wilkinson, and F. L. YagKee. " Resistance

800 900

500 600

of Materials to Attack by Liquid

MetiilH," ANL-4417, July, 1950; C. O. Nelson, "Summary Report of Reaction Tests of Various Materiala Tested with Sodium and Sodium Potassium," KAPL-557, June, 1951; Anton De S. Brasunus, '* Interim Report on Static Liquid-Metal Corrosion," ORNL- 1647, June, 1954.

f

Temperatures indicated

are approximate

only.

mately 40 between 150 and 500°C). Two effects associated with excessive oxide content in nonisothermal systems are mass transport' and plugging. Mass transport is the removal of metal from a surface at high temperature and its subsequent deposi tion on surfaces at lower temperature. Plugging is an accumulation of solid material, mostly oxide, which builds up in low-temperature regions of the system. Mass transfer of radioactive constituents of structural materials may be reduced by the addition of certain "inhibitors." For this purpose barium, calcium, mag nesium, antimony, and titanium have been found to be effective. 1.2

1.21

Static Seals. The component sections of large liquid-metal systems are joined by electric arc welding. The two principal requirements are that weldment be resistant to thermal shock and that the inner surface be smooth and

normally the

Closures


13-84

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

slag-free. The use of an internal inert-gas blanket has been found to be effective in Thin wall sections are normally inert-gas arc welded (helium reducing slag formation. or argon using a tungsten electrode). Brazing techniques may be used for the fabri cation of complex structures provided that brazing alloys containing silver, copper, and zinc are avoided. For small experimental systems flanged joints are often used. The ring-type flange with a mild-steel ring in a V groove has been used successfully at temperatures up to 500°C. Threaded joints are not satisfactory. 1.22 Conventional packing glands are suitable only for use at Dynamic Seals. relatively low temperatures and in situations where liquid-metal purity is not impor For the scaling of linear motion devices (valve stems, etc.) the metal-bellows tant. For increased reliability multiple bellows (i.e., one type of seal is in general use. bellows backing up another) is recommended. Tests have shown that bellows life is extended considerably by operation at elevated temperatures (~250°C). Permissible bellows flexure depends upon manufacturing methods, expected life, etc., but is usu Slow rotary motion ally in the range of 10,000 to several hundred thousand cycles. (wobble drive) may be obtained with a bellows if a crank movement is employed without transmitting a twisting motion to the bellows. Frozen-sodium seals may be formed by permitting sodium to freeze and form s seal between a moving and a stationary member. The seal* depends upon a thin layer of fluid sodium which is maintained in contact with the moving part by careful ambient-temperature control. Temperature control is rather critical, since too high a temperature causes excessive leakage while too low a temperature leads to seizing of the moving member. Extensive investigation of this type of seal is under way at present. 1.23 An effect associated with the action of sodium or Seizing (Self-welding). NaK on structural metals is that of seizing or self-welding. This may occur when two surfaces are maintained in intimate contact while submerged in sodium or NaK at ar. elevated temperature. It has been observed that certain combinations of metal surfaces are more susceptible than others. Pressure, temperature, and duration o: contact enhance the effect. 1.3

Pumps

1.31 Mechanical Pumps. Centrifugal-type mechanical pumps have been widely The significant differences in design involve the used for pumping liquid metals.4 methods used for sealing the pump housing. The simplest arrangement consists of The liquid an impeller mounted on the lower end of a long, vertical, overhung shaft. level is maintained so that the shaft bearings at the upper end are above the liquid and operate in an inert-gas atmosphere. A labyrinth is provided to reduce the dif fusion of liquid-metal vapor into the housing containing the bearings and drivint motor. The impeller shaft may be either direct- or belt-driven by a motor located in the inert-gas-filled enclosure at the upper end of the assembly. Liquid level in tbspump housing is controlled by maintaining a fixed liquid level in an overflow tank connected to the inlet side of the pump. Pumps of this type have been constructed

with capacities up to 500 gpm (Fig. 1). They have the advantage of ease of fabrica High-temperature operation tion and reliable operation at moderate temperatures. is made difficult by loss of bearing lubricant, diffusion of sodium vapor into the bearine and motor housing, and differential expansion effects. Completely sealed mechanical pumps are constructed by enclosing the pump impel ler and driving rotor of an induction motor in a common housing (Fig. 2). A thin annular shell is used to separate the motor armature from the field winding. Endazfi rotor pumps6 have the advantage of presenting a completely sealed system, but they Moreover, involve the serious problem of operating bearings in heated liquid metal. if the pump is required to move high-temperature liquid, the cooling of the motor field winding presents a problem of considerable difficulty. Pumps of this type have received much attention, and extensive effort has been applied to the solution of th* bearing problem, so far with varied results. Additional development will be required before such pumps are considered reliable for long-time operation.

Sec. 13-3]

LIQUID-METAL

REACTOR SYSTEMS

13-85

The difficulties associated with the enclosed rotor pump may be avoided by locating the impeller shaft bearings external to the pump housing and bringing the shaft out of the housing through a frozen-sodium seal. The temperature of the shaft housing


must be maintained within rather narrow limits. This may be accomplished by flowing an organic coolant at a controlled temperature through channels in the shaft Pumps of this type with capacities up to 1,200 gpm have been constructed.' enclosure. The high electrical conductivity of liquid metals 1.32 Electromagnetic Pumps. makes them suitable for pumping by electromagnetic means. Considerable effort has been expended in exploring the possibilities of electromagnetic pumps because

BY -PASS' DISCHARGE-

SUCTION

Fig.

1.

Liquid-metal mechanical

pump,

sump type, as used on

EBR-I.

the absence of both bearings and seals makes a pump of this type extremely attractive.

The operation of all types of electromagnetic pumps depends on the principle that a force is exerted upon a conductor carrying current in a magnetic field. Electromag netic pumps differ in the means provided for causing current to flow in the liquid metal that is to be pumped. Direct-current Conduction Pumps. The d-c conduction pump is the most direct The elements of this type of pump are illustrated in Fig. 3. A thinapproach.7'8 walled liquid-carrying duct with copper conductors attached to opposite sides is Current enter located in the magnetic field between the poles of an electromagnet. ing through the duct wall traverses the liquid that fills the duct and develops in it a longitudinal thrust. The pumping rate can be shown to depend on the current, mag netic-field intensity, and the geometry of the pump duct according to the following

relation:

13-86

MECHANICAL

DESIGN AND REACTORS

MOTOR FIELD WINDINGS

!

MOTOR WINDINGS OIL COOLED


MOTOR ROTOR AND BEARINGS COMPLETELY IMMERSED IN LIQUID METAL

PUMP DISCHARGE

PUMP SUCTION

Fio.

MAGNET

2.

Enclosed-rotor liquid-metal pump.

POLE —

PUMP DUCT

CURRENT

COPPER BAR CONDUCTOR

P»FLOW

Fio.

MAGNET

3. Direct-current electromagnetic

POLE

pump.

[SEC.

13

Sec. 13-3] Q

LIQUID-METAL

REACTOR

4(io«)g r

- 1.75(W)PS B

s

l IRb

SYSTEMS

13-87

(ltb + R.)

where Q = flow, gpm P = developed head exclusive of internal hydraulic losses, psi B - magnetic field intensity in the liquid between the magnet poles, gauss / = current fed into the duct, amp iS = width of duct in the field direction, in. R, = resistance of liquid filling pump duct in region between magnet poles, ohms Rb = combined effective resistance along all bypass current paths both in duct wall and in liquid beyond field at entrance and exit sections of duct, ohms The magnitude of R, may be calculated from the dimensions and resistivities of the Ri, is a complicated function of the duct geometry, fring liquid and duct material. profile. Its magnitude is determined ing field distribution, and liquid-velocity experimentally. The expression for the static (zero-flow) pressure developed by the pump is psi


The magnetic field due to the flow of current across the pump duct distorts the main field of the magnet. For efficient operation it is necessary to minimize this

Kio.

4.

Direct-current electromagnetic

pump with magnetic-field

compensation.

effect either by proper tapering of the magnet gap or by compensating for the field due to the current in the liquid by returning the current across t he main field through a conductor that is adjacent to the duct in the region in which pumping occurs (Fig. 4).

The

pump duct is fabricated from sheet or formed by pressing

The

type of field winding employed is dictated by the operating requirements of the Separately excited windings of the many-turn high-voltage type present an

thin-walled seamless

tubing into the desired cross section. Wall thicknesses of 0.025 to 0.0625 in. are normally used. Since high electrical resistivity as well as compatibility with sodium and NaK is necessary, the materials in general use are austenitic stainless steels and nichrome (80 Ni-20 Cr alloy). The heavy copper bars are attached to the duct walls by either silver or nickel-alloy brazing. A welded box-type attachment employing sodium as a liquid bond has also been used for large pumps and for those which are to operate at high temperatures. pump.

13-88

MECHANICAL DESIGN

AND REACTORS

[SEC.

13


insulation problem when high-temperature operation is desired. A series-type wind ing, on the other hand, consisting of only one or two turns of large-cross-section copper is readily insulated and is capable of very high temperature operation. Pumps with this type of field winding require no auxiliary cooling system. Figure 5 shows such a pump with a two-turn series field winding.

Fio.

5.

Direct-current electromagnetic

pump with two-turn field winding.

The current requirements of d-c pumps range from the order of 1,000 amp for small pumps (5 to 10 gpm) to many thousands of amperes for high-capacity pumps. The voltage drop across these pumps is in the vicinity of 1 to 2 volts. The large direct currents required may be supplied by either rectifiers or generators. Since the pump volt age requirement is low, the rectifier efficiency For is in the range of 20 to 40 per cent. higher efficiency a hornopolar generator*-11 employing NaK as a liquid brush offers a means of obtaining very large currents at effi ciencies in excess of 80 per cent. Generators of this type have been constructed that are capable of supplying currents in the range of 20,000 t o 500,000 amp, and machines of greater capacity are currently under construction. Pumps with capacities of a few to several hundred gallons per minute are in use, and much larger units are contemplated. Tne electromagnetic pump (Fig. 6) used in the primary circuit of the experimental breeder reactor (EBR-I) has a capacity of 500 gpm at Fig. 6. EBR-I electromagnetic pump, 25-psi head. Figure 7 illustrates the currentlocated in helium-filled enclosure. capacity characteristics of this pump, ani Fig. 8 shows the operating characteristics with NaK at 250°C. The efficiency of the pump (exclusive of current supply soure*-1 3i 300 gpm at a 40-psi head is 43 per cent. The advantages of d-c electromagnetic pumps are based on their inherent sim plicity of construction and practical absence of insulation difficulties. Reliable operation at high temperatures (800°C) for long periods has been achieved. The efficiencies obtainable vary from 15 to 20 per cent for small pumps to 40 to 50 per cent for large units. Over-all efficiencies including current-supply source and bos

Sec. 13-3]

LIQUID-METAL

REACTOR

13-89

SYSTEMS


losses are in the range of 10 per cent for small pumps to 40 per cent for large systems in which homopolar generators are used. Alternating-current Conduction Pumps. These are similar to d-c pumps except that the magnetic field reverses in synchronism with the current in the fluid.12 For maxi mum pumping, the field and duct currents 600, must be properly phased. This may be lb/mz accomplished by placing the field winding 500 in series with the path through the liquid.

The large current requirement is met by in ^30 b/in2 5 corporating a step-down transformer whose secondary winding is connected to the °- 300 pump duct." The magnetic circuit of the lb/in' transformer may have one leg in common with the field structure of the pump (see Fig. 9). Because of the complex current 100 and field interactions characteristic of this type of pump, no rigorous analysis of performance is possible. Moreover, power 14 16 18 20 CURRENT, KILOAMPERES losses are relatively high and efficiencies range from 2 to 20 per cent. No large Fio. 7. Current-carrying characteristics of EBR-I electromagnetic pump. pumps of this type have been constructed, but for laboratory applications they have found wide use because they can be operated directly from a single-phase 60-cycle power line. These pumps develop large currents directly in the fluid Induction-type Pumps. by electromagnetic induction. The duct containing the fluid is located in a region 60 1

JNPUT

50

1 POWER, KW

16.6

40 12.5 m ■3 30

o'

9.1

■a UJ

"

x 20 7 10

200

Fio.

of

8.

300 400 CAPACITY, GAL/MIN

Head-capacity characteristics

of

EBR-I

500 electromagnetic

pump.

changing field intensity and is so oriented that the induced currents react on the

main field and produce motion of the fluid. There are numerous geometric arrange ments possible. The simplest is found in the linear a-c pump'* in which the pump

duct

in the form of a flattened tube extends between two core sections which contain a three-phase a-c winding (see Fig. 10). The winding is similar to that of an induction motor stator except that the field structure is flat, so that a sliding rather than a rotating magnetic field is produced. Detailed analysis of the pump is similar to that of an induction motor except that the problem is more complicated because the current paths and field distribution are not sharply defined. Since the magnetic gap is large, field losses are high and the power factor is low. The pressure developed is roughly proportional to the relative speed of the liquid and the field. The core ends introduce harmonic components in the field that contribute power losses. A number of pumps of this type have been

MECHANICAL


13-90

DESIGN

AND REACTORS

[Sec. 13

Efficiency of the constructed with capacities in the range of 100 to 3,500 gpm.lsle larger pumps is 35 to 40 per cent at heads of 75 to 100 psi. For use at high tempera ture it is necessary to provide an auxiliary cooling system for the field winding. The induction pump is attractive be cause the large currents are developed directly in the liquid, thus avoiding the need for large current bus systems. Moreover, the power supply is usually 60cycle alternating current. The Einstein-Szilard pump11 is similar to the flat linear induction pump. In this type of pump the fluid flows through an annulus between an inner core section and an outer core containing a system of toroidal coils. A three-phase alternating current produces a moving field similar to that of the flat-type pump. In this case, however, the induced-current paths are circular, so that the edge losses of the flat -type pump are avoided. Fabrication problems have tended to retard the de velopment of this type of pump, and extensive operating experience has not yet been acquired. Another arrangement of the a-c induc tion pump consists of a helical thin-walltd duel located between a three-phase a-c Fio. 9. Alternating-current induction-type field structure (similar to the stator of an pump. electromagnetic induction motor) and a fixed center-core section. forces cause the fluid to circulate in the helical duct. Electrodynamic Fabrication difficulties and low efficiency have limited the use of this type of pump.

3

FIELD

PHASE WINDING CORE

LIQUID METAL

Fio.

10.

Alternating-current linear induction-type electromagnetic

DUCT

pump.

A closely related version of the helical pump is the flnt-lype rotnling-field arrange ment in which the liquid is driven in a spiral path adjacent to a disk-type stator. Only small pumps of this type have been constructed, and they have been found to be subject to severe vibration of the thin-walled diaphragm which forms one side of the A modification of this design employs a mechanically driven magnet fluid chamber. structure (either permanent or d-c electromagnets) that rotates in close proximity to the flat thin-walled fluid chamber. Again, low efficiencies and fabrication difficulties have discouraged development. Of the many forms of electromagnetic pump only the linear d-c conduction and linear a-c induction types have as yet been used in permanent installations. The d-c

Sec. 13-3]

LIQUID-METAL

REACTOR

SYSTEMS

13-91

pump, since it operates at low voltage, avoids insulation difficulties attendant to highOn the other hand, it requires a separate d-c temperature and radiation effects. supply and so lacks the convenience of the a-c pump, which is capable of direct opera tion from three-phase power lines. 1.4

Heat Exchangers


Heat Exchangers. 1.41 Metal-to-metal Heat exchangers employed for trans ferring heat from one liquid metal to another are normally required to operate at Extra precautions are usually taken to ensure leaktightness relatively low pressures. In addition, liquid-metal heat exchangers between primary and secondary circuits. are designed to withstand more severe thermal stresses than are common in conven tional heat-exchanger practice. Tube-and-shcll type exchangers are in current use. Plate-type designs have frequently been studied and are feasible because of the lowpressure requirements. It has been common practice to weld the heat-exchanger tubes into the tube sheet . All other joints are also welded, and a rigorous leak-detecting procedure is followed. tube-wall material is desirable, because in liquid-metal High-thermal-conductivity heat transfer a significant part of the total temperature drop occurs in the tube wall.

- COPPER (DIFFUSION BONDED TO NICKEL TUBES) GROOVE '/32" WIDE x 3/64" DEEP ENTIRE LENGTH (20 GROOVES EQUALLY SPACED)

Fig.

11. Steam-generator

tul>c, as used on

EBR-I.

Hairpin-type tube Shells are normally fabricated from austenitic stainless steel. construction has been common, although other geometries, such as those involving 90° tube bends, have also been used. Liquid-metal heat exchangers are frequently called upon to operate with much higher temperature differences than are common in water-type exchanger practice. Extra precautions are necessary, therefore, to avoid situations that aggravate thermal Careful attention to details stresses due to either thermal shock or thermal cycling. of structural design and the application of special techniques should be emphasized. Metal-to-air Heat Exchangers. Metal-to-air heat exchangers arc usually 1.42 of the finned-tube type with either natural-convection or forced-air cooling. Design considerations are conventional, since most of the temperature drop is on the air side. NaK rather than sodium is desirable in air-cooled exchangers because of its low freez ing point. If sodium or lithium is used, care should be exercised to avoid freezing of The result metal in some tubes while others in the same parallel bundle remain open. ing contraction of the cold tubes may produce serious thermal stresses. The design of metal-to-water or steam 1.43 Metal-to-water Heat Exchangers. exchangers is governed by the consideration that sodium, NaK, and lithium react violently with water. Consequently, every effort is made to ensure that incipient leakage from one system to the other is readily detectable. Tube- and shell-type exchangers with double-walled tubes and double-tube-sheet construction have been used. Metal flow is on the shell side, and water flow can be adjusted in vertical tubes so that the tubes contain only a film of water in contact with the tube inner wall. Thus the greater part of the tube volume is filled with vapor. This has the

13-92

MECHANICAL DESIGN

AND REACTORS

[SEC. 13

the quantity of water that is available to react with liquid metal in the event of tube failure. It also provides an expansion volume so that dangerously It should be noted that a great advantage of high pressure will not be attained. liquid-metal boilers over conventional flame-fired types is that tube burnout does not occur in the case of a dry tube, since the liquid-metal temperature is always far below the melting point of the tube material. Several procedures have been followed in fabricating double-walled tubes. Figure 11 shows a cross section of the EBR-I boiler-tube design. Inner and outer nickel tubes are diffusion-bonded to a heavy-walled copper tube which serves as both a thermal bond between the nickel tubes and a means for detecting leakage. The latter is accomplished by placing longitudinal channels in the outer surface of the copper tube and connecting them to headers at each end so that gas may be circulated. Leakage of sodium or NaK into a channel can be detected by the transport of sodium Another arrange vapor to a detecting device (gas flame or electrical-discharge tube). ment employs an annulus of mercury as the thermal bond between concentric tubes.


effect of limiting

Fio.

12. Valve

with double-bellows

seal.

Leakage of either liquid metal or water into the mercury annulus is detected by the A third variation of the double-tube design resulting change in mercury pressure. consists of inner and outer tubes separated by a very thin annulus of helium, which serves as both a thermal-transfer medium and leak detector. The spacing between the tubes is maintained by shallow ridges applied to the outside surface of the inner tube.

Double-tube Tubes are in all cases welded to the tube sheets or headers. are recommended, the space between being used for leak detection. 1.6

sheets

Valves

Since tightness is a prime requirement of liquid-metal systems, bellows-sealed valves have been used almost exclusively. The bellows are of austenitic stainless steel and are welded into the valve bonnet. In order to achieve greater reliability, two bellows are often arranged in scries. They may be located one above the other in the valve bonnet or, for greater compactness, may be placed concentrically. It is good practice to provide a leak-detection device that will function when the inner bellows develops a leak. This may be accomplished either electrically, by means of an insulated probe that is short-circuited when contacted by liquid metal, or by detecting a pressure change in the gas-filled space between bellows (see Fig. IT,

Sec. 13-3]


13-93

Experience has shown that bellows should not be mounted in a manner that will allow the flow of liquid to cause vibration but should be placed well up in the valve bonnet and in contact with static liquid only. The ambient temperature of the bellows may be quite high, extended tests having shown that a better life expectancy has been achieved for bellows operating at 250°C than for bellows operating at room temperature. The limited motion that the use of a bellows seal implies has restricted their use to globe-type valves, a Y valve being used where pressure drop must be minimized. Bellows-seal valves in sizes up to 8 in. are now in use and can be obtained commercially. The difficulty of applying a bellows seal increases rapidly as the size of the valve increases. For large valves frozen-sodium seals should be considered. It may even prove expedient to use packing-type seals with the packing maintained at a tempera ture much below that of the working fluid. Although leakproof seals have been obtained by the use of bellows, leakage across the valve seat has proved to be a problem in cases where the valve must operate at high temperature and is exposed to temperature cycling. Warpage and the restric tion of valve seats to materials that can withstand the attack of liquid metals at elevated temperatures have presented problems that have not yet been successfully solved. Consideration must also be given to the phenomenon of seizing (self-welding), especially in applications where the valve is to remain closed for extended periods at high temperatures.


1.6

Instrumentation

No special problems are present in connection 1.61 Temperature Measurement. with temperature measurement in liquid-metal systems. Thermocouples are gen erally used, and where precise measurement is not necessary, the junction may be welded or brazed to the pipe or tank surface. However, the use of wells is a requisite for accurate temperature determination. 1.62 Pressure. Since pressure gauges are normally located at the end of long, small-bore side tubes, special problems arise in the case of liquid-metal applications. The use of such extensions from the main system presents two difficulties: (1) Oxide tends to precipitate and plug small-bore tubes, which are at lower temperature and contain static liquid; (2) sodium and lithium solidify either directly from the liquid or from vapor that diffuses to colder sections of the side tube, so that eventual plugging occurs. Consequently, the extension to the pressure gauge should be maintained at a temperature equal to or above that of the main system and the gauge should be of large cross section. Bourdon-type pressure gauges are not satisfactory except for In applications that require remote indication,18 bellows or dia temporary use. Figure 13 phragm-actuated pressure transmitters have been found satisfactory. shows a bellows-actuated transmitter with a variable-inductance signal transmitter. 1.63 Flow Measurement. Liquid-metal flow rates can be determined by the same The orifice-type meter has been widely used and has means used for other liquids. been found to give readings of accuracy comparable to those obtained in aqueous No precautions peculiar to liquid-metal systems are necessary other than systems. those associated with the use of pressure gauges as discussed above. The high electrical conductivity of liquid metals has made possible a simple applica tion of electromagnetic induction for the measurement of flow." The emf generated by the passage of a liquid metal across a magnetic field of a few hundred gauss inten sity can be readily detected by merely attaching leads to the outside wall of a non magnetic pipe carrying the liquid as it passes between the poles of a permanent The variation of generated voltage with flow rate is a linear function, magnet.'0 Figure 14 shows a typical arrangement of a per regardless of field distribution. manent magnet mounted on a 4-in. -diameter stainless-steel pipe. Figure 15 shows a It is necessary to minimize thermoelectric typical calibration curve for this device. This can be accom potentials at the junctions between pipe wall and voltage leads. plished by employing leads that have been fabricated from the same material as the The leads are welded to the pipe at opposite points on a diameter. pipe wall. The permanent-magnet field structure should be so arranged that the magnets are


13-94 MECHANICAL DESIGN AND REACTORS

[Sec. 13


Sec. 13-3]

LIQUID-METAL

REACTOR SYSTEMS

13-95

It has been kept at as low a temperature as practical in order to minimize aging. found that maximum response is not reached until complete wetting of the inside This may require several days unless special care surface of the pipe has occurred. Careful degreasing followed by has been exercised in cleaning the interior pipe wall. Operation at high temperature greatly accelerates electropolishing is recommended. Calibration of the electromagnetic flowmeter against an orificethe wetting process. type meter can be made with the orifice meter in the same system or can be accom plished in an auxiliary system, after which the section of pipe containing the electro The emf generated can magnetic flowmeter is transferred to the permanent system. be calculated from a relation developed by Elrod11-*' in cases where a uniform mag For flowmeters mounted on pipes of 4-in-diamenetic field of known intensity is used. ter or over, it is usually not practical to provide a magnet of sufficient size to establish Normal practice is to calibrate the flowmeter against an orifice- or a uniform field. venturi-type meter. Since the response is linear, this is a relatively simple procedure. In principle, any device used for the level Liquid-level Measurement. 1.64 However, the determination of other liquids can be applied to liquid-metal use. electrical conductivity of liquid metals suggests unique methods of level detection not Float-type devices are generally not desirable applicable to conventional liquids. because they involve internal mechanisms operating directly in the liquid and are difficult to service. Bubbler-type instruments are not suitable because they introduce contamination and would be susceptible to plugging difficulties due to vapor transport. Differential-pressure methods are not attractive because pressure measurement itself is difficult in the case of liquid metals and thermal expansion makes temperature cor rections necessary. The simplest method consists of determining when an electrical CONSTANT VOLTAGE 60~AC SUPPLY contact is made at the liquid surface by a metal rod which is passed into the con I taining vessel through an insulating and : This arrangement is sealing bushing. : ^SINGLE LAYER : WINDING suitable for only temporary use because 1 c the insulation at the bushing eventually : NaK LEVEL short-circuits owing to the formation of a t : It is conducting liquid-metal deposit. : other useful, however, for calibrating ) .-IRON WIRE CORE : : Another device types of instruments. makes use of the decrease in resistance t | ^TANK WALL that occurs as the liquid metal rises : : around a conductor that extends into -•THIN WALLED | the vessel containing the liquid metal. STAINLESS STEEL : . Electromagnetic induction may be used TUBE 1 to detect liquid-metal level. One device :■ : . operating on this principle consists of a [ solenoidal winding (Fig. 16) that extends ; :; downward inside a thin-walled stainless steel tube closed at the lower end but . i : t; ■1 open to the outside at the upper end." An alternating current is passed through Fig. 16. Solenoid-typo electromagnetic level the coil, and its impedance is measured indicator, as used at EUIt-I. as a function of the height of the liquid The action is similar to that of a transformer in which the surrounding the tube. coil is the primary and the surrounding liquid constitutes a short-circuited secondary with variable coupling as the level changes. Impedance changes may be detected in terms of changing current through the coil with a constant voltage across it. When the device is operated on 60-cycle current, an iron core is inserted in the coil. At high frequencies the core may be omitted. The device has the advantage that the coil can be removed for servicing without opening the tank, but it is tem and must be calibrated with the aid of some other levol-detccperature-sensitive tion method. These difficulties may be avoided by using a small, movable search

13-96

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

coil which is raised and lowered inside a thin-walled tube. As the coil passes the surface of the liquid, a change in its effective inductance is readily detected. A motor drive for raising and lowering the coil in the tube together with a sclsyn posi tion indicator permits remote operation. This arrangement is self-calibrating, is not temperature-sensitive, and can be made to follow changes in liquid level automati It can be used to calibrate the fixed-coil level indicator discussed above if twin cally. tubes are placed in tanks in which liquid-level determination is desired. 1.7

Oxide Control

The increase in the rate of attack of liquid metals on container material with increase in oxide (Na20) content makes it necessary to maintain oxide concentration at a low level." Experience has shown that for stainless steel systems operating at tem peratures up to 600°C an oxygen concentration of less than 0.005 to 0.007 weight There are several methods of oxide determination, all of which per cent is acceptable. The principal difficulty arises in withdrawing require rather involved manipulations. HOT SODIUM FROM

SYSTEM

SODIUM RETURN TO SYSTEM


HEAT

ECONOMIZER

COOLING

COIL

STAINLESS STEEL MESH

PACKING

CRYSTALLIZING TANK

Fio.

17. Sodium

oxide cold trap with heat economizer.

a representative sample of liquid metal from the system. Normal procedure is to withdraw a small sample into a glass vessel which is either evacuated or contain* a pure inert-gas atmosphere. Separation of the metal from oxide is accomplished

or reaction with n-butyl by vacuum distillation," mercury and amalgamation," bromide." The components are then carefully weighed. Reduction of the oxide content of a liquid-metal system is accomplished by taking advantage of the fact that the quantity of oxide held in solution decreases rapidly with decrease in temperature."' A procedure known as cold trapping involves bypass ing a portion of the liquid into a region of reduced temperature and then passing it over an extended surface in the form of a wire-mesh element, after which it is returned to the main system. A heat-exchanger arrangement is often used that conserves heat by permitting the fluid leaving the main system to feed heat to the fluid return ing to the main system after having passed through the cold trap (Fig. 17). 1.8

Large Liquid-metal

Systems

The salient features of extensive liquid-metal systems result from the following considerations: Liquid metals are chemically active, and therefore special precautions must be taken to avoid contact with air and water. In addition, the liquid, if it is used as a reactor coolant, will be radioactive. Care must be exercised, therefore, :»


Sec. 13-3]

13-97

ensure that the system is tight to a degree heretofore common only in high-vacuum Moreover, because of the hazards involved in opening the systems to practice. atmosphere, it is important that the component parts (pumps, valves, instrumenta tion, etc.) are capable of long, trouble-free operation and require little if any servicing. Finally, if repairs do become necessary, provision must be made for effecting them with very little disturbance to the system. Draining and flushing must be possible, and means must be available for controlling any leakage that may result from opening the system. Figure 18 shows the arrangement of components in a typical experi mental system.

Operation at elevated temperatures requires that consideration be given to reducing thermal stresses to a safe value. Expansion joints and bends must be adequate, and EXPANSION

TANK

C LEVEL INDICATOR


WELL

HEAT TRANSFER SECTION T.C.4-V

i

-LEVEL INDICATOR WELL

Fiq.

DIFFERENTIAL PRESSURE

TRANSMITS 18. Typical liquid-metal experimental system.

fixed points must be carefully selected.

Since high temperature enhances

corrosion,

attention must be given to proper choice of system materials. In addition, provision must be made for maintaining the system at a temperature above the melting point of the working fluid. Electrical-resistance heating and electrical-induction heating When relatively low heat input is required, a are both used for pipes and tanks. spaced winding of heater cable consisting of niehrome wire with asbestos insulation confined in a woven-metal sheath is applied directly to the pipe or tank and then When higher heat flux is necessary, ceramic-sup covered with thermal insulation. ported heating coils (clamshells) may be clamped on the pipe or tank surface or metalclad heaters may be immersed in the liquid. Induction heating with 60-cycle current The procedure may be employed in cases where magnetic wall material is present. followed in this case is to apply a winding of approximately ten turns per foot of insulated copper wire over the outside of the thermal insulation covering the s3'stem. The advantage of this method is that the electrical circuit is entirely at low tempera ture, so that burnout is unlikely and servicing is simplified, since the electrical system is external to the thermal insulation. Thermal insulation of pipes and tanks should be of a type that does not react with


13-98

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

burning metal. Mineral-wool and magnesia-compacted types have proved tolerable in this respect. A tight outer metallic jacket is effective in restricting leakage tem It has been found desirable on large systems to insert a tracer wire at the porarily. inner surface of the insulation for the purpose of detecting leaks at an early stage by electrical short circuiting of the tracer wire by the liquid metal. Liquid-metal systems should be provided with a dump lank both as a matter of The safety and for convenience when adjustments of the system require drainage. A dump-tank capacity should be adequate to hold the contents of the entire system. filter of stainless steel wool or sintered stainless steel should be located in the charging line connecting the dump tank to the system. A separate line with a fast-acting motor-operated valve but without filters is recommended for rapid discharge of liquid metal from the system to the dump tank. Systems should be constructed so that it is possible to drain them completely. The absence of pockets and dead areas is extremely desirable, since they collect oxide and other impurities and retard their removal from the system. When electromagnetic pumps are used, it is particularly important to avoid situations in which gas is entrained in the liquid as a result of turbu lence in tanks or other locations in which a free liquid surface may exist. The very considerable thermal expansion of liquid metals requires that an expansion space he An expansion tank with an inert-gas provided so that high pressures will not arise. blanket above the liquid-metal surface is standard practice. Expansion tanks also should contain provision for locating the liquid level. Thorough precleaning of the entire system is necessary in order to remove organic material and scale, which would produce oxide upon reaction with the liquid metal. A degrcasing treatment is considered minimal preparation of a system that is to con tain liquid metal. Electropolishing, where practical, is desirable. The removal of all moisture by heating under vacuum is recommended. If the system cannot be evacuated, a prolonged flushing with a dry inert gas must be carried out. Liquid metal is usually transported to the system in a shipping container fron. Removal of liquid which it is transferred to the dump tank by inert-gas pressure. metal from the system is normally carried out as follows: The liquid is drained from the system into the dump tank or into auxiliary storage tanks. Small systems ths'. can be readily separated into sections can be disassembled and transferred to a metaldisposal area, where they are then flushed out with dry steam followed by water. Extensive systems and sections that are not adequately vented to prevent pressure After build-up can be cleaned by a prolonged flush of moisture-carrying inert gas. all activity has ceased, the system is flushed with dry steam followed by waterSmall and delicate items are best removed from the system and soaked in iso-propyl alcohol. Disposal of Alkali Metals. Liquid metal is often drained into flat steel tray? and Unless burned, either by applying a fine water spray or by heating with a gas torch. this process is carried out in a remote area, the hydroxide smoke should Vk> collected and dissolved in water with a gas scrubber. Large quantities of liquid metal may also be disposed of by ejection of a well-dispersed stream of metal several feet below the surface of a large body of water. 2

HEAVY-METAL SYSTEMS

The small nuclear capture cross sections of bismuth and lead make them attractive as possible coolants for thermal reactors. Mercury, with its greater cross section, is feasible but unattractive as a coolant for fast reactors. It was, however, the first liquid metal to be used as a reactor coolant (Los Alamos fast reactor Clementine'. The physical characteristics of the heavy metals that have been considered for reactor coolant use are given in Table I. Corrosion of container materials by lead and bismuth have not been studied as extensively as corrosion by the alkali metals. *~" Table 3 gives the approximate temperature limits for a number of materials. Serious mass-transport effects have been observed. Pickup of radioactive material from the

reactor core channels would defeat the advantage of low intrinsic activity in the liquidmetal coolant. Lead and mercury are naturally toxic, and bismuth soon becomes


Sec. 13-3] Table 3.

13-99

Resistance of Materials to Bismuth, Lead, and Lead-Bismuth Material

Max temp for long-term use, °Ct

Alloy*

Max temp for temporary use, °Ct

Lead

550 ;oo 600 800 800 1000 1000 600 900

Zirconium

1000 800 1000 1000

1000 1000

Bismuth


1000 650 750 500 500 700 1000 700

1000 650 750 1000 1000

Lead-Bismuth Alloy (Eutectio) 250 500 550 750

High-Cr steel

550

* "Liquid Metal Handbook," 2d ed.. NAVEXOS and Department of the Navy, June, 1952; Anton de Metal Corrosion." ORNL-1647. June, 1954.

600 700 1000 1000 500 800

P-733 (rev.). U.S. Atomic Energy Commission S. Brasunas, "Interim Report on Static Liquid

*f Temperatures indicated are approximate only.

dangerous owing to the formation of polonium by neutron capture. Consequently, containment problem is more severe than in the ease of alkali metals. The poor wetting qualities of bismuth, lead, and mercury cause erratic electrobehavior. Little or no experience with electromagnetic-pump magnetic-flowmeter operation has as yet been acquired. The circulating systems so far constructed have been of limited capacity, and emphasis has been on the study of corrosion effects rather than operational experience with large systems. The increase in volume upon freezing of some of the lead-bismuth alloys has introduced additional problems in

the

some

cases. 3

Liquid

HOMOGENEOUS METAL SYSTEMS*

metals have been proposed as media for transporting fuel through a reactor. fuel may be carried either in solution or as a slurry. Fluid fuel may be passed through the reactor at a rate sufficient to remove the heat energy developed, or a In either case a con separate non-fuel-bearing coolant may remove the fission heat. venient method of replenishing reactor fuel is provided and continuous chemical reprocessing becomes possible. Moreover, the effects of radiation on fuel structure are no longer of concern. Extensive investigation has been made of the properties of bismuth-uranium alloy systems in which the fission products are continually removed

The

*

See Sec. 13-4.

13-100

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

by a dual process involving vacuum degassing and fused-salt processing." The lim ited solubility of uranium and thorium in liquid metals restricts the application of fluid fuels of this type. In addition, the attack of these liquids on structural materials presents a formidable problem. Much higher concentrations of fuel can be obtained with metal or oxide slurries The principal problems are those of agglomeration and settling. The seriousness of erosion and corrosion problems resulting from the flow of liquid-metal slurries has not yet been investigated in other than laboratory-scale apparatus. 4

DESIGN CONSIDERATIONS OF LIQUID-METAL-COOLED

REACTORS

The salient problems that must be considered in the design of liquid-metal-cooled reactors are the following: 4.1

Integrity of the Reactor Coolant Circuit

Since the liquid-metal coolant will be not only chemically active but also extremely radioactive because of inherent activity or activity of entrained impurities, ii is extremely important that the system be initially leak-free and be capable of remaining so for long periods. Furt hermore, all components such as valves, pumps, instruments, etc., must be so constructed that essentially no servicing will be required that emails manipulation inside the primary system shielding.


4.2

Emergency Removal of Shutdown Heat

Provision must be made for the removal of fission-product heat from the reactor in the event that pumping ability is lost, either because of pump failure or interruption in pump power supply. This requirement may be met by (1) gravity flow of coolant through the reactor and heat exchanger from an elevated storage tank, (2) thermalconvection flow through reactor and heat exchanger, or (3) use of auxiliary pumps operating from a stand-by power source. 4.3

Fuel Elements

Current practice is to strive for a fuel-element design that permits high jacketsurface temperature with minimum internal fuel temperature in order to minimize thermal stresses and avoid, if possible, phase transitions in the fuel as reactor power is varied. Heat flux is often limited by the permissible At in the fuel rather than by the heat-transfer coefficients at the jacket-coolant interface. Coolant temperature rise through the reactor is restricted by several factors, including permissible thermal stresses occurring in the system as a result of sudden large changes in reactor power Coolant velocity in core channels is usually as high as is consistent with pressure drop and vibration limitations. Since high power density is important from an economic standpoint, every effort should be made to take advantage of the unique properties of liquid metals as reactor coolants. Release of fission products into the coolant is prevented by enclosing the fuel in a suitable jacket. For high-temperature operation the formation of low-melting-point alloys of fuel and jacket material must be considered. 4.4

Control

The control problem may be resolved into the nuclear effect of the control eleraer:-. and the method of moving the element in the control operation. Control may be effected by adjusting the quantity of neutron-absorbing material (poison) in the reactor, by adjusting the amount of neutron leakage from the reactor (reflector con trol), by adjusting the amount of moderator, or by adjusting the quantity of fuel Poison control has been universally used in thermal reactors, whereas reflector control has been used in fast reactors. Fuel control is to be preferred from a conversion-rati-i standpoint but is made difficult by the heat-removal requirements.


Sec. 13-3]

13-101

Control-drive mechanisms must satisfy the requirements of extremely high integ rity of operation and fast response time when effecting reactor shutdown and be Drive linkage must be incapable of producing sudden large increments in reactivity. essentially leak-free at the point where it enters the reactor vessel. Fuel Loading and Removal

4.5

Loading of new fuel into a reactor will, in general, require the use of a handling machine capable of inserting the fuel subassemblies into a predetermined position in the reactor core. A method for locking the assemblies in the core should be provided so that pressure and vibration cannot produce displacement. Removal of fuel from the reactor core requires that provision be made for cooling the fuel subassembly during its transfer from the reactor to the storage receptacle. Even though the transfer operation is normally sufficiently rapid so that no auxiliary cooling is required, the possibility of accidental holdup in the process probably will demand that some form of forced heat removal be provided. Subsequent removal of coolant adhering to the fuel assembly must be considered in the light of safely dis posing of the radioactive liquid metal. This may be accomplished by a system of holdup and concentrating tanks. 4.6

Reactor Tank Seals


Since it is necessary

to have movable members for control and fuel handling enter be provided to prevent the leakage of is in a high-temperature environment, bellows can be used, provided the extent of travel is not excessive. Bellows can also be used to a limited extent for rotational applications by means of a wobble-motion For extended longitudinal motion, conventional packing-gland type of mechanism.

ing the reactor vessel, a number of seals must inert gas and liquid-metal vapor. If the seal

seals located in regions of low temperature and available for servicing may be used. Another arrangement involves the use of frozen-metal seals. In cases where only intermittent motion is necessary, a normally solid metal (lead or tin) seal may be used with provision for electrical heating when motion is to occur. When continuous motion is required, frozen-sodium shaft seals may be considered. 4.7

In

Secondary Coolant Circuits

the interests of safety it has usually been considered desirable to avoid bringing

radioactive alkali-metal coolant from the reactor into the steam generator. Conse quently, a secondary nonradioactive circuit has been introduced to transfer heat TURBINE SUPERHEATER INTERMEDIATE

V

HEAT

EXCHANGER^

CONDENSER STEAM GENERATOR

REACTOR

PUMP

PRIMARY SYSTEM SHIELO

PUMP

PREHEATER ■ FEED WATER PUMP

Fio.

19. Renctor

primary and secondary liquid-metal coolant systems.

13-102

MECHANICAL

DESIGN AND REACTORS

[SEC.

13

from the primary circuit to the steam-generating system (Fig. 19). This, of course, involves the use of an intermediate liquid-metal-to-liquid-metal heat exchanger and additional pumps and other associated equipment. The secondary circuit should be operated at a somewhat higher pressure than the primary circuit, so that any leakage which may develop in the intermediate heat exchanger will not introduce radioactive metal into the secondary circuit. If the same fluid is used in both circuits, minor internal leakage in the heat exchanger may be tolerated.

REFERENCES 1.

2. 3. 4. 5.


6. 7.

"Liquid Metals Handbook,"

2d ed., NAVEXOS P-733 (rev.), 1952, and "Liquid Metals Handbook," Sodium-NaK Supplement, 3d ed., TID 5277, 1955. U.S. Govern ment Printing Office, Washington, D.C. Mausteller, J. W., and E. F. Batutis: Effect of Oxygon on Mass Transport of Stainless Steel Components, Mine Safety Appliance* Co. Tech. Rept. 36, March, 1955. Heywood, W. A.: "Frozen Sodium Shaft Seal," KAPL-1265, August, 1954. Clark, P. M. : " Mechanical Liquid Metal Pumps in the Alphaus (Genie) Heat Transfer SyBtem," KAPL-635, November, 1951. Leland, Marshall B.: Pumps, Bearings and Packing Shaft Seals for Use with Liquid Metals, Babcock
1953). 8. Cage, Jr., J. F.: Electromagnetic Pumps for High Temperature Liquid Metals, Machine Design, 85(3): 178 (March, 1953). 9. Barnes, A. H„ F. A. Smith, G. K. Witham: "Large Current Homopolar Generator," ANL-4361, October, 1949. 10. Vandenburg, L. P.: "Homopolar Generator Pumps for Liquid Metals," KAPL-590 (AECD-3368), August, 1951. 11. Watt, D. A.: "A Homopolar Generator for High Circuit, Low Voltage DC Supplv."

AERE-CE/R-821, November, 1951. _12. Werner, R. C, and J. C. King: Electromagnetic Flowmeters

and Electromagnetic Pumps for Sodium Potassium Alloys, Mine Safety Appliances Co. Rept., N6-ori-146, June, 1948. 13. Watt, D. A.: "Liquid Metal Pumps for Laboratory Use," AERE-CE/R-1089. March.

1953. *14. Barnes,

A. H., F. A. Smith, and G. K. Witham: "Electromagnetic Pump for Liquid Metals," ANL-4317 (AECD-3431), July, 1949. -15. Barnard, John, and C. D. Collins: "Test of 1200 GPM Linear AC Electromagnetic Pumps," AECD-3460, May, 1951. -16. Cage, J. F.: "Description of Test Results of a 400 GPM Liquid Metal Induction Pump," AECD-3459, October, 1951. 17. British Patent Application 303,065, Dec. 24, 1928. 18. Bialous, A. J.: "SIR Mark A Double Diaphragm Pressure Transmitter for the Primary Coolant System," KAPL-1213, October, 1954. 19. Kolin, A.: Rev. Sci. Jnstr., 16: 109 (1945). 20. Barnes, A. H., F. A. Smith, and G. K. Witham: "Electromagnetic Flowmeter for Liquid Metals," ANL-4092 (AECD-3407), September-November, 1947. 21. Elrod, H. G.: Babcock & Wilcox Co. Rept. 5206, June, 1950. 22. Elrod, H. G., and R. R. Fouse: An Investigation of Electromagnetic Flowmeters, Trans. ASME, 74: 589-594 (1952). 23. Shercliffe, J. A.: "Theory of D.C. Electromagnetic Flowmeter for Liquid Metals," AERE-X/R-1052, November, 1952. 24. Argonne National Laboratory Quarterly Report of Reactor Engineering Division for March to November, 1948, ANL-4238. 25. Walters, S. F.: Effects of Adding Oxygen to Sodium Flowing in a Stainless Steel Sys tem, Mine Safety Appliances Co. Tick. Rept. 6, September, 1950. 26. Argonne National Laboratory Quarterly Report July 1 to Sept. 30. 1954. ANL-5348. 27. Pepkowitz, L. P., and W. C. Judd: Determination of Sodium Monoxide in Sodium. Anal. Chem., 32: 1283 (October, 1950). 28. White, J. C, W. J. Ross, and R. Rowan, Jr.: "Determination of Oxygen in Sodium.'" ORNL-12S0, April, 1952. 29. Ratutis, E. F., 8. F. Walters, and J. W. Mausteller: Removal of Ns?0 from a NaK


Sec. 13-3] 30. 31. 32. 3.3.

34.

35.

13-103

System Using a Natural Circulation Cold Trap, Mine Safety Appliances Co. Tech. Rept. 21, March, 1953. Pray, H. A., R. S. Peoples, and W. K. Boyd: "Corrosion by Molten Bismuth, " BMI773, October. 1952. Grassi, It. C: "Final Report — Materials Subjected to Lead-Bismuth Allov," AECU2201, July, 1952. Baker, O. H.: Experimental Investigation of Corrosion and Erosion of Liquid Lead Systems, Babcock & Wilcox Co. Rept. 5248, July, 1953. Ferguson, K. M.: Observations on Alloys Exposed to Molten Lead and Additives in Oscillating Capsule Tests, Babcock 4 Wilcox Co. Rept. 5247, August, 1953. Shephard, O. C. : Investigations of Materials for Use in a Heat Transfer System Con taining Lead and Bismuth, Stanford Univ. Rept. 16, June, 1954. (See also earlier reports) . "Liquid Metal Fuel Reactor Processing Loops," part 1, BNL-322 (T-55), June, 1954.

BIBLIOGRAPHY Corrosion Batutis, E. F., S. J. Rodgers, and J. W. Mausteller: Influence of Oxygen on the Corrosion Rate of Type 347 Stainless Steel in 1000F Sodium, Mine Safety Appliances Co. Tech. Rept. 33, November,

1954.

Brasunas, A. de S. : Liquid Metal Corrosion, Corrosion, 9: 78-84 (March, 1953). Brewer, L., and J. L. Margrove: "On the Removal of Na*0 from Na by Distillation." UCRL-1241 (rev.). April, 1951. Cotheart, John V.: "Mass Transfer in Dynamic Lcad-Inconcl Systems," CF-52-1 1-1 16, Generated for wjivans (University of Florida) on 2015-09-23 02:59 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

November,

1952.

Cygan, R.: "Circulation of Lead-Bismuth at Intermediate Temperatures," NAA-SR-253, October,

1953.

Dana, A. W., O. H. Baker, and M. Ferguson: Erosion and Corrosion Studies of Liquid Metal Systems, Babcock A Wilcox Co. Rept. DC-52-5-19, August, 1952. Dana, A. W., O. H. Baker, and M. Ferguson: Investigation of Large Scale Dynamic Liquid Lithium Corrosion Apparatus, Babcock & Wilcox Co. Rept. DC-52-25-60, April, 1952. Also Rept. DC-27-45, May, 1952, and Rept. DC-52-28-140, August. 1952. Elgert, O. J., and C. J. Egan: "Dynamic Corrosion of Steel by Liquid Bismuth," CRD-TI74, August,

1952.

Elgert, O. J., and C. J. Egan: "Dynamic Corrosion of Steel by Liquid Bismuth," MTA-12, January, 1953. Klrod, H. G., R. R. Fouse, and P. B. Richards: Erosion and Heat Transfer with Molten Lithium, Babcock & Wilcox Co. Rept. 5217, April, 1951. Gross, M. R.: " Basic Information on the Bearing Properties of Various Materials in Liquid Metals," EES-9C(2)966 861, June, 1951 (U.S. Naval Engineering Experimental Sta tion): 9C(3)966 861, February, 1952. Holt, B. D.: "Determination of Hydrogen in NaK by the Method of Pepkowitz and Proud," ANL-4388, January, 1950. Jaffee, R. I.: "Gallium for Nuclear Reactors," AECD-3317, August, 1949. Johnson, H. A., J. P. Hartnett, and W. J. Clabough: "Design and Operation of 40 KW Lead-Bismuth Eutectic Heat Transfer Systems," AECU-2848, February, 1954. Koenig. R. F.: "Corrosion of Zirconium and Its Alloys in Liquid Metals," KAPL-982, October,

1953.

"Metallurgical Investigation

of Materials Subjected to Liquid Lead-Bismuth Allovs Environment," ORO-119 and ORO-I20, 1954. Nelson, C. O. : "Summary Report of Reaction Tests of Various Materials Tested with Sodium and Sodium-Potassium Alloy," KAPL-557, June, 1951. Shopard, O. C. : "Investigations of Materials for Use in Heat Transfer Systems Containing Lead or Bismuth." AECU-2549, April, 1953; AECU-2794, December, 1953. Vail. Donald B.: "Compatibility of Materials in Liquid Metal," KAPL-589, August, 1951. Wilkinson, W. D., and F. L. Yagee: "Attack on Metals by Lithium," ANL-4990, October, 1950.

Pumps " Development of Electromagnetic Pumps Suitable for Pumping Liquid Metals," NP-4892, June, 1952. Cage, Jr., J. F.: Electromagnetic Pumps for High Temperature Liquid Metal, Mech. Eng., 75(6): 467 (June, 1953).

Asti. J. F.:

13-104

MECHANICAL

DESIGN

AND REACTORS

[SEC. 13

Higgins, R. M.: New Bearing Made of Any Material Lubricated with Any Fluid, September,

Iron

Age,

1952.

General Bonilla, C. F., and B. Misra: "Boiling and Condensing of Liquid Metals," NYO-3152, April, 1953. Brooks, R. O., and A. L. Rosenblatt: Nuclear Power Plant Design and Performance of Liquid Metal Heat Exchangers and Steam Generators, Mech. Bng., 76(5): 363 (Slav, 1953).

Cottrell, W. B., and L. A. Mann: "Sodium Plumbing," ORNL-1688, August. 1953. Drugas, P. G., I. R. Rehn, and W. D. Wilkinson: "Resistivity of NaK," ANL-5115. October.

1953.


Jackson, C. B. (ed.): "Liquid-Metals Handbook," Sodium-NaK Supplement, 3d ed., TID 6277, U.S. Government Printing Office, Washington, D.C., July, 1955. Lyon, R. N. (ed.): "Liquid-Metals Handbook," NAVEXOS P-733 (rev.), 2d. ed., U.S. Government Printing Office, Washington, D.C., June, 1952. SherclifT, J. A.: "Theory of D. C. Electromagnetic Flowmeter for Liquid Metals," AEREX/R-1052, 1952. Trocki, T., and D. B. Nelson: Liquid Metal Heat Transfer System for Nuclear Power Plants. A Report on the Development of Suitable Heat Exchangers and Steam Generators, Mech. Eng., 76(6): 472 (June, 1953).

GAS-COOLED REACTOR SYSTEMS

13-4

BY Miles C. Leverett This section covers some problems peculiar to gas-cooled reactors and to equipment used in conjunction with them. The material is divided into three parts: (1) gascooled reactors, (2) once-through systems, and (3) recirculating systems.


1

GAS-COOLED

REACTORS

The first nuclear reactor to produce a measurable amount of heat was a gas-cooled reactor, the graphite uranium reactor built at Oak Ridge in 1943. This reactor is still in active operation at a steady heat rate of 3,500 kw. A second air-cooled reactor was constructed at the Brookhaven National Laboratory and began operation in 1948. It operates at a power of 25 to 30 Mw. In 1957 the Brookhaven reactor was reloaded with enriched uranium fuel in place of the original natural uranium. Several air-cooled reactors have been constructed in England. The first of these was BEPO, constructed at Harwell and operating at a power near 6,000 kw. Air-cooled reactors for the production of plutonium were later constructed in England at Windscale. An extensive program of building graphite natural uranium reactors, cooled with pressurized COj, and designed for electric power production, is under way in the The reactors reportedly in this program are listed in Table 1. United Kingdom. Table

Gas-cooled Power Reactors in the United Kingdom

1.

Designation

Electric power, kw

Date of operation*

46,000 46,000 46,000 46,000 46,000 46,000 46,000 46,000 150,000 150,000 137,000 137,000 137,000 137,000 160,000 160,000

1956 1957 1958 1958 1958 1958 1959 1959 I960 I960 1960 I960 1961 1961 I960 I960

Calder A- 2 Calder B- 1 Calder B-2

Hunterston-2 * Operation dates later than 1957 are as predicted

in 1957.

Several French power reactors of the air- or COr-cooled type have been announced also, as shown in Table 2. A 200,000-kw (electricity) reactor of the Calder type is projected for an Italian location also. The selection of a gas-cooled reactor for the first one at Oak Ridge was based pri

marily

on the inherent simplicity

of this type of reactor and the speed with which it 13-105

MECHANICAL DESIGN

13-106

AND REACTORS

[SEC.

13

could be built . The selection of an air-cooled reactor for Brookhaven was based upon the highly satisfactory experience with the Oak Ridge air-cooled reactor and upon the desire to achieve an operating reactor at Brookhaven as quickly as possible. Both reactors have been found extremely satisfactory for research and development work and for pilot production work, the purposes for which they were designed, even though they are somewhat limited by the fact that the thermal flux of neutrons in them is lower than is desired for much research and development work aimed at higher power reactors. The British selection of gas-cooled research and production reactors was based on the fact t hat such reactors are inherently nuclearly safe in the event of loss of coolant as opposed to certain types of liquid-cooled reactors, which increase markedly in reactivity if they lose coolant. This may not be a lasting objection, since it is responsibly indicated that British power-production reactors of the more distant future may be of the liquid-cooled type. Table Designation

G-1

G-2 G-3

EDF-1

French Power Reactors

Cuulant

Electric power, lew

Location

Date of operation*

Air CO, CO, CO, CO,

5,000 30,000 30,000 60,000 60,000

Marcoule Marooule Marcoule

1956 1957 1958 1959 I960

•Operation dates later than 1957 are as predicted use natural uranium fuel and graplute moderator.

Voine Voine in 1957.

All the French reactors luted in Tabl* 2

In the field of mobile nuclear power plants suggestions have been made to use gatcooled reactors in shipboard and aircraft applications. At least one gas-cooled reactor is known to be under development for shipboard use, and extensive development oi air-cooled reactors for aircraft use is going on. In 1956 an aircraft turbojet engine wa* operated on nuclear heat from an air-cooled reactor as a part of the United States Aircraft Nuclear Propulsion

program.

is

it

1.

is

is

is

is

of

is

is,

The design and operation of gas-cooled reactors present certain problems not found in reactors of other types. These arise in one way or another from the facts that parcooled reactors generally require substantial power for pumping the coolant through the reactor, they operate at temperatures higher than many liquid-cooled reactors they have inherently a high percentage of gas space in them, and heat transfer from fuel elements to coolant is relatively critical. The high pumping-power requirement makes good aerodynamic design essentia) Such devices as throttling the gas flow to those channels of the reactor in which the power generation is below the maximum will evidently conserve on pumping power However, even greater economy of pumping power can br to a certain extent. obtained by so designing the reactor that each cooling channel emits the same energy, One method of doing this that so t« by flattening the power in a radial sense. construct the reactor that the ratio of moderator to fuel greater near the outside The nuclear physical principles of this technique the reactor than near the center. are established. Another expedient, practiced at Oak Ridge for some time, to load the peripheral fuel channels with more fuel than those near the center. Because this results in longer loaded sections in the outer fuel channels, the air flow through these channels somewhat reduced and the power per fuel channel increased. More nearly uniform exit-air temperature achieved. This "dished" loading diagrammed in Fig. Warping and expansion due to temperature differences tend to be aggravated by the reactors usually operate. Alternate high-temperature levels at which gas-cooled heating and cooling of the reactor structure, as by starting and stopping at frequen; intervals, are particularly detrimental in this respect. Also, too rapid heating up or cooling down should be avoitled. Some operators limit the rate at which they increase is


EDF-2

2.

Sec. 13-4]

REACTOK SYSTEMS

GAS-COOLED

13-107

is

it

is

is,

the power in the reactor, and the rate of planned power decrease for this reason. The usual practices of good design of high-temperature equipment should be followed in That allowance must be made for the expansion of designing gas-cooled reactors. each heated component. Attempts to prevent such expansion by mechanical restraint usually result in failure of the component or of the restraining member. It also good practice to keep lineal dimensions of individual components small so that the amount of expansion in any one component will not be large. The smallness of heat-transfer coefficients to gases as compared with heat-transfer coefficient to most liquids tends to result in a design in which the temperature differences between the fuel element and the gas are relatively high. Thus, a given percentage error in estimation of the heat-transfer coefficient can produce a larger overtemperature of the fuel-element surface in the case of gas cooling than in the corre Moreover, customary in designing liquid-cooled sponding case of liquid cooling.

INSERT "C" FUEL CHANNEL GRAPHITE

X 7/7//////////////////M/////\ URANIUM

7////////////////////////m-j

//////)))))))))))))))))///m. AIR FLOW

////////////////////////////-/

U//////)))))))))))))///////7

7J/////))))))))))))))//////n >//////)))))))))))))))/J//fTT.

/////))))))))))))))))))////n ///tt)))))))))))))))))))///i.

////////////)))))))))))))rrn

1.

Fio.

"Dished" loading

of graphite reactor.

Used to flatten power radially.

is

is

it

is

a

a

is

is

if

is

heat exchangers to allow for a so-called "fouling coefficient." If the fouling or scale thus allowed for actually present, the over-all heat-transfer coefficient becomes less sensitive to variations in the film heat-transfer coefficients of liquid to the solid were absent. If actually than that an extra safety absent, the net result factor has been allowed in the heat-transfer coefficient. No such corresponding safety factor customary in designing gas-cooled heat-exchange surfaces. Thus, considerably more sophisticated heat-transfer design required in the case of gascooled reactors than in the case of liquid-cooled reactors. On the other hand, hot spots are of less serious consequence in gas-cooled systems than in liquid, since the thermal conductance of the fuel element itself will usually be large compared with that of the gas film, so that sideways drainage of heat from locally overheated region or locally poorly cooled region will be more effective than in the case of liquid systems. Good reactor-design practice requires that perturbations of the temperature of the fuel elements and other critical reactor components be in the nature of cold spots easy to arrange, since most of the per rather than of hot spots. Fortunately, this turbing factors such as control rods, supporting structure, and discontinuities in the it


//)))))))))))))))))))))V)///

13-108

MECHANICAL

DESIGN AND REACTORS

[Sec. 13

heat-transfer surface tend, respectively, to depress the local heat-generation rate, to drain heat away from the heat-generating part, and to increase the local heatAll these tendencies result in locally lower rather than higher transfer coefficient. temperatures. The presence of a high volume fraction of gas-filled space in the reactor tends to make the reactivity of the reactor sensitive to changes in gas pressure if the gas is an absorbing one like air (in which the nitrogen is a fairly strong neutron absorber), and operators of air-cooled reactors have learned that they can detect barometric pressure Furthermore, since changes fairly easily by changes in reactivity of their machines. the cooling paths in a gas-cooled reactor constitute neutron-leakage channels, an element of added complexity is introduced into the neutron analysis of the reactor. Characteristically, this makes the designer even more dependent upon adequate criti cal experiments than in the case of liquid-cooled reactors. In spite of the problems enumerated above, gas-cooled reactors have been found to In particular, the coolant have many advantages for both research and production. in an air-cooled reactor is not toxic, is only mildly radioactive, is nonflammable, and in the event of leakage. Obviously also, test or experimental noncontaminating equipment introduced into the atmosphere of an air-cooled reactor need not be made tight against the coolant, a considerable convenience in many cases. 2

ONCE-THROUGH SYSTEMS

Gas-cooled reactor systems may be classified according to whether the coolant through the reactor only once or is continuously recirculated. The reactor auxiliary equipment for handling the coolant in once-through systems is mainly of Although suggestions have conventional type and presents no unusual problems. been made that other gases might be so used, it appears that air is the only practical coolant for once-through use, and in this article it is assumed that air is the coolant. Such reactors may be operated either as power producers or as neutron producers. The latter are simpler and will be considered first. Figure 2 presents a diagram of a simple once-through system in which the only function of the coolant is to carry the heat away from the reactor in order to dissipate it. Such a reactor might be operated as a research tool or for the production of The principal elements of this system are plutonium or other radioactive materials.


passes

s

PREFILTER

POSTFILTER

Fio.

2. Simple air-cooled reactor system.

Sec. 13-4]

OAS-COOLED

REACTOR

13-109

SYSTEMS

is

is

it

is

if

it,

the prefilter, the reactor, the postfilter, the blowers, and the exit stack. Placing the blowers after the reactor has the desirable result of putting the reactor under subThis makes certain that any leakage around the reactor atmospheric pressure. structure will cause leakage of air into the cooling system rather than outward leakage of air, which may have become activated by passage through the reactor and may Because this feature is particularly contain radioactive dusts of various kinds. as for experi advantageous where the reactor structure has a number of holes in mental work, the reactors at Oak Ridge, Brookhaven, and Harwell are arranged in this fashion. On the other hand the blower placed after the reactor, its power increased by the fact that consumption operating in hot air. This has led to the installation of coolers between the reactor and blower in the Oak Ridge, Brook-

Table 3.

Typical Operating Characteristics of Reactor Air Systems Oak Ridge

System

pressure drop,

Brookhaven

135 12 175

% of atmosphere

2.1

335 12 333

Filters

Both prefilters and postfilters have

1

is

is

is

filter in. drop

a

a

6)

A

is a

is

is

it,

is

been found desirable in air-cooled reactor instal filter obviously desirable for the removal of any radio contained in the air. These may originate from dust ini tially contained in the air, may be drawn into the reactor from portions of the reactor or may itself abraded off by relative movement of its parts or by the air through enter the air stream by accidental release of materials (as from experiments) in the reactor during operation. The prefilter serves primarily to reduce the dust load on the postfilter. This desirable because the material that collects on the postfilter is at least in part radioactive and changing the filter, though not difficult, laborious compromise and to be avoided when possible. The fineness of the filter employed between desire for complete removal of any suspended solids in the air stream on the one hand and increased pressure drop and increased filter cost on the other hand. filter (Chemical Warfare Service No. that will retain 99.9 per cent of the particles is used in the more recent installation. When new and clean, such filters give pres " sure drop of about in. of water. They are usually protected by so-called "rough ahead of them, such as Fiberglas No. 50, which puts in a pressure drop of about of water. It considered advisable to change these filters when the pressure across them rises to a few times its initial value. This usually requires 12 to 18 months. At Oak Ridge the filter house so constructed that the filter cells can be isolated from the main air stream in order to change them. After the filter cell isol»te
lations. The postreactor active solids that may be

1


3

is

is

is

is

is

is,

a

is

substantial fraction of the reactor heat haven, and Harwell installations in which removed from the air stream. At Oak Ridge this cooling was accomplished for a time by spraying demineralized water into the air ahead of the postfilters. This At Harwell the heat transferred to water that system however, no longer in use. used for heating the buildings of the establishment. At Brookhaven the heat wasted. The requirement for this cooler after the reactor can be eliminated, and the power requirement of the blower can be further reduced by placing the blower ahead of the reactor. This was done in the production reactors at Windscale. In this case the reactor envelope essentially gastight, so that the escape of activated air or dust into the reactor building improbable. Table summarizes some of the characteristics of the Oak Ridge and Brookhaven systems as regards air flow, pressure drop, and temperature.

ter

13-110

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

remote control into a canal filled with water. The water provides shielding for the filter elements until they can be moved to suitable carriers for transportation to a At both Oak Ridge and Brook haven it has been found that the filters burial ground. become sufficiently radioactive to warrant enclosing them in a shielded compartment. This radioactivity is probably mostly from fission products contained in uranium, At oxidized off the accidentally exposed surfaces of fuel elements in the reactors. Windscale the postreactor filters were located on top of the exhaust stack, which is 404 ft high.


2.2

Air-moving

Equipment

Air-moving equipment of conventional types is suitable for use with air-cooled No special lubricant, bearing, or reactors. Centrifugal blowers are generally used. control problems exist, but if the blower is placed ahead of the reactor, care should be taken that blower lubricant is not introduced into the air stream in excessive quanti Not only would it be possible to build up a heat-blocking film on the fuel ele ties. ments if this were to occur, but it has been found that oily, greasy, or sticky surface? Thus, excess lubricant in th
Valves and Seals

Valves and seals for an air-cooled reactor system present no unusual problems. Valves of conventional design can be used on the main air stream, although at Oak A Ridge seal pits were used to isolate the individual fans in the original design. diagram of such a device is shown in Fig. 3. When the seal pit is full of water, the When it is empty, air passes passage of air from one side to the other is impossible. unimpeded.

Sec. 13-4]

GAS-COOLED

REACTOR

SYSTEMS

13-111

Though entirely workable, this practice has not been followed in later designs, ordinary valves being used instead.


2.4

Heat Recovery in Exchangers

Utilization of the heat carried in the reactor exit-air stream can obviously be accom plished by heat exchangers. Although such a heat exchanger is installed in the Brookhaven reactor system, no use is made of the heat. The water is circulated over a cooling tower and typically enters the heat exchanger at 86°F, and leaves it at 107°F. At Harwell, where the outlet temperature of the air from the reactor is 176 to 194°F, a water heater has been installed in the hotair outlet duct. This heater has a nomi nal heat-exchange rate of 2,000 kw and pro duces water at about 162°F which is used for heating buildings in the research establishment. No attempt was made to utilize the heat produced at the Windscale air-cooled production reactors, and, since the fans in this case were located ahead of the reactors, cooling of the air for power conservation is less important than at Harwell, Brookhaven, or Oak Ridge. Waste heat boilers may be introduced into the air stream for the production of steam from the heat of the air. Such an installation will, as for other heat exchangers, introduce a pressure drop into the system and will, therefore, increase the pumpingHowever, studies of waste heat boilers in other somewhat simi power requirements. lar systems indicate that reasonable economic balance can be expected if the boiler has an effectiveness ratio of 0.85 to 0.90 and imposes on the system an additional back Such an arrangement is particularly attractive pressure of about 5 in. of water. where large quantities of relatively low-temperature steam are required for process or Where higher pressure or highly superheated steam is required, the heating use. heat from the reactor may be used to preheat the feed water for a chemically fired boiler or to produce saturated steam, later to be heated in a chemically fired super heater. Still another possibility is that the hot air from the reactor be used as combustion air under a chemically fired boiler. Whereas in chemically fired systems it is neces sary (because of moisture condensation) to keep stack temperatures above certain permissible minimums (ranging from 175 to 260°F depending on the typo of fuel used), no such restriction is present in the case of air heated by nuclear reactors. Although no economic studies have been reported on the use of reactor-heated air for heating feed water, analyses of the corresponding chemical systems indicate that exit stack gas temperatures can reasonably approach within 100°F of the entering feed-water temperature. The details of utilizing the heat of the reactor exit air as steam or sensible heat in water can become extremely complicated. Since these details are not strongly influenced by the fact that the heat originates in a reactor, however, this subject will not be further pursued here. In any real case a detailed economic study taking into account the factors peculiar to the individual situation will be necessary. 2.6

It

Power Recovery in Gas Turbines

is evident that the hot air issuing from the reactor may, if at sufficiently high be expanded through a turbine, producing work. This work can be used first of all to drive the system blower or compressor, and the excess remaining can be usefully applied externally. Numerous applications of this principle, using chemical fuel as the source of heat, exist, for example, in turbojet aircraft engines (where the excess energy of the gas is converted to kinetic energy of the jet rather than to shaft power) in turboprop aircraft engines, in locomotive gas turbines, in gas turbines

pressure,

MECHANICAL

13-112

DESIGN AND REACTORS

[Sec.

13

PRE FILTER

I'iu.

COMPRESSOR

TURBINE"

4. Air-cooled reactor system

IIIIIIIIIIIUIH

with gas-turbine power recovery

FILTER

I

COMPRESSOR DRIVE SHAFT COMPRESSOR TURBINE

Fio.

5. Air-cooled reactor

system

POWElT" TURBINE

SHAFT WORK

with two-shafted gas turbine for power recovery.

it,

a

5. is a

a

it

is

it

is

in then expanded adiabatically through a turbine. When nuclear heating used instead of the burning of chemical fuel, the process, although thermodynamically unchanged, resembles more closely the Joule cycle, from which the Brayton cycle was derived. Gas turbines are characterized by relatively high power-weight ratios. shows the simplest type of gas-turbine cycle as Figure would be used in gasturbine nuclear power plant. Several other more complicated cycles have been found to have merit in chemically fueled systems. For example, the separation of the functions of driving the compressor and driving the external load by introducing second turbine has been found to have major advantages when the external load variable. Such a unit, as would be applied to reactor, diagrammed in Fig. In some cases the heat remaining in the exhaust gases after leaving the turbine can 4


in driving gas pipeline compressors, in peak-load power-generating installations, The substitution of nuclear for chemical heat in such a chemical processes, etc. So far, however, no installation based on this machine is thus a natural suggestion. principle has been announced except the turbojet operation noted in Art. 1. gas turbines operate on the Brayton cycle. In this cycle, air Thermodynamically, is first compressed adiabatically, then heated at constant pressure by burning fuel

gas-cooled

Sec. 13-4]

reactor systems

13-113

be used in a regenerator to preheat the air coming from the compressor, thus reducing Also the air can be extracted from the compressor when partly fuel consumption. compressed and cooled in intercoolers before being returned to the compressor to Intercooling reduces the work required to compress the air, continue compression. Furthermore, the residual heat in the thus increasing the useful output of the unit. gases exhausted from a gas turbine can be used to preheat boiler feed water, to make All these devices increase both steam, or as preheated air to be burned under boilers. useful work output and the efficiency of the over-all system, and all are as applicable to nuclear as to chemical gas turbines.

Simple-cycle single-shaft chemically fired gas turbines commercially available in sizes of 8,500 hp and up have efficiencies (based on lower heating value of natural gas fuel) of 21 per cent or more at 80°F compressor inlet temperature and 1,000-ft altitude. of 27 per cent or Regenerative-cycle two-shaft units have efficiencies more. Simple-cycle single-shaft gas turbines yield about 74 hp-sec of useful mechani cal work per pound of air handled. The regenerative two-shaft unit yields about In both cases turbine inlet temperature is about 1450°F. Both the 69 hp-sec /lb. efficiencies and work yield per pound of air handled would be somewhat lower if a This is primarily nuclear reactor were substituted for the chemical heat source. because of the extra pressure drop introduced into the system by the reactor. Usual gas-turbine design imposes a pressure ratio between turbine inlet and compressor exit of . . pressure Turbine inlet ~ „ . ■ = 0.95 approximately Compressor exit pressure


—

The pressure ratio between these points with the reactor system between them will almost certainly not be greater than about 0.90 and may be substantially less than this. For comparative purposes the pressure drops in the Brookhaven and Oak Ridge reactor systems are tabulated herewith in Table 4, even though these systems have little in common with power-producing systems. Table

Pressure Drops in Air-cooled

4.

(As per cent of external

atmospheric

Reactor Systems pressure)

Brookhaven

Inlet to reactor

0. 1

Across reactor

9.7 2.5

Blower to external atmosphere Net

Oak Ridge

2.4 7 8

-0.3

2.0

12.0

12.2

A gas-turbine reactor power plant will have its design 2.61 Design Criteria. characteristics established mainly by economic considerations. However, in lieu of These economic analysis two physical criteria seem likely to be of major importance.

are:

1. The work produced per pound of air handled 2. The over-all energy efficiency of the system

Of

these the former is probably much the more important,

since the size of the air-

handling equipment will largely determine its cost, and fixed charges based on first cost of the plant are the most important element in the cost of power from any atomic system.

The

system efficiency determines the rate of burnup of fissionable fuel and thus

How determines that part of the power cost due to supplying new fuel periodically. ever, the major fuel cost is not that of new fuel to replace burnup but that of reproc

MECHANICAL DESIGN

13-114

AND REACTORS

easing the (usually) much larger amount of partially reused. This cost is proportional to the quotient

[SEC.

used-up fuel so that it

13

can be

Amount of fuel in reactor Time between replacements of fuel charge

In gas-cooled reactors now known, the time between replacements of the

fuel charge

found to be determined by non-nuclear effects such as oxidation or mechanical failure, being frequently independent or nearly so of the actual amount of material fissioned. Thus, system efficiency may well be of minor significance in the over-all economic picture. For this reason a third criterion may be of major impor tance, namely, the life of the fuel elements against oxidation or mechanical failure Data on this subject are extremely scarce. is frequently

0123456789

10

II

U

CPR

Fio.

6.

Thermal efficiency vs. CPR for

5 per cent pressure

loss.

2.52 The design variables of major Relationship among Design Variables. importance in estimating the performance of a gas-turbine nuclear power plant w the following:

Compressor inlet-air temperature Compressor pressure ratio (CPR) 3. Pressure ratio between compressor exit and turbine inlet 4. Turbine temperature 7*< 1.

2.

if

is

S)

a

is

sli

In order to show how these variables are interrelated a study resulting in thecurvs of Figs. 6 through 1 1 has been made. These curves cover a range of variation in the above parameters except compressor inlet-air temperature. Other data sho* that the power output of gas turbine fairly sensitive to compressor inlet tempera ture, typically increasing by a factor of 1.5 when the inlet temperature drops from to 0°F. An efficiency increase by a factor of about 1.13 occurs at the same time. These curves show that the optimum compressor pressure ratio for a practical »nturbine nuclear power plant probably not less than about 7.5/1 1500oF air to the turbine can be attained. full investigation may well show even higher compress* pressure ratios to be desirable in order to keep reactor size (and hence, probably, fori The curves also show that thermal efficiency not strong investment) down. dependent on compressor pressure ratio. The desirability of turbine inlet temper* is

A


30


Sec. 13-4]

tures

gas-cooled

reactor

systems

13-115

well above 1200°F is evident, and it is likely that a practical simple-cycle airpower plant will require fuel elements capable of producing air at about 1200°F or more. In the derivation of these curves reasonably advanced component effi ciencies were used and allowances were made for leakage, air bled for cooling, and kinetic energy lost in the exit-air stream. Power used for auxiliaries (other than for moving the air) was, however, not subtracted from the output.

turbine

13-116

MECHANICAL

AND REACTORS

DESIGN

[Sec.

90

T

80

o z o u

13

1500

70

60

o

i

50

O

n. 0-

x i-

1200^ 30

u. ■< 20

5

10


900

II Fio.

9. Shaft horsepower

CPR

vs.

for

10 per cent pressure

12

loss.

30

Z

u u

25 1

o;

a

20

uj

15

U m

1S0O

,0

^S^wo 10

11

CPR

Fio.

10.

Thermal efficiency vs. 2.6

CPR

for 20 per cent pressure loss.

System Instrumentation

Gas-cooled reactor systems are provided with the usual instrumentation for meas urement and recording of temperatures, pressures, and air flows. Although it was formerly thought necessary to provide a venturi section for accurate flow measure ment, a calibrated Pitot tube in fixed geometry has been found adequate. Calibra tion is usually by a dilution method, introducing COi or NHt into the air stream at

gas-cooled

Sec. 13-4] known

reactor

systems

rates and measuring the resulting effluent concentration.

13-117 Air flow must

be

known in order to convert neutron-sensitive instrument readings into accurately absolute power levels. It is customary to measure the radioactivity of the exit gas in order to get an early indication of the escape of fission products into the gas stream. This is particularly important in air-cooled reactors, because discharge of excess radioactivity into the A number of exit-gas activity atmosphere can be dangerous to nearby persons. devices have been used. Large ionization chambers, perhaps several feet in length, are useful but reflect barometric and temperature density changes in the exit gases to an undesirable degree. Moreover, such a chamber traps a portion of any radio active dust carried in the exit stream ; this raises the background in the chamber and makes it insensitive to normal variations in activity even after the dust flow has been 90

no 1

i

70 15 60

iCl"""

a 1

«°

CL

*

(-

U•<

f

20

5 10

1200 .

90c

10

15

CPR

Fio. stopped.

11. Shaft

Probably

horsepower

vs.

CPR for

20 per cent pressure loss.

the most effective means of collecting samples

of particulate

activity carried in the air stream is a sticky surface, such as an oily rag, which can be counted after removal from the stream. A more elegant device has been used experi mentally at Oak Ridge. This device consists of a moving charged wire on which charged atoms, left when the fission gases decay, collect by electrostatic attraction.

The wire, carrying its load of decaying daughters out of the gas stream and into a counting chamber,

is

At

is

is,

of the fission gases, moves slowly where the rate of disintegration of its load is measured. Presumably, the activity of the collected atoms results almost exclusively from decay of fission gases, and the device therefore, well suited to the detection of ruptured fuel elements in the reactor. The early detection of ruptured fuel elements very desirable in air-cooled reactors. Brookhaven an elaborate system used that reveals leaks in the fuel-element casings by decrease in pressure of helium that fills channels left for the purpose in the uranium slugs. At Oak Ridge a program of regular and periodic inspection followed; an inspector looks through the fuel channel in the reactor and with expe rience can detect at an early stage the swelling of the cans that presages a can rupture. is


50 O

13-118

MECHANICAL DESIGN

Operators of air-cooled reactors become fuel-element failure.

AND REACTORS

very alert to the possibility

[Sec

13

and signs of

a

Radioactivity

2.7

During the course of operation of an air-cooled pile, radio 2.71 Decontamination. active deposits will gradually build up in equipment downstream from the pile. This is particularly true if there are fuel-element ruptures or if foreign materials are acci dentally released in the interior of the reactor as sometimes happens in a reactor used for experimentation. Since the equipment itself is not rendered radioactive, removal of the deposit is all that is necessary to remove the radioactivity. Unfortunately, this is not always easy, and in some cases it is more economical to discard the equip ment than to free it of its radioactive deposits. In other cases, particularly where the activity is short lived, it is feasible simply to wait until radioactive decay has taken


place.

Radioactive deposits accumulate most rapidly on surfaces that are rough, in areas of direct air impingement, and in oily, greasy, or dirty areas. Precleaning of equip Active ment with hot water and a good detergent will save much trouble later. oily or greasy deposits are best treated first with a good grease-cutting solvent and then with hot water and a detergent. Persistent activity on acid-resistant surfaces can sometimes be effectively removed by acid washes or high-velocity acid sprays, but such procedures are frequently disappointing. 2.72 Equipment Shielding. That the reactor must be shielded is obvious, and there is also a clear possibility that radioactive deposits may require shielding of However, experience seems to show equipment downstream from the reactor also. that only the downstream air filters actually need shielding. Only two components of normal air yield 7-active products on passage through the pile, argon (A41) and While these isotopes will always be present in the exit gases in oxygen (O13, N1"). quantities sufficient to prohibit human breathing of them without dilution, the quan tity of pile exit air contained in any reasonable sized piece of equipment will not con stitute a 7 source requiring shielding. The shielding required for the filter will, in general, be fairly modest. The require*! thickness of the shield is best computed on the basis of the quantity and activity of solids that might be carried into it from the reactor in the event of fuel-element failure. 2.73 Air Activation and Disposal. Even though the pile exit air contains do solids or foreign matter, it will contain certain radioactive materials, mainly A", O", All these have undesirable biological effects if inhaled in large concen N16, and C14. Ejected at tration, and it is customary to dispose of the exit air through a chimney. height, the gases become diluted to or below tolerance before reaching human beings The isotopes mentioned have the following properties: Table

5.

Gaseous Radioisotopes Normally in Reactor Exit Air Decay Isotope

A" 0"

N» C»

energy,

Mev

Half-life

109 min 29. 4 eea 7 . sec 5.600 years

J

Beta (max)

Gamma

1.25 4.5. 2.9 10 0. 155

1.37 1.6 6.1

None

The half-lives of O10 and N1" are so short that they will have decayed to negligible concentration by the time the diluted cloud reaches the ground. The Cu, though o: long half-life, is present in only very minute quantities and is taken care of by the dilution required for the A", which therefore controls. The stack height required is a function of the neutron flux and spectrum in the reactor, the residence time of the

Sec. 13-4]

GAS-COOLED

REACTOR SYSTEMS

13-119

At Oak air in the reactor, the air flow rate, and the local meteorological conditions. Ridge a 200-ft stack is used, at Brookhaven the stack is 320 ft, and at Windscale the stack is 404 ft high. Records of wind direction, velocity, air temperature, and atmospheric radioactivity in the vicinity of an air-cooled reactor should be kept for legal reasons. Measure ments of air temperature near the stack as a function of altitude up to a few hundred feet are also usually kept, since by this means atmospheric temperature inversions may be detected and, if desired, reactor operations correspondingly controlled. 2.74. Fission-product Release. Fuel-element failure is the largest potential source of radioactivity release from an air-cooled reactor, and, while highly reliable fuel elements have been developed, rare instances of failure have occurred. Early operations at Oak Ridge encountered this problem, and in 1957 one of the plutoniumproducing reactors at Windscale became overheated during a graphite-annealing operation and liberated substantial quantities of fission products to the atmosphere. The results of such incidents have been less troublesome than might be expected, and may be put in proper perspective by comparison with the contamination resulting from atomic bomb tests, which release much larger quantities of fission products. Power Requirements

2.8


Most of the power required in an air-cooled reactor system goes into driving the fans that move the air. As a matter of general interest the following figures are quoted for the Brookhaven and Oak Ridge reactors: Table 6.

Reactor

Power Required for Air -cooled Reactors Electric- power input, kw

Reactor heat rate, kw

690 5.390

3.500 24.000

Ratio reactor electric

heat energy

5. 1 4 5

Since both these reactors were built to supply neutrons for research purposes, it should not be assumed that the ratio of heat output to power input is necessarily representative of anything but a research-type reactor. 3.

RECIRCULATING SYSTEMS

Little information has been released on the equilibrium associated with gas-cooled Mention has been made of the extensive French reactors of the recirculating type. and British interest in closed-system reactors using air and carbon dioxide. Helium was seriously considered by the Manhattan Project in its early days as the coolant However, water cooling finally appeared more for plutonium-producing reactors. feasible and was chosen. 3.1

Types of Systems

Two main types of closed-circuit gas-cooled reactor systems may be distinguished, based on the manner of using the heat produced. These types are: 1. Waste-heat boiler type, in which the reactor exit gases pass through a boiler. The steam produced can be used in part to supply the driving energy for the gascirculating blowers. 2. Gas-turbine type, diagrammed in Fig. 12. In the gas-turbine system the coolant supplied to the cooler may be water and steam may be produced in the cooler. In either case much of the heat must be removed by a secondary coolant, such as air or water.

