Nuclear Engineering Modeling and Simulation

MODELING AND SIMULATION OF NEUTRON TRANSPORT IN NUCLEAR ENGINEERING “

” ”

A Hands-on, Easy-to-Understand Approach Approach

Zafar-ullah Koreshi 7/13/2014

A textbook for building the foundations for advanced applications in nuclear computation for design.

i

FOREWORD Computational Methods in Science and Engineering, have reached a very high degree of sophistication compared with classical solutions of the earlier days. In nuclear engineering, classical solutions like those from the Wiener-Hopf method and the first numerical solutions of the neutron transport equation equation with the finite-difference method and refinements refinements using the Discrete Ordinates  and Spherical Harmonics  methods. They now encompass sufficient depth in each deterministic approach as well as in the Monte Carlo approach to permit widespread generalization and application to other areas of science and engineering where the underlying transport phenomena are similar. The primary objective of computation in nuclear engineering is to obtain information on the flux of neutrons as a function of energy, position and time in nuclear systems which vary from small assemblies to large nuclear power reactors. All other quantities such as reactor power and thermal-hydraulic parameters parameters can then be determined. determined. At of end 2012 there were 437 operational reactors contributing to about 15% of the world’s energy requirements. Many of these reactors have ‘aged’; some are as old as 43 years and require ‘re‘re-engineering’ engineering’ for f or life extension, extension, while 140 have been permanently shut down. Public perceptions, perceptions, especially especially after Three Mile Island (1980), Chernobyl Chernobyl (1986), and Fukushima (2009) have affected the image of nuclear technology as an energy source. Yet, there is the possibility of a nuclear renaissance since this technology represents a reliable, safe and sophisticated high-density energy source. Thus, a total of 64 power reactors are under construction with the lead taken by Asia: China (26), Russia (11), India (7), Pakistan (4), Republic of Korea (2), and most recently, the United Arab Emirates (1). As a discipline, nuclear engineering has a future as good as any other advanced technology, and thus computational methods that address the transport of phenomena in a nuclear reactor will continue to lie at the heart of the field. Central to all such analysis is the neutron transport equation which its roots in the Boltzmann transport equation expressed by Ludwig Boltzmann (1844-1906) for the kinetic theory of gases. As a conservation equation, it can be interpreted as the zero-th moment of the Boltzmann equation. All methods thus applicable to the kinetic theory of molecules are thus

ii

equally applicable to neutral particle transport (neutrons) as well as to electromagnetic radiation (photons and thermal radiation) and charged particle transport (e.g. ions and electrons in plasmas). These notes arise from a formalization of computational methods taught to graduate students at Air University. The present text, along with other other reference material material on probability and statistics, statistics, and on Monte Carlo methods, was followed for the graduate program in the Mathematical Modeling and Scientific Computing Program (Fall Semester 2012) for the Elective course on Monte Carlo Methods. The MCQ’s, MCQ’s, Mid-Term and Final Examinations given to the students are attached at the end. I found that a large part of the course could be taught to the students with careful interjections of the nuclear engineering context. While I have made every effort to include what I feel is necessary knowledge for understanding neutron transport and mathematical methods and computation, I do not claim that it is complete. I have assumed knowledge and skills obtained from the usual five or six mathematics undergraduate courses. This book can thus be used for a course in a nuclear engineering program at the advanced undergraduate level if the heavy mathematics of transport theory is largely skipped, or at the graduate level in its entirety. It can be part of a useful collection for engineers and scientists entering the field from other disciplines. Nuclear engineers can skip the first chapter while mathematicians can skip the second chapter. The body of the text begins essentially from the third chapter. For professionals, professionals, it can serve se rve as useful introductory material for ‘tying -up’ neutron diffusion, transport and Monte Carlo methods. When compared with the existing text and teaching material in Monte Carlo methods, such as the first books by Handscombe, by Cashwell and Everett or by Spanier and Gelbard, it is more engineering-oriented and makes use of Matlab® which of course was not available to them. Compared with the book by Lewis and Miller, it is easier to follow. Standard nuclear engineering text books presently taught in universities ( e.g. Lamarsh, Henry, Duderstadt, Stacey etc.) cover more breadth and are undergraduate-focused. I can say that if I had a book like this when I started my B.Sc (Hons.) in Nuclear Engineering at Queen Mary College (now Queen Mary University of London, without the nuclear engineering program), what a difference it would have

iii

made! Afterwards at University of Wisconsin, Madison, I learnt Monte Carlo as an M.S. student from the late Prof. Charlie Maynard who would so affectionately take us all through the details with the Los Alamos MCNP manual pages in his hands. Remember those were the pre-PC days, and we had access only to mainframe computing. I was fascinated with the Monte Carlo method as it seemed so intensive and so capable of handling real-world problems without any geometry or model idealizations. Later on, at Cambridge with Jeffery Lewins, I went through the Ph.D. getting a grip on Monte Carlo perturbation theory. theory. In several discussions with Herbert Rief, at Ispra in Italy, I learnt the concepts; and turned out my first paper with a simple close-to-analytical paper presented at Budapest. My work-horses were MORSE, an Oak Ridge multi-group code, and the more versatile MCNP, a Los Alamos code, now running at Version 5. MCNP is a production code; code; running running it, they say, is almost almost as good good as carrying carrying out an experiment. experiment. This book goes beyond the traditional material in neutron diffusion and stochastic transport; it is aimed at training graduate students, as well as researchers, to ‘think Monte Carlo’, or analog simulation, and then write the equations rather than the other way round. The book closes, in Chapter 10 with an applications’ approach for the nuclear engineering practitioner practitioner involved with criticality criticality storage storage systems, systems, reactor core neutronics neutronics and neutronic neutronic applications applications in emerging emerging areas such as medical radiation physics. Generally, computational methods will continue to find more relevance with new developments in computer hardware. One such area is with Field Programmable Gate Arrays (FPGAs), which are also being initiated in the Mechatronics Engineering department; there is great scope for parallel computing with high speed-ups for large challenging simulation problems. Dr. Zafar ullah Koreshi Professor, Mechatronics Engineering Department Dean Faculty of Engineering, Air University, Islamabad July 2014

iv

Contents FOREWORD ............................................................................................................................................................................ ii

List of Figures ........................................................................................................................................................................ xii Notation ................................................................................................................................................................................ 16 1

Preliminaries: Prelim inaries: Nuclear Nuclea r Engineering Enginee ring .................................. ................. ................................... ................................... ................................... ................................... ................................... ....................... ..... 19 1.1

Introduction Introd uction ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ............................... .............. 19

1.2

Neutron Neutro n Flux .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ................................. ................ 2

1.3

Adjoint Flux ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................... .................... 2

1.4

Radiation Radiati on Dose .................................. ................. ................................... ................................... .................................. ................................... .................................... ................................... .............................. ............. 3

1.5

Neutron Neutro n Current ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... ........................... .......... 3

1.6

Nuclear Cross-section Cross-s ection ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... ................... 3

1.7

Ficks’ Law .............................................................................................................................................................. 12

1.8

Reaction React ion Rates .................................. ................. ................................... ................................... .................................. ................................... .................................... ................................... ............................ ........... 13

1.9

The Nuclear Fission Reaction React ion ................................... .................. ................................... ................................... ................................... ................................... ................................... ....................... ..... 13

1.10

Neutron Neutro n Slowing Down and Diffusion Diffusi on .................................. ................. .................................. ................................... .................................... ................................... ............................ ........... 19

1.11

Scattering Scatt ering ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................... .................... .. 21

1.12

Nuclear Reactors Reacto rs .................................. ................. .................................. ................................... ................................... ................................... .................................... ................................... ......................... ........ 24

1.12.1 1.13

Nuclear Nuclea r Reactor Reacto r Components Compone nts ................................... .................. ................................... ................................... ................................... ................................... ............................... .............. 25

Criticality Critic ality .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ................................... .................... .. 28

References ........................................................................................................................................................................ 31 Problems ........................................................................................................................................................................... 32 2

Preliminaries: Prelim inaries: Mathematical Mathema tical Foundations Foundat ions ................................... .................. .................................. ................................... .................................... ................................... ............................ ........... 35 2.1

Introduction Introd uction ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ............................... .............. 35

2.2

Ordinary Ordinar y Differential Differ ential Equations .................................. ................. ................................... ................................... ................................... ................................... ................................... .................... .. 35

2.3

Partial Partia l Differential Differ ential Equations ................................... .................. ................................... ................................... ................................... ................................... ................................... ....................... ..... 38

2.4

Integral Integr al Equations Equatio ns ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... ...................... ..... 39 v

2.5

Special Specia l Functions Functio ns.................................. ................. .................................. ................................... ................................... ................................... .................................... ................................... ......................... ........ 42

2.5.1

The Delta Function Functio n ................................... .................. .................................. ................................... ................................... .................................. ................................... ................................ .............. 42

2.5.2

Legendre Lege ndre Functions Functio ns .................................. ................. .................................. ................................... ................................... .................................. ................................... ................................ .............. 43

2.5.3

Bessel Besse l Functions Functio ns ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... ................... .. 44

2.6

The Laplace and Poisson Poisso n Equations .................................. ................. ................................... ................................... ................................... ................................... ............................... .............. 46

2.7

Integro-diffe Integr o-differentia rentiall Equations .................................. ................. ................................... ................................... ................................... ................................... ................................... ....................... ..... 48 Complex Integration ................................................................................................................................................. 50

2.8 ..................................................................................................................................................................................... 50 2.9

Numerical Numeric al Methods ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... ................... .. 52

2.9.1

The Finite-Differ Finite -Difference ence Method .................................. ................. ................................... ................................... ................................... ................................... ............................... .............. 52

2.9.2

The Finite Element Elemen t Method ................................... .................. ................................... ................................... ................................... ................................... .................................. ................. 53

2.10

Evaluation Evaluat ion of Integrals Integ rals .................................. ................. .................................. ................................... ................................... ................................... .................................... .................................. ................ 57

2.11

Probability Probabi lity and Statistics Statis tics .................................. ................. .................................. ................................... ................................... .................................. ................................... ................................ .............. 58

2.11.1

Random Processes Proces ses ................................... .................. .................................. ................................... ................................... .................................. ................................... ................................ .............. 58

2.11.2

Markovian Markov ian Processes Proces ses .................................. ................ ................................... .................................. ................................... ................................... ................................... ............................. ........... 58

2.11.3

Sample and Population Populat ion ................................. ................ ................................... ................................... ................................... ................................... ................................... .......................... ........ 58

2.11.4

Probability Proba bility Distribution Distr ibution Function Functi on ................................. ................ .................................. ................................... .................................... ................................... ............................ ........... 58

2.11.5

Random Numbers ................................. ................ .................................. ................................... ................................... .................................. ................................... ................................... ................. 62

2.11.6

Sampling Samplin g from PDFs .................................... .................. ................................... .................................. ................................... ................................... ................................... ............................. ........... 63

2.11.7

Sampling Samplin g from non-analytic non-analy tic PDFs ................................... ................. ................................... ................................... .................................... ................................... ......................... ........ 64

2.11.8

Kullback-Leibler Divergence for Uniform Random Numbers Numbers .................. ........................... .................. .................. .................. .................. .................. ......... 64

2.11.9

The Law of Large Numbers ................................. ................ ................................... ................................... ................................... ................................... ................................... .................... .. 66

