Pipeline Design for Water Engineers

PIPELINE DESl6N fOR WATER EN6IMEERS THIRD REVISED AND UPDATED EDITION

DEVELOPMENTSIN WATER SCIENCE, 40

OTHER TITLES IN THIS SERIES VOLUMES 7-3 ARE OUT OIF PRINT 4 J.J. FRIED GROUNDWATER POLLUTION 5 N. RAJARATNAM TURBULENT JETS 6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS 7 v. HANK AND J. SVEC GROUNDWATER HYDRAULICS 8 J. BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL AFRICA 9 T.A. McMAHON AND R.G. MElN RESERVOIR CAPACITY AND YIELD 10 G. KOVACS SEEPAGE HYDRAULICS 1i w.n. GRAF AND C.H. MORTIMER (EDITORS) HYDRODYNAMICSOF LAKES: PROCEEDINGS OF A SYMPOSIUM 12-13 OCTOBER 1978. LAUSANNE, SWITZERLAND 12 W. BACK AND D.A. STEPHENSON (EDITORS) CONTEMPORARY HYDROGEOLOGY: THE GEORGE BURKE MAXEY MEMORIAL VOLUME 13 M.A. MARlfJO AND J.N. LUTHIN SEEPAGE AND GROUNDWATER 14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE 15 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series) 16 W. BACK AND R. LETOLLE (EDITORS) SYMPOSlClM ON GEOCHEMISTRY OF GROUNDWATER 17 A.H. EL-SHAARAWI (EDITOR) IN COLLABORATION WITH S.R. ESTERBY TIME SERIES METHODS IN HYDROSCIENCES 18 J.BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL REGIONS 19 D. STEPHENSON PIPEFLOW ANALYSIS 20 I.ZAVOIANU MORPHOMETRY OF DRAINAGE BASINS 21 M.M.A. SHAHIN HYDROLOGY OF THE NILE BASIN 22 H.C.RIGGS STREAMFLOW CHARACTERISTICS 23 M. NEGULESCU MUNICIPAL WASTEWATER TREATMENT 24 L.G.EVERETT GROUNDWATER MONITORING HANDBOOK FOR COAL AND OIL SHALE DEVELOPMENT 25 W. KINZELBACH GROUNDWATER MODELLING: AN INTRODUCTION WITH SAMPLE PROGRAMS IN BASIC 26 D. STEPHENSONAND M.E. MEADOWS KINEMATIC HYDROLOGY AND MODELLING 27 A.M. EL-SHAARAWI AND R.E. KWIATKOWSKI (EDITORS) STATISTICAL ASPECTS OF WATER CIUALITY MONITORING - PROCEEDINGS OF THE WORKSHOP HELD AT THE CANADIAN CENTRE FOR INLAND WATERS, OCTOBER 1985 28 M.JERMAR WATER RESOURCES AND WATER MANAGEMENT 29 G.W. ANNANDALE RESERVOIR SEDIMENTATION 30 D.CLARKE MICROCOMPUTER PROGRAMS IN GROUNDWATER 31 R.H. FRENCH HYDRAULIC PROCESSES IN ALLUVIAL FANS 32 L. VOTRUBA, 2. KOS. K. NACHAZEL, A. PATERA ANDV. ZEMAN ANALYSIS OF WATER RESOURC_ESYSTEMS 33 L. VOTRUBA AND V. BROZA WATER MANAGEMENT IN RESERVOIRS 34 D. STEPHENSON WATER AND WASTEWATER SYSTEMS ANALYSIS 35 M.A. CELlA ET AL. COMPUTATIONAL METHODS IN WATER RESOURCES, VOLUME 1 MODELING SURFACE AND SUB-SURFACE FLOWS. PROCEEDINGS OF THE VII INTERNATIONAL CONFERENCE, MIT. USA, JUNE 1988 36 M.A. CELIA ET AL. COMPUTATIONAL METHODS IN WATER RESOURCES, VOLUME 2 NUMERICAL METHODS FOR TRANSPORT AND HYDROLOGICAL PROCESSES. PROCEEDINGS OF THE V11 INTERNATIONAL CONFERENCE, MIT, USA, JUNE 1988 37 D.CLARKE GROUNDWATER DISCHARGE TESTS: SIMULATION AND ANALYSIS

THIRD REVISED AND UPDATED EDITION

DAVID STEPHENSON Department of Civil Engineering University of the Witwatersrand Johannesburg, South Africa

ELSEVlER Amsterdam - Oxford - N e w York - Tokyo 1989

ELSEYIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 2 1 1, 1000 A€ Amsterdam, The Netherlands Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHINGCOMPANY INC. 655, Avenue of the Americas New York, NY 10017, U.S.A.

ISBN 0-444-87373-2

0 Elsevier Science Publishers B.V., 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./ Physical Sciences & EngineeringDivision, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred t o the publisher. No responsibility is assumed by the Publisher for any injury and/or damage t o persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

V

PREFACE TO FIRST EDITION

Pipelines lengths

are

and

being

working

are

essential

for

years

has

recently

to

constructed

pressures.

achieve economic

resorted

to

been

in

and

in

an

and

safe

semi-empirical

done

ever-increasing

Accurate

rational

designs.

design

bases

Engineers

formulae.

to r a t i o n a l i z e

effort

diameters,

design

have

Much

the

work

design

of

pipelines. This as

book

well

as

retaining

presenting

oook

or

hydraulic

subject

is

to

but

field.

a l so of

be

bring

useful

infancy

and

of

the

graphs

in

planning

of

features

manufacture, practice

design

of

which

large

at

the water

in

the It

are

should

stressed

of

may

suitable the

as

and

were of

In

are

and

the

is

techniques

to

the

Pipelines covered

as

and

many

half, book

t h i s book other

be

noted

that

covered pipes,

and v a r i o u s coatings.

opposed

by

will

fluids some

of

not

desk. to

of

The the the

computers.