MECHANICAL

13-120

DESIGN AND REACTORS

[SEC. 13

EE3E3

POWER TURBINE COMPRESSOR TURBINE

-;.

COOLANT

Fig.

12.

Heat from coolant may

be

as steam. 3.2

Gases Available

Many gases have been considered for closed-system gas-cooled reactors. In gen eral a suitable gas should be noncorrosive to pile and system materials, nonbazardous to handle, not strongly neutron-absorbing, not adversely affected by pile radiation, While and reasonably cheap. In addition, the gas should be an efficient coolant. cooling efficiency may be defined in various ways, one useful index is the quantity of heat energy that can be removed from the reactor by application of a given amount of It is clear that such an index is valid mechanical work in the gas pumping system. for comparison of gases only if evaluated so that the only difference between the Thus the pressure level, fuelcases being compared is the gas used as coolant. element and gas temperatures, reactor power density, and fuel-element geometry Then, for a differential length of coolant channel must be the same in the two cases. it may be shown that

_ E is proportional

=

heat removed pump work due to fuel-element flow friction

to

cJkM*

P*&

r»

P

\dQJ

= gas pressure at the point A = (fuel-element temperature) — (gas temperature), at the point T — gas temperature (absolute) at the point cp = specific heat of gas at constant pressure k = thermal conductivity of gas n ■» viscosity of gas M = molecular weight of gas In this formula dT is the rise in temperature of the gas as it flows distance dr through Leaving aside the reactor, and dQ is the quantity of heat picked up in so doing. questions of relative stability, cost, corrosivity, etc., this relation permits a com parison of gases as reactor coolants based on the same power density, pressure level, terminal temperatures, fuel-element temperature, and fuel-element geometry. In such a comparison dT/dQ by definition, independent of the gas used, as are

where

is,


recovered

Recirculating-type gas-cooled nuclear power system,

Sec. 13-4]

GAS-COOLED

REACTOR

13-121

SYSTEMS

P, and T. The only part of the formula dependent on the gas is the group Cp'kM'/it, and this may therefore be taken as the heat-transfer-effectiveness index of the gas. The expression for the heat-transfer-effectiveness index is derived from the usual heat-transfer relationships, Fanning's equation, the perfect-gas law, and the assumption that the 0.4 power of Prandtl number, (Cpn/k)0-*, is essentially the same for all gases. also A,

Table 7.

Heat -transfer Effectiveness Index of Various Gases (at 212°F and Gas

Hs

CH. NHi He

HjOt COi Air

k

Mol wt

Cp.

Btu/(°F)(lb)

2.016 16.03 17.03 4.003 18.02 44.01 29

»

3.5 0.60

1.45 1.33 1.16 1.41 0. 944 1.33 1.45

252 74 56 44 28 27 19

0 55 1.25

0.45 0. 22

0.25

1 atm)

*

yt

1.40 1.31 1.31 1.66 1.32 1.30 1.40

Another "efficiency index," similarly derived, is the product p'c,*. related to the index used above by noticing that

c,'kM* — ■ —— = U

cJM1

It

may be

k — Cv\l

is,

Since cPn/k is fairly invariant with the gas in question, it may be regarded as approxi mately constant, and the density p for perfect gases, proportional to M, whence

— k

c,»A/»

oc

cPn

p»C„>

is

is

io

Most of the gases likely to be considered are arranged in the same order of merit by these two criteria. of pump work evidently the best as regards the rat Of the gases listed, hydrogen at the other end of the scale. Except for helium all the to heat removed, and air gases listed exhibit some chemical attack of common structural materials at very high Such problems may well be of temperatures or present other materials problems. more importance in selection of coolant than the indexes discussed above in some While the radiation stabilities of CH< and NH! have not been established in cases. great detail, these gases are probably stable enough to be considered as reactor coolants. The cross sections of the elements in these gases for thermal-neutron reaction are:

N

0.33

Reaction

n,

7

C

He

barns

n, n, n, n,

y y y y

H

Cross section,

0

Element

A O


* Data taken mostly from W. H. McAdams, "Heat Transmission," 3d ed., McGraw-Hill Book Company, Inc., New York, 1954. t At low pressure. J Ratio of specific heats cf/c* at various temperatures from room temperature to 212°F. " Handbook of Chemistry and Physics," 38th ed., Chemical Rubber Publishing Company, 1956-1057.

0.0045 1.75 0.0002

0.62

MECHANICAL DESIGN

13-122

AND REACTORS

[Sec.

13

Although the thermal neutron absorption cross section of nitrogen is fairly large, it is not a major parasite of neutrons because when it is a component of a gas very little of it is present in the reactor at any one time. All the gases listed are nuclearly satisfactory coolants. In addition to the reactions indicated above an n, p reaction takes place on 016, leading to Nu. This reaction takes place at energies above 9.5 Mev and has a cross section of 0.014 millibarns. Helium, although attractive as a coolant, has the disadvantage that the ratio of its This leads to the requirement of high physical speeds or of specific heats is large. many stages in the helium compressor and turbine and a somewhat difficult machinerydesign

problem.


A preliminary

design study of a helium-cycle turbocompressor power plant has The recently been privately communicated by the American Turbine Corporation. cycle pressure is 1,000 psi at the reactor, the compressor compression ratio 2.44 (in 28 stages, divided into two groups with intercooling between groups), the turbine inlet temperature 1400°F, the compressor inlet temperature 90°F, and the plant output The plant has an over-all thermal efficiency of 40.4 per cent. 60 Mw of useful work. High-effectiveness heat exchangers are used as regenerators, intercoolers, and preA two-turbine design is used, the high-pressure turbine driving compressor coolers. This study the compressors and the low-pressure turbine supplying external work. gives emphasis to the performance advantages that result if a gas-cycle system is One of operated at high pressure, but there are also some troublesome problems. Internal thermal insu these is the design of high-pressure high-temperature ducting. lation in such ducting will probably be necessary, as will also external cooling of the duct walls. Sizable amounts of fission products in the circulating gas would consti Probably the development of fuel elements tute a distinct radiological hazard. adequate for such a reactor can be accomplished, and the general technical feasibility As yet no economic analyses have, how of such a power system may be satisfactory. ever, been announced, so that the attractiveness of this system cannot be judged. The British reactors cooled with COj will produce electricity and plutonium, and the economics of the power generation cannot be divorced from the price paid for plutonium.

REFERENCES 1. 2. 3.

"Harwell — The British Atomic Energy

Establishment," Her Majesty's Research Stationery Office, London, 1952. Hinton, Sir Christopher: "The Graphite Moderated, Gas Cooled Pile and Its Place a Power Production," Paper presented at Geneva Conference, August, 1955. "Combustion Gas Turbines," GEA-551GB, General Electric Company, Schenectady.

N.Y.,

1955.

4. Buckland, B. O., and D. C. Berkey: "Design Features of a 5,000 H.P. Gas Turbine.' Paper presented at annual meeting, American Society of Mechanical Engineers, Novem ber, 1951. Material," pp. 383 et seq., TID-5275, 5. "Research Reactors — Selected Reference United States Atomic Energy Program, Government Printing Office, Washington, D.C, 1955.

"The Reactor Handbook," vol. 2, Engineering, pp. 367 et seq., Gas Cooled Systems; pp. 991 et seq., Graphite Moderated Heterogeneous Reactors, AECD-3646, McGraw-Hill Book Company, Inc., New York, 1956. 7. Hinton, Sir Christopher: "Nuclear Reactors and Power Production," The James Clay ton Lecture, Feb. 20, 1954, Chartered Mech. Eng., 1: 235-251 (May. 1954). "Application of Gas Turbines to Nucleax Power 8. Grantz, H. E., and B. H. Neuffer: Plants," Paper presented at American Power Conference, Chicago, General Electric Company, Atomic Power Equipment Department, Schenectady, N.Y., March, 1955. 6.

U.S. Atomic Energy Commission:

13-6

KINDS OF REACTORS BY Herbert S. Isbin


1

REACTOR CLASSIFICATION

Nuclear reactors are classified in several ways depending upon the features empha sized. The following terms are used to characterize the reactor assembly: upon thermal, 1. Continuation of the neutron chain reaction dependent principally intermediate, or fast neutrons. Homogeneous 2. Fuel-moderator assemblies may be homogeneous or heterogeneous. reactors include slurry and suspension types, water boilers or aqueous-solution type (not to be confused with boiling-water reactors), and circulating aqueous solution reactors. Liquid-metal-fuel reactors may be of the homogeneous or heterogeneous type. 3. The fuel for the reactors may be natural uranium (containing 0.7 per cent U2"), enriched uranium (containing additional Us"), U236, U233, or Pu238. 4. Moderator materials include graphite, heavy water, light water, beryllium, beryl lium oxide, and hydrocarbons. 5. The type of coolant or coolant-moderator may be emphasized, such as sodium, pressurized water, and boiling water. 6. The method of confining the reactor core, such as a pool or tank type, is used in the descriptive titles of reactors. Reactor use is another basis for classification. 1. A controlled, critical, self-sustaining reactor assembly may be used as a mock-up for low-power or zero-power reactor development studies. (Only a few zero-power A reactor reactors or critical assemblies have been included in the reactor catalogue.) used as a test of a design is sometimes designated an experiment. 2. Reactors may be used for research, materials testing, or shielding studies, or as an experimental facility. 3. The production of plutonium may be a primary objective. 4. Converters produce one type of fissionable fuel and consume another, such as in Breeders consume and produce the system with Um fuel, producing Pu239 from U238. the same fissionable fuel. Examples include the use of Pu1" fuel and the production of Pu230 from U238 and the use of U233 fuel with a thorium breeding blanket to produce U233 from Th232. Breeding or conversion is termed successful if a net gain in fissionable fuel is obtained. The distinction in the terminology between breeders and converters need not be rigorous. 5. Power-producing reactors have been developed for stationary, mobile, and Submarine, aircraft, and ship reactors have been constructed or packaged units. designed. 2

NUCLEAR-REACTOR CATALOGUE

Tables 1 through 7 represent a catalogue of nuclear reactors that have been operated and of nuclear-reactor developments. The information tabulated in Tables 1 through 5 includes the following items: reactor name or classification and location ; power level (heat) and flux in neutrons per centimeter squared per second, designating average thermal flux if not otherwise noted; the description of the fuel, moderator, and fuel 13-123

13-124

MECHANICAL DESIGN

AND REACTORS

[SfiC.

13

core arrangement; reflector; shielding; over-all size; control methods; coolant, includ ing additional details on secondary systems if used; and additional notes, including start-up date. Table 6 summarizes reactor developments in the United States, and the tabu lations include reactors in operation not listed in Tables 1 through 5 and reactors under construction or authorized. A world-wide view of some of the advanced plans for reactors is summarized separately in Table 7.

Table

L

Thermal

Heterogeneous

Graphite-moderated

Natural-uranium

Reactors


CP- J. Uranium Graphite Pile (West Stands, Chicago) Hating: 200 watts (max) Fuel description: 6.2 tons natural uranium (2.060 lumps 2,-4-in. diam), 40.3 tons natural-uranium oxid? (UOi and UiOi) (various shapes, mainly 3>i-in. diam). Fuel lumps placed on 8>£-in. cubical spacing in a flattened ellipsoid of radii 309 and 388 am Reflector: 1 ft graphite Shielding: None Over-all site; 24*4 by 24M by 19 ft high; 365 tons graphite Control: 2 horizontal safety rods, 15 ft by 3>£ in. wide. steel-Cd sandwich; boron steel control rod and several Cd control strips (hand -opera ted) Coolant: None Operated Dec. 2, 1942. After initial operation, assembly dismantled Remarks: World's first reactor. (Mar. I. 1943) to form basis for CP-2 CP-2 (Palos Park, 111.) Rating: 200 watts (up to 2 kw, 3 X 10") Fuel description: 3,200 uranium metal lumps (10 tons), 2^-in. diam; 14.500 uranium oxide lumps (42 tons), 2^-in. diam; about 50 tons uranium; lumps spaced on 8>£-in. square lattice; fuel core 18 ft wide, 20 ft long, 19 ft high Reflector: 1 ft graphite Shielding: 5-ft concrete walls; 6 in. lead; 4 ft wood on top Over-all size: 30 ft wide, 32 ft long, 25}£ ft high; 472 tons graphite; total weight about 1.400 tons Control: Bronze strips oovered with Cd; 2 regulating, I shim, 3 safety Coolant: None Remarks: Operations started Mar. 20, 1943; in service until May, 1954

QLEEP,

Graphite Low Energy Experimental Pile (Harwell. England) Rating: 100 kw, 37 X 10" Fuel description: 13 tons uranium bars, 0.97-in. diam; 12 in. long, sprayed with 0.003 in. Al; 26 toes uranium oxide in Al cans 1.62-in. diam, 12 in. long, 0.01 in. thick: each can containing 6 paperwrapped UOi pellets, weighing 2.52 kg; line lattice with 7>i-in. pitch; fuel core 5.2 meters Ion* 3.05-meter radius; metal in central portions up to 1.9-meter radius Reflector: 2 ft graphite in shape of octagon Shielding: Concrete shield with 5-ft-thick walls and 4-ft roof Over-all size: 37-ft oube containing 505 long tons graphite Control: Rods containing Cd, 4 coarse control, I fine control, and 6 safety pressure, channels 1.85 in. square Coolant: Air, at least 5,000 cfm at subatmospheric Remarks: Jfc» — 0.002; max uranium oartridge temp, 60°C (1947) Oak Ridge Graphite Reactor (X-10 pile) (Oak Ridge National Laboratory, Oak Ridge. Tenn.) Rating: Increased from 1.000 to 3.500 kw, 1.4 X 10* per watt Fuel description: Uranium fuel slugs in 0.035-in. -thick 2S Al jackets, 0.060-in. end cap, Al-Si bonding. 1.1-in. diam, 4.1 in. long, 2.57 lb per slug; 39 to 54 slugs per channel; 1.248 fuel channels on 8-ia. rectangular lattice, 821 fuel channels used; 49.2 metric tons uranium; diamond-shaped channel. \$i in. square; fuel core about 20-ft oube Reflector: 2 ft graphite Shielding: 7 ft concrete Over-all size: 47 ft long, 38 ft wide, 35 ft high; 612.5 metric tons graphite Control: 4 safety (each 0.003 5Jfc), 2 shim (each 0.007 5*). 2 regulating (each 0.005 ik); back-up safetr consists of boron steel shot. Vertical rods 3>£ in* square, 8 ft long, contain layers of Cd encased in steel; horizontal steel rods 13^ in. square, 200 in. long, contain l}£ % boron. Coolant: Air, 7,300 Ib/min drawn through reactor, filtered, and discharged through 200-ft stack. Inlet temp 77°F. outlet temp 196°F Remarks: Max sheath temp 537°F, max moderator temp 292°F (1943)

BEPO, British Experimental Pile (Harwell, England) Rating: Increased from 4 to 6 Mw; central thermal flux ~ 1.4 X 10>* Fuel description: 0.9-in. diam, 12-in.-long uranium bars, encased in Al; 26 tons for criticality. 40 too* for full load; 888 channels used, 20 bars per channel, 7>£ in. between channels; cross soctaoa of empty channel 3>£ in.1; fuel core 10-ft radius, 20 ft long Reflector: About 3 ft graphite

Sec. 13-5] Table

1.

kinds of reactors Thermal Heterogeneous Graphite-moderated Reactors. ( Continued)

13-125 Natural-uranium

Shielding: 6-in. cast-iron plate, 6>£ ft (barytcs aggregate, p — 3.5) concrete on aides and 7>£ ft on top; 600 tons steel, 3.000 tons concrete Over-all size: About a 40-ft cube, containing 850 tons graphite Control: 4 horizontal and 10 vertical boron-carbide-filled hollow steel rods, 2-in. diam Coolant: Air; 180,000 cfm, filtered, and discharged from 200-ft stack; stopping power of filters down to 5 ft Remarks: Fuel elements operate at max temp of 250°C; max graphite temp I I0°C, exit-air temp 80°C; about 1.2 Mw of heat recovered in a foot-water system for heating a building (first practical utilization of heat, 1948)


Brookhaven Graphite Research Reactor (Brookhaven National Laboratory, Upton, N.Y.) Rating: 30 Mw; 4 X 10" Fuel description: Al-clad uranium metal rods placed on 8-in. centers; 60 tons uranium; 1.369 circular fuel channels circular cross sections of 36 cm1; 33 uranium fuel slugs, each 1.1-in. diam and 4 in. long, placed in an Al can 11 ft long with 6 longitudinal fins 0.6 in. high; graphite moderator in 2 right rectangular prisms. 12â by 25 by 25 ft separated by vertical opening (7 cm wide) for cooling air Reflector: 4.5 ft graphite Shielding: 6-in. iron plate, 4>£ ft heavy concrete (p —"4.5), 3-in. iron plate Over-all size: 38 by 55 by 50 ft high, 20,000 tons (including foundation), 700 tons of graphite Control: Steel horizontal rods containing 1££% boron, each 2 in. square, 25.5 ft long, entering from 2 adjacent corners in 2 arrays of 8 rods each; back-up safeties consisting of boron-steel shot to 4 tubes in graphite, trichlorobenzene to 11 tubes Coolant: Air, 300,000 cfm at aubatmospheric pressure; each half of moderator complex cooled separately, Pumping power 5 Mw; pressure drop 103 mm exit-air temp 330°F. Filters on inlet and exit air.

Hg system for channel-leak Remarks: Complicated, mechanical-electrical detection. He in fuel elements serves to indicate leak by pressure drop, provides an inert atmosphere for uranium, and improves of natural uranium fuel with MTR-type fuel plates bent into long Replacement heat transfer. troughs with 60° angle, 3 plates per element (1950)

Han ford (Hanford, Wash.) Rating: Classified Fuel description: Uranium fuel slugs 1.359-in. diam, up to 8 in. long, bonded in 2S Al can, 1.440-in. OD up to 8.70 in.; long, with Al end caps up to 0.350 in. thick. Slugs loaded in Al process tube (0.072 Process tubes placed in squarein. thick) with 2 bottom ribs, forming a 0.086-in. coolant annulus. lattice spacing Reflector: Graphite Coolant: Water Remarks: 3 Pu production reactors in 1945, and 5 additional units have been added; secondary recircu lating glycol system furnished heat to building Hanford 305 Test Reactor (Hanford, Wash.) Rating: 6 watts Fuel description: Uncooled graphite-moderated natural-uranium reactor Over-all size: About an 18- to 20-ft cube, weighing 2.000 tons Control: Control rod, shim rod, safety rod; BFi control system tested Coolant: None Remarks: Used to determine amount of reactivity that will be absorbed by a given amount of sample. Not sealed from atmosphere, .and ambient pressure affects reactivity Windecale (Scllafield, England) Rating: Classified Fuel description: Similar to BEPO, natural uranium, graphite- mode rated Reflector: Graphite Shielding: Biological concrete Bhicld, steel-plate thermal shield Control: 24 boron steel control rods, horizontally at right angles to fuel channels; shutdown Coolant: Air, inlet and exit filters, discharge through 400-ft stack Remarks: Two Pu production reactors (1950, 1951)

Colder Hall (Harwell, England) Rating: 182 Mw (heat), 39 Mw (net electrical) Fuel description: 130 tons natural uranium, 1,696 vertical fuel channels, 8-in. pitch. clad, 20 ft U in channel. diam, 3.5 ft long, magnesium-alloy Transverse fins. 21 ft high Reflector: 3 ft graphite Shielding: 7-ft concrete biological shield, 6-in. steel thermal shield Control: 60 coarse and 4 fine control rods; boron steel in stainless-steel sheath Coolant: COi coolant, 2.000 lb/sec, inlet pressure 100 psig, 284°F; exit temp 637°F

16 vertical

rods for

Fuel rods 1.15-in. Core 3 1-ft diam,

MECHANICAL DESIGN

13-126 Table 1.

AND REACTORS

[SEC.

13

Thermal Heterogeneous Graphite -moderated Natural-uranium Reactors. (Continued)

Remarks: Two reactors

for Calder Hall A (1956) optimized

2.0 in. thick

vessel 37-ft diam,

for Pu production.

Reactor pressure

0-1 (Marcoule, France) Rating: 40 Mw (heat),

5 Mw (electricity), 5.5 X 10" Fuel description: Uranium rods, 100 mm long, 26-mm diam, sheathed in magnesium in form of car tridges 3.8 meters long. Graphite Each sheath has 6 finlike protrusions, over-all diam 62 mm. channel 70-mm diam, 100 tons. 1.337 horizontal channels for each half pile (graphite stack in form of cylinder 8.84 meters long, 9.6-meters diam, divided by an 80-mm space in the middle) Reflector: 0.66 meter graphite Coolant: Air (atmospheric), exit temp 220°C. Heat exchange to water Remarks: Produces 15 kg/year Pu (1956)

Table

2.

Thermal Heterogeneous D20-moderated

Uranium Reactors


CPS

(Argonne National Laboratory, Chicago, 111.) Rating: 300 kw; 10" Fuel description: 120 (max 136) uranium metal rods 1.1-in. diam, 0.035-in. Al jacket. 6 ft long; 3 ton." natural-uranium metal; square lattice, 5?6 in. center line to center line; 6>£ tons DiO contained is 72-in.-diam, 105-in.-high Al tank Reflector: 2 ft graphite Shielding: 4 in. Pb-Cd, and 7-ft 8-in. octagonal concrete wall I 3 ft high; top shield contains Cd. I ft lead, and 4 ft wood and steel Over-all size: Octagonal 26 ft across, 14 ft high Control: Al-clad rods containing Cd metal; 2 control and 2 safety, signal-arm type; 2 vertical safety rods Coolant: 200 gpm DiO Remarks: World's first DiO reactor. May I 5, 1944. Helium sweep for dissociated DjO. Reconstructed as CP- 3' with enriched uranium alloyed with Al

CPS' (Argonne National Laboratory,

Lemont,

111.)

Rating: 300 kw; 3 to 4 X 10" Fuel description: 4.2 kg Um, about 122 fuel rods 2.16-cm diam, 1.68 meters long: highly enriched ia U!» Al alloy of 98 % Al and 2% U. 106 rods for criticality (3.8 kg Um) Reflector: See CP- 3 Shielding: See CP-3 Over-all size: See CP-3 Control: See CP-3 Coolant: See CP-3 Replaced by CP-5 in 1954 Remarks: Reconstruction of CP-3 (1950).

ZEEP (Chalk River, Canada)

Rating: 10 watts; 10s (30 watts max) Fuel description: Natural uranium metal in form of slugs, 1.285-in. diam, 6 in. long, jacketed with 25 Al "stockings," 1.295 ± 0.005-in. ID, 0.040 ± 0.003-in. wall, and 9 ft 6£{ in. long, holding 9 slo*:. total of 148 rods; square-lattice spacing of 6 in. (adjustable). 10 tons of DsO contained ia ■*« tank 6^4-ft diam. 8J.£ ft high Reflector: Graphite, 23-2 ft thick under tank, 3 ft around Shielding: Water in tanks, 3 ft thick Over-all size: About 25 by 15 ft high Control: 4 plates, 8 rods, Cd-coated stainless-steel jacket in Al, operating in vertical gap between eare and reflector

Coolant: None Remarks: 1945 ZOE (Chatillon, France) Rating: Increased from 5 to 150 kw, 2 to 3 X 1010to 10" (max) Fuel description: Originally, sintered UOs in tablets (p = 8.3 g/cm1) DjO (at 40 ± 6°C) 66-mm ID, and 180-cm effective height. 181-cm ID, 235.5 cm high. No. of fuel bars 67 (3.55 tons UOi), 18.6 cm). Uranium oxide replaced by metal rods 36.5-min diam Reflector: Graphite (diffusion length 45 cm). 15 mm DiO at bottom Shielding: Concrete 150 cm thick increased to 1.8 to 2 meters Over-all size: About 500-cm cube Control: Two Cd safety rods in center of pile; 3 Cd regulating plates Coolant: DiO circulated through external circulation system Remarks: Laplacian =» 5.8 meters"' changed to 8.0 meters"' (1948)

3 cm high stacked in Al tubes of contained in cylindrical Al set hexagonally (aides of bexagca (1.9 tons U) decreased to I 1 mm

between

tank and reflector

Sec. 13-5] Table 2.

kinds of reactors

Thermal Heterogeneous D20 -moderated Uranium Reactors.

13-127 (Continued)

Heavy-water Research Reactor (USSR) Rating: 500 kw Fuel description: Uranium-fuel rods suspended in DtO contained in 175-cm-diam tank, 195 cm high. Critical experiments performed included use of 2.2 and 2.8-cm-diam rods with a 0.1-cm Al envelope; square-lattice spacing from 63 to 162.6 cm; No. of rods from 86 to 292; critical DjO level from 120 to 181.6 cm; rod lengths about 160 cm Reflector: 100-cm-thick graphite reflector on sides and bottom Shielding: Concrete side shield 2.5 meters thick Over-all size: About 9-meter diam by 8 meters high Control: 4 Cd control rods Coolant: DsO circulated Remarks: He atmosphere above DiO (1949)

F-2 (Saclay, France) Rating: 1,500 kw, 7 X 10" (max) Fuel description: 136 uranium rods 2.6-cm diam, 215 cm long, triangular lattice with rod spacing of 15.1 cm. Fuel element consists of 4 concentric Al cylinders with innermost one as protective sheath 1 mm thick for U rod, gas-coolant space of 4 mm between innermost sheath and second Al cylinder (0.5 mm thick), gas insulating space of I mm between second and third Al cylinder (0.5 mm thick), and outer gas-coolant space 8.5 mm thick between third and fourth Al cylinder (1.5 mm thick). Total diam 6 cm. About 3 tons uranium. Fuel immersed in Al tank 258 cm high and 201 cm in diam, partly filled with DiO (about 6.3 tons). With DiO at 1794 mm, critical height (extrapolated) He = 224 ± 3 cm, critical radius (extrapolated) 238 cm', Re — 132 ± 2 cm; Af* B* (5.30 ± 0.15) X 10"* cm"" Reflector: Graphite 90-cm-thick outside tank and on sides and bottom; DiO on bottom I cm at periphery and 3 cm at center Shielding: Cast-iron thermal shield about 20 cm thick, concrete shield about 210 cm Over-all size: About an 8-meter cube Control: Cd plate moving between tank and reflector, 2 Cd rods in tank Coolant: Originally, recirculation of Na at 10 atm, flows down outer gas coolant space and up inner coolant space. System now uses COi under 7 atm Remarks: Modifications include cooling of DiO and alteration of the fuel rods to achieve higher powers — 2,000 about kw (1952)

-


-

JEEP

(Kjeller, Norway) Rating: Designed for 100 kw, increased to 350 kw; 10" Fuel description: 2,300-kg uranium slugs in Al tubes; 35.5 kg per rod; rods 25.4-mm diam, 300 mm long, placed on 180-mm centers. Fuel core 1.9 meters long, 7 tons DiO contained in 2-meter-diam tank. 65 to 76 fuel elements

Reflector: Graphite, 700 mm thick Shielding: Octagonal concrete shield, 2 meters thick on sides Over-all size: About 7J-£ by 7)4 by 5 meters high Control: 4 Cd plates 1,300 mm long, 350 mm wide, 1.7 mm thick, held between Al plates, placed between tank wall and reflector Coolant: 4 liters /sec of DjO; inlet temp 20°C, outlet temp 40°C Remarks: 1951 (Plan to increase power to about 800 kw)

NRX

(Chalk River, Canada)

Rating: 30 Mw, 6.8 X 10", rebuilt reactor up to 40 Mw Fuel description: 176 uranium rods, Al clad, surrounded by

two concentric tubes. Uranium rod diam Outer Al sheath ID 1.66 in.; thickness 0.040 in., uranium 1.36 in.; Al sheath thickness 0.079 in. assemblies hang vertically. DaO contained in calandria I0>£ ft high, 8^4-ft diam. Core contains about 10.5 tons natural uranium and about 18 tons DtO. Each fuel rod is 10 ft long, coolant annulus reduced from 0.100 to 0.071 in. Hexagonal lattice, 6**4 fl in., 199 positions in 22 rings Shielding: Graphite, 9 in. plus 2 ft (total 58 tons) Shielding: 7 to 8 ft concrete, cast-iron thermal shield (two 6-in. shields) air-cooled, 3 water-cooled shields on top and 4 on bottom Over-all size: 34-ft diam, 34 ft high Control: Varying DiO level, Cd and boron rods; 18 shutoff rods of boron-carbide powder in steel tubes, 1 control of Cd slugs in steel tube CToolant: HiO flows downward in inner annulus (air flows upward in outer annulus), 40°C temp rise; 95 % of heat removed by water DsO circulated to external heat exchanger; air cooling of graphite. flow in fuel channel. Max reflector temp 300°F 1952, December, following accident and severe con Remarks: Helium purge system. Shut down in tamination; operation restored in February, 1954. Plutonium and U2" (thorium oxy carbonate ring pellets loaded in annular in reflector) are produced (1947)

(Argonne National Laboratory, Chicago, 111.) I X 10" virgin). 1.000 kw, 3 X 10" (1 X 10" epithermal,

Itating:

Design is 4 Mw elements are curved sandwich plates, 0.05 cm fuel Al-U "'-alloy clad with 0.05 cm 12 plates mounted in a 3- by 3- by 24-in. fuel box. Alternate fuel element: concentric Normal assembly has 12 fuel boxes and 16 required at 4 Mw. DsO contained in Al tank

p'uel description: Fuel 72S-A1;

tubes.

MECHANICAL DESIGN

13-128 Table 2.

AND

Thermal Heterogeneous D20-moderated

REACTORS

[SEC.

Uranium Reactors.

13

(Continued,

Assembly contains about 1.6*0 g Fuel core about 2-ft diam by 2 ft high. 6-ft diam, 9>£ ft high. U»»and 7 tons DiO Over-all 10 ft high and 10-ft diam Reflector: 2 ft DsO and 2 ft graphite around and below tank. steel, 3M in- lead, 4 ft 8 in. limonite-iron concrete (p — 4.4 g/cm1) Shielding: >£-in. boral, H Over-all size: Octagonal, 20 ft across and 13>£ ft high (total weight 600 tons) Control: 4 shim-safety rods (Al-clad Cd, I by 5>i* by 60 in.), signal-arm type, operating between parallel rows of fuel assemblies; one regulating rod moving up and down outside the fuel assemblies; quickopening valve to drain DiO Coolant: About 1.200 gpm DsO flows up through fuel assemblies [about I meter /sec in central sections, DiO circulated, through max heat flux about 4.5 cal/(cm*)(sec)] and down in tank with a 5°C rise. Chilled water system for limiting external HiO heat exchanger and cooled from 126 to 118°F. temp rise and to increase reactivity. Thermal shield cooled by HsO. (Total shield heat load including graphite about 2 to 3 % of total.) Helium purge system, graphite reflector cooled with Temp rise in graphite 200°C. helium. Remarks: Reactor housed in an almost leakproof building (1954)


R-l, First

Swedish

Research Reactor (Stockholm,

Sweden)

Rating: 300-kw design, 600-kw operation, 3 X 10" (max), increased to 8.7 X 10" (max) Fuel description: 126 uranium fuel rods, arranged in a hexagonal lattice, spacing 145 mm, 29-mm diam. canned in reflectal (very pure Al alloyed with 0.5% Mg) by forcing through a die, reducing wall thickness of tube from 1.25 to 1.10 mm. DiO contained in 1.85-meter-diam and 2.54-meter-biga reflectal tank Reflector: 900-mm-thick graphite surrounding bottom and sides of tank Shielding: Biological shield of poured concrete with iron ore, p — 3.7, 1.8 meters thick, lined with s Cd-Al sandwich Over-all sise: About 1\$ by 7>£ by 6.8 meters Control: 2 security rods equivalent to about 2 % reactivity, 2 regulating plates between reactor tank and graphite reflector control about 0.6% reactivity 1,000 liters /min to air-cooled heat exchanger, about 0.001 of circulatios Coolant: DsO recirculated, passes through mixed-bed ion exchanger. Air used for moderator cooling, passed upward between reflector and tank and downward between graphite and shield Helium atmosphere above DsO and circulated through paQaRemarks: Reactor built underground. diura catalyst recombination unit (1954). (Plan to increase power level up to 1,000 kw witi. fourfold increase in flux) Dimple, Deuterium-moderated Pile Low Energy (Harwell, England) Fuel description: DtO contained in Al tank, 8^-ft diam, 10 ft high.

Variety of fuel elements

can be

used

Reflector: 3 ft graphite on sides and Shielding: 2 ft concrete on sides and Over-all sixe: About a 20-ft cube 2 banks Control: DiO level varied. Coolant: None Remarks: Britain's first DsO reactor

base top of Cd safety

tubes,

l>^-in. diam by 18 in. long

(1954)

DIDO (Harwell, England)

Rating: 10.000 kw, 10" Fuel description: 90% enriched uranium, DsO-cooled boxes, forming a cylinder 60 cm long, equivalent Reflector: 60 cm graphite Shielding:

Steel tank,

biological

and -moderated. 86-cm diam

U-Al alloy

plates arranged

concrete shield

Coolant: Forced upward flow of DiO Remarks: Pluto is also a 10-Mw high-flux DsO reactor (1956)

NRU (Chalk River,

Canada)

Rating: 200 Mw, 3 X 10" 200 fuel rods. and -cooled. Fuel description: Natural uranium, DsO-moderated I l-ft 6-in. diam Reflector: 12 in. HsO outside reactor tank Shielding: Steel thermal shield, 8-ft heavy concrete biological shield Coolant: DsO Remarks: (1957)

Reactor vessel, AL

CP-6, Savannah River Reactors (Aiken, S.C.) Rating: Production reactors Fuel description: DiO moderator, natural-uranium fuel. Fuel slugs 1-in. OD by 6 in. long, in I.OS-ia 4 coolant channel tubes joined to form qua trefoil process tubes. OD Al can. Tubes contain 3 ni* to center slugs

Coolant: DiO Remarks: Five reactors completed

by 1955

kinds of reactors

Sec. 13-5] Table 3.

Thermal Homogeneous H20-moderated

Enriched-uranium

13-129 Reactors

LOPO, Low-power Water Boiler (Los Alamos, N.M.) Rating: o watt Fuel description: Enriched-uranium sulfate solution, 580 g Um. 3.378 g U,M, 534 g S, 14,068 g O, 1.573g H; p *> 1.348 g/cra' at 39°C. Solution contained in 1-ft-diam Ms-in. -thick stainless -steel sphere ( 15 liters) Reflector: 3- by 3- by 6-in. bricks of BeO, p — 2.7; graphite on bottom Shielding: None Over-all size: About 3 ft square by 4 ft high Control: Cd cylinder, ^-in.-diam, 34 in. long, Cd safety curtain Coolant: Water. Max temp 39°C Remarks: World's first water-boiler reactor, 1944; replaced by HYPO

HYPO, High-power Water Boiler (Los Alamos, N.M.) Rating: 6 kw. 3 X 10" Fuel description: Enriched-uranium nitrate solution, 896.6 g TJ"*, 5.341 g U"8, 731 g N, 13,780 g O. 1,312 g H; 13.65 liters UOi(NOi)f6HiO in HiO, p 1.615, contained in l-ft-diam Me-in.-tbick stainless-steel sphere Reflector: 24 by 24 by 27 in. BeO surrounded by graphite to form a 60- by 48- by 60-in. rectangular parallelepiped Shielding: 4 in. lead. Ha in. Cd, 5 ft concrete Control: Rods containing Cd, 1 shim, 2 control, 1 safety Coolant: 50 gph water through 6- turn coil, 3-2-in. ID, 157 in. long. Max solution temp 185°F Remarks: About 50 cm*/scc of air used to sweep out Hi and Ot from decomposition of HiO. Replaced by SUPO (1944)


-

SUPO, Superpower Water Boiler (Los Alamos, N.M.) Rating: 45 kw, 1.7 X 10" max (estimated max intermediate flux 2.8 X 10"; fast flux, 1.9 X 10") Fuel description: Enriched-uranium nitrate solution, 88.7 ?c U as Um, 777 g UJ" for critical mass, 870 g U"5 used; p — 1.10, 12. 700-cm3 solution contained in l-ft-diam stainless-steel sphere Reflector: About a 55-in. cube of graplute Shielding: H in. of BiC in paraffin; 2 in. steel; 4 in. lead; 5 ft concrete Over-all size: About 15 by 15 by 1I ft Control: See HYPO, 2 additional control rods move in reactor core in reentrant thimbles Coolant: Water circulated through three 20-ft-long >i-in.-OD, M«-in.-ID stainless-steel tubes Remarks: 100 litere/min air circulation, recombiner for Hi and Oi; rate of radiolytic hydrogen evolution about 0.55 mole/kwhr (1951) WBN3, Water-boiler Neutron Source (North American Aviation, Inc., Downey, Calif.) Rating: 1 watt, 5 X 10* Fuel description: 1.5 lb uranium fuel in form of U*" enriched (90%) uranyl nitrate in HtO solution; solution contained in a t-ft-diam stainless-steel sphere He in. thick Reflector: Graphite, 5-ft diam and 6 ft high Shielding: 2 ft of concrete blocks Control: Two safety rods, coarse control, fine control, move in reflector Coolant: None 1.561 (leakage 0.360) (1951) Remarks: Calculated fc«

-

Livermore Water Boiler (Livermore, Calif.) Rating: 500 watte, 2.4 X 10" max thermal flux Fuel description: 14.524 liters HiO solution of UO18O4 containing 694.2 g U"'; 798 g- moles hydrogen; 420. oxygen; 5.64, sulfur; and 2.95, uranium. Solution contained in stainless-steel sphere, 12^-in.

OD,

0.06-in.

wall thickness

Reflector: Right cylinder of graphite, 5-ft diam by 5 ft high. (Brookhaven graphite with thermal diffusion length of 51.6 cm) Shielding: Graphite in cylindrical steel tank, 0.030-in. Cd shot and 5-tn. lead, 3-ft concrete blocks Over-all size: 24 by 12 by 12 ft high Control: 2 safety and 2 control rods. Rods are flattened stainless-steel tubing packed with 1 lb (control rods) or 2.2 lb (safety rods) boron carbide Coolant: Cooling coil in sphere, 6-turn helix of ^fa-in.-ID tubing. 160 in. long, distilled water used as coolant, and refrigeration unit used to reduce solution temperature. Avg fuel temp 65°F

Remarks:

NCSC,

Closed

gas-handling

system

(1953)

North Carolina State College Reactor (North Carolina State College. Raleigh, N.C.)

Rating: 10 kw, 5 X 10" Fuel description: 14 liters HiO solution

of UOiSO* containing 848 g U"6 (790 g for criticality), of 90% iaotopic enrichment, p = 1.08 g/cm1; solution contained in stainless-steel cylinder, Ms in. thick, in. high, 10^-in. diam Reflector: 20 in. graphite; 105 ft1, 5.4 tons; p = 1.65 g/cm» Shielding: 6 ft concrete, barytes coarse aggregate and colemanite fine aggregate, p — 3.4 g/cm1; 4 to 6 in. lead around graphite Dver-all size: Octagon, 17 ft across, 12 ft high

II

MECHANICAL DESIGN

13-130 Table

3.

AND REACTORS

Thermal Homogeneous H20-moderated Reactors. (Continued)

[SEC.

13

Enriched-uranium

Control: 2 control rods, stainless-steel tubes of 2.5 g/cm< sintered B*C powder, in reentrant sheaths 2 shim rods, 4-in.-wide Cd strips, periphery of reactor cylinder Coolant: HiO (refrigerated city water), 1-gpm flow through each of 4 helical coils of ii-in. stainless-steel tubing,

7-ft immersion

length

Remarks: Max reactor solution temp 80°C (1953). ARR,

Armour Research Reactor

Chicago,

111.)

stainless

steel

Reactor undergoing

(Armour Research

revision

Foundation of Illinois Institute of Technology.

Rating: 50 kw. Max thermal 2.2 X 107 per watt Fuel description: 1.430 g uranium (88.14% U"*) as uranyl sulfate solution in a 15-liter spherical core o: Reflector: 24 in. graphite or greater Shielding: Dense concrete 5 ft, lead 4 in. max, bismuth 6 in. max Over-all size: 18 by 16 by II ft Control: 4 identical safety and regulating rods, boron carbide in stainless steel, %-in. diam, 12-in. travel, 2% 6k each

Coolant: Distilled water, flow through stainless-steel coils, 10 sections in parallal, J-i-in. OD, hii-iz. wall, inlet temp 80°F, outlet temp I09°F. (Avg fuel temp 176°F) Remarks: Triple barrier for containment of radioactive gases: closed-cycle gas-handling system sur rounded by an all-metallic envelope and gastight reactor room (1956) Experiment Water Boiler (Atomics International, Canoga Park, Calif.) Rating: 50 kw, max 9 X 10" Fuel description: Uranium ( >90% Um) 1,680 g (critical mass) as uranyl sulfate in 12.3-in.

KEWBt Kinetic


steel sphere

Reflector: Graphite stacked to form 56-in. cube Shielding: 1 ft concrete, 6 ft earth Over-all size: 15 by 26 by 10 ft high Control: 4 regulator and safety rods, %-in.-diam by 16 in., boron carbide

jacketed

ID stainles»-

in stainless ated

1.7% 3Jt/rod

Coolant: Water-coolant flow through 90-ft stainless-steel J-i-in.-OD, by 0.028-in. wall coil in reactor core. Inlet temp 85°F, outlet temp 109°F (avg fuel temp I76°F) Remarks: Similar to ARF reactor but contains additional mechanism to introduce reactivity into cere in times from 0.040 to 5 sec (1956)

HRE-I, Homogeneous Reactor Experiment (Oak Ridge National Laboratory, Oak Ridge. Tenn.) Rating: 1.000 kw Fuel description: Enriched uranyl sulfate (enrichment > 90%) dissolved in distilled water (35 g/kt Boiling ce HaO) ; solution contained in an I S-in.-diam 347 stainless-steel sphere with ?i g-in. walls. fuel solution suppressed by maintaining a pressure of 1,000 psi (in pressurizer, solution heated v: Max fuel temp 482°F 545°F). Reflector: 10-in. layer of DaO, pressurized with helium to within ± 100 psi of fuel solution, temp rege lated to 350°F

Shielding: Reflector and core contained in an outer pressure vessel of forged steel, 39-in. ID with 3-ia-thick wall; 7-ft-thick concrete walls (barium sulfate ore as aggregate to increase p to 3.5) Control: Reactor was self -stabilizing as a result of its large negative temp coefficient (and mechanic safety devices found to be unnecessary). Safety measures included 2 safety plates, fall in 0.91 sec and are completely effective within 0.2 sec, reduce reactor temp from 250 to 243°C ; reflector may N? dumped within 6^£ sec, drops operating temp to 160°C; dilution of fuel requires 15 nun aed e normal shutdown procedure; reduction in power demand; and draining fuel solution Coolant: Reactor solution circulation 100 gpm; outlet temp 482°F and cooled to 410° in a C-tube best exchanger, generating 3,000 lb/hr steam in the shell side at 200 psia. Generated 150 kw of elec tricity. Canned-rotor centrifugal pump for circulating fuel solution. About 50 kw removed r~ recirculating DsO reflector through boiler-feed-water preheater Remarks: ( 1952) Gas evolution of 2Hj + Os at rate of 10 cfra (STP) and 20 cm »/day of nanon-proc*^ gases. About 40 kw liberated in flame recombiner, followed by a catalytic recombiner. Raitkactive gases adsorbed in activated carbon beds. Catalytic recombiner for Di and Oi gaê1 Radioactivity of fuel solution about 30 curies/cm.* Reactor dismantled in 1954

LAPRE-l, Los

I

Alamos Power Reactor Experiment (Los Alamos, N.M.) Rating: 2 Mw, fast 9 X IQ18, max 1.4 X 10"; thermal 1.7 X 10»», max 2.6 X 10»» Fuel description: Uranyl phosphate solution in water, 0.55 Af UOi and 7.5 M HiPO*. —90^ C5* 4.2 kg Ul" in core, 9 kg total. Core cylinder, 15-in. diam, 16 in. tall Reflector:

3 in. steel plus water

Shielding: 4 ft water plus 6 in. lead plus 5 ft water Over-all size: 22 by 22 by 22 ft Control: Shim: solution concentration; 4 regulating rods, 0.535-in. diam 16 in. long B,e cylinder. I-0-7V Safety: control rods, large negative temperature coeScwe^ in. diam 22 in. long B-10 cylinder. emergency rupture disk and tank

KINDS OF REACTORS

Sec. 13-5] Table 3.

13-131

Thermal Homogeneous H20 -moderated Enriched-uranium Reactors. (Continued)

Coolant: Forced fuel circulation through heat exchangers; 4.000 pai, 12 gpm feed-water pumps. Fuel temp 800°F (avg). Water-coolant inlet temp 70°F. outlet temp 800- F. Coolant tubes: 22 spiral coils, 34 ft long, Ms-in. OD, H-»n. ID Remarks: Low phosphate concentration, high- vapor- pressure solution. All components in contact with high-temperature fuel solution utilize- gold or platinum claddings ( 1956)

II

LAPRE-II,

Los Alamos Reactor Experiment (Los Alamos, N.M.) Rating: 0.8 to I Mw, fast 6 X 10". 9.5 X 10»a max; thermal 1.2 X 10", 1.8 X 10" max Fuel description: Uranyl phosphate solution in water, 0.3 M UOi and 17.5 M HsP04. ^-90% Um, -2.8 kg in core, ~4 kg total. Core cylinder, 15-in. diam, 24 in. tall Reflector: 11 in. graphite Shielding: 20 ft dirt Over-all size: 42-in.-ID, 20-ft-long steel cylinder Bet vertically in ground Safety: large negative temp coefficient auxiliary Control: Shim: solution concentration, graphite sleeve. reservoir tank Coolant: Natural convection of fuel solution, avg temp 800°F. Feed-water pumps at 150 psi. 6 gpm. Coolant tubes: 44 Bpiral coils, 21 ft long, Ms-in. OD, H-in. ID, 0.006-in. gold cladding low-vapor-pressure Remarks: High phosphate concentration, solution. To be used for long-term test of feasibility of uranyl phosphate solutions and use of gold-cladding corrosion protection (1956 scheduled)

Table

4.

Thermal Heterogeneous Enriched-uranium

Reactors


Pool and Tank Types

Bulk Shielding Facility (Oak Ridge National Laboratory, Oak Ridge, Tcnn.) Rating: 1,000 kw max, 1 X 10" (2 X 10" max available thermal flux. 1 X 10" max fast) 18 curved platen, each 3 in. wide, 24 in. long, and 0.06 in. thick, comprise a Fuel description: MTR type. 0.1 17-in. water space between plates. fuel box. Enriched uranium (>90% Um). About 140 g Al (including structural members) to HiO ratio is 0.7. Fuel boxes held in an Al UM* per fuel box. grid 5 in. deep. Critical mass about 3 kg Um in fuel core 12 by 12 by 24 in.; with allowances for other factors including beam holes, burnup, 3.5 kg; with BeO reflector, 2.4 kg U»" in fuel core 9 by 12 by 24 in.

Reflector: 10 cm BeO on 4 sides of active lattice or HtO Shielding: 16 ft of water above reactor, water and high-density concrete below reactor Over-all size: Pool 40 ft long, 20 ft wide, and 20 ft deep Control: Three \Vi- by 2>£-in. vertical B«C shim (and safety) rods. 1 stainless-steel regulator Coolant: HtO, convective flow Remarks: Addition of 50 ppm Na:CrO< to inhibit corrosion of Al ( 1950)

l>i- by 2>£-in.

LITR, Low-intensity Test Reactor (Oak Ridge National Laboratory, Oak Ridge, Tenn.) Rating: Increased from 500 to 3.000 kw; 2.9 X 10" in lattice, 4 X 10" in reflector, max Fuel description: 3.4 kg Um. MTR fuel assemblies (16 plates per fuel assembly). Core up to 15 by 27 by 24H in. long Reflector: Loosely stacked Be blocks, thickness 8 in. Shielding: Un mortared concrete blocks, except for outermost 1-ft layer. Min thickness 11.5 ft Over-all size: 26 by 26 by 22 ft high Control: 3 shim-safety rods containing a cadmium section and a fuel section, 1 control rod in reflector Hydroxide Water-air secondary heat exchange to remove heat. Coolant: HiO recirculation system. precipitations in system (Al and carbon-steel piping) removed by use of cellulose filters, pH of water 6.5; bypass line to demineralizer maintained between 5.5 and Remarks: Installation served as mock-up for MTR; converted to a training and research reactor (1950) MTR. Materials Testing Reactor (NRTS, Arco, Idaho) Rating: 40 Mw, 5.5 X 10" max, 3 X 10" max fast flux Fuel description: Fuel assembly is a 3- by 3- by 24-in. box containing 19 curved, vertical fuel plates (see Fig. 39 of Sec. 12-1). Ua" per plate increased from 140 to 200 g. Lattice configuration can be

varied from a 3 by 9 (27 fuel assemblies) to 5 by 9 array. About 4.5 kg IT1". Core held between 2 Al grids and contained in an At tank 54-in. diam with side extension to form a well 30 ft deep Al: HaO volume ratio 0.73 filled with water. Reflector: Primary reflector, Be 3 ft high, 54-in. diam in water zone; secondary reflectors, graphite in two zones outside water tank; pebble zone of 700.000 l-in.-diam balls to form 7-ft 4-in. square by 9 ft high, outer stack of graphite to 12 by 14 ft by 9 ft 4 in. high Shielding: Two 4-in. layers of steel thermal shield, 9 ft barytes concrete, wnter 15 ft above core and 5 ft

Over-all

size: 32H by 34 by 40?$ ft high

MECHANICAL DESIGN

13-132 Table

4.

AND REACTORS


Reactors.

[SEC.

13

(Continued)

Control: 7 vertical shim-safety rods, combinations of Al-Cd box and fuel element rods or beryllium rods. 2 vertical Al-Cd regulating rods in Be reflector Coolant: HsO recirculated, cooled by flash vaporization, 22,000 gpm, enters at I I5°F at top at 64 psi and About 1.8 per cent of heat generated in graphite and thermal shield. leaves at bottom at 128°F. air-cooled; 4% heat generated in Be reflector Fuel plates curved Remarks: Reactor used for study of materials exposed to intense radiations (1952). laterally on a 5.5-in. radius to strengthen them and to define direction of thermal strain OWR, Omega West Reactor (Los Alamos, N.M.) Rating: I to 5 Mw. Per Mw, fast 10"; thermal, I011 Max of 54 fuel positions. 19 by Fuel description: MTR-type fuel elements, >90% U"», ~3 kg U"'. Core at bottom of water-filled stainless-steel tank 8-ft diam, 25 ft high 28>£ by 24>£ in. -thick beryllium reflector Reflector: Water plus provisions for Shielding: Sides, 3 ft water plus 5 ft heavy concrete (280 lb /ft1); 17 ft water at top Over-all size: Modified octagon, 18 ft face to face Control: Eight 6- by %-in. vertical boron rods Max temp (estimated) sheath I69°F; moderator Coolant: Forced recirculation of water, 3.400 gpm. temp I59°F Remarks: 1956

RMF,

Reactivity Measurement Facility (Reactor Testing Station, Arco, Idaho) Rating: 100 watts max, 5 watts normally Fuel description: 30 MTR fuel assemblies installed in MTR canal Reflector: Water, ft Shielding: 14 ft water Over-all size: 8 by 10 by 18 ft deep Control: 1 rotary 6- by 25-in. boral paddle, 1 rotary 2 concentric air-filled cylinders with Cd quadrant*; safety boral and Cd curtains Coolant: Temp rise negligible Remarks: Cold clean reactor used for precise measurements in reactivity effects (1955)


l\i

3

Geneva Reactor Exhibit (Geneva, Switzerland) Rating: 10 (nominal) to 100 kw. 10" (at center) Fuel description: About 3.6 kgUl« (enriched to 20%) in 23 MTR fuel assemblies (18 plate* per assembly! (swimming-pool- type reactor); Al:HiO volume ratio 0.65; active lattice 15 by 15 by 24 in.

Reflector: HiO Shielding: HsO plus earth water Over-all size: Pool size, 10 ft in diam, 21 ft deep, 13,000-gal capacity of demineralized Control: 3 safety and control rods, boron -carbide, each about 2% 6k Coolant: HzO, natural convection Remarks: Built by United States for the International Conference on the Peaceful Uses of Aiox.; Energy, 1955. Automatic control from start to full power. Reactor sold to Switzerland, dis mantled (1955), and moved to Wurlingen in 1956 Experimental Nuclear Reactor (RFT) (USSR) Rating: 300 kw, 2X10* Fuel description: Active core approximated by a cylinder 40-cm diam, 50 cm high; core consists of 51 units with 24 units each containing 16 fuel elements (remaining units for control roils) ; eylindnc&J fuel elements, 9-mm OD, 50 cm high, square lattice with ~!8-mm spacing; fuel elements conta-a HaO moderator, reflector, and coolint. Core in SftO-asa10% enriched U*"; 3.5 kg U"» in core. Al and Alafie* diam water-filled Al tank, and fuel units fixed at top and bottom in guide grids. used as structural materials in active core and for canning fuel elements

Reflector: Water Shielding: Cast iron and water used as shielding Control: 3 borax-carbide safety rods, 1 steel rod M ix fuel surface temp 7^ Coolant: 240 meters'/hr water flow through core, enter at 30°C, 1°C rise. 52 unitRemarks: A 2-Mw thermal power nuclear reactor for research purposes h.is been designed. fuel elements 10-mm OD; fuel loading 4.5 kg U"*; water flow 900 cubic meters /hr; 8 boron e*K i> rods (3 safety,

5 shim) and

1 steel regulating

rod

Pennsylvania State University Reactor (University Park, Pa.) Rating: 100 kw. 0.7 X 10" Fuel description : MTR-type fuel elements with 10 plates per element. Plates 14.1% 90% U"*), clad with 0.02 in. Al, 15- by 12- by 24-in. core Reflector: Water, graphite (in Al cms) Shielding: Water, 19 ft water, or water, concrete, earth Over-all size: Pool 15 by 30 by 21 ft Control: 3 boron-carbide-filled shim and safety rods; one stainless -steel regulator Coolant: Natural convection of water Remarks: Patterned after the BSF (1955)

Al-U alloy (ab««

kinds of reactors

Sec. 13-5] Table

4.


13-133 Reactors.

(Continued)

NRR, Naval Research Laboratory Reactor (Naval Research Laboratory, Washington, D.C.) Rating: 100 to 1.000 kw 18-plate MTR-type fuel elements. Fuel description: Two basic core configurations, one for pool posi tion and for the main shield position. Core up to 21 by 21 by 24 in. high Reflector: Several-foot water reflector on all Bides or graphite-reflected on 3 vertical faces and waterreflected on remaining sides Boral, lead, barytes concrete Shielding: 16 ft water for vertical shield plus lead in top of fuel elements. used in the main shield position Over-all size: Over-all pool and shield, 49 by 32 by 28>j ft high Control: 3 boron-lead vertical shim rods, I stainless -steel vertical regulator Coolant: Water, natural convection Remarks: 1956


Ford Nuclear Reactor (University of Michigan, Ann Arbor, Mich.) Rating: 1 Mw, 1 X I0» Fuel description: BSF fuel elements, 18 curved plates per box. Each plate contains 14. 1 % U-Al alloy 2.6 to 3 kg U"». Core up to 24 by 30 by 24 in. (140 g U»»), clad with 0.020 in. Al on each side. Reflector: Graphite canned in Al and /or water Shielding: Top by 20 ft water, sides by 4 ft water plus 6H ft high-denBity barytes concrete Over-all size: Pool 10 by 27 by 27 ft Control: Three JfJ- by 2>^-in. vertical, B«C-Pb shim rods, one "H-by 2,1^-in. vertical, stain less -steel regulator rod Coolant: Either forced or natural convection; de mineralized water, max pool temp 90°F Remarks: 1956 Battelle Research Reactor (West Jefferson, Ohio) Rating: 1.000 kw, max fast 1.8 X 10", max thermal 1.4 X I0» 2.8 kg Um in uranium-aluminum alloy, 0.020-in. Al cladding, fuel-plate assemblies Fuel description: (MTR type) 3.160 by 2.996 by 24 in., core up to 18 by 27 by 24 in. Reflector: HtO or 3 in. graphite Shielding: Lead, water, barytes concrete Control: 4 boron carbide shim-safety rods, 1 stainless-steel regulator Coolant: Demineralizcd water, 900 gpm; inlet temp 75°F, outlet 85°F; moderator temp 75 to 85°F; max sheath

temp

147°F

Remarks: 1956 Trombay Pool Reactor (Trombay Island, India)

Rating: I Mw Fuel description:

25 to 30

MTR-type

Reflector: Water, graphite thermal Shielding: Water, 8.5 ft concrete Remarks: 1956

LIDO

fuel elements. 2-ft cube core column at one end of pool

(Harwell, England)

Rating: 100 kw Fuel description: Swimming-pool reactor, enriched-uranium Over-all size: Pool 28 by 8 by 24 ft deep Remarks: 1956

plate-type

fuel elements

Pressurized Water

SlW

STR), Submarine Thermal Reactor (Mark I, prototype, National Reactor Testing {Formerly Station, Aroo. Idaho; Mark II, S2W, USS Nautilus) Fuel description: Zirconium-sheathed enriched uranium Coolant: Pressurized HtO Remarks: World's first mobile reactor unit; first major use of zirconium (1953). (S2W core replaced 1957)

Boiling

Water

Borax-I, Rating:

I

Boiling-reactor Experiment (BER-I) (NRTS-ANL, Arco, Idaho) 1.200 kw, 10" description: 26 to 30 MTR-type fuel elements consisting of 18 plates per element, plate spacing 0.1 77 in., U"1 content per element 138.6 g, fuel-bearing lengths 23.6 in. high Reflector: 16 in. water. Reactor tank 4-ft diam, 13 ft high. Water to within 3 to 4>£ ft from top Shielding: Reactor tank contained in shield tank, 10-ft diam, partly sunk in the ground 5 Cd control rods in the form of plates C'oolant: Water, boiling condition from atmospheric to 130 psig Renutrks: Operated successfully under large power excursions but was destroyed in 1954 in a "runaway" simulated test with a power rise over 10' kw in 0.1 sec (1953)

Fuel

Control:

MECHANICAL DESIGN

13-134 Table 4.

AND REACTORS


Reactors.

[SEC. (Continued)

Borax-II (BER-II) (NRTS-ANL, Arco, Idaho) Rating: 6.400 kw, 3.5 X 10" 10 fuel plates per element, 0.324-in. plate spacing. Fuel description: Similar to Borax-I.

Um

loadings UBed, 93.4 and 157.3 g per element. Reflector: 7 in. water Shielding: 3 to 7 ft concrete Control: 1 cross. 4 blades, Cd Coolant: Max pressure rating of vessel increased to 300 psig Remarks: 1954 of two

Core,

13

Combination

65.5 by 70.4 by 61 cm high

Borax-III (BER-II1) (NRTS-ANL, Arco, Idaho)

Rating: 12.000 kw (heat): 2.300 kw (electrical), 3 X I0» Fuel description: Box-type fuel elements, with 24 plates, 60 mils thick, approx one-half as wide as Borax-II with 0.264-in. water channels, 1.27 in. wide, 26.8 in. long. Fuel boxes have upper and Core 54-cm radius, 65.5 cm lower extensions. TJa" content varied from 137 to 233 g per element. high

Reflector: 5 in. water Shielding: 3 to 7 ft concrete Control: 4 blade rods, H in. thick and about 1 ft wide; I cross; boral and hafnium Coolant: Water, 4I9°F, 300 psig; natural-convection flow Remarks: Prototype boiling- water-reactor power plant (1955)


8PBRT-I,

Special Power Excursion Reactor Tests-I (NRTS, Arco, Idaho) Rating: Large power excursions Fuel description: Modified MTR-type fuel elements. Each element contains 51 plates, 168 kg Um (stiffening members to increase strength of plates). Core tank 4-ft diam Reflector: Water Shielding: Core tank in 10-ft-diam pit tank Control: 5 blade-type control rods Coolant: Water (not pressurized) Remarks: SPERT-II designed for pressures up to 300 psi; SPERT-III for 2.500 pai (1955)

EBWR,

Boiling-water Experimental Reactor (Argonne National Laboratory, 111.) Rating: 20 Mw (heat). 4.6 Mw (electricity), I X I01* Fuel description: 112 fuel-element assemblies, each containing 6 plates 3% in. wide and 52 in. loisf Core about 4-ft diam, 4 ft high. About 75 kg U:". Enrichment, 1.44% U1". Plates clad with Zircaloy-2 Reflector: 18 in. water Shielding: in. steel, 3 in. lead, 7 ft 6 in. concrete Control: 5 hafnium control rods, 4 borated stainless-steel safety rods Coolant: 600-psig operating pressure, Bteam produced fed directly to turbine; natural circulation Remarks: (1956) Later tests to be with forced convection and new core with DiO

Water-Graphite-moderated

RPT, Reactor for Physical and Technical Investigations (USSR) Rating: 10 Mw, 8 X 10" (max) Fuel description: Central core consists of 37 cylindrical ducts, 54-mm diam, with 14-cm spacing, piereiac graphite layers, core 1-meter diam, I meter high. Tubes of Al alloy including fuel elements arc inserted in ducts; fuel elements are hollow cylinders containing enriched uranium, covered with as Al casing; spacer ribs on Al end caps and on Al belt in the central section. Both graphite and water act as moderator

Reflector: Core and graphite reflector fill a cylinder 240 cm high, 260-cm diam, 80 cm graphite on sides. Graphite, p <= 1.8 g/cm1, thermal-diffusion length —52 cm 60 cm on bottom. Shielding: Side shielding, iron layer/frame 2.5 cm thick, 320 cm concrete, p =• 2.4 g/cms; top, ISO ea graphite, 40 era Pb, 20-cm-thick iron slab Control: 2 automatic-control rods, 3 manual-control rods, 3 slowly moving automatic rods of boroa carbide,

2 systems of safety rods

Coolant: Distilled water enters annulus between fuel tube and duct and flows out through internal d-iet of fuel element; circulated in closed system; inlet temp 20 to30°C, outlet temp 55 to 65°; individaal flow to ducts measured, 6 metersVhr per duct Remarks: Use of He for heat transfer in graphite and to dry graphite if water should leak in. Calculated k with and without water is 1.59 ± 0.03 (1952) (Replica of APS)

APS, Atomic

Power Station (USSR) Rating: 30 Mw (heat), 5 X 10" (5 Mw electrical power) Fuel description: Enriched uranium (5% Um), graphite- and water- moderated. 128 fuel ehauel; pierce central part of graphite brickwork, forming a core 154 cm in diam and 170 cm high. Tfanwalled austenitic stainless steel used for fuel-element tube, 9-mra OD and 0.4- ram wall tfajcines*Fission-product retaining can 0.2 mm thick. Each channel is 65 mm in diam and contains 5 longitudinal holes. Water passes down central tube and returns up in the 4 outer tubes. Rhcmbfr lattice spacing at 12-cm pitch. Total uranium charge 550 kg

kinds op reactors

Sec. 13-5] Table

4.


13-135 Reactors.

(Continued)

Reflector; Graphite encased in a herraetical steel jacket, clearance allowed in graphite brickwork, and jacket filled with He or nitrogen. Side reflectors 80 cm thick; end, 60 cm Shielding: Side water shield 100 cm thick, concrete wall 300 cm thick, iron plate and covering for top shield Control: 18 boron carbide rods (water-cooled) for manual control; 4 auto ma tic -control rods in reflector, and 2 safety rods in active zone Coolant: Distilled water at 100 atrn pumped down through the tube and return upflow over surface of the uranium fuel elements. Recirculation with make-up system. Flow and temp recorded for each channel. Inlet temp I90°C, outlet temp 280°C. Heat exchange to generate steam at Separate cooling facility for graphite 12.5 atm. 270°C. Fuel element not bonded to jacket (1954) Remarks: Um burnup about 15 to 20%. Others


TTR, Thermal Test Reactor (Knolls Atomic Power Laboratory, Schenectady, N.Y.) Rating: 100 watts, 3.4 X I0» 20 fuel slug tubes, each 2-in. diam, 24 in. long, containing U-Al-alloy disks with poly Fuel description: ethylene spacers strung on e-in.-diam rod; disks (slugs) immersed in light paraffin-base oil con Tubes in cylindrical array in Al tank of 12-in. ID and 18-in. OD; tank filled tained in slug tube. 2.7 kg U"1 required for criticality; extra 0.1 kg provides about 0.6% excess reactivity with water. Reflector: 30-in. cylindrical ring of graphite Shielding: Reactor located in a room with 6-ft-thick concrete walls Over-all size: About a 5-ft cube Control: Coarse control, 6 Fe-clad Cd sheets, 4 in. wide, 18 in. long, at periphery of tank. Fine control, Safety, 4 Cd rods H-in. diam 2 Cd rods, 18 in. long, >£-in. diam, move between fuel slug tubes. Coolant: Original versions without cooling Remarks: Internal thermal column is a graphite cylinder 12 in. in diam, 18 in. high. Test hole in center of column. Several reactors of this type have been built (1951) General Electric Nuclear Test Reactor (Knolls Atomic Power Laboratory, Schenectady, N.Y.) Rating: Up to 30 kw, 10" Core size 18-in. diam and 20 in. long, placed horizontally. Fuel description: Improved version of TTR. disks, Al alloy containing 90% enriched VUb, clad with 2S Al, 16 fuel assemblies, 3-in.-diam 2.5 kg U»» Control: 6 shutdown sheets, 4 safety rods, 2 coarse rods, 1 fine servo rod Coolant: 50 gpm deionized water Remarks: Several nuclear test reactors have been built

Argonaut (Argonne National Laboratory, Chicago, 111.) Rating: 10 kw, 5 X 10" (max) Fuel description: 20% enriched UiOs-Al fuel plates, about 6-in. -thick annulus (internal thermal column 2-ft diam Graphite spacers between 5 ft square and 4 ft high. Reflector: Minimum 12-in. graphite reflector surrounding

4 kg Um, fuel boxes 24 in. long, placed in and 4 ft high) centered in a block of graphite fuel boxes;. water in annulus fuel annulus. Above and below fuel boxes

is 12-in. water reflector

Shielding: 7-ft concrete block Control: Fast moderator dump, 6 Cd blades Coolant: Water, natural convection or forced at ^3 gpm; exit temp 100°F Remarks: Inexpensive, flexible facility for teaching and research (late 1956} AGN-BOl

(Aerojet-General Nucleonics,

San Ramon, Calif.)

Rating: 100 milliwatts, max 4.5 X I0» Fuel description: About 3.000 g uranium

(20% Um) as UOi embedded lO-in.-diam by 9>j in. high, 11 kg polyethylene Graphite reflector, 20 cm thick

ethylene.

Reflector: Shielding:

in radiation-stabilized poly

Core

Core, reflector, and lead contained in steel tank. 10-cm-thick lead. provides 55-cm-thick water layer containing 3% HBOi Over-all size: 6J.£-ft diam by 9 ft high Control; 2 safety rods, 2 control rods, core fuse Coolant: None Remarks; 1956. (Several reactors built for teaching and demonstration)

BPR* Beryllium Physics Reactor (USSR) Kuel description: HiO solution of UaOs ( 10%

Outer water

tank

Um) in annular formed by two coaxial steel tubes, 13.4 mm diam by 0.2 mm wall and 9.0 mm diam by 0.4 mm wall, 214 g UsOi per element, height 960 mm. Rectangular lattice 107 Beryllium blocks form cylindrical core, 960 mm high, 1040 mm diam. by 64 mm for vertical channels. 6 channels, 14.5 ram diam arranged on a 20 mm radius for each 108 horizontal channels 31 mm diam. element with 17 mm diam channel in center. Central channels of vertical cells, horizontal channels and inner tubes of elements used to study effect of H2O, graphite. 11.73 kg Um for criticality, no reflector Reflector: Beryllium, 15.5 cm thick (366 uranium fuel elements, 6.66 kg Um for criticality) Control: 2 Cd control rods, 8.2 mm diam, 960 mm long. 8 Cd safety rods. Remarks: Thermal column may be used in center (1954)

13-136

MECHANICAL DESIGN Table

5.

AND REACTORS

[SEC.

13

Fast and Intermediate Reactors

Lot Alamoa Fast Reactor "Clementine" (Los Alamos, N.M.) Rating: 25 lew, 10" fast flux Fuel description: Rods of pure plutonium metal clad with steel, in 6-in. array, contained in a 6~in.-diam mild -steel pot; rods of clad natural uranium interspersed with the fuel rods Reflector: Around fuel pot, 6-in. -thick reflector of natural uranium built out of silver-plated blocks, 6 in. of steel followed by 4 in. lead to form 38-in. cube Shielding: Alternating layers of 3 in. of iron and masonite and later iron and boron-impregnated plastic, total thickness of 30 in. on 3 sides. 18 in. heavy aggregate concrete Over-all size: 11 by 15 by 9 ft high Control: 2 safety and 2 regulating rods in uranium reflector, natural uranium in lower section and B-10 in upper section, reactivity equivalent 0.004; safety block, large section of reflector, dropped to produce a reactivity change of 0.013 Coolant: Mercury, water used to remove heat from uranium reflector Remarks: World's first fast reactor (1946). Dismantled 1954 after failure of fuel element


EBR-I,

Experimental Breeder Reactor (Reactor Testing Station, Arco, Idaho) Rating: 1,400 kw, fast flux, center of core, I.I X I01*. (200 kw electrical) Fuel description: Close-packed array of rods, each containing UM* center section with top and bottom natural uranium blanket section. Fuel slugs (2 per rod), enriched to 90% Um. 4t^ in. long. 0.384-in. diani, jacketed in stainless steel. Provisions for 217 rods. (Rods not containing: I"*1* are loaded with natural uranium.) 48.2 kg UI1S for criticality, 52 kg used. Double-walled reactor tank with gas space used for leak detection and insulation Reflector: Natural-uranium blanket in two sections: first section, tightly packed array of rods 1 Mf-in diam, 20>« in. long, clad with 0.020-in. stainless-steel jackets. Core and inner blanket contained brick*, in a 15^ -in. -diam stainless-steel tank. Outer blanket consists of array of keystone-shaped forming a cylinder around sides (and bottom) of reactor tank. Bricks are natural uranium, jacketed in 0.020-in. stainless steel. Outer blanket and shield mounted on hydraulic elevator. IB in. graphite for reflector Shielding: Thermal-neutron shield, 6 in. iron (air-cooled), 9 ft ordinary concrete Control: 12 control rods of natural uranium in outer blanket, 8 for shutdown (0.2% 6k), 4 for control (0.1 % 6k), one bottom safety (0.07% Sk), 4^-in. blanket travel 0.89% 6k, complete removal of outer blanket,

8.9%

6k

Coolant: 292 gpm NaK, temp rise 88°C, 228°C in, flow upward in core and downward in first blanket. Central fuel rods at 357°C (internally). Outer blanket air-cooled, 6,000 cfh maintains brieks »i 200°C. Coolant adds a 14% outer blanket. 72% heat generated in core. 14% inner blanket, reactivity equivalent to 2 kg Um 200 kw produce4. Remarks: World's first production of electrical power from a nuclear reactor (1951). with 400-psi steam. Conversion rates determined by two methods as 1.00 ± 0.04 and !.0I ± 0.85. Second uranium core to be replaced by plutonium core. (First core produced over 3 X 10* kwhr Melt down of core, Nov., (955 of heat.) Fission gases collected in top section above fuel slugs. Mark III core (1957) 810 (Formerly SIR), Submarine Intermediate Reactor (Mark A. prototype. West Milton. B for USS Sea Wolf, S2G) Fuel description: Enriched uranium, beryllium moderator Coolant: Liquid sodium l-in. -thick steel sphere, gastight. Remarks: Prototype reactor enclosed in 225-ft-diam prototype has produced electrical power (1955)

N.Y.; Mari

Laad-basroi

Zero-Energy Fast Reactor (Harwell, England) Rating: 2 to 30 watts Fuel description: Cylindrical core with height » diam —15 cm. consisting of natural uranium acJ plutonium Reflector: Uranium Shielding: No shielding, but reactor in small concrete room Control: Control and safety rods consist of uranium rods moving vertically in channels around can uranium safety block can be pulled out. Fuel rods fitted with fusible link at the center Coolant: None Remarks: 1954 Zephyr,

BR-S, Experimental Fast Neutron Reactor (USSR) Rating: 150 kw, I0'*fast (max) Fuel Description: Plutonium rods, 10 mm diam, 130 mm long, enclosed in s.s. tubes of 0.3 mm walfe 108 plutonium and uranium rods. Fuel assembly in steel tube 130 mm ID. Reflector; Layer of uranium (700 mm OD), steel jacketed 35 mm diam rods; layer of copper (1000 rem OD) Shielding: HtO shield 500 mm thick, cast iron 400 mm thick, heavy concrete (p — 4.2 g/cra1) 1200 thick Regulating system and emergency shutdown mten Control: Copper-nickel alloy control elements. Coolant: Mercury coolant for c^re, reflector air-cooled Remarks: (In operation by 1956), reactor to be rebuilt

kinds of reactors

Sec. 13-5] Table

13-137

Reactor Developments in the United States

6.

Research, Pool, and Tank Types,

Testing Reactort

oak ridge research reactor (ORR): The ORR is a tank-type HtO- cooled and -moderated reactor MTR-type fuel element* are used with 19 fuel plates per element. designed for a heat output of 20 Mw.

ordnance materials research reactor (OMR): The reactor for the Watertown Arsenal Laboratory is to be used primarily for metallurgical research. The basis for the design is the bulk shielding reactor. The reactor is centered in a cylindrical room 60 ft in diam, and a gastight steel shell will enclose the entire structure.

the uvermore plates.

pool-type

reactor (LPTR)

: I Mw (heat).

90% enriched

U"*

is used for the fuel

(1958)

curtiss-wright research natural convection

and from

tower 8hieldi.no facility

reactor: The

reactor is a modified BSR reactor, 100 kw to 1 Mw with forced circulation.

(TSF):

(Oak

oround test reactor (GTR): (USAF,

from

0 to 100 kw with

Ridge National Laboratory, Oak Ridge, Tenn., 1954) Fort Worth. Tex.,

aircraft shield test reactor: (USAF, Fort Worth,

1953)

Tex.,

1954)

westinghouse TESTING reactor ( WTR) : The WTR is designed to provide irradiation exposures under conditions of high flux, high temperature, and high pressure. Heat output will be 20 Mw. 60 fuel assemblies together with 9 Cd-Al control rods and 7 high-pressure thimbles are placed in triangular array of close-packed tubes in a core 28 in. in diam by 36 in. high. The fuel assemblies are 2.5-in. 3 concentric uranium-aluminum, Al-clad cylinders. diam and contain HiO is used as the coolant and moderator.


nuclear engineering test facility (NETF): The Air

Force NETF is to be built for the Air Research and Development The 10-Mw reactor is a modified Command at Wright Air Development Center. ORR type and incorporates large irradiation facilities, including two 5-ft by 6-ft 8-in. by 10-ft cells. The reactor is to be housed in a pressuretight steel building.

engineering test reactor (ETR): The ETR is designed with large in-pile test facilities with high thermal- and fast-neutron fluxes for performing engineering tests on materials, coolants, fuel elements, components. and other nuclear-reactor The design heat output is 175 Mw, and the coolant-water inlet 49 MTR-type pressure is 200 psia, with 110°F inlet temperature and a 138°F outlet temperature. fuel elements are used. (1957) engineering

ETR

and

12.5 Mw.

and research reactor (ERR): The ERR incorporates some of the features of both the The reactor is a tank type, HiO-cooled and -moderated, with a heat output of

ORR.

medical-research reactor: The BNL medical-research reactor is a l-Mw (heat) Seventeen 140-g (Um) BSF fuel ele reactor designed for medical research and therapy. critieality Experi with a graphite reflector surrounding a 2-ft-diam core tank. ments are required for in., 4 4 holes, by mental beam will provide thermal-neutron fluxes from 10'° to I011 neutrons/ (cm9) (sec) at position of patient's head. The coolant-water flow up through the core is 600 gpm, and air is used to cool the graphite. brookhaven

MTR-type

Massachusetts institute of technologt reactor (M1TR): The MITR is a CP-5 reactor designed to operate at 1.000 kw. A medical therapy room is located directly beneath the reactor.

food-irradiation reactor design: The food -irradiation reactor is designed as a y source for use in treating foods and other materials. High radial neutron leakage is absorbed by an indium sulfate MTR-type fuel elements are used, and the water-coolant flow is solution which blankets the core. 2.420 gpm for a total heat output of 10.5 Mw. Aqueous-eolution

Type

homogeneous-reactor

test (HRT) (HRE-2): The HRT is a two-region, forced-circulation aqueous homogeneous reactor. A dilute solution of uranyl sulfate (about 90% Um) is used as the fuel, and in The design output is 5,000 kw heat and the first experiments only DtO will be used as the blanket. 250 kw electricity. The fuel inlet pressure is 2.000 psia, inlet temp 256°C, and outlet temp 300°C. Core ID 32 in., 14-in.-thick blanket. (1957) Pennsylvania advanced reactor (PAR); 70 laboratory reactor: Designed specifically

to 150 Mw (electrical),

uranium-thorium fuel.

5 for low-cost nuclear- reactor training and research, watts; 1-ft-diam stainless-steel sphere containing aqueous solution of enriched uranyl sulfate, surrounded by 6-in. -thick lead shield and placed in water-filled tank 8 ft high, 8-ft diam.

Heavy-water Moderated

cakolinas-viroinia -cooled.

east

nuclear power associated reactor:

central-florida

enriched uranium reactor.

oroup

reactor:

50-Mw (electrical)

17-Mw (electrical)

DiO- moderated,

DiO- moderated gas-cooled,

and

slightly

MECHANICAL DESIGN

13-138 Table

6.

AND REACTORS

Reactor Developments in the United States.

chugach electric association heactor: moderated, approz. 2% enriched uranium,

[SEC.

13

(Continued)

40-Mw (thermal), I0-Mw (electrical) 10,000 kg U.

sodium-cooled,

DiO-

Organic-moderated

organic-moderated reactor experiment (OMRE): The OMRE is an experimental facility designed to test and develop the technology employing an organic as a coolant, moderator, and reflector for a reactor. OMRE heat output 16 Mw, max fuel temp 824°F, coolant inlet temp 700°F. power-type Fully enriched UOt-stainless-steeJ Reactor core 22.5 in. square by 36 in. high. outlet temp 7I0°F. Batch purification clad on each side. fuel plates, 20 mils thick, 2.5 in. wide, 5 mils stainless-steel system for organic. Sodium-cooled

sodicm reactor experiment (SRE): The technology of sodium-cooled graphite-moderated reactors Max fuel temp 1200CF, Heat output 21 Mw, electrical output of 6 Mw. studied with the SRE. Graphite moderator avg fuel temp 750°F. Sodium coolant, inlet temp 500°F, outlet temp 960°F. in form of hexagonal prisms 11 in. across flats, 10 ft high, clad with 0.035 in. zirconium on sides, 0.10 in. Each graphite assembly in Cans placed 0.170 in. apart form coolant channel. on top and bottom. core contains a 2.80-in.-ID central zirconium tube in which the fuel elements are suspended. Fuel element is a cluster of 7 rods, each rod containing a 6-ft stack of uranium (2.778 weight %) slugs, Slugs in a stainless-steel jacket tube, 0.010 in. thick, with NaK to thermally 0.750-in. diam by 6 in. bond slugs. Rods spaced in cluster by 0.091-in.-diam stainless-steel wire wrapped on 6 outside rods. (1957) CONSUMERS public power

Detroit reactor: 245-Mw

(thermal),

75-Mw (electrical).

Gas-cooled


oab-cooled

reactor experiment (GCRE): Located at NRTS, Arco,

Idaho.

Preesuri zed-water

pressurized- water reactor (Shippingport, Pa.) (PWR): To produce 230 Mw heat. 60 Mw electricity power plant). HtO-cooled and -moderated, enriched uranium in clad plates and natural (full-scale uranium in tubes. Zirconium-uranium alloy plates clad with Zircaloy-2, 0.080-in. -thick plates. 0-080-ia. spacing for water coolant. UOi pellets in Zircaloy-2 tubing, about 10 in. long, 0.413-in. OD (0.02S-irt wall); 100 tubes assembled in a bundle, 7 bundles fastened in a stack to form a total height of 6 ft Reactor core is 6-ft- diam right cylinder, 6 ft high, containing both highly enriched "seed " and natural uranium blanket assemblies. Seed assemblies consist of the plates, and blanket assemblies contain 52 kg of enriched U"s in seed assemblies and 12 tons of uranium in form of 10: ia the tube bundles. blanket. Reactor vessel over-all height 33 ft, 9-ft diam, nominal wall thickness 8}£ in., 2.000 psm Water flow 45.000 gpm, inlet temp 508°F, outlet temp 542°F, 600 psia saturated steam produced. 24 hafnium control rods. army package power reactor i ( A PPR I) : The APPR-I is prototype of a reactor designed to hare aU components readily transportable by air and to meet the requirements of furnishing power at ren-.o*.? military bases. The reactor is a pressurized- water reactor with a 10,000-kw heat output and a 1.9C0-Lir electrical output. The water-coolant inlet pressure is 1.200 psi, inlet temp 432°F, outlet temp 450CF consolidated edibon thorium reactor (CETR): 500-Mw (thermal), 250-Mw (electrical, including HO-Mw conventional superheater). Reactor pressure 1.500 psig, 5I0°F outlet temp. About 90*~ U>" enrichment, 10,000 kg U and 8.275 kg Th. YANKEE ATOMIC ELECTRIC COMPANY REACTOR navy propulbion reactors: SSN586. 2 reactors for a radar picket submarine Additional 10 reactors for submarines 2 reactors for guided- missile cruiser Initial studies on 8 reactors for an aircraft carrier merchant ship: 74,000-kw

(thermal),

20,000 shaft hp, reactor

for N.8. Savannah.

Boiling-water

Dresden reactor, g.e. dual cycle boiling water reactor: 627-Mw (thermal), 180-Mw (electrical:. Steam pressure steam temperature, 1.000 psig (primary), 500 psig (secondary); 547°F (primary' 60 tons uranium. 470°F (secondary). 1.5% enrichment,

elk river reactor: 58-Mw 890 psig, temperature

53I°F.

northern btateb power controlled

recirculation.

(thermal), 22-Mw (electrical, including superheater). Secondary system steam at 600 psig, 825 i .

co.

reactor:

164-Mw

pacific gas & electric co. reactor: 165-Mw argonne boiling-reactor facility and the

EBWR.

(thermal),

66- Mw

(electrical,

Reactor pressjrc with superheater'

(thermal), 50- to 60-Mw (electrical).

(ARBOR): Facility will

extend studies

of the Borax

kinds of reactors

Sec. 13-5] Table ALPR,

in the United States.

Reactor Developments

6.