2.11.10

The Central Centra l Limit Theorem Theore m ................................... .................. ................................... ................................... ................................... ................................... ............................... .............. 68

2.11.11

Accuracy Accura cy and Precision Precis ion of Results .................................. ................ ................................... ................................... .................................... ................................... ...................... ..... 72

2.11.12

Applications of Boltzmann’s Entropy ........................................................................................................ 72

2.12

Entropy in Monte Carlo Simulation Simulat ion .................................. ................. ................................... ................................... ................................... ................................... ............................... .............. 74

2.13

Optimization......... Optimiz ation........................... ................................... .................................. ................................... ................................... ................................... ................................... ................................... ....................... ..... 77

2.13.1 2.14

Optimization Example: Volume Volume of a cylinder................. .......................... ................... ................... .................. .................. .................. .................. .................. ............... ...... 77

The Adjoint Operator Operat or ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... ................... .. 82

References ........................................................................................................................................................................ 84 Problems ........................................................................................................................................................................... 85 vi

3

The Neutron Diffusion Diffusio n Equation ................................... .................. ................................... ................................... ................................... ................................... ................................... .......................... ........ 86 3.1

The Conservation Conserv ation (Continuity) (Contin uity) Equation Equatio n................................. ............... ................................... ................................... .................................... ................................... ......................... ........ 86

3.2

One group Diffusion Diffusi on Equation ................................. ................ ................................... ................................... ................................... ................................... ................................... ....................... ..... 88

3.3

One-group One-gr oup Diffusion Diffusi on Equation Applications Applica tions ................................. ............... ................................... .................................. ................................... .................................... .................... .. 90

3.4

The One-group One-gro up Critical Critic al Reactor............... Reacto r................................ ................................... ................................... ................................... ................................... ................................... ....................... ..... 94

3.5

Multi-group Multi-gr oup Diffusion Diffusi on Equation Equatio n .................................. ................. ................................... ................................... ................................... ................................... ................................... .................. 100

3.5.1

Multigroup Multigr oup Criticality Critic ality .................................. ................ ................................... .................................. ................................... ................................... ................................... ........................... ......... 101

Multigroup constants for Pu 3.6

239

(barns) ....................................................................................................................... 101

The Adjoint Diffusion Diffusio n Equation ................................... .................. ................................... ................................... ................................... ................................... ................................... .................. 102

References ...................................................................................................................................................................... 103 Problems ......................................................................................................................................................................... 103 4

The Neutron Transport Transpor t Equation Equatio n .................................. ................. ................................... ................................... ................................... ................................... ................................... ........................ ...... 105 4.1

Introduction Introd uction ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ............................. ............ 105

4.2

Structure of the Neutron Transport Equation ................... ............................ .................. .................. .................. ................... ................... .................. .................. ................. ........ 105

4.2.1

An integro-differential form of the Neutron Transport Equation ................. .......................... ................... ................... .................. ................. ........ 106

4.2.2

The Integral form of the the Transport Transport Equation Equation ................. .......................... ................... ................... .................. .................. .................. .................. .................. ............. .... 111

4.3

Multi-group Form of the Integral Transport Equation................. .......................... ................... ................... .................. .................. .................. .................. ................ ....... 112

4.4

Exact solutions of the Transport Equation Equation ................... ............................ .................. .................. .................. .................. .................. .................. .................. ................... ............... ..... 114

4.4.1

The Classic Albedo Problem ................................... .................. ................................... ................................... ................................... ................................... ................................ ............... 115

4.4.2

Infinite Medium with a plane plane isotropic source ................. .......................... ................... ................... .................. .................. .................. .................. .................. ........... 116

4.4.3

Finite Sphere with a point isotropic source .................. ........................... .................. .................. .................. .................. ................... ................... .................. .............. ..... 119

4.5

Numerical Methods Methods for Solving the Transport Equation .................. ........................... .................. .................. .................. .................. ................... ................... ........... .. 125

4.5.1

The Discrete Discre te Ordinates Ordinate s Method ................................... .................. .................................. ................................... .................................... ................................... .......................... ......... 126

4.5.2

The Spherical Spheric al Harmonics Harmonic s Method ................................... ................. ................................... ................................... .................................... ................................... ....................... ...... 127

4.5.3

Other Deterministic Determ inistic Methods ................................... .................. ................................... ................................... ................................... ................................... ............................. ............ 129

4.6

Transport Theory in the approximation ......................................................................................................... 129

4.7

Applications Applicat ions of Transport Transpor t Theory .................................. ................. ................................... ................................... ................................... ................................... ................................ ............... 130


The Monte Carlo Method .................................. ................. .................................. ................................... ................................... ................................... .................................... ................................... ................. 133 5.1

The Monte Monte Carlo Method for for Deterministic Problems .................. ........................... .................. .................. .................. .................. ................... ................... .............. ..... 133 vii

5.1.1 5.2 5.2.1

MC Evaluation Evaluatio n of Integrals Integ rals .................................. ................. ................................... ................................... ................................... ................................... ................................... .................. 133 Importance Import ance Sampling ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... ................. 137 MC solution of Integral Integ ral Equations Equation s .................................. ................ ................................... ................................... .................................... ................................... ....................... ...... 138

5.3

Monte Carlo Simulation of Neutron Transport .................. ........................... .................. .................. .................. .................. ................... ................... .................. ................. ........ 139

5.4

Geometry Geomet ry Description Descr iption ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................ .............. 141

5.5

Estimators Estimat ors and Tallies ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................ .............. 147

5.6

Nuclear Data Processing Proce ssing .................................. ................. .................................. ................................... ................................... .................................. ................................... .............................. ............ 149

5.7

Sampling an Initial Initia l Source ................................... ................. ................................... .................................. ................................... ................................... ................................... ........................... ......... 149

5.8

Sampling the “Distance to Collision” .................................................................................................................. 150

5.9

Determining Deter mining the type of Event ................................. ................ ................................... ................................... ................................... ................................... ................................... ..................... ... 150

5.10

Determining the nuclide of interaction ............... ........................ .................. .................. ................... ................... .................. .................. .................. .................. .................. ............. .... 151

5.11

Processing Proce ssing a scattering scatte ring event .................................. ................. ................................... ................................... ................................... ................................... ................................... ..................... ... 151

5.12

Processing Proce ssing a fission event ................................... ................. ................................... .................................. ................................... ................................... ................................... ........................... ......... 152

5.13

Processing Proce ssing a capture captur e event ................................... .................. ................................... ................................... ................................... ................................... ................................... ........................ ...... 154

5.14

Processing an escape-from-system event ................ ......................... .................. .................. .................. .................. .................. .................. .................. ................... .................. ........ 154

5.15

Mean and Variance Varianc e ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... .................... ... 154

5.16

Batch, History, Random Walk and Events .................................. ................ ................................... ................................... .................................... ................................... .................... ... 155

5.17

Variance Varianc e Reduction Reduct ion Methods.................. Methods ................................... ................................... ................................... ................................... ................................... ................................... ..................... ... 156

5.18

Estimating Perturbations with Monte Carlo Simulation ................. ........................... ................... .................. .................. .................. .................. .................. ............. .... 156


Nuclear Engineering Enginee ring Computer Compute r Codes ................................... .................. ................................... ................................... ................................... ................................... ................................ ............... 161 6.1


6.2

Deterministic Deter ministic Codes ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... ................. 163

6.2.1

ANISN .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 163

6.2.2

DOT........................... DOT......... ................................... ................................... ................................... .................................. ................................... ................................... ................................... ........................... ......... 163

6.2.3

TORT ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................... .................. 164

6.2.4

PARTISN PARTI SN .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ............................. ............ 164

6.3

Monte Carlo Codes ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... .................... ... 164

6.3.1

MCNP .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 164

6.3.2

TART ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................... .................. 165 viii

6.3.3

MORSE ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 165

6.3.4

KENO ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 166

6.3.5

MONK .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 166

6.3.6

TRIPOLI TRIPOL I ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ............................. ............ 166

6.4

Typical Simulations.............. Simulat ions............................... .................................. ................................... ................................... ................................... .................................... ................................... ....................... ...... 167

6.4.1

One-dimensional One-dim ensional Calculations Calculat ions .................................. ................. ................................... ................................... ................................... ................................... ............................. ............ 167

6.4.2

Two-dimensional Two-dim ensional calculations calcula tions ................................... .................. ................................... ................................... ................................... ................................... ............................. ............ 169

6.5

Nuclear Data Libraries Librar ies ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................ .............. 171

6.5.1

ENDF/B Libraries Librar ies ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................ .............. 172

6.5.2

Nuclear Nuclea r Data Processing Proce ssing Codes ................................. ................ ................................... ................................... ................................... ................................... ............................. ............ 176

References ...................................................................................................................................................................... 176 7

Multigroup Diffusion Theory and Optimal Distribution ............ ..................... .................. ................... ................... .................. .................. .................. .................. .................. ........... 179 7.1


7.2

Theory ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................................... ..................... ... 181

7.3

Two-group Two-gro up Diffusion Diffusio n Theory: uniform core .................................. ................ ................................... ................................... .................................... ................................... ................. 181

7.3.1

Two-group Diffusion Theory: non-uniform core .................. ........................... ................... ................... .................. .................. .................. .................. ................ ....... 182

7.4

Goertzel’s Theorem ............................................................................................................................................. 184

7.5

Optimal Distribution: Pontryagin’s Maximum Principle ..................................................................................... 184

7.6

Results Result s ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................................... ..................... ... 189

7.6.1

Uniformly Unifor mly Distributed Distri buted Core ................................. ................ ................................... ................................... ................................... ................................... ................................... .................. 189

7.6.2

Non-uniformly Non-unifo rmly Distributed Distri buted Core .................................. ................. .................................. ................................... .................................... ................................... .......................... ......... 193

7.6.3

Non-uniformly Distributed Core with PMP ................... ............................ .................. .................. .................. ................... ................... .................. .................. .............. ..... 204

7.7

Conclusions Conclus ions ................................. ................ ................................... ................................... ................................... ................................... ................................... ................................... ................................ ............... 214

References ...................................................................................................................................................................... 216 8

Monte Carlo Simulation Simulat ion in Nuclear Systems ................................... ................. ................................... ................................... .................................... ................................... ....................... ...... 222 8.1


8.2

Monte Carlo Simulation in a Critical Sphere ................... ............................ .................. .................. .................. ................... ................... .................. .................. .................. ........... .. 222

8.3

The Godiva Experiment Exper iment ................................. ................ .................................. ................................... ................................... .................................. ................................... ................................. ............... 223

8.4

Mathematics and and Computation Computation of Criticality .................. ........................... .................. .................. .................. .................. ................... ................... .................. .................. ........... .. 223

8.5

Nuclear Data .................................. ................. ................................... ................................... ................................... ................................... ................................... ................................... ............................. ............ 223

8.6

Monte Carlo Simulation Simulatio n Results................. Results .................................. ................................... ................................... ................................... ................................... ................................... .................. 230 ix

8.7

Convergence and and Stationarity of MC Results Results ........... .................... .................. .................. .................. .................. ................... ................... .................. .................. ................. ........ 234

8.8

Monte Carlo Simulation in a Fixed-Source Non-multiplying Non-multiplying Sphere .................. ........................... ................... ................... .................. .................. ........... .. 237