design

deal

in

and and

detail

i t replace design

Emphasis

industrial

pub1 ications.

on

is

the

and

domestic

Although

directed

be of use to engineers i n v o l v e d as of

patents.

methods

structural

nor should

mind

hydraulics

with

does

Many

research.

the assistance of

in will

in t h e i r

calculators

The

other

students.

are s t i l l

in

engineer's

in

the book

further

concerned

the

to

techniques

background

of

the c i v i l

introduction

advanced

for

aim

bear

l a y i n g a n d operation, from

Although

the

problems

second

discussed.

many

concrete

leads

many

book

data.

post-graduate

prepared with

this

of

be

for

design methods

instances,

an

most

and

provide

computers

engineer,

piping

described

may

pipelines.

piping

is

proposed

book half

ancillary

of

many

the sound theoretical

book

acceptance

with

techniques in

d a t a on

under-graduate

modern

the

rational

such as mathematical optimization,

solution

first

It

contains

to

new

on

the most modern design

i3ecause of

methods

codes

some

material

approaches

engineer. also

the s u b j e c t s ,

The

pub1 ished

conventional

the

the

col lates

well the

as

solids

designs

These

stiffening

and

include pipes

and

techniques

types and

gases.

of

pre-

branches

VI

The S . I .

system of

metric u n i t s i s preferred i n

imperial u n i t s a re g i ve n a n d equations examples work in

text.

summarized general chapter.

at

these The the

for as

many they

The

end

appendix data.

problems a n d

instances.

of

the reader

often elaborate on

algebraic

references a r r a n g e d

other useful

in many

Most g r a p h s

a r e represented i n uni versa1 dimensionless form.

are given

through the

i n brackets

the book although

symbols

that in

gives

used

chapter

together

the order of further

i s advised

ideas not in

the

references

Worked to

highlighted

each

chapter

are

with

specific

and

subject matter i n the and

standards

and

PREFACE TO SECOND EDITION

The

gratifying

in small ations

response

amendments to

to

the f i r s t

the second

e d i t i o n of

impression,

t h i s book

resulted

a n d some major a l t e r -

i n t h i s new edition.

The

chapters

replaced

by

on

data

transport

more

of

relevant

solids

to

and

water

sewers

Thus

chapter on the effects of a i r i n water pipes i s included, chapter

on

pumping

systems

for

water

have

engineers.

new

as well as a

The

pipelines.

been

a

latter

was

reviewed by B i l l Glass who added many of h i s own ideas. There

are

additions and

u p d a t i n g throughout.

There

is additional

information on p i p e l i n e economics and optimum diameters i n Chapter 1 . A

comparison

Chapter and

2.

The

sewer

drainage

of

currently sections

flow

are

engineer

used

on non-circular

omitted.

and

‘Stormwater

Hydrology

introduction

to

water

friction

as

These

such

are

and

Drainage’

hammer

theory

sections

together.

An

on

structural

largely

covered

enlarged

design

section

on

is

now

pipe and p a r t l y

are

of

in

preceeds

the

the

made

full

in

pipes

interest

to

author’s

the book

1981).

A

basic

design

of

water

(Elsevier,

l i n e s i n Chapter 4 .

hammer protection of pumping a n d g r a v i t y

The

formulae

of

flexible

soil-pipe

pipes

are

interaction

brought

and

limit

states of f l e x i b l e pipes preceeds the design of s t i f f e n e d pipes. Although recognised needs

some

that

of

this

refreshing

the

new

edition

is

now

fairly

basic,

it

is

i s d e s i r a b l e for both the p r a c t i c i n g engineer who

and

the

student

p i p e l i n e design f o r the f i r s t time.

who

comes

across

the

problem

of

VIII

PREFACE

TO THIRD E D I T I O N

Recent research i n c a v i t a t i o n a n d flow control h a s prompted a d d i t i o n a l sections on pipes a n d make up

this.

There

secondary this

a r e also new

stress.

edition.

Some

sections on

Additional sections

supports

references

appearing

and

in

a

to

exposed

new

layout

p r e v i o u s editions,

noteably on p i p e network systems a n a l y s i s a n d o p t i m i z a t i o n have been ommitted

as

p a r a l l e l book

they

were

considered

‘Pipeflow A n a l y s i s ’

more

appropriate

in

b y the same p u b l i s h e r .

the

alJthOr’S

IX

ACKNOWLEDGEMENTS

The b a s i s for course of

Engineers although I the

my

01 i v e r ,

and

in

t h i s book was d e r i v e d from my experience a n d i n the

duties

the Hand Water

these

Engineers.

organizations

Board a n d Stewart, S v i r i d o v

The

may

extensive

therefore

be

knowledge reflected

of

herein

I am solely to blame f o r any inaccuracies o r misconceptions.

am g r a t e f u l twins

with

Consulting

during

to my wife Lesley,

many

a

lost

who,

weekend,

i n a d d i t i o n to looking a f t e r assiduously

typed

the

first

d r a f t of t h i s book.

David Stephenson

X

CONTENTS

CHAPTER 1

ECONOMIC PLANNING

.

.

.

lntroduct ion P i p e l i n e Economics B a s i c s of E c o n o m i c s M e t h o d s of Ana I y s i s Uncertainty in Forecasts Balancing Storage

CHA?TER 2

. . . . . .

.

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.

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.

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HYDRAULICS

The F u n d a m e n t a l E q u a t i o n s of F l u i d F l o w F l o w H e a d Loss R e l a t i o n s h i p s . . Empirical Flow Formulae 2 a t i o n a l Flow Formulae C o m p a r i s o n of F r i c t i o n F o r m u l a e M i n o r Losses . . P r e s s u r e and F l o w C o n t r o l i n P i p e s lntroduct ion T y p e s of V a l v e s I so l a t i n g V a Iv e s Control Valves C a v i t a t i o n in C o n t r o l V a l v e s . . I n t e r a c t i o n b e t w e e n C a v i t a t i o n a n d W a t e r Hammer P r e s s u r e s

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

CHAPTER 3

1 2 9 10 11 14

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. .