AROONNE LOW-POWER

13-139 (Continued)

REACTOR

Fast Power

fast breeder powrr reactor, APDA-I; 300-Mw (thermal),

APDA developmental

cul), sodium coolant, 800°F outlet temperature.

experimental breeder

coolant, steam produced uranium alloy.

Steam at 600 psia, 730°F.

100-Mw (electri2, 100 kg. 27 % enrichment,

Sodium primary REACTOR-ii (EBR-II): 60- Mw (thermal), 20-Mw (electrical). at 1250 psi and 850°F. Initial fuel, enriched uranium, and later, plutonium-

Others

los alamob moltcn molten

PLUTONIUM

Pu alloy.

PLUTONIUM-ENRICHED, TRANSIENT

reactor

GRAPHITE

EXPERIMENTS:

AND WATER REACTOR,

TREAT reactor experiment. LMFR:

About l-Mw (thermal); NaK coolant. AND

DlO

15 kg

POWER REACTORS

REACTOR TEST EXPERIMENT,

liquid metal FUEL bismuth, graphite moderator.

Table 7.

5- to 10-Mw (thermal), circulating fuel solution in

World-wide Reactor Developments

Argentina: To purchase

3-Mw swimming-pool reactor.

Australia: A 10,000-kw

DiO-tnoderated reactor under construction,

hipar (Dido type).


Belgium

beloian reactor-

I (BR- 1): A 3.5-Mw (heat), natural-uranium 18-cm lattice spacing. Al-clad fuel slugs (United States).

fuel,

graphite-moderated, air-cooled.

beloian

test reactor with an 6IH«-in.-diatii REACTOR-2 (BR-2): Designed as an engineering hole, Beryllium reflector. annular core from 3 to 6 in. thick and 30 in. high. HtO-cooled and -moderated. Heat output from 25 to 50 Mw. pressurized water reactor ordered. 1 1.5-Mw (electrical)

Brazil, Cuba, Mexico: Three power plants ordered 10 to 12 Mw (electrical): two 40- Mw single-cycle boiling-water type, one organic- moderated power reactor. A 5-Mw swimming pool reactor is to be located on the new campus of the University of Sao Paulo, Brasil. Bulgaria: 2-Mw Soviet reactor for research and isotope production. Canada: Pool-type research reactor to be built for McMaster University's Hamilton College in Ontario. 20-Mw (electrical) prototype DjO power reactor to be built (NPD-2). Pool Test Reactor (PTR), 100 watts, 1957. China: 2-Mw Soviet swimming pool reactor. 150-Mw power plant, COi-cooled, planned for the 5-year period 1956-1960. 2.000 kw, 10% enriched fuel in 823 rods, First reactor supplied by the Soviet Union. Tank type. 7.5 mm apart, contained in Al tubes 10-mm diam, 50 cm long.

Czechoslovakia:

Denmark: Danish Reactor No. I (DR-I) 500-w homogeneous solution reactor (1957). (DR-2) 5-Mw tank-type reactor. (DR-3) 10-Mw Pluto-type reactor (enriched uranium and DiO), materials testing.

Dominican Republic: Ordered a I3-Mw (electrical) pressurised- water reactor. Egypt: 2-Mw pool-type reactor from USSR. France: EL-3 is a materials testing reactor DiO-moderated and -cooled; 1.3% enriched uranium. 10 Airtight cylindrical metallic enclosure, 150-ft-diam. to 20 Mw, I0U thermal flux. facility. Prosperine is a small plutoniura Aquilon is a critical facility similar to the English Dimple water boiler. Two swimming-pool reactors for training and shielding research are planned. Mincrve is a lowpower reactor design for reactivity measurements. G-2 and G-3 are power and plutonium producers scheduled for operation by 1958. Each, naturaluranium fuel elements, COi-cooled at 15 atm, 150 Mw heat, 25 to 30 Mw net electricity. EDF-I is the fourth graphite reactor (G-l, G-2, G-3) but is designed principally as an electrical power producer; 300 Mw heat, 60 to 70 Mw electricity (net); scheduled for 1959. EDF-2. 100- to 200-Mw (electrical). Propulsion reactor designs are Reacteur Marin and Pile Chaudc. Q-244.

MECHANICAL DESIGN

13-140

Table 7.

AND REACTORS

World-wide Reactor Developments.

[SEC.

13

(Continued)

Hamburg; 50-kw homogeneous Germany (West): 5-Mw swimming-pool- type reactor for Kernenergie solution-type reactor for University of Frankfurt (1958); 1-Mw reactor for the Technische Hochschule '* Merlin" in Munich (1957); plans for a 40-Mw (electrical) reaotor for Dusseldorf City; 5.000-kw Karlsruhe reactor designed for natural swimming-pool reactor to be located in Northrhine- Westphalia. 15-Mw (electrical) boiling water reactor. 10-Mw Dido-type. uranium and DiO. Germany

supplied. Greece:

(East): 2-Mw

l-Mw pool-type

Soviet

research

reactor

ordered.

reactor

(1957).

100-Mw

experimental

power

reactor to be

Hungary: Soviet reactor for research and isotopes production to be built at Csilleberc. India:

CIR

Zerlina,

(Canada-India Reactor) NRX-type reactor; sea-water natural uranium, DtO reactor.

cooling,

10.000 kw.

zero energy,

Israel: l-Mw pool-type reactor ordered. Italy: 50-kw aqueous-solution-type reactor, l-Mw pool-type, and a 5-Mw heavy-water research reactor ordered. I34-Mw (electrical) Yankee Atomic Power type, and a I50-Mw (electrical) preasuriaed 200-Mw gas-oooled reactor ordered (1957). water reactor ordered. Japan:

DiO, natural uranium reactor

50-kw water-boiler-type reactor.

Purchased

a 10,000-kw

DtO

research reactor. The Netherlands: 20-Mw modified ORR-type reactor to be completed by end 1958. To be rebuilt for research and training. 10-kw swimming-pool reactor for 1957 exhibition. 250-kw reactor test of UOi fuel slurry. 10- to 20-Mw boiling- DiO reactor, Plans for nuclear ship propulsion.


Norway:

slightly enriched

uranium,

to furnish steam

at Haldec

Pakistan: Medium power research reactor; power reactor. Philippines: U.S. AEC to assist in providing reactor. Poland: 2-Mw Soviet research reactor (1957); design of second reactor started. Portugal: l-Mw swimming-pool reactor; Puerto Rico:

l-Mw pool-type

planning a 100-Mw

(electrical)

power station.

reactor.

Rumania: 2-Mw Soviet research reactor. Spain: 3.000-kw swimming-pool reactor. Sweden: R-2 to be a 30-Mw ORR-type reactor scheduled for 1958 operation at Studsvik. R-3a is a 76-Mw reactor design for central district heating. Natural uranium and DtO. R- 3b is * version of R-3 with provisions to produce M Mw (electricity). R-4 is a power reactor project for 100 Mw electricity, using natural uranium and DiO. Other central district heating reactors planned (a total of 6 by 1966) and one or two 100-Mw (electric ity) power stations. P-34, 10-Mw (heat), natural-uranium DiO- mode rated and -cooled reactor under ooaSwitzerland: struction. Relocation and rebuilding of the Geneva reactor exhibits. Power level to be increased to 1.000 k> Swiss reactor I. Turkey:

Plans to obtain research reactor.

United Kingdom

power plants: 300,000-kw (electricity) Calder Hall-type power station to be built at Brad well. Essex. Two reactors. Core 45-ft diaro, 31 ft high, 2,600 fuel channels, contained in 3-in. -thick steel pressure vessel 67-ft diam. Fuel slugs l-in. diam (natural uranium) sheathed in Mg alloy. CO* coolant at Refueling while in operation. 130 psi. 275,000-kw (electricity) Calder Hall-type power station (2 reactors) to be built at Berkeley Gloucestershire. Core 48-ft diam, 30 ft high, 3.000 vertical fuel channels. CO> gas at 125 pcd. 320,000-kw (electricity) Calder Hall-type power station (2 reactors) to be built at Hunterstoa Ayrshire, Scotland. Core 28 ft high, 50-ft diam, 3,288 fuel channels. COt at 150 psi. Atomic Energy Authority erecting two Calder Hall reactors at Calder B station and plans four addi tional reactors at Chapel Cross.

other reactors: Dounreay

reactor under construction at Dounreay in Northern Scotland. 60 M» (heat), core 2-ft diam, 2 ft long, sodium-cooled. Neutron shield 4 ft thick, graphite containing boroa. (ZEUS, zero-energy uranium system, test moiel of the DousRe actor housed in 135-ft-diam sphere. reay Reactor, 1955.) Medium-energy light-water-moderated industrial nuclear reactor (MERLIN). Research reactor 5 Mw, DiO, 20% enrichment. Zero-energy thermal reactor (ZETR) (1957).

kinds of reactors

Sec. 13-5] Table

7.

World-wide Reactor Developments.

Homogeneous reactor experiment. Initial operations with pLutoniutn, be changed from HsO to DiO (1956). Pluto, DiO enriched uranium engineering test reactor, 10-Mw (1957). DMTR, Dounreay Materials Testing Reactor, 10-Mw Pluto reactor. Neptune, zero-energy, HtO, naval research reactor (1957). Nero, U, Pu, graphite, research reactor, zero energy (1957).

13-141 {Continued) changed to Um.

Solvent to

USSR: Planning a 100- to 200-Mw (electricity) DiO power reactor with COi cooling. Boiling- DiO homogeneous reactor planned by I960. A 200-Mw (heat) reactor to power icebreaker Lenin. Pilot plants scheduled for 1959-1960 include HiO-moderated thermal reactor with direct feed of steam to turbines; homogeneous DtO-moderated thermal-breeder reactor with Um-Th*" cycle; thermal graphite- moderated and sodium-cooled reactor; fast sodium-cooled breeder reactor with PuM*-UMI cycle. Other reactors

power plants scheduled in 1956-1960 period include 200-Mw HiO-moderated and -cooled with 70- Mw turbines using saturated steam at 30 atm; graphite- moderated, steam- and HjO-cooled reactors producing steam at 90 atm. superheated to 480 to 500°C. Yugoslavia: Venezuela:

Research

reactor.

3,000-kw swimming-pool research reactor

Euratom, European Atomic Energy Community Luxembourg) plan 15,000 Mw (elect rioal) 1967.

ordered.

(France, Germany, Belgium,

OEEC (Organization for European Economic Co-operation), considering

Italy,

homogeneous

Netherlands.

aqueous reactor


development.

3

REACTOR

DESCRIPTIONS

Several unusual aspects of the designs and engineering developments of nuclear reactors are presented in the following descriptions, drawings, and flow sheets. Refer ence should be made to the Nuclear-reactor Catalogue for detailed descriptions of the reactors. 3.1

Graphite -moderated Reactors

The 26-ft graphite stack in the BEPO reactor was built using about 28,000 blocks by 29 in. was selected, of graphite.1 Even though a standard block size of 7J4 by

The graphite stack rests on a steel floor 1,500 different sizes were required. leveled to an accuracy of ±0.010 in., and the correct height of the stack was achieved to within ±0.015 in. Each graphite block was machined to an individual tolerance of ±0.0025 in., and a careful selection of under- and oversized blocks was maintained in the stacking operation. Extreme precautions were taken to avoid contamination of the graphite during the handling and stacking. In the several American reactors, the graphite blocks are about 4 in. square in cross section. The graphite stackings in the BEPO and the Oak Ridge reactors are shown in Figs. The Brookhaven reactor has a vertical gap (from 2^ to 1 and 2. >n-) which halves the reactor core. Cooling air is introduced into the gap and flows out in both directions. Some of the features necessary for a production-type reactor are illustrated in the schematic diagram of the Windscale pile,1 Fig. 3. Production reactors are among the largest reactors built, and because of their size, special problems arise in the design and construction of the foundations and structures. Monolithic structures are used. The reinforced walls of the biological shield of the Windscale pile secure the rigidity Special loading of the monolith and are adequate for handling the thermal stresses. and unloading facilities are provided, and in the latter case a water duct below the pile and an underwater railway are included. A continuous monitoring of the coolant air from the fuel channels is maintained to locate any possible failure of a fuel slug. Filters are provided for both the inlet and exhaust air to remove radioactive par ticulate matter. For example, the Oak Ridge paper filters remove 99 per cent of all Discharge of the air coolant is made from particles greater than 0.2 p in diameter. stacks as high as 200 to 400 ft. Sufficient dilution and residence in air must be pro some

13-142

MECHANICAL DESIGN

AND REACTORS

[SEC. 13

/


Fio. I. Graphite

stacking

in

BEPO

reactor.

(From Sir Christopher

Hinton, Re/. I.)

"

I

Fig. 2. Graphite stacking in Oak Ridge reactor. ntroductism (From Richard Stephenson. to Nuclear Engineering," 2d. ed., McGraw-HiXl Book Company, Inc., A'etr York, 1954. p. 73.) About 7,000 curies of 1 10-min radioargon vided for the radioactive gas constituents. is discharged per day from the Brookhaven reactor. One of Britain's first commercial-scale nuclear power plants utilizes a COj gas A schematic diagram is given in Fig. 4. coolant under pressure. 3.2

DjO Reactors

The CP-5 reactor is an advanced-type research reactor. Figure 5 is a pictorial view of the installation. More than 50 experimental holes have been provided, and some of these arc shown in the sections illustrated in Figs. 6 and 7. As shown in Fig. 8, a float arrangement is used to measure the D2O flow through each of the fuel

3.

diagram

Schematic

Hinlon, Ref. I.)

of the Windseale

pile, cross section.

Sir Christopher

(From

OVERHEAD CRANE R CHIMNEY FOR COOLING AIR

Dp

!''

tU

CHIMNEY FOR COOLING AIR HOT GAS DUCT

-HIGH PRESSURE STEAM_ -LOW PRESSURE STEAM :

t=

(

W-HOT GAS

!

STEAM TO CONVEN NTIONAL TURBINE GENERATING PLANT

HIGH PRESSURE STEAM DRUM

- LOW PRESSURE STEAM DRUM OVERHEZ IEAD CRANE ELECTRICALLY DRIVEN COOLANT CIRCULATINGBLOWER

I

SEP 1

I I

x

.


Fia.

13-143

KINDS OF REACTOUS

Sec. 13-5]

MOTOR GENERATOR

LTHERMAL SHIELD

Fio. lief.

4.

21.)

Schematic

diagram

of the Calder Hall reactor.

(From

Sir

Christopher

Hinton,

13-144

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

The upward flow causes the D20 in the fuel assembly to stand several The D2O system is shielded by 2 ft of concrete, inches above the level of the pool. and ventilation is provided in the isolated area as a precaution in case of a spill. Outside Minute amounts of tritium are produced under the irradiation conditions. filters Micrometallic the reactor, stainless-steel piping and equipment are used.


assemblies.

HEAVY HANDLING

Fio.

5.

WATE.I

SYSTEM

A pictorial representation

of the CP-5 reactor.

(From

J.

T. Wnlis, Ref. 70

and a resin column are installed in a secondary loop to maintain the purity of the D20. A flow sheet for CP-5 is given in Fig. 9, and the D20, H20, and gas systems A helium-gas system is employed for balancing pressure and for convey are shown. ing the deuterium and oxygen to the recombiner. The underground location of the R-l reactor is illustrated in Fig. 10a and a 1 section is presented in Fig. 106. Figure 11a is a vertical section of the JEEPi showing the major accessory equipment required, and Fig. 116 is a horizontal sec tional view. The cooling system and the details of a cell for the Saclay reactor are


18-145

ENCLOSEDSHELF-


MOTORPACKAGE FORNEUTRON COLUMNSHUTTER

y ^-GRAPHITE

■tV-.i--

STEa BILLETS -PNEUMATIC TUBES TANGENTBEAMHOLES . PLENUMCHAMBER

— PIPE TUNNEL-

Fid.

7.

Vertical section through the CP-5 reactor.

(From

J.

T. Weilte. Ref.

7.)

«S8$>

W>°~_.^**

$*«** Fio.

***&

8. Fuel-box assembly for CP-5. Convex plates are mounted in a fuel box. The flow into each box is regulated by use of an inlet orifice, and the flow rate is measured by a float. Coaxial fuel tubes have been used in place of fuel plates. (From T. Weill*. Re!- *•)

J.

13-146


18-147


Flo.

10a. Sketch of the underground location of the R-l (DiO) reactor, Sweden's first It is located in a solid rock excavation. (From Sigvard Eklund, Ref. 3.)

reactor.

Fio.

10b.

Vertical section of the R-l reactor. 13-148

(From Sigvard Eklund, Ref. 3.)

kinds of reactors

Sec. 13-5] shown in sections.

Figs.

12a

and

126

3.3

13-149

and Figs. 13a and 13b are vertical and horizontal

Aqueous -solution-type Reactors

A detailed drawing of the shield and reflector assembly and of the location of the recombiner assembly for the SUPO reactor is given in Fig. 14. Solution decomposi tion, release of fission-product gases, and entrainment are factors contributing to


0

12

1

1

3 I

4 METERS I

Fig. 1 la. Vertical section of the Kjeller reactor (Jeep). [From T. Brakslad, Chem. Eng. Progr. Symp. Series, 50(11), 184 (1954).)

J.

Barendregt and

K. H.

potential explosion and radioactive hazards. These dangers are avoided through the use of closed gas-circulating and recombination systems, with retention and disposal of the excess gas. Figure 15 is a schematic representation of the SUPO recombiner assembly.

The problems of handling much more complex for the

the decomposition and radioactive fission gases become high-specific-power homogeneous reactors. HRE-I was a homogeneous reactor experiment in which the fuel solution was circulated to an external heat exchanger. Gas separation was facilitated by a swirling motion which was imparted to the reactor solution and which formed a vertical cylindrical vortex

13-150

MECHANICAL

DESIGN

AND

[Sec.

REACTORS

13

of approximately J4_m- diameter. Hydrogen and oxygen gases were recombined in a flame recombiner followed by a platinized alumina catalyst. Fission-product gases were absorbed onto water-cooled activated carbon beds buried underground. Exit gases were diluted with 1,000 cfm ventilating air stream and disposed from a 100-ft stack. 3.4

Materials Testing Reactor

The MTR is a high-flux irradiation facility designed for studying the behavior of materials for high-specific-power reactors and for special radionuclide production. Nearly 100 experimental and instrument holes are provided. Two horizontal holes are as large as 8 in. in diameter, and three vertical holes are 12 in. square. Figures 16 and 17 illustrate the features of the MTR and indicate the location of a few of the

OPENING SHIELD


THERMAL C0O«»i

FAST NEUTRON OPENING

" 200x200

-8" TUBE

Fig. 1 16. Horizontal section of the Kjcller reactor (Jeep). K. H. Brakstad, Chem. Eng. Progr. Symp. Series. 50(11), 184

[From T. (1954).]

J,

Barendregt

and

Graphite pebbles were selected for the first zone of the graphite experimental holes. reflector for the following reasons: to relieve any expansion, contraction, or stress; to improve the heat transfer; and to facilitate the replacement of damaged and oxi dized material. The design and operation of the MTR, including the fabrication of the fuel plate sandwiches, have expedited the development of many similar reactors, including other research and testing reactors, pool and tank types. 3.5

Pool- and Tank -Type Reactors

A number of pool- and tank-type reactors have been built, and many new designs have evolved. Features include fixed and movable cores, with and without a partial graphite reflector and thermal column, and the use of forced convection flow at high Considerable flexibility is provided in the make-up of the core. powers. In the bulk shielding facility, the reactor is suspended from a movable bridge. The reactor can be moved and isolated at one end of the pool through the use of an alumi num gate. The remaining section of the pool can be drained to permit the installation

Sec. 13-5]

KINDS OF REACTORS

13-151

f

23

12

fc


PILE

A

I'm.

12a.

Cooling system for the Saclay reactor.

(From

J.

A

A

Yvon, Ref. 21.)

Fio. 126. Detail of a cell in the Saclay reactor: (1) Cold gas in, (2) plunger, (3) DiO, (4) thermal-insulation sheath, (5) uranium rod, (6) gas-flow adjusting nozzle, (7) hot gas out. (From J. Yvon, Ref. 21.)


13-152 MECHANICAL DESIGN AND REACTORS

[Sec. 13

Sec. 13-5]

KINDS OF REACTORS

13-153

GRAPHITE BORON CARBIDE AND PARAFFIN LEAD

EH

STEEL

CONTROL ROOS SPHERE AND STACK

CONCRETE BISMUTH

r~.i BORAX PARAFFIN

RECOMBINER ASSEMBLY SHIELD.LIMONITE ENO

^/ TOP, SOUTH

CANS-

GRAPHITE STACKING, SOUTH THERMAL COLUMN

LIMONITECANS

SHIELD, LEAD-SOUTH THERMAL COLUMN SHIELDS, BORON CARBIDE AND PARAFFIN CANS

BLOCKS MARKED "R'ARE REMOVABLE


Fio.

North-south vertical section of the Supo water boiler.

14.

Re/. 13.)

(From

L. D. P. King,

EXTERNAL CONDENSER TO EXHAUST

/

'/«" VERTICAL LINE TO TOP OF SHIELD COOLING WATER IN

OPEN LINE

V7 ^COOLING WATER OUT

'Vie' LINE TO EXHAUST STACK PRESSURE MANIFOLD PRESSURE

COUPLING

TUBES

, STACK

OR REFLUX CONDENSER

DESCRIPTION SIB5) 1 LOWER BUBBLER TUBES 2 UPPER BUBBLER 3 HIGH FLOWMETER 4

LOW FLOWMETER

5 e

BLOWER SEAL RECOMBINER INLET AIR INLET

7 8 9 10 11

RECOMBINER OUT (EXT COND INLET) STACK CONDENSER IN (EXT COND OUT) STACK PRESSURE STACK PRESSURE

12 13 14

STACK CONDENSER OUT SPARE (PLUGGED) LEAK DETECTOR-COPPER

Fio.

15.

Ref. 13.)

Schematic

diagram

of the Supo

recombination system.

(from L. D. P. King.

-32'-6-

r- FIRST FLOOR


j_

J

- - PEBBLE

Î^-^rt!]

B^« ' Fio.

16.

North-south vertical section of the

LEVEL

MTR.

(from

J.

DISCHARGE CUBICLE

R. Huffman, Ref. S.)

32'-6 THERMAL COLUMN

EXIT WATER DOWNLEG I INLET WATER

HR4

Flo.

HR3

17. Horizontal section of the

MTR.

13-154

(From

J.

REACTOR STRUCTURAL STEEL

R. Huffman, Ref. 8.)

Sec. 13-5]

KINDS OF REACTORS f— r-4"

• 9-4"

13-155

29-4

■21-7

-r-7"-

y.i

} t. tank! AND""

_

CLOSE5rAPP2Q>^|5l

\— REMOVABLE

ALUMINUM BARRIER

REACTOR MAY BE PLACED IN THIS POSITION FOR MEASUREMENTS WITH OTHER SECTION OF POOL DRAINED


REACTOR MAY BE LOCATED IN THIS SECTION OF POOL FOR MEASUREMENTONLY WHEN THIS SECTION IS FILLED " WITH WATER

<0 "in

3/4"2SFAL PLATE 1

t REACTOR

;i

REACTOR ADJUSTABLE ALONG THIS ONLY WITHIN

t

-K3'-6" EITHER WALL— p->.y

REMOVABLE

HIGH

DENSITY CONCRETE 2'-2

W

Fio.

of

BLOCK

18.

Bulk shielding facility.

(From

W.

M.

Breazeale, Ref. 9.)

shielding tests, including the adjustment of floor level if required.

drawing of the bulk shielding facility. 3.6

a

:

OPENING

Figure 18 is a

Other Thermal Enriched Heterogeneous Reactors

Several Russian reactors utilize enriched fuel with a graphite moderator and Hj() as A sketch of the atomic power station is given in Fig. 19. coolant and moderator. 3.7

Fast Reactor

Figure 20 is a sketch of the over-all arrangement of the EBR-I reactor, primary and Flow of NaK is by secondary liquid-metal loops, steam boiler, and turbogenerator. gravity from the supply tank through the reactor and heat exchanger to the receiver tank. Pumps are used to return the NaK to the supply tank. Shielding of the The radioactive coolant is provided through the use of cells as shown in the figure. secondary NaK coolant is pumped through the heat exchanger and steam generator and back to a suction tank. A horizontal cross section of the EBR-I is shown in Fig. 21. A schematic diagram of the Zephyr reactor illustrating the operating and loading positions is given in Fig. 22.


13-156

MECHANICAL

DESIGN

AND REACTORS

[SEC.

13

Fig.

19. USSR atomic power station: (1) Graphite brickwork, (2) lower plate, (3) upper plate, (4) fuel channel, (5) safety-rod channel, (6) automatic control rod. (7) ionization chamber, (8) side shielding (water), (9), (10) refrigerator, (11) inlet distribution, (12) outlet header, (13) cast-iron top shielding (14) cooled reflector stand. (From D. I. Blokhinttev and N. A. Ntkolayei', Ref. 21.)

SUPPLY

TAMK

TURBO-GENERATOR

CONTROL ROOM STEAM BOCER

EXCHANGER

HEAT

RECEIVING

TANK

ELECTROMAGNETIC PUMP

Fig.

20.

berger,

EBR-I

Ref. 18.)

plant layout, showing arrangement

of flow system.

(From H. V. Liekl**-

Sec. 13-5]

KINDS OF REACTORS


Flo. 21. Horizontal cross section at mid-plane F. W. Thalgott, and M. Novick, Re/. 21.)

of

EBR-I.

13-157

(From H. V. Lichtenberger

,

CORE (FIXED) FIXED ENVELOPE

-SHUT-OFF BLOCK SHAFT

OPERATING REACTOR

Fio.

22. Schematic Holmes, Ref. 21.)

CLOSED

diagram

REACTOR

of the Zephyr operating and loading positions.

OPEN

(From

J.

E. R.

REFERENCES Graphite-moderated

Reactors

Hinton, Sir Christopher: Nuclear Reactors and Power Production — I, Atomics, 6: 115 (Bepo, Windscale.) (1954). 2. Ramsey, M. E., and C. D. Cagle: Ten Years' Operating Experience of the ORNL Graphite Moderator Normal-uranium Reactor, Chem. Eng. Progr. Symposium Ser.. 50(11): 149 (1954). (Oak Ridge reactor.) 1.

MECHANICAL DESIGN

13-158

AND REACTORS

[Sec

13

D20 Reactors

J.

3. Eklund, Sigvard: The Swedish Reactor, Nuclear Energy, X: 93 (1954). (R-l.) 4. Goldschmidt, M., and F. Perrin: The Cooling of the Saclay Pile, Ckem. Eng. Progr. Symposium Ser., 50(12): 243 (1954). (Saclay.) Phyt.. 5. Johns, M. W., and B. W. Sargent: Photoneutrons in a Heavy Water Pile, Can.

J.

6. 7.

38: 136 (1954). (ZEEP.) Lundby, Arne: Operational Characteristics of a Heavy- Water Reactor, Chem. Eng. Progr. Symposium Ser., 50(12): 1 (1954). (JEEP.) Weills, J. T.: A Description of the Argonne National Laboratory Research Reactor CP-5, Chem. Eng. Progr. Symposium Ser., 50(11): 213 (1954). (CP-5.) Materials

8.

Testing Reactor

Huffman, John R.: "The Materials Testing Reactor AECD-3587, July 6, 1953. (MTR.) Swimming-pool

9. 10.

as

an Irradiation

Facility,"

Reactors

Breazeale, W. M.: "The New Bulk Shielding Facility at Oak Ridge National Labora tory," ORNL-991, May 8, 1951. (BSF.) Grigorieff, W. W. (ed.): " Proceedings of the University Research Reactor Conference.' AECU-2900, February, 1954.


Water Boilers Busey, Harold M., and R. Philip Hammond: Recombination of Radiolytic Gas for Aqueous Nuclear Reactors, Chem. Eng. Progr. Symposium Ser., 50(13): 27 (1954). 12. Flora, J. W., J. W. Shortall, and E. J. Strain: "Operating Characteristics of the Water 11.

13.

Boiler," LRL-151, June, 1954. King, L. D. P.: "The Los Alamos Homogeneous October,

14.

Reactor,

Supo

Model,"

LA-1301

1951.

King, L. D. P., R. P. Hammond, J. A. Leary, M. E. Banker, and W. R. Wykoff: Recombination System for a Homogeneous Reactor, Nucleonics, 11(9): 25 (1953).

Ga*

Homogeneous Reactor Experiment 15.

Beall, S. E., and C. E. Winters: The Homogeneous Progr., 50: 257 (1954). (HRE-I.)

Reactor Experiment. Ckem. Ew

Fast and Intermediate Reactors 16.

Holmes, J. E. R., D. D. McVicar, H. Rose, L. R. Shepherd, R. D. Smith, and A. M. Smith: Operational Features of Zephyr, /. Nuclear Energy, 1: 47 (1954); L. R. Shep herd, R. D. Smith, J. E. R. Holmes, H. Rose, and D. D. McVicar, Chem. Eng. Prop Symposium

17. 18.

Ser., 50(13):

191 (1954).

(Zephyr.)

Jurney, E. T.: The Failure and Disassembly of the Los Alamos Fast Reactor. Ckn* Eng. Progr. Symposium Ser., 60(13): 191 (1954). (Clementine.) Lichtenberger, H. V.: The Experimental Breeder Reactor, Chem. Eng. Progr. Sytstp*sium Ser., 50(13):

129 (1954).

(EBR-I.)

Other Reactor Developments Gill, Joseph P.: A High Performance Research Reactor, Chem. Eng. Progr. Symposiv* Ser., 60(12): 43 (1954). (HPPR.) 20. Went, J. J., and H. de Bruyn: Liquid-fuel Reactors with Uranium-Oxides, Chem, Eks(Suspension reactor.) Progr. Symposium Ser., 60(12): 120 (1954). 21. Peaceful Uses of Atomic Energy, Proc. Intern. Conf. in Oeneva, vol. 2, Research Reac tors; vol. 3, Power Reactors, August, 1955. 22. U.S. Atomic Energy Commission, "The Reactor Handbook," vol. 2. sec. 8, Reactor Designs, AECD-3646, McGraw-Hill Book Company, Inc.. New York, 1955. 19.

Sec. 13-5]

KINDS OF REACTORS

13-159

BIBLIOGRAPHY General references used in preparation

of Tables 1 to 8.


Benedict, Manson: Nuclear Reactors for Research, Chem. Eng. Progr., 51: 53 (1955). Charpie, R. A., D. J. Hughes, D. J. Littler, and M. Trocheris (eds.): "Reactors, Progress in Nuclear Energy Series II," McGraw-Hill Book Company, Inc., New York, 1956. Glasstone, Samuel: "Introduction to Reactor Engineering Principles," chap. 13, D. Van Nostrand Company, Inc., Princeton, N.J., 1955. Isbin, Herbert S.: Nuclear Reactor Catalog, Nucleonics, 10(3): 10-16 (1952); Supplement, Nucleonics, 11(6): 65-69 (1953); Ref. 21, vol. 3, 373 (1955). News items appearing in current issues of Chem. Eng. News, Nucleonics, Atomics, and Atomic Industrial Forum Memo, Nuclear Engineering, and The Soviet Journal of Atomic Energy.

REACTOR

13-6

BUILDING DESIGN BY

Alf Kolflat Building design must be adapted to each reactor type as well as to plant location, The etc., and it may take years before some degree of standardization is achieved. building design will also vary depending on whether the reactor is operated with its cooling system at atmospheric pressure or it is a power-generating reactor designed for The present status of reactor building design is discussed fairly high steam pressures. under four main headings depending on the general requirements. 1

BUILDING WITH NO GASTIGHT REQUIREMENT


Buildings with no provisions for the prevention of gas leakage have been used for many research reactors — swimming pool, CP-2, CP-3, X-10, BXL, and NRX — where

./RECEPTIONIST ENTRANCE

LOUNGE

Flo.

1.

Plan of Geneva reactor building.

there is little likelihood of overheating and substantial steam or gas formation. This type of construction has also been used for the British gas-cooled power reactor plants Basically, as well as the first 5,000-kw Russian pressurized-water atomic power plant. such construction is similar to that of any other industrial building. This type of design is also used for plants that have special gastight housing around the reactor in the interior of the building. An example is shown in Fig. 1 , which is a plan of the swimming-pool reactor building at the International Conference on the Peaceful Uses of Atomic Energy at Geneva, Switzerland (1955). The reactor is centrally located with space on all sides for observers, and a prominent position has been given the control room. Figure 2 indicates the building elevation. The ORR, a tank-type pool research reactor, is housed in a four-level mill-type 13-160

Sec. 13-6]

REACTOR

BUILDING DESIGN

13-161

building with structural steel frame and insulated metal-panel siding. The space requirements dictated a building 108 ft long with a central crane bay 60 ft wide and wings, one on two 20-ft-wide service The crane each side as shown in Fig. 3. bay height is 71 ft, and the service bays 1 1 1 All construction below are 36 ft high. £300000 doods A full base grade is reinforced concrete. ment contains the subpile room, reactor substructure, future cell areas, shielded pump, operating gallery, and ventilating Via. 2. Geneva building elevation. The first floor houses gen equipment. eral experimental areas, the second floor has a balcony access around the reactor pool and offices, and the third floor contains laboratories, shop areas, control room, and access around the pool. Another example of this type of construction is that housing the Sodium Reactor Experiment at Santa Susana, California, shown in Fig. 4. The reactor is located


If dun JrimL

GRADE

13-162

MECHANICAL DESIGN

AND REACTORS

[SEC.

13

below grade with the upper surface of the top shield at floor level in the reactor room. Piping and equipment are also below floor level in trenches with the exception of motors for pumps and control rods, which are located above the floor level for ease The center crane bay has a 75-ton bridge crane for handling leadof maintenance. shielded coffins used to remove fuel elements. The building walls are light concrete. The cost of structures for which no gas-leakage provisions have been incorporated is similar to that for ordinary building construction except for the reactor shielding. 2

GASTIGHT BUILDING WITH NOMINAL PRESSURE SPECIFICATIONS


Research reactors in which considerable experimentation is to be done and where the reactor is cooled by circulating cooling water are often housed in a gastight build ing or a separate reactor enclosure designed for a relatively low pressure differential, in the order of 2 to 3 in. of mercury.

Fia.

5.

CP-5 first

floor plan.

A good example of this type of construction is the research reactor CP-5 at the The building around this reactor is the first gas-tight Argonne National Laboratory. reactor building in the world. It was designed on the principle that, a second con tainer for fission products outside the reactor itself would afford a considerable meas ure of protection against dispersion of fission products that might follow a reactor incident. The gastight portion of the main building, shown in Figs. 5, 6, and 7. is circular in shape and is constructed of concrete. Access into or out of the gastiga; portions of the building is through either a personnel air lock with two sliding doors or a truck air lock with overhead doors. These doors are on each side of an air cham ber and are interlocked in such a manner that the building always remains sealed. Ventilation is provided by centrifugal fans which maintain a slight negative build ing pressure (approximately J^-in. water pressure) at all times. Air is introduced and exhausted through large concrete ducts, each of which contains dampers thai are automatically closed in the event of excessive radioactivity in the buUding air.


Sec. 13-6]

BUILDING DESIGN

REACTOR

-TRUCK

ENTRANCE

Fio. SCRUB TOWER-

13-163

7.

Elevation. VAPOR SPHERE

-FILTERS

30,000

CONTROL HOUSE-, CONTROL ROOM-

FT.3

CAPACITY

-NEOPRENE DIAPHRAGM

18" DUCTS

Flu.

8. Schematic

of vapor dome for CP-5.

Liquid seals are also provided in each duct to ensure the gastight feature. Exhaust air passes through a holdup room with a capacity of about 1-min air flow before passing to the exhaust stack. This air holdup allows the fan motors to be stopped and damp ers and seals closed automatically before radioactive air reaches the exhaust stack. Following a possible release of radioactivity in this building, the negative pressure is maintained in the building by a "vapor dome" with a floating diaphragm, princi pally as shown in Fig. 8. The vapor dome contains a large diaphragm attached to

13-164

MECHANICAL DESIGN

AND REACTORS

[SEC.

13

at the bottom of the steel hemisphere. The diaphragm will fall slowly to The space above the diaphragm U maintain a negative pressure in the building. connected by pipe to the building, and the space below is vented to the atmosphere. The volume is such that 1,000 ft'/day in-leakage could take place for 20 to 30 days after an incident. The tightness of a building with a vapor dome can be tested by closing the building in the same fashion in which it would be closed following a radioactive contamination and allowing the vapor dome to expand to its largest volume. This volume and the time required for its filling are simple measures of the leakage rate of the building when corrected for changes in the humidity, temperature, and atmospheric pressure.


a flange

Fig.

9.

Armour Research reactor housing.

All internal surfaces of concrete are painted with three coats of a vinyl resin pain", which seals the pores of the concrete and provides a surface that can be easily decon The normal building leakage approximates Lj taminated of radioactive products. per cent of its volume per day with a pressure differential of \& in. HtO. A different design was adopted for the housing around the research reactor installed This reactor, which is a solution type, occu at the Illinois Institute of Technology. pies part of a new building constructed to house research laboratories and offices. All reactor facilities occupy one end of the building as shown in Fig. 9. The reactor room itself is 72 by 48 by 30 ft high. It is enclosed on three sides by a 1-ft-thkk concrete wall with a course of face brick on the outside and on the fourth side by i 1-ft-thick concrete wall in which there are two 3^- by 7-ft gastight swinging doors arranged in an air lock for personnel access, a 6- by 7-ft gastight door for materia! handling, and gastight observation windows from the control room and other observsi ion points. A feature of this reactor is its self-contained design as indicated in Fig. 9. All radioactivity is confined within the shielding structure of the reactor, and no fumes.

Sec. 13-0]

REACTOR BUILDING

13-165

DESIGN

smoke, or any other material will be discharged from the unit. The reactor utility room below the reactor is isolated from the rest of the area by two gastight c,-iii. steel casing that forms doors and is completely lined with a gastight envelope a

:'i

gases,

GASTIGHT ENCLOSURES WITH HIGH PRESSURE SPECIFICATIONS it

Even though reinforced concrete has been used for moderately pressurized reactor would seem economical only when the pressure differential to be sealed buildings,

Y .

Flo.

10.

Keactor component

arrangement

of

HRT. a

is

is

It extremely difficult completely to avoid gas leakage through very low. against substantial reinforced concrete and cracks that might develop, particularly when pressure rise occurs which would further complicate the tightness problem. If particular building arrangement appears well adapted to concrete construction. a


3

a

1

is

a

a

is 4

and also aids in decontamination. Measurement of completed concrete reactor buildings approximately 12 in. thick indicates that the minimum leakage which may be obtained ft'/hr per 1,000 ft* of surface for pressure differential of J-fi-in. water pressure. Within limits, the wall increases directly in proportion to the pressure dif leakage through such ferential. The best result that can be expected per cent volume leakage per 24 hr for a pressure differential of J^-in. water pressure, varying with size and shape of concrete housing. The use of caulked joints in the building design should be reduced to minimum, as their sealing effectiveness changes with time. Special consideration may be given to the design of such joints, which will reduce the possibility of leakage. One method that has been applied successfully has been the addition of iron filings with the mastic material. Upon oxidizing, the iron filings increase in volume, effectively pressurize the mastic, and seal any cracks that might exist in the joint. The cost of this type of construction will be the same as that of comparable con crete work with the exception that special features for sealing any services passing through the gastight walls must be included, as well as the vapor dome and its acces sories in designs similar to that of CP-5.

13-166

MECHANICAL DESIGN

AND REACTORS

[Sec.

13

it might be better and more reliable to use a separate thin steel lining for tightness and concrete for strength. One design along this line is exemplified by the arrangement selected for the homo geneous reactor test (HRT) at the Oak Ridge National Laboratory. This required accessibility and flexibility because of the experimental nature of the unit, and pro visions were necessary for complete containment of the contents of the reactor system should a leak develop as well as the ability to flood the system with water for main tenance or replacement operations. The reactor shield pit is 54 by 30.5 by 25 ft and is lined with %-in. steel plate, all reinforced with structural members not shown in tht The roof of the pit is of highpicture to withstand an internal pressure of 30 psi. A completely welded density concrete 5 ft thick in two layers of removable slabs. steel sheet between the layers extends across the top of the pit to form a gastight lid Figure 10 shows a section of this construction. 4

PRESSURE -RESISTANT ALL-STEEL CONTAINMENT

HOUSING

For atomic power generation, an all-steel containment housing seems most advan Considerable variation existageous as long as pressure containment is required. in these designs depending on the amount of equipment that may become radioactiv?
and on maintenance considerations. Steel containment vessels have been utilized for several plants, two examples which will be described in general terms prior to considering more detailed desircriteria. West Milton Containment Vessel

Of historical significance and the largest containment vessel built up to this time 225-ft-diameter sphere that was constructed in 1953 at Knolls Atomic Po*-:: Laboratory at West Milton, N.Y., for the submarine intermediate reactor (SIB' This vessel was built to house an experimental nuckai project as shown in Fig. 11. submarine power plant. The volume of the sphere 6,000,000 ft*, and the spfcerwas designed to contain the gases resulting from an incident at a pressure of 15 ps at normal design stresses. is

is a


4.1

Fig.

11. West

Milton containment

vessel.

Sec. 13-6]

reactor building design

13-167

The sphere rests upon a bowl-shaped foundation 179 ft in diameter and 38 ft deep. This concrete bowl serves primarily to take the load of the installation inside the shell, which is carried on a compacted rock fill inside the shell. The weight of the shell above grade, of the shell insulation, and of wind and snow loads on the shell are supported on 26 tubular columns rising to the equator of the shell. These columns were kept adjustable during the construction phase and were checked frequently to be sure that they were carrying their proportional share of the load. The shell is relatively thin for a vessel of this size, with plates approximately 1 in. thick, but is entirely self-supporting for its own weight plus insulation, wind loads, and Access to the interior of the shell is attained snow loads, with no internal bracing. sphere. The through one of two air locks, each consisting of a 16^-ft-diameter spherical shape for the air locks was chosen to obtain circular intersections with a large ball in order to avoid points of stress concentration. The shell was constructed of ASTM A201 steel conforming to ASTM Specification A300 for normalizing and conformance with Charpy impact tests. Edges of the Any edge defect found was plates were inspected by magnetic particle examination. All traced by ultrasonic inspection, and the plate repaired or rejected if necessary. The sphere was pressure-tested with soap-bubble inspec welds were radiographed. tion of the welds, and the pressure held for 48 hr, with measurements of temperature and pressure taken to check the tightness of the structure.


4.2

EBWR

Containment Vessel

The design of the containment vessel for the experimental boiling-water reactor built at the Argonne National Laboratory represents an alternate shape to that con This vessel is a vertical right circular cylinder with an structed at West Milton. ellipsoidal bottom and a hemispherical head, designed for an internal pressure of The vessel has a diameter of 80 ft, 15 psig with normal pressure vessel stresses. an over-all height of 119 ft, and a capacity of 400,000 ft3. Owing to the proximity of a densely populated area, it wae felt desirable to locate the reactor below grade. Thus, the main operating floor of the plant is at grade level with about 53 ft or 44 per cent of the vessel underground. The cylindrical shape lends itself very well to designs providing sufficient strength to resist the pressure of the earth backfill. During construction, the vessel is supported approximately 4 ft off the bottom of the excavation by temporary structural-steel columns to allow for inspection while A concrete mat under the exterior of the vessel the vessel is tested for tightness. was eliminated by utilizing the concrete inside the shell as an inverted dome to dis The tribute the loads that are carried down through the interior walls and columns. 4-ft gap between the bottom of the shell and the bottom of the excavation was filled with plain concrete to serve as a medium for direct transmission of the vertical loads from the shell to the soil as shown in Fig. 14. The shell is % in. thick and is an entirely self-supporting structure for its own weight plus all external loads above grade with no internal bracing above grade. To with stand the compressive forces against the shell below grade due to the active soil pressure, a 2-ft-thick concrete wall lines the interior of the shell. All access into the shell is attained through one of two personnel air locks. The shell was constructed of ASTM A201 steel conforming to ASTM Specification A300 for protection against dynamic loading of the vessel under low-temperature Upon completion of the vessel, approximately 10 per cent of the welds conditions. were X-rayed and the vessel tested for tightness. This test involved maintaining the test pressure for 72 hr and resulted in calculated leakages of less than 500 ft'/day. 4.3

Shell Volume and Shape

If the coolant contained in the reactor system is suddenly released as the result of an accident, all the vapor formed by flashing should be contained within the shell. The curves of Fig. 12 give the number of cubic feet of free shell space required for

13-168

MECHANICAL DESIGN

AND REACTORS

[SEC.

13

each pound of water contained in a reactor system that may be flashed to steam. The pressures marked on the curves correspond to the saturation pressure in the pressurized-water or steam system for different conditions between 200 and 2,000 psia. These curves can be used directly when a boiling-water reactor is under consideration, but when a pressurized-water reactor is being considered, the pressure curve corre The abso sponding to the average water temperature in the system should be used. lute pressure that will result in the shell after complete flashing is shown as the abscissa. These curves indicate that up to 60-psia resultant shell pressure a marked

drop in volume is required per pound of water while for higher shell pressures

the


34,

0 20

30

ULTIMATE

Fio.

12.

40 50 60 70 80 90 100 110 120 PRESSURE IN CONTAINMENT SHELL PSIA

Containment-shell volume requirements.

curve flattens out. Approximately twice the containment volume is required kt 2,000-psi saturated pressure cycle as for 400 psi for the same final shell pressurr This figure also shows the temperature that the air-steam mixture will have in the shell after completed flashing before cooling occurs. For 30-psia shell pressure. » Tb» temperature of approximately 200°P will exist in the mixture after flashing. curves illustrate the condition immediately after flashing is complete and do not tab account of the cooling that occurs when the steam heats the reactor shell or other surfaces within the shell or any heat that may be added to the system by a chemki reaction following a rupture of the primary system. Where a steel shell withosn inside insulation is used, this cooling reduces the pressure fairly rapidly. The curves in Fig. 12 give the maximum pressure and temperature that willocr-T in the shell if no heat is transferred to the shell itself or to other interior surfaces dure* the short period when steam is flashed from the hot water. Certain tests, which have been made at the suggestion of the author, indicate its'


Sec. 13-6]

reactor building design

13-169

the actual pressure which occurs inside the steel shell is considerably lower than calculated when no heat was conducted to the shell. The tests during which the entire hot-water quantity was flashed in 5 to 8 sec indicate that a very active heat exchange takes place. Even in this short period, a considerable amount of heat is conducted to the shell and interior surfaces. These tests further indicate that it might be possible to reduce greatly or almost eliminate the pressure in the shell by the right placing and application of a sufficient amount of heat-absorbing material or water. Further reference is made to Proceed ings of the American Power Conference, vol. XIX, pp 651-659, 1957. Some of the shells constructed have been spherical in shape, and some have been cylinders with hemispherical and/or ellipsoidal end closures. The spherical shape is the most economical as far as steel-plate requirements are concerned. Spherical shells of 50-, 100-, and 200-ft diameter, respectively, require approximately 15 per cent less steel than cylindrical shells of 40-, 80-, and 160-ft diameter with hemispheri cal tops and bottoms and with the same cubic content as the spherical shells. There are, however, additional fabricating costs which partly offset these reduced steel The cylindrical vessels have the advantage that it might be easier requirements. to standardize on certain diameters and vary the height for the desired volume. Where concrete and other building work is to be done inside the shell, a cylindrical shell will result in simpler form work. The arrangements required for removal of fuel elements and the arrangement of control-rod mechanism for power reactors may require considerable building heights and fit better into a design with a cylindri cal than with a spherical shell. A number of small shells have been used that are connected with rather large pipes It is interesting to note that for a so that the combined volume will be available. given amount of vapor and fixed pressure the same amount of steel is required whether one large or a number of small shells are used. This is strictly true only for spherical but approximately correct for cylindrical shells. 4.4

Material for Shell

The steels used for ordinary pressure-vessel construction will be satisfactory. How ever, if there is a possibility of dynamic loading under low-temperature conditions, it is desirable to utilize steel conforming to the ASTM Specification A-300, which provides for heat treatment of the steel to improve its ductility at low temperatures and reduce the possibility of cold embrittlement. These considerations may be important even though the shell is insulated on the outside. SA201 and SA285 are generally satisfactory steels for use in constructing these vessels. The design of several of the steel shells that have been constructed to date has been based on the ASME Boiler and Pressure Vessel Codes, Section 8, Unfired Pressure Vessels, which require that the shell be designed with a safety factor of 4. Sugges tions have been made that the safety factor could be reduced to 2 because the full design pressure on the shell would occur only once while the Codes are prepared for a As long as it is deemed advis vessel that is stressed to its design limit continuously. able to use a containment shell, it would seem prudent also to design the shell on a fairly conservative stress basis. The shell will have special openings or configura tions that make it difficult to establish correctly the actual existing stresses at all points, and, as long as the art is new, it might be well to go slowly in lowering the A safety factor of 3 might be permissible where careful specified safety factors. fabrication and erection are done. The present codes also have requirements in regard to inspection of the vessel, etc., that may be difficult to follow in these designs, and it might be advisable to consider some revisions in the existing codes for these special enclosures. 4.6

Welding

The design of the joints The methods of fabrication are covered by the code. varies depending upon the thickness of the plates in the shell, and the welding pro cedures must be approved. Generally speaking, vessels with plate thicknesses of

MECHANICAL

13-170

DESIGN AND REACTORS

[SEC.

13

\i

1 in. or less are simple, while thicker plates are complicated by the additional stressrelieving and radiographic-examination requirements. For some shells, extensive X-ray surveys have followed the welding. In the case of the shell for the EBWR constructed at the Argonne National Laboratory, this X-ray survey covered approximately 10 per cent of the welds and involved a total of about 800 ft of radiographs. The cost of this radiographic survey was approxi mately $10,000 and was conducted over a period of 6 weeks. With an experienced fabricator and reasonable precautions, it would not seem necessary to subject the shell to an X-ray check except where there are complicated welding connections and as specifically required by the code. ASME Code for Unfired Pressure Vessels requires X ray of all joints in which the in. as W
\i

4.6

Test for Tightness

of the shell fabrication, leakage tests are conducted. These of pressurizing the shell to an agreed-upon value, usually below the design pressure, and soap or Frcon test on all welds, pressuriz The determination of the amount of ing the shell to the test pressure required. leakage requires considerable time and is complicated by the effect of temperature variations during the testing and the changes in the vapor pressure of the moisture in the shell during the testing procedure. The leakage can be determined by observing the pressure-volume-temperature In addition to relationship in the stored air and the application of the gas laws. the PVT calculations, the temperature determines how much the shell itself is affected For the EBWR, resistance thermometers were suspended in the air. during the test. and thermocouples were mounted against the shell inner surface for gas- and wall-tem Pressure measurements were made with mercury manome perature measurements. ters and barometers. Very accurate air-temperature readings are required. Aircirculating fans were located strategically within the shell to prevent stratification of the air. The air within the shell was kept saturated with water vapor. It was found that the leakage rate was less than the specified maximum of 500 fta/day for a 400,000-ft3 shell at 15 lb gauge. An error of J^°F or 0.03-psi pressure represents 2 days' leakage allowance. Upon

completion


tests should include a visual examination

4.7

Foundations

and Footings

The foundations can

be designed so that the equipment inside the containment independently of the containment vessel as shown in Fig. 13, or the loads of the internal equipment can be transferred to the foundations supporting both the shell and the internal equipment as shown in Fig. 14. A design as in Fig 13 will allow complete inspection of the containment vessel at any time following is* erection. A design as in Fig. 14 necessitates provision for temporary support of the containment vessel during its erection, inspection, and testing before any equipment or concrete is placed within the shell. vessel

is supported

V

4.8

Insulation

It is desirable to insulate the outside of the shell. This will eliminate rapid (tem The insulation perature changes and warping in the shell during daily operation. of a spherical shell is somewhat more complicated than a cylindrical shell. A wcjIIinsulated shell was obtained for EBWR by welding }£- by 3>£-in.-long studs to tfl\e shell, hot-spray-coating the shell with mastic, impaling factory-formed 3-in. insulating) blocks on the studs, and pressing the blocks firmly into the mastic with the joints\ After the insulating block has been placed, stud retainers should ' suitably caulked. be applied and trimmed flush with the surface of the insulation. The entire surface is then hot-spray-coated with mastic into which glass fabric should be embedded and is allowed to stand for 48 hr, followed by a final hot spray coating of mastic. Painting of the desired color may follow.

Sec. 13-6]

REACTOR

BUILDING

13-171

DESIGN

STEAM

GENERATOR


GRADE

STEEL SHELL SUPPORTS

Fig.

13.

Foundation design — independent

TURBINE

Fio.

14.

CONCRETE ANO MACHINERY SUPPORTS WITH FLEXIBLE AND TIGHT SHELL CONNECTION support

for container.

REACTOR

Foundation design — integral support for container.

13-172


[Sec. 13

Missile Protection

4.9

A missile shield may be provided to prevent possible explosion fragments from passing through the shell. Such missiles may originate in almost any portion of the The form that the primary missile pressurized system or from rotating equipment. protection takes may vary considerably, from a carbon-steel sphere surrounding the core pressure vessel to a laminated steel structure incorporated in the biological shielding. The latter design, which was used for EBWR, is such that the shield is Table

1.

Approximate Reactor Plant Space Requirements Research

reactors

Reactor room: Reactor room, including general experimental


and storage,

Installations

i

2

3

9,500 11,400

5,050 3,850

3,500

20,900 780 1,620 1,730 1,120 5,450

8,900 450 3,000 17,220 1,250 6,600

3,500 200 4,600 1,120 1,560 2,800

areas and reactor

Additional reactor auxiliaries and general experimental

Service

for Typical

areas, ft1. .

floor area, ft3

Power reactors

Plant capacity, elec, kw.

10

100,000

163,000 (nuclear)

11,500

68,000

80,000

80,000

4,070

20,100

2,300

18,200

22,800*

28,400

0.0407

0.123

0.200

0.268

0.285

0.158

1,230

3,100

Reactor with auxiliaries lo cated within containment vessel, turbine generator located external to contain ment vessel: ground Total projected area of containment vessel, ft* Area, ftVkw elec. capac

ity Control room, floor area,

3,750

ft«

12,500

5.000

4,660

5,030

0.374

1.005

Reactor with auxiliaries and equip turbine generator ment located within con tainment vessel: ground Total projected area of containment vessel, ft» Area, ft'/kw elec. capac

'

ity Control room, floor area, ft" 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Oak Ridge Research Reactor, Oak Ridge National Laboratory, Oak Ridge. Tenn. CP-5, Argonnc National Laboratory, Lemont, 111. Armour Research Foundation, Chicago, I1L Enrico Fermi Plant, Power Reactor Development Co., Monroe, Mich. Indian Point Station, Consolidated Edison Co., Buchanan, N. Y. BR-3, Mol, Belgium. PWR, Duquesne Light Co., Shippingport, Pa. Proposed Plant, Sodium Graphite, Consumers Public Power District, Hallam, Nebr. Dresden Plant, Commonwealth Edison Co., Dresden, 111. Proposed Plant, Boiling Water Experimental Boiling Water Reactor, Argonnc National Laboratory, Lemont, III.

* Figures are for reactor building, as there is no containment vessel as such.

250

500

REACTOR

Sec. 13-6]

BUILDING DESIGN

13-173

A possible rupture of the reactor pressure weakest in the downward direction. vessel would be directed downward, where a large mass of concrete is present that could absorb the energy of the missile without harming the containment vessel. Secondary protection may be afforded by lining the walls of the containment ves sel with reinforced concrete. Several buildings presently under construction will utilize 12 to 24 in. of reinforced concrete for this purpose. 6

PLANT AND LABORATORY

LAYOUT

Arrangement of equipment and service facilities varies with the reactor types and The arrangement should provide for easy access and for sufficient room For installations in which all or part of the to perform maintenance operations. equipment is to be housed in a gastight container, the size of the container may be governed by the amount of space occupied by the auxiliary equipment instead of volume required for gas containment. Space requirements for any individual installation will vary greatly, but as a general guide Table 1 has been prepared which indicates floor space provided for several research and power reactors. purposes.

6

BUILDING DETAILS


6.1

Air Locks

When continuous gastightness is prescribed, special building details are required. locks with two sets of tight doors for personnel must be designed. Doors must be interlocked in such a manner that only one of the two doors can be open at a time. For an emergency condition arising while one of the air-lock doors was open, or if an accident occurred to an operator on the inside with the inner door open, a second set of personnel air locks for emergency conditions has sometimes been provided. This second set of air locks should be reserved for emergency conditions only. The inter locking arrangement on the doors should be such that it can be controlled from the outside of the vessel under certain stipulated conditions.

Air

Fig.

15.

Gastight door, low-pressure

design.

13-174

MECHANICAL DESIGN

AND REACTORS

[SeC. 13


The air-lock designs are of two categories: those designed for a relatively low pres sure differential, which utilize an inflatable rubber seal, and those for high-pressure differentials, which are dependent upon a compressive force for sealing action. Typi cal designs for each of these systems are illustrated in Figs. 15 and 16, respectively. For personnel air locks, a 3 ft by 7 ft or a 3 ft by 6 ft door is satisfactory, and for handling material a 5 ft by 7 ft or a 6 ft by 8 ft door is often used. Larger openings become quite expensive for the high-pressure installations, although air locks of about 10 by 12 ft have been built for truck access in low-pressure installations.

Fig.

16.

Gastight door, high-pressure

design.

Provisions must be incorporated for handling fuel elements and other large equip ment through the shell. The opening in the shell for these purposes may be arranged without an air lock. In the case of large power reactors, this will be the preferred arrangement because of its lower cost. 6.2

Fuel -element Storage Pit

The choice of design of storage pits will have a marked influence on the reloading provisions for the reactor. In some instances, the reactor may be flooded, so fuel elements are handled under a sufficient depth of water to avoid the use of speriil " A fuel-element storage pit is then usually located adjacent, to the reactor coffins." and is equipped with special tools that permit removing the spent assembly from tbt reactor core, transferring it to the storage pit where it remains until radiation has dropped sufficiently to permit shipment to a reprocessing plant. Reactor sysrems not adapted for water flooding require special coffins designed for fueling and trans porting spent fuel elements to a storage pit. 6.3

Ventilation

The ventilation requirements and methods depend on whether or not the bufldioi is occupied during operation. The 200,000-ft3 low-pressure building for CP-5 is ventilated with 15,000 cfm of outside air corresponding to 4J.£ air changes per hour to

Sec. 13-6]

REACTOR

BUILDING DESIGN

13-175

maintain comfortable conditions within the structure. For a high-pressure building without attendants, consideration should be given to refrigeration and heating units in conjunction with a rather small quantity of outside air to maintain proper condi tions within the building. This arrangement avoids large openings in the shell that are required when outside air is used for cooling. The 400,000-ft' steel shell for EBWR uses four 20-ton air-conditioning units in conjunction with 3,000 cfm of outside air, corresponding to In all cases, exhaust air leaves air change per hour. through a duct system with holdup time to allow dampers to close before contami nated air can be discharged to the atmosphere. This holdup time varies considerably with each design and is about 1 min for the CP-5 building and 2 sec for the EBWR container. Filters for incoming air are used to reduce the quantity of dust introduced into the The types of filters used vary in each installation, depending on the building. quantity of air, from throwaway filters used with the air-conditioning system to electrostatic precipitators as employed for larger volumes of air. 6.4

Exhaust Stacks


Consideration should be given to the height and size of exhaust stacks through which contaminated gases are discharged in order to obtain dispersion and dilu tion. This study should include a review of the prevailing meteorological condi tions and the locations of existing population centers. 6.6

Water

Several of the steel containment vessels provide a large quantity of water to be This water supply may be used as a spray sprayed after a possible reactor incident. to decrease the pressure and temperature of a steam-air mixture resulting from the flashing of pressurized water into the containment vessel and to serve as a reservoir for a separate spray system to prevent overheating and vaporizing of the reactor core. Water should be supplied from a permanent overhead storage tank. The most economical arrangement incorporates an overhead storage tank at the top of The inside the containment vessel which utilizes the vessel as a part of the tank. tank, vented to the containment vessel, reduces the required elevation, as it does not have to oppose the pressure occurring in the tank in an emergency. The quantity of water required will vary with each design. In an overhead tank for the 5,000-kw EBWR installation 15,000 gal was used, and the 60,000-kw PWR installation was provided with 55,000 gal that could be pumped to the four contain

ment

shells. 6.6

Internal Surfaces

Care must be taken in the finishing of surfaces that may be exposed to radioactive in order to facilitate decontamination. All concrete or permeable contamination surfaces should be painted or coated, resulting in a surface that may be easily decon taminated by surface washing without extensive leaching. 6.7

Cable Openings

Special designs are required for passing electrical cables through the wall of the Leakage may occur between the cable and the vessel wall, through stranded conductors, and through the space between the cable and its insulation. To prevent leakage through the interstices of multiconductor cables, the cables Ttiay be assembled with suitable rubber fillers that, when the completed cable is vul canized, fill all spaces between conductors and between conductors and sheath. To prevent leakage through the conductor itself, cable conductors of size No. 6 and smaller may be furnished of the solid type. Conductors larger than size No. 6 may be of stranded construction. A section of the strands on both ends of these cables should be filled with solder and reinsulated to seal all spaces between strands effectively. ■vessel.

13-176

MECHANICAL DESIGN

AND REACTORS

[Sec. 13


A design used for sealing the numerous instrument leads consists of a well section of a cable pan welded to the vessel wall as shown in Fig. 17. This arrangement accom modates approximately 50 cables, and after installation of all the cables, the well is This compound expands slightly and fills all voids. filled with a cement compound. Spare gastight conduit sleeves should be provided to accommodate replacement or addition of cables that may be required after the installation is completed. With shells designed for relatively low-pressure containment, liquid seals may be They allow the greatest flexibility in design and of replacement used to advantage. or addition of cables.

;:

Fig.

17.

Cable seals.

Large electrical cables such as generator main leads may be brought through the wall in individual nonmagnetic steel sleeves welded to a nonmagnetic steel section of the shell. vessel

6.8

Drains and Sewers

Special consideration must be given to the methods of disposing of any wastes from This necessitates a careful analysis building in which radioactivity is present. of the drain and sewer system to ensure that lines which may contain radioactive wastes are not discharged into open systems in such amounts as to be deleterious. The actual form that the drainage system takes will vary with the particular circum stances at each individual installation; however, it may involve duplicate drain sys tems, extensive monitoring of waste activity, and on-site provisions for concentrating or diluting radioactive wastes prior to disposal. a

6.9

Miscellaneous

The general plant or building services will not be markedly different from those found in more conventional construction. The problems encountered in the design will generally be those associated with providing adequate shielding for personnel protection around certain equipment and provisions for maintaining equipment that These provisions will vary with each design, of course, but necessi is radioactive. tate consideration of the installation of remote-handling equipment, hot cells, and their attendant auxiliaries.

13-7

REACTOR CONSTRUCTION BY George A. Anderson

A successfully operating reactor is the result of combined efforts of specialists in all and engineering, with a considerable share of success due to careful efforts of the field engineers. Careful adherence to the specifications is essential, and the field engineer must always be on the alert for potentially troublesome items that may or may not be covered by specifications. This requires that the field engineer have considerable knowledge of nuclear reactors as well as experience in general field construction. The problems pertaining to reactor construction can be discussed in general, but ultimate solution can be obtained only after careful study of the specific reactor under A number of the components are common to all types but vary in consideration. design and may require considerably different erection techniques. Present state of development indicates two broad reactor classifications, namely, heterogene ous and homogeneous, depending on distribution of the fuel in the reactor core. Generally, construction problems of heterogeneous reactors may resemble those of a conventional power plant, while a homogeneous reactor resembles in some respects a chemical plant. In either case, supervision and workmanship must be at least as good as the best in present nonnuclear construction practice.'-"''^ The following topics discuss some of the deviations from nonnuclear construction practice in order to assist the constructor in avoiding some of the pitfalls encountered to date. Construction is assumed to be from that point at which design is complete and ready for fabrication.


fields of science

1

THE REACTOR

and drawings describe the plant and give detailed information of They generally place emphasis on those items which require special attention during construction. It must be understood, how ever, that, at the present stage of development, the reactor plant is seldom completely developed in specifications and drawings and requires full cooperation of all persons to give assistance during fabrication of components and construction. Specifications

components of the reactor system.

1.1

The Core

installed, and Core-support structure structures; e.g., controlrod channels in the core-support structure may be required to line up with bearings supported by the reactor vessel or by external structures. If all relating components ire not available, jigs or other inspection devices may be required to ensure a proper inal assembly. A careful study of clearances and tolerances" should indicate the equipment needed for assembly and inspection of components. The core structure is usually assembled in the pressure vessel before the vessel is nstalled at the site in order to simplify inspection. If not, procedures must be worked >ut for careful inspection of components to ensure satisfactory assembly in the field, t must be kept in mind that the parts must go together when made according to 13-177

Structural members must

be carefully inspected for possible flaws,

tested to make sure all members are in proper relationship. usually must be in some specific relationship to independent

MECHANICAL

13-178

DESIGN AND REACTORS

[SEC. 13

Hand fitting with drawings so replacement is possible with a minimum of trouble. files, grinders, or other tools in the field may make replacement impossible. Core structural materials are usually specified in detail, and in some cases the type of inspection is specified. ""The constructor should keep in mind that the core-sup port structure will be difficult to replace after operation has commenced and that a failure of these components or use of a wrong material may prevent satisfactory opera tion of the plant, •- Complete control of foreign material must be exercised in order to avoid possible Particular care must be taken to avoid inaccessible contamination of the system. pockets that cannot be properly inspected and cleaned. Specifications should indi cate potential danger points, but the field engineer must be on the alert for those areas which may have been overlooked as well as those indicated in the specifications. Unless permitted in the specifications, use of cleaning materials or other materials that may remain in the system must be approved by reactor specialists to be sure their possible presence will not cause trouble during operation. Initial loading of the fuel or fissionable material is not done until after completion The charging operation, or the critical experiment and trial operation of the plant. phase, is done by a highly specialized crew, ^~-

l

1.2


V

Reactor Vessel

'"'

Reactor vessels, as well as other components of the system, vary considerably in The constructor must be thoroughly familiar design for the various types of reactors. with the requirements of the vessel under consideration and be able to recognize the design requirements and methods of solution as presented by the specifications and The design may have been determined by any one or more of the following: drawings. Nuclear properties of the vessel materials resistance of the vessel materials 3. Fabrication limitations 4. Replacement possibilities 5. Applicable code 1.

2. Corrosion

■""

1.21 Procurement Problems. Frequently, considerable difficulty is encountered in obtaining special compositions of the material specified for the vessel, which may make it advisable to investigate delivery of the vessel very early in the construction Construction sequence may have to be altered, and special jigs, fixtures, program. and measuring devices may be required in order to avoid delay in construction due to a long vessel-delivery time. Considerable delay can be encountered by the rather thorough material inspection frequently required, particularly if the procedures are not understood and planned well in advance. It is not unusual to start special control of material at the steel null Serious delays were encountered in at least two and continue to the final product. cases where pipe was welded into the system in the field and was later found not to meet material specifications. This could have happened to various connections to the reactor vessel if proper care had not been exercised at the fabricator's plant as well as in the field. Leak testing of the reactor vessel, as well as all other components of the system, can It is advisable to include this step in the require considerable time and expense. early planning and do any pretest work that may be required in order to understand thoroughly the procedures specified and to reach agreement with the designers for Some special those points of the test program which may not be fully delineated. equipment may be required, both during shop fabrication and field erection, that As an example, a heliummay be rather difficult to obtain without ample lead time. leak-detector tost may cause delay in obtaining equipment and experienced personnel to conduct the test, in addition to possible delay in obtaining the necessary helium, especially outside the boundaries of the United States. ^~


Sec. 13-7]

REACTOR

CONSTRUCTION

13-179

1.22 Cleanliness requirements must be carefully followed throughout Cleaning. If they are not delineated in detail in the fabrication and erection of the vessel. specifications, the constructor should determine a proper cleaning sequence and obtain It is generally better to exercise careful control of each approval of materials to be used. step in fabrication and construction rather than attempt to clean after the plant is com pletely assembled. '-"Type of cleaning materials to be used is important, not only from the standpoint of effectiveness but because of the problem of removing the material and the potential trouble in the plant if the cleaning materials are not completely removed. An example can be cited where carbon tetrachloride was used in an HsO The carbon tetrachloride could not be completely drained, system to remove grease. Alcohol then was used to remove the so the system had to be partially dismantled. last traces of carbon tetrachloride, followed by a water wash to remove the alcohol. If this type of solvent was required, each component should have been carefully cleaned before assembly rather than waiting until after assembly. The constructor should understand the plan used for supporting 1.23 Alignment. the reactor vessel in order to ensure proper flexibility and relationship to fixed points of the system. The constructor generally must determine a method for alignment, In some based somewhat on construction and delivery schedule of components. cases, jigs or patterns are required in lieu of the vessel to locate pipes or experiment ports in the shielding. Another method may be to leave clearance around the pipe and port locations so that the shielding can be constructed first and the piping can be installed after Particular care must be taken in this latter case to fill the delivery of the vessel. clearance space completely in areas where shielding may be required. 1.24 Thermal Insulation.1- Reactors operating at elevated temperature require thermal barrier between the vessel and the biological shield. The thermal barrier Thorough is frequently a special type of insulation which must be carefully applied. inspection, and possibly test, is usually reqviired to ensure proper application of the If leak testing of the reactor vessel is required, the insulation insulating material. In this case, the constructor must leave cannot be installed until after the test. It must sufficient room for the test as well as application of the thermal insulation. be kept in mind that thermal insulation usually cannot be replaced once operation has started.

1.3

Control -rod Drive

The control-rod drive assembly varies from relatively simple units used on open pool research reactors to quite complicated units used on power reactors. All must be carefully handled according to the manufacturer's instructions and specifications. are precision mechanisms that must be carefully installed and thoroughly tested before plant start-up. Provision for Replacement and Maintenance. The rod drives must be 1.31 assembled and must function properly when all components are made exactly accord ing to drawing. No field fitting can be tolerated, because such work may make If difficulties are encountered, the designers replacement difficult or impossible. should be consulted for a solution and the necessary drawing revision made. Sufficient space should be allowed for maintenance and replacement. Control wiring should be placed where a minimum of interference is encountered. Assembly and disassembly should be tried to be sure the unit can be replaced. Various limit switches are usually required on the rod drive. These switches are adjusted during initial start-up, and possibly after operation when operating conditions are changed. They must be accessible for the required adjustment but made inaccessible by means of locks or other devices after the proper settings have been made. 1.32 Alignment. Control rods and their drive units require careful alignment with respect to the core structure and pressure vessel. If requirements are not clearly deline ated on drawings, the constructor must understand the design basis of the system and plan his construction sequence accordingly. Frequently, drive shafts are required to extend beyond fixed shielding or other permanent structure. Construction sequence may require the reference points to be established independently; e.g., the

They


13-180

MECHANICAL

DESIGN AND REACTORS

[SEC. 13

reactor vessel may not be completed at the time the shielding is constructed in the field. If possible, under these conditions, the constructor should try to install the components in a manner that leaves sufficient flexibility for final adjustment after all components are installed. 1.33 Testing. "'Rod drive units should be thoroughly tested before installation It is desirable to test the control rods, drives, or at least before reactor operation. and core assembly as a complete unit, particularly where rather close fits are required, All new designs of rod drives should be given life tests as in water-type reactors. on a prototype before final units are fabricated, generally requiring considerable time for development and test. The constructor should determine if this work has been completed when planning his schedule. «Preoperational testing of the plant should include replacement of a control rod and drive unit, particularly if radioactivity may complicate the operation. A number of reactor types have schemes for detaching and removing the control rods from the drives and require a removal operation as part of preoperational testing of the plant. ' -Control and shutdown of the reactor are related to rate of travel and neutron charac teristics of the rods. Speed characteristics can be readily determined during pre operational testing; neutron characteristics require careful material control and frequently an actual determination of neutron-absorbing characteristics. Control rods may also require corrosion testing as part of the acceptance. If clad, the rods may require testing to determine integrity of the cladding and also bonding of clad to the base material. Bond testing is quite difficult and usually quite time-con Rods may be hollow in addition to clad, injecting another test to determine suming. leak tightness of welds. Any one or all of these tests are in addition to the usual dimensional inspection, i 1.4

The Reflector

Reflectors made of solid material (e.g., graphite or beryllium), may be a source of trouble to the constructor unless he is thoroughly familiar with the particular material used in the reactor. The reflector region in a power reactor is usually less A power reactor generally has a relatively complicated than in a research reactor. simple reflector with no particularly difficult alignment problem. A research reactor always has a considerable number of holes penetrating the reflector, thus adding to the fabricating and assembly problem, in addition to the coolant holes required for both power and research reactors. Graphite and beryllium represent two extremes in solid reflecting material in that graphite can be machined with relatively inexpensive machining by a minimum of skilled help while beryllium requires special equipment and represents a health hazard to those handling this material. 1.41 The constructor generally has a choice of method for machining Graphite. graphite, usually dictated by the quantity of material. For a relatively small graphite zone, the method used in constructing the CP-5 reactor is good. The CP-5 graphite zone was machined to finish cross-sectional dimensions by the supplier and cut to shape as it was installed at the site. After all building and reactor concrete work was completed, the building was thoroughly cleaned. A wooden shod, approxi mately 12 ft square, was then moved into the building, and a fan and filter assembly was connected to filter and exhaust the air out of the shed. A band saw and circular saw were installed in the shed, with flexible hose connected from the tools to the exhaust fan to remove graphite dust during cutting operations. It was found that a skip-tooth blade was required on the band saw and carbide tips on the circular sawin order to get satisfactory life from the equipment. A sheet-metal pattern was made to show the location of all vertical experimental holes in the graphite, with pull tabs provided to indicate the level at which the various holes were to start. A large sheet of clean paper was laid on the floor, and inner and outer circles drawn to assist in laying the graphite blocks with a minimum of scrap. In this manner a layer of graphite blocks was laid out, inner and outer diameters scribed directly on the graphite, and hole locations made from the metal pattern. The individual blocks were then carried into the shed, cut to desired shape, and installed in the reactor. some

reactor construction

Sec. 13-7]

13-181

is

If

2 2.1

SHIELDING Thermal

Shield

used as containment vessel as The thermal shielding in some types of reactors In these cases, may have about the same degree of tightness as the well as a shield. similar leak-testing reactor vessel, and the constructor may be required to conduct procedure. This gen Positive cooling required in conjunction with the thermal shield. erally accomplished by coils around or in the thermal shield. The coils should be a line has been blocked carefully installed and leak-tested and checked to determine Failure of thermal-shield cooling passages may mean failure or partially restricted. of the entire plant, frequently with no way of correcting or replacing the system. Fabrication and erection present no particular problem; they generally require either setting rather heavy assemblies or fabricating the shield in the field. This concrete biological shield. There may be shield may serve as the inner form for differential expansion between thermal shield and the biological shield, requiring a As needed mastic coating on the thermal shield to allow the relative movement. basic seal points must be clearly delineated and the leak-testing sequence determined. a

if

is

is

a

it

a

is

L

2.2

Boral

if

;

is

is

Boral, a mixture of boron carbide and aluminum, clad with aluminum, frequently This material used in conjunction with the thermal shield. somewhat difficult to machine. Small holes should be punched rather than drilled large holes can usually be cut with fly-cutter turning at a very low speed (turned by hand possible). shear or sawed with Sheets can be trimmed with low-speed band saw. Ji-in.in., and somewhat less thick sheet can be rolled to a minimum diameter of about cracks arc not objectionable. Large sheets can be fastened to a steel thermal shield

if

A

8

a

a

a


is

a

is

is

is

is

is

a

a

A

)•£

Horizontal holes were located at the layers involved and cut as required. Particular elevations were established with a jig and hand woodworking router. Aluminum in. in diameter, was inserted in holes drilled between graphite blocks round stock, to hold the blocks in place. After the graphite was all installed, the inside diameter was machined by a wood router on an arm pivoted from a 3-in. pipe carefully located on the vertical center line of the reactor. Although satisfactory for the small amount of graphite required for CP-5, this system would not be practical for large graphite reactor. Brookhaven-type reactor has sufficient quantity of graphite to justify great deal more expense in tooling but generally does not require so many different shapes. The problem primarily one of machining to cross section, with cither corners trimmed or holes bored to form fuel and coolant passages. Installation must be carefully worked out correct and perpendicular holes are located where required. so the lattice spacing Beryllium. Beryllium metal 1.42 machined and sent to the site for assembly. anticipated Precautions taken in machining are well understood, and no difficulty done in when the work Precautions to be taken at the shop set up for this work. site, however, must be understood, and due allowance made in planning the job. , 1.43 Material Control. It generally advisable to require shoe coverings, gloves, Marking ink or crayon can and clean overalls for everyone handling the material. As much as be used only on approval in order to avoid serious contamination. possible, foreign materials should be kept out of the assembly area further to avoid The constructor should plan to have a greater than normal accidental contamination. number of showers and wash areas for personnel during this stage of construction. The method of inspecting this material must be fully delineated in the specification. If nuclear properties arc to be determined as part of material acceptance, considerable sufficient, the num a chemical analysis expense and delay may be encountered. ber of analyses per unit weight should be specified and type of analyses should be understood.

13-182

MECHANICAL DESIGN

AND REACTORS

[Sec. 13

with drive screws. Cartridge-propelled pins can be used if the steel is satisfactory in Cadmium plating may be required on the screws. composition and thickness. Specifications should define the amount of surface contact required between boral and thermal shielding if conduction to the thermal shield is the method used for cooling boral. This may require the inner thermal shield surface to be clear and rather smooth. Boral is somewhat easier to put on the inside of a curved surface if the boral is rolled to a slightly larger radius than the surface of the shield. If the design permits, it is easier to put boral strips over the joints between the boral sheet* than to try to trim and form scarfed joints between the sheets accurately. Holes for pipes and experimental ports can be handled in a similar manner where large holes are cut in the boral sheets, and rather close-fitting boral collars arc slipped in place to cover the gap.


2.3

Biological Shield

A shield consisting of alternate layers of steel plate and 2.31 Laminated Shield. masonite has been used. This shield is quite satisfactory but expensive. Small sec tions of this type of shield are occasionally used in particular areas where other types of shielding may not be sufficient, but seldom for entire reactors at the present time. Some trouble can be encountered if there is not a good fit between plates and not sufficient overlap between layers to be sure of avoiding a continuous seam from inner to outer layers of the shield. If y leakage is a problem, the steel may be replaced by lead, either as brick or as plate. ^ A satisfactory shield has been made by substituting water for the masonite, wi:h the alternate layers of steel plate. This type of shield should be much easier to con struct (without the masonite layers, but the steel may require cladding to prevent contaimin3ttBl>«
"

I bag Portland cement.

Magnetite

Ji-in.

punchings

K-in* punchings

..

Found* 94 298 336 . 125

Sec. 13-7]

REACTOR

13-183

CONSTRUCTION

The laboratory report for the final mix is given in Table 1. The particular mix was established after 15 to 20 trial runs with various combinations of material. Actual pouring techniques used for pouring the concrete depend to a considerable extent upon the amount of material to be placed. Â small reactor may have approxi mately 200 yd', while the large graphite reactors may require approximately 3,000 yd3. In addition, work schedules or sequence may require a number of small batches, with some time lag between pours. A typical example of a small batch pour is that used for the CP-5. In this case, no single pour was greater than 10 yd'. Very little expenditure for equipment could be justified for a pour of this magnitude, so only Wooden hoppers were made for steel scrap and equipment on hand could be used. limonite; a small tractor with scoop attachment was used to keep the hoppers full. 1. Typical Dense Reference number Date sampled Location of placement Batch number

Table

Proportions


Class concrete Cubic yards Cement Fine aggregate Coarse aggregate Coarse aggregate

Water Admixture Time sampled Weather Concrete

temperature

Slump Wet density

Appearance Storage Date of test Age of test Specimen number Dimensions Density, lb /ft" Compressive strength Remarks

Concrete Batch — Research Reactor Project 57L-492BC Apr. II. 1957 Center foundation to I ft below floor level 45 1-3. 17-5.32 by weight 2.500/3.000 at 28 days 1.8 (approx) 94 lb "Iron Clad" 298 lb magnetite 376 lb >i-in. punchings 125 lb ?i-in. punchings 4.9 gal

None

1:50 p.m.

Fair, 54°F 62°F 2>i in. 275.0 lb /ft*

Satisfactory Job, then laboratory fog room May 9. 1957 28 days 24B 6 by 12 in. 278.5 5.060 Strength good

24C 6 by 278.0 5,220

I2H

in.