8.9

Simulation Simulat ion Process Proces s .................................. ................. .................................. ................................... ................................... ................................... .................................... ................................... .................... ... 237

8.10

Matlab® Program for for Point Isotropic Source in a Finite Non-multiplying Sphere .................. ........................... .................. .................. ........... 239

8.11

Results Result s ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................................... ..................... ... 243


Monte Carlo Simulation compared with Deterministic Solutions .................. ........................... .................. .................. .................. .................. ................... ............... ..... 250 9.1

Exact Solutions .................................. ................. ................................... ................................... .................................. ................................... .................................... ................................... .......................... ......... 252

9.1.1

Transport Transpo rt Theory..................... Theory... ................................... .................................. ................................... ................................... ................................... .................................... ............................. ........... 252

9.1.2

Diffusion Diffusi on Theory ................................. ................ .................................. ................................... ................................... ................................... .................................... ................................... ................. 255

9.1.3

Monte Carlo Simulation Simulat ion ................................... .................. ................................... ................................... ................................... ................................... ................................... ..................... ... 257

9.2

Results Result s ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................................... ..................... ... 257

9.3

Comparison for for Finite Sphere with a Point Isotropic Source .................. ........................... .................. .................. .................. .................. ................... ............... ..... 266

9.3.1

Transport Transpo rt Theory Exact Solution .................................. ................. .................................. ................................... .................................... ................................... .......................... ......... 266

9.3.2

Diffusion Diffusi on Theory Exact Solution Solutio n ................................... .................. .................................. ................................... .................................... ................................... .......................... ......... 267


Monte Carlo Simulation: Practical Applications .................. ........................... .................. ................... ................... .................. .................. .................. .................. .................. ........... 282

10.1


10.2

Simple Assemblies Assembl ies ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................... .................... ... 283

10.3

Reactor Reacto r Core Modeling ................................. ................ .................................. ................................... ................................... .................................. ................................... ................................. ............... 287

10.3.1

Input File ................................... .................. ................................... ................................... .................................. ................................... .................................... ................................... .......................... ......... 288

10.3.2

Surrounding Surro unding cells ................................... .................. .................................. ................................... ................................... ................................... .................................... ................................ .............. 289

10.3.3

Source Sourc e Description Descr iption ................................... .................. .................................. ................................... ................................... .................................. ................................... .............................. ............ 290

10.3.4

Plotting Plotti ng the Geometry Geome try .................................. ................. ................................... ................................... ................................... ................................... ................................... ........................ ...... 291

10.3.5

Tally Cards ................................. ................ ................................... ................................... .................................. ................................... .................................... ................................... .......................... ......... 293

10.4

Nuclear Criticality Critic ality Safety Analysis .................................. ................. ................................... ................................... ................................... ................................... ................................ ............... 298

10.5

Perturbation Pertu rbation Calculations Calculat ions ................................... ................. ................................... .................................. ................................... ................................... ................................... ........................... ......... 298

References ...................................................................................................................................................................... 303 Problems ......................................................................................................................................................................... 303 x

Multiple Choice Questions .................................................................................................................................................. 305 Mid-Term Examination ....................................................................................................................................................... 309 Final Examination ................................................................................................................................................................ 311 Annex A Chandrasekhar’s Chandrasekhar’s H function .................................................................................................................................. 316 Annex B Matlab® Program Transport Theory ..................................................................................................................... 320 Annex C Neutron Flux (Transport Theory) .......................................................................................................................... 325 Annex D Matlab® Program One Group Monte Carlo .......................................................................................................... 326 Annex E Matlab® Program Finite Element Method ............................................................................................................ 336 Annex F (MCNP input and MATLAB Processing Files) ......................................................................................................... 357 About the Author ................................................................................................................................................................ 362

xi

List of Figures Figure 1-1 U238 total cross-section (ENDF/B-VII.0) .................. ........................... .................. .................. .................. .................. .................. .................. .................. ................... ................... ............ ... 5 Figure 1-2 U238 total cross-section: low energy 1/E behavior................. .......................... ................... ................... .................. .................. .................. .................. .................. .............. ..... 5 Figure 1-3 U238 total cross-section: resonance behavior ....................... ................................ .................. .................. .................. ................... ................... .................. .................. ............... ...... 6 Figure 1-4 U238 total cross-section: high energy behavior ................ ......................... ................... ................... .................. .................. .................. .................. .................. .................. ........... .. 6 Figure 1-5 Nuclear fission...................................................................................................................................................... 14 Figure 1-6 Fission fragment yield (Lewis, p.15)............... p.15)......................... ................... .................. .................. .................. .................. .................. .................. ................... ................... .................. ........... 15 Figure 1-7 Fission spectrum .................................................................................................................................................. 16 Figure 1-8 Fission spectra ..................................................................................................................................................... 17 Figure 1-9 Cranberg and Watt fission spectra .................. ........................... .................. .................. ................... ................... .................. .................. .................. .................. .................. .................. ......... 18 Figure 1-10 Neutron spectra in fast and thermal reactors .................... ............................. ................... ................... .................. .................. .................. .................. .................. ............... ...... 21 Figure 1-11 Elastic scattering ................................................................................................................................................ 22 Figure 1-12 Scattering angle in Lab and CM systems .................. ........................... .................. .................. .................. .................. .................. .................. .................. ................... ................. ....... 23 Figure 1-13 Reactor Core ...................................................................................................................................................... 27 Figure 1-14 Neutron multiplication keff ............................................................................................................................... 30 Figure 2-1 Bessel functions of o f the first and second kind ................. .......................... .................. ................... ................... .................. .................. .................. .................. .................. ............ ... 45 Figure 2-2 Neutron transport in a 1-D albedo problem .................. ........................... .................. .................. .................. .................. ................... ................... .................. .................. ............. .... 49 Figure 2-3 Some basic elements e lements in FEM: (i) 1-D linear element, (ii) 2-D triangular element, (iii) 2-D square element, (Lewis et al 2004) ............................................................................................................................................................................. 53

Figure 2-4 Triangular elements in a curved pipe ................... ............................ .................. .................. .................. .................. .................. .................. .................. ................... ................... ............. .... 55 Figure 2-5 Temperature contours in a curved pipe ...................... ............................... .................. ................... ................... .................. .................. .................. .................. .................. ............... ...... 56 Figure 2-6 Two-dimensional quadratic element with 8 nodes ................... ............................ .................. .................. .................. ................... ................... .................. .................. ........... 57 Figure 2-7 Numerical Integration .......................................................................................................................................... 57 Figure 2-8 Uniformly distributed random number .......................... ................................... .................. ................... ................... .................. .................. .................. .................. .................. ............ ... 59 Figure 2-9 Exponentially distributed random number ................... ............................ .................. .................. .................. ................... ................... .................. .................. .................. ............. .... 60 Figure 2-10 Exponentially distributed random variable from Matlab(R) .................. ........................... .................. .................. .................. .................. ................... .............. .... 61 Figure 2-11 Exponential pdf from Matlab(R) GUI ................................................................................................................. 62 Figure 2-12 KL distance for 5 bins ......................................................................................................................................... 65 Figure 2-13 KL distance for 10 bins ....................................................................................................................................... 65 Figure 2-14 KL distance for 20 bins ....................................................................................................................................... 66 Figure 2-15 Estimates of value of  ...................................................................................................................................... 70 Figure 2-16 Estimates of value of  (M=500) ....................................................................................................................... 71 Figure 2-17 Estimates of  (N=5000, 10000, 15000, 20000 for M=500) ..................... .............................. .................. .................. .................. .................. .................. ............ ... 71 Figure 2-18 Estimates of  .................................................................................................................................................... 72 Figure 2-19 A simplified picture of particles in a box. .................. ........................... .................. ................... ................... .................. .................. .................. .................. .................. ............... ...... 73 Figure 2-20 Absolute entropy S vs number of states N ........................ ................................. .................. .................. .................. ................... ................... .................. .................. ................ ....... 74 Figure 2-21 R, H contours for fixed surface area .......................... ................................... .................. ................... ................... .................. .................. .................. .................. .................. ............... ...... 80 Figure 2-22 Maximum cylinder volume for fixed surface area ................ ......................... .................. .................. .................. .................. ................... ................... .................. ............. .... 81 Figure 2-23 Area constraint for A=10 cm^2 and Volume contours ................ .......................... ................... .................. .................. .................. .................. .................. ............... ...... 82 Figure 3-1 A volume element ................................................................................................................................................ 87 xii

Figure 3-2 Escape probability in a graphite slab ................................................................................................................... 93 Figure 3-3 Neutron flux in a bare 'spherical' reactor ................... ............................ .................. .................. .................. .................. .................. .................. .................. ................... ................. ....... 96 Figure 4-1 Relaxation length in m.f.p.'s for isotropic scattering ........... .................... .................. .................. .................. .................. ................... ................... .................. .............. ..... 110 Figure 4-2 Albedo for monoenergetic neutrons for isotropic medium .................. ........................... .................. .................. .................. .................. ................... ............... ..... 116 Figure 4-3 Angular flux for c=0.2, infinite medium ................... ............................ .................. .................. .................. .................. .................. .................. .................. ................... .................. ........ 117 Figure 4-4 Angular flux c=0.4, infinite medium.................. ............................ ................... .................. .................. .................. .................. .................. .................. ................... ................... .............. ..... 118 Figure 4-5 Angular flux c=0.8, infinite medium.................. ............................ ................... .................. .................. .................. .................. .................. .................. ................... ................... .............. ..... 119 Figure 4-6 Total flux, infinite medium isotropic source ............... ........................ .................. .................. .................. .................. ................... ................... .................. .................. .............. ..... 119 Figure 4-7 Transport Theory flux in a finite sphere (c=0.3) ...................... ............................... ................... ................... .................. .................. .................. .................. .................. ........... 121 Figure 4-8 Transport Theory flux in a finite sphere (c=0.9) ...................... ............................... ................... ................... .................. .................. .................. .................. .................. ........... 122 Figure 4-9 Transport Theory flux in a finite sphere .................. ........................... .................. .................. .................. .................. .................. .................. .................. ................... .................. ........ 123 Figure 4-10 Transport Theory flux for values approaching c=1 ................... ............................ ................... ................... .................. .................. .................. .................. ................ ....... 125 Figure 4-11 Discrete ordinates with N=4,8,16,24 .................. ........................... .................. .................. .................. .................. .................. .................. .................. ................... ................... ........... .. 126 Figure 5-1 MC Integral Evaluation by Hit-and-Miss technique .............. ....................... ................... ................... .................. .................. .................. .................. .................. ............. .... 135 Figure 5-2 MC integration for f(x,y)=           ................................................................................................... 136 Figure 5-3 MC integration for f(x,y) = sin r / r ..................................................................................................................... 137 Figure 5-4 Neutron transport in matter .............................................................................................................................. 140 Figure 5-5 A typical complex geometry model ................................................................................................................... 141 Figure 5-6 Concentric spheres ............................................................................................................................................ 145 Figure 5-7 Ray-Plane intersection ....................................................................................................................................... 146 Figure 5-8 Ray intersection with a plane ............................................................................................................................ 147 Figure 5-9 Random walk of a neutron ................................................................................................................................ 148 Figure 5-10 Forward scattering of neutrons crossing a slab ........... .................... .................. .................. .................. .................. ................... ................... .................. .................. ........... .. 149 Figure 5-11 Chain of possible events .................................................................................................................................. 151 Figure 6-1 Neutron flux mapping in ITER ............................................................................................................................ 162 Figure 6-2 Evolution of MCNP ............................................................................................................................................. 165 Figure 6-3 Dose rates calculated with MAVRIC ................. ........................... ................... .................. .................. .................. .................. .................. .................. ................... ................... .............. ..... 166 Figure 6-4 One-dimensional ITER model ............................................................................................................................ 167 Figure 6-5 ITER radial model ............................................................................................................................................... 168 Figure 6-6 Streaming of 14MeV neutrons throug t hroug a duct in ITER ........... .................... .................. ................... ................... .................. .................. .................. .................. ............. .... 169 Figure 6-7 Neutron flux and gamma-ray heating in ITER................... ............................ .................. .................. .................. ................... ................... .................. .................. ................. ........ 170 Figure 6-8 ENBF/B-VII data retrieval page .................. ........................... .................. .................. .................. .................. ................... ................... .................. .................. .................. .................. ............. .... 172 Figure 7-1 Fluxes for first eigenvalue .................................................................................................................................. 189 Figure 7-2 Convergence of k_eff in the power iteration method .......... ................... .................. ................... ................... .................. .................. .................. .................. ............. .... 190 Figure 7-3 Convergence of group-1 fluxes (60,40 mesh intervals) ................ ......................... .................. .................. .................. .................. ................... ................... .............. ..... 191 Figure 7-4 Convergence of group-2 fluxes ..................... .............................. .................. .................. .................. ................... ................... .................. .................. .................. .................. .................. ........... 191 1 91 Figure 7-5 Convergence of fluxes in a critical sphere (60,40 mesh intervals, 20 iterations) ............... ........................ .................. .................. ............. .... 192 Figure 7-6 Critical configuration (60,40 intervals, 20 iterations) ................ ......................... .................. .................. .................. .................. ................... ................... ................. ........ 193 Figure 7-7 Variation of atomic ratio .................................................................................................................................... 195 Figure 7-8 Converged fluxes fo r the 21.9 cm reactor with W -U ratio of 880:349.21 .......... ................... .................. .................. .................. .................. ........... 195 Figure 7-9 Fluxes for two equi-volume zones ..................................................................................................................... 196 xiii