16 18 18 19 24 26 28 28 28 29 29 32 34

P I P E L I N E SYSTEM ANALYSIS AND DESIGN

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

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.

Network A n a l y s i s E q u i v a l e n t P i p e s for P i p e s in S e r i e s o r P a r a l l e l Loop Flow Correction Method The Node H e a d C o r r e c t i o n M e t h o d A l t e r n a t i v e M e t h o d s of A n a l y s i s iqetwork A n a l y s i s b y L i n e a r T h e o r y O p t i m i z a t i o n of P i p e l i n e Systems D y n a m i c P r o g r a m m i n g f o r O p t i m i z i n g Compound P i p e s T r a n s p o r t a t i o n P r o g r a m m i n g for L e a s t - c o s t A1 l o c a t i o n o f Resources L i n e a r P r o g r a m m i n g f o r D e s i g n of l e a s t - c o s t Open N e t w o r k s

. . . . . . . . .

37 37 38 40 41 43 44 45

.

.

48

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

52

.

.

.

.

XI

CHAPTER 4

WATER HAMMER AND SURGE

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R i g i d Water Column Surge Theory M e c h a n i c s o f W a t e r Hammer E l a s t i c W a t e r Hammer T h e o r y Method o f A n a l y s i s Effect of F r i c t i o n Protection of Pumping Lines Pump I n e r t i a Pump B y p a s s R e f l u x V a l v e Surge Tanks Discharge Tanks A i r Vessels In-Line Reflux Valves Release V a l v e s Choice o f P r o t e c t i v e D e v i c e

.

. . . . . .

CHAPTER 5

AI3

. . . . . . .

. . . . . . . .

IN P I P E L I N E S

Introduction Problems of A i r Entrainment A i r I n t a k e a t Pump Sumps A i r Absorption a t Free Surfaces H y d r a u l i c Removal of A i r H y d r a u l i c Jumps Free F a l l s A i r Valves Head Losses in P i p e l i n e s W a t e r Hammer

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

CHAPTER 6

58 60 64 64 68 69 73 76 77 79 85 89 91 93

. . . . . . . . . .

97 97 99 101 102 102 104 105 108 109

EXTERNAL LOADS

. . . . . .

. . . .

. . . . .

. . . . . . .

Soil Loads Trench conditions Embankment Conditions Superimposed Loads T r a f f i c Loads Stress Caused b y Point Loads Uniformly Loaded Areas . Effect o f R i g i d Pavements

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

. . . . . . .

.

113 113 116 120 121 121 122 123

XI1 CHAPTER 7

CONCRETE PIPES

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

The Effect of B e d d i n g Prestressed Concrete Pipes Circumferential Prestressing C i r c u m f e r e n t i a l P r e s t r e s s a f t e r Losses Circumferential Stress u n d e r F i e l d Pressure L o n g i t u d i n a l Prestressing L o n g i t u d i n a l Stresses A f t e r L o s s e s P r o p e r t i e s o f Steel and C o n c r e t e

CHAPTER 8

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160 162 165 172 173 173 174 175 176

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. . . . . .

. . . . . .

179 179 179 180 180 180

STEEL AND F L E X I B L E P I P E

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

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

I n t e r n a l Pressures Tension R i n g s to Resist I n t e r n a l Pressures D e f o r m a t i o n of C i r c u l a r P i p e s u n d e r E x t e r n a l L o a d Effect of L a t e r a l Support S t r e s s d u e to C i r c u m f e r e n t i a l B e n d i n g More General Deflection Equations S t i f f e n i n g R i n g s to R e s i s t B u c k l i n g w i t h no side support Tension R i n g s Stiffening Rings

CHAPTER 9

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

10

. .

142 143 146 149 151 152 156 156 156

SECONDARY STRESSES

Stresses a t Branches Crotch Plates Internal Bracing Stresses a t Bends T h e P i p e a s a Beam Longitudinal Bending Pipe Stress a t Saddles . . Ring Girders Temperature Stresses

CHA?TER

127 129 131 131 133 134 136 136

P I PES.

. . . . . . . . .

. . . . . . . . .

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. . . . . . . . .

. . . . . . . . .

F I TT I NGS AND APPURTENANCES

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

?ipe M a t e r i a l s Steel P i p e Cast I r o n P i p e Asbestos Cement P i p e Concrete P i p e Plastic Pipe

. . . . . .

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

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

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

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

L i n e Val ves Sluice Valves B u t t e r f l y Valves Globe Valves Needle a n d Control Valves Spherical Valves Reflux Valves A i r Valves A i r Vent Valves A i r Release Valves Thrust Blocks Forces Induced by Supports . L o n g i t u d i n a l Stress Temperature Stresses Forces at Bends L a t e r a l Movement Forces on Supports Unbalanced Forces Flow Measurement Venturi Meters Nozz I es Orifices Bend Meters Mechanical Meters Electromagnetic I n d u c t i o n Mass a n d Volume Measurement Te I erne t ry

CHAPTER 1 1

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182 182 183 184 184 185 186 186 186 188 190 194 195 195 196 196 197 197 198 198 199 1 99 200 200 201 201 201

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205 206 208 208 210 212 21 6 21 7 218 21a 222 223

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.

.

.

.

.

L A Y I NG AND PROTECTION

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

Selecting a Route L a y i n g a n d Trenching Thrust Sores Pipe Bridges Underwater Pipelines Joints a n d Flanges Coatings Linings Cat hod i c Prot ec t ion Galvanic Corrosion Stray Current E l e c t r o l y s i s Thermal I n s u l a t i o n

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.