A wheelbarrow scale was used to weigh the material before it was dumped into the The mixer, a standard 1-yd3 machine, had to have the mixing blades rein mixer. Approxi forced to withstand the greater impact and wear from the steel and iron ore. mately 0.1 yd' was mixed per batch. The steel scrap, limonite, and cement had to be put into the skip before it was dumped into the mixer in order to obtain uniform mixing. The limonite was stored on an open wood platform, and consequently there was considerable difference in water content in different parts of the pile of material. This required a continuous watch of water addition during mixing and loss of some batches due to excessive water. A fork truck with dump bucket was used to transport the mix to the reactor. Wooden troughs were used where placement was difficult, but a majority was poured rfireetly. The Massachusetts Institute of Technology reactor was also poured in small batches, so it also had the limitation on equipment cost. The magnetite ore was covered with canvas, so the water problem was avoided. In both the CP-5 and MIT reactors, a test pour of several cubic yards was made to test the mix and pouring procedure by filling special shielding plugs required for the finished reactor. Typical test results of this concrete are shown in Table 1. Large continuous pours have been successfully made at Hanford using the pre packed method. In this method, the coarse aggregate is prepacked in the forms and the void space filled with a special grout. This method minimizes segregation in a large pour by avoiding excessive drop of the mixture and gives a concrete of uniform density, even in restrictive areas. Coarse aggregate can be blended accurately with


13-184

MECHANICAL

DESIGN AND REACTORS

[SEC. 13

The fine grout is then available equipment and carefully placed in the reactor. pumped in through pipes, which direct it to the bottom. ' This technique has been used for the barytes-type shields in the MTR and ORR as well as the limonite-steel shields at Hanford. 2.33 Hole Alignment. Reactors frequently have access holes through the shield Location of these holes is usually rather precisely deter to or into the reactor tank. mined and may require considerable effort in the field to meet dimensional tolerances. Tubes or lines must be set according to some geometric pattern, which must be under If not delineated in the specifications and drawings, stood by the field engineer. reference points must be established by consultation with the designers. With this A system of wires and transits has been used for line-up with fair success. system, angular location of the various holes is generally determined from the reac Reference marks are made on tor center before any shield construction is started. The transit or taut wires can the floor, near the outer wall of the reactor building. then be used from the outer reference marks to check parallelism and vertical loca tion by setting a target at the reactor center. The hole liners can be located and securely fastened to inner and outer shield forms or walls and rechecked before pour ing the shield. A collimator has been used for somewhat more precise line-up than possible with the transit. In one case an aluminum spool was made to locate collimator and target Rubber O rings were in the respective centers of the reactor tank and port liner. The center tube used around the disks of the spool to center the spool in the liner. was accurately machined parallel to the O-ring grooves in the front and back disks. A target was inserted into the corresponding hole in the reactor tank, and the two A somewhat similar system was used for aligning about 2,000 tubes readily aligned. in front and rear faces of a large graphite reactor at a considerable time and cost saving over other methods that were tried. 3

CIRCULATING SYSTEM

A considerably excessive amount of time and money can easily be spent if the fabri cation and construction of the circulating system have not been carefully planned. For purposes of discussion, this article includes all piping and components of the various circulating systems required by a reactor. 3.1

Testing

Special examination of welds should cause no particular trouble, but leakage speci fications can be very expensive and time-consuming. Test procedures must be thoroughly understood and carefully planned before Component testing of pumps, valves, and pipe subassemblies piping is started. should be done in the shop in order to decrease the amount of testing and repair in the field. Frequently the entire primary system may have X-ray examination of all welds, requiring thorough planning and scheduling to make sure welds have not been In one case some pipes had to be chipped out of a concrete structure overlooked. because they had not been X-rayed, and other welds made at the same time were rather poor. Those welds in the buried pipes were found to be faulty and had to be repaired. v_Leak testing may require mass spectrometer examination, using helium as the test gas. This type of testing requires rather expensive equipment to be used in the field If freon or other materials are used for leak testing, their by trained personnel. possible effect on materials of the reactor system must be understood and they may be used only upon approval of the reactor designers, •t^-' 3.2

Cleaning

All sections of pipe and components must be carefully checked for extraneous material before, during, and after assembly. Generally, component cleaning before

Sec. 13-7]

REACTOK

CONSTRUCTION

13-185

is more feasible than complete cleaning after the entire system is assembled. case, fine mesh screens should temporarily be put into the circulating lines during initial testings of the system. Screens must be strong enough to withstand maximum pump discharge pressure so complete clogging will not cause the screens to come free and damage other parts of the system. Usually the screens can be removed after the initial test period and before the reactor is put into operation, and The coarse screens are generally sized to remove solid coarse screens substituted. assembly


In any

— material that could block fuel coolant channels. be carefully delineated in the specifications and con 'Cleaning requirements should sidered at every step of fabrication and construction. Stainless-steel castings may require particular attention to be sure all core sand or other extraneous material has been removed.1 - In one case, valves had to be disassembled in the field and the valve bodies given a 15 per cent HNOj, 5 per cent HF treatment at approximately 180°F to remove core sand. This was followed by some hand grinding. Marking ink and other materials used during fabrication of components can cause trouble if not carefully removed, preferably immediately after they are used in the shop. A low-meltingpoint alloy, high in antimony, was used in one case to fill a small pipe when a sharp bend was being made. The material was melted out, and the pipe given an acid cleaning and degreasing before it was installed in a reactor system. Approximately 6 months after the unit was put into operation, radioactive antimony became notice able in the equipment, and finally the unit had to be shut down and cleaned. "^Cleanli ness control can be more readily exercised if it is possible to complete the reactor building before starting assembly of the piping systemic Those lines which must be installed before or during concrete pouring must be kept sealed at all times. ;^-Final cleaning of the assembled plant depends upon the care taken with the com ponents and the amount of control possible during assembly. A water-cooled reactor may require no more cleaning than several days' circulation, using high-purity water. Analysis of the water at frequent intervals will determine the relative success of the materials-control procedure. An appreciable quantity of sand was found in one The sand had been overlooked during assem reactor during this phase of operation. bly of components, and a sandstorm during final assembly stages had probably added more to the system. Components of the system had to be taken apart to remove the sand, 3.3

Drying

Heavy-water systems Reactor systems may require drying in addition to cleaning. be free of ordinary water to prevent dilution of the 1 )2G. This was accomplished in the CP-5 by isolating the reactor tank from the piping system and circulating hot, dry air through the reactor tank. This was continued until the desired moisture The piping system, which could withstand a lower content of the air was reached. pressure than the tank, was dried by connecting a vacuum pump to the system and tracing all lines with a gas torch. This method is very effective if considerable care is taken to avoid water pockets and the heating is thorough.- One section of pipe did contain water, which was detected by noting that the pipe was very cold (the vacuum was low enough to freeze the water). Infrared heat lamps were distributed through out the equipment room to raise the ambient temperature to about 110°F before the vacuum drying operation was started. Occasionally the vacuum pump was turned off and pressure raised to atmospheric by feeding helium into the system. The pump was then started and exhausted through a cold trap to check moisture content. This drying procedure continued for 2 days. from oxygen as ( Sodium systems require a similar degree of drying and freedom well. Other reactor systems may have other particular requirements which must be thoroughly understood and carefully conducted.

must

3.4

Materials

Reactor systems frequently have very stringent material requirements which may

need

thorough materials control, starting at the source of supply and not ending until

13-186

MECHANICAL

DESIGN AND REACTORS

[SEC.

13


Too many of the existing reactor plants the plant is put into satisfactory operation. have had considerable trouble from careless material control, and at least two have encountered considerable expense in removing improper materials from piping systems. Temporary components that may be added by the constructor should be of proper materials and removed as soon as possible. In one case a mild-steel cover was used After leak testing, the cover was left on for for leak testing an aluminum tank. When the cover was removed, after preliminary operation prior to reactor operation. the test run, the system was found to be badly contaminated with iron and had to be partially dismantled for cleaning.

SECTION

14

ISOTOPES by

PAUL

C. AEBERSOLD, M.A., Ph.D., Assistant Director for Isotopes and Radiation, Division of Civilian Application, U.S. Atomic Energy Commission. CHARLES E. CROMPTON, Ph.D., Associate Technical Director, National Lead Company of Ohio, formerly Energy Commission.

GLENN CLEWETT, A.B., Nuclear

Co., formerly


Director,

Isotopes Division,

U.S. Atomic

Group Leader, Sterling Forest Laboratory, Union Carbide Director of Materials Chemistry Division, Oak Ridge

ARTHUR F. RUPP, B.S.Ch.E., Generated for wjivans (University of Florida) on 2015-09-23 03:00 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

Deputy

Services Superintendent,

Laboratory

Oak

Ridge

National Laboratory. FRANKLIN T. BINFORD, B.S., Senior Development Engineer, Reactor Operations Department, Oak Ridge National Laboratory. GEORGE A. GARRETT, Ph.D., Superintendent of Operations, Analysis Division, Oak Ridge Gaseous Diffusion Plant. C. P. KEIM, A.B., M.Sc, Ph.D., Director, Technical Information Division, formerly Director, Stable Isotopes Division, Oak Ridge National Laboratory. NORMAND, Ph.D., Stable Isotope Research and Production Division, Oak E. C. Ridge National Laboratory.

CONTENTS 14-1

ISOTOPES

AND THEIR USE

PRODUCTION OF RADIOISOTOPES BY ARTHUR F. RUPP AND FRANKLIN T. BINFORD 14-3

BY PAUL C. AEBERSOLD AND CHARLES E. CROMPTON 1 2 3 4

Radioisotope Distribution Industrial Applications Other Applications Isotope Licensing and Distribution References

PA OF. 14-2 14-3 14-10 14-12 14—14

1 2 3 4

Basic Concepts Chemical Exchange Other Processes Commercially Available Isotopes References

Irradiation 14-20 Chemical Processing 14-31 Radioisotope Separations Procedures. . . 14—32 Radioisotope Processing Facilities 14-30 Reference*

ISOTOPE

SEPARATION BY CHEMICAL EXCHANGE AND RELATED PROCESSES BY GLEN CLEWETT

14-2

PAor. 1 2 3 4

14-37

GASEOUS-DIFFUSION SEPARATION PROCESS BY GEORGE A. GARRETT

14-4

14-15 14-17 14-22 14-24 14-25

1 The Cascade References

14-38 14-43

THE ELECTROMAGNETIC 14-8 SEPARATION OF STABLE ISOTOPES BY C. P. KEIM AND C. E. NORMAND 1 The Calutron 2 Operation of the Calutron Separated 3 Survey of Iaotopea

Calutron References

14-1

14-44 14-50

in the

14-54 14-58

14-1

ISOTOPES AND THEIR USE BY

1

y

a

is

fi,

Before the advent of the nuclear reactor, the limited quantity and high cost o: radioisotopes then obtainable only from particle accelerators such as the cyclotroa restricted their availability to just a few investigators. Consequently, the important of radioisotopes as research and testing tools, although recognized, could not fully be exploited. The development of nuclear reactors for carrying out the U"* fission process under controlled conditions, however, afforded large quantities of low-cos: fission products, particularly radioisotopes of elements from atomic numbers 30 to 6-! as well as high neutron fluxes that could be utilized to produce desired radioisotope by irradiation of elements in the reactors. Although about 1,000 radioisotopes are known, physical considerations such is half-life and mode of production render only about 100 of them of practical valje. These comprise the "stock in trade" of the radioisotopes distribution progrsn. Others are produced in small quantity for special research use by groups located near the reactor or accelerator. The versatility of radioisotopes as tools in research, medicine, agriculture, and industry derives from the wide variety in their chemical and physical properties free Chemically they cover the entire periodic system plus ir.» which one can choose. The range of compounds into whose molecules the tracer atoms can be incorporated. and rays; X rays; and character of radiation varies over a range including a, Each of these positrons and, by secondary means, can be extended to neutrons. Equally important. th> wide energy range. radiations, in turn, obtainable over half-lives of radioisotopes cover a tremendous range from billionths of a second billions of years.

RADIOISOTOPE DISTRIBUTION a

Early in 1946 the Manhattan Engineering District established an Isotopes Bran.-fc The first announeemre: distribution program. at Oak Ridge, Tenn., to administer of radioisotopes available under the program was published in the June 14, 19#% The first purchases for industrial use, in rubber and hydrocarUa issue of Science. research, were authorized in the following month. Less than months following the start of the program, the Atomic Energy Com mission took over the administration of the atomic energy project from the Man In a short time isotope distribution became a wellhattan Engineering District. established and growing program. In the first year of the program, approximately 280 radioisotope shipments wermade to 83 institutions that used them primarily in fundamental research problem? During 12 years of distribution, approximately 105,000 shipments have been mjî from AEC production laboratories to over 3,400 institutions throughout the country Nearly half of these groups are medical institutions and slightly less than half art industrial firms. At the close of 1957, 60 countries outside the United States had In the early days of the radioisotope? received 7,623 shipments of radioisotopes. At the end of program individual shipments averaged about 30 millicuries. year? Today, excluding shipmeotthe average shipment was approximately 130 millicuries. 14-2 5

6


Paul C. Aebersold and Charles E. Crompton

Sec. 14-1]

AND THEIR

ISOTOPES

USE

14-3

If Co60 is included, the average is of Coeo, the average is about 1,400 millicuriea. about 10.9 curies. The sharp increase in quantity per shipment in part reflects commercial participa tion in the processing and redistribution of radioisotopes. The total number of isotope shipments received by ultimate users is several times greater than the 105,000 Many commercial firms purchase radioisotopes in bulk quantities and listed above. reprocess them into such forms as labeled compounds,* sterilized pharmaceuticals, and custom-manufactured radiation devices; thus, a single large unit of radioactivity from an AEC laboratory may represent hundreds of shipments to individual licensed customers. In addition, thousands of shipments have consisted of small generallySuch generally-licensed quantities are readily licensed quantities of radioisotopes. exchanged among institutions and arc not included in the total shipments mentioned above. 2

INDUSTRIAL APPLICATIONS Radiography

2.1


More than 450 firms are now employing radioisotopes for purposes of radiography (illustrated in Fig. 1). Reactor-produced Co'0, Cs137, and Ir1,s are now used for

advantages:

I -versatile 2- INSPECTION 3- VERY HIGH LOW COST

l ie. 1.

and

reliable inspection

WITHOUT DISMANTLING ACTIVITY SOURCES AVAILABLE AT MADE

Measuring thickness and defects of metal castings with radioisotopes.

inspecting metal castings and welds for possible flaws in much the same way that radium and X-ray devices have been used heretofore, but they are cheaper and easier to handle and often possess more suitable radiation energies. Savings in both material and weight are possible where castings or the joining welds In designing for the use of welded plate, nonradioare known to be free of flaws. graphed welds are rated as 80 per cent efficient, whereas the figure used for radio graphed welds is 90 per cent. 2.2

Gauging

Radioactive gauging is another large industrial use of radioisotopes, with some 500 The ability of nuclear radiation to penetrate companies employing such gauges. material and, by its degree of penetration or reflection, to furnish information regard ing the material has made possible a large variety of testing and measuring instruments. 2.21 Transmission Gauge. In the transmission-type gauge, illustrated in Fig. 2, the emergent radiation intensity falls off exponentially with absorber thickness:

/ * Some 1,000 "tagged" compound*

=

Itf-i"

are commercially

available today.

14-4

ISOTOPES

[Sec.

14


where /o is initial intensity, / is the emergent intensity, t is thickness, and m is the absorption coefficient corresponding to the particular material and the type and energy of radiation. For practical purposes the thickness of material actually is found from an empirical A given thickness of a material calibration curve of / versus t for the particular job. may reduce a beam of hard 7 rays (small n) by only an insignificant amount but mayabsorb about half of a soft beam (large u). Since sensitivity to slight variations in the thickness would be poor in the first case but good in the second, ^-thickness gauges They are being used have proved to be most popular in gauging thin sheet materials. routinely to gauge mass, density, or thickness in an unbelievably large number of situations, i.e., metal foil, artificial leather, tire fabric, boxboard, plastic sheet, paper, floor tile, adhesive tapes, ink films, and cigarettes. Refer to Tables 1 and 2. Strictly speaking, these gauges measure mass of material rather than dimension. Thickness is obtained on the tacit assumption that the density and composition of the The gauges can, therefore, measure density with equal material remain constant. The 0 source is, We also find many 7 "density" gauges used in industry. ease. however, the most versatile of all radiation sources now in use.

k«

advantages:

I

-

23Fig.

2.

RADIATION

RADIOACTIVE

SOURCE.

source selected to suit MATERIAL

NO CONTACT-NO

TEARING-NO RAPID AND RELIABLE

MARKING MATERIAL

Radioactive source for gauging thickness.

Compared with other elements, hydrogen has twice as many electrons per unit This absorption differential is exploited, weight and thus has a strong /S absorption. for example, in the hydrogen-carbon ratio gauge (shown in Fig. 3), which requires both measurement of density and ft attenuation. Beta-ray absorption by the sample increases with specific gravity and with the hydrogen-carbon ratio; variation is com pensated by the movable wedge absorber, the position of which is calibrated to give the desired ratio. The H:C ratio is important in determining the fuel value of a Table

1.

Approximate Range of Thickness Material

Aluminum

Measured

Density, g/cm"

Asbestos

2.7 2.5

Glass

8.9 2.7

Plastics Steel

(celluloid)

1.0 1.4 1.0

7.8

by Strontium Thickness,

in.

0.0002 0.075 0 002 -0.075 0. 0006 0 020 0.002 -0.075 0.005 -0.200 0.004 -0. 14$ 0.005 -0.200 0.0006 0.020 5-150 os/yd«

90

Beta Gauge

isotopes and theih use

Sec. 14-1]

Table 2.

14-5

Comparison of Process Control Process control, % (2 aigma limits, i.e., 95 % correlation)

Application

Industry

Conventional control (normally sampling

I'ourdrinier paper machine Tandem cold-roll mill

Metal* Rubber

Sheeting

calender

Tire fabric calender Calender


± 12 ± 10 ± 10

±6.5

Saturator dip coater Abrasive maker

method)

±7

Radiation

gauge with manual

control

± 10

±7 ±6 ±5.5 ±3.5

Radiation

gauge with automatic control

±3

±1.5 ±3.2 ±3.5 ±2.5

Automatic control provided an mprovement factor of 2 . 5 over conventional Automatic control provided an mprovement factor of 2 over ci nventional

hydrocarbon product and can be continuously monitored with the gauge for process control. Perhaps the chief advantage of radioisotopes for gauging is the range in pene trating power, allowing accurate measurement of many materials. The high-energy (1.33-Mev) y rays of Co60 are used routinely to measure J^-in. steel strip, and a The ultrasoft 500-millicurie source of CsU7 (0.6-Mcv) is used to gauge 0.2 in. of steel. (19-kev) 0 particles of tritium can measure the variations in 1-mil plastic sheets.

Fig.

3. Hydrogen-carbon

ratio gauge using 0-ray source.

No contact with the material is required, so that difficulties with easily marred or sticky materials are avoided. Where formerly it was necessary to stop the production machinery while a sheet product was measured mechanically, a fast-moving product can now be continuously monitored. In the brass industry (which has long faced a very difficult gauging problem) contact gauges proved unsatisfactory, particularly at high rolling speeds, because inaccuracy and marking of the product are undesirable. The successful use of fi gauges has increased production speeds and reduced scrap losses and off-specification products. 2.22 One of the interesting features of Thickness of a Coating by Reflection. j3 radiation is that quantitative information can be derived from the back-scattered

ISOTOPES

14~6

[Sec. 14

The source of radiation is shielded from the detector, and both are located radiation. in the same housing. The ability of a material to back-scatter 0 rays is a function of the density of inter acting electrons per unit volume as expressed by the atomic number. (The backtimes the mass absorption coefficient ji». scattering coefficient m, equals a constant The coefficients are expressed in terms of reciprocal milligrams per square centimeter, ranges between 2.5 and 3.0.) and the constant Through the use of Sr»°, Co»°, Cs1", Ru10«, and S" virtually the whole range of measurable thicknesses have been covered. The inherent accuracy of such a gauge is The ±0.05 per cent of sample thickness or ±0.01 mg/cm', whichever is larger. sensitivity of this type of gauge involves the factor [(Zb/Zc)" — 11, thus emphasizing that the atomic numbers of the backing Z\, and coating material Zc must be sufficiently different to result in distinctly different "infinite-thickness" detector currents.

/

/

Tracing with Radioisotopes

2.3

The ability to determine the presence, location, and amount of radioisotopes through For this reason their radiation renders them uniquely suited to tracing problems.

AUTOMATIC

TIMER-RECORDER


SEAL LEG

TIME RE0UIRE0 FOR BEAD TO PASS BETWEEN COUNTERS MEASURES FLOW RATE

I 23-

ADVANTAGES:

Fig.

4.

IMMEDIATE

INDICATION

NO INTERFERENCE

OF IMPROPER

WITH

POSSIBLE AUTOMATIC

OPERATION

PRODUCTION

ADJUSTMENT

OF

FLOW

RATE

Control of catalyst flow rate using radioactive beads.

they are widely used in industry for tracing materials both in bulk, as in fluid flow, Indeed, tracing is perhaps the and on the molecular level, as in product analysis. most important industrial use. The petroleum industry, for example, uses isotopes in oil-well development to find leaks, open formations, and permeable zones and to "tag" the acid used in acidizing wells. This same industry uses a radioactive compound to mark the interface between dissimilar products pumped through cross-country pipelines. Numerous In catalytic fluid-flow problems in the petroleum refinery are solved with tracers. cracking plants flow rate of solid catalyst particles is determined by timing the passage of an occasional radioactive bead (Fig. 4). 2.4

Measurement

of

Flow

Radioisotopes as just mentioned can be used to measure flow rate. The method of measuring flow rate by recording integral counts is illustrated in Fig. 5. Only one This detector is required, and the need for determining pipe volume is eliminated. use is based on the principle that the total number of y rays registered by a radiation detector on a pipe through which a definite quantity of radioactivity is passing is

Sec. 14-1]

AND THEIR

ISOTOPES

14-7

USE

inversely proportional to the flow velocity. Fewer counts are recorded when the The integrated count is independent of the variation in isotope passes rapidly. isotope concentration along the stream as long as the flow rate is constant. The validity of the method is due to the sensitivity of the radioisotope method, which permits measuring the longer tails of the build-up and fade-out curves that are undetectable with other tracers. To translate the total counts to an absolute deter mination of flow rate, it is necessary to calibrate the counting setup on a particular type and size of pipe involved. COUNTER OF (A) MILLICURIES RADIOISOTOPE INJECTED

SPREAD OF ACTIVITY "A" IS IMMATERIAL COUNTING TOTAL

RATE

COUNT

DEPENDS


Fia.

5.

ON TIME OF PASSAGE COUNTS =CONSTANT

DEPENDS

THEREFORE, FLOW

ON CONCENTRATION COUNTS/SEC =CONSTANT

X MC/GALLON

X SEC x MC/GALLON

IS GIVEN BY TOTAL COUNT GAL /SEC =CONSTANT X "A" x I /COUNTS

RATE

Measuring flow rate by integrated

count using radioisotope

©

tracer.

©

'—MACHINED CHIPS RADIOACTIVE TOOL USED FOR MACHINING

CUTTING TOOL IN IRRADIATED NUCLEAR REACTOR ADVANTAGES!

Fio.

I-

MACHINED CHIPS FOR MEASURED

RADIOACTIVITY

REPRODUCIBLE AND SENSITIVE THAN OTHER TESTS 2- FASTER AND MORE EFFICIENT 3- YIELDS KNOWLEDGE OF WEAR PROCESS

6. Measuring

MORE

cutting-tool wear and life by radioactivity tests. 2.5

Measurement of Wear

Considerable improvement in the sensitivity, speed, and accuracy of measurement metal wear has been realized through the use of metal parts made radioactive by neutron irradiation. By making approximately one atom in a billion radioactive the wear of piston rings amounting to 1/100,000 oz can be readily determined. In the machine-tool industry radioactive cutting tools are used to measure tool "life." Only 6 sec of actual cutting time is required, during which about 0.3 mK of wear takes place. Only about 10 millicuries represents the total activity of the cutting tool (Fig. 6).

of

2.6

In

Process Control

P" as

the textile-dye migration test, shown in Fig. 7, a few millicuries of are added to the predominant color of the textile pattern.

phosphate

soluble When the

isotopes

14-8

[Sec. 14


counter, suspended in the dye box containing the most sensitive color of the multicolor printing machine, starts to pick up radiation, it indicates that the stronger color has migrated from the fabric. Experience has shown that this technique of production This represents a real achievement control prevents "color soiling" of the fabric. in the textile industry, since the absolute elimination of soiling has seldom been achieved in the past without wasteful, too frequent replacement of dyes.

CRITICAL CONCENTRATION OF MIGRANT "PIRATE COLOR" REVEALED BY GEIGER COUNTER

Fia.

7. Detection

of dye migration with radioactive phosphorus, 2.7

P".

Processing

2.71 Acceleration of Chemical Reactions. Initiation of chemical reactions offers The often slow halogenation interesting possibilities in the chemical industry. reactions, for example, can be speeded up with nuclear radiation supplying the triggering energy. Oxidation reactions, another large group of important industm' processes, seem to rely in many cases on a chain reaction of free radicals in the present of oxygen. Radiation is effective in producing free radicals and may lead to easy oxidation of materials otherwise difficult to treat. Polymerization reactions may be enhanced by ionizing radiations. The rupturing of bonds that initiates the polymerization of many substances such as plastics and synthetic rubbers may be produced by radiation. New materials may thus be obtained that can be quite different from a product polymerized by chemical cat-sly&e2.72 Cross Linking of Polymers. A further result of radiation-induced bonl rupturing is the initiation of cross linking between neighboring molecules in polymers already formed, thus resulting in a quite rigid three-dimensional structure having different strength, melting point, and other properties (Fig. 8). Polyethylene con tainers, for example, which would melt under steam sterilization, can be conditioned by radiation-induced cross linking to withstand the treatment. Maehine .in-: instrument parts, also, can be fabricated from easily molded thermoplastic materal and then cross-linked by irradiation as a thermosetting material.J 2.8

Sterilization

Gamma radiation in large enough doses can destroy bacteria in a material without raising its temperature significantly. The absence of high temperatures is quite necessary in the processing of many items such as antibiotics and certain other drugs and this cold sterilization thus has great potential importance. In addition, large-scaV irradiation of perishable foodstuffs such as bananas, potatoes, and onions to inhihh

Sec. 14-1]

ISOTOPES AND THEIR USE

14-9

sprouting or spoilage holds interesting possibilities. Approximately 20,000 r delivered to potatoes inhibits sprouting for at least 18 months. The value of this type of radi ation use has been recognized since 1896. Several large research organizations arc now doing intensive research on cold sterilization with y radiation. This possibly large industrial use of radioisotopes is illustrated in Fig. 9. CROSS-LINKED POLYTHENE

HYDROGEN GAS

SOURCE

advantages:

?

+

RADIATION

i

polythene is: a-more heat resistant b-more elastic

-cross-linked

2-

c-less soluble

no heat or addition of foreign atoms -degree of cross-linking can be carefully controlled 4 -many new plastics can be synthesized

requires

3


Fio.

8. Activation of chemical

EFFECT OF RADIATION ON MICROORGANISMS (ASPERGILLUS NIGER)

STERILIZATION REMOVE

BLOCK

TO

FISSION PRODUCT

(MOLD

reactions. UNIT

REPLENISH -

SOURCES

NO RADIATION COUNT =45,000.000)

-

FISSION PRODUCTS

CONVEYOR EXIT

SPENT FISSION PRODUCTS

260,000 REP (MOLD C0UNT=0) ATOMIC

RADIATION

HAS

LARVAE IN PACKAGED

BEEN USED TO PRESERVE DRUGS-FOODS-KILL INSECT PRODUCTS-CONTROL PRODUCTION OF LIVESTOCK PESTS

Fig. 2.9

9.

Cold sterilization.

Electric Batteries

There are several methods for direct production of electricity from the energy released in the decay of radioactive atoms. This energy is extremely minute even when compared with that from conventional storage batteries but can be used for certain special applications requiring tiny amounts of electrical current. Beta rays ejected from an insulated radioactive source leave it with a positive The useful charge; this is the principle of the Moseley generator developed in 1913. industrial application of this principle has been realized in the development of one This high-potential, low-current battery is illustrated in type of atomic battery.

Fig.

10.

One type of battery takes advantage of the ion-multiplication principle to produce electricity from radioisotopes. The electricity is generated by the high-energy 0

14-10

ISOTOPES

[Sec. .r_f—


CHARACTERISTICS:

I- MAXIMUM

14

ANODE

CATHODE VOLTAGE

UP

TO 7kv.

2- APPROXIMATELY 1% EFFICIENT 3- 40mpo. AT ZERO VOLTAGE FOR lOmc 4- LONG USEFUL LIFE UNDER EXTREME 5- SMALL -LIGHT- INEXPENSIVE Fio. 10. Voltage production.

Sr90 CONDITIONS

These high-energy electrons are used to bombard a particles emitted from SrM. The pea-sized silicon P-N junction (the merging of two specialized types of silicon). P-N junction in turn releases about 200,000 slow-moving electrons for each high-speed electron striking it. About 10 millivolts is produced,* strong enough to cause an audible signal in a telephone receiver. The battery is about the size of a thimble and has a shelf-life of more than 20 years. 3

OTHER APPLICATIONS 3.1

Medical Applications

For the first time in medical and biological research, investigators have had » By suitable tool to study the behavior of a specific batch of atoms or molecules. comparing the behavior of radioisotope-labeled compounds in normal subjects with their behavior in those having disease, investigators find differences giving valuable clues as to the cause and cure of disease. In the field of medical diagnosis radioactive I"1 is being used routinely in severs! hundred hospitals to measure thyroid-gland activity (Fig. 11), brain-tumor locahxa-

MEDICAL

ACTION:

I

- DIAGNOSIS

23-

Fio.

11,

LOCATION

AND

TREATMENT OF HYPERTHYROIDISM

OF THYROID CANCER OFFSHOOTS (METASTASES)

TREATMENT OF THYROID CANCER AND METASTASES Radioactive iodine, I1", for diagnosing and treating thyroid gland disorders.

isotopes and their use

Sec. 14-1]

14-11

This same radioisotope is also widely used therapeutically tion, and blood volumes. in treating thyroid and cardiac disorders.

Radioactive P" is used therapeutically in about 500 institutions. It is considered the treatment of choice for polycythemia (overproduction of red cells) (Fig. 12) and offers some relief for chronic leukemia.

THERAPEUTIC ACTION!

I 23-


Fig. 12. Radioactive phosphorus, chronic leukemia.

ADVANTAGES:

Fio.

13.

PARTIALLY SELECTIVE UPTAKE SLOW PROTRACTED IRRADIATION INHIBITS BLOOD CELL PRODUCTION

P",

for treatment

of (a) polycythemia vera and (b)

I- EFFECTIVE

DOSE AT DEEP-SEATED TUMOR SMALL DOSE AT SURFACE 2- ALLOWS SELECTION OF IRRADIATION PATTERNS 3- ROTATION AND SHUTTER REMOTELY CONTROLLED

Rotational teletherapy,

-

using Coe0 gamma rays.

Radioactive Au1*8, injected in colloidal form into body cavities, such as those the lungs and abdomen, is useful in reducing the amount of liquid that collects in those areas in certain types of cancer. Very large sources of Co80 as "teletherapy" unite are being used in place of large X-ray machines for treating deep-seated cancer. Such units contain from 1,000 to 1 ,500 curies each, and the annual demand for this use alone approaches 300,000 curies

around

(Fig.

13). 3.2

of

Agricultural Applications

Radioisotopes have yielded extremely interesting and valuable results in the field Much new information has been obtained using radioactive P", Ca46, agriculture.

14-12

ISOTOPES

[Sec.

14

etc., to study the mechanism, rate, and extent of fertilizer uptake under varying conditions of cultivation and growth (Fig. 14). Some' 50 agricultural experiment " stations have programs in effect including studies such as migration habits of "tagged insects and fate of tagged insecticides.

HOURS


I -quick efficient 2- FOLIAR METHOD

i

way to apply fertilizer shows: 95% EFFICIENT-ROOTS 10% 3- CAN BE APPLIED WHEN MOST NEEDED Fio. 14. Fertilizer uptake by roots and foliage using radioactive phosphorus,

3.3

P".

Conclusion

It would be improper to conclude this brief resume' of isotope applications without giving due credit to t he versatility of radioisotopes as analytical tools. In all phases of academic, medical, and industrial endeavor radioisotopes find numerous application* in isotope dilution analysis, inverse isotope dilution analysis, and activation analysis. During the past several years the rate of growth of industrial applications of radio isotopes has increased rapidly, and there is every indication that future growth will occur with even greater acceleration. The savings to industry resulting from the use of isotopes have been estimated at $500 million annually. The future holds in store even more impressive and beneficial results from radioisotopes. 4

ISOTOPE LICENSING AND DISTRIBUTION 4.1

General Procedure

Distribution of AEC-supplied isotopes has been accomplished by means of both an authorization and a licensing procedure. In the case of radioactive isotopes, this procedure is designed to make the materials available under conditions that *1 A similar allocation procedure was used early m assure the AEC of their safe use. distributing concentrated stable isotopes because of the limited quantities of the materials available. The AHC's isotope licensing program is administered by its Division of Licensing and Regulation at Washington, D.C. Application forms, information on policies and procedures, and related services may be obtained from that division. Domestic distribution of radioisotopes includes elements with atomic number? 3 through 83 that can be produced in a nuclear reactor and that have half-lives sufficiently long to permit packaging and shipment. Limited quantities of tritium and Po210 are also available. A wide variety of concentrated stable isotopes is also available through sale or loan distribution. Applicants for radioisotopes submit Form AEC-313, "Application for Byproduct

isotopes

Sec. 14-1]

and

their

use

14-13

Material License," indicating the kind and amount of radiomaterial desired, the pro posed use of the material, experience and training of the user in radiation protection, and procedures and instrumentation to be used in assuring radiation safety. These applications are reviewed from the standpoint of adequacy of available facilities and the training and experience of personnel who will carry out the proposed work. The primary responsibility of the AEC is to determine that the radioactive materials will be used in a safe manner. Following review by the AEC, Form AEC-374, "Byproduct Material License," is issued. The applicant then submits a purchase order to the appropriate AEC radioisotope distributor or the private supplier of his choice. For procurement of stable isotopes from AEC laboratories, the customer should sub mit a completed copy of AEC Form 391, together with a purchase order, to the supplier. For procurement from non-AEC laboratories, the AEC Form 391 is unnecessary. 4.2

Types of Licenses


Most licenses are issued for the procurement of specific quantities of a single radioisotope; these are called specific licenses. However, under certain conditions, broad licenses are issued to large, experienced users of radioisotopes for the procure ment of large quantities of any reactor-produced radioisotope (of atomic number 3 to Broad licenses are 83 inclusive) and from any AEC national laboratory or supplier. available for (1) research and development activity, (2) large medical programs, and (3) commercial processing of radioisotopes for resale to other authorized persons. 4.3

Generally

Licensed Quantities

Sections 30.71 and 30.72 of Title 10, Code of Federal Regulations, part 30, lists certain devices and equipment incorporating byproduct material and small quantities of certain radioisotopes that are generally licensed. These generally licensed quanti ties may be procured from a distributor or supplier without the necessity of filing an "Application for Byproduct Material License." These quantities may not be combined or altered in such a way as to increase their radioactivity. Furthermore, these generally licensed quantities may not be adminis tered externally or internally to a human being except as permitted by a valid byprod They must be used, stored, and handled in accordance with uct material license. AEC radiological health-safety standards. 4.4

Available Materials and Services

Radioactive and stable isotopes are sold (or loaned) by the Commission through its national laboratories or by authorized suppliers. AEC national laboratories are the primary isotope producers or manufacturers. Suppliers are persons or firms who further process and resell materials usually obtained from AEC national laboratories. The basic products available from Commission facilities are:

AEC

1. Unprocessed irradiated

materials of targets furnished by the applicant b. Routinely irradiated batches or "standard reactor units" 2. Chemically processed radioisotopes* a. From target materials 6. From uranium (fission products) a. Service irradiations

Catalogues or brochures, with price lists, describing available isotopes and services may be obtained from a number of private and commercial suppliers as well as the following AEC national laboratories: * Processed

forms.

radioisotopes

are available from Commission distributors primarily

in simple

chemical

14-14

ISOTOPES

Argonne National Laboratory University of Chicago P.O. Box 299 Lemont, 111. Attn: Special Materials Dept.

[Sec.

14

Mound Laboratory Monsanto Chemical Company Miamisburg, Ohio Attn: Operations Division Reactor Testing Station Phillips Petroleum Company

National Brookhaven National laboratory Associated Universities, Inc. Upton, Long Island, N.Y. Attn : Isotopes and Special Materials

P.O. Box 1221 Idaho Falls, Idaho Attn: Atomic Energy Division

Oak Ridge National Laboratory Carbide & Carbon Chemicals Company P.O. Box X Oak Ridge, Tenn. Attn: Radioisotope Sales Department

REFERENCES The following references include general information for administrative or management personnel as well as some technical references for the many types of industrial uses o: radioisotopes. Bradford, John R.: "Radioisotopes in Industry," Reinhold Publishing Corporation, New York, 1953. 2. "Business Opportunities in Atomic Energy," a Forum Report, available from Atom* Industrial Forum, Inc., New York, May, 1954. " Isotopes, An Eight Year Summary," AEC, 1955, Superintendent of Docuroem3. U.S. Government Printing Office, Washington 25, D.C., price $2.00. 4. How Isotopes Work for Refiners, Oil Gas J„ 66 (38), Sept. 23, 1957. Applications of Radioactive Tracers, Second Pacifa 5. Guinn, V. P., et al. : Refinery-Scale Area National Meeting, ASTM, Los Angeles, Sept. 17-21, 1956. 6. Radioisotopes Help Solve Refinery Problems, Oil Gas J., 64 (69) : 99-103, Aug. 27. 19S< 7. Colishman, E. L., and R. F. Fish: Radiation Polymerization Makes Stronger Poly


1.

styrene, Nucleonics, 15 (10), October, 1957. 8. Savage, M. W., and L. O. Bowman: Radioactive Tracer Measurements of Enginebearing Wear, SAE National Fuels and Lubricants Meeting, Tulsa, Nov. 8—9, 1956. U.S. AEC: The Economic Aspects of Radioi»oto(9. Libby, W. F., Commissioner, Utilization, UNESCO Conf. on Radioisotopes, Paris, France. Sept. 17, 1957. Am. Dieiel. Assoc., 33 : 33-36. 1957 10. Wasserman, R. H. : Radiation Processing of Foods,

J.

11.

Urlain, W. M.: Sterilization Through Radiation, Milk Plant Monthly. 45 (2):

30. 12,

1956.

12.

Manowitz, B. : Atomic Energy and the Packaging Industry, Prc-Pack-Age, 1956.

13.

Foster,

George B.: Industrial Thickness Gages, Nucleonics,

14

(5):

ftf>

9 (6):S

67. 1956.

9.

14-2

ISOTOPE SEPARATION BY CHEMICAL EXCHANGE AND RELATED PROCESSES BY Glen Clewett BASIC CONCEPTS

All processes considered here have in common the development of a concentration difference between an enriched stream and a depleted stream and the multiplication of this concentration difference by countercurrent flow of the two streams. An arrangement of many countercurrent stages is called a cascade. Perhaps the simplest example of this is a continuous-feed distillation column with continuous tops and bottoms withdrawal. In fact, distillation can and has been used for isotope separation in many cases. The concentration difference existing between the enriched stream and the depleted stream after a single contact is usually expressed in terms of the separation factor a defined as follows: a

=

*/(1^>

- X)

x/(l

(1) w

where y = mole fraction of desired isotope in the upstream, i.e., Moles desired isotope Moles desired isotope + moles undesired isotope in the upstream irrespective of the chemical form or the presence or absence of inert diluents x = mole fraction of desired isotope in the downstream, irrespective of chemical form, etc. The minimum number of stages to achieve a given degree of separation is given by Fenske's equation \[yD(\ -xB)]/lxB(l yD)]\ ,„,

-

_ln

In a

is

is

where yo = tops concentration (or distillate) Xb = bottoms concentration When a is near unity as it usually is in isotope separation work, In a is commonly and replaced by a — 1 or termed the enrichment factor. When the enrichment small, isotopic concentrations in the two streams are nearly equal, which factor allows us to substitute xd for yD with negligible error. Thus

a

—

In 1

Nmin =

—— Xfl/(1

—

xB)

An actual producing plant must have more than the minimum number of stages. example, a so-called "ideal"* plant (see Cohen V and Shacter and Garrett5)

*

An ideal gaseous diffusion cascade may be denned as one in which there unequal concentration. Superscript numbers refer to References at end of subsection.

14-15

is

For

t


1

no mixing of streams

of

14-16

ISOTOPES

[Sec.

14

must have twice the minimum. A "square" plant * might have about 1.5 or 1.6 times the minimum number. With a fixed withdrawal rate of product at a fixed concentration, the interstage flow in the body of the cascade must exceed a certain minimum value. This is some times expressed as the minimum ratio of interstage flow to product flow, or (.V/D)**. Shacter and Garrett* have shown that this can be expressed in terms of other process variables as follows: (yD xn)[l + (« l)z„] „ m


(V\ \D/rain

-

(a

-

l)(l„)(l

-

Xn)

This gives (V/D)mm at the nth stage in terms of yi,, a — 1, and xn which is the mole fraction of the desired isotope in the downstream at the nth stage. In comparing isotope separation processes, it is usually necessary to consider to someThe processes we are considering can consume degree the energy requirements. energy in at least three ways. Some of the processes require energy to establish a concentration difference between the enriched and depleted streams, e.g., gaseoj; diffusion and thermal diffusion. All separation processes require energy input to maintain interstage flow. All require energy in various degrees to reflux the process streams at each end of the cascade, e.g., the reboiler and the condenser on a distillation column. These energy requirements for isotope separation processes are usually very great and become an important consideration in comparing processes. In recognition of this fact, Benedict3 has shown how it can be used as both a qualitative and quantitative basis for comparison. In order to compare net energy requirement-of different processes, it is important to have a common measure of the amount w useful energy lost in them. He employs the concept of "net work" defined as the difference between the useful work that might have been done in a thermodynamicauy perfect engine by the energy chargeel to the process and the maximum work ths; could be done in such an engine by the energy removed from the process. The degree of reversibility of net work in a separation process, or the extent to which the net work may be made to approach the minimum required by thermodynamics to effect On this baj separation, then provides a basis for classifying separation processes. processes can be classified as potent ially reversible, partially reversible, and irreversible Table Type of process

Column

1.

Classification of Separation Processes*

Potentially reversible

stage . . .

* M. Benedict,

Absorption

Distillation Reversible absorption (Countercurrent gas cen

Solvent extraction packed column

in

Multistage compression distillation

Solvent extraction rate units

in sepa

trifuge)

Separate

Partially reversible

Trans.

Irreversible

Thermal diffusion Countercurrent

eear"--

fuge

Aieotropic distillation Extractive distillation

Am. Innt. Chem. Engrt.. 43 (2) (February, 1947).

diffusion Mass diffusion Electrolysis

Gaseous

(With permission.)

In order to achieve significant enrichment, it is necessary to multiply the singlestage enrichment many times by connecting the stages together to form a ca«:3«' In some processes, such as distillation or chemical exchange, it is relatively easy w In fact, the cascade can be simply a column for such diffe-rentB* connect the stages. However, other processes such as electrolysis or gaseous diffusion require processes. stage* special interstage connections. This degree of complexity of connecting provides another basis for comparison. Using these two qualities, Benedict3 has made a qualitative comparison number of processes in a chart that is shown as Table 1.

of

a

large

* A "square" plant has uniform magnitude of interstage flow of isotopes from end to end in caox-i? • to an "ideal" plant which has less interstage flow in the stages at the product end than in tb< near the bottom end and is thus said to be a tapered plant.

Sec. 14-2]

isotope separation by chemical exchange

14-17

With equal separation factor, moving from processes located at the lower right in the chart to processes located at the upper left gives decreasing costs in general. The foregoing concepts provide sufficient basis for considering some of the advan tages and disadvantages of some of the more common isotope separation schemes. 2

CHEMICAL EXCHANGE

In searching for new chemical-exchange processes or evaluating known processes, it has been found to be convenient to define certain criteria of a successful chemical These might be expressed as follows: process. 1. The element should be distributed between two phases. 2. Rapid isotopic exchange must take place between the chemical forms in the two phases. 3. The equilibrium constant of the isotopic exchange reaction must be other than unity; i.e., the isotopes must differ in their NH3 equilibrium distribution between the two phases. 4. Means for achieving total reflux at NH4NO3 the top and bottom of the plant must be V —EXCHANGE COLUMN provided.

r


Examination of one of the first systems and a very successful one devised by Thode and Urey,* the enrichment of Nls shown

schematically in Fig. 1, indicates the essen tial nature of these criteria. The exchange reaction involved here is

w-

DIRECTION

OFN15 ENRICHMENT

N'5H, + N»H,+ — N'«H3 + Nl5H,+ "1

with

a calculated

—

equilibrium constant of NoOH I NM.NOjI-* 1.035 at 298°K. This equilibrium constant I I SOLUTION ENRICHED HT is equal to the separation factor for the I (g-ljo I INN'5CANBESAMPLED\* 3 system provided the following two criteria HERE j are met: lMfluxjectjonj 1. The chemical species involved in the Kio. 1. Thode and Urey system for N" exchange reaction must contain or exchange enrichment. only one atom of the isotopic element. (In this example NHS and NH(+ each contain only one atom of nitrogen and thus the ex change reaction meets this criterion.) 2. There must be no net transfer of the two chemical species across the boundaries of the two phases. (In the example shown, all the ammonia gas should remain in the gaseous phase and all the NH(+ should remain in the liquid phase.) In actual operation, the observed separation factor is nearer 1.02 (enrichment factor of 0.02) because of the solubility of ammonia gas in the aqueous phase. Similar isotopic exchange reactions have been found for a number of other low-atomic-weight elements, such as hydrogen, lithium, carbon, oxygen, and sulfur. Isotopic exchange reactions involving isotopes of the heavier elements exhibit much smaller enrichment In fact, it is questionable if separation by chemical exchange may be prac factors. tical for elements above 40 or 50 mass units because of the unfavorable enrichment However, if the desired isotope is relatively abundant and the required factors. degree of separation is not too great, then a low enrichment factor may not be an insurmountable obstacle. Nevertheless, further complications may arise; e.g., the separation plant may take too long to arrive at a steady state. It is of some interest to the nuclear engineer to explore the relationship of these many factors. It is not within the scope of this paper to derive all the various formulas and equations, but rather to present a few with some discussion and give examples of the usefulness of each.

\

14-18

ISOTOPES 2.1

[Sec.

14

Single-stage Separation Factor

Each chemical separation process is based upon an isotopic exchange reaction The equilibrium constant for this reaction is related to the single-stage separation factor for the process. Urey and Greiff,s in 1935, calculated constants for several exchange reactions involving some of the light elements and predicted the feasibility Later, Urey6 extended these calculations and Bigeleisen and Mayer1 of enrichment. have pointed out several simplifying cancellations that take place in calculating ratios of partition functions for isotopic equilibria. It is particularly important to note that the masses of the isotopes cancel out completely for all elements except hydrogen. Thus, the simple square root of the ratio of the masses of the isotopes (or molecules in some cases) does not give the separation factor for chemical-exchange equilibria ii it does for gaseous diffusion, molecular distillation, and other processes based upun the average velocity of the molecules. The ratio of the square root of the masses It almost invariably gives too high a factor for chemical separation processes. rather the nature of the chemical binding in the species involved in the exchange that Bigeleisen and Mayer7 express it qualitatively as determines the separation factor. follows: "To get a large separative effect, it is necessary to equilibrate one compound which has high frequencies and large frequency shifts on isotopic substitution with another compound (preferably the gaseous atom) with low frequencies and small frequency shifts." They also present simplified equations that can be useful in estimating separation factors for isotopic exchange not involving hydrogen. A typical exchange reaction may be written


j

aA, +

bB,âA,

+ bB,

where A and B are molecules having one element as a common constituent and ibf subscripts 1 and 2 indicate that the molecule contains only the light or the heavyisotope, respectively. Then the equilibrium constant for this reaction is given by

R

m

(Qa./Qa,)' (Qa./Qs,)'

where the Qs are the partition functions of the molecules. Only the ratios of partitka functions enter into these equilibrium constants, and this fact makes it necessary ;c consider only these simple ratios here. I. Ratio of partition functions for isotopically substituted molecules. If we rrprpsent these ratios as /a and fa, then Bigeleisen and Mayer have shown that certain simplifications and cancellations in the ratios of vibrational partition functions (heavy molecule over light molecule) lead to the following expression:

where

= ratio of symmetry numbers (heavy molecule over = ratio of partition functions Gu = M (\/u) + l/(e1) Au = change in u upon isotopic substitution

S/S'

/

-

-

hcu u =

kT

h = c = u = k = T =

Planck's constant, erg-sec speed of light, cm/sec vibrational frequency in wave numbers, cm"1 Boltzmann constant, crg/deg absolute temperature

light molecule)

Sec. 14-2]


14-19

Note: Refer to original paper by Bigeleisen and Mayer for the derivation of this equation, for the role played by symmetry numbers, and for a table of values of the function G for values of u from 0 to 14. , The separation factor is related then to the equilibrium constant as follows:

If

K

where

(S/S')f,

=

^ Jb

(4)

we concern ourselves with molecules containing only one atom of the element isotopic substitution, the symmetry numbers cancel and this reduces to

undergoing

v _

—

fA JB

Thus, if the proper vibrational can be calculated using Eq. (4). Example:

frequency data are available, the separation factor

Calculate the separation

CO

factor at 27°C represented by the exchange reaction

+ C"Os


given the following vibrational frequency

C'«0

^±

C"0< C'"Oi

+ C>»Oj

data:*''

2167.4 2119.2 1351.2 1351.2

C'K)

C"0

672.2(2) 653. 1(2)

2396. 4 2328.4

The equilibrium constant for the exchange reaction can be expressed as the quotient of the ratios of partition functions:

"_ Q(C"O,)/Q(C"0,)

A'

Then fco

Q(C»0)/0(C'»0)

/coi fco

= 1 + 2G„, Ah,

ut =

hcu

Ac (2 167 .4)

kT

kT

10.40*

G„, = 0.4038

Aui /oo fco, u, Au, GU1

u, An,

C,

fco,

K

"

he —

he Aw

kT

= 0.231

(48.2)

= 1 + (0.4038) (0.231) = 1.093 = 1 + = 0

ZC, Au,

and thus

= 3.23 = 0.0916

C,

Am will vanish

= 0.2.316 = 11.50 = 0.326

- 0.413

= 1 +
-

^co*

1.177

fco

1.093

-

1.077

* In actual calculations the arithmetic mean of u for the heavy and light molecules would be preferable greatest accuracy, although negligible error arises from the use of either one or the other.

for

isotopes

14-20

[Sec. 14

A calculation by Urey* that includes anharmonicity corrections gives 1.086. II. Ratio of partition functions for symmetrical molecules of heavy atoms. If the central atom of mass M or M + AM is symmetrically surrounded

by A

identical atoms of mass m, an approximation (also derived by Bigeleisen and Mayer i is useful in the absence of a complete knowledge of the vibrational frequencies of the molecules. ,.. , . AMm ... where AM = difference in mass between isotopes of heavy element m = atomic weight of atoms symmetrically bonded to central heavy element M = mass of heavy element N = number of atoms bonded to central heavy element Ui = hcui/kT and a>i is totally symmetric or "breathing frequency" Example: Calculate the ratio of the partition functions for 0"0<~ anrl temperature if the symmetrical vibration frequency is 935 cm-1.

f" AM

1 +

CP'Ot"

at

AMm

Sm. UlW

= 2

m = 16

M1

N

=

-

approximately 35 X 37, which we shall Ac(935)


kT

ui*

R

use

4

4.49

= 20.16

-

1.083

A calculation by Urey$ that includes anharmonicity corrections

gives 1.0847.

In most actual isotope separation problems, the dearth of spectroscopic data make? the foregoing equations virtually useless unless one is willing to make extremely bold Sometimes data on molecules of similar siz* guesses about vibrational frequencies. and structure can be helpful in this regard. Despite the rarity with which these equations can be used, they do, nevertheless, give some mathematical basis for predictions that are needed at times. 2.2

Interstage Flow and Reflux Rate

In all chemical-exchange isotope-separation

processes, one of the most important is the magnitude of the interstage flow at the feed point in relation to the product flow. By referring to Fig. 1 showing the NHa — NH(+ exchange column, we can soon see the reason for this. In this chemical-exchange system, the ammonium ion moves down the system in the liquid phase. By treatment with sodium hydroxide, the ammonium ion is converted to ammonia gas, which is then sent back up the column countercurrent to the liquid phase. This reflux operation leads to a stoichio metric consumption of sodium hydroxide. In a similar way, reflux must be accom plished at the other end by combining NH, with nitric acid to form NHiNOi for return as the downstream liquid phase. The net reaction is the neutralization of nitric arid with sodium hydroxide. The interstage flow at the feed point thus determines the rate of consumption of chemicals. The minimum consumption of chemicals to produre product at a fixed rate and concentration can be calculated from Eq. (3). The actual ratio of interstage flow to product flow must be greater than this minimum and in i square plant might be expected to be about 1.3 to 1.4 times the minimum. Although tapering the plant toward the product end, i.e., using successively smaller column?, has certain advantages in lowered holdup and shortened equilibrium time, it cannot reduce the above minimum chemical consumption. This chemical consumption represents the major energy requirement of this process. In all chemical-exchange isotope-separation processes, the chemicals used at one end can be obtained from the other if work is done upon them. This statement may aspects

Sec. 14-2]


be clarified by reference to Fig. 2 showing an between Reflux NH8 and (NH4)2CO,.

accomplished by treating the aqueous with Ca(OH)2 slurry to liberate The CaCO, can NH3 and form CaC03. then be heated to form C02 and CaO. The COj is used at the top of the cascade to react with NHj to re-form (NHiJ.COj. The CaO is slaked to form Ca(OH)2 slurry for recycle to the reflux. Thus, the chemical cycles can all be closed if we add energy such as wc have done here in the form of heat. Usually, it is less costly to buy the chemicals from a large efficient producer than to regenerate the chemicals in this way in a parasitic plant that is small and has high overhead charges. Since the reflux chemicals can easily account for one-third to one-half of the total cost of the operation, including amortization when carried out on a large scale, an example of the method of esti mating this might be helpful.

N"

14-21

enrichment plant using the exchange

i»

WASTE REFLUX


stream

(NHJjCOj-

(NH^COj-"'

Flo. 2. Ammonia, ammonium system for N15 enrichment.

Example: Calculate the approximate chemical cost per pound of 95 per cent by the process shown in Fig. 1, assuming a separation factor of 1.02 and

N" enriched

0.95

VD

a *!

1.02

0.0037 (normal

abundance

1.3(V/£)mi. (a

-

-

of

N»)

-xj)

0.95

(V/O).ot

carbonate

- 0.0037

(0.02) (0.0037) (0.9963) 12,835 (1.3)(12,835) 16,686

Thus, 16,686 lb moles each of NaOH and HNOi is required per pound mole of 95 per cent N" product. This is equivalent to about 667,000 lb of NaOH and about 1,050,000 lb of HNOi per pound of 93 per cent Nls product. Assuming 3 cents/lb for NaOH and 4 cents/lb for HNOa, then 667,000 X 0.03 1,050,000 X 0.04

20,000 42,000 862,000

or

02,000/15

=

/lb

mole

approximately $4,100/lb

N"

Thus, the chemicals for reflux will cost $62,000 per pound mole of 95 per cent N", or approximately $4,100 per pound of N". 2.3

Equilibrium

Time

An isotope plant starting up with normal inventory must use the net transport for inventory build-up for a certain period of time before product level material can be It drawn off at a steady rate. This is called the equilibrium time for the cascade. A cascade having an unreason is a useful concept in evaluating separation processes. Since, as Professor Urey has said, it is ably long equilibrium time is impractical.

14-22

[Sec. 14

ISOTOPES

relatively easy to design chemical-exchange isotope-separation systems that might take centuries to come to equilibrium, it is an aspect of the problem that should not be Calculation of the equilibrium time will sometimes reveal a process as neglected. being impractical that in other respects seems sound. The equilibrium time is essentially the time required to produce enough enriched material to fill the cascade, and we can arrive at this by sizing the cascade and dividing Since we are not interested in by the production rate as described by J. Shacter.' designing the whole cascade to find its capacity, let us examine a simpler procedure. The equilibrium time for an "ideal" cascade designed to produce comparable product can be calculated by the use of the equation by Cohen1: (2/i/(')[QVp Vn time = Eq. Hl¥1„

- 2NPN0

+ No) In— («,/£.) Np — No

-

2(A'„

- A".;]

(6j

where h = holdup to throughput ratio or processing time per stage = (a — l)/2 € = and a are as previously defined) = mole fraction of desired isotope in the product Np No = mole fraction of desired isotope in the feed Rp = A'„/(l Np) R„ = AVU No) In most cases an actual cascade will have greater holdup and a longer equilibrium Thus, the use of the above equation will give only a time than an ideal cascade. probable minimum time. Nevertheless, this will usually be sufficient to allow for the rejection of most systems that are completely impractical. Using the NHj — N'Hr system again as an example, the equation above reduces to

(


-

Eq. time = ?? (6.48)

(7)

t*

Under normal conditions, we can assume that

h

will be about

1

min and

< will

be about

0.01 ; thus

Eq. time = 12.96

X

10*

min or 90 days

Were the enrichment factor to be one-tenth as great, however, it would take the plant one hundred times as long to come to equilibrium, or about 25 years. Similarly, if the rate of isotopic exchange between NH3 and NHt+ were slow enough to require contact of the phases for an hour at each stage (and with the original assumed enrichment factor of 0.01), then the equilibrium time would be about 15 years.

2.4

General

In spite of the many at tractive features of chemical-exchange systems, one striking disadvantage is their lack of versatility. Only very rarely is a particular process feasible for the enrichment of the isotopes of more than one element. Even then, the efficiency is very low for the enrichment of the isotopes of the second element. (For example, the heavy water distributed by the AEC is partially enriched in O" by accident rather than design.) For maximum efficiency, a process must be designed specifically for each element But then the chemistry of many (other than those described in Sees. 14-4 and 14-5). of the elements is such as to seem to preclude the possibility of developing any satisfactory chemical-exchange process. Thus, very often one is forced to consider other processes for a particular isotope. 3

OTHER PROCESSES

Of the many types of processes for separating isotopes t hat remain to be considered (other than those described in Sees. 14-4 and 14-5), two have received considerable attention. These are distillation and thermal diffusion. A detailed discussion of

Sec. 14-2]


14-23

However, there arc certain these schemes is beyond the scope of this article. general features of each which are so important that they deserve to be mentioned.

Distillation

3.1

Distillation has one very decided advantage over the chemical method in its closer The very high chemical costs for reflux approach to thermodynamic reversibility. found in the chemical method arc obviated in distillation and replaced by a problem The great simplicity of reflux by condensation at one end and in heat conservation. boiling at the other justifies the consideration of distillation for isotope separation the enrichment factor is usually wherever it is technically feasible. Unfortunately, very small and the resultant high ratio of interstage flow to product flow [see Eq. (3)] This large plant with its high capital cost often leads to an objectionably large plant. loads the product with a high fixed cost. It is important to note that if the enrichment factor is small, the length (number of stages) and width (interstage flow) are both inverse functions of the enrichment Thus, the sizes of the enrichment sections of two plants factor (approximately). that arc alike in all other respects arc inversely proportional (approximately) to the square of the enrichment factors. Size

1

Size

2

_

(a, (a,

-

1)' 1)'

...

(


This rule

is also generally used in comparing complete plant sizes. Because of the greater size of the reflux operations in relation to the enrichment sections in a chemical-exchange plant, the above approximate rule is less applicable to Nevertheless, the chemical-exchange-plant comparisons than to distillation plants. simplicity of the rule leads to its extensive use in estimating relative sizes of conceptual chemical-exchange plants. 3.2

Thermal Diffusion

The term thermal diffusion is generally used with reference to the countercurrent In this multistage thermal diffusion columns developed by Clusius and Dickel.10 type of apparatus, separation is obtained at each stage by virtue of the sharp tem perature gradient set up in the working gas or liquid by the hot and cold walls of the Countercurrent flow or convection currents are also achieved by these hot column. and cold walls of the column. The stage-wise degradation of energy occurring in thermal diffusion has led to its classification as an irreversible process. This is in contrast to distillation and chemical Indeed exchange, where the stage-wise separation is based upon reversible equilibria. the consumption of energy is the chief reason why thermal diffusion cannot often One area where the process can be compete with other isotope-separation methods. Perhaps not so readily useful is the separation of the isotopes of the rare gases. recognized is the value of thermal diffusion in producing, in the laboratory, small This arises by virtue of quantities of enriched isotopes for experimental purposes. the simplicity of both construction and operation of the apparatus. Power costs do not loom so large on such a small operation despite the unchanged proportionate relationship. Except for these two types of applications, rare gases and small-scale separations, thermal diffusion represents a seldom-used tool. 3.3

Miscellaneous

Of the other processes for separating The electrolysis separation phenomena. of the large kinetic isotope effect at the highly successful, but its usefulness seems current gas centrifuge is based upon the produced in a rapidly rotating cylinder.

Processes

isotopes, some utilize unusual or unique of water to enrich deuterium makes use The process is well known and cathode. limited to water at present. The counterseparation obtained in the centrifugal field The enrichment factor is given by

isotopes

14-24 «

-i

= <**

- MQ*

2RT

[Sec. 14 K '

Mt

= two molecular masses in the gas mixture being centrifuged V = peripheral velocity of the cylinder, cm/sec R = gas constant, ergs/(mole)(deg) T = absolute temperature Thus, in separating Cl,H» and C"H( in a 3-in.-diameter bowl at 66,000 rpm at normal temperature, the factor would be

where Mt and

a_1

2Rf— =

(l)[(3)(3.14)(2.54)(66,0O0)/60]» (2) (8.3

X

10') (300)

= 0.014

It should be noted that the enrichment factor is proportional to the difference in molecular weights and will thus have as high a factor for isotopes of heavy dementi as for isotopes of light elements if the same difference in atomic weight exists. Whether or not this potential application to the separation of heavy isotopes is the only impor tant virtue of the gas centrifuge seems to await further development and improvements in mechanical design.


4

COMMERCIALLY

AVAILABLE ISOTOPES

At present there are relatively few commercially available isotopes made by the methods covered in this article. This situation will undoubtedly change as the demand increases for larger quantities of separated isotopes. Research quantities of the stable isotopes are generally produced at present by electromagnetic separation as described in Sec. 14-5. 4.1

Deuterium

Deuterium in the form of DjO is being produced in many countries by methods similar to those which have been described in this article. During World War II, Enrich DsO was produced in this country by distillation followed by electrolysis.*1 ment by chemical exchange between hydrogen gas and water was accomplished at Trail, British Columbia. No information is available on the current production method or methods used in this country. The AEC price of DjO is $28/lb. The Stuart Oxygen Company of San Francisco is the licensed distributor for heavy water and deuterium gas for the AEC. The price per gram of heavy water varies from $0.20 to $1, dependent upon the size of the order and the specified ampoule sixe, which may vary from 1 to 100 g. Deuterium gas sells for $1 /liter (NTP) for the first 100 liters. Purchasers of deuterium (or O18) must first apply for an AEC allocation by sub mitting a "Stable Isotope Request" (Form AEC-100) to the AEC, P.O. Box E, Oak Ridge, Tenn., Attn: Isotopes Division. 4.2

Carbon 13

Isotope concentrates of C" made by fractional distillation of CO have been on sale in England by the Isotope Division of the Atomic Energy Research Establishment. The British product is sold in the United States by the Isomet Corporation, Palisades Park, N.J. The basic form of the material is BaC03 in which the carbon is 65 to 70 atom per cent C13. It is offered at a price of $438 per gram of contained Cu. Other forms than BaCOa can be made available. Isotope concentrates of C" made by the chemical exchange between HCN and KCX have been placed on sale in the past in this country by Eastman Kodak Com pany, Rochester, N.Y.

Sec. 14-2]

isotope separation by chemical exchange 4.3

14-25

Nitrogen 15

William Spindel and T. I. Taylor have recently originated and operated success fully an N" enrichment process based upon the exchange between HN03 and gaseous oxides of nitrogen." Nitrogen 15 enriched by the Spindel-Taylor process is offered for sale by the Isomet Corporation, Palisades Park, N.J., as HN03 in which the Forms other than HNO», such as NH3, N2, NO, nitrogen is 95 atom per cent Nls. NOj, and various salts, are said to be also available. Eastman Kodak Company, Rochester, N.Y., has, in the past, marketed an N" concentrate prepared by chemical exchange between NHj and NH4+. 4.4

Oxygen 17 and 18

Water enriched in O" and O18 by a process of fractional distillation is offered for Atomic Energy Research Establishment. Their catalogue dated July, 1954, lists Ou at approximately 6 per cent as H20: £110 per gram of O18. It is stated that the O17 in this material is about five times enriched. The Stuart Oxygen Company distributes O17 and O18 enriched water under contract with the AEC. The oxygen in this water contains about 0.11 per cent of O17 and about 1.5 per cent of 0". The price per gram is the same as the price listed under deuterium and requires the same AEC "Stable Isotope Request." Concentrates made in Israel of O18 containing up to 90 atom per cent O" and about 1 J.^ per cent O17 are for sale in the United States by the Isomet Corporation, Palisades Park, N.J. A recent price is $575 per gram of H20 containing oxygen of the above description.


sale by the British

. 1.

K.: "The Theory of Isotope Separation," National Nuclear Energy vol. IB, McGraw-Hill Book Company, Inc., New York, 1951. Shacter, J., and G. A. Garrett: AECD-1940, 1948. Benedict, M.: Tram. Am. Inst. Chem. Bngrs., 43 (2): 41-60 (February, 1947).

Cohen,

Div.

2. 3. 4. 5. 6. 7.

REFERENCES

Ill,

Series,

Thode, H. G., and H. C. Urey: J. Chem. Phys., 7: 34-39 (1939). Urey, H. C, and L. J. Greiff: J. Am. Chem. Soc., 57: 321-327 (February, 1935). Urey, H. C.: J. Chem. Soc., pp. 562-581, 1947. Bigcleison, J., and M. J. Mayer: J. Chem. Phys., 16 (5): 261-267 (May, 1947). 8a. Hcnberg, G.: "Molecular Spectra and Molecular Structure," Diatomic Molecules, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1939. G.: "Infrared and Raman Spectra of Polyatomic Molecules," D. Van b. Herzberg, Nostrand Company, Inc., Princeton, N.J., 1945. c. Hibben, J.: "The Raman Effect and Chemical Applications," Reinhold Publishing Corporation, New York, 1939. 9. Shacter, J.: "Rapid Estimates of Limits for Net Transports and Equilibrium Time," Report K-1044, Carbide and Carbon Chemicals Company, Oak Ridge, Tenn., Aug. 3, 1953.

(Unclassified.)

10. Clusius, K., and G. Dickel: Z. physik. Chem., 44: (1942). 11. Selak, P. J., and 12. Spindel, W., and

J.

397

(1939), 48: 50 (1940), 52: 348

Finke: Heavy Water, Chem. Eng. Progr., 50 (5): J. Chem. Phys., S3: 981-982 (1955).

T. I. Taylor:

221

(May, 1954).

BIBLIOGRAPHY A Bibliography Begun, G. M., and R. E. Allen: " Isotope Separation and Isotope Exchange. of Unclassified Literature," United States Atomic Energy Commission Report TID3036, Office of Technical Services, Department of Commerce, June, 1954.

14-3

PRODUCTION OF RADIOISOTOPES* BY

Arthur F. Rupp and Franklin T. Binford 1

IRRADIATION

Radioisotopes are produced chiefly by the exposure of target materials to the neutron field of a nuclear reactor, although some are produced in particle accelerators. f-1 1.1

Target reactor. Generated for wjivans (University of Florida) on 2015-09-23 03:00 GMT / http://hdl.handle.net/2027/mdp.39015011142083 Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google

because

Methods of Irradiation

materials are enclosed in aluminum containers before insertion in the necessary, the target is first scaled in a quartz ampoule (glass is not used of high neutron absorption), and the ampoule is supported in the aluminum container by quartz-wool wadding. The target containers are generally extruded tubes closed with a crimped cover (Fig.

If

1).

In a graphite reactor the small con tainers are inserted in holes in long rec tangular graphite blocks, or "stringers," that are pushed into the side of the re actor; the larger containers are pushed into special holes from which part of the graphite has been removed. Xo elabo rate tools are used for manipulating the stringers; they are merely pushed in with -MAY BE BULLET NOSE poles and pulled out into long lead ■MUST NOT LEAK UNDER 5PSI AIR PRESS. "coffins" after irradiation (Fig. 2). Special short-term irradiations may r* Fia. 1. Crimp top can. made in capsules that are moved by gas pressure in and out of the reactor through a smooth aluminum tube. The targets should not volatilize, decompose appreciably, or produce materials that will attack the container. When they can be obtained with acceptable purity, solid Stable oxides and carbonates are sL-o metallic elements are generally used as targets. employed. Compounds containing elements such as chlorine (which yields S", P**, and CI") are usually avoided unless multiple production is desired. Salts such as nitrates (which decompose to nitrites and oxygen) are also usually avoided when possible. Organic compounds can be irradiated but may char at reactor temperatures and often suffer considerable radiation damage, thus producing gas pressure and an unusable mixture of compounds. In general, materials that are thermally stable are also fairly stable under neutron bombardment. The generation of excessive heat in samples under irradiation is usually not a problem in the ORNL graphite reactor (X-10) unless some fissionable material is * This article is reproduced, with some adaptation and with permission of the American Institute Appl. Pfeyj., 14: fi of Physics, from an article of the same title and by the same authors published in (September, 1953). t Superscript numbers refer to References at end of subsection.

/.

14-26

production of radioisotopes

Sec. 14-3]

14-27

present; y heating at the fluxes available is insufficient to raise the temperatures materially above that of the surrounding graphite. Adequate cooling must be provided, however, for irradiations in higher fluxes. 1.2

Production Rates


Radioisotopes are produced in a reactor mainly by simple neutron capture, by neutron capture followed by decay to another species, by neutron-proton reaction,

Flo.

Fio.

3.

2.

Method of removing radioisotopes

from X-10 reactor.

IRRADIATION TIME /HALF -LIFE Growth of primary radioactive product during irradiation.

occasionally by a neutron-a reaction, and by the fission of uranium. equation is

A(l)

=

P(l

-

The production

e-*«)

where A(t) is the activity produced, t is the exposure time, X is the decay constant of the nuclide produced, and P is the rate of production, which is dependent upon the nuclear characteristics of the target and upon the value of the neutron flux. The A plot of maximum activity available as t approaches » is P, the saturation activity. 1 — e x< as a function of the ratio of irradiation time to half-life is given in Fig. 3.

isotopes

14-28

If A (0

is expressed

[Sec. 14

in curies per gram-atom of target material, A(t) = (l.C3/W>

•

-

10")(1

e~x<)

/

where is the isotopic abundance of the target isotope in the target element, a is the (isotopic) activation cross section in square centimeters, is the neutron flux in neutrons per square centimeter per second, and the factor (1.63/a<£ • 10") is equal to P, the production rate. In this equation, it is assumed that the amount of target material remains essen This assumption is valid for nearly all tially unchanged throughout the irradiation. However, some target materials exposed in fluxes of 10'* neutrons/(cm*)(sec) or less. isotopes such as Eum and SmU9 have cross sections of several thousand barns,* ami the burnup of the target material in long or intense irradiations must be consider**!. In this case

m

=

}J3f*4k

\

—

101,(e-^,

o

_

e-Xl)

The burnup of the radioisotope itself must be considered in a few cases where thtradioisotope produced has a large neutron-capture cross section. The production equation becomes Px o

A

where a' is the capture cross section of the radioisotope being formed. This effect need be considered only when
A(t)

= 0.866WV(1

-

e-*«)

where A(l) is the number of curies per kilogram of fuel at time t, W is the power output in watts per kilogram of fuel in the region under consideration, and r is the fractional fission yield of the nuclide sought. A desired radioisotope is sometimes produced by the decay of a primary radio isotope. The activity produced is given by

Mt)

= z

P

r

A2 — Al

[Xt(l

- e-*'0 - X,(l -

e-XH)]

where P refers to the production rate of the primary radioisotope and subscripts 1 an J 2 refer to the primary and secondary products, respectively. The ratio of the second ary to primary activity is given by

A^t)

Xj

In

Fig. 4, the reciprocal R of this expression is plotted against irradiation time for the production of Zr^-Nb"6. When t becomes large compared with the longer half-life, the ratio As(t)/A,(l) For the case where X* approaches unity. \i (i.e., the half-life of activity 2 is much greater than that of activity 1), then

«

1,£P(1 on the other hand, if Xi

»

Xi, then

4iSP(l * 1 barn = 1 X 10"" cm".

-

- e-M)

e-x>')

=

Ai


Sec. 14-3]

14-29

It often happens that more than one radioisotope of the same element is formed For example, 53-day Sr88 and 25-year Sr90 are produced in fission. during irradiation. The ratio of these two activities as a function of time is plotted in Fig. 5 as determined from A,(l)

"_
- e-*-') -

«-*•<)

which clearly approaches oifi/a^fi as a limit with increasing i. Ai and At are the activities of the longer and shorter lived products respectively, while fi and/2 are the corresponding isotopic abundances in the target. 1.3

Complex Production Chains


The calculated production rates are often in error because of self-absorption of neutrons by relatively large pieces of target material. Large absorbers in the vicinity of the sample under irradia tion may also have a great effect on the activation because of the localized flux depression. Therefore, even though calculations can be used as a guide, it is often necessary to chart the flux zones in the reactor with monitors (for example,

80 120 160 200 240 IRRADIATION TIME.(DAYS)

Fio.

4.

activitv

280

Ratio of Zr»' activity to Nb»5 function of irradiation time.

as a

2

4 6 8 10 12 IRRADIATION TIME, MONTHS

14

Fig.

5. Ratios of radioisotopes of same ele ment formed during irradiation.

thin films of cobalt) and to determine the activation rates experimentally with the actual material under consideration. From a production standpoint, radioisotopes fall into two broad classes: those which occur alone, free from significant contamination with other radioisotopes of the product element and free from daughter activities, and those which occur in conjunc tion with radioisotopes of the product element other than the desired activity or which Many of the fission products fall into this have one or more daughter activities. second

class.

In the first

class are found those radioisotopes which are produced from monoisotopic targets, those from targets having significant cross section for only one radioisotope-producing reaction, those from targets in which any extraneous reactions are extremely short-lived, and those which are the last radioactive product of short lived parents in a decay chain. Separation of such products is accomplished by ordinary chemical means. The separation of products in the second class, however, has certain intrinsic diffi culties — as an example, the system 8r8,-Sr'°, produced in fission (half-lives 53 days and

[Sec. 14

ISOTOPES

14-30

As shown in Fig. 5, the ratio Sr,0:Sr" is small for periods of 25 years, respectively). Thus, after a 12-month irradiation that are Bhort compared with the half-life of Sr"°. irradiation period, this ratio is only about 0.03 and the activity present is primarily After the target has been removed from the reactor, however, the ratio SrM:Sr" Sr8'.

increases by a factor of 10 in approximately 6 months, so that after 18 months the This illustrates one method of separating two ratio Sr90:^8' has increased to 30. The shorter isotopic products; it is applicable only when the half-lives differ greatly. lived material can be economically separated only if it is produced at a much faster rate than the material with the longer half-life. In the case of the Fe"-Fe" system (half-lives 2.9 years and 46.3 days, respectively), circumstances are not so favorable for the separation of the short-lived product. Because in natural iron Fe6< is approximately 18 times as abundant as Fess, the ratio of the long-lived Fe" to the short-lived Fe" is quite high, even for short periods of irradiation, as can be seen in Fig. 5. It is still possible to obtain Fe" relatively free


DECAY OF Zr 95

DECAY OF Nb 95

20

40 ELAPSED

Fig.

6.

mixture.

60 TIME,

80

20 4060 10 30 50 70 90IIOI3OI5C TIME IN TIME AFTER REMOVAL {DAYS!

100

MONTHS

Fraction Zr" present in

Zr"-Nb"

Fig.

REACTOR (DAYS) 70

7. Growth and decay of during and after irradiation.

Zr"

and

Nb"

It is impossi of Fe" by performing long irradiations and allowing the Fe" to decay. ble, however, to obtain ratios Fe":Fe" less than about 1.7 if normal iron is used as the target. Better results are obtained by the use of target material in which the Fe" content has been enriched elect romagnetically. In the case where the product itself decays to a radioactive daughter, the two can be separated chemically; however, the daughter will eventually grow back. If the daughter is very short-lived compared with the parent, the two activities will be essentially the same and the mixture will decay with approximately the half-life On the other hand, if the daughter is long-lived compared with the of the parent. parent, the parent will eventually decay completely, leaving only the daughter activity. The case of

the daughter and parent having comparable half-lives, with the daughter being shorter-lived than the parent, is illustrated in Fig. 6, which shows the fraction of Zr" present in a mixture of Zr" — > Nb" (half-lives 65 days and 35 days, respec tively) as a function of decay time. Several different starting mixtures are shown. Regardless of the fract ion of Zr" present (provided it is not zero), all mixtures approach the limit of 0.317 as / increases. Figure 7 shows the actual growth and decay of the mixture during and after irradiation. In addition to interference due to the simultaneous production of radionuclides of the same elements, stable elements are also produced during the fission process as a

Sec. 14-3]


14-31

The presence of these stable result of decay chains that end in stable isotopes. isotopes materially affects the specific activity (activity per unit weight of element) Thus very high specific activity Im cannot be made using of the separated product. long-irradiated uranium because of the accumulation of I127 and I159. Cs1" is encum bered with comparable amounts of Cs1" and Cs1", and Sr30 contains comparable Some of the stable isotopes produced by fission and decay are also amounts of Sr". reactivated by neutron capture. 1.4

"Carrier-free" Separations

Separation and purification of carrier-free radioelements require special precautions in only microgram amounts and must be separated from gram amounts of target material and milligram amounts of extraneous impurities; for example, P" weighs 3.5 /ig/curie and must be separated from a kilogram of sulfur. A large part of a carrier-free product undergoing processing may easily be lost by adsorption on the walls of the vessels and filters or on an inconspicuous bit of However, with the use of techniques such as ion exchange, solvent extrac "crud." Small amounts tion, or controlled adsorption, quite pure products can be obtained. of chemical contaminants are necessarily picked up from processing chemicals and equipment and are usually those ever-present elements found in most solutions: Ca, Mg, Si, Al, Na, and Fe. Occasionally Pb is found, but most preparations to which the specification could apply are kept below 10 ppm of heavy metals as detected by standard tests. The total nonvolatile solids in carrier-free preparations of short half-life materials (which have very little mass per unit of activity) may be kept down to between 0.001 In the effort to obtain carrier-free materials of high and 0.1 per cent in most cases. activity per unit weight, which are useful in studying radiation characteristics of radioisotopes, fairly large amounts of activity (e.g., several curies) should be processed, because experience has shown that about the same amount of extraneous solids per batch is obtained whether the amount of activity is small or large.


because the product element may be present

2

CHEMICAL PROCESSING 2.1

Processing Required

The purpose of chemical processing is to isolate the product element, free from radioactive and inactive chemical contaminants, and to put it into a readily usable form. One or more of the following chemical processes are generally used for sepa rating and purifying radioisotopes: distillation, gasification, ion exchange, precipitation and coprecipitation, solvent extraction, adsorption, electrochemical separation, and In general, only the simplest chemical compounds arc made paper chromatography. at ORNL; further processing to make specific preparations needed for particular Most preparations are applications is done by the user or by a commercial supplier. simple solutions that may be easily and accurately dispensed; chloride in a slight The excess of hydrochloric acid is the type of product solution usually made. chemistry of chlorides of most elements is well known; excess hydrochloric acid may be removed by volatilization or neutralized to produce NaCl, which is required in many solutions used for injection into the body. 2.2

Chemical Purity of Target Materials

Target materials that are apparently quite pure often are found to have con taminating radioactivities after neutron irradiation, caused by the unsuspected presence of impurities with large neutron cross sections. An example would be the large amount of CoM found upon irradiation of reagent-grade nickel, which is pure enough for ordinary chemical work. Although it is preferable and often possible to

purify

target

materials

before

irradiation,

from the targets after irradiation in order to

induced activities are often removed use the isotopic-dilution techniques in

14-32

[Sec. 14

ISOTOPES

which inactive carriers are added and to detect easily the contaminating means of their radioactivity. 3

RADIOISOTOPE SEPARATIONS PROCEDURES

Brief descriptions of the chemical processing used at important radioisotopes are given below. 3.1


elements by

ORNL

for some of the more

Carbon 14

Carbon 14 is produced by the irradiation of beryllium nitride that is first compressed into dense pellets. The nitrogen atom is the target for the N14(n,p)C14 reaction, and beryllium nitride was chosen because it contains a large proportion of nitrogen and because the beryllium absorbs few neutrons. Chemical stability is also somewhat The target is dissolved in better than that of other nitrides that might be used. sulfuric acid and hydrogen peroxide, and the gases that contain the C14 are swept out in a stream of pure nitrogen. The gas evolved has a variable composition and con tains organic compounds such as methane, as well as COj. These compounds an? oxidized to COj over hot CuO and then absorbed in highly purified NaOH solution. BaC14Oi is precipitated by the addition of Ba(OH)s, washed, dried, and assayed for The average isotopic ratio, (CI4/total C) X 100, of the product is ~12 per shipment. cent, with a maximum at the present time of 80 per cent. Process-development work currently is largely directed toward the isolation of the various organic compoundfrom the gas stream and the reduction of carbon contamination in the target matem. in order to produce C14 with higher isotopic ratios. 3.2

Phosphorus 32

Kilogram amounts of highly purified elemental sulfur are irradiated to produ-e PM, S"(n,p)P". The carrier-free Psl is extracted from molten sulfur by weak HXOi. Because sulfur melts above 120°C, this operation is necessarily done under pressure The extract must b* in an autoclave to prevent boiling of the aqueous phase. purified to separate P"0<3_ from sulfate, nitrate, silicate, and contaminating cations such as Fe, Cr, Ni, Mg, Pb, Ca, and many other lesser components. Even highly purified sulfur contains organic matter that is also very troublesome when it is The purification steps are carried out entirely in Pyrex glass or quart: extracted. The extract liquor is collected in a Stang precipitator. Lanthanum equipment. carrier is added, and La(OH)j is precipitated with ammonium hydroxide; PO,3"" coprecipitates as LaPO<. The precipitate is dissolved in weak acid and passed throueh a cation exchanger to remove La3+, thus leaving only H+, Cl~, and P043_ ions in the effluent. The effluent is evaporated to dryness and taken up in weak HC1 as a produit. Although carrier-free P" can be made in this way, it is the usual practice to add 25 of inactive phosphorus carrier per millicurie of P" in the first evaporation step to reduce losses by adsorption. 3.3

The first I"1 distributed

Iodine 131

in the radioisotope

program was made

bv irradiating

tellurium metal [Te13°(n,->-)Te131 -» I1"]. The demand increased rapidly beyond the production practical by this method, and fission-product I1" (2.8 per cent fission yield) is now separated from uranium metal. Uranium that has had a short residence in the reactor (~00 days) is dissolved, and the iodine is volatilized under controlled conditions by steam-air sparging. The condensate and alkaline-scrub liquor capture the iodine; then the liquors are further processed to concentrate and purify the I131 by distilling iodine from an acid-hydrogen peroxide solution and receiving the distillate in an alkaline solution. The distmate

Sec. 14-3]

OF RADIOISOTOPES

PRODUCTION

14-33

is then treated in another distilling apparatus by first oxidizing the iodine to iodate with KMnO<. The oxidation step is followed by distillation, in which the volatile impurities are removed, and the iodine remains in the distilling flask as the nonvolatile iodate. The iodate is then reduced to elemental iodine by phosphorous acid, and the iodine is quickly distilled over into alkaline sodium sulfite. This concentrated product is diluted, assayed, and prepared for shipment. 3.4

Sulfur

36

The KC1 is Sulfur 35 is produced by fast-neutron irradiation of KC1, Cl"(»»p)S36. dissolved in water that contains a trace of hydrogen peroxide to ensure oxidation of all — . A trace of P" is present sulfur to SO< [as a result of Cl3S(n,7)P"l and is removed The solution is passed by passage through a bed of bright aluminum shavings. through a cation-exchange bed where the potassium is removed and the HC1 and HjS"Oi are allowed to pass into the effluent. The effluent is distilled to remove HC1; — which remains adsorbed on the walls of the distillation flask, is removed by the S350» , boiling with weak HC1.