Figure 7-10 Converged fluxes for Water-U235 atomic ratio 310:1291.73 310 :1291.73 ................... ............................ .................. .................. .................. ................... .................. ........ 197 Figure 7-11 Finite-difference convergence .................. ........................... ................... ................... .................. .................. .................. .................. .................. ................... ................... .................. ........... .. 198 Figure 7-12 Flat thermal flux with infinite water reflector .................... ............................. ................... ................... .................. .................. .................. .................. .................. ............. .... 200 Figure 7-13 Fast and thermal fluxes with uniform fissile loading .................. ........................... .................. .................. .................. .................. ................... ................... .............. ..... 201 Figure 7-14 7 -14 Fast and thermal fluxes from non-uniform fissile loading of Fig. 7-13 .................. ........................... ................... ................... .................. .............. ..... 201 Figure 7-15 Non-uniform fissile loading in a 3-zone equi-volume core.................. ........................... .................. .................. .................. .................. ................... ............... ..... 202 Figure 7-16 Effect of a step distribution on the critical mass ................... ............................ ................... ................... .................. .................. .................. .................. .................. ........... 203 Figure 7-17 Fast and thermal flux ....................................................................................................................................... 205 Figure 7-18 variable atomic ratio ........................................................................................................................................ 209 Figure 7-19 Variable atomic distribution (3-zone PMP) .................. ........................... .................. ................... ................... .................. .................. .................. .................. .................. ........... 209 Figure 7-20 Variable atomic ratio ....................................................................................................................................... 211 Figure 7-21 variable atomic ratio ........................................................................................................................................ 212 Figure 7-22 Variable atomic ratio ....................................................................................................................................... 212 Figure 7-23 Variable atomic ratio 400, 635 ................. .......................... ................... ................... .................. .................. .................. .................. .................. ................... ................... .................. ........... .. 213 Figure 7-24 Variable atomic ratio 500, 570 ................. .......................... ................... ................... .................. .................. .................. .................. .................. ................... ................... .................. ........... .. 213 Figure 8-1 ENDF/B-VII 20-group cross-sections of U235 ................... ............................ .................. .................. .................. ................... ................... .................. .................. ................. ........ 224 Figure 8-2 ENDF/B-VII 20 group cross-sections of U235.................. U235........................... .................. ................... ................... .................. .................. .................. .................. .................. ........... 225 Figure 8-3 ENDF/B-VII 20-group cross-sections of U238 ................... ............................ .................. .................. .................. ................... ................... .................. .................. ................. ........ 225 Figure 8-4 ENDF/B-VII Fission cross-sections of U235 and U238 ................. .......................... ................... ................... .................. .................. .................. .................. ................ ....... 226 Figure 8-5 Difference in ENDF/B-VII total cross-sections of U235 and U238 ................... ............................ .................. .................. .................. .................. ............. .... 226 Figure 8-6 Neutrons sampled as a function of number of neutrons emerging from fission .................. ........................... .................. .................. ........... 227 Figure 8-7 Watt fission spectrum (a=0.7, b=1) .................. ............................ ................... .................. .................. .................. .................. .................. .................. ................... ................... .............. ..... 228 Figure 8-8 Sampled Watt fission spectrum (a=0.7, b=1, N=100000) ................. ........................... ................... .................. .................. .................. .................. .................. ........... 228 Figure 8-9 Sampled Watt fission spectrum (a=0.7, b=1, N=100000) ................. ........................... ................... .................. .................. .................. .................. .................. ........... 229 Figure 8-10 Watt spectrum and sampled spectrum spect rum (a=0.7, b=1, N=100000) .................... ............................. ................... ................... .................. .................. ........... .. 229 Figure 8-11 keff for 100 neutrons, 20 generations .............................................................................................................. 231 Figure 8-12 keff for 50 neutrons, 30 generations ................................................................................................................ 231 Figure 8-13 keff for 50 neutrons, 40 generations................................................................................................................. 232 Figure 8-14 keff for 10% perturbation in material density .......... ................... .................. ................... ................... .................. .................. .................. .................. .................. ................ ....... 232 Figure 8-15 keff for 90% enrichment .................................................................................................................................... 233 Figure 8-16 MC results for a one-group Godiva simulation.................. ........................... .................. .................. .................. .................. ................... ................... .................. .............. ..... 234 Figure 8-17 MC one-group Godiva relative error .................. ........................... .................. .................. .................. .................. .................. .................. .................. ................... ................... ........... .. 235 Figure 8-18 Three histories for a point isotropic source at center of a non-multiplying sphere..................... sphere.............................. .................. ........... 237 Figure 8-19 Monte Carlo flux in a finite sphere .................................................................................................................. 244 Figure 9-1 Flux (exact solution) in slabs of Al, Fe, U, Au, B and Gd. .................. ........................... .................. .................. .................. .................. ................... ................... ........... .. 258 Figure 9-2 One-speed flux in an infinite medium (Ganapol) ................. .......................... ................... ................... .................. .................. .................. .................. .................. ............. .... 259 Figure 9-3 One-speed flux in a 1-D slab .............................................................................................................................. 260 Figure 9-4 One speed flux in a 1-D slab (high-c media) ...................... ............................... ................... ................... .................. .................. .................. .................. .................. ................ ....... 260 Figure 9-5 One-speed flux in 1-D slab (DT vs P1) ................... ............................ .................. .................. .................. .................. .................. .................. .................. ................... ................... ........... .. 261 Figure 9-6 One-speed flux in Gd: DT, P1 and P2 comparisons.................. ........................... ................... ................... .................. .................. .................. .................. .................. ........... 261 Figure 9-7 Collision density due to a mono-directional point source incident from left...................... left............................... ................... ................... ........... .. 262 xiv

Figure 9-8 Comparison of collision density: Monte Carlo vs Diffusion Theory ............. ...................... .................. ................... ................... .................. ................. ........ 263 Figure 9-9 Collision density: MC (1000X5) (1000X5 ) vs Diffusion Theory in an Fe cube ................... ............................ .................. .................. .................. .................. ............. .... 264 Figure 9-10 Collision density: MC(2000X5) vs Diffusion Theory in an Fe cube .................. ........................... .................. .................. .................. .................. ............. .... 264 Figure 9-11 Collision density: MC(1000X5) vs DiffusionTheory in a Boron cube .................. ........................... .................. .................. .................. .................. ........... 265 Figure 9-12 Flux estimates: Diffusion and Transport comparisons .......................... ................................... .................. .................. .................. .................. .................. ............. .... 265 Figure 9-13. Transport Theory flux ..................................................................................................................................... 267 Figure 9-14. Diffusion Theory flux ....................................................................................................................................... 269 Figure 9-15. Diffusion theory in a finite sphere ...................... ............................... .................. .................. ................... ................... .................. .................. .................. .................. .................. ........... 270 Figure 9-16. Transport and Diffusion fluxes ........................................................................................................................ 272 Figure 9-17 Ratio of Transport to Diffusion Flux ............. ....................... ................... .................. .................. .................. .................. .................. .................. .................. ................... .................. ........ 273 Figure 9-18 MC flux vs Diffusion flux in a 10-equivolume 10 -equivolume sphere .................. ............................ ................... .................. .................. .................. .................. .................. ............. .... 274 Figure 9-19 Neutron flux MC TLE compared com pared with Diffusion Theory in 30 equi-volume regions of aluminum .................. .................. 275 Figure 9-20 Monte Carlo compared with Transport Theory (infinite medium) and Diffusion Theory .................. ........................... ............. .... 276 Figure 9-21 Transport Theory (asymptotic) compared with Diffusion Theory ................... ............................ ................... ................... .................. .................. ........... .. 277 Figure 9-22 Ratio of asymptotic transport flux to diffusion flux ............ ..................... .................. ................... ................... .................. .................. .................. .................. ............. .... 277 Figure 9-23 9 -23 Kullback Leibler Divergence vs Sample Size N for 10 spatial bins...................... bins............................... .................. .................. .................. .................. ........... 280 Figure 10-1 keff in a Godiva sphere .................................... .................. ................................... .................................. ................................... ................................... ................................... ........................... ......... 284 Figure 10-2 Flux in a Godiva sphere .................................................................................................................................... 284 3