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XIV

CHAPTER 12

PUMP I NG I NSTALLAT I ON5

I n f l u e n c e of Pumps in P i p e l i n e Design T y p e s of P u m p s P o s i t i v e Displacement Types C e n t r i f u g a l Pumps T e r m s and D e f i n i t i o n s Head Total Head Net P o s i t i v e S u c t i o n Head S p e c i f i c Speed Impeller Dynamics Pump C h a r a c t e r i s t i c C u r v e s ivto t o r s Pumpstat ions

. . . . . . . . . . .

. .

. . . . . . . . . . .

228 228 228 229 231 23 1 231 232 233 234 236 239 240

GENERAL REFE2ENCES AND STANDARDS

. . . . . .

242

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250

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252 253 254 255 256

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258 260

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BOOKS FOR FURTHER READING

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APPEND I X

. . . . . . . .

Symbols for p i p e f i t t i n g s Properties of p i p e shapes P r o p e r t i e s of w a t e r P r o p e r t i e s of p i p e m a t e r i a l s Conv ers i o n f a c t o r s

AUTHdR SUBJECT

INDEX INDEX

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. . . . .

. . . . .

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1

CHAPTER 1

ECONOMIC PLANNING

I NTRODUCT ION

Pipes

have

been

used

for

many c e n t u r i e s

for

transporting fluids.

The Chinese f i r s t u s e d bamboo p i p e s t h o u s a n d s o f y e a r s ago, p i p e s were

unearthed a t

were

in

England.

that

pressure

used

however, used

extensively

were f i r s t

grade

diameters square

spiral

small

and

3

to

welded

became

available

is

still

and

working

The

Welding

and

for

the

Cast

design

of

pipelines

with

10 Newtons have

Reliable

per been

welded

or

a l s o made

asbestos cement,

this

was pipes

facilitating

techniques

pipelines

iron,

i n c r e a s i n g use o f

P i p e l i n e s a r e now

conditions.

pipes

iron

Steel

circumferentially

p r e - s t r e s s e d concrete, varying

cast

used.

pressure over

manufactured.

suit

of

t h e e n d of t h e l a s t c e n t u r y ,

longitudinally

to

advent

manufactured. and

p i p e s to b e m a n u f a c t u r e d .

claywares,

the

large r o l l i n g m i l l s has enabled

be

enabling

were

Century

c e n t u r i e s wood-stave

with

large bore pipelines.

metres

in r e i n f o r c e d concrete,

and

19th

later

only

pipelines

steels a n d

over

millimetre

perfected

was

introduced towards

c o n s t r u c t i o n of high

It

the

in

In

Pompeii.

and lead

flow

plastics formulae

century,

thereby

a l s o p r o m o t i n g t h e use of p i p e s . Prior fluids

to

this

century

transported

common

means

for

Liquid chemicals

being

pumped

now o v e r world.

by

water

a n d sewage

pipeline.

transporting

and

solids

gases

in slurry

The g l o b a l

kilometres

of

e x p e n d i t u r e on

pipelines

and

oils

form or

t h r o u g h p i p e l i n e s on ever

two m i l l i o n

were p r a c t i c a l l y

Nowadays

over

the o n l y

are long

the

distances.

in containers

i n c r e a s i n g scales.

most

are

also

There a r e

pipelines i n service throughout

the

p i p e l i n e s i n 1974 was p r o b a b l y o v e r

5 5 000 m i l l i o n . There

are

many

advantages

of

pipeline

o t h e r methods s u c h a s r o a d ,

rail,

waterway

(1)

the

most

Pipelines

are

often

( c o n s i d e r i n g e i t h e r c a p i t a l costs,

(2)

Pipelining

costs

are

not

very

transport

compared

with

and air:-

economic

form

of

transport

r u n n i n g costs o r o v e r a l l c o s t s ) . susceptible

to

fluctuations

in

2 prices,

since

the major cost

i s the c a p i t a l o u t l a y and subsequent

o p e r a t i n g costs are r e l a t i v e l y small. Operations

are

not

to

susceptible

attendance i s r e q u i r e d .

labour

disputes

as

little

Many modern systems operate automatical-

lY. Being

hidden

beneath

the

ground

a

pipeline

will

not

mar

the

n a t u r a l environment.

A b u r i e d p i p e l i n e i s reasonably secure against sabotage. A

pipeline

is

independent of

external

influences

such

as

traffic

congestion and the weather. There

is

normally

no

problem of

r e t u r n i n g empty

containers

to

the source.

It

is

relatively

easy

to

increase the c a p a c i t y of

a pipeline by

i n s t a l l i n g a booster pump.

A b u r i e d p i p e l i n e w i l l not d i s t u r b surface t r a f f i c a n d services. ( 1 0 ) Wayleaves

for

pipelines

are

usually

easier

to

obtain

than

for

than

for

roads and r a i l w a y s .

(11)

The

accident

rate

per

- km

ton

is

considerably

lower

other forms of t r a n s p o r t .

(12) A

pipeline

can

cross

rugged

terrain

difficult

for

vehicles

to

cross. There are of course disadvantages associated w i t h p i p e l i n e systems:The

initial

capital

uncertainty

in

expenditure

the

demand

i s often

some

large,

degree

of

involved

in

so i f there i s any

speculation

may

be

necessary. There

is

often

a

high

cost

( e s p e c i a l l y long fuel

lines).

Pipelines

used

(although

cannot

be

there

are

for

more

multi-product

than

filling

a

one m a t e r i a l

pipelines

operating

pipeline

at on

a

time batch

bases).

There solids,

are

operating

problems

associated

with

the

pumping

of

such as blockages on stoppage.

I t i s often d i f f i c u l t

to locate leaks o r blockages.

P I PEL I NE ECONOM I CS

The main cost

of

a p i p e l i n e system i s u s u a l l y that of the p i p e l i n e

3 itself.

The

gravity

pipeline

systems b u t

cost

is

fact

in

practically

as the adverse head

cost

for

increases so the power

the

only

and

pumping s t a t i o n costs increase.