3.6

Fission Products

After uranium and plutonium have been removed, fission-product mixtures are fractionated by ion-exchange or precipitation procedures into the following groups: alkaline earth (Sr8».»°, Ba14°), rare earth (Ce'4>.»44, Nd1", Pr'«, I'm1", Y81, and Eul"), The gases I"1, alkali (Cs1"), anionic (Ru103'108), and radiocolloidal (Zr»-Nb»6). Xe131, and Kr" are removed during previous processing. The rare-earth fraction is further fractionated by using ammonium citrate complexing agent on long, heated, ion-exchange columns that have 500 to 1,000 theoretical plates. The citrate fraction containing the pure radioelements is metathesized to the The alkaline earths are also fractionated by chloride by ion-exchange techniques. ion-exchange methods or purified by classical separations in concentrated nitric and hydrochloric acids. Cs1*7 is purified by ion exchange or successive cocrystallizations with ammonium alum. The Zr"-Nb95 system, which exists in colloidal form in strong acid solution, is separated by adsorption on silica gel and elution in oxalic acid complexing agent. Further purification is accomplished by ion-exchange methods, in which oxalic acid in various strengths is used as the complexing agent. Ru106 is separated and purified primarily by oxidation with sulfuric acid and The oxidation is followed by distillation into HC1 and hydrogen peroxide KMn04. A number of distillations might be necessary to achieve the desired purity. solution. 3.6

Neutron-Gamma

{n,y) Activated Products

An example of n,-y-activated products is Ca" produced by irradiation of CaCOj fCa44(n,7)Ca46]; since gram amounts are used, fairly common purification methods can be used. After the irradiated compound is dissolved in HC1, the solution is first scavenged with iron and rare-earth hydroxides and then the heavy metals are removed by sulfide precipitation. The calcium is precipitated from 80 per cent HNOj, dis The carbonate is dissolved in solved, and then reprecipitated as the carbonate. dilute HC1, and the solution is analyzed and assayed for shipment. To produce high-specific-activity

is

used.

material, target CaCOi, electromagnetically

enriched in

Ca44,

The foregoing is typical of the chemical processing methods used in this work — that of following classical methods whenever possible and introducing newer methods, such The as ion exchange and solvent extraction with selective chelation, when necessary. purification procedure selected depends upon the basic chemistry of the element involved, the nature of the target compound, the probable kind and amount of impuri ties, and the amount of radiation to be encountered in processing.

isotopes

14-34 Table

1.

Review of Radioisotope Produc

Nuclide

Half -life

Na"

12 . 46 years 53 . 6 days 2.4 X 10« years 5,568 yearn 2. 6 years

Na" pii

15.06 hr 14.3 days

H»

Be' Be10

C'«

tion method

Li-n-a Li-p-n Be-n-7 N-n-p Mg-d-ar

Na-p-d Na-n-7 S-n-p

[Sec. 14

Preparations

and Methods

activity (unite /g) attained at ORNL Specific

2.59 curica/cm' 8TP Carrier- free m curie 1. 2 curies 1 curie 1 millicurie 2 curies Carrier- free

Principal purification methods

used

Gasification and absorption Coprecipitation (Fe1*) Precipitation Gasification Ion exchange and precipitation Precipitation (chloride) Precipitation (chloride) Precipitation [La(OH)i] and ion exchange

P-n-7 Cl-n-a 25 days 67. 1 days

CI" A" A"

3 . 8 X 10" years 35 days ~265 years 1. 5 X 10' years 12.4 hr 152 days

K*°

K«

Ca«

S-n-p Cl-n-p S-n-7 Cl-n-7 Ca-n-a K-n-p K-n-7 K-n-7 Ca-n-7 Sc-n-p


Unprocessed Precipitation [La(OH)>] and ion

20 millicuriee

Distillation Ion exchange

Carrier-free

exchange

P"

S«

50 millicuriee

Carrier-free 100 millicuriee

Unprocessed

0. 60 mitlicuries Carrier- free

Distillation (HC1) Adsorption

1 curie 6. 5 curies

Precipitation (chloride) Precipitation (chloride) Precipitation (nitrate, carbonate],

Carrier-free

Chelation, solvent extraction,

ion exchange

ioo

exchange

Carrier-free

Chelation, solvent extraction Ion exchange (citrate)

15 curies

Precipitation

Sc"

v«

85 days 16 days

Cr»»

27 . 8 days

Mn»'

6 . 0 days

Mn"

290. 7 days

Fe"

2. 94 years

Sc-n-7 Cr-d-a Ti-p-n Cr-n-7 V-p-n Cr-d-2n Cr-p-n Cr-d-n Cr-p-n Fe-n-7

45. 1 days

Mn-p-n Fe-n-7

> 5 curies

Solvent extraction (dichiorodiethrl

Co-n-p

100 inillicuries

Solvent extraction (dicbjorodiethrl

Co-d-2p

1 curie

Solvent extraction (dichlorodiethyl

Fe-d-n Fe-d-n Ni-n-p Co-n-7 Ni-n-7 Ni-n-7 Cu-n-7 Cu-n-7 Zn-n-7 Cu-p-n Zn-p-n Ga-n-7 Oe-n-7

Carrier-free Carrier- free Carrier-free

Ion exchange Ion exchange Solvent extraction Precipitation Ion exchange Ion exchange Ion exchange Precipitation

Fe"

Co" Co"

270 days 72 days

Co"

5. 27 years 8X10* years 85 years 12 8 hr 12.8 hr 250 days

Ni" Ni"

Cu" Cu" Zn«

Ga" Ga" Ge" Ge" As" As"

77.9 hr

As76

26.8hr

I4.3hr

114 days

12 hr 76 days 18 days

Se7*

38 hr 120 days

Br"

35.87hr

Aa"

60 curies

Carrier- free

Ion exchange

(oxalate)

Carrier- free

Ion exchange

(oxalate)

> 5 curies

Solvent extraction (dichlurodiethvi ether)

ether)

ether)

60 curies

Low 500 inillicuries 6. 5 curies > 3.000 millicuries 900 millicuries

ether)

Carrier-free Carrier-free

Ion exchange

2 curies

Unprocessed Unprocessed Unprocessed

Cle-p-n

Carrier-free Carrier -free Carrier-free

As-n-7 Ue-n-7-0

Carrier- free

Distillation Distillation Distillation Distillation Distillation

Cie-H-7

Ge-d-n Ge-d-n

8 curies

Se-n-7 As- p-n

> 5 curies

Br-n-7

2 curies

Se-p-n

Ion exchange

unit unit unit


Sec. 14-3] Table

Nuclide

Review of Radioisotope Preparations and Methods.

1.

Half -life

Produc tion method

Kr"

10. 27 years

Rb»

19. 5 days

Sr"»

63 . 8 days

8r"

53 days

Sr"

19.9 years 2. 54 days

Y" yii

Zr»

Nb"

Tc"Tc"

Mo" Ru"


Ru>»"

59. 5 days 65 days 35 days 60 days 2. 1 X I0» years 2. 79 days 2.44 days 39. 8 days

Ru"1

1 year

Pd"» Pd"» Pdiw

17 days

7X10*

80 miUiouries/em*

V-f

220 inillicurics Carrier-free Carrier-free

Kr-n-> Rb-n-7 Rb-d-2n Rb-p-n Sr-n-7

V-f V-f

Sr".0 Y-n-y

v-f V-f

Zr-n-v Zr»,0 Mo-p-n

V-f

Mo-n-7 Ru-n-T

V-f

Ru-n-7

V-f years

Ru-n-7 Pd-n-7

V-f

In""

Pd-n-T Pd-p-n Ag-n-7 Pd.fi Cd-n-7 Cd-n-7 In-n-7 Cd-p-n

Sn»«

112 days

Sn-n-7

8b'»

2 .8 days 60 days 2. 7 years

Ag"°

Ag»' Cd"»Cd»»

Sb"<

Sb'» Te""»

58 days 33. 5 days

Te'" I in

77 hr 60 days 1. 56 X 10' years 8. 08 days

Tci»-

VTill

Xe'«

5. 27 days

Cs1"

La"«

9. 6 days 2. 3 years 30 years 11. 52 days 9. 5 years 12.8 days 40.0 hr

Ce"'

32. 5 days

Cs"<

Cs'« Ba"' Ba'" Ba'"

Ce'" priit Pr»«

Pr"'

282 days 19.2 hr 19 2hr 13. 95 days

Specific activity (units /g) attained at ORNL

U-/

13.6 hr 40 days 270 days 7 . 6 days 43 days 53 hr 49 days

Ag>"

14-35

In-p-n Sb-n-7 Sb-n-7 Sn"»,0

V-f

Sbi",/}

V-f

Te-n-7

V-f

Te-d

V-f V-f

Te>".fi

Carrier-free 20 millicuriea

Carrier-free Carrier-free Carrier-free

900 millicuriea

Carrier-free Carrier-free 0. 05 millicurie Carrier-free Carrier-free Carrier-free > 10 millicuriea >2 millicuriea >5 curies /g 30 millicuriea > 5 curies /g 30 millicuriea Carrier-free 2 curies 1 curie

Carrier- free 210 curies 200 millicuriea 1 curie Carrier-free 20 millicuriea Carrier-free > 1000 millicuriea 2,587 millicuriea Carrier-free Carrier-free Carrier-free Carrier-free

0.16 millicurie Carrier-f ree Carrier-free Carrier-free Carrier-free Carrier-free

Carrier-free

Ba-n-7-fl Cs-n-7

Carrier-free 30 curies

V-f

Carrier-free 0. 14 millicurie

V-f

Carrier-free

Ba,0 La-n-T

V-f

Ce-n-7

u-/

Pr-n-7 Pr-n-7

u-/

Principal purification methods

used

Low-temperature fractionation Unprocessed

unit

Precipitation (in HNOi) Precipitation (in HNOi) Precipitation Ion exchange Precipitation (in HNOi) Ion exchange Unprocessed unit Ion exchange Adsorption and ion exchange Unprocessed unit Precipitation (MnOj) Precipitation Precipitation Distillation Distillation (RuO«) Unprocessed unit Distillation (RuO«) Unprocessed unit Unprocessed unit Precipitation

Carrier-free

V-f

Ba-n-7 Ba-n-7

STP

(Continued)

>5 millicuriea

Carrier-free 1 curie Carrier-free 75 millicuriea Carrier-free >5 curies 10 curies


Ion exchange Precipitation Precipitation Precipitation Precipitation Distillation Distillation Precipitation Unprocessed

unit

Distillation Distillation Distillation Unprocessed unit Low-temperature fractionation

Cocrystallisation (alum) Precipitation Precipitation Ion exchange Ion exchange Unprocessed unit Ion exchange Unprocessed unit Ion exchange Ion exchange Unprocessed unit Ion exchange Unprocessed unit

isotopes

14-36 Table

Nuclide

1.

Review of Radioisotope Preparations

Half-life

Produc tion method

Nd'« Pm"'

2. 6 years

Sm'" Sm'" Eu'» '" Eu'»

73 years 47 hr 13—16 years 1. 7 years

Gd>"

236 days 73. 5 days 9 . 4 days 129 days 45 days 115 days 73. 2 days 24.1 hr 97 days 16 days 74. 37 days 19 hr 2.69 days 3.15 days

Tb'" Er1"

Tm"» Hf>"

Ta'« Wiu W'« Os1" 0s»>

Ir'" It'"

Au'M

Au»


11.3 days

HgmHg»'

Hgtoa.sot XI Jo*

UXl

23 hr 2.71 days 45 days 4 years 24. 1 days

U-/

Nd-n-T

U-/ Nd"',0 U-/

Sm-n-7 Eu-n-Y U-f Sro'".0 Gd-n-Y Td-n-T Er-n-7 Tra-n-T

Ilf-n-T

Ta-n-T W-n-y

W-n-r

Os-n-y Os-n-y Ir-n-7 Ir-n-7 Au-n-7 Pt-n-7

Specific

[Sec. 14 and Methods.

activity

(unite /g)

attained at ORNL

Carrier-free 3. 5 millicuriea Carrier-free Carrier-free Carrier-free > 10 curies 5 curies

(Continued)

Principal purification methods used

Ion exchange Unprocessed


Ion exchange Ion exchange

unit unit



unit

1 curie 1.700 tniilic.urics 1,000 millicuries

Unprocessed

unit

>3 curies

Precipitation Unprocessed

873 millicuries > 10 curies 30 curies 30 curies Carrier- free

Distillation Chlori nation C hi ori nation Extraction Extraction

unit

Pt'»0

Hg-Ft-Y

1 curie

Precipitation

Hg-n-7 Tl-n-7 Natural

830 miUicuries 500 millicuries

Precipitation Precipitation Ion exchange

Carrier-free

Some of the impurities usually found with several radioisotopes are CoM in Fc"H, Ni", Fe» in Cr", Ba1" in Ca« HP" in Zr», Sm1" in Eu1"1", Rb«« in K« rare earths in Sc", S" in P", and Rb" in Cs1". Table 1 gives some data on many of the other radioisotopes produced at ORNL. The specific activities indicated are not necessarily the best ones obtainable, but they represent the maximum that has been attained for materials distributed in the radio isotope program. Co"» in

4

RADIOISOTOPE PROCESSING FACILITIES

Other than the nuclear reactor itself, the main parts of the radioisotope production plant are the facilities for chemically processing, storing, and packaging radioactive of material. Also included are the facilities for the disposal or decontamination radioactive wastes or chemical equipment. 4.1

Types of Equipment

Equipment for processing ranges from the simplest laboratory bench apparatus to complex, heavily shielded, remote-control equipment, and the type used generally depends on the amount of material being processed and the intensity and the kind of Radioisotopes for which the demand is small and the radia radiation encountered. tion level is low are processed in temporary setups of standard laboratory equipment, the so-called "beaker and tong" method. On the other hand, I"1, which is produced on a regularly scheduled multicurie basis and involves processing large quantities of Radio highly radioactive material, requires a fairly large, industrial type of unit. chemical processing is based upon three types of radiation: low, intermediate, and high level. Within these general groups are many subgroups for which several of the

Sec. 14-3]


14 37

criteria are production requirements, toxicity of the radiochemical, type and com plexity of the process, general safety considerations, and relation to other processes.

REFERENCES


1. Harrison, W. M., and J. G. Hamilton: Chem. Act., 49: 237 (1951). 2. Coryell, C. D., and N. Sugarman: "Radiochemical Studies: The Fission Products," National Nuclear Energy Series, McGraw-Hill Book Company, Inc., New York, 1951. 3. Keyes, R. : "Estimation of Self-absorption in Neutron-irradiated Samples," AECD3000, 1950. 4. Holmes, D. K.: "Problems in Neutron Population in Localized Absorbers," ORNL 56-1-141.

14-4

GASEOUS-DIFFUSION

SEPARATION

PROCESS

BY George A. Garrett Large-scale separation of Us" from natural uranium for industrial and military The first such purposes is effected in plants utilizing the gaseous-diffusion process. plant was put into operation at Oak Ridge, Tenn., in 1945. Several additional plants have been built at Oak Ridge since that time, and plants have been erected at Paduc&h, Ky., and Portsmouth, Ohio. The gaseous-diffusion plants at Oak Ridge and Paduc-ari are operated for the Atomic Energy Commission by the Carbide and Carbon Chemirat Company, and the Portsmouth plant will be operated by the Goodyear Atomic Corporation.


1

THE CASCADE

In the gaseous-diffusion process gaseous uranium hexafluoride is caused to diffuse The fraction passing through the barrier is slightly through a porous barrier. enriched in the lighter isotope, U*", and the fraction remaining is correspondingly The process is repeated through a large number of stages in scries until the depleted. desired degree of enrichment and depletion is achieved. The series arrangement of stages is referred to as a cascade of stages. 1.1

The Cascade Principle

The basic relationships of the multistage, or cascade-type, processes apply equally to the gaseous-diffusion process and to many other separation processes, the most

J} Feed

Enriched

Depleted ~~7~\

-W

stream

Fia. I.

Gaseous-diffusion

sireom

stage.

familiar being fractional distillation. Figure 1 shows the essential features of a gaseous-diffusion stage. In order for the process to operate successfully, the hole* through the barrier must be of the same order of magnitude as the mean free path of the gas molecules. For the present purposes the gas, uranium hexafluoride, may be regarded as consisting of two components: U"'F«, whose molecular weight is 349. and U,38F6, whose molecular weight is 352. Separation is accomplished by virtue of the fact that the lighter molecules, which have a shorter mean free path than the heavier ones, diffuse through the barrier more rapidly than do the heavier ones. Consequently, the gas that diffuses through thf barrier is richer in the lighter molecules than the gas remaining on the high-pressure side of the barrier. 14-38

gaseous-diffusion separation process

Sec. 14-4]

14-39

Figure 2 shows the arrangement used to make gaseous diffusion a continuous multistage, or cascade, process. The feed stream entering a stage is a mixture of the depleted stream from the stage above and the enriched stream from the stage below. Fresh uranium hexafluoride feed is introduced at an intermediate stage, product enriched in the lighter molecule is withdrawn from the top stage of the cascade, and depleted material, called waste, is withdrawn from the bottom stage. The portion of the plant above the feed point is called the enrieher, and the part below is called the

r

*■ Product

Tqp__ —

stoge

1. Stage

enrichment factor: x and y are

mole fractions of U"6

lol

-yj

Ln+I

(L„+l-

Yn+I

»n+2

Stoge_ri+J

Fxf

Yn


-

=

=

3. Combining 1 and 2c:

"1 L. ,-P

1

yn-i

Ly.

(c)

Stage n

L„-r

X*+l

— Xn

Ua-

I»n

(b)

Feed-

=

Ik *w

- (P/L) 1

1

l)x„(l -xn)

dx _Bpttpm stage

*.)

Pxp + Wxw (L P)xn+i + Pxp (From net flow across AA') (6)

xn +l

Xn

(a

2. Material balances: (a) F = P + W

U

(Ln-P)

1

Vn — Xn +

l)x(l

x)

- r (zf P

-Xn) -

x)

The basic enrichment equation Waste

(C)

Fio. stripper.

2.

Multistage process of gaseous diffusion.

Also shown in Fig. 2 arc the basic relationships, the definition of the stage separation factor.

which consist of material

balances and

1.2

Stage Separation Factor

The stage separation factor is the isotopic ratio of U"' to US3» in the enriched stream leaving a stage divided by the corresponding isotopic ratio in the depleted stream from the stage. If y„ is the concentration of U"'Fe in the enriched stream leaving stage n, xn the concentration of U"*F6 in the depleted stream, and a the stage separation factor, then the relationship defining a is !/./(!

- V.)

Z./U

Xn)

-

CD

which is the first equation given in Fig. 2. When defined in this manner, the stage separation factor is the exact analogue of relative volatility in fractional distillation. Similarly, the equations derived below have exact analogues in fractional distillation. The theoretically ideal separation factor at a point along the barrier has the value V352/349, or 1.0043. With the stage separation factor defined in terms of the depleted stream, which has a concentration lower than the average concentration along the barrier, the maximum theoretically ideal stage separation factor is about

isotopes

14-40

[Sec.

14

In the numerical examples presented below, a constant stage separation facto: 1.006. of 1.0043 is assumed. 1.3

The Enrichment

Equation

The fundamental steady-state enrichment equation derived in Fig.

^ an

-

-

(a

2

is

- - £ (xF - x)

l)i(l

X)

L

(21

where x is the Um concentration expressed as mole fraction, a the stage separation factor, P the product rate, L the interstage flow rate, and xp the U"* concentration of The rates P and L may be regarded as either moles of hexaftuoride per the product. unit time or atoms of uranium per unit time. In this form Eq. (2) is applicable to differential processes as well as those in which The thermal diffusion process and the countercurrent centri the stages are discrete. fuge are differential processes.*'1 1.4

Minimum Number of Stages and Minimum Interstage Flow Rate

v Nmia =

(a -

,

1

In

1)

xp/{\ ——

xF/(l

-

—

Xp)

Xf)

(3>

= 1.0043, xp = 0.80, xF = 0.0071 (the natural abundance of U»»), A*min is 1,472. dx/dn is set equal to zero in Eq. (2), a formula is obtained for the miniruuir> interstage flow rate that must be provided in order that there be any enrichment at the point in the cascade where the concentration x occurs. This formula is

For a

If

P

_

r = -. i-min

(a

-

rr 1)

xp —

x(l

-x

- x)

(4,1

Substitution of the values given above shows that, at the feed point, Lm-,„ = 26,157?. In other words, the flow rate of uranium through the barrier of the first stage must be at least 26,157 times the product rate. These two formulas make it evident that a gaseous-diffusion cascade must contain many stages if it is to accomplish a very high degree of enrichment, and that large amounts of power and barrier area must be provided to give the high flow rates required. 1.5

"Ideal" Cascade

Superscript numbers

refer to References

xn ,

-" 1

--"-+i 1

-_•

A much better picture of a gaseous-diffusion cascade can be obtained from con sideration of the so-called "ideal cascade." This cascade has been described by Cohen.1 An ideal cascade may be defined as one in which there is no mixing of streams of unequal concentrations. Mixing of unequal concentrations obviously results in inefficiency in a separating cascade. It can be shown that for a constant value of a an ideal cascade has the minimum barrier area and the minimum power consumption. In Fig. 2 it is evident that the requirement of "no mixing" implies that y. = x»+j. The definition of a results in the equation

*


Some insight into cascade separation requirements may be gained from consideration If P is set equal to zero in Eq. (2) and the equation is integrate; of two special cases. from xf, the feed concentration, to xp, the product concentration, a formula results for the minimum number of stages required in order that the product concentration xp be reached. This formula is

xn

at end of subsection,

(5)

Sec.

If

14-4]


14-41

the enrichment per stage is constant, it may be deduced that

-

1

X„+,

Additional approximations

Fig.

- X.

1

to those used in the derivation

similar

2) give the equations

-

x„+i

-"

x„

and

2/1

V

£J=J x(l 2

=

^ an

x„(l

:—

(6)

of Eq. (2) (see

- x„)

(7)

- x)

(8)

—

1

- -*In ^^ a zf/(1

1

NM

Xf/(1

—

xp\

It should be noted that since a is so nearly unity in uranium isotope separation, the above equations are all essentially exact in spite of the approximations. Integrating Eq. (8) results in the formulas xr)

= 2Nmia

(9)

and also a

-

- Xf)

is

if

is

last formula gives the stage number at the point in the ideal cascade where the true for both enriching and stripping stages concentration x occurs. The formula the feed stage designated the zeroth stage, with the stripping stages denoted by negative numbers. Combining Eqs. (2) and (8) gives the formula for interstage flow rates in the enricher of the ideal cascade as follows: X)

—

2Lmin

(11)

1.6

— x

a -

1(1

—

(x

- xw)

(12)

.._.

xir

—

Practical

(!>>)

x);

2W 1

t I>i
L^

- - l)x(l - x) (a

an

and

—

portion of the plant the equations analogous to (2) and (11) are,

^

In the stripping respectively,

J?*'~\ a x(l 1

=

Lide.,

i

Cascade

3

1.

1.

is

is

1

in The formulas given above indicate the great importance of the quantity a — Although a differs from unity by only a cascade processes for isotope separation. In the ideal plant both the extremely important. small fraction, that fraction number of stages and the interstage flow rate required at a given concentration are inversely proportional to a — approxi Thus the physical size of ideal plants mately inversely proportional to the square of a — For illustrative purposes, the formulas have been applied to an ideal cascade producing a product of 80 per cent U"6 concentration with the waste being stripped shows curves of con Figure concentration one-half that of natural uranium. to It centration and interstage flow rate (per unit product rate) vs. stage number. 34.4 million times the product rate, noted that the total flow rate in the cascade illustrating again that Um enrichment by gaseous diffusion requires a large amount of power. The ideal cascade not a practical one from the standpoint of construction, even requires less power and less barrier area than any other cascade, since each though

it

is

is

is

a


This

14-42

ISOTOPES

[Sec.

14

However, it is feasible to construct s is different from neighboring stages. practical plant whose performance closely approximates that of an ideal plant. Tbt design of a practical plant often results in one that has several different sizes of stages" The usual result is that the practical plant has fewer stages than the ideal plant, but, o; course, it requires more power and barrier area. The original gaseous-diffusion plant at Oak Ridge cost about 500 million dollars.1


stage

Stage

Fig.

3.

number

Flow rate and concentration vs. stage number in "ideal" plant. 1.7

Some Advantages of Gaseous Diffusion

Since gaseous diffusion has proved to be the most economical and successful prof* * for producing U"5, it is appropriate to note some of the advantages of this proec* over its competitors. The chief competing processes, all of which have had success, trials in separating uranium isotopes, are thermal diffusion, the countercurrent centri fuge process, and the electromagnetic process. It is less econom*** Thermal diffusion is probably the simplest of all the processes. than gaseous diffusion for large-scale Um enrichment because of its low separate:: The net resx" factor and the relat ively low flow rates attainable in a single column. For a smaU-sreJe is that too many columns and too much steam are required. In iV" separation job these disadvantages do not necessarily rule the process out. there may very well be applications where thermal diffusion is preferable to gaseo_diffusion. As compared with the electromagnetic and centrifuge processes, the gaseoo-diffusion process offers two great advantages, simplicity and flexibility. Relative!speaking, gaseous diffusion is a simple process that runs continuously with very lit" The process machinery is fairly conventional, and there are no severey attention. restrictive requirements as to operating personnel. 1 As regards its flexibility, a very great range of operating conditions is possible. t* of practical gaseous-diffusion plant designed for one set operating conditions can operated under greatly different conditions without too great a loss in efneke v Interstage flow rates can be adjusted over a wide range within a group of stages ti are of the same physical size by simple adjustments in operating pressures. At a fa< product concentration, the product rate can be varied merely by changing the f«*


Sec. 14-4]

14-43

Since there are present in the plant rate. Product concentration may also be varied. all concentrations from the waste concentration to product concentration, it is possible to withdraw such materials if the need is sufficiently great to counterbalance the corresponding loss in top production.

REFERENCES 1.

Cohen,

Div.

K. : "The Theory of Isotope Separation," National Nuclear Energy vol. IB, McGraw-Hill Book Company, Inc., New York, 1951.

Ill,


2. Hogerton,

J.

F.

: Eng.

News-Record,

December,

1945,

p. 135.

Series,

14-5

THE ELECTROMAGNETIC SEPARATION OF STABLE ISOTOPES BY C. P. Keim and C. E. Normand 1


1.1

THE CALUTRON

Origin of the Electromagnetic

Process

The separation of isotopes by electromagnetic methods can be traced back to the pioneering work of Dempster*'1 (1918) and Aston' (1919) and to the even earlier work of J. J. Thomson8 (1913). Out of the work of these men, together with further development at the hands of such workers as Bainbridge, Bleakncy, Jordan, Nier, and Mattauch, has come an impressive and varied array of instruments in which isotope* can be electromagnetically separated.4 For 20 years or more, mass spectrometers, as instruments of this type are called, were developed and used almost exclusively to detect, identify, and measure the relative abundance of the isotopes of the various Not until about 1940, when a pressing need arose for U*" in experimental elements. and even larger quantities, was any serious effort made to adapt the principles of the mass spectrometer to the separation of isotopes in substantial quantities, e.g., milli grams, grams, or even kilograms. 1.2

General Specifications of the Calutron

As is now well known, a group under the direction of E. O. Lawrence at the Uni versity of California Radiation Laboratory undertook the adaptation of mass spec trometers to the separation of isotopes in quantity and developed, during 1941 and 1942, a production-type mass spectrometer to which the name calutron has been The first U"' to be made available for military applications was produced applied.* in calutrons at the Y-12 plant in Oak Ridge, Tenn., and some of these calutrons, modified in various ways to meet the requirements for processing of different elements, have been in use since November, 1945, as basic equipment in the isotope separation program there.8'7 In principle, there is no essential difference between the calutron and a laboratory The similarity is particularly striking if a comparison is made mass spectrometer. with some of the 180° focusing instruments designed by Dempster in his early work. All such instruments are designed to separate atoms of different mass, i.e., isotopes, of the element being processed. Ions of the element are produced in an ion source, accelerated by an electric field, and projected as a beam of essentially monoenergetic particles across a magnetic field. Ions having different masses are thereby con strained to move along different paths —circles of different radius if the magnetic field is uniform — and can be separately measured or collected on suitably placed collectors. Differences between the calutron and the laboratory mass spectrometer arise almost Since the mass spectrometer wholly from the vastly different ion currents dealt with. is used only for detection and measurement, ion currents are often of the order of 10-lt amp, but the calutron, in order that it may produce usable quantities of collected isotopes, is designed to operate at currents of 10~2 to 10" ' amp. Figure 1 shows schematically the arrangement of ion source, accelerating electrodes, * Superscript numbers

refer to References

at end of Bubsection.

14-44

Sec.

14-5]

ELECTROMAGNETIC

SEPARATION

Fig.


1. Schematic diagram of calutron showing separation focusing of each isotope beam.

14-45

of two isotopes and approximate

-MAGNET YOKE

MAGNET

COILS

ION SOURCE

LINERSCHEMATIC 2 4" RADIUS

SEPARATOR

Fio.

STANDARD 2.

TYPE

Cutaway view of 24-in. -radius calutrons.

and isotope collectors in the vacuum tank of a calutron. Both the mass separative action and the first-order focvising effect of this type of separator are also indicated. It will be noted that both maximum mass separation and optimum beam focusing occur at a point 180° along the ion path from the source. Both of these effects are obviously essential in an isotope separator designed for high ion beam intensities. The Oak Ridge calutrons are of two sizes — size being usually designated by the maximum radius at which ions can be collected. The smaller calutrons are of 24-in.,

isotopes

14-46 Table

[Sec.

Major Specifications

1.

14

of Calutrons Type

Alpha (large) Nominal size — maximum ion collection Magnet gap dimensions, in.

radius,

in.

48

Height. 96 Depth. 72 Width, 24

Maximum magnetic flux, gauss Magnet generator rating Vacuum

tank

3,500 250 volts, 1.200 amp 8.840-312 Four 20-in. oil diffu sion Four 6-in. oil booster One 300-cfm mechanical 15.000 0.015-0 030

volume. 1-ft1 equipment

Vacuum pumping


Vacuum pumping speed, 1/sec Operating vscuum range, microns, Maximum arc length, in Number arcs per source unit Accelerating voltage, kv

llg

Beta

-i.jll 24

Height. 72 Depth. 60 Width. 12 7.000 240 volts. S4M amp 2.4 50-8* 5 Two 20-in. Mil diffu sion

Two 8-in. oil One 300-cfm mechanical 5.500 • 015 0 030 7 I 2 35

It

1-2-4 35

the larger of 48-in. radius. Figure 2 is a cutaway view showing the principal The 48-in. units are similarly arranged. ponents of two 24-in. calutrons. Major specifications for calutrons of the two sizes are listed in Table 1. 1.3

The Calutron Equation

The basic equation expressing the separative action of the calutron follows i If m is the ately from the elementary equations of ion motion across a magnetic field. mass and e the charge of an ion, and if v is the velocity imparted to it in falling through a potential difference V, then Ve = Mrnv* (li

If now the ion continues moving with this velocity across a magnetic field of uniform strength H, it will follow a circular path of radius R such that ~

Combining

li

(

Eqs. (1) and (2) to eliminate the ion velocity H*R*

_

2V

v,

in

13

e

where H is expressed in oersteds, R in centimeters, m in grams, and V" and e art in For the particular case of singly charged ions of absolute electromagnetic units. volts, Eq. (3) atomic weight M, accelerated through a potential difference of reduces to

V

HR

= 144

y/MV

<4)

Thus, an ion of any atomic weight M can be made to follow a path of any chosen radius ft by a suitable choice of magnetic field strength H or accelerating voltage V. Suppose, for example, that one wishes to determine the appropriate magnetic field strength for collecting singly charged, 20-kv ions of atomic mass 13 at a 25-cni i Direct substitution in Eq. (4) gives tion radius.

H X and as the required field strength.

25 = 144

H

V 13

X

2

— 2,938 oersteds

X

10*

electromagnetic

Sec. 14-5]

separation

Once H and V have been chosen, ions of different masses paths having different radii Ri and Si such that

14-47 Mi

and

M % will

follow

It has been found by experience in electromagnetic isotope separation that in passing from one element to another of significantly different atomic mass, it is usually better to change the magnetic field while keeping the radius and accelerating voltage essen However, minor adjustments of beam radius to assure optimum tially constant. collection during a run are made by small variations of the accelerating voltage. Vacuum Requirements

1.4

1.6

Ion Source

a

3

a

is

a

a

is

is

a

a

is

is

a

is

is

a

is

a

is

a

a

it is

is

to produce an intense, well-defined The basic function of the calutron ion source beam of high-velocity, essentially monoenergetie ions of the element being processed. In order to do this, necessary that the element or one of its compounds be fed continuously as vapor to an arc maintained by stream of 100- to 200-volt electrons heated tantalum filament. emitted from Except in those rather rare cases where the element or suitable compound gaseous or highly volatile at room temperature, vapor supplied by controlled heating of the chosen charge material contained in " metal " bottle and enclosed in an oven that an integral part of the source assembly. of Acceleration the ions obtained by maintaining the whole source assembly some 35 kv positive with respect to ground. Ions that emerge from the arc chamber slit in its front surface pass, in turn, through openings in an accelerating through electrode (usually run at some few kilovolts negative) and a grounded electrode. The use of an accelerating and a grounded electrode permits some degree of control of beam intensity and definition and still projects the ions into the field-free vacuum space at ground potential and with velocities appropriate to the source voltage. Figure a cutaway view showing the principal components of a typical ion source. One of the major variables encountered in separating the isotopes of different elements that of the source temperature required to vaporize the different charge materials and to prevent condensation of material in and about the arc chamber. Operating temperatures ranging from room temperature up to some 2800°C have been encountered. No single source has yet been devised capable of operating over such wide temperature range. It has been found necessary, therefore, to develop sources of at least four distinct types, each characterized by different operating temperature range. These four source types are referred to as external-feed, medium-temperature, high-temperature, and very-high-temperature sources. The external-feed source, as the name implies, used with those elements or com pounds which are gaseous or which vaporize freely at room temperature.' The charge material contained in cylinder or bottle outside the vacuum chamber and fed as a gas or vapor through suitable control valve and metal tube directly into the arc chamber or into the vapor manifold connecting the charge oven with the arc chamber in a standard medium-temperature source. Little or no heating of the source unit required except as possible aid to outgassing during the startup period. is


is

is,

In order that ions may follow simple orbits from source to collector, the separation must be carried out in a vacuum such that the ion mean free path is greater than the This distance traveled by the ions in passing from the source to the collector. Each requires operating pressures of the order of 1 X 10~6 to 5 X 10-' mm of Hg. therefore, assembled in a vacuumtight tank of welded steel construction, calutron and these tanks are continually evacuated during operation by oil diffusion pumps. Each of the smaller calutrons equipped with two 20-in.-diameter diffusion pumps, each backed by an 8-in. booster pump. The two 8-in. booster pumps are backed, in turn, by a single 300-cfm mechanical vacuum pump. The pumping systems of the larger calutrons are similar except that four 20- and four 8-in. pumps are provided for each tank.8

14-48

ISOTOPES

[Sec. 14


The medium-temperature sources are heated by resistance-type (Calrod) heaters cast into the copper charge oven and arc castings.10 The normal operating tempera ture range of these units is about 300 to 550°C, the lower limit being fixed by the onset of instability in the temperature-regulating equipment and the upper limit by the service range of the heaters. By the addition of water cooling to the chargeoven casting and the insertion of heat baffling between the oven and the arc castings, regulated oven temperatures as low as 50°C have been attained. Also, by the applica tion of heat shielding to the heater castings, the upper temperature limit has been raised to 650°C. In the high-temperature sources, copper castings arc replaced by graphite structures and the calrod heaters by strip heaters of graphite supported in Lavite insulators." With adequate heat shielding, units of this type have been operated with good tem perature regulation at temperatures ranging from less than 600 to above 10OO°C. The very-high-temperature source was developed specifically for processing metals of the platinum and palladium groups.11 Heating is obtained by bombarding a

Fig.

3. Cutaway view of typical ion source.

graphite block that contains both the charge material and the arc chamber with beam? With adequate heat shielding, units of this of high-voltage (about 30 kv) electrons. type have been operated at temperatures ranging from approximately 1700 to 2800T, 1.6

Isotope Collectors

The A different isotope collector assembly is required for each element processed. number of collector pockets depends on the number of isotopes to be collected and The spacing between adjacent pockets depends on the may vary from two to ten. percentage difference between masses and may be as much as 7 in. between Li* and Li7 pockets at 48-in. collection radius to as little as 0.16 in. for Pb,0T and Pb,0» pockets at 24-in. collection radius. Collector pockets are usually of graphite or copper, although other metals, e.p.. Metal pockets are usually silver, stainless steel, and aluminum, have been used. water-cooled and arc used whenever high ion-beam intensity or volatility of the product makes such cooling necessary. Both graphite and metal pockets are made in different sizes and wall thickness to conform to the space available and to accom modate beams of different intensities. A standard type of collector has been developed in which any required number of pockets of the desired size and material can be assembled in a supporting frame with appropriate spacings between pockets. The whole pocket assembly is covered by a graphite faceplate in which beam-defining slits have been cut to match the resolved A typical collector assembly is beam pattern and the collector pocket openings. shown in Fig. 4.

Sec. 14-5]

ELECTROMAGNETIC SEPARATION

14-49

The facts that the pockets and defining slits are curved and that the defining faceplate cuts the vertical plane of the source at an angle of approximately 45° are results of an inhomogeneity introduced into the magnetic field of the calutron by magnetic shims designed to improve the focus at the collector of ion beams that diverge at the source. Each pocket is electrically insulated from adjacent pockets and from the supporting frame and the defining faceplate, thus permitting individual isotope beams to be


monitored during collection. Usually, all the isotopes of an element are collected at the same time; however, space limitations at the collector sometimes make it impossible to install a full com plement of collector pockets. This situation arose in the collec tion of the isotopes of samarium11 (Sm144, Sm147, Sm118, Sm1", Smls°, Sm1", and Sm1"). In this case, two different collectors were used. One rejected the Sm147 and Sm1" isotopes; the other rejected Sm148 and Smlso. On the other hand, there have been a few occasions on which isotopes of more than one element were collected simultaneously. When the nickel isotopes were collected, using NiClt charge, intense copper ion beams also Since these appeared owing to chlorination of copper parts of the source structure.

FlQ.

4.

Collector assembly.

two elements are of adjacent atomic mass, additional pockets were installed in the collector and the copper isotopes (Cu63 and Cu88) were collected, along with those of nickel (Ni68, Ni»°, Ni81, and Ni65). 1.7

Beam Resolution

The ions collected are usually the singly charged positive ions of the element being Ions of many other species — multiply charged and complex ions of the processed. ;lement sought, ions of other fragments of the charge compound, and extraneous ions rom residual gases and materials of construction —always appear as ion "side bands." figure 5 is a magnetic scan of some of the ions produced when selenium oxide is The preponderance of Se+, 0+, and C+ ions over all vaporized as charge material. >ther species and also the sharpness of focus of the Se+ beams are clearly shown. Figure 6 is an oscillographic scan, on an enlarged scale, of five of the seven mercury sotope beams (Hg"18 and Hg1" were off the screen to the right) and shows the degree To attain this resolution in the case of mercury13 and to >f mass resolution obtained. irevent serious contamination of the collected isotopes by the diffusion of neutral aercury vapor into the collector pockets, it was necessary to operate t he calutron with he tank liner refrigerated to — 50°C.14 In order to retain mercury in the collector lockets, these were also refrigerated, and they were, moreover, cast from silver in rder that the retention might be further enhanced by amalgamation. Examples of till other cases in which retention of the collected element was enhanced by allowing S to combine with the material of the pocket or with some material placed in the ocket are oxygen with copper to form Cu20, sulfur with copper to form CuS, chlorine ,-ith barium and magnesium, and bromine with cadmium.16

ISOTOPES


14-50

SELENIUM

Fio.

Oj

5. Magnetic scan of the ion beams from SeOi charge.

NATURAL ABUNDANCE

s

MASS NUMBERS

Pio.

Se

6. Oscillographic scan showing 2

2

g

oj *?

3

£

o 5

8

£

resolution of five of the seven isotopes of

mercury.

OPERATION OF THE CALUTRON 2.1

Charge Selection

Preliminary to the actual processing of any element, in the calutron is the problem of selecting, and often of preparing, a suitable charge material. The first requirement of a suitable charge material is that adequate vapor for maintaining the ionizing arc be supplied by the material when heated to temperatures attainable in the source.

electromagnetic separation

Sec. 14-5]

14-51

Other desirable characteristics are simplicity of molecular structure, chemical stability, If the and freedom from destructive and deposit-forming products of decomposition. element itself has a suitable vapor pressure, its use is favored over that of a compound, The above criteria are useful since extraneous ion side bands are thereby reduced. in guiding the search for a satisfactory charge material; however, the selection is based ultimately on the performance of the material in the calutron. favored Table 2 presents a summary of elements processed, operating temperature ranges, and collector materials. Table 2.

charge

materials,

Charge Materials, Operating Temperatures, and Types of Collectors for Stable Isotope Separations Approximate

Element

Charge

LiCl

material

Li Bed,

plus

BC1, Vitrogen Vlagneaium


Sulfur

COi Ni

Oi Mg SiCl. CS.

BCli K

Ca Vanadium

TiCU VF. CrCli FcCh

NiCl,

CuiCU Zn

Gal, GeCU SeOi

tubidium trontium irconium 'alladium ilver

SrBn Rbl Sr

ZrCl.

MoCli

Ru Pd AgCl Cd

Inl

ntimony 'cllurium

SnCl. SbiOi TeCl. Ba

LaCli CeCh

:i rnariu in

!afniuiu antalum

Latinum [ercury hallium ead

NdCli

SmCli HfCl«

TaCli

WCU ReiO?

Ir Pt IlgCli

HI

PbCI.

metal

Collector material

oven temp. °C

575 250

t

t t t

525 X X 260 200 600 X 700 550 550 600 400 475 120 X 150 700 600 600 250 75 2600 1750 650 300 250 X 400 200 725 600 625 650 700 150 80 150 200 2600 2150 80 350 460

Copper* Graphite Copper Copper Graphite with IT to form nitride Copper or graphite with fine Cu to form oxide Copper Copper Copper with fine Cu to form sulfide Copper with Ba or Mg turnings Copper Copper Graphite Graphite Graphite Copper or graphite Graphite Graphite Copper Copper Graphite Copper Copper containing Cd turnings Copper Copper Copper Graphite Graphite Graphite Copper Copper Graphite Graphite Stainless

steel

Aluminum Graphite Graphite Copper or graphite Graphite or copper Copper or graphite Copper Carbon Copper Graphite Graphite (iraphite Silver Copper Copper

* All copper collectors are water-cooled. t Pile-produced Be10 separated from natural Be'. from an external charge bottle X Vapor pressure at room temperature sufficient for supplying the arc

isotopes

14-52 2.2

[Sec.

14

Operating Cycle

The calutron-operating cycle may be divided into three periods: start-up, innap. and shutdown. Start-up is that time, usually a few hours or less, during which the system is bein: evacuated, the source is outgassed, the accelerating voltages are brought to their desired values, and the required charge-oven temperature is reached. Innage is the productive period during which ion beams are received at the col lectors. This period may vary from a few hours to several days, depending greatly The ion-beam intensities at the various pockets are on the element being processed. monitored at regular intervals throughout this period, at first, for comparison with the relative natural abundances of the isotopes being collected, thereby making sure that the correct masses are being collected in appropriate pockets, and, later, as a means of maintaining optimum production and as a basis for estimating the amount and purity. of material collected. Production is optimized by adjustments of various source and collector variables. Control variables include charge temperature, are voltage and current, accelerating voltage, accelerating electrode position, and collector orientation. The shutdown period begins with the turning off of the accelerating voltage and coot inues until the tank is brought to atmospheric pressure.


2.3

Run Termination

Operation of the calutron through the above operating cycles is termed a run. Runs are terminated for a variety of reasons, some of the more common being chans exhaustion, filament burnout, electrical shorts at source or collector, the formation of obstructing deposits in the arc region, and erosion of collector pockets. Occasional terminations result from an almost endless list of other causes. Generally, however the cause of run termination lies in the source rather than in the collector. It is usuil practice, in such cases, to repair or replace the source without disturbing the collector. Thus the same collector is often used throughout a number of consecutive runs. At the conclusion of a collection, or whenever a collector is to be replaced, it b removed from the calutron and disassembled, with every precaution being taken to avoid loss or contamination of the collected material. Individual collector pockets ait then wrapped, labeled, and delivered to the chemical laboratory for product recovery. 2.4

Collector Retention

The amount of useful product recovered from a collector pocket is usually less than the amount estimated on the basis of monitored ion current. The ratio of the recovered to the estimated collection, expressed as a per cent, is called the retention. This factor varies widely (from a few to nearly 100 per cent) with the different ele ments, different collector pockets, and different modes of operation. Although it is known that low retention is likely to result from high beam intensity, inadequate pocket cooling, and high volatility of the collected element, only through actual operating experience can a reliable value be assigned to the retention to be expected in the collection of a given element or isotope. 2.6

Estimating

Collection Rate

The rate at which material arrives at the collector in the form of ions can be calcu One ampere-hour of singly lated directly from elementary electrochemical theory. charged elemental ions is equivalent to 0.0373 g-atoms of the element. Then for an average total collector current of 100 ma, the rate of collection is 3.73 X Af mg/hr, For lithium (.1/ = 6.94) the collection where M is the atomic weight of the element. rate at 100 ma total ion current is about 26 mg/hr, and the material collected is divided between the Lie and Li' pockets in about the ratio of their natural abundances, Atthosnmeeurrent, thecollectionratesfor molybdenum (3/ = 95.9o> i.e., 7.5 to 92.5. and for lead (A/ = 207.2) would be 358 and 773 mg/hr, respectively. The total ion

SK .I.

14-5]

ELECTROMAGNETIC

SEPARATION

14-53

currents actually attained may vary considerably during any run, and the average current over different runs varies widely from element to element. The 100-ma value assumed in the above examples is not unusual, however, and it is interesting to note that this current is equivalent to about one billion (10') ions moving from the source to the collector every billionth (10~9) of a second. Any accurate estimate of the rate at which a given element or isotope can be processed in the calutron depends, therefore, on a knowledge of the average ion current and the retention that can be expected. Any estimate of these quantities, except on the basis of previous experience, must be considered as highly speculative.


2.6

Recovery and Purification

of Isotopes

The immediate products of calutron separation as contained in the various collector They must be removed from the pockets, chemically puri pockets are of little use. fied, and converted to some stable chemical form suitable for storage and ultimate use. Samples from each lot of material must also be prepared for spectrochemical and mass analyses. Moreover, all this must be done without loss or isotopic contamination of the collected product. Various methods and combinations of methods are required for removing different products from the collector pockets. Scraping, dissolving, and distillation are methods commonly used. Whatever the method of removal, numerous impurities are carried along with the collected isotopes. These must be removed by whatever The impurities normally found can be traced to a chemical methods are applicable. number of sources — materials of the collector pockets, materials of the ion source (particularly those parts exposed to contact by the charge vapor), and impurities in Some of the impurities probably find their way into the the charge material itself. collectors as neutral atoms or molecules; others undoubtedly arrive as ions, e.g., doubly charged ions of an element or complex having twice the mass of the collected isotope, doubly charged strontium in calcium, hafnium in zirconium, and sodium chloride in nickel. Chemical refinement of isotopes is much more than a quantitative analysis of the It is more nearly analogous to the preparatlL. vK a highly refined recovered material. chemical, but with the added requirements that all steps of the operation must be In many cases, quantitative and applicable to very small amounts of materials. refinement is not unlike the quantitative recovery and purification of a rare element found as a minor constituent in a complex ore. 2.7

Spectrochemical

Analyses of Isotopes

The ultimate goal of chemical recovery and refinement is the quantitative conversion of the collected isotopes to a useful form as free as practicable of all chemical impurities To guide the chemists in evaluating their and without loss of isotopic enrichment. purification procedures and to supply users of the isotopes with indispensable informa tion, both spectrochemical and mass analyses are made of the refined products. Spectrochemical analysis has been adopted as the method of analysis for chemical The method is rapid and sufficiently versatile to be applicable to any impurities. It requires only small amounts of material and a minimum of sample isotope. preparation and handling, and it gives low limits of detection for the impurities sought. It has not been found practicable to set up rigid specifications or maximum limits of A compromise must often be made between the contamination in isotope products. time and effort involved in repeated treatments and the material used in repeated analyses on the one hand and the known effects of certain elements on nuclear reac tions to which the isotopes will be subjected on the other hand. 2.8

Mass Analyses of Isotopes

Results of these Mass spectrometric analyses are made of all isotopes collected. analyses are required by users of the products, and they also serve as a guide in

isotopes

14-54

[Sec. 14

In most cases, the element is first converted to a developing separation techniques. compound that has been found most suitable for isotopic analyses — usually an oxide or halide. The recent development of a type of mass spectrometer ion source in which the element or compound being analyzed is vaporized from a heated filament of tungsten or iridium has greatly simplified the mass analysis problem by extending the list o: elements and compounds that can be readily analyzed and by simplifying the chemis try of sample preparation and the mechanics of charge introduction. 3

SURVEY OF ISOTOPES SEPARATED IN THE CALUTRON


3.1

List of Separated Isotopes

The isotopes that have been separated in the calutrons and the isotopic enrichment? that have been obtained are listed in Table 3. Higher isotopic purities than those listed can be obtained for many of the isotopes if the importance of the product war rants the additional effort that would be required. Of the 63 elements known to have stable isotopes, all but 12 have been separated in the calutrons. No attempt has been made to separate the isotopes of hydrogen, and only token collections of nitrogen and oxygen have been made, partly because of collector-retention problems but mainly because these elements are being satisfactorily separated by other methods. Separations of helium, neon, argon, krypton, anc xenon await the development of satisfactory methods of collecting more than mere trace amounts of these inert gases. The heavier rare earths (europium, dysprosium, erbium, yttrium, and lutetium) will be separated as soon as suitable charge material? are available. The separation of osmium has been postponed because of toxicity hazards. Solutions of the inert-gas-collection problem and that of rare-earth charge production are progressing satisfactorily and appear to present no insurmountable difficulties. Osmium can be separated whenever the need for its isotopes is such as to warrant the considerable time and effort required. 3.2(,

,.^pme Observations

Regarding Calutron Operation

Certain general observations regarding the calutron separation of isotopes can be made on the basis of experience in collecting nearly 1,000 lots of separated isotopes of some 50 elements. The halides stand out as materials most frequently found to be satisfactory a? charge materials, although oxides and the elements themselves are sometimes preferrf-i One, two, or four ionizing arcs may be run in a single source assembly with a corre sponding increase in production rate, provided there is not too much overlapping of the separated isotope ion beams at the collector. Failures of the tantalum filaments that supply electrons for maintaining the ionizing Both positive-ion bombard arc are the most common single cause of run termination. ment and chemical attack appear to contribute to these failures. The percentage of material vaporized from the charge bottle and arriving as ions at the collector ("process efficiency") varies from a few up to as high as 50 per cent, depending on the element being processed. Graphite collector pockets are preferable to metal pockets provided the collected Water-coo.Vd isotopes have a low vapor pressure and water cooling is not required. copper pockets have proved superior in preventing loss from the collector of isotopes Under the most favorable conditions, actual retention having high vapor pressure. at the collector approaches 100 per cent. Because of the complexity of the calutron separating process, it is not generally possible to predict the performance characteristics of one element from those of a related element. 3.3

Examples of Separations

The widely differing performance characteristics illustrated by a few examples.

of different

elements may well be

Sec. 14-5]

ELECTROMAGNETIC Table 3. Iso

Element

tope mass 6 7 10

II

Carbon

Oxygen Magnesium Silicon

Sulfur


Chlorine

Titanium

Vanadium Chromium

Nickel

Copper Zinc

Gallium Germanium

...

12 13 14 15 16 17 18 24 25 26 28 29 30 32 33 34 36 35 37 39 40 41 40 42 43 44 46 48 46 47 48 49 50 50 51 50 52 53 54 54 56 57 58 58 60 61 62 64 63 65 64 66 67 68 70 69 71 70 72 73 74 76

14-55

SEPARATION

Isotopes Processed and Isotopic Enrichments Attained in the Calutrons Natural

Enriched

abundance. %

abundance. %

7.52 92.48

99.4

18.98

81.02 98.89 1.

II

99.64 0.36 99.76 0.037 0.204

* * * * *

92.27 4.68 3.05 95.06 0.74 4.18 0.016 75.4 24.6

99.4 68.6 72.6 98.5

93.08 0.0119 6.91 96 97

99.96 7.8 99.2 99.98 82.5

22.1 20.7

0.88 95.7

72 1

98.0

0.0033 0. 185 7.95

10.2 84.3

4.31

83.76 9.55 2.38 5.84 91.68 2.17 0.31

84.4 82.8 99.2 79.4 84.7 22.8 99.98 89.5 99.97 95.2 89.0 96.7 99.9

67.76 26.16 1.25

83. 1

1.16 69. 1

30.9 48.89 27.82

4.14

18.54

0.617 60.2 39.8 20.55 27.37 7.61

36.74 7.67

(7

Zirconium

Molybdenum

Ruthenium

Palladium

Cadmium

87.3

86.0 99.9 98.5

3.66

Rubidium

66 3

2.06

5.34 0.24 99.76

Bromine

98. 1

0. 145

73.45 5.51

74 76 77 78 80 82 79 81 85

Selenium

89.9 97.8 99.99 7.52

99.6 92.3

7.75

tope mass

99.97

78.60 10. II 11.29

0.64

Iso

Element

96.8 97.4 99.7 98.2 93.4 93.8 62.6 95.5 48.4 98.4

Indium

Tin

98.1

91.4 94.9 78.0 95.8 81.0

Antimony Tellurium

..

84 86 87 88 90 91 92 94 96 92 94 95 96 97 98 100 96 98 99 100 101 102 104 102 104 105 106 108 110 107 109 106 108 110 111 112 113 114 116 113 115 112 114 115 116 117 118 119 120 122 124 121 123 120 122 123

Natural abundance, %

0.87 9.02 7.58 23.52 49.82 9. 19

50.52 49.48 72. 15 27.85

0.56 9.86 7.02 82.56 51.46 11.23 17. II 17.40

2.80 15.86

9.12 15.70 16.50

9.45 23.75 9.62 5.47 1.84 12.77 12.56 17.10

31.70 18.56

0.96 10.97 22.23 27.33 26.71 11.81

51.35 48.65 1.215

0.875 12.39 12.75 24.07 12.26 28.86

7.58 4.23 95.77 0.95 0.65 0.34 14.24

7.57 24.01

8.58

32.97 4.71

5.98 57.25 42.75

0.089 2.46 0.87

Enriched abundance, % 33.1

88.5 91.7 98.4 98.4 89.9 90.5 96.8 96.0 89.6 63.7 89.0 73.1

99.7 98.7 88.0 95.8 97.9 89.5 95.5 84.9 91.3 92.0 89.6 96.3 93.0 77.3

65.05 91.18 88.9 91.13 94.2 98.24 34.3 63.2 78.2 82.3 94.2 91.4 96. 1

99.5 32.9 24.8 70.0 64.5 83.5 54.1

94.2 71.2 65.4 99.9 72.5 50.0 17.7

92.6 83.8 96.2 83.6 98.2 88.9 95.0 99.4 96.7 22.3

86.2 60.9

14-56

ISOTOPES Table

Ibotope mane

Element

Tellurium (cnnt.)

Lanthanum

....


Neodymium. . . .

Samarium

Gadolinium.

. .

3.

Natural %

4. 61 6.99 18.71 31 79 34. 49 0. 101 0.097 2.42 6. 59 7.81 11.32 71.66 0.089 99.911 0. 193 0.250 88. 48 11.07 27. 13 12.20 23.87 8.30 17. 18 5.72 5.60 3. 16 15.07 11.27 13.84 7.47 26.63 22.53

0.20 2. 15 14.73 20.47 15.68 24.87 21.90

Enriched

Element

abundance. %

83.9 87.9 95 4 96.5 97.8 27. 5 12.0 51.4 67.3 50.0 43.6 98 0 1.7 99.96 30.0 13. 1 99.7 90. 1 93.9 83.9 95.7 78.6 95.6 89.9

94.8

85.7 85. 1 76.0 81.5 74. 1 93.9 99. 1 15.0 33.2 72.3 80. 2 69.7 92.9 95.4

Tantalum Tungsten

Rhenium

Platinum

Mercury

Thallium Lead

Iso

Natural

tope mass

abundance. %

174 176 177 178 179 180 180 181 180 182 183 184 186 185 187 191 193 190 192 194 195 196 198 196 198 199 200 201 202 204 203 205 204 206 207 208

0. 18 5. 15 18.39 27.08 13.78 35 44 ~0 . 01 99.99 0. 135 26.4 14.4 30 6 28.4 37.07 62.93 37.3 62.7

0.012 0.78 32.8 33.7 25.4 7 23 0. 146 10.02 16.84 23 13 13 22

29.80 6.85 29 70 1 23 22 52

50 50 48 6 6 3

... * No mass analysis

14

Isotopes Processed and Isotopic Enrichments Attained in the Calutrons. (Continued)

abundance,

124 125 126 128 130 130 132 134 I3S 136 137 138 138 139 136 138 140 142 142 143 144 145 146 148 150 144 147 148 149 ISO 152 154 152 154 155 156 157 158 160

[Sec.

Enriched abundance. %

10 1 59.5 62.9 84 8 53 3 94.0 0 21 • 9 0 94.3 86 2 95.7 97.9 85 4 98.2 85.9 89. 1 0.8 13 9 65.1 60 1 65 9

tit

8.4 79 1 73. 1 91.3 71 8 98.3 89.2 86.0 98.7 27. t 81 0 66 < 96 6

-

made.

3.31 Lithium. Separation of the isotopes of lithium proved difficult in spite of the very favorable percentage mass difference. First, difficulty was encountered in maintaining the ionizing arc with the accelerating voltage applied. The introduction of a small amount of nitrogen as a "support" gas served to stabilize the arc and cor rected this difficulty. Then lithium chloride charge material, although it yielded an

abundance of lithium ions, reacted directly or through its products of dissociation with heated copper and possibly other components of the source assembly. Such reactions not only were destructive of source parts but also led to the formation of compounds that deposited in the ion exit slit of the ionization chamber and caused eventual run termination. The dissociation of lithium chloride into LijCl* and Cl~ ions is believed to contribute to this deposition. The addition of a few per cent lithium metal to the lithium chloride charge eliminated this trouble. As difficulties were overcome, the length of runs increased, the total ion current to the collector reached a value of several hundred milliamperes, and Li* and Li7 of very high isotopic purity were obtained. 3.32 Potassium.1* Potassium was processed primarily to produce enriched K«*>. With continued improvements in equipment and techniques, K4' was produced

Sec. 14-5]

electromagnetic separation

14-57


having greater than 10 per cent isotopic purity as compared with 0.012 per cent In a later collection, potassium chloride charge was used in natural abundance. which the K40 content had been approximately doubled by irradiation in the Chalk River Pile [K"(n,y)K,° reaction]. From this collection, several milligrams of 22 per cent K<0 were obtained. Iron. Iron isotopes, Fe'4 (22 per cent natural abundance) and Fe" (0.3 per 3.33 cent natural abundance), were enriched to produce high-purity, nonradioactive iso topes that were then irradiated in the Oak Ridge Pile to produce radioactive Fe" and

Fe" of high specific activity. Specially purified graphite collector pockets were used in this collection to avoid contamination of the product material by traces of natural iron in the graphite. Ruthenium, Platinum, Iridium, and Palladium.11 Because all the com 3.34 pounds of these elements decompose on heating, the metals themselves were used as charge material and their processing entailed the development of an ion source in which the metals could be vaporized directly into the ionizing arc. This required source temperatures ranging from about 1800 to 2800°C, whereas the upper tem To attain perature limit of previously designed sources had been some 1000°C. these high temperatures, both the charge oven and the arc chamber were greatly reduced in size, heating was by high-energy-electron bombardment, and heavy heat With the use of this very-high-temperature source, many shielding was applied. milligrams of enriched isotopes have been collected. At an ion beam strength of only 10 ma as much as 50 mg of platinum isotopes was collected hourly. 3.36 Rare Earths. Separation of the isotopes of the rare-earth elements of atomic numbers 57 to 71, inclusive, presents the basic problem of obtaining several hundred A chemical grams of relatively pure halides of each of the rare-earth elements. separation of the rare-earth elements, using monazite and gadolinitc ores, has pro vided adequate charge material for separation of the isotopes of lanthanum, cerium, neodymium, samarium, and gadolinium. Dysprosium and europium charge materials are now in hand, and the remaining elements will be available with continued effort. A continuous extraction system, employing the solvents tributyl phosphate and nitric acid,17 has been applied to separating large quantities of rare earths, and best conditions are being determined for the concentration of each element. Ion-exchange concentration is sometimes used for final chemical purification of the enriched isotopes. From the foregoing examples, it is evident that special problems are encountered in While they successfully separating the isotopes of most elements in the calutrons. enrich the isotopes of many elements, considerable research is done in chemistry and physics as well. 3.4

Uses Made of Separated Isotopes

The principal uses of enriched stable isotopes fall into three main categories: (1) as basic material of research, (2) in isotope-dilution analysis, and (3) as starting materials for the production of certain radioactive isotopes. In the last category, Fe64 and Fe*8 are supplied on a recurrent basis for irradiation in the Oak Ridge Pile to produce Fe" and Fe" of high specific activity. Recent studies of the technical and economic feasibility of using certain other stable isotopes for production by cyclotron bombardment of radioactive isotopes not otherwise available have led to the initiation of such a program. Production of Mn" from jnriched Cr", I1" from enriched Tem, and I"0 from enriched Te"° has already been ;arried out. As a by-product of this program, facilities and techniques have been leveloped for preparing targets of enriched isotopes for cyclotron and other types of >ombardment. The uses made of enriched isotopes in basic nuclear research are of a variety that an best be illustrated by typical examples:''7'14 (1) the determination of masses and ctivities of radioactive isotopes from studies of irradiated stable isotopes, (2) 0- and -decay investigations, (3) neutron-cross-section measurements and the location of esonances, (4) determination of nuclear spins and magnetic moments, (5) studies of yperfine structure and isotope shifts, (6) isomerism and isobarism, (7) determination f nuclear-energy levels, (8) photoneutron- and photoproton-emission studies, (9)

ISOTOPES

14r-58

[Sec.

14

superconductivity effects, (10) as tracers and in isotope-dilution analyses, and (11) assignment of radioactivity to specific isotopes and in studies of other phenomena characteristics of nuclei of specific isotopes. Since enriched isotopes were first made available for research purposes, more than 400 papers and abstracts have appeared in the scientific literature reporting on research in which these isotopes were used. Contributions of the stable isotope separation program to research are not restricted to the uses made of the calutron-enriched isotopes, however; research in the field of gaseous discharge, developments of ion sources, applications of mass spectrometry and optical spectroscopy, and advances in inorganic chemistry have contributed not only to the separation of isotopes but in much broader fields as well.

REFERENCES J.:

1. Dempster, A. Science. 62: 559 (1920); Proc. Natl. Acad. Sci., 7: 45 (1921): Pkyi Rev., 11 : 316 (1918) ; 18 : 415 (1921) ; 19 : 431 (1922) ; 20 : 631 (1922) ; : 209 (1923 2. Aston, F. W. : "Mass Spectra and Isotopes," 3d ed., Edward Arnold & Co.. London.

SI

1933.

3. 4.


5.

6. 7.

J. J.: "Rays of Positive Electricity," 2d ed., Longmans, Green & Co.. Ltd London, 1921. Urey, H. C: Repts. Progr. in Phys., 6: 48 (1939). Wakerling and Guthrie: "Electromagnetic Separation of Isotopes in Commercial Quantities," National Nuclear Energy Series, Div. I, vol. 4, McGraw-Hill Book Com pany, Inc., New York, 1951. Keim, C. P.: Ann. Rev. Nuclear Sci., vol. 1, 1952. Keim, C. P.: Enriching Stable Isotopes Electromagneticallv, J. Appl. Phys., 24 : 1555 Thomson,

(1953).

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Normand, C. E., et al. : "Vacuum Problems and Techniques," National NucleaEnergy Series, Div. I, vol. 11, McGraw-Hill Book Company, Inc., New York, 1950. Bell, W. A., and L. O. Love: "The Electromagnetic Concentration of Carbon-12.' ORNL Report Y-813, 1961. Love, L. O., W. A. Bell, and H. J. Buttram: "Electromagnetic Concentration of tfcStable Isotopes of Zinc," ORNL Report Y-693, 1950. Love, L. O., and W. A. Bell: "The Electromagnetic Concentration of Samarium. AECD-3502, 1950. Bell, Jr., W. A., L. O. Love, and C. E. Normand: "A 2800°C Calutron Ion Source Unit and Its Use in Separating the Isotopes of Pd, Pt, Ru and Ir," ORNL 1767, 1954. Love, L. O., and W. E. Leyshon: "The Electromagnetic Concentration of the Stable Isotopes of Mercury," AECD-3501, 1951. Duncan, F. R., and W. G. Cobb: "Refrigeration System Used in Mercury Isotope Collections," ORNL Report Y-538, 1949. Buttram, H. J., L. O. Love, and W. A. Bell: "The Electromagnetic Concentration of Bromine," ORNL Report Y-671, 1950. Love, L. O., and W. A. Bell: "The Electromagnetic Concentration of Potassium-IO ORNL Report Y-623, 1950. " Topp, A. C, and Boyd Weaver: Distribution of Rare Earth Nitrates between Tribusvl Phosphate and Nitric Acid," ORNL Report ORNL-1811, 1954. Keim, C. P.: "The Electromagnetic Separation of Stable Isotopes from 1945 to 1951 " ORNL Report Y-733.

INDEX

A

Acoustic velocity,

Abel's equation, 3-108 Abrasion, 10-75, 10-76 Absolute convergence, 3-95 Absolute counting, of activities, 8-32 particle,

B-28, 6-29

Absolute systems, 1-134 Absolute thermal neutron 6-104

flux, measurement

of,


Absorbed dose, definition of, 7-24 Absorber, central, in bare reactor, 6-109 two-group solution for, 6-67 to 6-69 lumped, in infinite medium, 6-105 to 6-110 in regular array, 6-108, 6-109 permissible,

in reactors

(table),

Absorption, of light, 4-19

13-11

in

reactor theory, excess, 6-79 relative, 6-79 Absorption area of control rods, 8-17, 8-19 Absorption cross section (see Cross sections,

ab

sorption) Absorption edges, 4-59 Acceleration in magnetic field, 6-124, 6-125, 14-46 Accelerators, 6-123 to 6-139 mechanics of particle acceleration in, 6-124, 8-125

particles from (see specific particle) types of, 6-123 betatron,

6-131 to 6-133

Cockcroft-Walton, 6-125, 6-126 cyclotron, 6-130, 6-131 f-m, 8-134, 6-135 electron, linear, 6-129 electron synchrotron, 6-135 FFAG, 8-137 fixed-field alternating gradient, 6-137 linear, electron, 6-129 heavy particle, 6-128. 6-129 proton, 6-127, 8-128 magnetic, alternating gradient, 6-136, 6-137 "betatron" oscillations in, 6-130 constant-gradient, 6-130 to 6-132 field index, 6-129 phase stability in, 6-133, 6-134 ruicrotron, 6-133 proton linear, 8-127, 6-128 proton synchrotron, 6-135, 6-136 pulsed, for neutron production, 6-75 synchrocyclotron, 8-134 to 6-155 synchrotrons, alternating gradient, 6-136, 6-137 Van de Graaff. 6-125, 6-126 Accounting, power plant, 13-96 to 13-99 Acoustic impedance, 10-157 table, 10-58 Acoustic tests, 10-157 to 10-159 gauging, 10-165

10-157 table, 10-58 Actinium, chemical properties of, 11-43 Activation by neutrons, 7-7 to 7-14 activation data (table), 7-8 to 7-12 (formulas), continuous 1-37, 1-38, 7-13 cross sections for (aee Cross sections, activation) 7 rays from (table), 7-74 to 7-76 intermittent (formulas), 7-14 neutron detection by, 6-102 of reactor coolants (aee Radioactivity, of reactor coolants) Activation analysis, chemical, 11-56 Actuators, control-rod, 8-69 to 8-75 (See also Control-rod drives) Adjoint equation, 3-109, 6-20 Adler. F. T., 8-14 Advisory Committee on Reactor Safeguards, 8-76 Age (table). 6-87, 6-88 calculation of, for mixtures (example), 6-98 effective, 6-46 measurement of, 6-107, 8-108 for metal- water mixtures (table), 6-88 summary, 1-35 (See also Slowing-down area) Aggregation, signs of, 3-2 Agricultural research, use of radioisotopes in, 14-11, 14-12 Air, physical properties of (tables), 3-2, 9-3 radioactivity of, from air-cooled graphitemoderated reactors, 13-118, 13-119 from reactors, calculation of, 1-40, 1-41, 7-19 Air conditioning of reactor buildings, 13-175 Air-cooled reactors (see Power reactors, gascooled) Air monitors, 7-45 Air-wall chambers, 6-6 Aircraft nuclear propulsion (ANP) reactor (table), 13-53 AISI types of stainless steel (table), 10-60 Aitkin's method of interpolation, 3-34 Aitkin's process, 3-39, 3-47 Albedo, 6-17, 6-18, 6-41, 6-42 d'Alembert's formula, 3-116 Algebra, 3-2 to 3-62 Algebraic equations (see Equations) Alkali metals, experimental systems for, 13-96 to 13-98 as reactor coolants, calculation of radioactivity in, 7-20, 7-21 summary, 1-40, 1-41 (See also Power reactors, alkali-metal-cooled; Sodium; Sodium-potassium alloy) Allotropy, theory of, 10-12, 10-13 Allowable leakage of coolant, magnitude of, 13-5 Allowed transitions, nuclear, 4-69 Alloys (see High-temperature alloys; Metallic state; specific metals) a, of fuels (see Neutron regeneration for fuels)

INDEX

2 a counting (see Counting, a-particle) a decay (see Radioactivity, decay, a) a emitters, handling (see Radioactive materials.

Arithmetic progression, 3-6 Army Package Power Reactor (APPR-1). 11-36.

or-induced reactions.

ARR, Armour


handling of)

4-80 a-particle counting (see Counting, a-particle) a particles, ionization by, 5-3, 6-4, 7-5 nuclear properties of, 4-57 range of (see Range) spectroscopy of, 6-29 to 6-31 Alternating-gradient synchrotron, 6-136, 6-137 Aluminum, corrosion of, 10-50 by liquid metals, 10-20, 10-21 mechanical properties of (tables), 10-48 to 10-51 physical constants of (table), 10-39 relaxation length, for fast -neutrons, 1-35 for 7 rays, 1-35 properties of, 9-113 thermal-stress for water reactors, 13-31 Aluminum alloys, composition of (table), 10-47 corrosion of, 10-50 mechanical properties of (tables), 10-49 to 10-51 physical constants of (tables), 10-48 with uranium (see Uranium alloys) Aluminum cladding alloys, 10-185, 10-186 Aluminum-water chemical reaction, 8-79, 6-85 Aluminum-water mixtures, nuclear age in, 6-88 Amberlite resins, 11-93 Americium, chemical and physical properties of, 11-44 Americium isotopes, formation of, 11-9 Amplifiers, for ionization pulse counters, 6-12 in reactor control, linear, 8-56, 6-57 AIA, AIB. 8-56 logarithmic, 8-60 period, 8-61 Amplitude, definition of, 1-16 Analog computer, reactor (see Simulators for re actor dynamics) Analog computing machinery, 3-146 to 3-148 Analog simulator for reactor system, 8-89 to 8-102 (See also Simulators for reactor dynamics) Analysis, isotopic, 14-53, 14-54 analysis) radiochemical (see Radiochemical Analytic continuations (mathematics). 3-85 Analytic functions, 3-82 to 3-84 (See also Functions) Angular distribution in scattering, 4-100, 4-101 Angular measures, conversion factors for. 1-145 Angular momentum (spin), 4-43 to 4-45 quantization of, 4-20 Annihilation radiation, 4-55 ANP (aircraft nuclear propulsion) reactor (table), 11-53 \ntimony, fission-product, chemistry of, 11-46 physical properties of, 11-5 Antimony-beryllium source, 8-25 Application of reactors, classification by, 11-6 table, 11-7 (See also Power reactors) Aqueous shielding, 7-83 Aqueous solutions, radiation damage to (table), 10-131 to 10-133 Area, conversion factors for units of (table), 1-143 Argon, physical properties of (table), 11-5 thermal conductivity of, under high pressure (table), 9-4 viscosity of, under high pressure (table). 9-3 Argonaut, description of, 13-135 Argonne Boiling Water Reactor Facility (ARBOR), description of, 18-138 Argonne National Laboratory reactors (see Reac tors, particular) Argument, definition of, 8-16

13-37 description

of, 13-138 Research

Reactor (*«* Reactors,

particular)

Arsenic, fission- product chemistry of, 11-46 physical properties of (table). 11-5 Associative law (mathematics). 8-9 Asymptotes, 8-32 Asymptotic cylinders, 8-25 Asymptotic reactor equation, 6-51 Atom, Bohr model of, 4-10 to 4-14 Rutherford model of, 4-10 shell structure of, 4-13, 4-27 Thompson model of, 4-10 Atom displacement, by cyclotron particle*. 10-100, 10-101 by radiation, 10-102 Atomic Energy Commission literature, avail ability of, 1-71, 1-72 information on, 1-72 to 1-74 Atomic mass unit (amu), mass and energy equiv alents of (table), 1-153 approximate values of (table), 1-13 Atomic number Z in nuclear theory, 4-33 Atomic numbers, table of, 1-14 Atomic Power Development Associates (APDAi (see Reactors, particular, Enrico Fermi) Atomic Power Station (APS). USSR, description of, 13-134 Atomic radius, 10-6, 10-7 Atomic theory, 4-2 to 4-33 Atomic weights, table of, 1-14 Attenuation, neutron (see Shielding, neutron at tenuation) Attenuation coefficients (see Shielding, >-ray at tenuation) Attenuation kernel, 7-93 (See also Shielding, geometric configuration?) Auger electrons, 7-61 Augmentation distance t, 6-37 Augmented matrix (mathematics), 3-36 Austenitic steel (see Stainless steels) Autocatalytic reactors, 11-20 Automatic control of reactors, 8-30 to 8-41 (See also Control of reactors) Automatic period control of reactors. 6-30 Autoradiography, 10-165 Avogadro's number, value of. 1-13. 1-154 Axes, translation of (mat hematic8-9 H Bachmann

charts,

9-94

Background radioactivity, natural (table). T-Sn Backward operator, 8-33 Baling of radioactive wastes, 11-146 Ball-joint manipulators, 7-121, 7-132 Ball-nut control-rod drive, 8-69 Bards well reactors, capacity and date of operatiâ of, 18-105

Barium, fission-product

chemistry of, 11-45 properties of (table). 11-5 Barn, definition of, 1-153, 4-71, 6-32 Barrier, Coulomb, 4-33 Barytas concrete (see Concrete, heavy) Battelle Research Reactor, description of. IS-SSc 14-9, 14-10 Batteries, radioisotope, Bearings, in high-temperature water (table*'. 10-80 to 10-82 Beds of particles, fluid flow through, 9-47 Bellows seals for control-rod drives, 8-69 physical

INDEX Bellows valves, for alkali metal reactors,

13-92.

13-93 for water reactors, 13-27 Bends, pressure losses at (table), 9-37

BEPO

(see Reactors, particular) Berkelium, activation cross section of, 1-22 chemical properties of, 11-44 Bernoulli method (mathematics), 3-47. 3-48 Bernoulli principle (fluid flow), 9-28 Bernoulli's equation (mathematics), 3-107 Beryllia, corrosion of, by liquid metals, 10-20, 10-21

by sodium, 12-71 by water, 11-74 cost of (table), 12-7(1 effect of radiation on, 10-115, 10-116 mechanical properties of (table). 10-54, 10-58 properties of (table), 2-9, 13-70 moderating nuclear properties of (tables), 2-9, 12-70 physical properties of (tables), 10-53, 11-70, 12-74

radiation damage to, 10-115, 10-116 thermal stress properties of (table), 9-113 Beryllia- uranium oxide, radiation damage to. 10-121

Beryllium, corrosion of, 10-52, 10-53, 12-74 by liquid metals, 10-20 cost of (table), 11-70, 11-74, 11-107 properties


mechanical

of (tables),

10-51, 10-52

moderating properties of (tables), 1-9, 11-70 nuclear properties of (tables), 1-9, 11-70 physical properties of (table), 11-70, 11-74 as source of neutrons in reactors, 8-27 toxicity of, T-57 for water reactors, 13-31 Beryllium carbide, mechanical properties of (table),

10-54. 10-58, 10-59

physical properties of, 10-53 Beryllium oxide (see Beryllia) Beryllium oxide-uranium oxide, radiation damage to, 10-121

Beryllium Physics Reactor (BPR), description of, 13-135

Besael functions, 8-89 to 3-91 To(x) and I\(x) (tables of), 1-106 to 1-109 auxiliary functions for, 1-115, 1-116 Jo(x) and J\(t) (tables of), 1-94 to 1-99 auxiliary functions for, 1-114 Ka(z) and JCiO) (tables of), 1-110 to 1-114 auxiliary functions for, 1-114 to 1-116 .Vou) and Ni(x) [see Yo(x) and Yi(x)\ Fo(z) and Yi(x) (tables of), 1-100 to 1-105 auxiliary functions for, 1-114 Bessel s inequality, 3-103 0 activities, absolute counting of, 9-32 0 counters (see Detectors) 0 counting (see Counting, 0- particle) 0 decay (see Radioactivity, decay, 0) $ emitters, handling (see Radioactive materials, handling of) 0 energy from fission, delayed, 1-4 (See also Fission products) 0 function, 3-87 0 gauge (tables), 14-4 to 14-6 0 particles, average energy of, 7-33 for gauging, 10-162 to 10-164 ionization by, 7-5 mathematics of, 9-3 range of (see Range) spectroscopy of, 8-31 to 9-35 (See also Counting, 0-particle) spectrum of, 4-68, 4-69 stopping of, theory of, 4-63, 4-64 0 rays (see 0 particles)

Betatron. 9-131 to 9-133 Betatron oscillations, 9-130 Binary scale in computing, 8-144 to 3-145 Binary system (mathematics), 3-5 Binder method for unsteady heat flow, 9-98 to 9-99 Binding energy, 1-35 to 1-36, 4-40 semiempirical formula for, 4-48 Binding fraction, 4-41 Bingham bodies, flow of, 9-40 viscosity of. 9-26 to 9-27 Binomial distribution, 3-61 Binomial expansion, 3-5 to 3-6 coefficients

of (table),

1-126

Biological effects of radiation (see Dose; Dose rate; Health physics) Biological shield, concrete, iron ore fur. procure

ment and storage of, 13-182 mix for (table), 13-182 to 18-183 pouring of, 13-183 to 13-185 steel punchings for, procurement and clean ing of, 13-182 laminated construction, use and fitting of, 13-182 Biphenyl (see Diphenyl) Bipolar coordinates, 8-32 Bismuth, liquid, corrosion by (tables), 10-21. 13-99 physical properties of (table). 9-18, 13-81 physical properties of (table). 11-5 price of, 11-107 Bismuth-lead alloy, liquid, corrosion by, 10-21, 13-99 physical properties of (table), 10-81 eutectic (table), 9-19 as reactor coolant, comparative table, 12-58 Bismuth-phosphate separation process, 11-61 to 11-63 Bismuthate process, 11-61 to 11-63 Bivectors, 3-11, 8-12 Black absorbers, flux distributions around, 8-106 to 9-108 Black-body radiation. 1-59, 4-4 Black boundaries, 9-36 Blatt, 10-103 Blowers, for air-cooled reactors, 18-110 mechanical, dimensionless coefficients of. 9-48, 9-49 BNL 325 (table), 1-15 to 1-23 maximum permis Body burden of radioisotopes, sible. 7-38 to 7-41 Body-centered cubic (bec) structure. 10-6 Bohr atom. 4-10 to 4-14 Bohr frequency condition, 4-11 Bohr magneton. 4-22, 4-23 Bohr theory of the hydrogen atom. 4-11, 4-12 Boiler feed pumps, price of (table), 11-134 Boiler-plant equipment, cost of, 11-97 Boilers for reactors (see Steam generators) Boiling, 9-66 to 9-68 effect of, on pressure loss (table), 9-41 heat transfer by, 9-66 to 9-84 confined natural-convective, 9-74 to 9-76 forced convective, 9-76 to 9-83 nucleate, 9-76 to 9-83 saturated, film, 9-83 nucleate. 9-80 to 9-83 subcooled. film, 9-83 mixed. 9-80 free convective, 9-69, 9-70 nucleate, 9-70 to 9-73 saturated, film, 9-73, 9-74 mixed, 9-73 nucleate, 9-70 to 9-73 subcooled, 9-69. 9-70

INDEX Boiling, heat transfer by, in fuel solution (««e in homogeneous reactors, below) in homogeneous- reactors, 9-83, 9-84 rate of bubble growth in, 9-84 superheat to initiate, 9-84 time delay in bubble formation. 19-47, 13-48 natural convective (tee free oonvective, above)

Boiling aqueous

homogeneous

reactors,

13-46,

13-60

Boiling point of elements (table), 11-6 to 11-8 Boiling water, separation factor for, 13-5 Boiling-water reactor (tee Power reactors) Boltzmann constant k. value of, 1-13. 1-154 Boltzmann equation, 6-3 one-velocity, 6-5 Bonds, valence, 10-4 Boral, fabrication and fitting of, 13-181. 13-182 properties and cost of, 10-71 BORAX I, II, III, boiling reactor experiment, de scription of, 13-133, 13-134 Boron, boiling point of, 11-6 as control material (table), 10-88. 10-191 heat of fusion of, 11-6 neutron shielding properties of, 3-34. 7-77 physical properties of (table), 10-70


price of, 13-107

Boron carbide, 12-74 to 13-76 corrosion resistance of, 13-76 cost of, 12-76 moderating properties of B'UC (table), physical properties of, 12-74 to 12-76 table,

12-70

12-70

thermal stress properties of (table), 9-113 Boron chambers (see Detectors) Boron fluoride chambers (see Detectors) Boron shot for emergency shutdown, 8-46 Boron stainless steel, mechanical properties of (table), 10-70 Boron steel, mechanical properties of (table), 10-70

Boundaries (in reactors), between media, gray, 6-37

6-36 6-37 to 6-38

Boundary conditions in reactor theory, 6-15 Boundary layers in turbulent flow, 9-35 Boundary point in set theory, 8-77 Boundary-value problems, 8-118 to 3-122 BR-2, Experimental Fast Neutron Reactor, de scription of, 13-136 BR-3, plant space requirements for, 13-172 Bragg curve, 5-3 Bragg-Gray equation, 7-26 Branch point, 8-84 Brayton cycle, 12-3 to 12-5 Breeding, 12-80 Breeding gain, formulas for, 12-83 general formula for, 1-28 Breeding ratio, in fast reactors. 12-25 formulas for, 12-82, 12-83 Breit-Wigner formula, 4-75, 4-93 for single-level resonance, 4-79, 5-77, 5-78 Bremsstrahlung, 4-55, 7-78 Brillouin zones, 10-8 Brinkroan, J. A., 10-101, 10-102 British experimental pile (see Reactors, particular,

BEPO)

Bromine, fission-product, chemistry of, 11-48 physical properties of (table), 11-5, 11-90 physical properties of Bromine pentafluoride, (table), 11-90 Bromine trifluoride, physical properties of (table), 11-90

Brookhaven Medical Research Reactor, deeerij.-tion of, 13-137 BNL, Brookhaven National Laboratory reactor* (see Reactors, particular) BNL 325 (tabic;), 2-15 to 2-23 Brooks, IL. 10-102 Bubble chambers, 6-20, 6-21 Buckingham r theorem, 9-58 Buckling (in reactors), 6-52 critical, 6-20 definition of, 5-108 geometric,

6-20

definition of, 5-108 material, 6-20 measurement of, in exponential experiments. 6-117, 5-118 (See also Physics of critical reactors) Budan's theorem, 3-43 Build-up factor, definition of, 7-61 dose rate (table), 7-66 to 7-68 energy absorption (table), 7-66 to 7-71 (See also Shielding, 7 -ray attenuation) Building facilities for radioactive work, 7-48. 7-49 Buildings for reactors, air conditioning of. 13-17". air locks for, 13-173, 13-174 airtight (see gastight, below) through walls of. 13-175. cable penetrations 18-176 chimneys for, 13-175 conventional, 13-160 to 13-162 drains for, 13-176 emergency cooling of, 13-175 exhaust stacks for, 13-175 fuel-element storage in, 13-174 gastight, high-pressure, 18-165 to 18-173 air locks for, 13-174 factor of safety for, 18-169 foundations for, 13-130 insulation of, 13-170 leak testing of, 13-170 missile protection for, 13-172 pressure requirements of. 13-167 to 13-1^ rate of pressure build-up in, 18-168, 13-16$ shape of, 13-169 size of, 13-167 to 13-169 for, 13-169 steel specifications volume of. 18-167 to 13-169 welding of, 13-169, 13-170 low-pressure, 18-162 to 13-165 air locks for. 13-173 internal finishes for, 13-175 for water reactors. 13-30 pressure requirements sewers for, 13-176 site selection for, 12-154 space requirements (table). 13-172 stacks for, 18-175 ventilation of, 13-174, 13-175 water sprays for, 13-175 (•Seealso Containment) Bulk shielding facility (see Reactors, particular1 Buoyancy, 9-90 thermal, 9-38, 9-39 Burden of radioactive materials in body, maxi mum permissible, 7-38 to 7-41 Bureau of Labor, cost-of-living index, 12-91, IS ■■■•. Bureau of Standards, availability of publication* of, 1-72 Burnout, 9-79 orificing to prevent, 9-44 to 9-46 Burnup, attainable, 12-109

conversion factors for units of, 1-28 conversion to Mwd/tonne, 10-117 effect on reactivity, 12-82

INDEX Burnup, in production of radioisotopes, range for power plants

14-28 (table), 12-108 12-1 11, 12-112

By-product power reactors.