Figure 10-3 UO2 sphere 70% enriched, den=10.9 g/cm , radius 200 kg ........................................................................... 285 Figure 10-4 Flux in a bare UO2 sphere ................................. ................ ................................... ................................... ................................... ................................... ................................... ........................ ...... 286 Figure 10-5 Flux in a UO2 sphere (in xy plane at z=0) ................................................ ............................... ................................... .................................... ................................... .................... ... 286 Figure 10-6 Hexagonal array of fuel assemblies in a reactor core .................. ............................ ................... .................. .................. .................. .................. .................. ............. .... 287 Figure 10-7 A typical PWR fuel assembly ............... ........................ ................... ................... .................. .................. .................. .................. .................. ................... ................... .................. ................. ........ 292 Figure 10-8 A 3X3 assembly ................................................................................................................................................ 293 Figure 10-9 Collision estimate of  .................................................................................................................................. 295 Figure 10-10 FOM of average  ...................................................................................................................................... 296 Figure 10-11 Neutron flux in a 3X3 PWR pin cell ...................... ............................... .................. .................. .................. .................. .................. .................. .................. ................... .................. ........ 297 Figure 10-12 Neutron flux in a 3X3 PWR pin cell c ell (xy plane) ....................... ................................ .................. .................. .................. ................... ................... .................. ................. ........ 297 Figure 10-13 Effect of perturbation in material density .................... ............................. .................. .................. .................. ................... ................... .................. .................. ................. ........ 299 Figure 10-14 1 0-14 Perturbation Pe rturbation estimates using derivative sampling and one-group diffusion ................. .......................... .................. .................. ............. .... 301 Figure 10-15 Perturbation estimates for J .................. ........................... .................. .................. .................. .................. ................... ................... .................. .................. .................. .................. ............. .... 302 Figure 10-16 4S-type core arrangement .................. ........................... .................. .................. .................. .................. ................... ................... .................. .................. .................. .................. ................ .......302

xv

Notation

u

lethargy

u

direction cosine in  direction

v

speed

v

direction cosine in  direction

w

direction cosine in  direction

w

statistical ‘weight’

zo

extrapolation distance

A

relative atomic mass

A

area

B

buckling

c

number of ‘secondary’ neutrons

d


d

track length

f

fission



thermal fuel utilization



probability distribution function

g

distribution function

h

distribution function

k

Boltzmann constant



material buckling



effective multiplication of a finite system



geometrical buckling



effective multiplication of an infinite system



collision operator



rest mass       (electron, proton, neutron)

D

diffusion coefficient

n

neutron density (neutrons m )

-3

̂

unit vector

p

resonance escape probability

q

slowing-down density

̅

position vector

s

scattering

t

time



diffusion time



moderation time

u

atomic mass unit

Kullback-Leibler divergence  

derivative operator

E

energy



energy recoverable from fission



cumulative distribution function

H

entropy



Chandrasekhar’s function



Bessel function of the first kind

J

joule

16

⃗

neutron current



associated Legendre polynomial



Bessel function of the first kind

Q

quality factor

K

kelvin

R

reaction rate



Bessel function of the second kind

R

relative standard deviation

̃

extrapolated radius

 kernel of integral operator L

diffusion length

S

source



integro-differential operator

S

entropy



integro-differential adjoint operator

 ̿

scattering matrix in multi-group diffusion



operator

T

temperature

M

atomic mass



matrix in multi-group diffusion equation

 

fission matrix in multi-group diffusion equation



transport operator



thermal migration area

U

internal energy

N

no. of neutrons

V

volume

N

statistical sample size

V CM CM

velocity in center-of-mass system

N

number density

W

Watt



Bessel function of the second kind

X

random variable



Avogadro’s Number



spherical harmonics

P

phase space

Z

atomic number

P

probability



probability

Greek (lower-case)



fast non-leakage probability



alpha radiation



thermal non-leakage probability



beta radiation



Legendre polynomial



gamma radiation

17



delta function



fast fission factor



eta: neutrons emitted per neutron absorbed

Greek (upper-case)



orthogonal angle



neutron current



inverse of diffusion length



Hamiltonian



mean free path



macroscopic cross-section



Lagrange multiplier



macroscopic absorption cross-section



 (cosine of orthogonal angle of scattering)



macroscopic fission cross-section



number of fission neutrons/fission



macroscopic scattering cross-section



transport ‘relaxation’ length



auxiliary functions



random number



solid angle



density



microscopic cross-section

Other Abbreviations



standard deviation

CFR

Code for Federal Regulations

 

variance

eV

electron volt



neutron age

ENDF

Evaluated Nuclear Data File



azimuthal angle

Gy

Gray (absorbed dose)



neutron flux

Rad

absorbed radiation dose



adjoint neutron flux

Rem

radiation equivalent man



complimentary solution

Sv

Sievert



particular solution

USNRC United States Nuclear Nuclear Regulatory Commission



fission spectrum



collision density



eigenfunction



18

1 Preliminaries: Nuclear Engineering 1.1 Introduction This chapter is intended to give a basic foundation of concepts and quantities on which the body of knowledge in nuclear engineering, both the underlying nuclear physics and the nature and scale of nuclear technology, is based. It only gives a cursory insight; more can be learnt during the lectures, going through the exercises and problems, and of course reading through the material given in the references. Before reading the introduction, it is advisable to review some basic nuclear physics. A simple picture of an atom, with a -27

dense nucleus consisting of neutrons with rest mass  1.67482 X 10 -27

1.67252 X 10

kg (1.008665 u) and protons (rest mass 

kg (1.007277 (1.007277 u) surrounded by electrons (rest mass  9.109 X 10

th

-31

kg ). ). The basic mass unit 1u 1 u is

12

defined as 1/12 the mass of an unbound carbon C atom. Let us consider all atoms to have a relative mass  ≡ / . Essentially then, neutron transport takes place in matter as neutrons go about interacting with atoms which can be as light as hydrogen (A~1) on one end, to heavy atoms such as uranium (A~238). Within the nucleus, the nucleons (neutrons are protons) pair up with opposite spins and thus are bound inside. The binding energy is the difference between the massmass energy of constituent nucleons     and the nuclear massmass -energy    . As an example, the binding energy of a tritium nucleus, 13 , i.e. 1 i.e. 1 proton and 2 neutrons, is calculated from the difference between its mass -energy of constituent nucleons [.00 .0078 785 5   .008665 .008665]9.5

9.5

 

 

 87.9885 87.9885  and its nuclear -mass energy .060 .06050 50 ∗

 809.450575 809.450575 , i.e., i.e., 8.4813 MeV. An often quoted figure is the average binding energy per nucleon

which in this case is 2.81 MeV/nucleon. Nuclei, like electrons, have excited states and decay by emitting alpha, beta or gamma radiation. Finally, neutrons can have a number of reactions with nuclei two important ones being fission, which is the breakingbreaking-up of heavy nuclei into lighter nuclei, and fusion which is the fusing or joining of light nuclei into heavy nuclei. Both these reactions make the system ‘move’ towards an increase in the binding energy per nucleon and are 19

The methods considered in the following chapters will give the reader a capability will be developed to carry out detailed whole-core computations and determine the neutron flux, the subsequent reactor power, fuel arrangement, and related nuclear reactor design parameters.

References 1. Cacuci, D.G., Editor, Handbook Editor, Handbook of of Nuclear Engineering Engineering , Nuclear Engineering Fundamentals, Springer Science and Business Media LLC2010. 2. Cullen, D.E., Brown, P., Lent, E., MacFarlane, R., McKinley, S., Criticality Calculations using LANL and LLNL Neutron Transport Transport Codes, UCRL-TR, Lawrence Livermore National Laboratory, 2007. 3. Duderstadt, J. J. and Hamilton, L. J., Nuclear Reactor Analysis, Analysis, John Wiley & Sons, 1976. 4. Etherington, H., Editor, Nuclear Editor, Nuclear Engineering Engineering Handbook , New York: McGraw-Hill, 1958. 5.

Gantayet, L. M., Editor, BARC Highlights, Chemical Sciences and Engineering, Bhabha Atomic Research Centre, Mumbai, India. http://www.barc.gov.in/publications/eb/golden/chemical/toc/chapter6/6_3.pdf

6. Glasstone, S. and Sesonske, A., Nuclear Reactor Reactor Engineering Engineering , Princeton, N.J., Van Nostrand, 1963. 7. Harmon, C.D. II, Busch, R D., Briesmeister, J. F., and Forster, R. A., Mendius, P. W., Ed., LA-12827-M Manual, Criticality Calculations with MCNP TM: A Primer, Los Alamos National Laboratory, UC-714, Issued: August 1994. 8. Henry, A. F., Nuclear F., Nuclear Reactor Reactor Analysis, Analysis, Cambridge, MA, MIT Press, 1975. 9. Isbin, H.S., Introductor H.S., Introductoryy Nuclear Nuclear Reactor Theory, Theory, New York: Reinhold, 1963. 10. Lamarsh, J.R. and Baratta, A. J., Introduction to Nuclear Nuclear Engineering Engineering , 3rd Ed., Prentice Hall, 2001. 11. Lamarsh, J. R., Introdu R., Introduction ction to Nuclear Nuclear Reactor Reactor Theory Theory,, AddisonAddison-Wesley Publishing Company, 1972. 12. Lewis, E.E., Fundamenta E.E., Fundamentals ls of Nuclear Nuclear Reactor Reactor Physics Physics , Academic Press, 2008. 13. Murray, R.L., Introduction R.L., Introduction to Nuclear Engineering , 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1961. 14. Reuss, P., Neutron P., Neutron Physics, Physics, EDP Sciences, 2008. 15. Stacey,W.M., Nuclear Stacey,W.M., Nuclear Reactor Physics Physics,, John Wiley & Sons, 2001. 31

16. Stephenson, R., Introductio R., Introduction n to Nuclear Nuclear Engineering Engineering , 2nd Ed., New York: McGraw-Hill, 1958. nd

17. Shultis, J. K., and Faw, R. E., Fundamentals of Nuclear Nuclear Science Science and Engineering Engineering , 2 Edition, CRC Press/Taylor & Francis, 2008. 18. Weinberg, A.M., and Wigner, E.P., The Physical Theory of Neutron Chain Reactors , Chicago: University of Chicago Press, 1958. Nuclear Data Data Center, Center, Brookhaven Brookhaven National National Laboratory, Laboratory, http://www.nndc.bnl.gov/exfor/endf11.jsp 19. National Nuclear 20. Encyclopæd Encyclopædia ia Britannica Britannica Online, Online, s. v. "nuclear reactor", accessed March 31, 2013, http://www.britannica.com/EBchecked/topic/421763/nuclear-reactor/45774/Coolant-system .

21. National Institute of Standards and T echnology echnology (NIST), U.S. Department of Commerce,

http://www.physics.nist.gov/cgi-

bin/Star/compos.pl?matno=121

Problems 1. Given atomic fractions: U 234 (0.0057%), U235(0.72%), and U238(99.27%), find the average atomic weight and the corresponding weight percentages 2. Calculate atomic densities for the following: a) Natural uranium uranium with b) Bare Pu

    

239

and atomic fractions U 238 0.992745 U235 0.007200.

metal delta phase 100% Pu239

    

.

c) Given the following data for the fast critical assemblies Godiva, Jezebel and Jezebel23 (Cullen et al, 2007) determine the weight fractions of each of the materials listed.

Model

Godiva

Jezebel

Jezebel23

Radius (cm)

8.7407

6.3849

5.9838

Density (g/cm3)

18.74

15.61

18.424

32

Composition

U235

4.4994e-2

Pu239

3.7047e-2

U233

4.6712e-2

(atoms/barn-cm)

U238

2.4984e-3

Pu240

1.7512e-3

U234

5.9026e-4

U234

4.9184e-4

Pu241

1.1674e-4

U238

2.8561e-4

Ga69

8.26605e-4

U235

1.4281e-5

Ga71

5.48595e-4

Mass (g)

52419.98

17019.77

16534.98

3. Calculate atomic densities of the fuels a) UO2 of density 10.5

 

b) PuO2 of density 11.46

with a U

235

enrichment of 17%.