1.1

Table

indicates

some

relative

costs

for

typical

installed

p i p e l i nes. With the

the

time

year,

economic

of

writing

and

particular increase

instability pipeline

relative the

faster

costs

cost

of

than

those

and rates of

costs for

may

different

petro-chemical of

inflation prevailing at

increase

by

materials

materials

concrete

20% o r more per

for

will

such

instance,

vary.

as

PVC

In may

so these f i g u r e s

should be inspected w i t h caution. TABLE 1 . 1

Relative P i p e l i n e Costs Bore mm

Pipe M a t e r i a l

150

6 7

PVC Asbestos cement Reinforced concrete Prestressed concrete M i l d steel High tensile steel Cast i r o n <:,,-s,

1 500

450

-

23 23 23 33 28 25

10 11 25

80

-

90

150

- 180 - 120 -

100

90

75

indicates not r e a d i l y a v a i l a b l e .

1 u n i t = d;/metre

i n 1974 under average conditions

The components m a k i n g up the cost of a p i p e l i n e v a r y widely from situation

to

situation

but

for

water

pipelines

in

Supply of p i p e

- 55%

(may

Excavation

- 20%

open

country

and

t y p i c a l conditions are as follows:reduce

as

new

m a t e r i a l s a r e developed)

may

(depends reduce

excavation

as

on

terrain,

mechanical

techniques

im-

prove)

- 5% (may increase w i t h

L a y i n g and j o i n t i n g

bour costs)

- 5%

F i t t i n g s a n d specials Coating and w r a p p i n g Structures ( v a l v e chambers,

anchors)

- 2% - 2%

la-

4

L

10

a L 0

r

m ._ U

5 Water

- 1%

hammer protection

Land acquisition, protection,

access roads, cathodic

security

structures,

.

- 1%

fences

Engineering and survey costs

- 5%

Admi n i s t r a t i ve costs

- 1%

Interest d u r i n g construction

- 3%

Many factors have to be considered in s i z i n g a p i p e l i n e : pumping

mains the flow

0.7

to

m/s

about

2

1 m/s

for

velocity

depending

m/s,

low pressure

at on

the optimum diameter flow

and

working

of 100 C/s

heads and a flow

for

of water,

and may be even h i g h e r f o r h i g h e r pressures.

factor

and

cost

power

v e l o c i t y o r conversely

000 C / S

1

of

and

structures

for

influence any

from It

is

increasing

about 400 m

pressure heads at

also

the diameter

varies

pressure.

to 2 m/s

a flow

For water

The c a p a c i t y

the

optimum

p a r t i c u l a r flow.

flow

1.1

Fig.

i l l u s t r a t e s the optimum diameter of water mains f o r t y p i c a l conditions. I n planning scale costs.

of

a

p i p e l i n e system i t should be borne i n mind that

operation

By

of

doubling

a

pipeline

the

head remaining constant, hand

the

cost

delivered which to

install

to

1/3

large

considerable

the

doubles

of

multi-product

a

of

pipe,

on

effect

other

factors

the c a p a c i t y increases six-fold.

approximately

decreases

justifies

has

diameter

the

that

original.

lines.

diameter

so

Whether

main

at

It it

the

the is

is

unit

such

as

On the other

cost

this

per

scale

i n fact

outset

the

the

unit effect

economical

depends

on

the

f o l l o w i n g factors as well as scale:Rate of low

growth i n demand ( i t may be uneconomical

capacity

factors

during

initial

years).

to operate a t

(Capacity

factor

is

the r a t i o of a c t u a l average discharge to design c a p a c i t y ) . Operating to

factor

maximum

depend

on

(the ratio

throughput the

rate

i n s t a l l i n g storage a t Reduced

power

costs

of

during

of

average the

draw-off

throughput

same and

at

period), can

be

any

time

which

will

improved

by

the consumer's end. due

to

low

friction

losses

p i p e l i n e i s -not o p e r a t i n g a t f u l l c a p a c i t y . C e r t a i n t y of f u t u r e demands. V a r y i n g costs w i t h time ( b o t h c a p i t a l

and o p e r a t i n g ) .

while

the

(6)

Rates of

(7)

Physical d i ffi cu l ti e s

interest and c a p i t a l in

availability.

the construction of

a

second

pipeline i f

required. The

optimum

factors,

not

rate

cost

of

design

p e r i o d of a p i p e l i n e depends

least b e i n g the r a t e of inflation,

c e r t a i n t y of f u t u r e demands In

waterworks

pipelines

for

throughput size

10

of

demands

it

up

has

10

to

and h i g h growth rates, so

the p i p e l i n e ,

years.

Longer

to

the r a t e of

that

planning

been

30

to

are

economic

to

scale and

economic

years

hence.

to

For

c a p a b i l i t i e s may

supplementation may stages

of

loans and the

growth,

found

technical

number

1973).

(Osborne a n d James,

practice

a

interest on c a p i t a l

addition

in

on

normally

size large

l i m i t the

be r e q u i r e d w i t h i n justified

for

small

bores and low pressures. It

may

Where

not

always

pressures

are

be

high

it

is

lay

economic

a

to

uniform

bore

pipeline.

reduce the diameter

and

consequent1 y the wal I thickness. In there

planning

a

may

a

be

Alternative economic.

trunk

main

number

layouts

with

of

should

progressive

possible

be

decrease

combinations

compared

before

i n diameter

of

deciding

diameters. on

the

most

Systems a n a l y s i s techniques such as l i n e a r programming and

dynamic programming are i d e a l l y suited f o r such studies. Booster pumping high

the

stations

along

pumps a t input

may

the

entire

the design

installed

end,

though

line.

stage

along

lines

instead

of

additional

By

providing

for

intermediate

instead of pumping to a h i g h head

the pressure heads

thicknesses may b e minimized. even

be

high pressure head a t the i n p u t end and m a i n t a i n i n g a

a

pressure

booster at

pump

to

and consequently

the p i p e w a l l

There may be a s a v i n g i n o v e r a l l cost,

pumping

stations

are

The

required.

booster

stations may not be r e q u i r e d f o r some time. The c a p a c i t y booster this

is

increase

pumps not

of

the p i p e l i n e may often be

a

at

later

always

stage

economic.

approximately

with

although The

the

power losses increase considerably The diameter be

selected

pipeline

cost

by

of an

a

square

increases

with

of

should losses the

be

r e a l i s e d that

along flow,

a

pipeline

consequently

f o r h i g h e r flows.

pumping main economic

it

friction

increased b y i n s t a l l i n g

to convey a known discharge can

comparison increasing

of

alternative

diameter,

whereas

sizes.