C C (center -of -mass) system, 4-30 Cable penetration through reactor buildings, 13-175. 13-176 Cables for detectors, 8-52 Cadmium, boiling point of, 11-5 chemistry of fission product, 11-64 as control material (table). 10-188 heat of fusion of, 11-5 physical properties of (tables), 10-70 thermal stress properties of (table). 9-113 measurement, detector for Cadmium-difference thermal neutrons, 5-102 Cadmium ratio, measurement of, 6-81 Cadmium-silver alloys, thermal stress properties of (table), 9-113 Cadmium-silver-indium for control rods, 10-191 Calcium, physical properties of (table), 11-5 Calcium-45, use of, in agricultural research, 14-11 Calculus, differential and integral, 3-64 to 8-75 of variation. 3-137 to 3-140 Calder Hall reactors (see Reactors, particular) Californium, chemical properties of, 11-44 cross section of, 2-22 Calutrons, 14-44 to 14-58 analysis of product of, 14-53, 14-54 mass spectrometric, 14-53, 14-54 spectrochemical, 14-53 beam resolution in, 14-40, 14-50 charge selection for (table), 14-50, 14-51 collectors for, 14-48, 14-40, 14-51, 14-54 equations of, 14-46 ion source for, 14-47, 14-48, 14-54 isotopes available from (table), 14-54 to 14-57 operating cycle for, 14-52 operation of, 14-50 to 14-54 process efficiency of, 14-52, 14-54 production rate of, 14-52, 14-53 purification of product of, 14-53 recovery of isotopes in, 14-53 retention in collectors of, 14-52 specification of (table), 14-44 to 14-46 vacuum in, 14-47 Canals. 7-117, 7-118 fuel discharge into, 7-137. 7-138 viewing in, 7-123 Cancer, radioisotope treatment of, 14-10, 14-11 Canned-motor control-rod drive. 8-73, 8-74 Canyon, 7-124 Canyon crane, 7-136 Capability, definition of, 13-91 Capacity, of electrical generating unit, definition of, 12-01 of nuclear power plants. 12-00, 12-100 (See al»o Nuclear power) Capacity factor, definition of, 12-01, 12-94 Capitalization of power plants (nee Nuclear power; Power, from fossil fuel) Capsule in-pile tests, 5-140 Capture, radiative, 4-05 to 4-09 description of, 4-72 Capture cross sections (see Cross sections) Capture 7 rays (table), 7-74 to 7-76 Capture resonances, theory of, 4-95 Carbon, physical properties of, 11-5 (See alto Graphite) Carbon-13, availability and production of, 14-24 Carbon-14, availability and production of, 14-32 in reactor coolants, radioactivity, 7-19, 7-20

5

Carbon dioxide, physical

properties

9-4 as reactor

of (tables),

coolant, radioactivity of, 7-20 Carbon-nitrogen cycle, 4-90 Carlson, Sn method of, 6-13 Carnot cycle. 12-3, 12-5, 12-123, 12-126 Carrier, 10-50 holdback,

10-61. 11-50, 11-60. 11-61

Carrier-free separations

in production of radio 14-31 Cartesian coordinates, 3-32 Casks in chemical processing, 11-138 to 11-140 Castle type manipulators, 7-121, 7-134 Catalogue of reactors, fast and intermediate re actors, 13-136 graphite-moderated natural-uranium reactors , 13-124 to 13-126 heavy-water reactors, 18-126 to 13-128 homogeneous aqueous reactors, 13- 120 to 13- 1■' I1 recent foreign reactors, 13-139 to 13-141 recent United States reactors, 13-137 to 13-139 water- mod era ted reactors, 13-131 to 13-135 Catalysis by radioisotopes, 14-8 Catalytic recombiner system for homogeneous aqueous reactors, 13-60, 13-61 recombiners, 13-68 to 13-70 Cataractogenesis by radiation, 7-28 Catcher foils, 5-105 Cauchy convergence, 3-75 Cauchy-Goursat theorem, 3-82 Cauchy-Kowalewski theorem, 8-115 Cauchy problem, 8-115 Cauchy-Rieraann differential equation, 3-82 Cauchy "s integral formula, 3-83 Caustic Boda (see Sodium hydroxide) Cavitation, 9-48 Cay ley-Hamilton theorem, 3-52 Cent (reactivity), conversion factors for, 8-8 Center-of-mass (C) system, 4-30 Central absorber in bare reactor, 6-109 two-group solution for, 6-67 to 6-69 Central-difference formula, 3-34 Central-difference operator, 3-33 Central field, 4-21 Central quadric (definition), 3-28 Central-station power (see Nuclear power; Power, from fossil fuel; Power reactors) Centrifugation, 11-104 Centroid, 3-28 Ceraments (see Cermets) Ceramic fuels, radiation damage to, 10-121 Ceramics, 10-68, 10-60 Cerenkov radiation, 4-64, 4-65 Cerium, in glass, for radiation resistance, 10-124 physical properties of (table), 11-5 Cermet, wear resistance of Metamic LT-1, 10-77 Cermets (table), 10-69 Cesium, chemistry of fission product, 11-45 as control material (table), 10-188 Cesium-137, for radiography, 14-3 for thickness gauge, 14-5, 14-6 isotopes,

CETR (Consolidated Edison Thorium Reactor), 12-96. 13-138. 13-172

Chain reaction, 6-33, 6-34 Chalk River reactor, NRX (see Reactors, ticular, NRX) Chamber

par

(«ee Detectors)

Channel energy in two-body collision.

4-72 Channels, in parallel, fluid flow in, 9-44 to 9-46 in reactor control, 8-57 to 8-61 Chapel Cross, A-l, A-2, B-l, B-2, capacity and date of operation, 13-105 Characteristic of logarithm. 3-4

INDEX

6

Chemical separation processes, ma ten* la handling in, 11-138 to 11-140

Characteristic curve (mathematics), 8-113 Characteristic equation, mathematics, 3-45 reactor theory, 6-53 (See also Physics of critical reactors) symmetric matrix, mathematics, 3-51 Characteristic function, mathematics, 8-51 Characteristic value, mathematics, definition, 3-29

Characteristic vector, definition, 3-20 Characteristic X rays, 4-55 Charged-particle emission, neutron-induced,

4-80

to 4-82

Charged-particle reactions, 4-80 to 4-82 Charged particles, ionization by. mathematics

of,


6-2 to 6-4 stopping of, theory, 4-02 to 4-65 Charging and discharging from reactors {ace.Fuel handling) Charpit's method, 3-113 Chart of nuclides, 1-15 to 1-19 Checkout of reactor, 8-82 Chelates, definition, 11-71 Chelation, definition, 11-71 Chemical analysis (see Radiochemical analysis) Chemical dosimeters, 7-47 Chemical-exchange isotope separation, 14-15 to 14-22 for C", 14-24 for deuterium, 14-24 equilibrium time, 14-21, 14-22 for N1* separation, 14-17, 14-25 and 018, 14-25 for reflux in. 14-20, 14-21 separation factor, 14-15 single-stage, 14-18 to 14-20 stages, 14-14 minimum number of, 14-15, 14-10 Chemical processing in production of radioisotopes

(table), 14-31 to 14-30 Chemical reaction of Al, Zr, or U with water, probability of, 18-34 Chemical scale, definition of, 1-153 Chemical separation processes, casks for, 11-138 to 11-140

centrifugation, 11-104

corrosion of equipment in, 11-128 to 11-131 (table), 11-132, critical mass considerations 11-133 decladding for, 11-112, 11-113 design of joints in equipment for, 11-130, 11-131 dissolution in, 11-112 to 11-114 distillation, fractional (see fractional distilla tion, below) 11-104 to 11-106 evaporation, filtration. 11-103, 11-104 fractional distillation, 11-83 to 11-91 principles of, 11-83 to 11-89 nitric acid recovery in, 11-89 solvent recovery in. 11-89 uranium hexafiuoride process, 11-90 to 11-91 fuel dissolution, 11-112 to 11-114 instrumentation for, 11-114 to 11-122 ion exchange, 11-91 to 11-97 dual bed in, 11-95 in fuel reprocessing, 11-96 resin performance. 11-96, 11-97 mixed bed in, 11-95 resins for (table). 11-93 theory of, 11-91, 11-92 maintenance of equipment for, 11-137, 11-138 decontamination in (table), 11-138 materials for equipment used in (table), 11-128, 11-129

off-gases in, 11-113, 11-114 precipitation, 11-58 to 11-65 bismuth phosphate process, 11-61 to 11-63 theory of, 11-59, 11-60 pyrometallurgical, 11-97 to 11-102 electrorefining of uranium, 11-99. 11-100 fused salt extraction, 11-100 molten metal extraction (tables). 11-100. 11-101 slagging, 11-98. 11-99 thermodynamic properties of impurities (table), 11-98, 11-99 zone melting, 11-101, 11-102 sampling of radioactive materials in. 11-138. 11-139 shielding of equipment for, 11-135 to 11-137 solvent extraction, 11-65 to 11-83 equipment for, 11-76 to 11-79 flow sheet. 11-72 to 11-76 performance data, 11-79 plutonium distribution coefficient. 11-71 principles of, 11-65 to 11-71 processes. 11-80 to 11-83 Purex, 11-82 Redox, 11-82 Thorex. 11-82 salting, 11-70 solvents for (table). 11-64, 11-70 to 11-72 uranium distribution coefficient. 11-71 ventilation of plant for. 11-134, 11-135 waste disposal from, 11-140 to 11-146 permissible radioactivity in waste (tables). 11-141 Chicago Pile. CP-1. CP-2. CP-3, CP-3'. CP-5 <*~ Reactors, particular) Chimneys for reactor buildings. 13-175 Chlorine, physical properties of (table). 11-5 Chlorine trifluoride, physical properties of (taSlr*. 11-90 Choleski method, 8-38 Choppers, fast, 6-74 to 6-76 slow, 6-86 to 6-88 Chromium, physical properties of, 11-4 Chromium alloys (see High -temperature aBoys) Chromium plate, wear resistance of. in high-tem perature water, 10-76 Chugach Electric Association reactor. 18-138 Circular functions, tables of, cos x and sin z. 1-80 to 1-83 tan x, 1-84, 1-85 Circulating fuel reactors (see Power reactor?. homogeneous aqueous) Circulation, thermal, 9-90 City of Piqua reactor, cost of, 18-96 Clad (see particular material) Clairaut equation, 3-108 Classification of reactors (see Reactors) Cleaning of reactor systems, 13-184, 18-185 Clementine, Los Alamos Reactor, 13-98 description of, 18-136 Closures, for alkali-metal systems. 18-83, 18-84 for pressure vessels, 13-14, 13-15 Cloud chambers. 6-18 to 6-20 diffusion-chamber type, 6-20 type, 6-19, 6-20 expansion-chamber

Clouded crystal ball, 4-95 Coal, 13-84 to 18-88 (See also World energy resources) Coatings, for reactor building interiors. 1S-17S thickness gauging of, by radioisotopes. 14-S to 14-6

INDEX


Cobalt, as control material

(table), 10-188 physical properties of (table), 11-5 thermal stress properties of (table), 9-113 Cobalt-60, in medical therapy, 14-11 for radiography, 14-3 for thickness gauge, 14-5, 14-6 Cobalt alloys (see High-temperature alloys) Cockcroft-Walton accelerator, 5-125. 5-126 as source of fast neutrons, 5-58 Code in digital computers, 3-143 8-77 Codes for reactor construction, Coefficient, Fourier, 8-102 (table), in 10-79 of friction water void, safety aspects of, 8-80 Coffins (casks) in chemical processing, 11-138 to 11-140 Cold-water accident. 8-88 Collision, elastic, 6-31 inelastic, 6-61 Collision density, 6-43 Collision processes, 4-28 to 4-33 Collisions, number of, in slowing down in modera tors (table), 2-10 two-body, 4-30 Columbium (eee Niobium) Combination (mathematics), 3-58 Commutative law, 3-2, 3-7, 3-9 Comparator in reactor control system. 8-30. 8-38 to 8-40 Compensated ion chamber for reactor control, 8-51, 8-66 to 8-07 Compensating devices in reactor control systems, 8-41 Complete integral, 3-112 Complex numbers, 3-15 to 3-18 polar form, 8*16 rectangular form, S-lfi Complex variables. 3-80 to 3-86 Components, mechanical, in high-temperature water. 10-80 to 10-82 Compound nucleus, 4-73 to 4-76 decay of, 4-75 to 4-76 Compressible fluids, maximum flow velocity in (table). 9-43. 9-44 Compton scattering, 4-6. 4-60, 4-61, 6-4 in shielding, 7-61 Computers, analog, reactor (are .Simulator for re actor dynamics) digital, 3-142 to 8-146 Concentration, conversion factors for units of (table), 1-152 materials maximum permissible, of radioactive (see Maximum permissible concentration) Concentration-cell corrosion, 11-123 Concrete, heavy, barytes type (table), 7-112 prepack placement of, 13-184 relaxation length of, for fast neutrons and y rays, 2-35 iron ore, procurement and storage of, 18-182 limonite type, prepack placement of, 13-184 mix for (table). 13-182. 13-1*3 prepacked, 18-183. 13-184 shielding properties of (tables), 7-114, 7-115 steel punchings for, procurement and clean ing of, 13-182 shielding properties of (table), 7-112 thermal stress properties of (table), 9-113 Concrete aggregates for shielding (table). 7-113 Condensation of vapors, 9-65, 9-66 Condensers, steam, price of (table). 13-133 Conductance, heat, 1-60 Conduction (see Heat conduction) Conductivity (see Electrical resistivity; Heat

conductivity)

7

Confluent bypergeometric functions, 3-88 Conformal mapping, 3-85 Conic function, definition of, 3-30 Conies, 3-28 to 3-32 Conjugate complex number. 3-17 Conjugate to vector, definition of, 3-29 Consolidated Edison Indian Point reactor (see Re actors, particular) Construction of reactors, biologic >1shield, con crete, iron ore, procurement and storage of, 13-182 mix for (table). 13-182, 13-183 pouring of. 13-183 to 13-185 steel punchings for, procurement and clean ing of, 13-182 {See also Concrete) laminated construction, use and fitting of, 13-182 circulating systems, cleaning of, 13-184, 13-185 drying of, 13-185 testing of, 13-184 codes for. 8-77 control-rod drives, alignment of, 13-179, 13-180 installation of, 13-179, 13-180 of, 13-170 interchangeability testing of, 13-180 core, assembly of, precautions in, 18-177, 13-178 hole alignment, 13-184 materials control, 13-185. 13-186 permit for, 8-76 reactor vessel, alignment of, 13-178, 13-179 cleaning of, 13-179 materials control for, 13-178 thermal insulation installation, 13-179 reflector, beryllium, 18-181 graphite, machining and stacking of, 13-180, 13-181 Consumers Public Power District (sodium graph ite) reactor, plant space requirements for, 18-172 Containment, 8-84, 8-85, 12-19 of reactor coolant water (table), 13-5, 13-6 secondary (see Buildings for reactors)

Contamination (see Radioactive contamination) Contiguous functions, 3-88 Continued fractions, 3-97 Continuity, equation of, 3-118 of functions,

3-79. 8-80

Continuous processing in homogeneous

reactors,

safety aspects of, 8-84 Continuous slowing-down model, 6-6 Continuum region, 4-77 Contour integral, 3-81 Contraction pressure losses in fluid flow (tables), 1-52, 9-36, 9-37 Contravariant representation, 8-14 Control of reactors, boiling, dual system, 12-148, 12-149 EBWR system. 12-147 control criteria. 8-40 graphite, 13-19 homogeneous aqueous, 13-73, 13-74 liquid-metal-cooled. 13-100. 13-101 loop comparator, 8-30 materials (see Neutron, absorbers for) period, 8-21 to 8-32 (See also Period) shutdown, abnormal, 8-43 to 8-46 alarms, 8-43 cutback, 8-43 scram, 8-43, 8-44 emergency, 8-46 routine, 8-42

INDEX

8 Control

of reactors,

startup,

to criticality, 8-2 1 to


t-29 initial. 8-25 to 8-27

source for. 8-25, 8-49, 8-50 subsequent to initial, disturbing influences. 8-27 8-27 photoneutrons, at operating level, automatic control, 8-32 to 8-41 manual control, 8-32 to power level, 8-29 to 8-31 automatic period control, 8-30 combination period and level control, 8-30 manual control, 8-29 period permissive control. 8-29, 8-30 (See also Control system of reactor) Control rod, calibration of, 8-113 to 8-115 drop experiments, 5-114 effectiveness of, 8-12 nonlinear, 8-24 interaction of, 8-15 location of, in water reactors, 18-14 regulating rods, 8-21 safety rods (scram rods), 8-21 shadowing of, 6-114 shapes of, 12-75. 12-76 shim rods, 8-21 theory of, 8-12 to 8-19 boundary conditions, 8-12, 8-13 central cylindrical rod, 8-13. 8-14 in bare reactor, 6-109 two-group solution for, 6-67 to 6-69 effective radius. 8-13, 8-16 to 8-18 multiple-rod interaction, 8-15, 8-16. 8-18, 8- 19

noncylindrical, 8-16 to 8-19 off-center cylindrical rod, 8-14, 8-15 worth. 8-13 to 8-15 thermal distortion of, 9-112, 9-114 (See also Neutron, absorbers; Physics of noncritical reactors) Control-rod drives, 8-67 to 8-75 alignment of, 13-179, 18-180 installation of, 13-179, 13-180 interchangeability of, 13-179 requirements of, 8-68 seals for, 8-68, 8-69 testing of. 13-180 types, 8-69 to 8-75 ball-nut. 8-69 canned-motor, 8-73, 8-74 hydraulic-piston, 8-75 inverted, 8-72, 8-73. magnetic-jack, 8-75 piston, 8-72 rack-and-pinion, 8-70, 8-71 Control system of reactor* 8-21 to 8-4H automatic operation, 8-32, 8-33 comparators, 8-38 to 8-40 networks, 8-41 compensating interlocks, 8-43 manual operation, 8-32 natural frequency of, 8-41 poison feedback, 8-37, 8-38 scramming circuits, 8-44 transfer functions, 8-33 to 8-39 (See also Control of reactors; Control-rod drives; Physics of noncritical reactors) Convection, combined free and forced, 9-63 table, 9-95 forced, normal to tubes, screens and beds, 9-61 table, 9-63 streamline flow in channels (table), 9-60 to 9- 62

forced, turbulent flow in channels (tables). 9-58 to 9-60 free (natural), in channels (table), 9-61 to 9-64 external (tables), 9-63. 9-64 Conventional power plants (tee Nuclear power. Power, from fossil fuel) Convergence, Cauchy, 8-74 Conversion, 12-80 effect of lumping on. 12-10, 13-11 internal, elementary description of. 7-4 of uranium salts to metal, 11-109 to 11-112 Conversion factors for units, derivation of, 1-4,:! Convection,

1-43 for engineering units, short table of. 1-12 list of tables of, 1-3 reactivity (table), 1-46 Conversion gain, formulas for, 12-82, 18-83 general, 1-28 Conversion ratio, formula for. 12-82, 12-83 general, 1-28 Converters, for fast neutron production, 4-91 for in-pile engineering tests, 6-141 for production of fission neutrons, 6-71, 6-7.! Coolants, classification of reactor by (table). 12-7 12-8 comparison of (tables), 9-90 to 9-93. 12-5* u 12- 60 film temperature drop in, 9-1 1 gases (see Gases) leakage of water, allowable (table). 13-4 to 13- 6 liquid metals, physical properties of (table). 13-81, 18-82

monitoring (see Monitoring) pumping power of, 9-90, 9-91 radioactivity of (see Radioactivity,

of reactor coolants) temperature rise in, 9-84, 9-85 (See also Power reactors, gas-cooled) Cooling, emergency, for reactors, 8-80 of irradiated fuel, 12-100. 12-101 shutdown, for air-cooled reactors. 18-110 for liquid-metal reactors, 13-100 for water reactors, 18-29, 13-30 Copper, physical properties of (table). 11-5 thermal stress properties of (table), 9-113 Core of reactor, construction of. 13-177. 13-17& Corrosion, 11-122 to 11-132 of fuel reprocessing equipment. 11-128 to 11-131 in homogeneous aqueous reactors. 13-57 1* by liquid metals, 10-21 heavy metals (table), 13-99 by lithium (table), 13-83 of materials in reactor systems, 10-7-? to 10-75 mechanism of, 11-122 to 11-127 by sodium, 10-20, 10-21 of beryllia, 12-74 of graphite, 12-70 to 12-74 table. 13-83 by sodium-potassium, 10-20. 10-21 table, 13-83 testing, 11-131, 11-132 types of, 11-131 by water, of beryllia, 18-74 of beryllium, 12-74 of metals and alloys (table), 10-75 permissible rate for metals, 10-74 of structural materials, 10-73 to 10-75 (See also Structural materials, in hign-teaperature water) Corrosion rates, conversion factors for (table). 1-152 Cos 0. cos 9 (tee Cosine of scattering ang4e)

INDEX Coi x, table of, 1-80 to 1-83 Coah x, table of, 1-00 to 1-02 Cosine integrals, S-05 Cosine of scattering angle, average, definition of, 2-12


fast, cos B, 2-28, 2-30 thermal, for the elements, 1 — cos 9 (table), 2-13, 2-14 Cost of living index. Bureau of Labor, 12-01, 12-02 Cost of power {see Nuclear power; Power, from foasil fuel) Coulomb barrier, 4-33 Countable, set, in mathematics, 3-77 Counter displays in reactor control, 8-57 Counters (see Detectors) Counting, o-particle, 6-28 to 9-31 absolute, 6-28, 6-20 detectors for, 6-28, 6-20, 11-53 energy spectrum in, 6-20 to 6-31 magnetic analysis, 6-30 pulse-height analysis, 6-20, 6-30 range measurement, 6-30, 6-31 in radiochemical analysis, 11-53, 11-54 0-particle, 6-31 to 6-35 absolute, 6-32 detectors for, 6-31, 6-32 energy speotrum in, 6-32 to 6-36 absorption analysis, 6-33, 6-34 magnetic analysis, 6-34 pulse-height analysis, 6-34, 6-35 analysis, 11-54, 11-55 in radiochemical electron (eee JS-particIe, above) fast-neutron, 6-42 to 6-40 detectors for, 6-42 to 6-40 by fission, 6-48 by long counter, 6-40 by proton recoil, 6-42 to 6-48 by threshold reactions, 6-48, 6-40 7-rayt 6-35 to 6-40 detectors for, 6-35, 6-36 energy spectrum in, 6-35 to 6-30 magnetic analysis, 6-30 6-30 pair spectrometers, methods, 6-30 photographic counters, 6-37, 6-38 proportional scintillation counters, 6-38, 6-30 analysis, 11-53 to 11-56 in radiochemical 6-41, 6-42 resonance-neutron, thermal -neutron, deteotors for, 6-40, 6-41 foil activation, 6-41 X-ray (see y-t&y, above) Counting-rate meter, 8-57 Covalent crystals, effect of radiation on, 10-123 Covariant representations, 3-14 Cramer's rule, 3-13 Crane, canyon, 7-136 Credible accident, 8-84 Crevice corrosion, 11-123, 11-124 Critical approach procedures, 6-110 to 6-112 Critical assemblies, experiments with, 6-108 to 6-117 control-rod calibration, 6-113 to 6-115 rod drop experiments, 6-114 danger coefficient measurement, 6-115 migration area measurement, 6-115 reactivity measurement, 6-112, 6-113 coefficient, 6-115 temperature instrumentation for, 6-100 Critical buckling, 6-108, 6-20 Critical condition, 6-53 (See alto Physics of critioal reactors) Critical equation, 6-10

*

See insert Revised

Cross-section

Data.

9

Critioal experiments, 6-100, 6-110 (See also Critioal assemblies) Critical mass in fuel reprocessing (table), 11-132, 11-133

Croloy (see High-temperature alloys) Cross linking of polymers by radiation, 14-8 3-15 Cross product (mathematics), Cross sections, absorption, correction for Maxwell-Boltzmann distribution, 1-30, 2-14, 2-23, 6-40 to 6-50 for \/v absorbers, 6-50 correction for non-l/o absorbers, factor (table, BNL 325), 2-15 to 2-23* Westcott values for fuels (table), 1-21, 1-22* 1-30, 2-23, 2-24, correction for temperature, 6-40 to 6-50 Westcott correction for fuels (table), 1-21,* 1- 22 definition of, 2-12 factor for, 2-15 to 2-23* fast, for Pu»» Th«» U»\ U'«, 2-7 measurement of (see measurement of, below) of mixtures, 1-31 nuclide chart, 1-15 to 1-10 1 Mev-noutron, for nuclides (table), 2-28, 2- 20 for threshold reactions, 2-30 30 kev-neutron, for nuclides (table), 2-28 two-group fast, 1-35 2-12 to 2-27 2200 m/sec-neutrons, of elements (table with other properties), 2-13 to 2-23 of fission products (tables), 1-23, 2-10. 2-11 of nuclides (table, BNL 325), 2-15 to 2-23 of U"», UJ» Pu»», Westcott values (table), 1-21,* 1-22 activation, definition of, 2-12 of fissionable and fertile materials (see absorp tion, above; radiative capture, below) measurement of (see measurement of, below) 2200 m/sec neutrons, of elements (table, BNL 325). 2-15 to 2-23 of nuclides (table, BNL 325), 2-15 to 2-23 table with decay data, 7-8 to 7-12 capture (see absorption, above; radiative cap ture, below) definition of, 4-41 differential scattering, 4-20 elastic scattering (see scattering, elastic, below) fission, 4-71, 6-54 correction for Maxwell-Boltzmann distribu tion, 1-30, 2-14, 2-23, 6-40, 6-50 factor, correction for non 1 /v absorbers, BNL 325 (table), 2-21,* 2-22* Westcott values (table), 1-21,* 2-22 1-30, 2-24 Correction for temperature. Westcott corrections (table), 1-21,* 1-22 epithermal, U»» (table), 2-11 fast (table), 2-11 for fertile materials, 2-7 for fissionable materials, 2-7 Np"7, Pu>", PuM0, Th"». Um, UJ3>, U"9 (table), 2-7 as function of energy, Um, 2-6 measurement of (see measurement of, below) of mixtures, 1-31 in natural uranium equilibrium spectrum, 2-8 nuclide chart. 1-15 to 1-10 " " for 2200 m/sec neutrons, BNL world best values (table), 2-21,* 2-22* Westcott values for U»», Um, Pu«» (table). 1-21, 1-22 hydrogen, at high energy, 7-81

/

/

/

INDEX

10

(ff

scattering, inelastic scattering inelastic, below) inelastic transfer, fur nuclides (table), 2-33 macroscopic, 1-31 of mixtures, 6-32 of selected elements (table), 1-24, 1-26 of, 6-55 measurement activation, principles of, 6-56, 6-57, 6-89 to 6-93 fast-neutron, 6-01 angular distribution, 6-63 to 6-65 capture, 6-71 fission, detectors for, 6-63, 6-69 inelastic scattering, 6-65 to 6-68 neutron sources for, 6-70 to 6-72 nonelastic scattering, 6-65 to 6-68 reaction (n,p), (n,a), (n,2n), (n,y), 6-68 to 6-72 in reactor, 6-70 to 6-72 total. 6-61 to 6-63 free atom, 6-99 partial, 6-55, 6-77 (intermediate-energy), resonance-neutron velocity selectors for, accelerators pulsed, 8-75 6-73, 6-74 crystal monochromator, detectors for, 6-76, 6-77 fast choppers, 6-74 to 6-76 time-of-flight selectors, 6-74 to 6-77 thermal-neutron, absorption, 6-93, 6-94 danger coefficient method, 6-93 6-94 mass spectrometer, oscillator method, 6-93, 6-94 activation, for Maxwell-Boltzmann distri bution, 6-89. 6-90 activation analysis, 6-92, 6-93 detectors for, 6-90 to 6-92 /factors for non-l/i» absorbers, 6-89,*


Cross sections,

6-90* second-order reactions, 6-90 total, 6-55 for Maxwell-Boltzmann distribution, 6-94 transmission, 6-94 to 6-96 microscopic, definition of. 6-32 of moderators (see Moderators, nuclear data

for)

nonelastic scattering, definition of, 1-32, 6-55 for high-energy neutrons, of nuclides (table), 2-32 nuclide chart, 1-15 to 1-19 partial, definition of, 4-71, 6-54 measurement of, 6-55, 6-57 radiative capture, for Au1", Pu«', Tb"». UUi. U*» 2-7 definition of, 2-12 in natural uranium equilibrium spectrum, 2-8 1 Mev-neutron, for nuclides (table). 2-28, 2-29 theory of, 4-98, 4-99 30 kev-neutron, for nuclides (table), 2-28 (See also absorption; activation, above) reaction, measurement of, 6-68 to 5-70 removal (table), 7-81 to 7-83 scattering, definition of, 6-54 differential, 4-29 elastic, epithermal, definition, 2-12 of the elements (table), 2-13, 2-14 fast, 2-28 thermal, of the elements (table), 2-13. 2-14 inelastic, definition of, 6-54, 6-55 fast, 2-28 for fission neutrons, of nuclides (table). 2-33 measurement of, 6-65 to 8-68

* See innert Revised

Cross-section

Data.

Cross sections, scattering, of mixtures, 1-31 thermal, of the elements. 1-13, 2-14 total, fast, 2-28 self-absorbed, 6-26 self-shielded. 6-26 threshold reaction for fission neutrons, of nu clides (table), 2-30 of thorium (see Thorium) total, definition of, 4-71. 6-54 fast-neutron, 2-28 measurement of (see measurement of, ooor*) transfer, inelastic, for fast neutrons, of nuclide(table). 2-33 transmission, measurement of, 5-94 to 5-96 transport, 1-33. 1-34. 2-28. 2-30 fast, 2-28 to 2-30 for Oodiva spectrum, 1-31 for 1-Mev neutron (table), 1-30 two-group, fast, 1-35 of uranium (see Uranium; Uranium-233; Cra nium- 235; Uranium-238) Crout's method. 3-38 Crystal cutoff, 5-99 Crystal effects in cross-section measurement. 6-*s 6-73, 6-74 Crystal monochromator, Crystal structure, 10-5, 10-6 Cubic equations, illustrative example of solution of, 1-66 to 1-68 by Horner's method, 1-66 to 1-68 Curie, definition of, 7-7 Curium, chemical properties of. 11-44 cross section of, 1-22 isotopes of, formation of, 11-9 Curl, 8-125 Current density, neutron, 6-35 Curve, slope of, 3-43 Cutback, reactor power, 8-43 Cyclic plastic strain, 9-115 Cyclotron, 5-130, 5-131 dose measurement from, 10-96, 10-97 experiments, 10-98 high-temperature 10-97, 10-98 experiments, low-temperature as source of fast neutrons, 6-58 Cyclotron particles, atom displacement by, 10-lpr Cylinders, heat conduction in (srt Heat conduc

tion) Cylindrical control rod, effective radius of. 5-13 Cylindrical coordinates, 3-32 Cylindrical functions (see Bessel functions)

d'Alembert's formula, 3-116 Damage, radiation (see Radiation damage)

Danger coefficient, 6-93 measurement of, 5-115 Dead time, in 0-particle counter. 6-31 in Geiger-Muller counters. 5-15 De Boer type process for fuel reproeesninc, 11-10^ De Broglie wavelength, 4-15 to 4-17 of neutrons, formula for, 1-13. 1-156 Debye temperature, 10-10 Decay (see Radioactivity, decav) Decladding, 11-112, 11-113. 11-101 Decontamination, 7-48 to 7-50 agents for (table), 11-137, 11-138 of equipment in gas-cooled resvetora. 13-1 1* Decontamination factor, 11-58 Deentrainment of water from steam. 18-47 dE/E spectrum, 6-73 Deformation, mechanism of, 10-13 to 10-16 Degeneracy of energy levels. 4-23 Degenerate surface, 3-24 Del (differential operator), 3-125

INDEX Delayed

Detectors, ion chambers,

eriticality, 8-8

neutrons, 8-2 to 8-5 data of Hughes et al. (table), 8-3 data of Keepin et al. (tables), 8-3 to 8-6 summary table, 1-46 effect of, on reactor kinetics, 6-111, 8-10, 8-1 1 effective value of delayed fraction, 8-10 energy of, 2-3 in homogeneous reactors, loss of, 18-21, 19-45 mean energy, definition of, 8-2 mean life, definition of, 8-2 precursors of, 2-3 relative importance of, 8-113 shielding against, 7-85. 7-92 (See also Physics of noncritioal reactors) Delta (Dirac) function, 8-93, 8-94 De Moivrc's theorem, 8-22, 8-23 Densitometers, 7-48 Density, conversion factors for units of, 1- 141» of the element* (table), 11-5 to 11-7 Density gauges, radioisotopes for, 14-4 Depository libraries for AEC literature, 1-71 , Delayed

1-72

Derivatives (mathematics), partial, 8-64

3-64

table, 8-69 rule of signs, 3-43

Descartes'


Detectors, bubble chambers, 6-20. 8-21 cloud chambers, 8-18 to 8-20 diffusion chamber type, 6-20

chamber type, 6-19, 5-20 of reactor con measurement stants, catcher foils, 6-105 fast neutron flux, 6-104 thermal neutron flux, absolute flux, 6-104 counters, 6-104 foils (table), 6-102 cadmium difference, 6-102 counting of, 6-102, 6-103 fission chambers, 6-68 to 6-69, 8-55 to 8-66 for cross-section measurement, 6-68 pulse. 8-55, 8-56 for thermal neutron counting, 6-41 Geiger-Mliller counters, for ff counting. 6-31 analysis, 11-54 in radiochemical characteristics, operational, 6-17 construction of, 8-16, 6-17 dead time in, 6-15, 6-31 efficiency of, 6-18 for y-ray counting, 6-36 gases for, 5-15, 6-16, 6-18 life of, 5-17, 6-18 in radiochemical analysis, 11-54 5-16 self-quenching, ion chambers, 6-5 to 5-7 air-wall chamber, 5-6 for 0 counting. 11-54 compensated, for reactor control, 8-51, 8-66, 8-67 cosmic-ray background in, 6-6 current, B"Fi, 8-58, 8-62 testing of, 8-65 to 8-67 current from, 5-6 saturation, 6-7 electrometers for, 5-7 to 8-1 1 fission, 8-64 to 8-66 for 7-ray counting, 5-35 to 6-36 gases for, 6-7 for thermal neutrons, 6-40, 5-41, 8-58 to 8-66 for X-ray counting, 6-35, 6-36 gases for, 6-7 insulating material for, 8-10 expansion

for experimental

11

pulse, for a counting analysis, 11-63 in radiochemical BF>, 8-53 to 8-55 radiation damage to, 8-66 fission, 8-55, 8-56 gases for, 5-11 in, 6-12, 5-13 microphonics principles of, 5-10 to 5-12 pulse amplification for, 6-10 to 5-12 for reactor control, 8-53 radiation damage to, 8-56 radiation damage to (table), 10-145 in radiochemical analysis, 11-53 to 11-55 saturation ion current in, 8-7 7-46 tissue equivalent, ionization chambers (see ion chambers, above) emulsions, 6-22, 5-23 photographic proportional counters, 5-13 to 8-15 for a counting, 5-29 for 0 counting, 5-32, 6-34 for 7-ray counting, 6-36 to 5-38 gas amplification in, 6-13 gases for, 6-14 pulse shape of, 5-14 for thermal neutrons, 6-40 to 6-41 B»« Ft, 8-53 to 8-55 for time-of-flight measurement, 6-77 uses of, 6-14 to 6-15 pulse chambers (see fission chambers, pulse; ion chambers, pulse, above) for reactor control, 8-53 to 8-56 cables for, 8-52 fission pulse chambers, 8-55 to 8-56 ion current chambers (B'°Fi and fission), 8-58, 8-62 to 8-67 testing of, 8-65 to 8-67 location of, 8-51 to 8-52 proportional pulse chambers, B10Fi, 8-53 to 8-55 pulse chambers, BFi, 8-53 fission, 8-55 to 8-56 scintillation counters (see Scintillation counters) for thermal-neutron cadmium-difference measurement, 5-102 (See also Counting; Instrumentation, nuclear) 8-11 to 3-13 Determinants. solution of linear algebraic equations by, ele mentary description of, 1-70 to 1-71 illustrative example of, 1-71 Deuterium, 14-24 as neutron source in reactors, 8-27 DiO (see Heavy water) DjO cooled reactors (see Power reactors, watercooled; Research reactors) Deuteron, magnetic moment of, 4-50 nuclear properties of, 4-57 quadrupole moment of, 4-50 D(d,n)He* reaction, 6-60 reactions, 4-81 Deuteron-induced Diamond, effect of radiation on, 10-123 DIDO, description of, 13-128 Dielectric, radiation damage to (table), 10-145 Difference equation, linear homogeneous, 3-44 non homogeneous, 3-45 Differential, of the dependent variable, 8-66 total, 8-66 Differential equations, 3-106 to 8-124 degree of, 3-106 order of, 3-106 ordinary, 3-107 first-order, 8-107, 3-108 linear, first-order, 3-108 to 3-110

12

1NDKX


Differential equations, ordinary, linear, secoudorder, 3-110 to 3-112 numerical Bolution of, 3-122 to 8-124 partial, 3-112 to 3-117 boundary- value problems, 8-118 to 3-122 Laplace transform, 3-121, 3-122 separation of variables, 3-119 to 3-121 elliptic, 3-115, 3-116 first-order, 3-112, 3-113 hyperbolic, 3-116, 8-117 Laplace and related equations, 3-117, 8-118 numerical solutions, 8-123, 8-124 partial differential equations, 3-123, 3-1 24 parabolic, 3-117 second-order, 3-113 to 3-115 solution by analog computer, 8-148 (See also Calculus, of variation) Differential scattering cross section, 4-29 Differential transformers for in-pile measure ments, 10-93 Differentiation (tables), 3-64 to 3-66, 8-69 of an integral, 3-65 numerical, 3-74, 3-75 Diffusion, anisotropic, in uranium, 10-118 effect of radiation on, 10-109 of neutrons (see Physics of critical reactors) in solids, theory of, 10-13 Diffusion afea, thermal, calculation of. 6-86, 6-87 summary, 1-34, 1-35 Diffusion cloud chambers, 5-107 Diffusion coefficient, evaluation of, 6-35 fast, calculation of, 6-87 of moderators (table), 1-20 numerical example, 6-98 summary, 1-34 thermal, calculation of, 6-85, 6-80 measurement of, 6-107 of moderators (table), 1-20 numerical example, 6-98 summary, 1-34 Diffusion constant (see Diffusion coefficient) Diffusion cooling, 6-107 Diffusion equation, 3-117, 6-5, 6-35 to 6-41 (See also Physics of critical reactors) Diffusion kernels. 6-39 equation for, 6-9 Diffusion length, 6-36 fast, of moderators (tables), 1-20, 8-9 numerical example of calculation for metal-DtO mixture, 6-98 thermal, of materials (tables), 1-20, 2-9, 6-86 measurement of, by thermal column, 6-105 to 5-107 Digital computing machinery, 3-142 to 3-146 Dilatant plastics, viscosity of, 9-26, 9-27 Diluents for fuel, physical and mechanical proper ties of, 10-38 to 10-54 Dimension systems, 1-134 to 1-139 comprehensive, 1-139 electrical, 1-137 to 1-139 English (table), 1-136 mechanical, 1-136, 1-137 metric (table), 1-135 thermodynamic, 1-137 change caused by radiation (see Dimensional Radiation damage) Dimensionality, 3-8 numbers, definitions of, 9-31 Dimensionless table of, 9-32 Dimple, deuterium-moderated pile, low-energy, description of, 13-128 Diphenyl, physical properties of (table), 9-22 radiation damage to (table), 10-137

Diphenyl, as reactor coolant, 11-57 to 1S-5& comparative table, 13-59 Dipole moment, 4-4 Dipole radiation, 4-4 Dirac delta function, 3-93 Dirao equation, 4-24 Direct conversion of fission energy. 18-2, 18-3 Direot cycle (see Nuclear power) Directrix, definition of, 3-30 Dirichlet's problem, 3*115 factor, 6-77 Disadvantage for resonance escape probability, 6-82 self-absorbed cross section, 6-26 Discharge from reactors (see Fuel handling) Discoloy (tee High -temperature alloys)

Discontinuity (see Functions) Discontinuous reactor regulating

system, 6-32.

8-33

Discriminant of quadratic equation. Discriminators, 8-57

"Dished"

loading,

3-43

13-106

Disintegration (see Radioactivity) Dislocation mechanism of deformation, 10-14, 10-15 Dispersions,

fuel (tobies), 10-183 to 10-185. 11-61 radiation damage to, 10-120, 10-121 in-pile measurement of, 10-93 Displacement, operator (mathematics), 3-33 Displacement Displacement spikes, 10-102 Disposal of solid radioactive waste, 7-54 Dissolution of spent fuel, 11-112 to 11-114 Distillation, 11-83 to 11-91 fractional, 11-83 to 11-91 theory,

11-83 to 11-89

equilibria, 11-83 to 11-89 rectification, 11-86 to 11-89 simple distillation. 11-85, 11-86

for isotope

separation,

11-23

Distortion, thermal. 9-112, 9-114 (See also Thermal stress) 8-60 to 8-62 Distributions (mathematics), Distributive laws, 3-2, 3-7, 3-9 Div (divergence), 8-125 Divergence

theorem

(Gauss),

8-126

Divided differences, 3-34 Dollar (reactivity), 6-113 conversion

factors

for, 8-3, 6-9

Dominant energy (see Shielding, neutron attenu ation) Doolittle solution of linear algebraic equation*. 3-38

illustrative example of, 1-69, 1-70 Doppler effect, 6-78. 6-79. 6-26. 6-27. 18-20 cause of, 6-78 safety aspects of, 8-88 Dose (radiation), absorbed, definition of. T-24 accumulated, 7-27 air, definition of, 7-25 definition of, 7-24 exposure, definition of, 7-24 fatal, for chemical and nuclear poison* (tables) 8-77 from y rays, 3-35 calculation of, T-65, 7*66 tables, 7-67, 7-68 from internal radioactive particles, 7-51, T-S2 maximum permissible, external. 7-29 to 7-32 internal, 7-38 to 7-45 particular radioisotopes (table), 11-141 measurement of, from cyclotron beam, 10-96. 10-97 in reactor, 10-92 from neutrons, distribution in tissue (ta±4e) 7-35

INDEX


Dose (radiation), rhythmic effect of, 7-27 tissue, definition of, 7-25 units of, 7-65 summary (table), 1-42 X-ray (table), 7-37 (See also Health physics) Dose build-up factor (tables), 7-66 to 7-68 Dose rate (radiation), build-up factor in attenu ation (tables), 7-66 to 7-68 from delayed neutrons in aqueous shield, 7-85 effect of, on injury, 7-27 from fission source, in aqueous shield, 7-82 to 7-84 in hydrogenous shield, 7-82, 7-83 from internal radioactive particles, 7-51, 7-52 maximum permissible, basic, 7-20 continuous exposure, 7-29 for minors, 7-29 from neutrons (table), 7-84 from neutron-attenuation secondary y rays, 7-89, 7-90 occupational, 7-30 to population at large, 7-29 from reactor neutrons, calculation of, in shields. 7-82 to 7-85 (See also Health physics) Double precision in computing machine, 3-144 Doubling time, 12-83, 12-84 general formula for, 1-28 Dounreay reactor, 18-140 Dowex resins, 11-93 Dowtherm, physical properties of, 9-22 radiation damage to (table), 10-137 (See also Diphenyl) Drag coefficient, 9-46 deanition of, 9-32 Drains for reactor buildings, 13-176 Dresden (see Reactors, particular)

Drives, control-rod (see Control-rod drives)

Dry

box, 11-135

Drying, reactor systems, 13-185 Dual-cycle boiling reactors, 13-42 (See also Power reactors, water-cooled) Dual-purpose reactor, 12-111, 12-112 Dulong-Petit specific heat, 10-9 Duolite resinB, 11-93 Dynamic condenser electrometers, 5-8, 5-9 Dynamic impedance of chambers for reactor con trol, 8-66, 8-67 Dynamic viscosity (see Viscosity) Dynamical variable, 4-19 Dysprosium, as control material (table), 10-180 density

of, 11-5

Dysprosium foils for heterogeneous

lattice,

8-120

K

EBR-I, EBWR

II

(see Reactors, particular) (see Reactors, particular) Eccentricity, definition of (mathematics), 3-30 Economics of nuclear power (see Nuclear power) Eddy-current testing, 10-160, 10-161 Eddy diffusivity, 9-34 EDF-I, (table), 13-106 Effective radius of control rods, 8-16 Effective resonance absorption integral, 8-82, 8-48 measurement of, 6-82, 5-83 Effective multiplication factor, 6-110 Effluents, radioactive, from reactors, 7-50 airborne particles, 7-50 A" 7-50 (See also Radioactive wastes)

II

13

Eigenfunotions in quantum mechanics, 4-19 Eigenvalue, definition of, 3-29 Eigenvalue problem, in quantum mechanics, 4-19 in wave mechanics,

4-19

Eigenvector, definition of, 3-29 Einstein-Sxilard pump, 18-90 Einsteinium (element 99), chemical

properties of, 11-44 cross section of, 2-22 Elastic collision, 6-42 Elastic scattering, 4-95 to 4-99 description of, 4-72 (See also Cross sections, scattering) Electric generating industry (see Nuclear power; Power, from fossil fuel) Electric moments of nuclei, 4-49, 4-50 Electrical analogs, for heat flow, 9-99 to 9-103 homogeneous, 9-99 to 9-101 lumped, 9-101 to 9-103 for streamline fluid flow, 9-31 Electrical equipment, accessory, in power plant, cost of, 12-98 Electrical gauging, eddy-current, 10-166 resistance methods, 10-166 Electrical properties, effect of radiation on, 10-105 resistivity (table), 10-108 (See also Radiation damage) Electrical resistance, in-pile measurement of. 10- 93 Electrical resistivity, effect of radiation on (table), 10-108 (See also Radiation damage) of water (table), 13-11 Electrical standard units, 1-141, 1-142 conversion factors for (table), 1-152 Electrical systems, radiation damage to (see Radi ation damage) Electrical tests, 10-159 to 10-161 eddy-current methods. 10-100, 10-161 resistance methods, 10-159, 10-160 (See also Electrical gauging) Electrodeposition in radiochemical analysis, 11- 53 Electrodynamics, quantum, 4-19 Electromagnetic flowmeter, 11-120, 11-121 radiation, 4-2 to 4-9 Electromagnetic classical theory of, 4-2 to 4-4 Electromagnetic separation of isotopes. 14-44 to 14-58 (See also Calu Irons) Electrometers, 5-5 to 6-10 6-8, 6-9 dynamic-condenser, electron-tube, 6-8 electrostatic, 8-7 insulating materials for, 6-10 pocket fiber, 7-47 quartz-fiber, 6-7 vibrating-reed, 6-8, 6-9 Electron capture, elementary description of, 7-3, 11-5 Electron configuration of elements (table), 10-3 Electron constants, energetics of, 4-67 value of (table), 1-154 Electron counting, 6-31, 6-32 {See also Counting, 0- particle) Electron linear accelerator, 6-129 Electron spin, 4-14, 4-24, 4-25 and relativity, 4-24, 4-25 Electron synchrotron, 6-135 Electron-tube electrometer, 6-8 Electron volt, definition of, 1-153 Electronic charge, magnitude of, 1-13 Electronic systems, radiation damage to (see Radiation damage)

14

INDEX

Electrons, ionization by, mathematics of, 6-3 mass and energy equivalent of (table), 1-155 approximate values of, 1-13 properties of, 4-55 wavelength of (table), 1-156 Electrorefining of uranium, 11-99, 11-100 Electrostatic electrometers, 6-7 Electrostatic precipitators for reactor building's 13-175 Element 99 (einsteinium), chemical properties of, 11-44 cro8s section of, 2-22 Element 100 (fermium), chemical properties of, 11-44 Elements, absorption cross sections of, for 2,200m-sec neutrons (table, BNL 325), 2-15* to


2-23* activation cross sections of, for 2200 m/sec neutrons (table, BNL 325), 2-15 to 2-23 fission cross sections of, for 2200 m/sec neutrons (table, BNL 325), 2-21,* 2-22* isotopic abundance of (table, BNL 325), 2-15 to

2-23* macroscopic nuclear properties of (table), 1-24, 1-25 nuclear data, guide to, 1-4 for thermal neutrons, 2-13, 2-14 periodic chart of, 11-4 periodic system in relation to Bohr atom, 4-13 physical properties of (table), 11-5 to 11-7 shell structure of (table), 10-2, 10-3 table of atomic numbers and atomic weights, 1-14 Elk River reactor (nee Reactors, particular) Ellipse, 3-30, 3-31 Ellipsoid, definition of, 3-30 Elliptic integral*. 3-95 Emergency cooling (gee Shutdown cooling) Emergency shutdown of reactors, 8-46 Emission of light, 4-19 Emulsions, photographic, for particle detection, 6-21 to 6-23 Endoergic reaction, 4-42 Endothermic reaction, 4-42 Energy, binding, 2-35, 2-36 conversion factors for units of (table), 1-147 and mass, 4-9, 4-40 to 4-42 equivalents (table), 1-155 of neutrons, 2200 m/sec, 1-13 Energy-absorption build-up factor, in >-ray attenuation (table), 7-06, 7-69 linear attenuution coefficient of (table), 7-62,

'

7-64 (See also Shielding, >-ray attenuation) Energy levels, atomic, 4-12 in nucleus, 4-52 Energy loss per collision, 6-42, 6-43 Energy resources (see World energy resources) Engineering testing in reactors, 6-140 to 6-145 dynamic, 6-141 to 6-145 cycling tests, 6-141, 6-142 loops, 6-142 to 6-145 static, 5-140, 6-141 tests, 6-140. 6-141 before-and~after in-pile measurements, 6-141 Enrichment, classification of reactor by (table), 12-6 to 12-8 in fast reactor, 12-19 Enrico Fermi sodium-cooled fast reactor (set Reac tors, particular) Enthalpy -entropy diagram, for air. 12-129 for steam, 12-123 to 12-125 Entire function, 3-84 Enumerable (set), 3-77 * See insert Revised Cross-section Data.

Enumeration, bases of, 3-4, 3-5 Epi cadmium fission fraction, 5-119 Epicadmium l/t> absorption in U"*. 6-119 < (tee Fast effect) Equations, algebraic, 3-35 to 3-55 homogeneous, 3-35 non homogeneous, 3-35 solutions of (see Solution* of equations) summary of, 1-63 to 1-71 systems of, 8-50, 3-51 theory of, 3-41 to 3-45 (See also Differential equations) Equi-convergence theorem, 3-104 Equivalent bare dimension, 6-69

Equivalent diameter (in fluid flow), 9-33 Erbium, as control material (table), 10-189 density of, 11-5 Error integral, 8-94 table of, 1-124 to 1-126 Error-signal amplifier, 8-33 Erythema caused by radiation, 7-28 Escape probability, resonance (see Resonance escape probability) Essential singularity, 3-84 ETR (see Reactors, particular) Euler equations, 3-137 differential, 3-110, 3-138 Euler method, modified, of solving differential equations,

3-122

Europium, as control material (table), 10-188 density of, 11-5 Evaporation, 11-104 to 11-106 of radioactive wastes, 11-145 Exact differential equation, 8-108 Excess absorption, 6-79 Excess multiplication factor, 2-110 Excess reactivity (see Physics of noncritie*! reaetore)

Exchange forces, 4-47 Excitation as cause of injury, 7-23 Excited state, atomic, 4-12 Exclusion area, 8-85 obsolete formula for, 8-86 Exclusion principle, 4-27, 4-44. 4-45 identical particles and, 4-26. 4-27 Pauli, 4-13, 4-14 Exhaust stacks for reactor buildings, 18-175 Exoergic reaction, 4-42 Exothermic reaction, 4-42 between coolant and fuel material, 12-19 Expansion, coefficient of, of elements (table), U-5 to 11-7 of functions, 3-95 to 3-106 table of, 3-99, 8-100 of piping, allowance for, 13-31, 13-32 thermal, theory of, 10-11 Expansion chambers, cloud, 6-19, 6-20 Expansion pressure losses in fluid flow (table). 1-52, 9-36, 9-37 Expansion theorem. 3-104 Experimental facilities in reactors. 10-83 to 10-92 10-96 to 10-99

Experimental neutron physics (»« Neutron physics)

Experiments in reactors (see Irradiation testing, Neutron physics) Exponent, definition of, 3-3 Exponential experiments, 6-117 to 8-119 buckling measurement, 6-117, 6-118 extrapolation distance measurement, 5-118 migration area measurement, 6-118 reflector savings measurement, 6-1 18 Exponential functions, e*. e~*. table of, 1-86 to 1-89

INDEX Exponential and related integrals, 8-94 ourves of, 7-102 tables of, 1-119 to 1-122 Exposure, exposure rate (see Dose; Dose rate) External dose, maximum permissible, 7-29 to 7-32 Extract, aqueous, definition of, 11-68 organic, definition of, 11-65

Extraction, fused Extraction stage,

salt, 11-100 11-66 Extrapolated end points, 8-16 Extrapolation, 3-33 Extrapolation distance, «, 8-37, 8-12 to 8-13 from experiments, 8-118 by thermal column, 8-105

/*

factor, in cross-section measurement, 8-90 for non-1 i' absorbers (table, BML 325), 8-15 to 8-23*

f-m cyclotron,

8-134, 8-135

Face-centered cubic (fee) structure, 10-6

Factor

theorem,

3-41

Factorial function xt, table of, 1-117, 1-118 Factorials and their reciprocals, table of, 1-126 Fallout particles from testing of nuclear weapons


7-51

Fanning factor, definition of, 9-32 formula for, 9-32 Faraday constant, value of, 1-154 Fast choppers, 8-74 to 8-76 Fast diffusion coefficient, constant (tee Diffusion coefficient)

15

Fermi energy of metals, 10-6, 10-7 Fermi plot, 8-33 Fermi reactor (see Reactors, particular, Enrioo Fermi) Fermium (element 100), chemical properties of, 11-44

Ferritio steels (tee Stainless steels) Fertile materials, characteristics affecting selec tion of, 12-10 nuclear properties of, guide to, 1-4 physical and mechanical properties of 10-17 to 10-38

Fiber electrometers (sec Electrometers) Pick's law of diffusion in solids. 10-13 Field, central, 4-21 Field index in magnetic acceleration, 8-129 Film badge, 7-47 Film boiling (see Boiling) Film coefficient. 9-57 (See alto Boiling; Condensation of vapors; Convection) Filters, for air-cooled reactors, 13-109, 13-1 10 changing,

13-109

Chemical Warfare Service. Fiberglas, 13-109

13-109

life of, 13-109

location of, 13-110

in gas-cooled reactors, shielding foi , 13-1 18 for graphite reactors, 13-141, 13-142 for reactor buildings. 13-175 Filtration, 11-103, 11-104 Fine structure, relativistio, hydrogen atom. 4-12, 4-25

Fine structure constant, 4-25 Fissile materials (see Fuel) Fission, asymmetry of, 4-88

Fast effect t, 8-22 calculation of, 8-84, 8-85 numerical example, 8-98 summary, 1-33 of, 8-120, 8-121 in uranium, 8-85 Fast fission effect < (see Fast effect) Fast flux, absolute value of, 8-59 formula for, 1-27 Fast lifetime in moderators (table), 2-10 Fast neutrons, 8-57 to 8-72 cross-section measurement of (see Cross sec tions, measurement of) definition of, 8-53 detection of (see Counting) from H"(l,n)He«, 8-72 monitoring for, 7-40 radiation damage to covalent and organic mate rials by, 10-126 (See also Radiation damage) relaxation lengths for (table), 2-34, 8-35 sources of, 8-57 to 8-61 Fast reactors, constructional features of, 13-155 to 13-157 neutron spectrum in (tables), 2-30 to 2-32 (See aleo Power reactors; Reactors, particular) Fatal dose for chemical and nuclear poisons (table), 8-77 Fatigue, theory of, 10-1*) thermal. 9-115 Fatigue cracking, 10-76 Fatigue failures in reactors, htisards of, 8-79 Feather's rule, 7-5 Feed pumps, price of, 12-134 Fermi age, 8-45 equation for, 8-6 theory of, 8-54 (See alio Age; Physics of critical reactors) Fermi constant, 4-69 Fermi-Dirac statistics, 10-6 * See insert Revised Cross-section measurement

Data.

0 energy from, delayed, 2-4 cross sections for (eee Cross sections, fission) data for, summarized, 2-2 to 2-5 energy from (table), 2-2 to 2-5 direct conversion of, 12-2 to 12-4 energy distribution of (table), 2-2 energy equivalents of, 1-26 (See alio Fission products, heat generation

from) from, delayed, 2-4 prompt, 2-4 T rays from, delayed, 4-84 prompt, 4-84 y energy

table.

7-72

light particle emission in, 2-4 neutrons from, delayed (see Delayed neutrons) origin of, 4-84 prompt, formula, 2-2, 2-3, 7-78 spectrum of (table). 7-91

spontaneous, 4-87 rates of (table), 2-5 thresholds for, 2-8 theory of, 4-86 theory of, 4-82 to 4-88

Fission chambers (see Detectors) Fission detectors (see Detectors) Fission fragments, stopping power for, 4-63 Fission-neutron sources for experiments, 8-70 to 8-72

Fission products, 14-33 absorption cross section of (tables),

1-23,*

8-10, 2-1 1»

adsorption of, in homogeneous

uqueous reac tors, 18-71 to 13-73 chemistry of, 11-44 to 11-60 composition of, during decay, 11-28 to 11-31 cross section of, 1-23.* 2-10,* 2-11*

INDEX

16

decay data for (table), 7-16, 7-17 135-day irradiated, 11-25 by nuclides (tables), 11-28 to 11-31 decay formulas, 7-14, 7-15 decay schemes and nuclear data for (table). 11-12 to 11-24 delayed neutrons from (see Delayed neutrons) equilibrium concentration of nuclides in reactor,

Fission products,

11-11


in fast reactors, effect on neutron economy, 13-19 y rays from, 4-84, 7-72, 7-73 heat generation from, 135-day irradiated, 11-26, 11-27, 11-33 0 heat by nuclides (table), 11-32 7 heat by nuclides (table), 11-32 Unterniyer-Weills formula, 7-15, 7-17 table for, 1-39 Way-Wigner formula, 7-15 metallurgical effects of (table), 1O-102, 10-103 yields and atomic volumes (table), 10-103 nuclear data for, iodine- 135 (tables), 2-10, 2-11 (tables), 1-23.* 2-10,* 2-11* samarium-149 stable (tables), 1-23,* 2-11* xenon-135 (tables), 1-23, 2-11 poisoning of, effect on start-up, 8-27 to 8-29 radiation damage to coolant and organic mate rials by, 10-128 range of, 2-34 release of, 13-119 at Oak Ridge, 13-119 at Windscale, 13-119 spectrum of, during decay, 11-28 to 11-31 storage of, 8-87 Fission pulse chambers («ee Detectors) Fission recoil effects, 10-102 Fission yields (table). 11-9 to 11-24 curve for U1" 4-84 materials (see Fuel) Fissionable Fissionable nuclides, guide to nuclear data for, 1-4

Fissium, composition of, 10-175 uranium alloys of, 10-175, 10-176 uranium-plutonium alloys of, 10-175, 10-176 Fittings, pressure losses in (table), 9-36, 9-37 Fixed-field alternating gradient accelerators

(FFAG),

5-137

Fixed-point computing machine, 3-144 Flame recombiner, for homogeneous aqueous reac tors, 18-60 to 13-62, 13-68 to 13-70

Floating-point computing machine, 3-144 Flocculation of radioactive wastes, 11-145 Flow (see Fluid flow) Flow net for streamline flow, 9-31 Flowmeters, for air, calibration of, 18-116, 13-117 electromagnetic, for liquid metals. 13-93 to 13-95

for radioactive solutions.

11-117 to 11-120 Flowsheets, solvent extraction, 11-72 to 11-76 Fluid flow, 9-24 to 9-49 blowers for. 9-48, 9-49 in complex circuits, 9-38 distribution in parallel channels, 9-44 to 9-46 orifices and nozzles, 9-44 to 9-46 heat transfer in (see Heat transfer) 14-6, 14-7 measurement of, with radioisotopes, orificing of, general considerations. 2-17 past external objects, 9-46, 9-47 pressure loss (see Pressure loss in steady fluid flow) pumps for, 9-48, 9-49 of slurries, 13-76 to 18-78 similitude tests, 9-104 to 9-108 * See insert Revised Cross-section

Data.

Fluid flow, streamline, buoyancy effects in. 9-38 in irregular channels, 9-29 isothermal, 9-29 to 9-31 no niso thermal, 9-31 pressure loss in, 9-29 to 9-31 summary of, 1-50 to 1-53 theory of, 9-27 to 9-49 turbulent, 9-31 to 9-40 boundary layers in, 9-35 of compressible fluids, 9-34 numbers for, 9-32 dimensionless eddy diffusivity in. 9-34. 9-35 initiation of, 9-35 nonisothermal, 9-33. 9-34 velocity distribution in. 9-34, 9-35 two-phase, 9-40 to 9-43 stability of, 9-44 to 9-46 unsteady, 9-39. 9-40 units of, conversion factors for, 1-146 Fluid friction (see Pressure loss in steady fluid flow)

Fluidization of particulate solids. 9-47 Fluids, compressible, maximum velocity id (table), 9-43. 9-44 viscosity of, 9-25 to 9-27

Fluorine, physical properties of (tables), 11-5, 11-90

Flux (neutron), definition of, 8-54, 8-31 distribution of, around black absorbers, 6 6-108 (See also Physics of critical reactors) fast, absolute value of. 6-60 "' loading and power conversion factor for. ll 12-79 measurement of, 6-102 to 6-105 prompt jump. 1-48. 6-115, 8-9, 8-10 resonance to thermal ratio, measurement of. 6-81 to 6-83 thermal, relation to power. 6-59 two-group, absolute value of, 6-60 summary of formulas for, 1-36 Flux-power relations, formula for, 1-26. 1-27

for U"» (table), 1-27 Focus, definition of, 3-30 Fog flow, 9-41 to 9-44 maximum mass velocity for (table), 9-43. 9-44 half life), 4-60 ft, fr, value (comparative Foils, activity calculation of. 6-103 catcher, 6-105 counting of, 6-102 to 6-104 dysprosium, 6-102 for heterogeneous lattice, 8-120 gold, 6-102 indium, 6-102 in resonance- to-thermal flux measurement, to 6-83 cadmium, 5-81, 6-82 indium, 6-81, 6-82 saturated activity of, 6-103

Forbidden transitions, atomic. 4-13 nuclear, 4-69 Force, conversion factors for units of (table). 1-145 standard unit of. 1-140 Force field, scattering in, 6-28 Forced convection (see Convection, forced:) Foroes, atomic, 10-4 nuclear, 4-45 to 4-48 Ford Nuclear Reactor, description of. 13-133 Forward operator, 3-33 Fossil fuels. 12-84 to 12-88 (See also Power; World energy resources)

INDEX


Four-factor formula, 6-75 faat effect, 6-22 (See also Fast effect) of, 6-24, 6-25 inaccuracies numerical example of, 6-96 to 6-98 resonance escape probability, 6-22, 6-23 (See aho Resonance escape probability) thermal utilization, 6-23, 6-24 (See also Thermal utilization) Fourier- Bessel expansion, 3-106 Fourier constants, 3-102 Fourier integral theorem, 3-12W Fourier number, definition of, 9-32 Fourier series, 3-105 cosine, 3-105 sine, 3-105 Fourier theorem, 3-105 Fourier transforms, 3-94. 3-129, 3-130 finite, 3-131 table of, 3-130 Fractional distillation, 11-83 to 11-91 Fracture, by thermal stress, 9-114, 9-115 Fredholm equations, kernel of, 3-131 Fredholm method, 8-134 to 3-136 Free-atom cross sections, 6-99 Free convection (tee Convection, free) French power reactor program (table), 13-106 Fresnel integrals, 3-95 Fretting, 10-74 of materials (table). 10-80 Friction, coefficient of, in water (table), 10-79 effect of corrosion in water on, 10-79 fluid (see Pressure loss in steady fluid How) Frog suits, 7-119 Frost test, 10-153 Fuel (fuels), characteristics affecting selection of (table), 13-9, 13-10 classification of reactor by mobility of (table), 13-7, 13-8 consumption of, conversion factor for (table), 1-27, 13-78 depletion of, effect on startup, 8-27 diluents for, physical and mechanical properties of, 10-38 to 10-54 dispersions of, in metals (tables), 10-183 ta 10-185 fossil, 13-84 to 13-88 (See alto Power; World energy resources) handling of (see Fuel handling) management of, 13-82 metallurgical properties of, summarized, 1-4 mobile, general considerations, 13-17 to 13-18 nuclear properties of, summarized, 1-4 physical and mechanical properties of, 10-17 to 10-38, 10-168 to 10-185 processing of, by remote control, 7-124 radiation damage to (see Radiation damage) recycling of, cost of, 13-100 to 13-102 reprocessing of (see Chemical separation processes) reserves of (see World energy resource*) safe handling of, 7-55 storage of, for homogeneous reactors, 13-62 thermal stress properties of (table), 9-113 transportation of, criticality hazards in, 8-77 (See also Nuclear power) Fuel elements, 13-59 clad for (see particular material) design of, for DiO moderated reactors, 18-22 fabrication of, criticality hazards in, 3-77 by remote control, 7-124 failure of, monitors for, 18-11 heat conduction in (see Heat conduotion) materials for (general), 13-60 to 13-62

17

Fuel elements, radiation damage to, 13-66 to 13-68

requirements for liquid- metal-cooled

reactor*, 13-100 shapes for, 13-62 to 13-66 castings, 13-64 extrusions, 13-R4 pins, 13-62 plates, 13-62. 13-63 rods, 13-62 slugs, 13-62 tubes, 13-62 wire, 13-63, 13-64 storage of, under water, 13-174 temperature in, 9-85 to 9-88 thermal distortion of, 9-112 to 9-1 14 Fuel handling, 7-55, 7-56, 7-137 in canals, 18-32 for homogeneous aqueous reactorB, 13-62 from reactor, machines for, 7-137 to 7-139 in graphite reactors, 13-21 in water reactors, loading, 13-33 unloading, 18-33, 13-34 methods of, 7-117 to 7-125 Fuel rods, thermal distortion of, 9-112 to 9-1 14 Function theory, 3-75 to 8-95 Functions, 3-78 to 3-106 analytic, 3-82 to 8-84 approximate representation of, 3-32 to 8-35 of complex variable, 3-80 to 3-82 conformal mapping of, 8-85 continuity of, 3-79, 8-80 definitions of, 8-78 elementary symmetric, 8-42 infinite products, 8-97, 3-98 limits of, 3-79, 3-80 residues, 3-84, 8-85 Rieraann surfaces, 3-85, 8-86 series expansions, Bessel, S-106 convergence of, 8-95, 3-96 Fourier, 8-105 Legendre, 3-105 orthogonal, 3-101 to 3-106 power series (table), 8-98 to 8-101 Sturm-Liouville, 3-101 to 8-106 singularities, 3-84 special, 3-86 to 8-106 Bessel, 8-89 to 3-91 tables of, 1-94 to 1-116 beta, 3-87 cosine integrals, 3-95 elliptic integral, 3-95 error, 3-94 table of, 1-124 to 1-126 exponential integrals, 8-94 tables of, 1-119 to 1-122 Fresnel integrals, 8-95 gamma, 3-86 table of, 1-117 to 1-118 hypergeometric, 3-87 to 3-89 incomplete gamma, 3-94 Legendre, 3-91, 3-92 table of. 1-123 logarithmic, 3-94 orthogonal polynomials, 3-92, 3-93 polygamma, 3-87 sine integrals, 3-95 Fundamental set, 3-109 Fundamental standards of measurement, 1-139 Fundamental theorem, of algebra, 3-41 of convergence, 3-96 of integral calculus. 3-68 Fused salts, extraction by, 11-100 fuel purification by, 11-100

INDEX

18

10-131 fused salts, radiation damage to (table). to 10-134 tut reactor coolants, 12-57 comparative table. 12-59 Fuses for reactor safety, 8-81 Fusion. 4-88 to 4-90 latent heat of, of elements (table),

11-6 to 11-8

reacting

radiation decomposition

as reactor coolant, tS-57 Galvanic corrosion, 11-123, 11-124 »■ Gamma function, 3-86, 3-87


*\

incomplete, 3-44 table of, 1-117, 1-118 ,-v (y,n) reactions, 4-91 y rays, absorption by air, 7-6 attenuation of (see Shielding) counting of {see Counting) detection of. 5-35, 6-36 (See also Detectors) dose from, 2-35 emission of, theory, 4-70 from fission, delayed, 4-84 prompt. 4-84, 7-72 for gauging, 10-162 to 10-164 handling emitters of (see Radioactive materials, handling of) heating by (see Shielding) for inspection, 10-154 to 10-157 ionization by, 7-6 radiation damage to covalent and organic ma

'X

terials by, 10-127 2-34, 2-35 relaxation lengths for (table), sources of, 7-72 to 7-78 activated materials (table). 7-74 to 7-76 capture y rays, 7-77 fission (table), 7-72 decay-energy groups, 7-73 fission-product inelastic scattering, 7-77 neutron capture (table), 7-77 spectroscopy, 6-36 to 6-40 suppression of (table), 7-77 (See also X rays) , shielding (see Shielding) Garaow factor, 4-33 Garabedian, H. L., 8-16 Gas amplification factor, 5-13 Gas centrifuge for isotope separation, 14-23 of, 1-154 Gas constant, universal, value Gas radiation of heat, 1-59. 1-60 recombiners, for homogeneous aqueous reac

Gas

tors, 13-68 to 13-70 systems for, 13-60 to 13-62 Cias separators for homogeneous aqueous reactors. 13- 66, 13-67 Gas systems, auxiliary, for water reactors. 13-32 Gas turbine, efficiencies of, 13-113 Gas turbine cycle, 12-3 to 12-5 of, 14-42, Gaseous diffusion process, advantages 14- 43 cascade, ideal, 14-40 practical, 14-40

process,

cascade, principle

of.