, with weight fractions of

c) U-10 wt% Zr alloy powder of density 15.48





0.118055 and



0.881945

with uranium consisting of 17wt% U 235 and the rest

238

U

.

4. Calculate atomic densities of the reflectors a) Beryllium metal density 1.85

  

b) Beryllium Oxide with a density of 3.01

  

, and weight fractions 4Be 0.360320, 8O 0.639680

5. Find the atomic densities in Boron Carbide given its density

 

 

(weight fraction

=0.217390) Atomic Weight=55.24, Answer: N(B 4C)=0.0277, NboronNat=0.1108, N(

N(





=0.782610,

=0.02205,

=0.08875, N(C)=0.0277

6. Find the atomic densities in a solution of UO 2F2 with a uranium enrichment of 5%, density of U 235 of 0.04 g/cc and a given ration of hydrogen to fissile atoms (H/X) of 500.U(4.89)O2F2 500.U(4.89)O2F2 solution N235=1.0889e-4, N238=2.0909e-3, N238=2.0909e-3, Nf=4.3996e-3, Nf=4.3996e-3, Nh=5.7058e-2, Nh=5.7058e-2, No=3.2929e-2, No=3.2929e-2, Ntotal=9.6586e-2 Ntotal=9.6586e-2 7. Calculate the atomic densities in the following structural materials a) stainless steel consisting of Fe with 18% chromium by weight, 8% nickel and 0.08% carbon, find the atomic densities. 33

b) aluminum (Atomic weight 26.9815, density

 

8. Determine the fast reactor macroscopic absorption cross-section

)

 of uranium consisting of 17 wt% U



235

in

uranium . Use the data in Table 1.3. 9. Determine the thermal macroscopic cross-section of UO 2 fuel pellets with 3wt% enriched fuel. 10. Determine the absorption mean mean free path of a fast neutron travelling in boron with



        ,

. Compare this with the corresponding mean free path for a thermal neutron for which

.

What possible use of boron could this have for a nuclear reactor? 11. For the reactor of Example 1.7, find the infinite multiplication



assuming that



, so that

   

for a

fuel to moderator mass fraction 1:80. Now determine the concentration of normal enrichment B 4C (

   ,

) at which the reactor



would fall to 1.00.

34

2 Preliminaries: Mathematical

Foundations 2.1 Introduction This chapter reviews the mathematical foundations and knowledge required for understanding the formulation, performing a simulation, simulation, and coding for numerical numerical implementation implementation of problems in nuclear engineering. The material is presented in the following order: general mathematics, mathematics, mathematics mathematics specific to the neutron diffusion and transport equations followed by mathematics, including probability and statistics, for Monte Carlo simulation. The purpose is, again, to provide a comprehensive review of the mathematics applicable to nuclear engineering, usually spread over different subjects, in one resource available for study before or during the phase when problems are encountered. Neutron diffusion diffusion and transport transport can be expressed expressed by ordinary differential differential equations, equations, partial partial differential differential equations, equations, integral integral equations and integro-differential integro-differential equations. We therefore discuss underlying solution techniques which will be used in later chapters.

2.2 Ordinary Differential Equations Ordinary differential equations (ODE’s) are used to represent systems with one independent variable such as the spatial variable



, and can, for many cases, be solved exactly. ODE’s can be first -order, second-order, or more generally



-

order where the order represents the order of the higher derivative of the dependent variable with respect to the independent variable. It can be shown that



-order ODE’s can be reduced to



first-order first-order ODE’s which, which, when linear,

are easily cast as state-space equations amenable to standard algebraic methods of linear algebra. The formulation of neutron diffusion can be expressed in the form of a second-order ordinary differential equation under a set of assumptions which we will examine in a later chapter on diffusion. Let us therefore review a method of obtaining an exact solution of a simple form of such an equation: the linear non-homogeneous form

35

            Eq. (2.1) also represents the 1-D motion in the mass-spring-damper problem, when and





(2.1)



represents the displacement



,

represents time , where the restoring force of the spring is represented by Hooke’s Law and an external time -

dependent force



is applied. The procedure of solving this equation is to seek a complementary solution

     

the homogeneous equation, and a particular solution



from

from the inhomogeneous term, and express the solution as

(2.2)

Thus, in operator form

The operator expressed as

̂   ∑ 

(̂  ̂  )         

is treated as a scalar and roots are obtained for

. For roots

(2.3)

 

, the solution is

. For the particular solution

  (̂  ̂  ) 

(2.4)

The solutions can thus be obtained after the usual factoring and algebraic manipulation. Constants in the solutions can then be determined from the given boundary conditions which can be Dirichlet (with the dependent variable specified on the boundaries), Neumann (with the normal derivative specified on the boundaries) or mixed Cauchy boundary conditions. Exercise 2.1 Find the exact solution of the equation

         36

for a planar a planar source at



 li→  /   ⁄ -

-

emitting neutrons cm 2 s 1 With the boundary conditions: conditions: (i) Dirichlet b.c: finite flux

and (ii) source Neumann condition:

;

). Answer :

   /

  

,

.

Example 2.1 Find the exact solution of the equation:

       for a planar a planar source at

  li→  /   ⁄

emitting neutrons cm-2 s-1 With the boundary conditions: conditions: (i)

and (ii) source condition:

;

       

,

). First obtain the complementary solution using the

homogeneous equation

The roots are

       ,

̂      , giving the complementary solution

    

For the particular solution

    (̂   )(   ̂    )( )   which can be written in partial fractions as

   (̂ )  (̂ )  and expressed as

Now expanding expanding

   ( (̂)  (̂)  (̂) ̂  (̂)  ⋯    , gives

The solution is thus 37

and the coefficients

 

   osh    sinh    can be found from the boundary conditions. Using the boundary conditions gives

||/   sinh[ osh[  ]

2.3 Partial Differential Equations Partial differential equations (PDE’s) represent systems where the differential equation expresses a system with two or more independent variables such as spatial dimensions



, orthogonal and azimuthal angles

  





, energy , and time .

A second-order PDE

                    at a point

 

, is classified as elliptic, parabolic or hyperbolic if

 

is less than, equal to, or greater than zero

respectively. The solution procedure is usually to convert PDE’s to ODE’s and then solved to obtain exact analytical solutions where possible, or a system of algebraic equations equations

 ̿̅ 

where

 ̿

̅

is is a matrix, is is the vector being solved, and



is a known

‘force’ vector. The solution is thus obtained, from standard numerical techniques, such as Gaussian elimination, Gauss Siedel iterative methods, LU decomposition methods etc., by inversion of the matrix

 ̿

giving giving

̅  

.

Consider the first-order partial differential equation

             ∫      

where

(2.5)

. A solution for this can be obtained by multiplying both sides by the integration factor

and integrate over :

38

             

(2.6)



where , a constant of integration, can be obtained from the given boundary condition. Exercise 2.2 Using the classification criterion specified above, we can classify the PDE:

with



; thus

   D          

, and the equation, which will be introduced in the following chapter as the

timetime-dependent neutron diffusion equation, is a parabolic PDE. This implies a certain kind of boundary conditions: a Dirichlet or Neumann b.c. on an open surface, to be specified for a stable solution. Similarly, elliptic PDE’s require Dirichlet or Neumann boundary conditions on a closed surface surrounding the region of interest, while hyperbolic PDE’s require Cauchy boundary conditions on an open surface.

2.4 Integral Equations Integral equations are classified according to the limits on the integral, the occurrence of the unknown function and the homogeneity of the equation. Some equations with their classification are listed in Table 2.1.

39

Table 2.1 Classification of Integral Equations

Classification

Equation

Homogeneous/ Fredholm/Volterra

Type I/II Inhomogeneous

                                               

Fredholm

Homogeneous

I

Fredholm

Homogeneous

II

Fredholm

Inhomogeneous

II

Volterra

Homogeneous Homogeneous

I

Volterra

Homogeneous Homogeneous

II

Volterra

Inhomogeneous Inhomogeneous

II

40

In the above equations,



is called the “kernel of the integral operator” . One way of solving integral equations is by

integral transforms e.g. for e.g. for Fredholm integrals, when

   

, the Fourier transform can be used while for

Volterra equations, the Laplace transform can be used. Exercise 2.3

i-

Solve the Volterra equation



  ∫  

, given the boundary condition

  

by

differentiating differentiating w.r.t .

    ∫    

ii-

Solve the Volterra V olterra equation

iii-

The integral equation for Chandrasekhar’s Chandrasekhar’s

by taking the Laplace transform.

function appears in the exact solution of the neutron transport

equation which will be used in a later chapter. How would you classify this equation?

           

(2.7)

An important integral equation for neutron transport In the previous section, we saw the integral equation of a Volterra form obtained from a partial differential equation

           



When this equation is integrated integrated over , the equation can be written in operator form as

  ̂     

(2.8)

̂          

(2.8a)

where

and 41

30

25

) m c ( 20 e m u l o V15 m u m10 i x a M

3

5

0 10

15

20

25

30

35

40

45

50

Surface Area (cm2) Figure 2-22 Maximum cylinder volume for fixed surface area

Figure 2.23 shows the height-radius permissible contour with volume contours for three cases viz half the maximum volume,

 

, the maximum volume, and twice the maximum volume. We see the volume contours shift towards the

right touching each other tangentially at the optimal value for the height for which the difference in the two values

  

goes to zero at the optimal radius. This represents the geometrical interpretation of the Lagrange multiplier as the

point where both volume volume and surface surface area contours contours are tangential to each other. other.

81

20

) m15 c ( t 10 h g i e 5 H

Area constraint V = 0.5V

o

V=V

o

V = 2V

o

0 0. 3

0. 4

0. 5

0. 6

0. 7

0.8

0. 9

1

1. 1

1. 2

0. 9

1

1. 1

1. 2

Radius (cm) 10

A 10

5

0

H V H 10-5 X 10

-10

0. 3

0. 4

0. 5

0. 6

0. 7

0.8

Radius (cm)

Figure 2-23 Area constraint for A=10 cm^2 and Volume contours

2.14 The Adjoint Operator Mathematical operators have “adjoints” adjoints”. Thus an operator

  ̂       would have an adjoint operator written as

̂       The adjoint of a derivative is its negative. A ‘forward’ equation of equation of the form

̂  82

would have a ‘backward’, or ‘adjoint’ form ‘adjoint’ form expressed as

̂   Both equations would be solved with the appropriate boundary conditions. In nuclear engineering,



is the adjoint neutron flux and is also called the ‘importance’ function. For a critical react or the

diffusion equation reads

̂   where

The adjoint operator

̂

̂     is defined as

 ⁄ ̂ ⁄ ̂ ⁄ ⁄

(2-57)

where u and v are any two functions which vanish at the physical boundaries. The above is also written in the form of an inner product as

( ̂) ̂ 

(2-58)

( ̂)  ̂ 

(2-59)

or equivalently as

For the second-order second-order ‘diffusion’ operator given in Eq,(2-x), Eq,(2 -x), it is readily shown, by carrying out an integration by parts, that the operator is self-adjoint, i.e. i.e.

83

The solutions,

  

̂  ̂

(2-60)

, to both the forward and backward homogeneous equations respectively, are thus

proportional to each other.