The

power cost

7

TOTAL COST

5

DIAMETER

1.2

Fig.

O p t i m i z a t i o n of d i a m e t e r of a p u m p i n g p i p e l i n e .

C

COST I N CENTS

P E R m3

Cl c2

C

4 =

01

Fig.

7 -3

Optirriization of

02

Q3

throughput

0,

+

0;

Discharge

0

for certain diaweters

8 overcoming

in

power Thus

costs by

costs,

friction

increase

adding

one

least-cost chapter.

steeply

together

obtains system

There

reduces

a

can

will

as

the p i p e l i n e

pipeline and

curve be

On

correspondingly.

such

selected.

Figure

An

b e a h i g h e r cost

v a l u e of

1.2,

example

other

hand

in d i a m e t e r .

i s reduced

the present

as

the

operating

from

is given

which later

the

in the

the g r e a t e r the design d i s c h a r g e

rate. If

at

some

capacity

of

diagram

such

a

stage

later

pumping as

different

possible

kilolitre

or

it

is

system,

Figure

it

in

1.2

is plotted

to

increase

i s convenient the

throughputs,

similar,

desired

form

the cost,

to

of

the

throughput

replot d a t a from a

Figure

1.3.

Thus

for

now e x p r e s s e d i n c e n t s p e r

a s the ordinate w i t h a l t e r n a t i v e

(real)

pipe diameter a parameter. It any

can

be

demonstrated

pipeline

annual

is

basis)

a

minimum

is

twice

the

that

the

when

cost

the

annual

per

unit

of

p i p e l i n e cost

cost

the

of

throughput

for

(expressed on

power

an

in o v e r c o m i n g

friction. in cents p e r c u b i c metre of water

T h u s t h e cost + C2(d)

C,(P) c =

is

61 wHQ + C 2 ( d )

C, -

(1.1)

Q

= c,w

+

2gdA

Hs)

+

CZ(d)/Q

- C2(d)/Q2

2C1 wHf/Q minimum C i.e. P of

C Z ( d ) = 2 C,(P) i s power

water,

H

friction,

Q

diameter

d,

(1.2)

requirement,

is is

the

total

pumping

C,(P)

is

proportional

to wHQ.

head,

subscript

rate,

C2(d)

t h e cost

of

is

power

s

w

refers

the

cost

i s the u n i t weight to

of

static a

and f

to

pipeline

of

( a l l costs converted

to a u n i t

time b a s e ) . (In a given

similar

diameter

maximum

if

a v a i table).

the

manner penstock friction

it

c a n b e shown supplying

head

loss

a is

that

the

power o u t p u t of a

hydroelectric one

third

of

station the

total

is

a

head

9 R e t u r n i n g to F i g u r e 1.3,

(1)

At

(2)

At

any

particular

the f o l l o w i n g w i l l

throughput

Q,

there

t h i s d i a m e t e r t h e c o s t p e r ton of

at

if

not

throughput

was

t h r o u g h p u t Q2.

some the

same

as

is a

c e r t a i n diameter

b e a m i n i m u m ( i n t h i s case D

w h i c h o v e r a l l costs w i l l

further

be observed:

2

1.

throughput could be reduced

increased.

Costs

would

be a minimum

Thus a p i p e l i n e ' s optimum throughput

the

throughput

at

for

which

it

is

the

is

optimum

diameter.

If Q

(3)

Q4

were

1

increased

+

Q1

=

pipeline

Q3

(with

so t h a t

total

throughput

a

not

to

install

D3 )

but

to

increase

optimum

the p i p e w i t h

through

a m o u n t Q3

an

by

i t m a y b e economic diameter

D2,

diameter

Q4C4

i.e.

second

the

flow

i s l e s s t h a n QIC1

+ Q3C3. (4)

a

At

later

pipeline

stage

the

when

justified

is

it

throughput

through

the

to

construct

overloaded

second

a

line

could

be

reduced. The

power

increasing most

cost

pipe

per

unit

diameter

economic

to

of the

so

increase

increases w i t h size (White,

additional

throughput

corresponding

throughput

decreases

likelihood

through

an

of

it

with being

existing

line

1969).

B A S I C S OF ECONOMICS

Economics designs. some

is

used a s a b a s t s f o r c o m p a r i n g a l t e r n a t i v e schemes o r

Different

rational

comparison interest

schemes m a y

form of

is

the

rate

discount

on

loans

r e q u i r e a discount reflect

a

b e more

time

The cash compared basis.

For

( i t s present

rate

which

redemption from

preference,

may

be

funds.

the

in

National

form

of

projects

the may

the p r e v a i l i n g interest r a t e ,

whereas

private

in the a c t u a l c a s h flows,

organizations

a n d consequently

to

will

use the

interest rate. i.e.

those

Thus a l l

instance

of

flows,

with

or

cash flows necessitating

The c r u x o f a l l methods o f economic

r a t e different

rate

interested

real borrowing

have different

comparison.

one

of

cash

flows

pound

value)

payments another

this

may

received year

if

and

by

returns,

bringing

be discounted next it

year

of

them

is

could earn

one scheme may b e to

a

common

to t h e i r p r e s e n t the

same

5% i n t e r e s t

as

if

time

value.

fl/l.05 invested

10 this a

year.