14-38. 14-39 enrichment equation of, 14-40 14-40 stage separation factor of. 14-39. disposal system for homo Gaseous fission- product geneous aqueous reactors. 13-62 r <"■ as reactor coolants, comparative

Gases

12-60 effectiveness

per 100 ev, 10-131 factor in homogeneous reactors, 13-58, 13-59 G-l, G-2, G-3 (see Reactors, particular) Gadolinium, as control material (table), 10-188 density of, 11-5 (Jailing, 10-74, 10-75 Gallium, chemistry of fission product. 11-45 corrosion by, 10-21 physical properties of (table), 11-5 liquid (table), 9-18

G, molecules

Gaseous diffusion

indexes, 13-121

radioactivity in, 7-19. 7-20 viscosity of (formula). 9-25 Gaskets, radiation damage to (table),

10-144 to

10-147 Gauging, 10-162 to 10-166 by autoradiography, 10-165 by back scattering. 10-164 to 10-164 beta emitters for (table), 10-162

10-166 eddy-current, electrical resistance, 10-166 10-162 to 10-164 gamma emitters for (table), magnetic, 10-165, 10-166 by Magne-Gage. 10-165. 10-166 14-3 to 14 by radiation absorption (table). radioisotopes for (tables), 14-3 to 14-6 by reflection of radiation, 10-165

ultrasonic. 10-165 by X-ray fluorescence, 10-165 X rays for. 10-162 to 10-165 3-123 Gauss-Seidel method, 3-39. Gauss theorem, 3-126 3-61 Gaussian density function. elimination, 3-37 Câussian integration, double, 6-13 Gaussian slowing down. 6-54 Qaut*sian for. in high-temperature wad Gears , materials to 10-82 (taMes), 10-80 3-93 GegenbauCJCk.P°1ynomiala' Detectors. , Geiger- M Oiler Geiger counted («« counters) ^ *"66

Geiger-Nuttall nh(&

Geiger plateau. 5-1.^ Tes»t Reactor. General Electric Nuc"iear tion of, 13-135 al. definition of. 12>9i Generating unit, electric* 'ctor «*fety. 0-87 Geology in relation to re:. ••20 buckling. 5-11>m8< Geometrical '■ . Geometry. 3-2 to 8-62 product 11-4* Germanium, chemistry of fissicsj" H-5 physical properties of (tablevi Giorgi system of units, 1-138. 1-lV "I9 lO-l-'-i Glass, cerium in. for radiation resisjtt*nce> 10-f.M effect of radiation on (table), 7-l^fc?shielding properties of, 7-122 GLEEP, description of, 13-124 Glove boxes, 7-125 to 7-127 procedure. 7-127 glove-replacement lock, 7-123 with interconnecting pouch, transfer procedure, 7-127 GM counters (see Detectors. Geiger-Muller Godiva (see Reactors, particular) equation. 6-7 Goertzel-Selengut Gold, as control material (table), 10-189 physical properties of (table), 11-5 Gold-198 in medical therapy, 14-11 Graeffe's method. 3-48 to 3-50 illustrative example of, 1-64 to 1-66 simplified description of. 1-63, 1-64 | Graetz number, definition of, 9-32 ' Grain boundaries, effect on deformation. 10-15 Gram-rad, definition of, 7-24 Gram-rep, definition of. 7-24 definition of, 7-24 Gram-roentgen, Graphical solution for heat flow. 9-98. t-W*

\

INDEX


Graphite, 12-70. 12-72 to 11-74 boiling point of, 11-5 corrosion of (tables), 10-59, 12-7 J to 11-74 by liquid metals, 10-21 by sodium. 11-72 to 11-74 coat of (table). 11-70 heat of fusion of, 11-5 kinds of (table), 10-110 machining and stacking of, 18-180, 13-181 mechanical properties of (tables), 10-5(1 to 10-59 melting point of, 11-5 moderating properties of (tables), 1-9, 11-70 nuclear properties of (table), 1-9 physical properties of (tables), 10-54. 10-56. 10-57, 11-70 price of. 11-107 radiation damage to, 10-109 to 10 ! in stacking of, 13-141. 13-142. 13-180, 13-181 thermal conductivity of, directional (table). 10-110 thermal stress properties of (table). 9-1 M uranium impregnation in. 10-185 for water reuctors, 13-31 Graphite- mod era ted reactors (see Power reactors; Research reactors) Graphite-uranium dioxide (graphite-UOt), 10-21 radiation damage to (tables) 10-121, 10-122 Graphite-UiO* bodies, impregnated, effect of irra diation (table), 10-12 Grashoff number, definition of, 9-32 Gray boundaries. 3-37 Great Britain, gas-cooled power reactor program in (table). 13-105 Greek alphabet, 1-11 Green's theorem, 3-126 Greuling equation, 6-9 Grid ionization chamber. 3-11 Groeber charts, 9-94 Ground state, atomic, 4-12 Grouts for shielding (table), 7-1 13 Growth, radiation (nee Radiation damage; constant, 10-10 Gruneisen's Guard ring, 3-10 Guard tut>es in proportional counters. 6-14 Guides, materials for. in high-temperature water (table), 10-80. 10-82 Gurney-Lurie charts. 9-94 11

// particles (hyperons), 4-58 H'U.nJHe* reaction, fast ncutroiut from. 3-72 Hafnium, boiling point of, 11-5 as control material (table). 10-189

heat of fusion of, 11-5 price of, 11-107 properties of (tables). 10-70 thermal stress properties of (table). 9-1 13 * Half-life, 4-34. 7-4 of activation products (table, BNL 326), 1-15 to 1-23 biological, 7-41 of 0 decay, 4-00 comparative, effective in body, 7-41 of naturally radioactive nuclides (table. BNL 325), 1-15 to 1-23 Half-thickness of materials. 1-35. 11-1341 conversion factors for, 1-44 Half-thickness value (HTV). definition of. 11-69 Halide leak test, 10-150 Halogen compounds, physical properties of (table), 11-90 Ilamjltonian. //, of particle system. 4-20

19

Hamiltonian function, 3-139 Hamilton's equation. 3-139 Hamilton's principle. 3-139 Handling of radioactive materials (•« Fuel han dling; Radioactive materials) Hanford reactors, fuel handling in, 7-137 Hanford 305 test reactor, description of. 13-125 Hankel functions, 3-89 Hankel transforms, 3-89 Hardening of reactor neutron spectrum, 3-25 in absorption cross-section measurement, 3-95. 5-90 Hardness, effect of, on wear in water. 10-18 Harmonic function, 3-83 Harnack's theorem, 8-116 Hastelloy (see. High-temperature alloys) Haynes alloys (see High-terapcrature alloys) Hazards, reactor (see Safeguards) Health physics, 7-22 to 7-59 beryllium handling, 7-57 decontamination, 7-48 to 7-50 effluent disposal, 7-50 to 7-55 7-55, 7-56 fuel handling, injury, cause of, 7-22, 7-23 of, 7-27 extent of radio maximum permissible concentration active materials, in air. general value 7-55, 11-141 (tables). particular radioisotopes (table), 11-141 water, general value (tables), 7-44, 11-141 in particular radioisotopes (table), 11-141 maximum permissible dose (radiation), exter nal, 7-29 to 7-36 internal. 7-38 to 7-45 rate for neutrons (table). 7-84 summary, 1-42, 1-43 monitoring, 7-45 to 7-48 natural background. 7-36 to 7-37 shipping of radioactive materials, 7-55 toxicity, comparison of nuclear and chemical poisons, 8-77 7-23 to 7-25 units of measurement, summary table, 1-42 waste disposal, 7-50 to 7-55 (See also Dose; Dose rate) Heat, conversion factors for units of (table) . 1- 147 from fission products (see Fission products) of fusion, of elements (table). 11-5 to 11-7 specific (see Specific heat) standard units of (table), 1-140. 1-141 Heat capacity, conversion factors for units of (table), 1-151 conversion factors for unit* of Heat conductance, (table). 1-151 Heat conduction, in cylinders, 9-56 to 9-57 clad fuel elements, simple case (table), 1-59 homogeneous (summary table), 1-55, 1-56 multilayer, with surface-film coefficients (summary table), 1-55, 1-58 differential equation of, 3-U7 relations of, 9-52 to 9-53 fundamental in slabs or plates, 9-54 to 9-56 clad fuel elements, simple cases (table), 1-59 homogeneous (summary table), 1-54 multilayer, with surface-film coefficients (summary table), 1-55, 1-58 in spheres, 9-57 homogeneous (summary table). 1-57 multilayer, with surface-film coefficients (summary table), 1-55, 1-58 Heat conductivity, conversion factors for units of (table), 1-151 effect of radiation on (see Radiation damage) of the elements (table), 11-5 to 11-7

INDEX

20

Heat conductivity, of gases (Sutherland formula). 9-53 of metal*, 9-53, 9-54 effect of radiation on, 10-100 uf packed beds (table), 9-54 theory of. 10-11, 10-12 of two-phase mixtures, 9-54 Heat convection (see Convection) Heat exchangers, 18-67 for alkali-metal reactors, 18-91. 13-92 for water reactors. 18-27, 18-28 (See alao Nuclear power, condensers for) Heat flow, simulated tests, 9-104 to 9-108 steady-state, electric analogs for, 9-90 to 9-103 numerical solutions for, 9-95, 9-96 unsteady-state, 9-93 to 9-103 analytic solutions for, 9-93 to 9-95 electric analogs for, 9-101 to 9-103 graphical solution for, 9-98, 9-99 numerical solutions for, 9-95 to 9-98 (See alao Boiling; Condensation of vapors; Con vection; Heat conduction; Heat gener ation) Heat flux, conversion factors for units uf (table), 1-151 Heat generation, distribution of, in reactors, 9-51. 9-52 in fluids, 9-88, 9-89 by >-ray attenuation (see Shielding, >-ray


attenuation)

internal, conversion factors for units of (table), 1-151 by neutron attenuation (see Shielding, neutron

attenuation)

Heat radiation, theory of, 4-4, 4-5 by gases, 1-59, 1-60 by surfaces, 1-59 definition of, 12-91 Heat rate (turbine-generator), Heat removal from reactor, 12-13 to 12-17 flow orificing, 12-17 heat transfer to coolant, 12-14, 12-15 heat transport by coolant, 12-14, 12-15 heat transmission through fuel elements, 12-13, 12-14 hot-spot factors, 12-16, 12-17 Heat transfer, by boiling, time delay in bubble formation, 18-47, 18-48 12-14 to coolant, general considerations, effectiveness of gases (table), 18-121 from slurries, 18-76 to 18-78 Heat- transfer coefficient, 9-57 over-all, 1-60 (See alao Boiling; Condensation of vapors;

Convection)

in fuel elements, general con Heat transmission siderations, 12-13, 12-14 coolant, general considerations, Heat transport, by 12-14, 12-15 heat generation in fluid. 9-88, 9-89 temperature rise of coolant, 9-84, 9-85 Heating by y- and neutron attenuation (aet Shielding) Heavy concrete (see Concrete) Heavy isotopes, formation of (table), 11-8, 11-9 Heavy-particle linear accelerators, 5-128, 5-129 Heavy particles, radiation damage by, to covalent and organic materials, 10-128 Heavy water, concentration of, in water, 8-27 critical properties of (table), 9-15 density of, at saturation (table), 9-16 dynamic viscosity of (table), 9-17 effect of isotopic purity on nuclear properties of, 1-19

latent heat of vaporisation of (table),

9-16

Heavy water, moderating

properties of (tables), 2-9, 12-70 effect of H«0 on, 1-19 nuclear properties of (table), 2-0 effect of HiO on, 1-19 price of, 12-107 as reactor coolant, 12-56 comparative table, 12-58 specific heat of (table), 9-17 thermal conductivity of (table), 9-17 vapor pressure of (table), 9-15 Heavy- water- moderated reactors, constructional features of, 13-142, 18-144 to 18-149 Heine-Borel theorem, 8-78 Helium, physical properties of (tables). 9-5, 11-5 as reactor coolant. 18-122 comparative tables, 12-58, 12-60 radioactivity of, 7-20 Helium-cycle turbocom pressor power plant. 18-122 Helium leak test, 10-150 to 10-152, 18-184 Holm hoi ti equation, 3-117 harmonics, 6-12 Hemispherical Henry's law, 11-83 Hcrmite polynomials. 8-93 Hermitian, 8-53 Hermitian kernel. 3-136 Heterogeneous reactor classification (table). 12-v 12-7 HETS, 11-66 Hexagonal close-packed (hep) structure, 10-6 High-density concretes for shielding (•«- Concrete? High-energy nuclear reactions. 4-82 High-temperature alloys, 10-63 to 10-06 composition of (table), 10-64 corrosion resistance of, to liquid metals, 10-21 to sodium and sodium potassium. 10-20 high-temperature properties of (tables). 10-H3 10-66 mechanical properties of (table). 10-66 physical properties of (table). 10-66 Hilbert-Schmidt theory. 3-133, 3-136 Holdback carrier, 11-61 Hole theory of positron, 4-56 Homogeneous reactor experiment {*ee Reactors

particular. HRE)

Homogeneous

reactor

ticular, HRT)

teat (see Reactors, par

Homogeneous reactors, boiling in. 9-83 to 9-84 aqueous (ace Power reactors, homogeneous aqueous) classification of (table), 12-6, 1S-7 containment for, 8-84, 0-85 liquid-metal (see Power reactors, heavy -metal safety aspects of, 8-80 temperature distribution in, 9-88, 9-80 Homogeneous slurry reactors, 12-46 Horner's method, 8-46 illustrative example of, 1-66 to 1-68 Hot caves, 7-128, 7-129 Hot cells, 7-128, 7-129 viewing in, 7-121 to 7-123 Hot channel, safety aspects of, 8-83 Hot-channel factors (table). 9-86 to 9-88 Hot laboratories, design of, 7-48 to 7-50 Hot-spot factors, general considerations. IS- 16 12-17 Hot spots. 9-87. 9-93 in gas-cooled reactors, 13-107, 13-108 Hottell charts, 9-94

HRE HRT HTV

(see Reactors, particular) (see Reactors, particular) (see Half-thickness value) Hunters ton reactors, 18-105

INDEX Hurwiti, H. Jr.,

and G. M. Roe, 8-16 Hydraulic- piston control-rod drive, 8-76 Hydraulic radius, 9-33 Hydroduct, 11-4 Hydrogen, cross section at high energy, 7-81 physical properties of (tables), 9-5, 9-6, 11-6 slowing-down integral equation, 6-6 Hydrogen-3 (see Tritium) Hydrogen atom, Bohr, 4-12 mass and energy equivalents of (table), 1-155 relativistio fine structure of, 4-25 Hydrogen fluoride, physical properties of (table), 11-90 Hydrogen spectrum, 4-11 fine structure of, 4-12 Hydrology in relation to reactor safety, 8-86, 8-87 Hydrostatic test, 10-150 Hyperbola, 3-30, 3-31 Hyperbolic functions, identities, 3-20 tables, of cosh x and sinh x, 1-90 to 1-92 of tanh x, 1-93 Hyperboloid, definition of, 3-30 Hyper geometric functions, 3-88 Hyperons (H particles), 4-58 Hyperplane, equation for, normal form of, 3-26 Hyperquadrics, 3-28 to 8-30 HYPO, 13-129


I Identity matrix, 3-10 Identity theorem, 3-83 Illinois Institute of Technology reactor (see Heactors, particular, ARR) Illumination of hot cells, 7-121 Impedance, dynamic, of chambers for reactor con trol, 8-66, 8-67 Impregnated graphite, 10-185 Incineration of radioactive wastes, 11-145 Inco (see High-temperature alloys) Incoloy (tee High-temperature alloys) Inoonel, physical and mechanical properties of, 10-19 (See alto High-temperature alloys) Indirect cycle, 13-6 Indium, chemistry of fission product, 11-45 as control material (table), 10-88 foils of. 8-81 to 8-83, 6-102 physical properties of (table), 11-6 Industrial Committee on Reactor Location Prob lems, 8-76 Inelastic scattering, 4-99, 4-100 cross sections (see Cross sections) definition of, 4-72 description of, 4-72 y rays from, 7-77 by lattice vibration, 5-99 measurement of (see Cross sections) Inelastic transfer cross sections for nuclides (table), 3-33 Infinite multiplication constant (see Multiplica tion factor) Infinite products, 3-97, 8-98 Inflection of curve, 3-43 Inhour, conversion factors for units of, 1-46, 8-1 13, 8-9 Inhour equation, 1-47, 8-113, 8-6 (See also Period) Inlet loss (see Pressure loss in steady fluid flow) Inner product, 3-15, 8-101 In-pile measurements, 10-92, 10-93 In-pile tests, 10-92, 10-93 cycling tests, 6-141 loops, 6-142 to 6-145

21

In-pile tests, static, 6-141 (See also Irradiation testing) Inspection (see Gauging; Testing, nondestructive) Instability, from reactivity coefficients, 8-80 of two-phase

flow, 9-44

Instrument chains, instrument channels (see In strumentation, nuclear, for reactor control) Instrumentation, for air, flowmeters, calibration of, 13-116, 13-117 for alkali metals, flowmeters, electromagnetic. 13-93 to 13-95 orifice, 13-93 liquid-level measurement, 13-95. 13-96 pressure measurement, 13-93 for critical assemblies, 6-109 non-nuclear, for steam power plants. 13-151 to 18-153 nuclear, for reactor control, 8-50 to 8-67 amplifiers, 5-56 to 6-61 channels, 8-50, 8-53. 8-57 to 8-61 counter range, 6-50, 8-53 counting rate meter, 8-57 detectors (see Detectors) displays, 8-57 period range, 8-50 power range, 8-50 for radioactive solutions, 11-114 to 11-122 flow measurement, 11-117 to 11-120 liquid-level measurement, 11-120 to 11-122 pressure measurement, 11-116. 11-117 temperature measurement, 11-115, 11-116 for water reactors, monitoring, 13-34 power level, 13-35 water level, 13-35 Instruments, radiation- measuring (see Detectors) Insulation, electrical, for in-pile measurements, 10-93 radiation damage to (table), 10-144, 10-145 for reactor building, 13-170 for reactor vessel, 13-179 Insulator materials for ionization chambers and electrometers, 5-10 Insurance against nuclear hazards, 13-110, 13-11 1 Integers, 3-2 Integral equations, linear, 3-131 to 3-137 Integral transforms, 3-127 to 3-131 Integrals (tables), 3-68 to 3-74 complete, 3-112 definite, 3-67 improper. 3-67 indefinite, 3-67 particular, 8-112 properties of (tables), 3-68 to 3-74 singular, 3-113 Integration, 3-66 to 3-75 numerical, 3-74, 3-75 Simpson's rule, 1-28 to 1-29 Integration factor, 3-108 Intercom system, reactor, 8-83 Interface measurements, 11-120 to 11-122 Intergranular corrosion, 11-123 to 11-125 Interior point (in set theory), 3-77 Interlocks, 8-82 neutrons, definition of, 5-53 Intermediate-energy (See also Resonance neutrons) Internal conversion (IC), 4-70 elementary description of, 7-4 Internal dose (radiation) (see Dose) Interpolation, 3-33 Aitkin's method of, 3-34 polynomial, 3-32 Interstitial, activation energy for movement of, 10-104 Interstitialcy vacancy pairs, 10-101, 10-102

INDEX

22


inventory charges on nuclear fuel, 12-106, 12-109 Inverse, definition of, 3-1 0 Inverse trigonometric functions, S-20 Inversion integral, 3-129 Inverted control rod, $-72, 8-73 Inverted plastics, viscosity of, 9-26, 9-27 Investment costs of power plants (see Nuclear power; Power, from fossil fuel) Iodine, chemistry of fission product, 11-48 physical properties of (table), 11-6 Iodine- 131, medical application of, 14-10, 14-11 production of, 14-32 Iodine- 135 as precursor of Xe1!i, 2-10 Ion-chamber electronic meters (see Electrometers) Ion chambers (see Detectors) Ion current chambers (see Detectors) Ion exchange, 11-91 to 11-97 dual bed in, 11-95 in fuel reprocessing, 11-96 resin performance of, 11-90, 11-97 mixed bed in, 11-95 analysis, 11-52 in radiochemical resins for (table), 11-93 theory of. 11-91, 11-92 for waste disposal, 11-145 Ion pair, energy to produce, 7-22 Ionic crystals, 10-105 radiation damage to, 10-105, 10-123 Ionic resins, 11-93 Ionization, by a particles, 5-4, 7-5 by 0 rays. 7-5 as cause of injury, 7-22 by charged particles, 5-2 to 6-4 mathematics of, 5-2 to 5-4 by -y rays, 7-6 by neutrons, 7-5 by particles, 7-5 to 7-7 efficiency of, 7-22 by photons, 5-4, 6-5 by positrons, 7-5 by protons, 5-4 specific, 5-2 to 5-4, 7-22 for particles in air (table), 7-34 formulas for, 7-33 by X rays, 7-6 Ionization chamber (see Detectors) Ionization energy, 5-3 Iridium, as control material (table), 10-189 isotopic separation of, 10-57 physical properties of (table), 11-6 Iridium-192 for radiography, 14-3 Iron, isotopic separation of, 14-57 properties of (table), 9-113 thermal-stress Iron ore for biological shields, procurement and storage of, 13-182 Irradiation, methods of, in production of radio isotopes, 14-20. 14-27 Irradiation testing, a particles for. 10-96, 10-97 cyclotron particles, displacements by. 10-100. 10-101 cyclotrons for, 10-90 to 10-99 dose measurement from, 10-96. 10-97 10-98 at high temperatures, at low temperatures, 10-97, 10-98 particles from, 10-96 to 10-101 deuterons for, 10-96, 10-97 electrons for, 10-99, 10-100 facilities for, 10-83 to 10-92, 10-96 to 10-99 in-pile measurements, 10-92, 10-93 10-93 of displacement, of dose (radiation), 10-92 of electrical resistance, 10-93 of pressure, 10-93 reactor interlocks for, 10-95 * See insert Revised Cross-section Data.

Irradiation testing, in-pile measurements, of strain, 10-93 of temperature,

10-92

10-93 to 10-95 postirradiation measurements, activity of materials in (table), 10-94 machining for, 10-95 of mechanical properties, 10-95 of metallographic properties, 10-93. 10-94 shielding for (table), 10-93, 10-94 protons for, 10-96, 10-97, 10-100 reactors for, 10-83 to 10-92 Armour research reactor, 10-91. 10-92 BNL (table), 10-89, 10-90 CP-5 (table), 10-91 LITR (tables). 10-84, 10-88 MTR (tables). 10-83 to 10-88 NRX (table), 10-90, 10-91 X-10 (table). 10-84. 10-89 specimens for, 10-92 Van de Graaff machine for, 10-99 Irrationals, 3-2 Isobars, 4-34 Isolation area, 12-19 Isomeric transition, elementary description of. 11-8 Isomerism, nuclear. 4-70 Isomers, 4-70 Isoperimetric problems, 3-140 Isothermal How, streamline, 9-29 to 9-31 (See also Convection; Fluid flow, streamline. Pressure loss in steady fluid flow) Isotope separation, 14-38 to 14-43 centrifuge process of. 14-42 process of, 14-15 to 14 - chemical-exchange comparison of processes of, 14-42 deuterium, 14-28 distillation process of, 14-23 electromagnetic process of, 14-44 to 14-58 gas centrifuge process of, 14-23 to 14-24 gaseous diffusion process of, 14-38 to 14-43 methods of (table), 14-16 thermal diffusion process of, 14-23, 14-42 '■Seealso Calutron; Chemical-exchange :*--*■: separation; Gaseous diffusion reparation process) Isotopes, definition of, 4-33 dilution methods of analysis, ll-5ti heavy, formation of (table), 11-8, 11-y stable, availability of (table). 14-54 to 14-57 sources of. 14-24. 14-25 uses of, 14-57, 14-58 (See also Radioisotopes) Isotopic abundance, 4-35 to 4-38 of the elements (table BNL 325), 2-15 to 1-23* Isotopic analysis, 14-54, 14-55 Iteration, 3-36 for solution of equations, 3-46 partial differential, 3-123 .1

j

factor, definition of, 9-32 Jacobi polynomials, 3-93 Jacobian matrix, 3-50

JEEP,

description of, 13-127 sectional drawings of, 13-149. 13-150 Jongenburger, 10-103 Jordan normal form of matrix, 3-52 Jump, prompt, 1-48. 6-115. 8-9. 8-10

K k,

k infinity, calculation of (•*« Four-factsr for inula)

I

INDEX

K capture

(electron capture), elementary tion of, 7-3, 11-5 particles, 4-58

K

Kernels,

descrip

6-4

attenuation, 7-93 (See also Shielding, geometric diffusion, $-5 table,

configuration*)

6-31*

slowing -down, 6-4, 6-46 synthetic, 6-4. 6-56, 6-57 table of, 6-46 transforms (table), 6-56 Kinetics, reactor (see Physics of noncritical reactors)

Kirchhoff's law applied to fluid flow, 9-38 Krypton, chemistry of fission product, 11-45 physical properties of (table), 11-7

kT

neutrons,

4-69

Kurie plot, 6-33


L (laboratory)

system,

4-30

Laboratories, hot, design of. 7-48 to 7-50 Lagrange equations of motion, S-13tt Lagrange function, 3-139 Lagrange identity, 3-110 Lagrange linear equation, 3-112 Laguerre polynomials, 3-93 Laminar flow (see Convection; Fluid flow, stream line; Pressure loss in steady flow) LAMPRE, Los Alamos Molten Plutonium Reac tor Experiment, brief description of, 13-33 Land, cost of, for nuclear power plants, 18-96 Lanthanum, physical properties of (table), 11-6 in separation processes, 11-63 to 11-65 Laplace and related equations, 3-83, 3-117 Laplace transforms, 3-94, 3-121. 3-122, 3-127 to 3-129 table of, 3-128

Laplacian in various coordinate systems, 3-125. 3-126

LAPRE I, II,

Los Alamos Power Reactor Experi

ment. 13-130, 13-131 Latent heat of fusion of elements (table), 11-5 to 11-7 Latent images, eradication of, 6-22 Latent vector, 3-52 definition of. 3-29 Lattice arrangement (sec Lumping) Lattice stresses around fission product atoms. 10-102 Lattice vibrations and inelastic scattering, 6-99 Latus rectum, definition of, 8-31 Laurents series, 3-83 Ixsad. corrosion by, 10-21 liquid, physical properties of (tables), t-19. 13-81 physical properties of (table). 11-6 resistance of materials to corrosion by (table), 13-99 thermal stress properties of (table). 9-113 Lead-bismuth alloy, eutectic, physical properties of (table), 9-19 liquid, corrosion by, 10-21, 18-99 physical properties of (table), 10-81 as reactor coolant (comparative table), 18-58 Leak detection in homogeneous reactors. 18-74. 18-75 Leak testa, 10-150 to 10-152 Leakage, from DiO reactors. 13-39 leak detectors for, 13-32 of neutrons (see Neutron leakage)

•>n rate from reactor buildings. Armour In stitute, 18-165

Leakage

CP 5. 13-164 EBWR, 13-170

Least squares, method of, 8-36 Legendre functions, 8-91, 8-92 Legendre polynomials, table of, 1-123 Legendre

series. 3-105

Length, conversion factors for units of, 1-143 standard unit of, 1-139 LET (Linear energy transfer), 7-26, 7-27 Lethal dose of chemical and nuclear poisons (table),

8-77..

Lethargy, 6-43 definition of, 6-8 Leukemia caused by radiation, 7-28 Level parameter, resonance, 6-77 Level width in resonance neutron absorption, 5-78 License, reactor construction, 8-76 Licensing of radioisotopes, 14-12, 14-13 LIDO, description of, 13-133 Life, lifetime (see Neutron lifetime) Life span reduced by radiation, 7-28 Light, absorption and emission of, 4-19 Light water (see Water) Limit point, 3-77 Limits of functions, 3-79, 3-80 Line, forms of equations for, 3-26 to 3-28 Linear A-C pump, 18-89. 18-90 Linear accelerators, 6-127 to 6-129 Linear amplifier in reactor control, 8-56 Linear attenuation coefficient (see Shielding,

7-ray attenuation) Linear combination, 3-7 Linear dependence, 3-8, 8-13 vectors,

3-7

Linear energy transfer (LET), 7-26, T-27 Linear equations, systems of, 8-35 to 8-41 Linear independence, 8-8 Linear integral equations, 8-131 to 8-137 Linear preamplifier (in reactor control). 8-56 Linearly independent functions, 8-102 Liouville normal form, 3-104 Liouville's theorem, 3-84 Liquid-drop model, 4-48 of fission,

4-87, 4-88

Liquid-fuel fast reactors (see Power reactors) Liquid-level measurement in radioactive liquids, 11-120 to 11-122

Liquid-metal coolants, comparative table of, 13-56. 18-58, 13-59

physical properties of (table), 18-81, 18-82 Liquid-metal fuel as heat-transfer medium, com parative table, 18-59 Liquid-metal homogeneous reactors (ere Reactors, particular. LAMPRE, LMFR) Liquid-metal reactor, submerged oharging and discharging, 7-139 Liquid-vapor mixtures, pressure loss in flow of (table)

9-40 to 9-43

Liquid waste (see Radioactive wastes) Liquids, radiation damage to (see Radiation dam age)

viscosity of, 9-26 LITR (see Reactors, particular) Lithium, as control material (table),

10-188,

10-191

isotopic separation of, 14-56 liquid, physical properties of (tables), 9-19, 13-81 as neutron absorber, 8-34 neutron shielding properties of, 7-77 physical properties of (table), 11-6

INDEX

24 Lithium, resistance of materials

to corrosion by 13-83 LiT(p.n)Be' reactions, 5-59 Litbium-7 as reactor coolant, 16-57 table, 12-59 comparative Li verm ore water boiler, description of, IS- 129 LMFR reactor, 12-47 to 12-51. 12-55 Load factor, definition of, 12-91 Loading, dished. 18-106 flux, and power conversion factors, 12-78, 12-79 Loading machine, reactor (see Fuel handling) Local boiling (see Boiling, heat transfer by) Location of reactor, safety considerations of, 8-85 to 8-87 Loci, 3-24 to 3-35 Logarithm, Briggsian, definition of, 3-4 common, table of, 1-76, 1-77 definition of, 3-4 natural (Naperian) definition of, 3-4 table of, 1-70 to 1-80 Logarithmic amplifier for reactor control, 8-60 Logarithmic energy decrement, 6-43 Logarithmic integral, 3-94 Long counter, 5-49, 5-68 Loops, in-pile-testing, 6-142 Lorentz contraction, 4-7 Lorentz transformation, 4-7 Los Alamos fast reactor "Clementine," descrip tion of, 13-98 Loschmidt number, value of, 1-154 Low Intensity Test Reactor (tee Reactors, partic ular, LITR) Lower triangular matrix, 3-37 Lubricants, radiation damage to (table), 10-139 Lumped absorbers, 6-105 to 6-110 in infinite medium, 6-105 to 6-110 in regular array, 6-108, 6-109 Lumped resonance absorbers, 6-20 Lumping, effect on conversion, 12-10, 12-11 effect on reactivity, 12-10, 12-11 Lutetium, physical properties of (table), 11-6


(table),

M McCabo-Thiele method, 11-86 Mach number, definition of, 9-32 Machining of radioactive specimens, Maclaurin's series, 3-83 Macroscopic cross sections, 1-31

10-95

of mixtures, 6-32 of selected elements (table), 1-24, 1-25 Magic numbers, 4-37, 4-50, 4-61 10-165, 10-166 Magne-Gage, Magnesium, corrosion of, by sodium and sodiumpotassium, 10-20 liquid, physical properties of (table), 9-20 mechanical properties of (table), 10-18 physical properties of (tables), 10-18, 11-6 for water reactors, 18-31 Magnesium oxide, corrosion of, by liquid metals, 10-20, 10-21 Magnetic accelerators (tee Accelerators, types of) Magnetic amplifiers, in reactor control, 8-40 safety circuits, 8-44, 8-45 Magnetic analysis of electrons, 6-39 Magnetic field, acceleration in, 6-124, 6-125. 14-46 Magnetic gauging, 10-165, 10-166 Magnetic-jack control-rod drive, 8-75 Magnetic moment, 4-23 of complex nuclei, 4-50 of deuteron, 4-50 of neutron, 4-49, 4-50

Magnetic moment, of proton,

of nuclei,

4-49

4-49

Magnetic quantum number. 4-23 Magnetic spectrometer, for a particles.

6-90 for 0 particles, 6-34, 6-35 for 7 rays, 6-39 Magnetic spectroscopy of particles, 6-30. 6-34 Magneton. Bohr, 4-23 nuclear, 4-43 Maintenance, of fuel reprocessing plants, 11-137. 11-138 in homogeneous aqueous reactors. 13-74, 13-75 of reactor, safety aspects of, 6-83, 6-64 Major axis of ellipse, 3-32 Man, standard, 7-38, 7-39 Manganese, physical properties of (table). 11-6 Manipulators, 7-120, 7-121, 7-130 to 7-134 ball-joint, 7-121 castle type of, 7-121 for handling radiactive material, 7-120. 7-121, 7-130 to 7-134 master-slave, 7-120, 7-130. 7-131 rectilinear electric, 7-120, 7-121 Mantissa of logarithm, 3-4 Manual control of reactors, 8-29, 8-32 Manual discharging of fuel into canal. 7-137 Many-body problem, 4-25 Schrodinger equation for, 4-26 Mascoule (eee Reactors, particular, G-l, G-2, G-3) Maritime reactors, 12-11 Martensitic steels (see Stainless steels) Mass, conversion factors for unite of (table). 1-14-5 and energy, 4-9, 4-40 energy equivalent of (table), 1-155 nuclear, semi-empirical formula for, 4-4S, 4-49 standard unit of, 1-139 Mass attenuation coefficients, 7-64 table of, 7-63 Mass defect, 4-40 Mass number, definition of, 4-34 (historical), 4-34 Mass spectrograph Mass spectrometer, for cross-section measure ment, 6-94 for leak testing, 10-150 to 10-152 in reactor systems, 13-184 Mass spectrometric analysis of isotopes. 14—53 14-54 Mass stopping power, 7-25 Mass transfer, 9-103, 9-104 Master-slave manipulator, 7-120. 7-130. 7-131 Master-slave servo manipulator, 7-131 Material buckling. 6-20 numerical example of calculation of. 6-99 Materials, for fuel reprocessing equipment ' tabled . 11-128 to 11-131 for homogeneous aqueous reactors, 13-57, 13-5$ neutron-absorbing (see Neutron, absorbers fori plastic, viscosity of, 9-26, 9-27 properties of (table), 9-113 thermal-stress 9-114 for water-cooled reactors (table), 13-30 to 13-32 window, 7-121, 7-122

Materials Testing Reactor, MTR (see Reactors. particular, MTR) Matrix (matrices), algebra of. 3-8 to 3-11 Jordan normal form of, 8-52

orthogonal, 8-14 symmetric, positive definite. 3-14 semidefinite, 3-14 transformations of, 3-8 to 3-1 1 upper triangular, 3-37 Matrix mechanics, 4-14 Matrix solutions, for reactors, numerical example of, 6-103 to 6-105

INDEX Matrix solutions, for reflected reactors, two-group

reactor theory, 6-70 to 6-74 Maximum credible accident, 8-84 Maximum modulus, principle of, 3-83

Maximum permissible concentration

of radio active materials, in air, general value (tables), 7-44, 11-141 particular radioisotopes (table), 11-141 water, general value (tables), 7-44, 11-141 in particular radioisotopes (table), 11-141 Maximum permissible dose (see Dose) Maximum permissible flux, neutron (table). 7-35 for various types of ionization radiation (table), 7-32 to 7-36 Maxwetl-Boltzmann distribution, 6-83, 6-84. 6-94. 6-48, 6-49 deviation from, 6-50. 6-51 most probable velocity in, 6-83 (See also Cross sections)

Maxwell distribution (see Maxwell-Boltzmann distribution) Mathematical constants, short table of, 1-12 table of, 1-75


Mathematical tables, list of, 1-2 Mean free path, definition of, 6-32 (See also Transport mean free path) Mean lifetime in radioactive decay, 4-34, 7-4 Mean-value theorems, 6-08 Measurement of cross sections (see Cross sections, measurement of) Mechanical properties, effects of radiation on (see Radiation damage) of the elements, 11-5 to 11-7 Mechanics, quantum, 4-14 to 4-33 wave, 4-14 Medical diagnosis, use of radioisotopes for, 14-10, 14-11 Medical therapy, use of radioisotopes for, 14-10, 14-11 Megawatt day (Mwd), fissions equivalent to, 1-20 per tonne, conversion units for, 1-28 Meltin transform, 13-131 Melting of fuel elements, 8-78 Melting point of the elements (table), 11-5 to 11-7 Merchant ship Savannah, reactor for, 13-138 Mercury, as control material (table), 10-189 corrosion by, 10-2 physical properties of (tables), 9-20, 11-6, 13-81 as reactor coolant (comparative table), 12-58 Meromorphic function, 3-84 Meson theory of nuclear forces, 4-53, 4-54 Mesons, 4-53 and hyperons, 4-57, 4-58 Metal-water reaction, 8-79, 8-85, 12-19 Metallic state, allotropy, 10-12, 10-13 alloying, 10-8. 10-9 atomic forces, 10-4 atomic radius, 10-6, 10-7 Brillouin zones, 10-8 crystal structure, 10-5, 10-6 deformation, 10-13 to 10-15 diffusion, 10-13 electron configuration of atoms, 10-2 to 10-4 electron mobility, 10-6 to 10-8 Fermi energy, 10-6, 10-7 heat conductivity, 10-11, 10-12 physical properties, 10-9 to 10-13 rupture, 10-15, 10-16 specific heat, 10-10, 10-11 thermal conductivity, 10-11, 10-12 thermal expansion, 10-11 valence bonds, 10-4

Metallographic inspection of radioaetive

25

speci mens, 10-93 to 10-95 Metals, liquid, corrosion by, 10-20, 10-21 structure of (set Metallic state) Meteorology in relation to reactor safety, 8-86 Metric systems (table). 1-135 Metrics, 3-13. 3-14 Michelson-Morely experiment, 4-6 Microphonics in ionization pulse counters, 6-12, 6-13 Microscopic cross section, definition of, 6-32 (See also Cross sections) Microtron, 6-133 Migration area, from exponential experiments, 6-115 measurement of, 6-118, 6-119 from period measurements, 6-115 Minimum equation, 3-52 Minimum polynomial, 3-52 Minor, of determinant, 3-12 Minor axis of ellipse, 3-32 Mirror systems for viewing, 7-123 MITR (see Reactors, particular, Massachusetts Institute of Technology reactor) Mixer-settler, 11-77 Model tests for fluid and heat flow, 9-104 to 9-106 Moderators, classification of reactors by, 12-7, 12- 8 comparative properties of (tables), 12-68 to 12-76 nuclear data for (tables). 1-19 to 1-20, 2-8 to 2-10 guide to (table), 1-4 thermal Btress properties of (table), 9-113 Modulus, definition of (mathematics), 3-16 Molar volume of gas, value of, 1-13, 1-154 Mother chart, for air, 12-129 for steam, 12-123 to 12-125 Molten-metal extraction, 11-100, 11-101 Molten plutonium alloy reactor, 12-33 Molybdenum, boiling point of, 11-6 corrosion resistance, to liquid metals, 10-21, to sodium and potassium, 10-20 fission-product, chemistry of, 11-47, 11-48 high-temperature properties of, 10-67 heat of fusion of, 11-6 mechanical properties of, 10-67 modulus of elasticity of, 10-19 physical constants of, 10-67 price of. 12-107 thermal-stress properties (table), 9-113 Molybdenum alloys, high- temperature properties of, 10-66, 10-68 Molybdenum silicide, mechanical properties of (table), 10-54 physical properties of (table), 10-54 Molybdenum-uranium alloys (see Uranium alloys) Moment, 3-59, 8-60 magnetic (see Magnetic moment) Monitoring, of coolant in gas-cooled reactors, 13- 117 fixed instruments for, 7-45 detectors, 7-45 for fuel failure, 13-11 graphite- moderated water-cooled reactors, 13-21 for, 7-45 to 7-47 instruments personnel, 7-47 chemical dosimeters, 7-47 film badges, 7-47 pocket condenser chambers, 7-47 pocket fiber electrometers, 7-47 survey instruments, 7-46, 7-47 for water reactors, 13-34, 13-35

INDEX

26

Monochromatization of thermal neutrons (see Thermal neutrons, velocity selectors for) Monochromator, crystal, 5-73, 5-74 Monte Carlo method, 6-13 to 5-15 Morera's theorem, 3-82 Most probable energy, 5-49 Most probable velocity in Maxwell distribution, 5-83

MPC MTR

(see Maximum permissible concentration) (see Reactors, particular) to (see Cosine of scattering angle) Multigroup calculations, 6-88 to 6-05 Multigroup equation, 6-8 Multigroup theory, 6-55 inelastic transfer cross sections in. 2-23 Multiple-angle formulas, 3-10 Multiple roots, 3-43 Multiplication, 5-111 subcritical, 6-21. 8-26 Multiplication constant (see Multiplication factor) 6-51, 6-75 Multiplication factor, k, effective, 6-110 excess, 6-110 subcritical, summary, 1-48 Multipole, 4-4 Multipole expansion, 4-3 Multivectors, 3-11 to 3-13 Mwd/tonne, conversion factors for, 10-117


N (n, 2n) reaction, 4-101, 4-102 NaK (see Sodium -potassium alloy) National Bureau of Standards, availability of publications of, 1-72 Natural background, radiation (table), 7-35 Natural circulation, 9-90 Natural convection. 9-61 to 9-64 Natural gas, reserves of, 13-84 to 13-88 Natural logarithm of complex number, 3-19 Natural resources (see World energy resources) Natural uranium (see Uranium) Natural uranium neutron equilibrium spectrum (table). 3-32 Navier-Stokes equation, 8-118 Nebraska Sodium Graphite Reactor, cost of, 12-96 Negative periods of reactor, 8-9 Neodymium, physical properties of (table), 11-6 Neon, physical properties of (table), 11-6 Neoprene, radiation damage to (table), 10-140, 10- 146 Neptunium, chemical and physical properties of, 11- 43, 11-44 cross sections of, 2-22 isotopes of, formation of, 11-0 Neptunium- 237, in natural uranium equilibrium spectrum, oy (table), 2-8 nuclear properties of, at high neutron energy in fast reactor, c/ (table), 2-7 spontaneous fission rate of (table), 2-5 for 2 Mev neutrons, oy, 2-8 Net, flow, for streamline fluid flow, 9-31 Net capacity, definition of, 12-01 Neumann function, 3-89 tables of. 1-100 to 1-105 Neumann's problem, 3-115 Neumann's series, 3-133 Neutrino, detection of, 4-84, 4-85 nuclear properties of, 4-56 theory of, 4-68, 4-60 Neutron (neutrons), absorbers for, 10-60 to 10-71, 10-188 to 10-191

Neutron (neutrons), absorbers

for, mechanic*] properties of (tables), 10-70, 10-71 nuclear properties of (tables). 10-187 to 10-189 physical constants of (table), 10-70 silver-base alloys, 10-190, 10-191 properties (table), 9-113 thermal-stress activation by, 5-56, 5-57 attenuation of (see Shielding, neutron attenua

tion)

capture of, theory, in oontinuum region, 4-98 in high-energy region, 4-99 in resonance region, 4-98 counters for (see Counting) counting chains for (see Instrumentation, nuclear) cross sections for (see Cross sections) delayed (see Delayed neutrons) detectors for reactor control, 8-53 to 8-56 transfer functions of, 8-40 (Sec also Detectors) fast (see Fast neutrons) fission (see sources of. Mow) ionization by, 7-5 leakage of, 5-60 magnetic moment of, 4-49, 4-50 mass and energy equivalents of (table), 1-155 maximum permissible flux for (table). 7-35 nuclear properties of, 4-57 production of, in reactors (see Neutron r*««en eration for fuels) prompt, from fission (see Fission, neutrons fromrelativistic definition of, 5-53 resonance (see Resonance neutrons t shields for, in reactors (see Thermal shield » shielding of (see Shielding, neutron attenuation) slowing down of (see Physics of critical reactors, neutron slowing down) purposes. 4-91 sources of, for experimental fission, delayed (table). 7-02 energy of (table), 7-92 prompt, 7-91 spectrum of. formula, 2-2. 7-78 table, 7-91 photoneutrons (table). 7-92, 7-93 prepared sources (tables), 7-92 spectrum of, classification of reactors by (table 12-7 dE/E, 5-53 in EBR I (table), 2-31

equilibrium, in natural uranium (table)

2-32 fission and capture cross sections, in. S-8 in fast reactors (see Fast reactors) from fission, formula for. 3-2, 7-8 table, 7-91 in Godiva (table), 2-31 in Zephyr (table). 2-32 survey instruments for, 7-46 temperature of, 3-24 thermal (see Thermal neutrons) thermal detection of, activation method* ol. 5-102 transmutation of nuclides (table), 7-7 to 7-14 2200 m/sec, energy of, 1-156 temperature of, 1-13 table. 1-156 wave properties of, 4-92 wavelength of (table), 1-156 Neutron age (see Age) Neutron balance (table), 13-9, 13-80 to 1S-&4 in fast reactor (tables), 12-22 to 13-47 for reactor types (table), 13-9 targe thermal (table), 13-80

II

INDEX


Neutron Neutron Neutron Neutron

current density, $-35 density, 5-83 diffusion (tee Physics of critical reactors) economy, 12-80 to 12-84 summary of formulas relating to, 1-28 Neutron filters, 6-85, 5-86 Neutron flux (see Flux) Neutron-gamma (n,y) activated products, produc tion of, 14-33 Neutron generation (see Neutron regeneration) Neutron generation time (see Neutron lifetime) Neutron ion current chambers for reactor control (see Detectors, for reactor control) Neutron leakage, 6-60. 12-11 to 12-13 rate, 6-35 (See also Physics of critical reactors) Neutron lifetime, 6-119 delayed, mean, 8-2 effective. 6-116, 8-6 fast, in moderators (table). 2-10 formula for, 6-112 mean, 8-6 safety aspects of, 8-80 thermal, measurement of, 5-105 in various reactors (tables), 12-20 to 12-22 (See also Physics of noncritical reactors) Neutron physics, experimental, with faxt neutrons, 5-58 to 5-72 (See also Fast neutrons) with resonance neutrons, 6-72 to 6-83 with thermal neutrons, 6-83 to 5-99 Neutron reactions, theory of, 4-91 to 4-102 Neutron regeneration for fuels, a, ij, r (table). 2-5* measurement of r for fast neutrons, 5-69 for mixtures. 6-75, 6-76 numerical example, 6-96 Westcott values (table), 1-22* "world consistent" values (table), 1-26* Neutron resonances, theory of. 4-93 to 4-99 Neutron-sensitive current chains for reactor con trol, 8-57 Neutron slowing down (see Physics of critical reactors, neutron slowing down) Neutron sources (see Neutron, sources of) Neutron spectroscopy (see Cross sections, meas urement of) Newtonian fluid, viscosity of, 6-25 formula, 8-34 Newton's backward-difference formula, 3-33 Newton's forward-difference Newton's identities, S-45 Newton's method, 3-46 ' illustrative example of, 1-69 Nickel, physical properties of (table), 11-6 Nickel alloys (see High-temperature alloys) Nimonic (see High-temperature alloys) 19-9DL, 19-9DX (see High-temperature alloys) Niobium (columbium), boiling point, 11-5 11-46, 11-47 chemistry of fission-product, corrosion resistance to liquid metals, 10-21 heat of fusion (columbium), 11-5 properties of (table). 10-67 high-temperature mechanical properties of (table), 10-67 modulus of elasticity of (table), 10-19 physical constants of (table). 10-67 price of. 12-107 Niobium clad, 10-186, 10-187 Niobium-uranium alloys (see Uranium alloys) Nitric acid recovery in fractional distillation, 11-89 Nitrogen, neutron shielding properties of, 7-77 physical properties of (tables), 6-6, 9-7, 11-6 thermal conductivity under pressure (table), 9-4 Nitrogen-15, availability and production of. 14-25 * See insert Revised Cross-section Data.

27

Nitrogen-16 in reactor coolants,

radioactivity

from, 7-19. 7-20 Noise in reactor control channels, 8-56 Non-autocatalytic reactors, 12-20 Noncritical reactors (see Physics of noncritical reactors) Nondestructive testing (see Testing, nondestruc

tive)

Nonelastio cross section, 2-32, 6-55 Nonelastic scattering, measurement of. 6-65 to 5-68 (See also Cross sections, scattering, inelastic) Nonisothermal flow, streamline, 9-31 turbulent, 9-33 (See also Convection; Fluid flow; Pressure loss in steady fluid flow) factor for (table BNL 325). Non-l/i> absorbers, 2-15 to 2-23* Westcott cross sections for (table), 1-21, 1-22* Nonparametric representation, 3-24 Nonsingular transformation, 3-11 Nonvolatile solids, 14-31 Nordbeim, L. W., and R. Scalleter. 8-14 Normal density function, 3-61 Normal distribution function, 3-61 Normal equation, 8-36 Northorn States Power Company reactor, descrip tion of, 13-133 Nozzles, for channels in parallel. 9-44 to 9-46 pressure losses at (tables), 1-52, 9-37 (See also Pressure loss in steady fluid flow) NRU. description of, 13-128 NRX reactor (see Reactors, particular) NSA (Nuclear Science Abstracts), 1-72 NTR, arrangement drawing of. 13-39 v (see Neutron regeneration) Nuclear constants for reactor design, absorption cross sections (see Cross sections, absorption) absorption integral (see Resonance absorption integrals) age (see Age) for fuels) a (see Neutron regeneration

/

cosine of scattering ments (table),

angle, 1-cos 6 for the ele 2-12 to 2-14

cos 9 for fast neutrons, 2-28, 2-30 delayed neutrons (see Delayed neutrons) diffusion (see Diffusion coefficient; Diffusion length) (see Thermal utilization) factor, in cross-section measurement, 6-90 for non l/v absorbers (table. BNL 325), 2-23* fast neutron data, 2-27 to 2-33 fission data (see Fission) guide to, 1-3, 1-4 iaotopic abundance of the elements (table), 2-15 to 2-23 logarithm of energy £ (see Slowing-down con stant) macroscopic and microscopic cross sections (see Cross sections) measurement of, 6-101 to 5-122 neutron generation from fission (see Neutron regeneration for fuels) for non l/v absorbers (table, BNL 325), 2-23* /factor in, cross-section measurement, 5-90 resonance absorption integral (see Resonance absorption integrals)

/

/

slowing-down constant (see Slowing-down constant)

slowing-down power, 2-9, 12-70 thermal diffusion length (see Diffusion length) Nuclear data, up-to-date sources of, 1-73 Nuclear disintegration, description of, 4-71

INDEX

28


Nuclear fission (see Fission)

Nuclear forces, 4-46 to 4-48 meson theory of, 4-53, 4-54 Nuclear fuel (see Fuel; Nuolear power) Nuclear isomerism, 4-70 Nuclear magneton, 4-43 Nuclear photographic plate emulsions, 6-22 Nuclear plant equipment (see Nuclear power) Nuclear power, boiler feed pumps for, price of, 11-134 buildings for (see Buildings for reactors) burnup for, range of (table), 12-108, 12-109 condensers for, double-tube sheet, 12-143 duplex tubes for, 12-143 price of, for straight condensing turbines (table), 12-133 seals for, 12-142 tube failures in, 12-143 control of load for, 12-146 to 12-150 dual system, 12-148, 12-149 EBWR system, 12-147 cost of, equipment, miscellaneous, 1-27, 12-98 fixed charges (table), 12-109 fuel fabrication (table), 12-109 fuel inventory (table), 12-106, 12-109 insurance, 12-110, 12-111 maintenance (table), 12-109 operation (table), 12 109 predicted (table), 12-109 in PWR, 12-108 reprocessing (table), 12-109 in United Kingdom, 12-108 cost per kilowatt of plant capacity, accessory electrical equipment, 12-98 account* 310 to 312, 314 to 317, and 343, 12-96 to 12-99 fixed charges, range of (tables). 12-108. 12-109 initial fuel charge, 12-98, 12-99 land and land rights, 12-96 miscellaneous power-plant equipment. 12-98 published (table), 12-90 range (tables), 12-99, 12-100, 12-108, 12-109 reactor system, 12-97 station equipment, 12-99 12-96, 12-97 structures and improvements, units, 12-97, 12-98 turbogenerator water reactors as function of size, 12-99 cost per kilowatt-hour for fuel, effect of burnup on, 12-100 effect of conversion ratio on, 12-104 fabrication cost, 12-105, 12-106 range (tables), 12-108, 12-109 future production, 12-108 initial charge, 12-98 inventory charges, 12-106 range (table), 12-109 reprocessing cost, 12-105, 12-106 range (tables), 12-108, 12-109 direct cycle, corrosion and wear in (table), 12-144 to 12-146 materials for, 12-144 to 12-146 recovery system for leakage from, 12-143, 12-144

shielding of turbine equipment in, 12-154 feed pumps, for straight condensing turbine plants, price of (table), 12-134 feedwater treatment, 12-150, 12-151 fuel for, burnup obtainable, 12-109, 12-110 cooling after use of, 12-100, 12-101 cost of, fabrication, 12-102, 12-104 to 12-106, 12-110

Nuclear power, fuel for, cost of, inventory charges, 12-106

plutonium, credit for, 12-104 reprocessing, 12-100 separation, 12-105. 12-106 shipping, 12-101 thorium, 12-104 uranium, enriched, 12-103 natural, 12-103 credit for. 12-104 uranium-233. 12-103 uranium-235. reenrichment of, 12-102, 12-103 reprocessing of, 12-18 to 12-19. 12-101. 12-102, 12-110 decladding. 12-100, 12-101 dissolving. 12-100, 12-101 head-end treatment. 12-100. 11-101 pyromctallurgical, 12-101, 12-102 separation, 12-100, 12-101 tail-end treatment, 12-100, 12-101 storage after use of. 12-100, 12-101 future predictions, 12-10S to 12-110 12-109, 12-110 geographic considerations.

instrumentation of (see Instrumentation)

insurance of, 12-110, 12-111 moisture in steam, 12-134 to 12-130 effeota of, 12-135 to 12-137 initial separation of, 12-134 intermediate separators for. 12-137 to 12-139 price of, 12-139 moisture catchers for, 12-135. 12-138 in straight condensing turbines (table). 12-138 plant, operation and maintenance of. 11-153, 12-154 seals for feed pumps, 12-140 to 12-142 turbine, 12-140 to 12-142 specific power for. range (table), 12-108. 12-109 steam condensers, price of (table), 12-133 steam generators for, control load in. 11-144 tc 12-150 dual oycle system. 12-148 EBWR system. 12-147 maintenance of, 12-153 steam-turbine generators for, air-drying system for, 12-143, 12-144 control of load in, 12-146 to 12-150 feed pumps for, price of (table), 11-134 seals for, 12-141 heavy-water recovery system for. 12-143. 12-144 instrumentation of, 12-151 to 11-153 materials for steam system of, 11-144 to 12-146 moisture in (see moisture in steam, ooosr) operation of. 12-153. 12-154 preferred standards (table). 12-130 radioactivity in. of direct-cycle turbine. 13- S to 13-10 regenerative cycle for, 12-134, 11-136 efficiency improvement in, 12-134. 11-136. 12-137 seals for, 12-140 to 12-144 straight condensing, condensers for. price of (table). 12-133 dimensions of, 12-135 efficiencies of, 12-131, 12-132 feed pumps for, price of (table). 11-134 heat rates of, 12-132, 11-133 moisture in steam (table), 11-138 valve glands for, 11-141 valves for. 12-141, 11-142

INDEX Nuclear properties, of concrete, barytes, relax ation length, for fast neutrons, 2-35 for 7 rays, 2-35 Brookhaven, relaxation length, for fast neu trons,

Be*(7,n)Bed,

3-33

forward, 3-33 reactor,

2-36

4-91

in heavy-water Um reactor (table), 2-3


Operating costs of power plants (see Nuclear power; Power, from fossil fuel) Operators, backward, 3-33 central -difference, 3-33 difference,

2-35

for y rays, 2-35 of moderators (see Moderators) relaxation length, for fast neutrons, for 7 rays, 2-35 Nuclear radius, 4-43 Nuclear reactions* 4-38 to 4-40 D(d,n)He», 5-60 H*('.n)He<, 5-72 Li7(p.n)Be». 5-59 photon-induced, 4-72 7,n reactions,

29

photohssion, 4-85 as source of fast neutrons, 5-59 to 5-61 T(d,n)He*. 5-60 T(p.7i)Hc>. 5-59, 5-60 theory of, 4-73 to 4-80 thermonuclear, 4-88 to 4-90 threshold, cross sections for fission neutrons (table), 2-30 in reactors, 5-70, 5-71 at very high energies, 4-82 Nuclear reactors (see Reactors) Nuclear runaway, consequences of, 6-78 to 6-80 Nuclear Science Abstracts, 1-72 Nuclear spin, 4-43 to 4-45 Nuclear temperature, 4-74 Nuclear theory, 4-33 to 4-54 Nucleate boiling (see Boiling) Nucleon, 4-34 definition of, 4-57 Nucleus (nuclei), basic properties of. 4-33. 4-34 collective model of, 4-53 liquid-drop model of, 4-87, 4-88 Rutherford model of, 4-10 shell model of, 4-50 energy levels in, 4-52 magic numbers, 4-50. 4-51 Nuclides, 4-34 chart of, 1-15 to 1-19 cross sections of (see Cross sections) formed in fission (see Fteshnrpf oducts) isotopic abundance of, in the elements. 2-15 to 2-23 Numbers, pure imaginary, 5-13 real, 3-2, 3-76 Numerical differentiation, 3-74, 3-75 Numerical integration, 3-74, 3-75 by Simpson's rule, 1-29 Numerical solution of differential equations, 3-122 to 3-125 Nusselt number, definition of, 9-32 Nyquist plot, 8-37

6-82 to 8-83

Oppenheimer-Phillips process, 4-81 Orbital electron capture, elementary

Order-disorder in alloys, effect of radiation on, 10-109

Organic compounds for production of radio isotopes,

14-26

Organic coolants, diphenyl (table), 9-22 polyphenyls, properties of, 13-48 Organio materials, radiation damage to (see Radiation damage) as reactor coolants, 12-57 to 12-59 Orifices, for channels in parallel, 9-44 to 9-46 pressure losses at (tables), 1-52, 9-37 ORR, Oak Ridge Research Reactor (see Reactors,

particular) Orthogonal functions, 8-101 Orthogonal polynomials, 3-92. 3-93 Orthonormal basis, 3-14 Orthonormal set, 3-102 Orthonormal vector, 3-101 Oscillation theorem, 3-104 Oscillator, for absorption crotw-section

measure ment, 6-93, 5-94 Osmium, physical properties of (table), 11-7 Otto cycle, 12-4, 12-5 Outer product, 3-15 Over-all heat transfer coefficient. 1-60 OWR, description of, 13-132 Oxidation, of high-temperature alloys, 10-63 to 10-66 of stainless steels (table), 10-63 Oxygen* physical properties of (table), 11-7 total cross section of, at high energy, 5-62 Oxygen-17 and -18. 14-25 Oxygen-19 in reactor coolants, radioactivity from, 7-19, 7-20

Pi,

P», P%, approximations, 6-10, 6-11 P-2 reactor, description of, 18-127 drawings of, 13-151. 13-152 P-wave meter, 8-87 Packing fraction /, 2-35, 2-36 Packings, radiation damage to (table), 10-147 Paint for reactor building interiors, 13-175 Pair production, 4-61 in shielding, 7-61 theory of, 4-56 5-39 Pair spectrometers, Palladium, chemistry of fission product, 11-48. 11-49

O rings, radiation damage to (table), 10-146 Oak Ridge National Laboratory reactors (set Reactors, particular) Off-center control rod. worth of, 6-14 Off-gases in fuel reprocessing, 11-113, 11-114 Oil

shales,

12-84, 12-88

Oils, radiation damage to (table), (See also Radiation damage) OMRE. description of, 13-138 One -group theory, 6-54 modified,

6-55

10-145

One-velocity Boltzmann equation, 6-6

description

of, 7-3

isotopic separation of, 14-57 physical properties of (table). 11-6 Parabola, 3-30, 3-31 Paraboloid, 3-30 Parallel channels, fluid flow in, 9-38 Parallel flow, 9-38. 9-44 to 9-46 orificing in. 9-44 to 9-46 (See also Convection; Fluid flow, streamline; Pressure loss in steady flow) Parallelepiped, rectangular, physics solution for. 6-58, 6-59. 6-66

Parametric representation, Parity. 4-26

3-24


30

INDEX

Parse val's theorem, 3-103 Partial cross sections, ff-54, 5-56 definition of, 4-71, 5-54 measurement of, 5-55, 5-77 Partial derivative, 8-64 Partial differential equations (see Differential equations) Particle acceleration (see Accelerators) Particles, mass and energy equivalents of (table). 1-155 radioactive, airborne, from reactor, 7-50 from processing operations, 7-50, 7-51 wave properties of, 4-15 Particular integral, 3-112 Partitioning, method of, 3-38 Pauli exclusion principle, 4-13, 4-14 Peat, 13-84 Peclet number, definition of, 9-32 Penetrant tests, 10-152, 10-153 Pennsylvania State University reactor, descrip tion of, 13-132 Period (reactor), 6-113, 6-114, 8-7 to 8-10 stable (asymptotic), 6-113, 8-7 approximate formulas for, 8-8 to 8-10 summary, 1-47. 1-48 negative, 8-9 prompt critical, 8-8, 8-31 with relative importance factors. 5-112. 6-113 startup, 8-25 subcrrtical, 8-22, 8-32 summary, 1-47, 1-48 Period amplifier, for reactor control. 8-61 Period demand signal, 8-30 Period permissive control of reactor. 8-29. 8-30 Periodic chart of the elements, 11-4 Periodic functions, 3-19 Periodic system, atomic, 4-13 of the elements, 4-27 Periscopes, 7-123 basic optical systems for, 7-136. 7-136 Permutations, 3-58 Permutit cation exchanger, 11-93 Perturbation method, applied to atomic theory. 4-27, 4-28 Perturbation theory (reactor), 6-21. 6-117 to 6-120 Petroleum, 12-84 to 12-88 (See also World energy resources) Phase oscillations in proton synchrotrons. 5-136 Phase shifts in scattering process, 4-29 Phase stability in constant-gradient magnetic ac celerators, 5-133, 5-134 Phase transformation (metallurgical), effect of radiation on, 10-109 Phosphors, efficiencies of. 5-25 5-23 inorganic, organic, 5-24 Phosphorus (yellow), physical properties of (table), 11-6 Phosphorus-32, production of, 14-32 use of, in agricultural research, 14-11 for medical therapy, 14-11 for process controls, 14-7, 14-8 Photodidintegration, 4-82 description of, 4-72 Photoelectric effect, 4-5, 4-59, 4-60 in shielding, 7-61 Photofission, 4-85 Photographic emulsion for particle detection, 5-21 to 6-23 latent images in, eradication of, 5-22, 5-23 range in, 5-23 types of, 6-22 Photomultipiier circuits. 5-27. 6-28

Photo multiplier tubes, 6-25 to 6-28 characteristics of, 6-25, 6-26 magnetic sensitivity of, 6-2«> spectral sensitivity of. 6-25 circuits for, 6-27, 6-28 light collection from, 6-26, 6-27 Photon, ionixation by, 6-4, 5-5 properties of, 4-55 wavelength of (table), 1-156 Photon attenuation (see Shielding. T-ray atteuua

tion) Photoneutron reactions, with beryllium, threshold energy

for, 2-3

with DiO, 2-3, 2-4 threshold energy for, 2-3 yield in DiO reactor (table), 3-3 Physical constants, fundamental, short table of.

1-13 table of, 1-154 Physical properties of the elements (table). 11-5 to 11-7 Physical scale, definition of, 1-153 Physics, experimental (see Experimental physic* ^ Physics of critical reactors, absorber, central, in bare reactor, 6-109 two-group solution for, 6-67 to 6-69 lumped, in infinite medium, 6-105 to 6-110 in regular array, 6-108, 6-109 bare reactors, solution of, 6-57 to 6-69 cylinder, finite, 6-58 infinite, 6-58 parallelepiped, retangular. 6-58, 6-69 slab, infinite, 6-57, 6-58 sphere, 6-58 characteristic equation, 6-53 for Fermi age theory. 6-54 for Gaussian slowing down, 6-54 for modified one-group theory. 6-55 for multigroup theory. 6-55 for one-group theory, 6-54 summary. 1-35 for two-group theory, 6-54. 6-55 diffusion theory, 6-5 boundaries, 6-15 to 6-18 Albedo, 6-17, 6-18. 6-41. 6-42 extrapolation distance, 6-16. 6-37. 6-36 Serber-Wilson method, 6-17 diffusion kernel equation. 6-9 diffusion kernels (table), 6-39 Fermi age equation, 6-6. 6-7 Greuling equation, 6-9 hemispherical harmonics method. 6-12 multigroup equations, 6-8. 6-9 Pi, Pi. Pi, approximations. 6-10, 6-11 reactor equation, 6-35 to 6-41 for monoenergetic neutrons. 6-35 Selengut-Goertzcl equation, 6-7 solution of reactor equation, diffusion kernels, 6-38. 6-39 for localized sources, 6-38 to 6-41 spherical harmonica method, 6-10. 6-11 fast fission effect e (see Fast effect f) lumped absorber, 6-105 to 6-110 central, in bare reactor, 6-109, 6-110 in infinite medium, 6-105 to 6-109 regular array of. 6-108, 6-109 multigroup theory, 6-88 to 6-94 equations for, 6-88, 6-89 solution for bare reactor, 6-89 to 6-94 neutron slowing down, 6-42 to 6-48 with absorption, 6-47, 6-48 age, Fermi, definition of, 6-6 collision density, 6-43. 6-44 continuous model. 6-6

INDEX Physics of critical reactors,

neutron slowing down, energy loss per collision. 6-42, 6-43 Fermi age approximation, 6-45 to 6-47 Gaussian, 6-54 kernels, 6-45, 6-46 number of collisions during, t-10 Selengut-Goertzel equation, 6-7 slowing down density, 6-43 definition of, 6-6 slowing down distribution, 6-44 time for, 2-10 6-43 general formula for, 6-6 perturbation theory, 6-21, 6-117 to 6-120 reactor equations, 6-18 to 6-21 adjoint equation, 6-20, 6-21 critical equations, 6-10, 6-51 to 6-57 6-52 asymptotic, diffusion equations for various group model**, 6-19 perturbation theory, 6-21 wave equation, for homogeneous bare reac tors, 6-57 to 6-59 for reflected reactors, two-group, 6-00 to 6-70 matrix solutions of, 6-70 to 6-74 (See also Four-factor formula) resonance escape probability, 6-22, 6-80 to 6-83 transport theory, 6-3 Boltzmann equation, 6-3, 6-4 boundary conditions, 6-15 to 6-18 extrapolated end points. 6-16 Serber- Wilson method, 6-17 hydrogen equation, 6-5 Monte Carlo method, 6-13 to 6-15 one-velocity Boltzmann equation, 6-5 Sn method, 6-13 Selengut-Goertzel equation, 6-17 space independent integral equations, 6-4 two-group theory (diffusion), 6-60 to 6-70 for central absorber, 6-67 to 6-69 cylinder, finite, with end reflector, 6-65 with radial reflector, 6-61 to 6-64 matrix solutions, 6-70 to 6-74 for criticality, 6-70 to 6-72 for flux distribution, 6-72 to 6-74 numerical example for reflected cylindrical reactor, 6-95 to 6-105 age, 6-98. 6-99 buckling, 6-99 critical bare reactor, 6-99 critical reflected reactor, 6-99 to 6-103 diffusion area, 6-98, 6-99 diffusion constants, 6-98 fast fission factor, 6-98 matrix solution, 6-103 to 6-105 multiplication constant, 6-98 regeneration factor, 6-96 resonance escape probability, 6-97 thermal utilization, 6-96, 6-97 rectangular, 6-66 for parallelepiped, for sphere, 6-66, 6-67 Physics of noncritical reactors, 6-110 to 6-120 control-rod theory, 8-12 to 8-19 delayed neutrons, 8-2 to 8-5 data tables, 8-3 to 8-5 summary table, 1-46 effect of energy of, 8-10, 8-11 mean energy, definition of, 8-2 mean life, definition of, 8-2 energy -dependent effects in, 6-10, 8-11 fast fission, effect of, 8-11


t

Physics of noncritical

31

reactors, inhour equation, 6-113. 8-6 summary, 1-47 kinetic equations, 6-111 to 6-113 lifetime, neutron, effective, formulae for, 1-47, 6-116, 6-117, 8-6 by perturbation theory, 6-119 multiplication of subcritical reactor. 6-111 multiplication factor, excess, 6-110 neutron lifetime (tee lifetime, above) one-group diffusion theory, 8-5 to 8-7 reactivity, 8-6 period, 6-113, 6-114, 8-7 to 8-10 stable (asymptotic), 6-113, 8-7 approximate formulas for, 8-8 to 8-10 summary, 1-47, 1-48 negative, 8-9 prompt critical, 8-8, 8-31 with relative importance factors, 6-112, 6-113 perturbation theory, 6-117 to 6-120 first-order, two-group, 6-117 to 6-119 neutron lifetime, 6-119 in lumped reactor, 6-120 nonuniform perturbation of bare reactor. 6-119 uniform perturbation of bare reactor, 6-1 19. 6-120 pressure coefficient of reactivity, 8-12 prompt jump, 1-48, 6-115, 8-9, 8-10 reactivity, 6-110, 8-6 units of, 6-113. 8-8, 8-9 conversion factors for, 1-46 stable period (tee period, above) temperature coefficient of reactivity, 8-11 two-group theory, reactivity &k/k, 8-10 numerical example of calculation of. 6-99 to 6-103 pressure coefficient of, 8-12 temperature coefficient of, 8-11 Pipes, optimum size of, 9-36 to 9-38 Piping, expansion stresses in, 13-31, 13-32 Piston control-rod drive, 8-72, 8-73 Pitting corrosion, 11-123, 11-124 Planck, hypothesis of, 4-5 Planck's constant, value of, 1-154 Planes (mathematics), 3-26 to 3-28 Plant factor, definition of. 12-91, 12-94 Plastic materials, radiation damage to, 10-141 to 10-143 viscosity of, 9-26, 9-27 Plat*, thermal conduction in (tee Heat conduc tion, in slabs or plates) Plateau, Geiger, 6-17 Platinum, isotopic separation of, 14-57 physical properties of (table), 11-6 Plutonium, chemistry of, 11-37 to 11-40 (See also Chemical separation processes) physical constants of (tables), 10-22, 10-37, 11-38 physical properties of (table), 11-38 thermal stress properties of (table), 9-113 Plutonium -23 9, burnup rate of, 1-27 handling of, 7-56 health hazards of, 10-37, 10-38 nuclear properties of, at high neutron energy, in fast reactors, a«, as, (table), 2-7 30 kev and 900 kev, v (table), 2-7 at intermediate neutron energy, a as a func tion of energy (table), 2-6 in natural uranium equilibrium spectrum oj (table). 2-8 prompt-neutron spectrum (formula), 2-2, 2-3

INDEX

32 Plutonium-239, nuclear properties of, sponta neous fission rates of (table), 2-5 tabular guide to data, 1-4 at 2200 in/see (table),
Westcott values «V #/ (table), 1-21.* 1-22 "world consistent" values, a, n, <■. 1-26
radiochemical analysis of, 11-52 Plutonium-240, nuclear properties of, at high neutron energy, in fast reactors, of (table), 2-7

in natural-uranium equilibrium spectrum (table), 2-8 spontaneous fission, rate of (table), S-5 Plutonium compounds (table), 11-39 Plutonium isotopes, formation of, 11-9 radioactive properties of (table), 11-38 Plutonium oxide-uranium oxide (PuOt-UOt),


properties

of, 10-181

Pocket oondenser chamber, 7-47 Point coordinate system, 3-8 Poisson distribution, 3-fil Poisson equation, 3-117 Poisson's integral, 3-116 Polar coordinates, 8-32 Polar kernel, 8-136 Pole, 3-84 Pollution, 8-87 Polonium, melting point of (table), 11-6 Polygamma function, 8-87 Polymers, cross linking of, by radiation, 14-8 Polynomial equations, Graeffe's solution for, 3-48, 3-49 illustrative example

of, 1-64 to 1-06

simplified description of, 1-63, 1-64 Horner's method, 3-40 illustrative example of, 1-66 to 1-68 Polynomials, orthogonal, 3-92, 8-93 (See alao particular polynomial) Polystyrene, moderating properties of (table), 12-70

Polythene, cross linking of, by radiation, 14-9 Pool boiling (see Boiling, free convective) Pool-type reactor (see Swimming-pool reactor) Ponchon-Savarit, 11-86 Portable survey instruments for monitoring, 7-46, 7-47

Posttronium, 4-56 Positrons, and Dirac equation, 4-25 ionization by, 7-5 properties of, 4-55 theory of, 4-56 Postirradiation measurements, 10-92 to 10-95 Potassium, iaotopic separation of, 14-50, 14-57 liquid, physical properties of (tables), 9-20, 13-81 physical properties of (table), 11-6 price of, 12-107

Potassium-sodium alloy (see Sodium-potassium alloy) Potential barrier, for fission, 4-87 penetration

of, 4-32

Potential scattering, 4-94, 4-96 Power, conversion factors for units of (table), 1-148 and fuel consumption, 12-78 nuclear, 1-155 electric (see from fossil fuel, below; Nuclear power) from fossil fuel, cost, of accessory electrical equipment, 12-98 accounts 310 to 302, 314 to 316. and 343 12-96 to 12-99

* See insert Revised Cross-section Data.

Power, from fossil fuel, cost, boilers. 12-97 capitalization, 12-93 fixed charges,

12-93

of fuel (table), 12-95 investment (tables), 12-93, 12-94 of land and land rights. 12-96 power-plant equipment. of miscellaneous 12-98

operating (table), 12-94. 12-95 of station equipment, 12-99 of structures and improvements. 12-VKi, 12-97

total (table), 12-95 of turbogenerator unita. 12-97, 12-98 definition of terms, 12-91 development of, since 1920, 12-91 to 12-92 consumption of, per capita, 12-91. 12-92

cost of, 12-91, 12-92 size of plants, 12-94 station heat rate, 12-91 to 12-93 plant factor, 12-94, 12-95 nuclear (see Nuclear power) Power cycles, for nuclear plants, 12-114. 12-11-5 thermodynamics of. 12-2 to 12-6 Power density, conversion factors for units of (table), 1-151 formulas relating to, 12-79 Power-flux relations, formula for, 1-26. 1-27. *-£&12-78, 12-79 Power plant (see Nuclear power; Power, from fossil

fuel) Power reactors, alkali-metal -cooled, 12-35 to 12-37, 13-80 to 13-98

control means for, 18-100. 13-101 experimental systems for, 18-98 to 18-101 fast (see fast, below)

fuel element requirements of, 18-100 graphite- moderated, effect of lumping on. 12-10, 12-11 heat exchangers

for, 12-9 1, 13-92 instrumentation for, flowmeters. 13-93 to 13-95

13-95, 13-9* 13-93, 13-94 measurement. temperature measurement, 13-93 intermediate, small (table). 12-52. 12-53 materials for (table), 13-83 oxide control system for, 13-96 properties of alkali metals. 13-81. 13-82 pumps for (see Pumps) seals for, dynamic, 13-84. 13-101 static 18-83, 13-101 secondary system for, 13-101. 13-102 graphite- moderated. 13-35 sodium-cooled shutdown cooling in, 13-100 steam generator for, 13-91, 13-92 valves for, 18-92, 18-93 boilers (see Steam generators) boiling homogeneous, 12-46. 13-60 boiling-water (HiO and DjO), deentrainment of water in, 13-17 design features of (table), 12-43 direct cycle, 13-42 to 13-46 dual cycle, 18-42. 13-43 DtO, 18-44, 13-45 efficiency of, 18-3 flashing, 13-42 heavy water, 18-44, 13-45 pressure in, 13-3 recirculation in, 13-46, 13-47 separation factor in, 13-5 steam slip in, 13-46 time delay in bubble formation in. 13-47, liquid-level measurement. pressure

18-48

INDEX boiling-water (HiO and DsO), direct cycle, turbine contamination in (table), 13-8 to 13-10


Power reactors,

general characteristics of (table), 12-39 to 12-44 small (tables), 12-51 steam purity in (table), 12-40, 12-41 voids in, 12-40 in, 12-42 water decomposition (See also water-cooled, below) comparison of types (tables), 12-23, 12-24 fast, liquid fuel, 12-32, 12-33 (tables), 12-22 to 12-27 sodium-cooled graphite-moderated, 12-33 to 12-37 liquid-metal homogeneous, 12-40 to 12-51 pressurized-heavy-water, 12-38, 12-39 fast, constructional features of, 13-155 to 13-167 of (tables), 12-23 to general characteristics 12-27 design specifications for (tables), 12-23, 12-24 neutron balance of (tables), 12-22 to 12-25 nuclear parameters of (table), 12-26 neutron spectrum in (tables), 2-30 to 2-32 structural materials permissible in (tabic). 12- 11 gas-cooled, 13-105 to 13-122 for aircraft propulsion, 13-106 British program, 13-105 closed-cycle, 13-119 to 13-122 dished loading for. 18-106, 13-107 expansion of, 13-106, 13-107 gas turbine cycle for, 13-1 LI to 13-1 17 cycle efficiency, 13-113 to 13-117 helium for, 18-122 turbine efficiency, 13-113 gases for, 13-120 to 13-122 effectiveness indexes, tables of, 13-121 helium for, 13-122 18-106, 13-107 general considerations, graphite-moderated, 12-34. 12-35, 13-141, 13- 142 heat transfer in, 13-107 hot spots in. 13-107, 13-108 once-through systems (open-cycle), 18-108 to 18-119 (See also Research reactors) recirculating systems, 18-119 to 13-122 for ship propulsion, 18-106 ANP, 12-53 water-moderated, heat exchangers (see Heat exchangers; Steam generators) heavy-metal, 13-98, 13-99 homogeneous, 12-46 to 12-51, 13-99. 13-100 LMFR, 12-47 to 12-50, 12-53 physical properties of heavy metals, 13-81, 13-82 (See also alkali-metal-cooled, above) heavy water (see boiling-water, above; pressurised-water, belotc) homogeneous aqueous, 13-49 to 13-79 of, 13-49, advantages and disadvantages 13-50 boilers for, 13-67, 18-68 boiling, 13-60 of, 13-50 classification constructional features of, 13-149, 13-150, 18-153 control of, 18-73, 13-74 cooling system for, 13-60 corrosion in, 18-57, 13-58 feed pumps for, 13-70 to 13-72 absorbers, 18-71, 13-73 fission-product

33

Power reactors, homogeneous aqueous, fuel handing system for, 13-62 fuel storage, system for, 13-62 tanks, 13-70 fuels for, 13-50, 13-51 slurries, thorium, 18-57 uranium, 18-53 to 13-50 thorium nitrate, properties and phase sta bility of, 13-56, 18-57 thorium phosphate, 13-57 thorium slurries, 18-57 fluoride, 13-57 oxide, 18-57 uranium slurries, 13-53 to 18-56 many] fluoride, corrosion by, 13-58 properties and phase equilibrium of, 13-52 uranyl nitrate, corrosion by, 13-58 properties and phase equilibrium of, 13-51, 13-53 uranyl phosphate, corrosion by, 13-58 properties and phase equilibrium of, 13-52 gas recombiners for, 13-68 to 13-70 in HRT, 13-68 systems for, 13-60 to 13-62 gas separators for, 13-66, 13-67 gaseous fission product disposal system, 13-62 adsorbers for, 13-71, 13-73 of, 12-44 to 12-46 general characteristics small (table), 12-52 heat exchangers for, 18-67, 13-68 leak detection in, 13-74, 13-75 maintenance of, 13-74, 13-75 materials for, 13-57, 18-58 pressure vessels for, 13-63 to 13-66 pressurizers for, 13-68 in HRT, 13-68 pumps, feed, for, 13-70 to 13-72 radiolytic decomposition in, 13-58 catalysts to limit, 13-59 rate of, 13-58 uranyl peroxide formation, 13-58, 13-69 reactor vessels for, 13-63 to 13-66 in HRE, 13-65 one-region, 13-65 two-region, 13-63 to 13-60 stability of, 8-37 steam generators for, 13-67, 13-68 valves for, 13-71, 13-72 homogeneous slurry, 13-46 liquid -metal-cooled (see alkali -metal -cooled, above) liquid-metal-fuel (see heavy-metal, above) NaK-cooled (see alkali-metal-cooled, above) organic -cooled, 13-48 properties of poly phenyls, 13-48 small (table), 12-53 pressurized -water, efficiency of, 18-3 of, 12-37, 12-38 general characteristics heavy water, 12-38, 12-39 pressure in, 13-3 pressure-tube type, 18-3, 18-4, 18-18 to 13-25 DiO moderated, 13-21 to 13-23 fuel design for, 13-22 general arrangement of, 13-21, 13-22 HiO cooled, 13-22, 18-23, 13-45 lattice spacing in, 13-22 leak detectors for, 13-32 leakage from, 13-39 tube size for, 13-22

INDEX

34

Power reactors, pressurized-water, pressuretube type, graphite- moderated, 12-35, 13-18 to 18-21

control system for, 13-19 graphite for, 13-19 HiO cooled, 13-22, 13-23, 13-45 loading,

13-20, 13-21

minimum size of, with natural uranium, 13-18


monitoring, 13-21 pressurizing vessels for, 18-26, 18-27 small (tables), 13-51, 13-36. 13-39 tank type, 13-3, 13-14 to 13-17 for DtO, 13-16, 13-17 of, 18-14, 13-15 general arrangement supercritical pressure, small (table), 13-52 below) (See also water-cooled, sodium-cooled (see alkali-metal-cooled; fast, above) space requirements for (table), 13-172 steam generators for (see Steam generators) supercritical water, small (table), 13-52 water-cooled, 13-3 to 13-48 auxiliary systems for, 13-32 to 13-35 carbon dioxide, 13-32 cooling, auxiliary, 18-32 to 18-33 disposal, 13-33 dump, 18-33 fuel handling, 13-32 helium, 13-32 boilers for, 13-27 boiling (see boiling- water, above) buildings for, pressure requirements of. 13-30 capability of (table), 13-5 cooling of, shutdown. 13-29. 13-30 DiO moderated (see pressurised -water, above) graphite moderated (see pressurized-water, above) heat exchangers

for, 13-27, 13-28

instrumentation for, monitoring, 13-34, 13-35 power-level,

13-35 water-level, 13-35 leakage, allowable (table), 13-4 to 18-6 materials for, 13-31 piping expansion in, 13-31, 13-32 pressure-tube type (see pressurized-water. above) pressure vessels for. closures for, 13-14, 13-15 seal welding of, 13-14, 13-15 radiation damage to, 13-13, 13-14 size limitations of, 13-12, 13-13 thermal shields for, 13-13, 13-14 thermal stress in, 13-13 pumps for (see Pumps, for water) purity of water for, 13-6 to 13-8 relief valves for, 13-29 rupture diaphragms for, 13-29 seals for. 13-23. 13-24 size of core for (table), 13-5 steam generators for, 13-27, 13-28 surge tanks for, 13-26, 13-27 tank type (see pressurized-water. above) valves for, 13-27 water for, ion exchange purification of, 13-9, 13-11, 13-12 purity of, 13-6 to 13-8 Power requirements of air-cooled reactor system* (table), 13-119 Power series, 3-98 to 3-100 table, 3-99, 3-100 Prandtl number (table), 9-14 definition of, 9-32 Praseodymium, physical properties of (table), 11-6

Preamplifier, for ionization pulse counter. 8-12 linear, in reactor control, 8-56 PRDC fast reactor (see Reactors, particular. En rico Fermi) Precipitation, 11-58, 11-65 in radiochemical analysis, 11-51 from solid solution, effect of radiation on 10-109

theory of, 11-59, 11-60 Pressure, conversion factors for units of (table), 1-147

in water reactors,

13-3 Pressure coefficient of reactivity, 8>-12 Pressure drop in homogeneous aqueous reactor vessels. 13-65 (See also Fluid flow; Pressure loss in steady fluid flow) Pressure gages (see Pressure measurements) Pressure loss in steady fluid flow, of Bingham bodies, 9-40 past external objects, 9-46, 9-47 through beds, 9-47 through tube banks, 9-47 of slurries, 9-40 streamline, 9-29 to 9-31 buoyancy effects, 9-38, 9-39 in complex circuits, 9-38 in eccentric annulus, 9-29, 9-31 effect of wall roughness, 9-31 inlet loss (table), 9-30 in irregular channels, 9-29, 9-31 9-31 non isothermal, in regular ducts (table). 9-29. 9-30 use of electrical analogs, 9-31 use of flow net, 9-31 turbulent, buoyancy effects in, 9-33, 9-39 at changes of section (tables), 1-52. 9-36, 9-37

in complex circuits. 9-38

fluids, 9-34 of compressible at fittings, (table!, 9-36. 9-37 friction in ducts, (table), 9-30. 9-32 to 9-35 nomsothcrmal, 9-33, 9-34 at nozzles (tables), 1-52, 9-37 at orifices (tables). 1-52, 9-37 in parallel channels, 9-46

transition from streamline, 9-35 two-phase (table). 9-40 to 9-44 bulk boiling. 9-41 to 9-44 local boiling. 9-40, 9-41 for alkali metals. 19-93 Pressure measurements, in-pile, 10-93 for radioactive solutions. 11-116. 11-117 Pressure Pressure

tests, 10-150

transmitter, 13-94 Pressure vessels, 13-12 closures for, 13-14, 13-15 seal welding of, 13-14, 13-15 for homogeneo'as aqueous reactors, 13-63 to 13-66 neutron shields for. 13-13, 13-14 radiation damage to. 13-13. 13-14 size limitations of, 13-12, 18-13 thermal stress in, 13-13, 13-14 Pressurized-water reactor (see Power reactors.

pressurized-water; Research reactors) for homogeneous aqueous reactor*.

Pressurizers, 13-68

for water reactors, 13-26. 13-27 Primitive (mathematics). 3-67 complete,

3-109

Principal quantum number, 4-13 Probability, 3-55 to 3-62 Probability function, table of. 1-124 to 1-136


INDEX Process control, use of radioisotopes for, 14-7, 14-8 Process instrumentation (see Instrumentation) Production of radioisotopes (tee Radioisotopes) Products, infinite. 8-97. 3-98 polynomial, 3-5 Program in digital computers, 3-143 Progression. 3-6 geometric, 3-6 harmonic, 8-0 Prompt critical period, 8-8. 8-31 Prompt criticality, 8-8 Prompt drop, 8-10 Prompt fission neutron spectrum (table), 7-91 formula. 3-2, 3-3 Prompt generation time (see Neutron lifetime) Prompt jump. 1-48, 8-115, 8-9, 8-10 Prompt neutrons, formula for, 3-2, 8-3. 7-78 Bpectrum of (table), 7-91 Prompt rise, 1-48, 6-115, 8-9, 8-10 Proper value, definition of, 8-29 Proper vector, definition of, 3-29 Properties, physioal, of elements (table). 11-5 to 11-7 Proportion, mean, definition of, 3-3 Proportional third, definition of, 3-3 Proportional counters (aee Detectors) Proportional reactor regulating system, 8-32, 8-33 Protactinium, chemical properties of, 11-43 melting point of (table), 11-6 Protactinium-231, nuclear properties of, for 2 Mev neutrons, a/, 2-8 Proton-induced reactions, 4-80 Proton linear accelerator, 8-127, 5-128 Proton-proton chain, 4-90 Proton recoil for neutron counting in gases. 8-42 Proton synchrotron, 8-135, 5-136 Protons, ionization by, 5-2 to 5-4 magnetic moment of, 4-49, 4-50 mass and energy equivalents of (table), 1-155 neutron capture by, 4-98 nuclear properties of. 4-56, 4-57 range of, 2-34, 5-4 Pseudoplastic materials, viscosity of, 9-26. 9-27 PuOt-UO», properties of, 10-181 Pulse amplification. 5-12 Pulse amplifiers in reactor control, 8-56. 6-57 Pulse ion chambers (see Detectors, ion chambers, pulse) Pulse shape in proportional counters, 6-14 Pulsed accelerators for neutron production. 6-74 Pulsed column, 11-78 Pumps, for alkali metals, electromagnetic, 13-85 to 18-91 alternating-current, conduction, 13-89 induction, 13-89 to 13-91 direct-current, conduction, 13-85 to 13-89 mechanical, canned-rotor type, 13-84. 13-85 frozen sodium seals for, 13-85 sump type, 13-84, 13-85 for homogeneous aqueous reactors, cannedrotor, 18-68 circulating, 18-68 feed, 13-70 to 13-72 9-48, 9-49 mechanical, cavitation in. 9-48 dimensionless coefficients of. 9-48, 9-41) for water, 13-24 to 13-26 canned-rotor, 13-24 to 18-26 gas-seal, 13-25, 18-26 seals for, 13-24 submerged, 13-25. 13-26 Purex process, 11-82 PWR (see Reactors, particular)

35

Pyrometallurgical fuel reprocessing,

11-97 to

11-102

Q Q, of reaction, 4-42 Q, unit of energy, 13-85». world input in, 12-85 Quadratic equation. 3-45 formula for, 1-63 Quadrics. 3-28 to 3-30 Quadrupole. 4-4 Quadrupole moment of deuteron, 4-50 Quantization, 4-12 of angular momentum, 4-20 rules of, 4-12 space, 4-21 4-19 Quantum electrodynamics. Quantum mechanics, 4-14 to 4-33 origins of, 4-14 Quantum number, 4-12, 4-21 magnetic, 4-23 principal, 4-13 Quantum theory, origin of, 4-4 Qviartz, effect of radiation on, 10-123 Quartz-fiber electrometer, 5-7 K

R-l

(aee Reactors, particular) Rabbit for radiation testing, 5-140, 6-141 Rack-and-pinion control drive. 8-70. 8-71 Rad, definition of. 7-24. 7-65 Rad equivalent man (rem). 7-24, 7-65 Radian measure, 3-19 conversion

factors for (table).