References 1. Abramowitz, M., and Irene A. Stegun, I.A., Handbook of Mathematica Mathematicall Functions with Formulas, Formulas, Graphs and and Mathematical Mathematical Tables, Tables, Dover Publications, New York, 1972. 2. Bell, G. I. and Glasstone S., Nuclear S., Nuclear Reactor Reactor Theory Theory,, Robert E. Kreiger Publishing Company, New York, 1979. 3. Brown, F. B., A Review of Monte Carlo Criticality Calculations – Convergence, Bias, Statistics, International Conference on Mathematics, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, New Your, May 3-7, 2009. 4. Densmore, J. D., and Edward W. Larsen, E. W., Variational Variance Reduction for Criticality Calculations using Monte Carlo Adjoint Fluxes, Nuclear Mathematical and Computational Sciences: A Century in Review, A Century Anew, Gatlinburg, Tennessee, April 6-11, 2003. 5. Duderstadt J. J. and Hamilton, L. J., Nuclear Reactor Analysis , Analysis , John Wiley & Sons, 1976.. 6. Henry, A. F., Nuclear F., Nuclear Reactor Reactor Analysis, Analysis, Cambridge, MA, MIT Press, 1975. 7. Kroese, D. P., Cross-Entropy Method, University of Queensland, Australia. [email protected] 8. Lamarsh, J. R., Introdu R., Introduction ction to Nuclear Nuclear Reactor Reactor Theory Theory,, AddisonAddison-Wesley Publishing Company, 1972. 9. Lewis, R. W., Nithiarasu P., and Seetharamu, K. N., Fundamentals N., Fundamentals of the Finite Finite Element Element Method Method for Heat Heat and Fluid Flow, Flow, John Wiley & Sons Inc., 2004. 10. Lux, I., and Koblinger, L., Monte L., Monte Carlo Carlo Particle Transport Methods: Methods: Neutron Neutron and Photon Photon Calculations Calculations,, CRC Press, Inc., 1991. 11. Shi, Bo., “Entropy-based Diagnostics of Criticality Monte Carlo Simulation and Higher Eigenmode Acceleration Methodology” Methodology ”, M.S. Thesis, Georgia Institute of Technology, August 2010.

84

12. Stone, M., and Goldbart, P., Mathematic P., Mathematicss for Physics Physics I , Pimander-Casaubon, 2000-2008. 13. Ueki, T., and Brown, F.B., “Stationarity “ Stationarity and Source Convergence Diagnostics in Monte Carlo Criticality Calculation, Nuclear Mathematical and Computational Sciences: A Century in Review ”, A Century Anew, Gatlinburg, Tennessee, April 6-11, 2003.

Problems

1. Estimate the integral

  ∫   with

2. Use the linear congruential random number generator generate random numbers.

 1      

as the estimator and

as the pdf. with g=20, with g=20, c=1, c=1, S o=1, p=100 to p=100 to

3. In the previous question, increase the value of g to g=40 to g=40 and and comment on the period of the random numbers. What general conclusion can be drawn? 4. Sample random numbers from a linear pdf

  

in the range



.

85

3 The Neutron Diffusion Equation This chapter considers bare nuclear systems with nuclear fuel materials such as U

235

and Pu

239

as well as non-fissile

materials such as aluminum, beryllium, boron, carbon (graphite), and iron used for containment, reactor operations, neutron moderation and reflection as well as for structural applications. Neutron diffusion theory is used here for obtaining the neutron flux and associated reaction rates (absorption, scattering etc) etc) in systems which can be specified using regular geometry (slab, sphere and cylinders). Diffusion theory can also be applied to irregular geometries using numerical methods such as finite-difference and finite-element methods discussed in the previous chapter. With further simplifications simplifications in the physics of neutron transport, the ‘simple’ simple’ neutron diffusion equation has the advantage that it can be readily solved analytically and permits considerable insight into the shape and magnitude of the neutron flux, and subsequent reaction rates. Useful engineering design information can thus be extracted as will be shown in following sections.

3.1 The Conservation (Continuity) Equation The deterministic models that describe neutron behavior in a system are based on the diffusion equation, the integro differential Boltzmann equation and the integral equation. A diffusion equation can be derived from first principles by writing a conservation equation in a volume element. Consider the volume element in Fig.3 -1.

86

Figure 3-1 A volume element

The change in neutron population can be written as

  ∆ ∆   ̅    ̅  ⊥  ΣΔΔ

(3.1)

Using a Taylor series firstfirst -order expansion for the current, we can write

 ̅  ⊥ ≈  ̅  ⊥ ∇∙ ̅∆∆

(3.2)

so that the diffusion equation becomes

Using Fick’s law

 ̅ ∇ ∇

  ∇∙ ̅  Σ 

(3.3)

, the diffusion equation reads

   D∇  Σ 

(3.4)

87

3.2 One group Diffusion Equation The time dependent diffusion equation, for a one -group energy model, can be solved analytically for simple problems in slab, cylindrical and spherical geometry. Although the oneone-group model is too simple for practical application, it gives considerable insight into important neutron characteristics of nuclear systems. For systems which are predominantly ‘one -group’ i.e. i.e. when the neutron spectrum is predominantly predominantly ‘fast’ or ‘thermal’, it is used with the appropriate ‘averaged’ nuclear nuclear data to obtain the neutron flux and associated quantities such as criticality parameters and thermal power. The neutron diffusion equation, Eq. 3.4, is a second -order ODE which can be solved given two boundary conditions: Neumann, Neumann, Dirichlet or mixed, depending on the physical conditions. A bare assembly, assembly, for example, will have the condition of vanishing flux at the extrapolated boundary (Dirichlet boundary condition) while a reflecting surface will have a zero fluxflux-gradient, or zero current, (Neumann boundary condition) condition. An interface will have both flux and current continuity conditions. The solutions, of the steadysteady -state homogeneous neutron diffusion equation, in slab, spherical and cylindrical geometries for some special cases are given in Table 3.1.

88

Table 3.1 Diffusion Theory neutron flux for some simple cases

Geometry

Medium

Source

Boundary conditions

Flux

  /   l→im  /               sih[| h[  ||// cosh[  ] l→im  /        cosh/     Σ        cosh                           Σ cos                cos  cos  cos    ̃ ̃ ̃ ̃        4/    lim→4    Planar source at

Infinite



emitting neutrons cm

Finite flux

-

Source condition

2

s-1

Infinite slab

Planar source at

(in y and z)

emitting neutrons cm

of thickness

2

s-1

Source condition

Infinite slab

Uniformly distributed

of thickness

sources emitting -3

neutrons cm s

Slab

Infinite

critical slab

-

-1

at

SourceSource-free,

is the reactor power,

reactor of

is the energy

thickness

recoverable from

and

fission (200 MeV)

at

SourceSource-free,

Bare cubical

is the reactor power,

reactor of sides

is the energy

recoverable from

at

fission (200 MeV)

Sphere

Infinite

Point isotropic source

Finite flux

at

Source condition

emitting neutrons s-1

89

SourceSource-free,

Bare spherical reactor of radius



 

is the reactor power, is the energy and

recoverable from

of radius



at





emitting neutrons s-1

and

Uniformly distributed



sources emitting -3

neutrons cm s

-1



Infinite

Cylinder

cylinder

 ()    ()    ()  

  4sih    sih  sih   Σ  sih   Σ  4

Finite flux

Finite sphere of radius

   4Σ si/ 

fission (200 MeV) Point isotropic source

Finite sphere

 ()  

Finite flux

Finite flux and

Finite flux Bare critical reactor and

3.3 One-group Diffusion Equation Applications The neutron diffusion equation, based on Poisson’s Poisson ’s equation, is mathematically much simpler than the integro -differential transport equation and hence is used for simple problems. Even then, numerical solutions are required for large problems such as “whole“whole -core” reactor design. In the multi -group form, the diffusion equation is used to obtain ‘firs t estimates’ and can serve as a useful step for providing guesses to full transport calculations. A limitation of diffusion equation is near sources and boundaries where the angular flux requires detailed consideration as provided, for example, in the discrete ordinates and spherical harmonics methods. The problems given here from one-group diffusion theory are useful to develop an understanding of the flux and current defined in the earlier sections of this chapter.

90

2a in the x-direction x-direction and infinite in the y y and z and z directions, with uniformly Example 3.1 In a graphite slab of thickness 2a



distributed sources emitting neutrons cm-3 s-1 the neutron flux is given by

     coshcosh/    Σ

Obtain an expression for the current density in the slab and calculate the escape probability for a slab of thickness 1 mean free path. The current density (Eq. 1.12) is obtained by Fick’s Law:

from which the escape probability

 si  h    ∇ ∇   cosh    si  h            cosh        si  h        cosh   →   →       Σ

is the fraction of neutrons leaking out of the system across the surface

From the ‘right’ surface surface the probability is thus

The total escape probability from the system is

From L’hospital’s rule it is readily seen that in the limit The diffusion length is defined as



.

; due to the symmetry of the system

,

(for

).

and the diffusion coefficient is given by

91

     Σ                oms cm

The first step is to find the atomic density

of graphite:

The next step is to get the macroscopic cross-sections: macroscopic absorption cross-section

macroscopic scattering cross-section

macroscopic transport cross-section

Σ       4 4    cm1 Σ       4    cmcm1 Σ    ̅            cm1

These are used to find the diffusion coefficient and the diffusion length:

diffusion coefficient coefficient

diffusion length

  1  1    cmcm   √   √ 1 4 cm

and




  1    cmcm

The mean free path , though not required in this example, is also determined:

  1  1   11    cmcm     si  h 4  si  h     4     cosh    ⁄ cosh  4

For a graphite slab of thickness 1 mean free path, the escape probability is thus

92

Thus in this case there is a 99.85% probability that a neutron will escape from the surfaces. Figure 3.2 shows the escape probability in the above graphite slab as a function of its thickness. The probability is ‘high’ since graphite is a weak absorber and is thus used for moderating neutrons neutrons in a reactor.

1 0.9 0.8 y t i l i 0.7 b a b o r P0.6 e p a c s 0.5 E

0.4 0.3 0.2 0

10

20

30

40 50 60 Thickness Thickness (mfp)

70

80

90

100

Figure 3-2 Escape probability in a graphite slab

3.3.1.1

Flux in an infinite medium

For a point source in an infinite medium, show that the (thermal) neutron flux is

where

  

, the thermal diffusion length.

      4

Exercise 3.1

Using the data in Table 33 -2 (Lamarsh and Baratta, p.254), and referring to Example 5.6 (Lamarsh and Baratta, p.255) fill in the columns for thermal neutron flux





incident on a person standing between a point source of strength

neutrons/s and the permissible time a radiation worker may remain at this point if the maximum permissible dose 93

(100 mrems/week) is not to be exceeded. Assume that a thermal flux of 260 neutrons cm -2 s -1 gives 1 mrem hour -1. How would the permissible time change if the incident flux consisted of fast (

≫

) neutrons?

Table 3.2 Thermal data (at 20 C) for Diffusion Theory neutron flux: Infinite Medium calculation

   g∙cm cm cm1 cm Density

Moderator

  

*

Graphite

1.00

0.16

0.0197

2.85

1.10

0.87

9.3 X 10-5

97

1.85

0.50

1.04 X 10-3

21

1.60

0.84

2.4 X 10



*

containing 0.25 weight/percent

59



/ ∈              1/                4   

Plot the thermal flux and permissible time as a function of temperature

with

-4

for

and

for

if the variations are as follows:

and zero otherwise.