It

definite

annual

to a71

r i s the

t o meet

paying

into a

from a

expenditure

interest

repayments

rate.

Provision

loan over

i s made f o r

s i n k i n g fund w h i c h a l s o c o l l e c t s

at

the

e n d of

each

year

required

in n y e a r s i s

i n t e r e s t r a t e on t h e p a y m e n t s i n t o t h e s i n k i n g f u n d . i s R,

t h e i n t e r e s t r a t e o n t h e toan

R(l+r)"+

capital

a certain

load by

The

to amount

where

usual

period at

r e p a y i n g the interest.

is

If

then the t o t a l a n n u a l payment i s

r-R (1.4)

( 1 + r )n-t Normally

the

interest

rate

on

the

loan

i s equal

to

the

interest rate

e a r n e d b y t h e s i n k i n g fund so t h e a n n u a l p a y m e n t on a l o a n of

El

is

(1.5)

Conversely

the

present

year over n years

v a l u e of

a

p a y m e n t of

S1 a t

t h e e n d o f each

IS

The p r e s e n t v a l u e of a s i n g l e amount of f l i n n y e a r s i s

Interest on

loans,

various

tables

and

are

available

the present

interest

rates and

for

determining

v a l u e s of

annual

redemption

the a n n u a l

payments o r

periods

(lnstn.

of

payments

returns, Civil

for

Engs.,

1962)

Methods of A n a l y s i s

Different engineering may

b e compared

and

incomes

value

for

schemes r e q u i r e d

economically

associated

comparison,

with

the

a

in a

to meet

n u m b e r of

t h e same o b j e c t i v e s

If a l l

ways.

scheme a r e d i s c o u n t e d

analysis

is

termed

a

to

payments

t h e i r present

present

value

or

11 discounted

cash

flow

incomes of d i f f e r e n t return

method.

o r g a n i s a t ions

analysis.

On

The

where

latter tax

is

Present

most

returns

such cases i t i s suggested that i s obtained.

the

other

schemes a r e compared,

value

and

hand

this

frequently profits

if

annual

i s termed used

feature

net

the r a t e of by

private

priminently.

In

the assistance of q u a l i f i e d accountants

comparisons

are

most

common

for

public

u t i I i t ies.

A

form

benefit/cost of

a

water

of

economic

analysis.

scheme,

for

supplies,

instance

a

popular

certain

although t h i s

domestic supplies. a r e attached

analysis

An economic benefit

economic

is difficult

Those schemes w i t h

highest

priority.

the

in

States

to a l l

value

is

attached

in

the

schemes

mutually

to

case of

benefit/cost

are

is

products

to e v a l u a t e

the highest

Where

United

i s attached

values

exclusive

such as i s u s u a l l y the case w i t h p u b l i c u n t i l i t i e s the scheme w i t h largest

present

requirements of

value a

of

net

town f o r

benefit

is

adopted.

instance were f i x e d ,

It the

the

total

least-cost

the

water supply

scheme would be selected f o r construction.

Uncertainty i n Forecasts Forecasts of

demands,

i n v a r i a b l y clouded w i t h

whether

they

uncertainty

be f o r water,

and

risk.

a n a l y s i s i s r e q u i r e d f o r each possible scheme, a n y p a r t i c u l a r scheme w i l l by

their

probability

for

be the sum of a

number

uncertain

forecasts,

but

his

analysis

a

are

probability

the net

the net

b y 5 to

(1971) recommends under-designing for

i.e.

possible

of

o i l o r gas,

Strictly

benefit of

benefits mul t i pl i ed demands.

Berthouex

10% f o r p i p e l i n e s to a l l o w does

not

account

for

cost

inflation. An a l t e r n a t i v e method of a l l o w i n g f o r or

interest

c a p i t a l cost

discount

scheme,

rate:

increasing

uncertainty

the

rate

i s to a d j u s t

will

which would be p r e f e r a b l e i f

the

favour future

the

a

low

demand

were uncertain.

Example

A consumer r e q u i r e s 300 P/s of increase

his

economic

l i f e of h i s f a c t o r y ) .

consumption

to

600

water P/s

for

for

a

5

years further

then

25

plans

years

to

(the

12 He

draws

75% of

for

the

time

every

day.

Determine

the

economic diameter and the number of p i p e l i n e s r e q u i r e d . i s s u p p l i e d b y a p u b l i c body p a y i n g no t a x .

Power

0.5

for

p/kWhr,

which

includes

an

allowance

on a s i n k i n g f u n d i s 10% p e r annum, of

The water

costs

a

flat

operating

and

The interest r a t e on loans ( t a k e n over 20 y e a r s ) a n d

maintenance.

cost

most

pipelines,

pumps

and

a n d the

power

6%

is

to 5300 p e r

pumpstations costs amount

rate of

inflation in

p.a.

incremental

Pump

kW,

and

(including

a n allowance f o r standby p l a n t ) a n d pump e f f i c i e n c y i s 70%. The effective discount r a t e may be taken the r a t e of i n f l a t i o n , x

+

1.10/1.06

The s u p p l y

O/s,

one

be made t h r o u g h one

P/s,

installed

Similarly

d e l i v e r i n g 300 P / s . for

two

or

five

a l t e r n a t i v e diameters pipeline.

since E l

interest

rate

less

f l

t h i s year i s worth

51 x 1.04 next year.

could

h a n d l i n g 600

4% p.a.,

i.e.

as the

years

i s made

large

pipeline

pipelines

after

the

each

was

capable

comparison

table for

made f o r

of

300

delivering

A

other.

i n the f o l l o w i n g

an a n a l y s i s

two

a

of

single

pipelines

each

i n d i c a t e d an optimum diameter of 600 mm

This

each p i p e l i n e a n d a

5 7 500 p e r 100 m.

smaller

total

present

Thus one p i p e l i n e ,

value for

both p i p e l i n e s of

800 mm diameter,

w i l l be the

most economic solution.