1-145

Radiation, black-body, 1-59. 4-4. 4-5 ferenkov, 4-C4, 4-65

detection of (see Detectors) effect of, on chemical reactions. 14-8 electromagnetic. 4-2 to 4-9 of heat, by gases, 1-59. 1-60 by surfaces. 1-59 ionizing effect of, 6-2 to 6-4 natural background (table), 7-36 nuclear (see Radioactivity) reduction of life span caused by. 7-28 thermal (see heat, above) units of (nuclear), 7-6, 7-7. 7-23 to 7-25 Radiation damage, to aqueous solutions (table), 10-131 to 10-134 to beryllium oxide, 10-115. 10-116 thermal resistivity (tables). 10-116 to components (tables). 10-141. 10-144 to 10-148 to covalent crystals, 10-123 to elastomers (table). 10-135. 10-140 to electrical systems (table), 10-141. 10-144 to 10-148 to electronic systems (table). 10-141. 10-144 to 10-148 to fused salts (table), 10-131 to 10-134 to glosses, 10-124 to graphite, 10-109 to 10-116 dimensional change, 10-110 to 10-112 mechanism of, 10-111 to 10-115 modulus of elasticity change, 10-115 recovery of, 10-104. 10-111 to 10-116 stored energy . 10- 110 to 10-115 thermal conductivity change. 10-112, 10-113 to human body (see Radiation injury to the body) to ionic crystals, 10-123 mechanism of damage in. 10-105

INDEX

36


Radiation damage, to liquids, 10-131 tu 10-136 energy absorption, calculation of, 10-126 to 10-134 energy to break a bond, 10-131 O, 10-131 summary of method of calculation, 1-62. 1-63 mechanism of, in solids, 10-09 to 10-105 atom displacements, 10-99 to 10-105 fission-product atoms (table), 10-102, 10-103 in graphite, 10-111, 10-115 interstitialcy vacancy pairs, 10-101, 10-102 ionic crystals, 10-105 recovery, 10-104 temperature ranges for (table), 10-104 spikes, 10-102 in uranium, 10-118. 10-119 to optical materials, 7-123 to organio fluids (table), 10-134 to 10-148 to organic materials, 10-134 to 10-148* energy absorption, calculation of, 10-129 to 10-134 energy to break a bond, 10-131 mechanism of, 10-126 to 10-129 summary of methods of calculation, 1-62, 1-63 to packings. 13-23 to plastics (table), 10-141 to 10-143 to pulse chambers, 8-56, 8-59 recovery from, mechanism of, 10-104, 10-106 temperature ranges for (table), 10-104 to semiconductors, 10-122, 10-123 to solids, 10-83 to 10-125 mechanism of, 10-99 to 10-105 to structural metals, 10-105 to 10-109 electrical properties, 10-105 resistivity (table), 10-108 heat conductivity, 10-105 mechanical properties (table), 10-105 to 10-107 metallographic condition, 10-105, 10-109 thermal conductivity, 10-105 thermoelectric power (table), 10-108 to systems (table), 10-144 to 10-148 to Teflon, 18-23 10-122, 10-123 to transistors, electrical resistance (table), 10-123 to uranium, 10-117 to 10-119, 12-66. 18-67 growth coefficient, G (tabic), 10-117, 10-118 to uranium alloys, 10-119 to 10-121, 11-67, 12-68 aluminum-base, 10-120, 10-121 beryllium-base, 10-121 uranium-base, 10-119, 10-120 to uranium dioxide, UOi, 10-121, 12-68 dispersion, in aluminum, 10-121 in stainless steel (table), 10-121 to uranium-dioxide— beryllia, UOi-BeO, 10-121 to uranium-dioxide-graphite (tables), 10-121. 10-122 to water (see Water, radiolytic decomposition of)

Radiation decomposition in homogeneous

aqueous reactors, 13-58, 13-59 catalysts to limit, 13-59 rate of, 13-58, 13-59 uranyl peroxide formation, 13-58, 13-69 (See also Water, radiolytic decomposition of) Radiation detection {see Counting; Detectors) Radiation dose, 7-65 maximum permissible, summary, 1-42, 1-43 units of, summary table, 1-42 (See altto Dose; Dose rate)

Radiation fatalities, 7-27 Radiation growth (see Radiation damage) Radiation injury to the body, 7-22 to T-45 linear types. 7-25 to 7-29 threshold types, 7-27 to T-29 Radiation monitoring, 7-45 to 7-48 Radiation sickness, 7-27 Radiation testing (see Irradiation testing) Radiative capture, 4-95 to 4-99 cross sections for (see Cross sections, radiativ* capture) description of, 4-72 Radicals (mathematics), 3-4 chemical, as cause of injury, 7-23 Radicand, definition of, 3-4 Radioactive contamination, 13-5 airborne, remedial measures for, 7-53 precautions against, 7-52 Radioactive decay (see Radioactivity) Radioactive effluents, airborne, from processing plants,

7-50, 7-51

from reactors, 7-50 {See also Radioactive wastes; Radioactivity ) Radioactive materials, handling of. 7-55. 7-11? to 7-142

in canals, 7-117 canyon crane for, 7-136 in caves, 7-118, 7-119, 7-128. 7-129 in cells, 7-118, 7-119. 7-128. 7-129 enclosures for, 7-125 to 7-129 fuel (tee Fuel handling) 7 and 0 or a emitters, 7-119 7 emitters, 7-118 7-free materials,

7-118

in glove boxes, 7-136 in hoods, 7-118 in hot caves, 7-118, 7-119, 7-128, 7-129 maintenance of equipment in processing plants, 7-123, 7-124, 7-136, 7-137 manipulators for, 7-120, 7-121, T-130to 7-134 reactor, fuel servicing (see Fuel handling t ventilation, 7-119, 7-120 viewing, 7-121 to 7-123, 7-134 to 7-136 shipping of, 7-55 Radioactive particles, internal dose, dose rate from, 7-51 to 7-52 Radioactive turbine contamination with directcycle boiling reactors (table). 13-8 to 13-10 Radioactive wastes, airborne, 7-50, 7-51 from chemical separation plant, 11-144, 11-14: disposal of. 11-140 to 11-146 baling, 11-140 to 11-146 evaporation, 11-145 fiocculation, 11-145 gases in chemical reprocessing, 11-113. 11-114

from homogeneous aqueous reactors. 13-62 fission-product adsorbers for, 13-7 1 to 13-73

incineration, 11-145 ion exchange,

11-145

solid, 7-54, 7-55 solidification, 11-146 evaporation of (table), 11-104 to 11-106 liquid, from processing operations. 7-53, 7-54 from reactor, 7-52, 7-63 storage of, 7-63. 7-64

from production facilities, 11-144 to 11-145 from research laboratories (table). 11-142 to 11-144 storage of, 7-53, 7-54 treatment of, 11-145. 11-146 Radioactivity, activation and decay summary (table). 1-37, 1-38

formulas,

INDEX


Radioactivity, background calculatioo of, 7-4, 7-5

(table),

7-36

summary (table), 1-37, 1-38 of coolants (see reactor coolants, below) decay, a, 4-65 to 4-67 elementary description of, 7-2 energetics of, 4-66 theory of, 4-66, 4-67 0, elementary description of, 7-2, 7-3, 11-8 energetics of, 4-47 Sargent curves, 4-67 theory of, 4-68, 4-69 of compound nucleus, 4-76, 4-76 of fission products (see fission products, below) nature of, 4-34 decay chains, calculation of, 7-4, 7-5 decay constant X, definition of, 4-34 decay y rays from fission products, 7-72 decay schemes, theory of, 4-70 of fission products, 7-14 to 7-17 decay data for (tables), 7-16, 7-17. 11-12 to 11-24 energy group data, 7-75 heat generation from, 135-day irradiated. 11-26, 11-27, 11-33 0 heat by nuclides (table), 11-32 7 heat by nuclides (table), 11-32 Untermyer-Weilla formula, 7-15. 7-17 table for, 1-39 Way-Wigner formula, 7-15 (See also Fission products) formulas for decay, 7-4, 7-5 summary (table), 1-37, 1-38 of graphite from reactor (table), 10-94 half-life, definition of, 1-36, 7-4 of nuclides (tables), 7-8 to 7-12, 11-12 to 11-24 intermittent activation, calculation for. 7-1 4 mean life, definition of, 1-37, 7-4 natural background (table), 7-36 of naturally radioactive nuclides, 4-34 to 4-38 of reactor coolants, 1-40, 1-41. 7-17, 7-18 air, 7-19, 7-20, 7-50 13-118, 18-119 graphite-moderated, alkali metals, 7-20, 7-21 bismuth, 7-21 boiling water, 7-19 carbon dioxide, 7-20 helium, 7-20 lead, 7-21 lithium, 7-21 potassium, 7-21 sodium. 7-21 summary tables, 1-40, 1-41 water, 7-17 to 7-19, 18-5 boiling, 7-19 of structural metals from reactors (table). 10-94 types of, elementary description of, 7-2 to 7-4 Radiochemical analysis, 11-50 to 11-57 activation analysis in, 11-56 activity measurement in, 11-55, 11-56 counting techniques for, 11-53 to 11-55 isotope dilution methods of, 11-56, 11-57 radiometric analysis in, 11-57 separations in, 11-51 to 11-53 analysis) Radiochemistry (see Radiochemical Radiographic inspection, 10-154 to 10-157 gauging, 10-162 to 10-164 Radiography, use of radioisotopes for, 14-3 Radioisotope separations procedures (tee Radio isotopes, production of) Radioisotopes, availability of, 14-13, 14-14

37

Radioisotopes, distribution of, 14-2, 14-12 to 14-14 licensing use of, 14-12, 14-13 production of, 14-26 to 14-37 burnup, of product, calculation of, 14-78 of target, calculation of, 14-28 carbon- 14, 14-33 carrier-free separations, 14-31 chemical processing in (table), 14-31 to 14-36 complex chains in, 14-29 to 14-33 equipment for, 14-36, 14-37 facilities for, 14-36, 14-37 from fission products, 14-33 iodinc-131, 14-32, 14-33 methods of irradiation, 14-26, 14-27 mono iso to pic targets, 14-34 (n,7) products, 14-33 phosphorus-32, 14-32 poly iso topic targets, 14-29 to 14-33 rate of, calculation of, 14-27 from fission, 14-28 saturation activity in, 14-27 specific activities (table), 14-34 to 14-36 sulfur-35, 14-33 summary table, 14-34 to 14-36 target material for, 14-26 mouoiso topic, 14-27 polyisotopic, 14-29 to 14-33 self-absorption in, 14-29 toxicity of, compared with chemical poisons (tables), 8-77, 8-78 use of, 14-3 to 14-12 for accelerating chemical reactions. 14-8 14-11, 14-12 for agricultural applications, for batteries, 14-9, 14-10 for C:H ratio measurement, 14-4, 14-5 for cross linking of polymers, 14-8 for flow measurement, 14-6, 14-7 for gauging (table), 14-3 to 14-6 14-10, 14-11 for medical applications, for process control, 14-7, 14-8 for radiography, 14-3 for sterilization, 14-8, 14-9 for tracing, 14-6 to 14-8 for wear measurement, 14-7 Radiolytic decomposition of water (set Water) Radiometric analyses, 11-57 Radium, physical properties of (table), 11-6 Radius, of convergence, 3-99 nuclear, 4-43 Radius vector, definition of, 8-16 Radon, physical properties of (table). 11-6 Raffinate, definition of, 11-65 Ramsauer effect, 6-11 Random variable, 8-59 Range, of a particles, in air, 3-34, 6-3, 6-4. 7-5, 7-34 high-energy, in metals (table). 10-98 in materials, 2-34 in soft tissue, 7-34 of 0 particles, in air, 2-34. 7-5, 7-34 in aluminum, 2-34 in soft tissue, 7-34 of deuterons, high-energy, in metals (table), 10-98 of electrons (see 0 particles, above) of high-energy particles in metals (table), 10-98 of protons, 2-34. 6-4 high-energy, in metals (table), 10-98 Rankine cycle, 12-3, 12-6, 12-123, 12-124, 12-126 to 12-137 Raoult's law, 11-83 Rare-earth metals, properties of (table), 10-71 Rare earths, as control materials (table), 10-188, 10-189. 10-191

INDEX

38 Rare earths, fission-product, 11-45, 11-40 isotopic separation price of, 12-108

of, 14-57

Rated capacity, definition of, 1S-91 Ratio, definition of, 3-3 Rational numbers, 3-2, 3-75 Rayleigh-Jeans law, 4-5 RBE (see Relative biological effectiveness) Reaction cross sections, measurement of, 6-68 tu 6-70


Reactions, nuclear (see Nuclear reactions) Reactivity, 6-110, 8-6 change, with burnup, 13-82 with time in natural uranium reactors, 12-11, 12-12 effect of lumping on, 12-10, 12-1 1 effect of step increase of. 6-79 equation for, with relative importance factors, 6-112 measurement of, 6-112, 6*113 pressure coefficient of, 8-12 of reflected reactor, numerical example of, 6-99 to 6-103 temperature coefficient of. 8-11 units of, 6-113, 8-8. 6-9 conversion factors for, l-4ti Reactivity coefficient measurement. 6-93, 6-115 Reactor constants (see Nuclear constants for reac tor design) Reactor Hazard Evaluation Staff. 8-75 Reactor period (see Period) Reactor Safeguard Committee. 8-76. 12-7 (See also Safeguards) Reactors, autocatalytic, 12-20 buildings for (sec Buildings for reactors) catalogue of. 13-123 to 18-159 (See also Catalogue of reactors) charging and discharging devices for i*er Fuel

handling) classification of (table), 12-6 to 12-8, 13-123 by application (table), 12-6, 12-7 by coolant (table), 12-7, 12-8

(table), 12-6 to 12-8 enrichment fuel, 12-8 heterogeneous (table), 12-6, 12-7 homogeneous (table), 12-6, 12-7 mobility of fuel (table), 12-7, 12-8 moderators (table), 12-7, 12-8 neutron spectrum (table). 12-6, 12-7 construction of (see Construction of reactors) control of (see Control of reactors) coolants for (see Coolants) core construction of, 13-177 to 13-178 critical (see Physics of critical reactors) distribution of heat generation in, 9-51, 9-52 for power and isotope produc dual-purpose, tion, 12-111. 12-112 lis element of control system, 8-33, 8-35 to 8-38 factors influencing selection of, 12-8 to 12-22 corrosion of fuel elements. 12-18 fertile materials, 12-10 (tables), 12-9. 12-10 fuel characteristics fuel mobility. 12-17, 12-18 heat removal, 12-13 to 12-17 flow uriffcing, 12-17 heat transfer to coolant, 12-14, 12-15 through fuel element, heat transmission 12-13, 12-14 heat transport by coolant, 12-14, 12-15 hot-spot factors, 12-16. 12-17 operating pressure and temperature, 12-16 lumping, effect of, on conversion, 12-10, 12-1 1 on reactivity, 12-10, 12-11 neutron balance (table), 12-9 by by as as by by by

Reactors, factors influencing selection of, repro

cessing of fuel, 12-18 to 12-19 safety, 12-19 to 12-22 containment, 12-19 delayed neutrons, 12-20 to 12-22 neutron lifetime (table), 12-20 to 12-22 temperature coefficient, 12-19 to 12-20 void coefficient. 12-20 structural materials, permissible quantities of (table), 12-11 fast (see Power reactors, fast) fuels for (see Fuels) hazards in (see Safeguards) kinetics of (sec Physics of noncritical reactors; neutron lifetime (see Neutron lifetime) non-autocatalytic. 12-20 noncritical, 6-110 (See also Physics of noncritical reactors) operating pressure of, general considerations, 12-16 operating temperature of, general consider ations, 12-16 operation of (see Control of reactors; Control systems of reactors) particular, AGN 201. description of. 13-135

ANP, Aircraft Nuclear Propulsion (tabled 12- 53

APDA, Atomic Power Development

Associ ated (see Enrico Fermi, below) Package Power Reactor. 13-3*. 13-37 description of, 13-138 APS. Atomic Power Station, description oi 13- 134

APPR, Army

ARBOR. Argonne Boiling Reactor Facility description

of, 13-138

Argonaut, description of, 13-136 ARR, Armour Research Reactor, building for. 13-164. 13-165 description of, 13-130 experimental facilities in, 10-91, 10-92 plant space requirements for. 13-172 Bard well 1, 2, capacity and date of operate t of, 13-105 Battelle Research Reactor, description of 13-133 BEPO, British Experimental Pile, descrip tion of, 13-124 graphite stacking in. 13-141, 13-142 power, 13-105 Borax I, Boiling-reactor experiment, descrip tion of, 13-133 description of. 12-134 Borax II, Borax

III.

BPR, Beryllium Physics Reactor, descrip tion of, 13-135

BR-2, Experimental Fast Neutron Reactor description

of, 13-136

BR-3, plant space requirements for, 13-172 British Experimental Pile (see BEPO, abme) Brookhaven Graphite Research Reactor,

BNL,

air cooling

in, 13-1 10. 1S-116

air system for, 13-109 building for, 13-160

date of, 13-105 description of, 13-125 table. 12-55 experimental facilities in (table). 10-89 10-90 filters for, 13-110 monitoring for fuel-element failure. 13-117 power of, 13-105 power requirements for, 13-105 pressure drop in, 13-113 shutdown cooling in, 13-110

INDEX


Reactors, particular, Brookhaven Medical Research Reactor, description of, 13-137 bulk shielding facility, description of, 18-131 sectional drawing of, 13-155 table, 12-55 Calder Hall reactors, A-l, A-2, B-l, B-2, building for, 13-160 capacity and date of operation of. 13-105 description of, 13-125 design data for (table). 12-34, 12-35 CETR, Consolidated Edison Thorium Reac tor (see Consolidated Edison Indian Point, below) Chalk River (see NRX, below) Chapel Cross, A-l, A-2, B-l, B-2, capacity and date of operation of, 13-105 Chugach Electric Association Reactor, de scription of, 13-138 City of Piqua reactor, cost of, 12-96 Clementine (see Los Alamos Fast Reactor, below) Consolidated Edison Indian Point reactor, cost of, 12-96 description of, 13-138 plant space requirements for, 13-172 Consumers Public Power District (sodium graphite) reactor, plant space requirements for, 13-172 CP-1, description of, 13-124 CP-2, building for, 13-160 description of, 13-124 CP-3, arrangement drawing of, 13-40 building for, 13-160 description of, 13-126 CP-3' (CP3 modified), arrangement drawing of, 13-40 description of, 18-126 fuel element for, 12-64 CP-6, building for, 13-162 to 13-164. 13-175 description of, 13-127 table, 12-55 experimental facilities in (table), 10-91 fuel elements for, 12-69 graphite for, machining and stacking of, 13-180, 13-181 plant space requirements for, 13-172 shield, pouring of, 13-182 to 13-184 CP-6 (see Savannah River Reactors, below) DIDO, description of. 13-128 Pile, Low Dimple deuterium-moderated Energy, description of. 13-128 Dounreay, description of, 13-140 Dresden, description of, 13-42 to 13-44 plant arrangement of, 12-115 to 12-117 plant space requirements for. 13-172 EBR-I, Experimental Breeder Reactor, core, composition of (table). 2-31 description of. 13-136 heat exchangers for, 18-91, 13-92 horizontal section of, 13-150 neutron spectrum in (table), 2-31 plant layout of, 13-156 pumps for, 13-88, 13-89 electromagnetic, 13-88. 18-89 mechanical, 13-85 13-91, 13-92 for, steam generator EBR II, fuel elements for. 12-65 EBWR, Experimental Boiling Water Reac drawing of, 18-47 tor, arrangement building for, 13-167 to 13-173, 13-175 condenser vents for. 12-143. 12-144 DiO recovery system for, 12-143, 12-144 description of, 13-134 fuel elements for, 12-66, 12-67

39

Reactors, particular, EBWR, load control in. 12- 146, 12-147

plant space requirements

of, 18-172 sectional elevation of, 12-40 in, 12-151 water decomposition EDF-1, 2, (table). 13-106 Elk River Reactor, cost of. 12-90 description of, 13-138 Enrico Fermi sodium-cooled fast reactor, building for, 12-32 cost of, 12-96 description of (table), 12-27 to 12-32 fuel elements for, 12-62, 12-65 plant space requirements of, 13-172 ETR, Engineering Test Reactor, description of, 13-137 table, 12-55

Fermi (see Enrico Fermi, above) Ford Nuclear Reactor, description G-l. description of, 18-126

of, 13-133

table, 13-106 G-2, G-3 (table), 13-106 General Electric Nuclear Test Reactor. description of, 13-135

GLEEP, Graphite Low Energy Experimental

Pile, description of, 13-124 Godiva, core composition of (table) 2-31 neutron

spectrum

in (table)

2-31

transport cross sections in (table) 2-31 Hanford, biological shield, placement of. 13- 183

description of, 18-125 fuel element

for, 12-63

Hanford 305 Test Reactor, description of, 13-125

Harwell, heat recovery from, 13-109, 13-111 Hinkley Point 1, 2, capacity and date of oper ation of, 13-105 Homogeneous Aqueous Reactor (see HRE, below) Homogeneous

HRE,

Reactor Experiment, description of, 13-130

table, 12-52 flame recombiner for, 18-69 reactor vessel for, 13-65 HRT. homogeneous reactor test, building for, 12-166 pressurizer for, 13-68 steam generator for, 13-67 Hunterston 1. 2, capacity and date of opera tion of, 13-105 HYPO. High-power Water Boiler, descrip tion of. 13-129 JEEP, description of, 13-127 sectional drawings of. 13-149, 13-150

KEWB, Kinetic Experiment Water Boiler, description of, 13-130

LAMPRE, Los Alamos Molten Plutonium

Reactor Experiment, brief description of, 12- 33

LAPRE-I, LAPRE-II,

Los Alamos Power Reactor Experiment, description of,

13- 130, 13-131

LIDO, description of, 13-133 LITR, Low-intensity Test Reactor,

descrip

tion of, 13-131 experimental facilities in (table), 10-84, 10-85

Livermore Water Boiler, description of. 13-129

LMFR, Liquid-metal-fuel Reactor, tion of, 12-47 to 12-51 table, 12-53

descrip

INDEX

40


Reactors,

particular* Los Alamos Fast Reactor, "Clementine," 13-98

description of, 13-136 Massachusetts Institute of Technology Reac tor, MITR, description of, 13-137 shield, pouring of, 13-182 to 13-184 Merchant Ship Savannah, description of, 13-138 MTR, Materials Testing Reactor, arrange ment drawings of, 13-37, 13-38 description of, 13-131 table, 12-55 experimental facilities in (tables), 10-83 to 10-88 flux distribution in, 10-87, 10-88 fuel element for, 12-68 fuel handling in, 7-137 sectional drawings of, 13-154 Marcoule (we G-l, G-2, G-3, above) NCSC, North Carolina State College Reac tor, description of, 13-129 Nebraska Sodium Graphite Reactor, cost of 12-96 Northern States Power Company Reactor, description of, 13-138 NRR, Naval Research Laboratory Reactor. description of, 13-133 NRU, description of, 13-128 NRX, arrangement drawing of, 13-34 description of, 13-127 table, 12-55 experimental facilities in (table), 10-90, 10-91 fuel-element tube for, 13-35 NTR, arrangement drawing of, 13-39 Oak Ridge Graphite Reactor (see X-10, below) OMRE, Organic Moderated Reactor Experi ment, description of, 13-138 ORR, Oak Ridge Research Reactor, biologi cal shield, placement of, 13-183 building for, 13-160, 13-161 description of, 13-137 OWR, Omega West Reactor, description of, 18-132 P-2, description of, 13-127 drawings of, 13-151 to 13-152 Pennsylvania State University Reactor, description of, 13-132 PRDC fast reactor, Power Reactor Develop ment Company (see Enrico Fermi, above) PWR, Shippingport Pressurized Water Reac tor, description of, 13-138 emergency cooling of building for, 13-175 fuel elements for, 12-61, 12-62 plant arrangement of, 12-115 plant apace requirements for, 13-172 power cost from, 12-108 R-l, building for, 13-148 description of, 13-128 vertical section of, 13-148 RFT, Experimental Nuclear Reactor, description of, 13-132 RMF, Reactivity Measurement Facility, description of, 13-132 RPT, Reactor for Physical and Technical In vestigations, description of, 13-134 Russia (see USSR, below) Saclay, description of, 13-127 drawings of, 13-151, 13-152 Santa Susana (see SRE, below) Savannah River Reactors, description of, 13-128 fuel elements for, 12-63

Reactors, particular, SC-WR, Supercritical Water Reactor (table). 12-53 SIR, Submarine Intermediate Reactor, (S1G), building for, 13-166. 13-167

description of, 13-136 table, 12-52, 12-53 SPERT-1, Special Power Excursion Reactor Tests 1, description of. 13-134 SRE, Sodium Reactor Experiment, building for, 13-161 description of, 12-35 to 12-37, 13-138 fuel element for. 12-65 STR, Submarine Thermal Reactor (S1W). description of. 13-133 table, 12-51 SUPO, Superpower Water Boiler Reactor, description of, 13-129 table, 12-55 recombining system for, 13-153 vertical section of, 13-153 SlG (see SIR, above) S1W. Submarine Water (see STR. above)

TREAT, Transient Reactor Test Experi ment, 13-139

Trombay Pool Reactor, description of, 13-133 TTR, Thermal Test Reactor, description of 13-135

USSR, first power reactor, schematic vertie*i section of, 13-156

Heavy-water Research Reactor, descrip tion of, 13-127 Reactor, building for. Pressurized-water 13-160 in USA, other, 13-137 to 13-139

WBNS. Water-boiler Neutron Source. description

of, 13-129

Windscale, blower position in system of. — 13-109 description of, 13-125 location of filters in, 13-110 vertical section of plant for, 13-143 world developments in, 13-139 to 13-141 X-10, Oak Ridge Graphite Reactor, air ooz ing in, 13-110 air system for, 13-109 date of, 13-105 description of, 13-124 table, 12-55 experimental facilities in (table). 10-84 10-89 Biters for, 13-109, 13-110 graphite stacking in, 13-141. 13-14monitoring of air in, 13-117 power of, 13-105 power requirements for, 13-119 pressure drop in, 13-113 seal pits in, 13-110, 13-111 shutdown cooling for, 13-1 10 Yankee Atomic Electric Companv Reactor 13-138 cost of, 12-96 ZEEP, description of, 13-126 ZEPHYR, Zero Energy Fast Reactor description of, 13-136 ZOE, description of. 13-126 physics of (see Physics of critical reactor* Physics of noncritical reactors^ power (see Power reactors) power cycles for, direct, 12-6 indirect, 12-6 link, lt-6 intermediate as radiation sources, 10-83 to 10-92 relief valves for water, 13-29 for remote locations, 13-111

INDEX Reactors, research (see Research reactors) safety of, comparison of fast and thermal reac tors (tables). 12-21. 12-22 (See also Safeguards) scramming of, 8-43 to 8-46 selection of, 12-s to 12-22 servicing of. 7-124, 7-125 for ship propulsion, 12-11 1 shutdown of, 8-42 to 8-46 devices for, 8-82 startup of (nee Control of reactors) structural material permissible in (table), 12-11 transfer functions of, 8-33 to 8-39 vessels for (nee Pressure vessels) Reciprocal, definition of, 3-10 Reciprocal coordinate systems, 3-14 for homogeneous aqueous reactors. Recombiners 13-68 to 13-70 Recovery from radiation damage, 10-t04, 10-105 Recovery time in Geiger-M tiller counters, 8-15 Rectification. 11-86 to 11-89 Rectilinear coordinates, 3-32


Rectilinear electric manupulator. 7-131 Redox process, 11-82 Reduced mass, 4-30 Reed electrometer, vibrating, 5-8, 5-9 Reenrichment of fuel, 12-102, 12-103 Reflector, construction of, graphite, machining and stacking of, 13-180. 13-181 beryllium, 13-180, 13-181 function of, 6-60 Reflector savings, 6-62, 6-69 from exponential experiments, 5-118 Refractaloy (see High-temperature alloys) Regeneration factor (see Neutron regeneration) Regulating rods, 8-21 (See also Control rod; Control-rod drives) Relative absorption, 6-79

Relative biological effectiveness 7-27

(RBE).

7-25 to

definition of, 7-25, 7-65 dose, from radioactive particles. 7-51 table. 7-27 Relative roughness, definition of, 9-32 Relativistic fine structure of hydrogen atom, 4-12, 4-25 Relativistic mechanics, 4-7 to 4-9 Relativistic neutrons, definition of, 5-53 Relativity, Bpecial, mechanics of, 5-124 theory of, 4-6 to 4-9 Relaxation, method of, 3-39 numerical solution by, 3-123 Relaxation lengths, for fast neutron*, 2-34, 2-35 for y rays, 2-34, 2-35, 11-136 conversion factors for, 1-44 Relaxation method for heat flow, 9-97 Rem (roentgen equivalent man, rad equivalent man), definition of. 7-24, 7-65 Remainder theorem, 3-41 Remote control in processing and refabrication of fuel. 7-124 Remote-control handling (see Radioactive ma terials, handling of) Remote maintenance, 11-137 Removal cross sections, effective (table), 7-81 to 7-83 Rep (roentgen equivalent physical), definition of, 7-23, 7-24 Replacement spHces, 10-102 Reprocessing of fuel, cost of, 12-100 to 12-102 general considerations, 12-18, 12-19 (See also Chemical separation processes)

41

Research reactors, air-cooled graphite-moderated, activity of air from (table), 18-118. 13-119

blowers for, 18-1 10 chimneys for, 13-118, 13-119 constructional features of, 13-141 to 13-142 decontamination of equipment in, 18-118 filters for, 13-109, 13-110 graphite stacking for, 13-141. 13-142, 13-180. 13-181 heat utilization from, 13-111 instrumentation for, 13-116 flow meters for, 13-116 monitoring of air for radioactivity in, 13-117 power requirements for, 13-1 19 seals for, 13-110 shutdown cooling for, 13-110 valves for, 13-110 comparison of types of (table). 12-54, 12-55 heavy-water (see water-cooled, below) space requirements for (table), 13-172 DjO moderated, 13-16, 13-17, water-cooled, 13-142 to 13-149 DjO cooled, 13-37. 13-38 HiO cooled, 13-22. 13-23 leakage from, 13-39 homogeneous aqueous, HiO moderated, 13-35 to 13-150, 13-154. 13-155 swimming-pool, 13-35, 13-36 water boilers, 13-50 reactor vessel, 13-66 reactors (table), 13-129, 13-130, 13-153 Reserves of nuclear fuels. 12-88. 12-89 (See also World energy resources) Residue theorem, 3-84 Resins, ion-exchange (table), 11-93 to 11-95 performance in fuel reprocessing, 11-96, 11-97 Resonance, 4-75 Breit-Wigner formula for Bingle-level, 5-77. 5-78 measurement of, 6-77 to 5-80 Dopplcr effect, 6-78, 6-79 moderator resonance constants, 6-83 theory of, 4-93 Resonance absorption integrals (tables), 2-24 to 2-27, 6-47, 6-48 definition of, 5-80 effective, 6-48 for lattices, 2-26. 2-27 measurement of, 6-82, 5-83 for homogeneous mixtures, 2-24 to 2-27 for lattices, 2-26. 2-27 measurement of, 5-81 to 5-83 parameters for, p, it/ A, X<> for ThMI, U111 (table). 2-27 It/ A for uranium mixtures (table), 2-27 Resonance activation integrals of materials used in neutron detection (table), 6-41, 6-42 Resonance capture, theory of, 4-95 Resonance constants for natural uranium. 6-82 Resonance disadvantage factor, 6-82 Resonance escape probability, 6-22 calculation of, 6-80 to 6-83 for homogeneous reactor, 6-80, 6-81 for lumped reactor, 6-80 to 6-83 numerical example of. 6-67 summary, 1-33 Resonance integral (see Resonance absorption integrals) Resonance neutrons, 6-72 to 6-83 definition of, 5-53 detection of, 5-41, 5-42 (See also Cross sections) Resonance region, theory of, 4-79, 4-80 Resonance scattering of neutrons. 4-94

INDEX

42 Reynolds number,

definition of, 9-32 description of, 1S-132 Rhenium, &b control material (table). 10-180 physical properties of. 11-6 Rhodium, chemistry of fission-product. 11-48. 11-49 as control material (table). 10-188 physical properties of (table), 11-6 Rhythmic effect of radiation dose. 7-27 Riccati equation, 3-108 Riemann sphere, 3-80 Riemann surfaces, 3-85, 3-86 Ring geometry in scattering cross-section meas urement, 6-64 Rise, prompt, 1-48, 6-115. 8-9 to 8-10 Roentgen, definition of, 7-7, 7-23 Roentgen equivalent man (rem), definition of, 7-24 Roentgen equivalent physical (rep), definition of, 7-23, 7-24 Rolle's theorem, 3-44 Roots, characteristic, 3-51 latent, definition of, 3-29, 3-51 to 3-55 lower bounds for, 3-44 upper bounds for, 3-44 Rotameters, 11-118 Roughness, in fluid flow, 9-31 for commercial materials, 9-32 relative, definition of, 9-32 wall, in streamline flow. 9-31 Routine in digital computers, 9-143 Rubber, radiation damage to (table), 10-140 Rubidium, chemistry of fission-product. 10-45 physical properties of (table), 11-6 Runge-Kutta method, 3-123 Rupture, theory of, 10-15 Rupture diaphragms for water reactors, 13-29 Russian reactors (see Reactors, particular, USSR) Ruthenium, chemistry of fission-product, 11-48. 11-49 of, isotopic separation 14-57 physical properties of (table), 11-6 Ruthenium-106 for thickness gauging, 14-6 Rutherford atom, 4-10 Rutherford scattering, 4-10n. R.vberg constant, in Bohr theory, 4-11 value of, 1-154


RFT,

S s wave, 4-77 Saclay reactor, 13-127, 13-151. 13-152 Safeguards (reactor), 8-76 to 8-88, 13-19 to 13-22 cold-water accident, 8-88 containment, 8-84, 8-85, 13-19 control safety requirements, 8-81. 8-82 interlocks, 8-82 delayed neutron contribution to, 13-20 to 13-22 storage, 8-87, 8-88 fission-product location, 8-85 to 8-87 geology, 8-87 hydrology, 8-86, 8-87 meteorology, 8-86 seismology, 8-87 loss of coolant, 8-80 maintenance, 8-83, 8-84 neutron lifetime (table), 3-20 to 8-22 nuclear runaway, consequences of, 8-78 to 8-80 contributing causes of, 8-78 to 8-80, 8-88 stabilizing influences on. 8-79, 8-80, 8-88 operational requirements, 8-82 to 8-84 functions of operators, 8-82 functions of supervision, 8-83 regulating bodies. 8-76 to 8-88

Safeguards (reactor), temperature coefficient 13-19, 13-20 toxicity of reactor products. 8-77. 8-78 void coefficient. 13-20 Safety circuits, 8-44 to 8-46. 8-59 Safety rods 8-21 Salting agent, definition of. 11-86 Salts, fused, radiation damage to (table). 10-13! to 10-134 Samarium, as control material (table). 10-188 physical properties of (table). ll-*i Samarium- 149. absorption cress section of (tables), 1-23. 3-10, 3-11 concentration of. 8-27 to 8-29 fission yield of (tables). 1-23, 3-10. 3-11 poisoning, 8-27 to 8-29 Sampling in fuel reprocessing, 11-138. 11-13$ Sargent curves, 4-67 Saturation activity, 6-56 Saturation current in ionizing chamber*. 6-7 Savannah River Reactors, 13-63. 13-128

SC-WR, Supercritical Water Reactor (table;.

13-55 Scalar product, 3-13, 3-125 Scalars, 3-7 algebra of, 3-2 to 8-12 Scandium, physical properties of (table). 11-6 Scattering, charged particles in force field. 4-2S to 4-33 Compton (photons). 4-60, 4-61, 6-4 cross sections (see Cross sections, scattering! neutron, angular distribution in, 4-100. 4-101 measurement of, 6-63 to 5-fi5 elastic, 4-95 to 4-99 description of, 4-72 inelastic, description of, 4-72 measurement of, 6-65 to 6-68 theory of, 4-76 Scattering amplitude, 4-28, 4-98 Scattering length, 4-96 Scattering resonance, 4-95 Scavenger, 11-52, 11-60 Schmidt lines, 4-50 Schmidt method for unsteady heat flow. 9-99. 9-3S Schrodinger constant, 1-154 Schrddinger equation, 4-17 to 4-19 for many-body problem, 4-25. 4-26 Schrodinger wave function, 4-16 Schwarz-Christoffel transformation. 3-85 Schwarz's inequality, 3-68 Scintillation counters, for electrons. 9-32 for 7 counting in radiochemical analvsi*. 11-M. 11-55 for 7-ray spectroscopy, 6-38 phosphors for (table), 6-23, 6-24 efficiencies of, 6-23 photomultiplier tubes for, 6-25 to 6-28 for thermal neutrons, 6-41 for time-offlight measurement. 8-77 Scram, 8-43 to 8-46 Scram rods. 8-21 Scramming circuits, 8-44 to 8-46. 8-59 Scrubbing, definition of, 11-66 Seals, for alkali-metal systems, dynamic. 1S-S3 13-84, 13-101 static, 13-83, 13-84, 13-101 for control-rod drives, 8-68, 8-69 for feed pumps. 13-141 for gas-cooled reactors. 13-110 seal pits, 13-110, 13-111 for liquid-metal systems, dynamic. 13-10; static, 13-101 radiation damage to (table). 10-144 10-14* 10-147

INDEX Seals, for steam turbine*, 12-140 to 11-142 for water, 18-23, 13-24 labyrinth, 13-23, 13-24 packings,

13-23

Secondary container (see Buildings for reactors) Seidel method, 3-39 Seismology, 8-87 Seits, F., 10-101 Seizing of parts in sodium and NaK, 13-84 Selection of reactors (see Reactors, factors influ encing selection of) Selection rules in atomic electron structure, 4-13 4-26

Sclengut-Goertzel equation, 6-7 Selenium, chemistry of fission- product, 11-47 physical properties of (table), 11-6 Self-absorbed cross section, 6-20 Self-adjoint equation, 3-106 Self-protection of foils. 6-81 Self-quenching counter, 6-16 Self-shielded cross section. 6-26 S«>lf-welding in sodium and NaK, 13-84 Semiannual reports of USAEC to Congress, 1-74 Semiconductors, radiation damage to, 10-122. 10-123

Separation energy, 4-41 Separation factor, in boiling water, 13-5 in isotope chemical exchange, 14-18 Separation processes (see Chemical separation processes)


Separation of variables in boundary- value prob lems, 3-110 to 3-121

Sequence, 3-6 of rational numbers, 3-75 Serber-Wilson boundary, 6-17 Series,

8-6, 3-95 to 3-106

alternating, 3-95 infinite, 3-95

table of, 3-99, 3-100 (See also Functions) Set theory, 3-76 to 8-78 Sewers for reactor buildings, 13-176 Shadow cone in scattering cross-section measure ment, 6-64 Shadow scattering. 4-101 Shadow shield in reactor building, 8-85 Shaft seals for control-rod drives, 8-68, 8-69 Shell structure of atoms, 4-13, 4-27 of the elements (table), 10-2, 10-3 nuclear, 4-37 theory of nucleus, 4-50 to 4-53 Shield thickness for Co" and Cs137 sources (table), 7-68 Shielding, 7-60 to 7-116 biological (see Biological shield) calculation of, summary, 1-43 to 1-45 of chemical processing equipment, 11-135 to 11-137 concrete (see Concrete) against delayed neutrons, 7-85, 7-92 -ray attenuation, attenuation coefficients, 7-61 to 7-64 energy absorption, linear, 7-64 mass (table). 7-63, 7-04 total linear. 7-61, 7-64 for heavy metaU (table), 7-112 total mass (table), 7-62, 7-64 for heavy metals (tables), 7-112 attenuation processes, Compton scattering, 4-6, 4-60, 4-61, 6-4, 7-61 pair production, 4-56. 4-61, 7-61 photoelectric effect. 4-5. 4-59, 4-60. 7-61 build-up factor, 7-61 analytic expressions for (table). 7-99. 7-100

43

Shielding, v-ray attenuation, build-up factor, dose rate (table), 7-66 to 7-68 energy absorption, 7-66 to 7-71 table,

7-69

for complete shields, 7-70, 7-71 cylindrical source, 7-105 to 7-107 disk source, 7-104 geometry of, 7-60, 7-69 to 7-71 heat generation in, 7-66 lead thickness for Co", Cs"', 7-68 line source, 7-104 mechanism of, 4-58 to 4-62 plane source, 7-104 slab source. 7-104, 7-105 sources of y rays (see y rays, sources of) spherical source, 7-107 to 7-110 geometric configurations, 7-93 to 7-103 disk-source transformations, 7-93 to 7-95 quadric-source transformations, 7-96, 7-97 mathematical functions for, 7-102 to 7-103 transformations, 7-93 to 7-95 plane-source point-source transformations, 7-93 to 7-97, 7-101 spherical-shell transformations, 7-95 volume-distributed sources, 7-97 to 7-99 neutron attenuation, 7-78 to 7-90 by aqueous materials, 7-82 by boron, 7-77 by capture, 7-79 dominant energy, 7-80, 7-81 dose rate from secondary gammas, 7-89, 7-90 effective removal cross sections (table), 7-81 to 7-83 by hydrogen, 7-81 heat generation by, 7-85 to 7-89 by hydrogenous nonaqueous materials, 7-82 by inelastic scattering, 7-79 by lithium, 7-77 nitrogen for, 7-77 by nonhydrogenous materials, 7-84, 7-85 by particle reactions, 7-79 by water (table), 7-85 in postirradiation testing, 10-93, 10-94 of reactor building, 8-85 source geometry (see geometric configurations. above)

tenth thickness of materials (table), 11-130 thermal (tee Thermal shield) windows for, 7-121 to 7-123, 7-134. 7-130 materials for, 7-122, 7-123 radiation damage to (table), 7-122 Shielding materials, 7-109 to 7-115 concretes (table), 7-112 aggregates for (table), 7-113 barytes (tables). 7-112, 7-114, 7-115 cost of (tables). 7-114. 7-115 criteria for. 7-109 to 7-1 11 grouts for (table), 7-113 high-density (tables), 7-114, 7-115 half thickness of, 11-130 heavy metals (table), 7-112 (table), 7-111 hydrogenous n, or absorbers (table), 7-111 relaxation length (tables), 8-34, 3-35. 11-136 conversion factors for, 1-44 tenth thickness of, 11-136 thermal stress properties of (table), 9-113 for windows, 7-122, 7-123 radiation damage to (table), 7-122 Shim rods, 8-21 (See also Control rod) Ship propulsion, nuclear, 13-111 Shipping, of irradiated fuel, cost of, 13-101 of radioactive materials, 7-55

INDEX

44 Shipping, (if radioactive wastes, 7-55 Shutdown of reactor, 8-42 to 8-46 (See also Control of reactors) Shutdown cooling, 8-80 of air-cooled reactors, 13-110 of irradiated fuel, 18-100, 12-101 of liquid-metal-cooled reactors, 13-100 of water-cooled reactors, 13-29, 13-30 Shutdown devices, 8-82 Shuttle for radiation testing, 5-140, 5-141 Sigma

bus, 8-45, 8-46

Silica gel cation exchanger, 11-93 Silicon, physical properties of (table), 11-6 radiation effect on electrical resistance of, 10-125

Silicon carbide,

mechanical

properties

of (table),

10-54

physical properties of (table), 10-53 Silicon-uranium alloys (see Uranium alloys) Silicones, radiation damago to (table), 10-39 Silver, chemistry of hssion-product. 11-45 as control material (table), 10-188 physical properties of (table), 11-6 thermal stress properties of (table), 9-115 Silver-cadmium alloy, thermal stress properties of (table), 9-113 Similitude tests for fluid and heat flow, 9-104 to 9-106


Simply connected

region (mathematics), 3-81 Simpson's rule, 1-28, 1-29, 3-75 Simulator for reactor dynamics, 8-89 to 8-102 analog setup, nonlinear, 8-94 results from, 8-100 to 8-102 simplified linear, 8-99 to 8-102 computer components for, 8-97 to 8-99 computer equations for, 8-92, 8-93 system equations, for heat exchanger, 8-95 to 8-97 for heat generation, 8-93 to 8-95 for mechanical equipment, 8-97 for moderator and reflector, 8-97 for piping, 8-95 Simultaneous equations (see Systems of linear algebraic equations) Sin x, table of, 1-80 to 1-83 Sine integral, 3-95 Singular transformation, 3-11 Singular integral, 3-113 Sinh z, table of, 1-90 to 1-92 SIR, S1G (see Reactors, particular) Skin erythema caused by radiation, 7-28 Slabs, heat conduction in, 9-54 to 9-56 clad fuel elements, simple cases (table), 1-59 homogeneous (summary table), 1-54 multilayer, with surface film coefficient (sum mary table), 1-55, 1-58 Slagging in pyrometallurgical processes, 11-98, 11-99 Slip (in metal deformation), 10-13. 10-14 Slip flow in boiling fluids, 9-41 Slow choppers, 8-86, 6-87 Slowing down of neutrons, 6-42 to 6-48 Slowing-down area, 6-87, 6-88 summary, 1-35 (See also Age) Slowing-down constant £, average value of, for mixtures, 1-32, 6-43 definition of, 3-12, 6-43 of the elements (table), 2-12 to 3-14 Slowing-down density, 6-43 Slowing-down distribution, 6-44 Slowing-down kernels, 6-4, 6-46 convolution of, 6-56, 6-57 Slowing-down measurements. 5-105 to 6-108

Slowing-down power (tables), S-9, 13-70 Slurries, aqueous, of thorium dioxide. 13-57 of thorium fluoride, 13-57 of uranium dioxide, 13-53 to 13-56 of uranium trioxide, 13-53 to 18-55 preparation of, 13-54, 13-55 viscosity of, 13-54 13-56 of uranyl phosphate, flow of, 9-40 heat transfer correlation for, 13-78 as heat transport media, comparative table, 13-59 physical properties of (tables), 13-76. 13-77 Slurry reactors, 12-46 Sn mothod, 6-13 Sodium, 12-56, 12-57 air reaction with, 12-19 corrosion by (tables), 10-20, 10-21. 12-33 liquid, physical properties of (table). 9-21, l3-£materials for use with, 13-83 physical properties of (table), 11-6

price of, 12-107 as reactor coolant, comparative table, 12-58 radioactivity of, 7-20 resistance of materials to corrosion by (tablri 13-83 Sodium-cooled reactors (see Power reactors.

alkali-metal -cooled) Sodium hydroxide, liquid, physical properties ■»;' (table), 9-18 moderating properties of (table), 12-70 as reactor coolant, 12-57 comparative table, 12-59 Sodium -potassium alloy, corrosion by, 10-20. 10-21, 13-83

liquid, physical properties of (table), 13-32 materials for use with, 13-83 physical properties of (tables). 9-21. 13-82 price of, 12-107

radioactivity of, 7-20 as reactor coolant, comparative tabic, 12-5* Sodium Reactor Experiment (see Reactors, partwular,

SRE)

Solar energy, 12-90 Solid radioactive waste (see Radioactive castor Solubilizing agent, 11-61 Solutions of equations, oubic equations, illustra tive example of, 1-66 to 1-68 Doolittle method, 3-38 illustrative example of, 1-69. 1-70 Graeffe's method, 3-48, 3-49 illustrative example of, 1-65. 1-66 simplified description of, 1-63, l-*»4 Horner's method, 3-46 illustrative example of, 1-66 to 1-68 polynomial equations, Graeffe's meth.*d. 3-4S 3-49

illustrative example of, 1-64 to 1-6*1 simplified description of, 1-63, 1-64 Horner's method, 3-46 illustrative example of, 1-66 to 1-68 quadratic equations, formula for, 1-63, 3-4-3 systems of linear algebraic equations. 3-35 to 3-41

illustrative examples of, 1-69 to 1-71 transcendental equations, Newton's method. 3-46, 3-47

illustrative example of, 1-69 successive linear interpolation method, S-4fi illustrative example of, 1-68. 1-69 two-graph method, illustrative example of. 1-68, 1-69

Solvent extraction, definition of, 11-65, 11-66 in radiochemical analysis. 11-52

INDEX


Solvent recovery (in fractional distillation), 11-80 Solvents, physical properties of (table), 11-64 for uranium extraction (table), 11-64 Sources (neutron), 7-92 fast, for cross-section measurement, converter, 5-70 to 5-72 H*((,n)He* reaction, 5-72 pbotoneutr on-induced, 8-27 for reactor startup, 8-25, 8-49. 8-50 efficiency of, 8-25 location of, 8-26 Po-Be, 8-25 Ra-Be, 8-25 Sbi**-Be, 8-25 strength of, 8-26 Space requirements for reactor (table), 18-172 Spallation, 4-82 Special Hilbert space, 8-101 Special materials, cost of, 18-106 to 18-108 Specific heat, conversion factors for (table), 1-151 Debye theory of, 10-9 Dulong-Petit rule for, 10-9 of the elements (table). 11-5 to 11-7 of solids, theory of, 10-9, 10-10 of water (table), 9-11 Specific ionization, 5-2 to 5-4, 7-22 of particles in air (table), 7-34 formulas for, 7-33 for specific ionization, 7-33 Specific power, formulas relating to, 18-79 range for power plants (table), 18-108 Specific speed, 9-48 Specimens for irradiation testing, 10-92 Spectral functions (mathematics), 3-129 analysis of isotopes, 14-53 Spectrochemical Spectrograph, mass (historical), 4-34 Spectrum, neutron (see Neutron, spectrum of) Spectrum hardening in reactors, 6-25 SPERT-1 (Special Power Excursion Reactor) description of, 13-134 Sphere, heat conduction in, 9-57 homogeneous (summary table), 1-57 multilayer, with surface film coefficients (summary table), 1-55, 1-58 physics solution for, (see Physics of critical reactors) Spherical coordinates, 8-32 Spherical-harmonics treatment of diffusion equa tion, 5-10 Spikes, displacement, 10-102 replacement, 10-102 thermal, 10-102 Spin, angular momentum, 4-43 to 4-45 electron, 4-14 nuclear, 4-43 to 4-45 Spin-dependence of nuclear forces, 4-47 Spin-orbit coupling, 4-51 Spiral monochromutor, 5-88 fission, 4-87 Spontaneous rates of, 8-5 Square-rooting method, 8-38 Stable isotopes, availability of, from calutron (table), 14-54 to 14-57 carbon- 13, 14-24 deuterium (as heavy water). 14-24 nitrogen-15, 14-25 oxygen-17 and oxygen-18, 14-25 uses of, 14-57. 14-58 Stable reactor period (see Period) Stacks for reactor buildings. 13-175 Stage, ideal, 11-66 theoretical, 11-66 Stainless steels, AISI types (table) 10-60 corrosion, in liquid metals, 10-21

45

in bodium, 10-20 in sodium -potassium, 10-20 in water, 10-C3 (See alto Structural materials, in high-tem perature water) for fuel reprocessing equipment (table), 11-128, 11-129 mechanical properties of (tables), 10-62 oxidation of (table), 10-63 physical constants of (tabic). 10-61 thermal stress properties of (tabic), 9-113 for water reactors, 18-31 (See also High-temperature alloys) Standard deviation, 8-59 Standard man, 7-38, 7-39 Standards, of measurement, 1-139 to 1-142 for reactor construction, 8-77 Stanton number, definition of, 9-32 Stark effect, 4-23 Stars, energy production in, 4-89 Startup (see Control of reactors) Stationary states. 4-12 of a single particle, 4-20 Statistical reactions, 4-74 Statistics, 3-55 to 8-62 of spin, 4-44, 4-45 Steam, deentrainment of, 13-46, 18-47 dynamic viscosity of (table), 9-8 maximum mass-velocity of (table). 9-42 to 9-44 Prandtl number of (table), 4-9 specific heat of, 9-9 specific volume of, 9-7 thermal conductivity of (table), 9-8 Steam condenser (see Nuclear power, condensers) Steam electric plant (see Nuclear power; Power from fossil fuel) Steam generators, for alkali-metal reactors. 13-91. 13-92 for homogeneous aqueous reactors. 13-67. 13-68 for HTR, 13-67 for water reactors, 13-27, 13-28 DiO. 13-28 Steam rates (table), 18-128 theoretical, 18-124 to 18-126 Steam tables, 18-118 to 13-123 saturated. 18-119, 18-120 18-121, 18-122 superheated. Steel, thermal stress properties of, 9-113 (See also Stainless steel) Steel punchings, cleaning of, 13-182 of, 13-182 procurement Stcfan-Boltzmann constant, value of. 1-154 Stefan-Boltzmann law, 4-5 Stellar energy, 4-88 Sterilization by radiation, 14-8, 14-9 Stern-Gerlach experiment, 4-23n. 4-24 Stewart. J. C, and J. H. Smith, 8-18 Stirling cycle, 18-4 Stokes theorem, 3-126 Stopping power, mass, 7-25 theory of, 4-63 Storage, of fission products. 8-87 of irradiated fuel, 13-100. 18-101 of radioactive wastes. 7-53 Stored energy of graphite caused by radiation, 10-110 to 10-115 Straggling, in cloud chamber. 4-63 Straight line, forms of equation for, 3-27 Strain, cyclic, plastic, 9-115 in-pile measurement of, 10-93 Strain gauges, resistance, for in-pile measurement, 10-93

Stainless steels, corrosion,

46

INDEX


Streamline flow, isothermal, 9-29 to 9-31 nonisothermal, 9-31 (See also Convection; Fluid flow, streamline; Pressure loss in steady fluid flow) Stress, thermal (see Thermal stress)

Stress corrosion, 11-123 to 11-125 Stripping (chemical), definition of, 11-06 in rectification, 11-80 Stripping reactions (nuclear), 4-81 Strontium, fission -product, chemistry of, 11-45 physical properties of (table), 11-6 Strontium-90 for thickness gauges 14-5, 14-6 Structural materials, 10-60 to 10-69 in high-temperature water, 10-73 to 10-82 corrosion resistance in (table) 10-73 to 10-75 effect of gases on, 10-73 to 10-74 selection of materials for, 10-74. 10-75 fretting in (table), 10-74 galling in, 10-74 water conditions, effect of, 10-73, 10-74 wear resistance in (tables), 10-74 to 10-82 abrasion, 10-75, 10-76 bearings, materials for (tables), 10-80 to 10-82 corrosion, effect of, 10-79 fatigue cracking, 10-76 friction, effect of, 10-79 galling, 10-74, 10-75 gears, materials for (table), 10-80 to 10-82 hardness, effect of, 10-78 plastic deformation, 10-76 selection of materials for, 10-76 to 10-82 surface finish, effect of, 10-78 to 10-79 tests for, 10-76 to 10-82 permissible in reactors (table), 12-11 reactors, 13-26 for Bodium-cooled thermal stress properties of (table), 9-113 Structures and improvements, cost of, for nuclear power plants, 13-96 Sturm-Liouville systems, 3-103 Sturm's theorem, S-43 Subcooled boiling (see Boiling) Subcritical multiplication, 8-21, 8-26 Subcritical multiplication factor, summary. 1-48 Subcritical period, 8-22, 8-32 Submarine Intermediate Reactor (SIR) (see Reac tors, particular)

Submarine Thermal Reactor (STR) (are Reactors, particular) Successive substitutions, method of, 3-134 Sulfur, physical properties of (table). 11-6 Sulfur-35, production of, 14-33 for thickness gauging, 14-6 Summation (sigma) bus, 8-48 Superpower Water Boiler Reactor (SUPO) (eee Reactors, particular) Supervision of reactor operation. 8-83 Surface absorption integral (eee Resonance ab sorption integrals) Surface finish, effect of, on wear in water. 10-78 Surface leakage from insulators, 5-10 Surge tanks for water reactors, 13-26. 13-27 Survey instruments for monitoring, 7-46. 7-47 viscosity of, 9-26, 9-27 Suspensions, Sutherland formula for heat conductivity of gases, 9-53 Swimming-pool reactor, 13-54 constructional features of, 13-150, 13-155 Symmetric kernel, 3-136 Symmetry of loci, 3-24 Synchrocyclotron, 6-134, 5-135 Synchrotron, alternating-gradient, 5-136, 5-137 Synergism in radiation injury, 7-26 Synthetic division, 3-44

of linear algebraic equations, 3-35 to 3-41 illustrative example of solution by determi

Systems

nants,

1-70. 1-71

by Doolittle method, 1-69. 1-70 by elimination, 1-69, 1-70 Szilard, Einstein-Szilard pump, 13-90

Tan x, table of, 1-84 to 1-85 Tanh z, table of, 1-93 Tank-type water reactor (ace Power reactors; Re search reactors)

Tantalum, as control material (table), 10-189 physical properties of (table), 11-6 Tar sands, 13-84. 13-85, 13-87. 1S-88 Targets for production of radioisotopes, 14V-26 chemical purity of, 14-32 Taylor's series, 3-83 Technetium, melting point of (table). 11-7 Telegraph equation, 3-117 Television, use of, for viewing radioactive mate rials. 7-123

Teller, Edward, 8-76 Tellurium, fission-product, chemistry of, 11-47 physical properties of (table), 11-7 Temperature, conversion table for. 1-150 in fuel element, 9-85 to 9-88 hot spot, 9-93

in liquid fuel. 9-88, 9-89 neutron, 3-24 nuclear, 4-74 rise in coolant, 9-84, 9-S5 standard units of, 1-140 Temperature coefficient, effect of, on reactor safety, 13-19, 13-20 on subcritical multiplication, 3-27 measurement of. 5-115. 6-116 negative, safety aspects of. 8-80 of reactivity, 8-11 Temperature correction for absorption cross sec tions, 3-23 Temperature difference in fuel elements. 1-59, 9-85 to 9-88 (See also Heat conduction) Temperature measurement, effect of radiation oa thermocouples, 10-108 of radioactive solutions, 11-1 15. 11-116 reactors, 10-92 in Tensor forces, 4-47 Tenth-thickness value of materials (tables). 8-So 11-136 conversion factors for, 1-44 Terbium, physical properties of (table). 11-7 Testing, engineering, in reactors. 5-140 to 5-14.5 mechanical, of radioactive specimens, 10-1*5 neutron ion current chambers for reactor coo tool, 8-63 nondestructive, acoustic tests (table), 10-157. 10-158 electrical tests, 10-159 to 10-161 eddy-current methods, 10-160. 10-161 resistance method. 10-159, 10-160 frost test, 10-153 gauging. 10-162 to 10-166 leak tests, 10-150 to 10-152 penetrant tests. 10-152, 10-153 pressure tests, 10-150 to 10-152 radiographic inspection. 10-154 to 10-157 energy of source for (table), 10-157 thermal teBts, 10-153. 10-154 ultrasonic tests (table). 10-157 to 10-159

visual inspection, 10-150 radiation (eee Irradiation testing)

INDEX Textile printing, use of radioisotopes

for, 14-7,

14-8

Therapy, use of radioisotopes for, 14-10, 14-11 Thermal circulation, 9-00 Thermal column, 4-91, 5-105 for measurement, of extrapolation distance. 5-105 of thermal of thermal


Thermal Thermal Thermal Thermal Thermal Thermal

diffusion length, 5-105 to 5-107 neutron

lifetime,

5-105

conduction (see Heat conduction) conductivity (see Heat conductivity)

cross sections (see Cross sections) cycling of uranium, 10-118, 10-119 diffusion for isotope separation, 14-23 diffusion area, calculation of, 6-86, 5-87 summary, 1-34, 1-35 Thermal diffusion coefficient (see Diffusion coeffi cient) Thermal diffusion constant (see Diffusion coeffi cient) Thermal diffusion length (see Diffusion length) Thermal distortion, 9-112. 9-114 Thermal expansion, theory of, 10-11 Thermal fatigue, 9-115 Thermal insulation, for reactor building, installa tion of, 13-170 for reactor vessel, 13-179 Thermal neutron flux, absolute, measurement of, 6-59 counters for measurement of, 5-104 Thermal neutron properties of the elements (table), 2-13, 3-14 Thermal neutron reactions, measurement of, 5-89 to 5-99 Thermal neutrons, beams for experiments, 5-84, 5-85 coherent scattering of, 5-83 definition of, 5-53 detectors for, 5-40, 5-41. 6-102 to 5-104 (See also Detectors) energy distribution of, 6-4S to 6-51 experiments with, 6-83 to 5-99 Maxwell-Boltzmann distribution of (see Max

well-Boltzmann distribution) monitoring for, 7-46 monochromatization of (see velocity selectors.

below) opticul effects, 5-83 radiation damage to covalent and organic mate rials. 10-27, 10-28 sources of. 6-83 to 6-86 velocity selectors for, filters, 5-85, 5-86 slow choppers, 6-86 to 5-88 5-88, 6-89 spiral monochromator, Thermal radiation, 4-4, 4-5 Thermal resistivity, effect of radiation on m < Radiation damage) Thermal shield, boral fabrication and fitting, 13-181. 13-182 cooling system for, 13-181 function of, 7-60 leak testing of, 13-181 in pressure vessels, 13-13, 13-14 Thermal-siphon boiling. 9-74 Thermal spike, 10-102 in uranium, 10-118 Thermal stress, 9-110 to 9-116 in bars, 9-111 fracture by, 9-114. 9-115 order of magnitude of, 9-110. 9-111 in plates, 9-111 in pressure vessels, 13-13 properties of materials (table), 9-113, 9-114 significance of calculated, 9-114 * See insert Revised Cross-section Data,

Thermal stress, simple shapes, summary

47

of (table), 1-61, 1-62 in solid cylinders, 9-111 in tubes, 9-111. 9-112 Thermal test reactor (TTR) description of. 18-135 Thermal utilization, /, 6-76 calculation of, for homogeneous reactor. 6-76 for lumped reactor, 6-76 to 6-79 measurement of, 5-119, 6-120 numerical example of calculation, 6-96, 6-97 summary, 1-32 Thermocouples, effect of radiation on (table), 10-108 for radioactive solutions. 11-115. 11-116 in reactors. 10-92 Thermodynamic properties, of air, 19-126 enthalpy-entropy diagram. 11-129 of aqueous fuels, 13-75, 13-76 slurries, 13-76 to 13-78 of steam. 19-118, 13-119 Mollicr diagram, 13-124. 13-125 tables, 12-120 to 13-123 Thermodynamics, definitions, 12-116, 12-118 of power cycles. 12-2 to 12-6, 12-119 to 12-126 Thermoelectric power, effect of radiation on (table), 10-108 Thermographic tests, 10-154 Thermonuclear reactions, 4-88 to 4-90 Thickness gauges, use of radioisotopes for (tables), 14-4 to 14-6 Thixotropic materials, vicosity of, 9-26. 9-27 Thomas meter, 11-119 Thompson atom, 4-10 Thompson cross section, 4-60 Thorex process, 11-82 Thoria. 11-42 slurries of, 13-57 Thoria-uranium dioxide (ThOi-UOi). 10-181 Thorium, 11-40 to 11-42 chemistry of. 11-40 to 11-42 corrosion of, 10-36, 10-37 by air. 10-36, 10-37 10-36 by sodium-potassium, by water, 10-36, 10-37 as fuel material, 12-60 health hazards of, 10-33 manufacture of. 10-35, 10-36 mechanical properties of (table), 10-33 to 10-35 nuclear properties of (see Thorium-232) physical constants of (table). 10-22. 10-33. 11-41 physical properties of (table), 11-40, 11-41 reserves of, 12-85, 12-89 sources of, 11-40 thermal stress properties of (table), 9-113 Thorium-232. nuclear properties of, absorption cross section ("world consistent value") (table), 2-21* fast fission cross section (table), 2-8 guide table, 1-4 scattering cross section, 2-5 spontaneous fission rate (table). 2-5 Thorium alloys, 10-36, 10-37 Thorium compounds, 11-42 Thorium dioxide, 11-42 ThOi-UOi. 10-181 Thorium fluoride slurries, 13-57 Thorium isotopes, radioactive constants of (table), 11-41 Thorium nitrate solutions, properties and phase stability of, 13-56. 13-57 Thorium oxide (see Thorium dioxide)

INDEX

4S

Threshold, for fission, theory of, 4-86, 4-87 for gas amplification, 5-13 Threshold detectors for neutrons, 5-49 Threshold energy, for atom displacement, 10-99 for nuclear

reactions,

4-73

photoneutron reactions, 3-3 Threshold reactions, cross sections for fission (table), 2-30 5-70, 5-71 Thulium, physical properties of (table), 11-7 Time, conversion factors for units of (table), 1-145 standard unit of, 1-139, 1-140 Time-of -flight measurement for inelastic neutron scattering, 5-67 Time-offlight velocity selectors, 5-74 Timken alloy (see High-temperature alloys) Tin, corrosion by, 10-21 fission-product, chemistry of, 11-46 liquid, physical properties of (table). 9-22 physical properties of (table), 11-7 Tissue dose, definition of, 7-25 Tissue-equivalent ion chamber, 7-46 Titanium, 10-66. 10-67 corrosion resistance to liquid metals, 10-21 properties of (table). 10-67 high -temperature mechanical properties of (table), 10-67 modulus of elasticity of (table), 10-19 physical constants of (table), 10-66, 10-67 thermal stress properties of (table), 9-113 Tolerances for radioactive nuclides (see Dose; Dose rate) Tolerances, for chemical and radioactive poisons (table), 8-77 (See also Dose; Dose rate) Total cross section, definition of. 4-71. 5-54 measurement of, 5-61 to 5-63 (See also Cross sections) Toxic materials (see Dose; Dose rate; Radioactive


neutrons in reactors,

materials) Tracers, 14-6 to 14-8 Transcendental constants, 1-75 short list of (table), 1-2 Transcendental equations, 3-50 solution of, by Newton's method, 3-46, 3-47 illustrative example of, 1-69 by successive linear interpolation, 3-46 illustrative example of, 1-68, 1-69 by two-graph method, illustrative example of, 1-68 Transfer cross sections (table), 3-33 Transfer function, 8-33 to 8-39 Transformation, point, 8-10 shielding, 7-93 to 7-97 Transforms, Fourier, 8-94, 3-129, 3-130 finite, 3-131 table of, 8-130 Hankel, S-130 kernel of, 3-127 Laplace (see Laplace transforms) Mellin, 3-131

Transient Reactor Test Experiment (TREAT*, 13-139

Transition probabilities, 4-19, 4-28 Transitions, atomic, allowed, 4-13 forbidden.

4-13

Translation of axes, 3-9 Transmission coefficient, nuclear, 4-32, 4-78 Transmission cross sections, measurement of, for M axwell-Boltzraann distribution, 5-94 to 5-96

Transmission gauges, use of radioisotopes

for, 14-3

to 14-5

Transmutation of nuclides bv neutrons 7-7 to 7-14

(table),

Transparent shielding materials,

physical proper ties of. 7-122 radiation damage to, 7-122 Transport cross sections (see Cross sections, trans port) Transport kernels, 6-5 Transport mean free path, X, (thermal) of ma

terials (table), 8-86 summary, 1-33 to 1-34 Triangles, solution of, 8-21 to 3-24 Trichlorobenzene for emergency shutdown, S-4s Trigonometric functions, 3-16 to 8-19 Trigonometric identities, 3-17 to 3-19. 3-23. 8-24 Tritium, for thickness gauges, 14-5 T(d.n)He«, 5-60 T(p.n)Hs, 5-60 Triton, nuclear properties of, 4-57 Trivectors, 3-11, 3-12 Trouton's rule, 11-87 polynomials. 3-92 Tschebyscheff TTV (see Tenth-thickness value) Tube banks, flow through. 9-47 Tube-type water-cooled reactors (see Power r»? tors, pressurized -water, pressure-tube type1 Tumors caused by radiation, 7-28 Tungsten (wolfram), physical properties of (table). 11-7 Turbines (see Nuclear power, steam-turbine geaerators for) Turbogenerator units, cost of, 13-97 Turbulence, initiation of, in fluid flow, 9-35 Turbulent flow (see Convection; Fluid flow, toarbulent; Pressure loss in steady fluid flow) Two-body collisions, 4-30 Two-graph solution for transcendental equation? illustrative example of, 1-6S Two-group theory, 6-64 Two-phase flow, pressure loss in (table), 9-40 » 9-43 V

UHN (uranyl nitrate),

11-37, 11-70 (See also Uranyl nitrate solution) Ultrasonic tests, 10-157 to 10-159 gauging, 10-165 Uncertainty principle, 4-16 Underwater viewing, 7-123 coefficients, method of. 8-6 Undetermined USSR reactors (see Reactors, particular. US£R Unit of radiation exposure. 7-23 to 7-25 Unit systems, 1-134 to 1-139 comprehensive, 1-139 electrical, 1-137 to 1-139 English (table). 1-135 mechanical, 1-136. 1-137 metric (table), 1-135 thermodynamic, 1-137 Unitary matrix, 3-53 Kingdom, gas-cooled power reactor pro United gram in (table), 13-105 United States (see Atomic Energy Commissaoa. Bureau of Labor; Bureau of Standards) Unity, roots of, 3-23 Unloading systems (see Fuel handling) Unsteady fluid flow. 9-39. 9-40 Urania (see Uranium dioxide) UOt (see Uranium dioxide) UiO>. physical properties of, 13-77 Uranium, chemistry of, 11-34 to 11-37 (See- also Chemical separation processes' compounds of (table), 11-35 to 11-37 corrosion of, by air, 10-28 by steam, 10-28

INDEX corrosion of, by water, 10-28, 10-29. 12-61 of, 11-99, 11-100 electrorefining handling of, 7-56 health hazards of, 10-24 isotopes, radioactive constants of (table), 11-34 isotopic abundance of (table), 2-22 manufacturers of, 10-25 to 10-28 mechanical properties of (tables), 10-24 to 10-28 nuclear properties of, natural, at thermal neu tron energy (table),
Uranium,

Uranium-238, nuclear properties of, spontaneous fission rate of (table), 2-5 summary table, 1-4 at 2200 m/sec, 0*.
with fissium, Fs, 10-175, 10-176 gamma-phase. 10-169 to 10-172 with molybdenum (tables), 10-170, 10-172. 10-176

with niobium (table),

of)

radiation damage to, 10-117 to 10-119 radioactive constants of (table), 11-34 resonance parameters for, 2-27


kev and 900 kev, y (table), 2-7 v in fast reactors,
10-169, 10-176 and mechanical properties of, 10-18 with plutonium, 10-176 radiation damage (sec Radiation damage, to uranium alloys) with silicon, 10-174, 10-175 thermal stress properties of (table), 9-113 uranium-base, 10-168 to 10-176 a phase, 10-168 to 10-169 5% Zr. 1.5% Nb, 10-169 corrosion resistance of, 10-169, 12-61, 12-62 EBWR fuel elements 10-169 heat treatment of, 10-169 manufacture of, 10-169 1 phase, 10-172 to 10-175 3.8% Si. 10-174, 10-175 hardness of. 10-170 physical constants of (table), 10-170 45 to 55% Zr, 10-170 to 10-174 corrosion resistance of, 10-173, 10-174 fabrication of, 10-173 physical constants of (table), 10-170 Zircaloy clad for, 10-173 with fissium (tables), 10-175, 10-176 with fissium and plutonium (tables), 10-175, 10-176 y phase, 10-169 to 10-172 8 to 12 % Mo, 10-169 to 10-172 fabrication of, 10-172 hardness of, 10-170 mechanical properties of (tables), 10-171 to 10-173 physical constants of (table). 10-171, 10-172 Zircaloy clad for, 10-171, 10-172 8 to 12% Nb, 10-169 to 10-172 hardness of, 10-170 physical constants of (table), 10-170 Zircaloy clad for, 10-172 radiation damage to, 10-119 to 10-121 zirconium-base, corrosion resistance of, 10-171, 10-174 density of (table), 10-31 fabrication of, 10-173 heat conductivity of (table), 10-32 high-temperature tensile properties of (table), 10-32 thermal stress properties of (table), 9-113 Zircaloy clad for, 10-174 4% U, physical properties of (table). 10-18 radiation damage to, 10-121 10% U, mechanical properties of (table), 10-173 22% U, mechanical properties of (table). 10-174 41% U, mechanical properties of (table), 10-173 50% U, physical properties of (table), 10-170, 10-171 physical

sources of, 11-34

thermal cycling of, 10-118, 10-119 thermal stress properties of (table), 9-113 Uranium -233, burnup rate of, formula for, 1-27 delayed neutrons from fission of, data of Hughes et al (table), 8-3 ' data of Keepin et al (tables), 8-3 to 8-8 — of, high energy, at neutron 30 properties nuclear

49

-

^

INDEX

50


Uranium carbide, 10-182 dispersed in graphite, 10-185 Uranium compounds, 10-24, 11-35 to 11-37 properties of (table), 10-24 refractory, 10-181, 10-182 Uranium dioxide (UO»). 10-177 to 10-181

aqueous slurries of, 18-53 to 18-56 ceramic, radiation damage to, 10-121 dispersions of, 10-183 to 10-185 in aluminum, 10-183 in ceramic materials, 10-181 in graphite, 10-185 in iron. 10-183 to 10-185 mechanical properties of (tables), 10-184, 10-185 radiation damage to, 10-121, 10-122 in stainless steel, 10-183 to 10-185 mechanical properties of (tables), 10-185 fuel elements, fabrication of, 10-180 corrosion of, 10-181 powder, production of, 10-177 properties of (table) 10-178, 10-179 properties of, lt-60 sintered, production of, 10-177, 10-178 physical properties of (table), 10-22 properties of (table), 10-178 to 10-180 in PWR, 12-01. 18-62 Uranium dioxide-alumina (UOt-AljOi), 10-181 Uranium dioxide-beryllia (UOi-BeO). 10-181 radiation damage to, 10-121 Uranium dioxide-magnesia (UOi-MgO), 10-181

Uranium dioxide-plutonium dioxide (UO»PuOi), 10-181 Uranium dioxide-thoria (UOj-ThO?), 10-181 Uranium dioxide-zirconia (UOj-ZrOi), 10-181 Uranium fluoride, conversion of, to metal, 11-111. 11-112

production of, 11-109 to 11-112 Uranium fuel materials, 18-60, 18-61 (See also Uranium; Uranium alloys; Uranium dioxide)

Uranium hexafluoride,

in fractional distillation,

11-90. 11-91 physical properties of (table), 11-90 Uranium isotopes, formation of, 11-9 Uranium metal, production of, 10-17, 10-23, 11-109 to 11-112 conversion of uranyl nitrate to fluoride, 11-109 to 11-112 conversion to metal, 11-111, 11-112 ore concentration, 11-106 to 11-108 ore purification, 11-108, 11-109 Uranium ore concentration, 11-100 to 11-108 Uranium oxide (see Uranium dioxide) Uranyl ammonium phosphate (UAP), 11-63 Uranyl fluoride solution, 18-52 corrosion by. 13-58 properties of, 18-52 Uranyl nitrate (UHN), 11-37, 11-70 Uranyl nitrate solution, corrosion by, 13-58 phase equilibrium of, 13-53 properties of, 13-52 Uranyl peroxide from radiolytic decomposition, 13-58 Uranyl phosphate solution, corrosion by, 18-58 phase equilibrium of, 13-55 properties of, 13-52 slurries, 13-56 Uranyl sulphate solution, corrosion by, 13-57 as heat transport medium, comparative table, 13-59 phase equilibrium of, 13-51, 13-54 physical properties of (table), 18-75, 18-76 properties of, 18-51, 13-52

Uranium trioxide, aqueous slurries of, 13-53 h yd rated, physical properties of, 13-77 Utilization, thermal (see Thermal utilization) Utilization factor, definition (in power genera tion), 12-91

average absorption cross section of. 6-50 1 (v law, theory of, 4-96 Vacancy, activation energy for movement of, 10-104 Vacancy pairs, 10-101 Vacuum tubes, radiation damage to (table), 10-146 Valence bands, 10-4 Valence electrons, 4-27 Valves, for alkali-metal reactors, 13-92, 18-03 for direct-cycle steam turbines, 12-141, 12-141' Johnson type, 12-141. 12-142 for gas-cooled reactors, 13-110 for homogeneous aqueous reactors, 18-71. 18-72 pressure losses at (table), 2-37 for water reactors, 13-27 bellows, 13-27 relief, 13-29 Vanadium, physical properties of (table), 11-8 price of, 12-127 van Arkel-de Boer process, 11-100 Van de Graaf machine, 8-125, 6-120 as source of fast neutrons. 6-58 Vapor dome. 13-163, 13-164 Vapor-liquid mixtures, pressure loss in flow of (table), 9-40 to 9-43 Vaporizing of fuel elements. 8-78 Variation, calculus of, 8-137 to 8-140 Vector analysis, 3-125. 3-126 Vector product, 3-15 Vectorial coordinate system, S-S Vectors, 3-2 to 3-15 algebra of, 3-6 to 3-8 arithmetic, 3-8 characteristic, 3-52 geometric, 3-8 latent, 3-51 to 3-55 proper, 8-52 Velocity, conversion factors for units of, 1-146 of light, value of, 1-13. 1-154 Velocity distribution in turbulent flow, 9-34 Velocity selectors, for intermediate-energy neu trons, crystal monochromator, 6-73. 6-74 fast choppers, 6-74 to 6-76 resolution of, 6-75 pulsed accelerators. 6-75 time-offlight, '6-74 to 6-70 for thermal neutrons, filters, 6-85, 6-86 slow choppers, 6-86, 6-87 spiral monochromators, 6-88. 6-89 Ventilation, of chemical separation plant.-. 11-134 11-135 for control of radioactivity, 7-119 of fuel reprocessing plants. 11-134. 11-135 of reactor buildings, 13-174. 18-175 Vertex, 3-31 Vibrating-reed electrometer. 6-8. 6-9 Viewing, of radioactive materials. 7-121 to 7-123. 7-134 to 7-136 underwater, 7-123 Viscosity, of Bingham bodies, 9-20. 9-27 conversion factors for units of (table). 1-14V definition of. 9-25 of dilatant plastics, 9-20. 9-27 of gases, formulas, 9-25

l/» absorbers,

INDEX Viscosity, of inverted plastics, 9-2H, 9-27 of liquids, 9-26 of Newtonian fluids, 9-26 to 9-27 of plastics, 9-20, 9-27 bodies, 9-26, 9-27 of pseudoplastic of slurries, 13-77, 18-78 of suspensions, 9-26 of thixotropic substances, 9-26, 9-27 Visoous flow (see Convection; Fluid flow, stream line; Pressure loss in steady fluid flow) Visual tests, 10-150 Void coefficient, 19-20 safety aspects of, 8-80 Voine reactors, 13-106 Volatilisation, in fuel reprocessing, 11-101 in radiochemical analysis, 11-52 Volterra equations, 8-131 Volume, conversion factors for units of (table), 1-144 von Karmiin number, definition of, 9-32

W Wall effect in ionization chambers, 6-6 Warping of fuel elements, 8-78 Waapaloy (see High-temperature alloys) Waste, aqueous, definition of, 11-65


organic,

11-1.5

radioactive (see Radioactive wastes) Water, concentration of heavy water in, 8-27 corrosion in (see Corrosion ; Structural mate rials, in high-temperature water) critical properties of (table), 9-9 density of, from 0 to 100°C (table). 1-49 (See also specific

volume,

18-70

pure, pH of (table), specific resistance

18-11 of (table), 13-11 radiolytic decomposition of, in EBWK, 10-151 mechanism of, 10-131, 10-132 in reactor systems, 10-32 to 10-34, 10-73, 10-74 (flee also Radiation decomposition in homo geneous aqueous reactors) aa reactor coolant, 12-56 comparative table of, 12-58 effect of impurities on radioactivity of, 7-18, 7-19 relaxation length, for fast neutrons (table). 2-35 for 7 rays (table), 2-33 shielding properties of (see Shielding) specific heat of (table), 9-11 specific volume of (table). 9-10 thermal conductivity of (table), 9-12 Water-aluminum mixtures, age in, 8-88 Water boilers. 13-50, 13-60 particular reactors (table), 18-129. 13-130. 13-153 reactor vessel. 18-66

Water circulation in boiling-water reactors. 13-47

13-46

Water coolant systems, radioactivity in, 7-17 to 7-19, 13-5

boiling, 7-19 Water-cooled reactors search reactors)

(nt Power

reactors;

Re

Water-metal mixtures, age in (table), 6-88 Water-metal reaction, 12-19 Water-moderated reactors (ere Power reactors;

Research Water power,

Wave equation, 3-117 for bare reactor, 6-58 Schrodinger, 4-17 Wave function, 4-15 Wave mechanics, 4-14 Wave number, 4-3 Wave-particle dualism, 4-5 Wave properties of neutrons, 4-92 Wavelength of neutrons, 1-13, 1-156 Wear, of structural materials (see Structural materials, in high-temperature water) use of radioisotopes for measurement of. 14-7 Weierstrass-Bolzano theorem. 8-78 Wein's displacement law (table), 1-154 Weisbach friction factor, 9-32n. Westcott cross sections, for U1". Pu>»

1-21,* 1-22 for Xe'», 1-23* Wheeler, A., 8-16 Width of level, resonance (table),

reactors) 12-85, 12-86. 12-90

Water-zirconium mixture age in, 6-88 * See insert Revised Cross-section Data.

V'.

J.

(definition), 6-78 Wiedemann-Franz law, 10-11 Wigner formula, for effective resonance absorp tion integral, 2-27 for resonance escape probability. 2-24 Wilson-Serber boundary, 6-17 Wind as source of energy, 12-90 Windows, shielding, 7-121 to T-123, 7-134. 7-136 materials for, 7-121 properties

of, 7-122

radiation damage to, 7-122 Windscale (see Reactors, particular) Wire screens, flow through, 9-47 Word in digital computers, 8-143 Work, conversion factors for units of (table),

below)

dynamic viscosity of (table), 9-13 heavy (see Heavy water) moderating properties of (tables), 1-20, 2-9

51

1-147

World energy resources, 12-84 to 12-88 fossil, coal (table), 12-84, 12-86 to 12-88 miscellaneous, 12-85 natural gas (table), 12-84 to 12-86, 12-88 oil (table), 12-84 to 12-88 oil shales (table), 12-84, 12-87, 12-88 peat. 12-84 petroleum (table), 12-84 to 12-88 tar sands (table), 12-84. 12-85, 12-87. 12-88 United States consumption of, rate of, 12-85 to 12-87 total to date, 12-85 world consumption, 12-85, 12-87 world reserves, 12-85, 12-87 hydroelectric, 12-85, 12-86, 12-90 miscellaneous, 12-90, 12-91 nuclear, abundance of, 12-85 cost of concentrates, 12-89 reserves of, 12-88 to 12-89 solar, 12-90 wind, 12-90 Wronskian, 8-109

{ (see Slowing-down constant) X-10 (see Reactors, particular) X-ray attenuation (see Shielding. >-ray attenua

tion) X-ray detection. 6-35, 6-36 (See also Detectors)

X-ray spectroscopy, 5-36 to 5-40 (See also Counting, 7-ray) X-ray shielding (see Shielding, y attenuation)

X

rays, bremsstrahlung,

7-78

characteristic, 4-55, 7-78 common exposures (table), 7-37 continuous, 7-78 for gauging, 10-102 to 10-165

INDEX

52 X

rays, ionieation by, 7-6 10-154 to 10-157 for inspection, (See alao y rays) chemistry of, 11-45 Xenon, fission-product, physical properties of (table). 11-7 Xenon-135, absorption cross section of (tables), 1-23.* 2-10,*

a-n*

concentration of, 8-27 to 8-29 2-10, 2-1 1 fission yield of (table), 1-23. poisoning, 8-27 to 8-29

Zircaloy clad, 10-171 to 10-174 Zircaloy 2, composition of, 10-44 corrosion of, 10-45, 10-46 physical properties of (table), 10- IS thermal stress properties of (table), ft-113 Zircaloy-2 clad (tables), 10-167, 10- Mf Zircon, effect of radiation on, 10-123 Zirconium, corrosion of, 10-44 to 10 i* in air, 10-44 in high-temperature water, 10-75 in liquid metals, 10-21. 10-45 in water, 10-45 111*, fissi on- product, chemistry uf. manufacture of. 10-40, 10-43 to 10-45 mechanical properties of (tables) 10-40 to

Y Y-12 plant, 14-44 Reactor, Yankee Atomic Electric Company 13-138 cost of, 12-96 Yield, fission (table), 11-9 to 11-24 fission curve, 4-84 of reaction, definition of. 4-71 Young's modulus of graphite, effect of radiation on, 10-111, 10-115 Yttrium, density of (table), 11-7 chemistry of, 11-45 fission-product, physical properties of (table), 11-7 Yvon, method of, 6-12


Z Zeem&n effect, 4-23 ZEPHYR, Zero Energy Fast Reactor, 13-136 Zero-leakage components for reactors, 13-5 Zero power assemblies (see Critical assemblies) chemistry of, 11-45 Zinc, fission-product, physical properties of (table), 11-7 Zinc bromide, shielding properties of, 7-122

stability of, 7-122 * See inaert Revised Cross-section

tt3e7RR5

20I3IBN

10-44 of (table). 10-39 price of, 13-107 production of, 10-38. 10-39 analysis of, 11-52 radiochemical thermal stress properties of (taVJe). 9 : I for water reactors. 13-31 Zirconium alloys, 10-40 to 10-47 corrosion of, 10-44 to 10-46 in air, 10-44 in water, 10-44 to 10-46 mechanical properties of. 10-40 to 10-44 (See also Zircaloy 2) Zirconium clad, 10-186 Zirconium-uranium alloys (aee Uranium aikiyi Zirconium-water mixtures, age in. 0-88

physical constants

Zirconium-water reaction, t-79, 8-85 ZOE, description of, 13-126 Zoe resins, 11-93 Zone melting, fuel purification 11-102

Data.

6305

by. 11-101.



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