3.4 The One-group Critical Reactor When the source is given by equation can be written as

where the material buckling

  Σ 

, with the infinite system multiplication factor

∇     

 

, the diffusion

(3.5)

is

94

   

(3.5a)

The flux can be obtained by solving Eq. 3.5 and applying the appropriate boundary conditions. For a bare sphere, the flux is

where the geometric buckling

  ⁄

   si    4Σ

(3.6)

, and the constant is determined determined from the power of the reactor



:

Example 3.2 Consider a spherical “reactor” of radius R=8.7407 cm operating at a power of 100 W. Find the constant A in A in

the expression for thermal flux (Eq.3.6).

Use the data:

Σ    cmcm1        cmcm

. The recoverable energy from fission



is

assumed to be 200 MeV. Then

     1  44    11  ∙  and since

     the flux is

    11 si  ∙  The flux thus has a magnitude determined by the operating power and a shape determined by the ‘buckling’, or the curvature. Figure 3.2 shows the neutron flux in this reactor for two operating powers 100 Watts and 200 Watts; the higher 95

magnitude clearly corresponds to the higher power while the shape is the same. It is important to note that the flux is finite at the surface and vanishes at the extrapolated boundary which in this case is at r = ±9.7619 cm.

10

10

x 10

9 8 )

1 -

s

2 -

Power 200 W

7

m c s 6 n o r t u e 5 n ( x u l 4 f n o r t u 3 e N

Power 100 W

2 1 0 -10

-8

-6

-4

-2

0 Radius (cm)

2

4

6

8

10

Figure 3-3 Neutron flux in a bare 'spherical' reactor

For criticality, the material and geometrical bucklings are equal:

    

, and thus Eq.(1.27) incorporating incorporating the

thermal non-leakage probability, is written as

                

(3.7)

(3.7a)

(3.7b)

This expression, simple as it is, can be used to gain considerable insight into a simplified reactor core ‘design’ which is demonstrated in the following example. 96

Table 9.5. Cell balances using diffusion theory flux

Sources

Sinks



Cell





 

Total In





 

Total Out

1

1

0

1.0000

0.1467

0.8533

1.0000

0

2

0

0.8533

0.8533

0.0590

0.7943

0.8533

0

3

0

0.7943

0.7943

0.0411

0.7532

0.7943

0

4

0

0.7532

0.7532

0.0317

0.7215

0.7532

0

5

0

0.7215

0.7215

0.0256

0.6958

0.7215

0

6

0

0.6958

0.6958

0.0213

0.6745

0.6958

0

7

0

0.6745

0.6745

0.0181

0.6564

0.6745

0

8

0

0.6564

0.6564

0.0155

0.6409

0.6564

0

9

0

0.6409

0.6409

0.0134

0.6276

0.6409

0

10

0

0.6276

0.6276

0.0116

0.6160

0.6276

0

279

10

-1

Bins=10

10

-2

) h | | g ( D

10

10

-3

-4

10

1

10

2

10

3

10

4

10

5

N

Figure 9-23 Kullback Leibler Divergence vs Sample Size N for 1 0 spatial bins

References 1. Bell, G. I. and Glasstone S., Nuclear Reactor Theory, Theory, Robert E. Kreiger Publishing Company, New York, 1979. 2. Clark, M, Jr., and Hansen, K. F., Numerica F., Numericall Methods of Reactor Analysis, Analysis, Academic Press, 1964. 3. Ganapol, B. D. and Parsons, D. K., “A heterogeneous medium analytical benchmark”, Los Alamos National Laboratory, 2008. 4. Ganapol, B. D., “Analytical benchmarks for nuclear engineering applications – Case studies in neutron transport theory”, Nuclear Energy Agency, NEA/DB/DOC (2008)1, OECD 2008. 280

5. Henry, A. F., Nuclear F., Nuclear Reactor Reactor Analysis, Cambridge, Analysis, Cambridge, MA, MIT Press, 1975. 6. Lamarsh, J. R., Introduction Introduction to Nuclear Reactor Theory, Theory , AddisonAddison-Wesley Publishing Company, 1972.

Problems

1.

In section 9.1 the eigenvalues  are defined by the transcendental equation (Eq. 9.3a). Calculate  for

 = 1.0, 1.2, 1.4 and comment on the ei genfunctions. 2.

.2, 0.4, .4, 0.6,0. 0.6,0.8 8, 1.0 and comment on your results, Repeat the analysis for Fig. 9.17 using values of for  = 0.2, especially in the limits of low c and the case c=1.

3.

Repeat the Kullback -Leibler Divergence estimates given in Fig. 9.23 for the number of bins varying from 2 to 20 and 20 and comment on the convergence.

281

10 Monte Carlo Simulation: Practical

Applications 10.1 Introduction This chapter considers full-scale state-of-the-art computation used for the simulation of nuclear systems including essentially nuclear criticality systems and nuclear reactor core analyses. While diffusion theory (Chapter 3) presented a somewhat simplistic modeling of neutron transport it was useful to enable simple solutions which revealed considerable detail of the neutron flux and subsequent reaction rates in idealized geometries and collision models. A better representation was the Boltzmann formulation of the neutron transport equation (Chapter 4) which incorporated a higher level of mathematical complexity but provided far more accurate results than the diffusion model, although in idealized geometry. It was however an elegant formalism which stands to this day in spite of the computational capability that enables full-scale modeling of real engineering systems. It was then seen that numerical methods such as finite-difference and finite-element methods enabled more modeling detail to be considered. That was called the “deterministic” approach in contrast to the “stochastic” Monte Carlo approach which was seen to be far stronger in terms of more realistic modeling of the underlying physics as well as the engineering design of systems. The state-of-the-art in nuclear engineering thus recognizes Monte Carlo to be one of the best available tools for the design and performance analysis of operational nuclear systems. And, in that domain, it is the MCNP code of Los Alamos National Laboratory, which over several decades of continuous improvement, based on very detailed experimentally obtained and validated nuclear cross-section data, is extensively quoted as the most reliable code in neutronics extending into radiation physics, medical

282

imaging, oil-well logging and a number of other derivative areas. Several other codes were discussed in chapter 6. The underlying mathematics of the Monte Carlo methods was reviewed in Chapter 2 and simple applications were demonstrated in chapter 8. A “unified picture” of neutron transport modeling, covering deterministic and stochastic approaches, was given in chapter 9. The analyses were restricted to simplified modeling and geometry to highlight the theoretical foundations. Real-life problems are complex in several ways and require sufficiently realistic modeling to give useful results. This level of modeling is provided by MCNP which will be be used in this chapter chapter to demonstrate demonstrate practical practical applications. applications. The problems considered for demonstration are the Godiva assembly discussed in previous chapters, a Pressurized Water Reactor (PWR) core with a standard 17X17 fuel assembly, and nuclear criticality safety assemblies. All validations of results presented in earlier chapters can easily be carried out as will be shown for the case of simplified one region problem that was used in previous chapters to compare diffusion, transport, and simple Monte Carlo analyses. analyses.

10.2 Simple Assemblies This section re-considers the Godiva assembly (described in detail in earlier chapters), and a bare sphere of uranium oxide. The multiplication (   and flux  are shown in Figs. 10.1 and 10.2 respectively for a Godiva sphere of 3

density 18.74 g/cm and mass 52.44 kg as as given in Section 8.4. The average of the collision, absorption and track-length estimators estimators for 1000 neutrons per cycle and 1000 cycles   is 0.992265 ±0.0006. The flux is predominantly predominantly ‘fast’ with the with the average neutron energy of 1.483 MeV for for neutrons causing fission. Of the total fissions, 5.43% fissions were caused by neutrons in the range 0.625 evev- 100 kev and kev and the rest by neutrons over 100 kev. kev. The average fission neutrons produced per neutron absorbed including capture and fission was 2.3261 while the average number number of neutrons produced per fission was 2.597. 283

Figure 10-1 keff in a Godiva sphere sphere

-3

x 10 x 10

5

-3

4.5

6

4

5

) s - 4 2 m c 3 / n ( x 2 u l F 1

3.5 3 2.5 2 1.5

0 10 5

10 5

0

Y (cm)

0

-5

-5 -10

-10

X (cm)

1 0.5 0

Figure 10-2 Flux in a Godiva sphere 284

For a bare sphere of UO 2 of the PWR fuel type, Fig. 10.3 shows the collision estimate for  in a 70% enriched sphere of density  = 10.   16.3621 cm radius cm radius and total weight of 200kg 200 kg . The neutron flux in the sphere is shown in Figs. 10.4 and 10.5. The central peaking of the flux is evident, as well as the finite flux at the boundaries consistent with the exact solutions discussed earlier.

3

Figure 10-3 UO2 sphere 70% enriched, den=10.9 g/cm , radius 200 kg

285

-3

x 10 -3

x 10

2

2.5 2

1.5

2

m c 1.5 / n x 1 u l F

1

0.5 0 20

0.5

0 Y (cm) -20

-20

-10

0 X (cm)

20

10

0

Figure 10-4 Flux in a bare UO 2 sphere

-3

20 15

x 10

2

10 1.5

5 0 1

-5 -10

0.5

-15 20

10

0

-10

-20 -20

0

Figure 10-5 Flux in a UO 2 sphere (in xy xy plane at z=0) 286

y(jset)=A(2,icount); kset=jset; z(kset)=A(3,icount); % only one value flux(iset,jset)=A(4,icount); error(kset)=A(5,icount); % not used end end for i=1:xsets for i=1:xsets for j=1:ysets for j=1:ysets fprintf (gid,'\n (gid,'\n %6.3f %6.3f %6.3f %12.4e %12.4e',x(i),y(j),z(j),flux(i,j),error(j)); %12.4e' ,x(i),y(j),z(j),flux(i,j),error(j)); end end surf(x,y,flux); fclose(fid); fclose(gid);

361

About the Author

Zafar ullah Koreshi [B.Sc.(Hons), Nuc. Engg., M.S., Nuc.Engg., Ph.D. Nuc. Engg] is Professor and Dean, Faculty of Engineering, Air University, Islamabad. With a cumulative research, industrial and academic experience of about 30 years, Dr Koreshi has remained involved with a diverse engineering profession spanning nuclear engineering, mechanical engineering, and more recently inter -disciplinary ‘mechatronics’ engineering.

His research work has been in Monte Carlo simulation for neutron transport with applications in nuclear systems, in mechanical mechanical engineering areas covering process simulation with applications in solar thermal energy storage for small power plants, and more recently in the optimal control of mobile robots.

Dr Koreshi has had a long association with the International Conference on Nuclear Engineering (ICONE) hosted jointly by the American Society of Mechanical Engineers (ASME), Chinese Nuclear Society (CNS), and the Japanese Nuclear Society (JNS). He has also been co -Chair of Track 14: Reactor Physics, Neutronics and Computations Sessions of ICONE -19 (held in Japan) , ICONE-20 (held in USA), ICONE-21 (Chengdu, China) and is Track Co-Chair for Track 11, ICONE-22 held in July 2014 in Prague, Czech Republic.

Dr Koreshi is life member of the Pakistan Nuclear Society, member of the Pakistan Engineering Council (NUC -05), and member of the American Society of Mechanical Engineers.

362

Nuclear Engineering Modeling and Simulation

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