Note

that

pipeline,

the

although

analysis it

was

is

independent

assumed

that

of

the

pressure

length

was

continuous low-pressure p i p e was a l l t h a t was r e q u i r e d . protection

costs a r e assumed

a n a l y s i s i s also

incorporated

independent o f

the p i p e

in

the c a p i t a l

such Water

cost

loan period,

the

of that

a

hammer

here.

The

a l t h o u g h the

r e s u l t s would be s e n s i t i v e to change i n the interest o r i n f l a t i o n rates. Discount

5,

25

factors

and

30

were obtained years

periods.

from

present

v a l u e tables

Uncertainty

was

not

4% over

for

allowed

for

but

would f a v o u r the two smaller p i p e l i n e s . Another

i n t e r e s t i n g p o i n t emerged from

p i p e l i n e was demand increase,

installed

increasing it

would

from

initially,

300

to

due

600

be more economic

to

a

P/s, to

the a n a l y s i s : high then

boost

the

I f a 600 mm

uncertainty if

the

of

the

demand

did

pumping

head

and

13 Sol u t i o n : 1.

I n s i d e D i a .mm

2.

FIOW

3.

e/s

5.

7.

e.

loss kw/100 m = (3.)xQ/70

0.14

0.55

0.06

0.24

0.03

0.12

0.02

0.07

0.60

4.,72

0.26

2.06

0.13

1.03

0.09

0.60

3900

31000

1700

13500

Annual cost

850

6600

590

3900

pumping

silo0 m

20

155

Equiv.Capita1 c o s t of p u m p i n g over 5 years = (6Jx4.452

90

--

Equiv .capital c o s t of p u m p i n g o v e r 25 y e a r s .

( 6 . )x15.622

-

24-30

e

67

40

-

-

1060

4

33

20

-

520

-

3

20

10

-

-

310

P r e s e n t v a l u e of p u m p i n g cost

2 070

= (8.)/1.170

10.

600

600

Energy requirem e n t s kW h r / y r / 100 m

=

9.

900 300

300

Power

= (4.1~8760 x 0.75

6.

800 300 600

600

H e a d loss

m/100 m 4.

700

600_ _ ~ _ _

N O

900

260

440

Cost of p u m p s etc.

f/100 11.

12.

13.

m : = ( 4 . ) x300

Present v a l u e of pump cost = (10.,)/1.170 for second s t a g e P i p e l i n e cost f/100 m TOTAL

180

1410

80

620

40

310

30

180

180

1200

80

530

40

260

30

150

3600

4200

7140

5750

4800

5400

5560 ::

5850

COST

f / 1 0 0 m f o r 300 G 600 e / s 7. '9. + 1 1 . t i 2

(least cost)

14

_ - -

-

Rate

m’ls

Reservoir empty

cumulative demand =/Qdt

Reservoir f u l l = c a p a c i t y required

24h I

1.4

Fig.

pump

Graphical c a l c u l a t i o n of r e s e r v o i r capacity

total

flow

rather

than

the

pipeline indicated

by

600 mm

one

a

through provide

comparison

the a

of

the

(57 140/100 m )

line

one

second present with

through two 600 mm lines ( 5 7 500/100

600

existing

600

mm

value

the

diameter

mm

pipeline. of

present

This

is

pumping through value

of

pumping

m).

BALANCING STORAGE

An

aspect

system

is

reservoir

industrial time mean may

of

water

day.

annual be

provide

which

six

deserves

close a t t e n t i o n

storage.

Demands such as those f o r

fluctuate

Peak-day

the season,

the day

demands a r e sometimes

demand whereas

peak draw-off

times

for

pipeline

the

mean

capacity

b a l a n c i n g r e s e r v o i r s are the head of

with

in p l a n n i n g the p i p e l i n e

to

a

day.

meet

the

of

domestic the

i n excess of

week

and and

twice

the

from r e t i c u l a t i o n systems

It

would

peak

be

uneconomic

draw-off

rates,

to

and

n o r m a l l y constructed at the consumer end ( a t

the r e t i c u l a t i o n

system)

to meet

these peaks.

The storage

c a p a c i t y r e q u i r e d v a r i e s i n v e r s e l y w i t h the p i p e l i n e c a p a c i t y . The b a l a n c i n g storage and

pipeline

capacity

requirement

may

for

be determined

any

known draw-off

w i t h a mass flow

pattern diagram:

15 Plot

cumulative

curve

plot

pipeline. period.

a

line

Move Then

draw-off with

this

the

over slope

line

period

a

equal

down

till

maximum o r d i n a t e

just

touches

between

the

the b a l a n c i n g storage r e q u i r e d (see F i g . An

economic

storage

comparison

capacity

for

is

time,

and a b o v e t h i s

the discharge capacity

to

it

versus

of

the

the mass draw-off

two

lines represents

1.4).

necessary

to

determine

a n y p a r t i c u l a r system (Abramov,

the

optimum

1969) b y adding

t h e cost o f r e s e r v o i r s and p i p e l i n e s and c a p i t a l i z i n g running c o s t s f o r different cost

is

c o m b i n a t i o n s and c o m p a r i n g them, selected.

It

is found

that

the system w i t h

t h e most economic

least total

storage capacity

v a r i e s f r o m one d a y ' s s u p p l y b a s e d o n t h e mean a n n u a l r a t e f o r s h o r t pipelines

to

Slightly than

more

450

reserve

mm

two

supply

day's

storage

may

diameter).

storage

should

be

long

for

b e economic

In a d d i t i o n

a

provided;

up

for

pipelines

(over

60

km).

small-bore

pipelines

(less

certain to

amount

of

emergency

12 h o u r s d e p e n d i n g u p o n

the a v a i l a b i l i t y of maintenance f a c i l i t i e s .

REFERENCES A b r a m o v , N., 1969. M e t h o d s o f r e d u c i n g p o w e r c o n s u m p t i o n in p u m p i n g w a t e r . I n t . W a t e r S u p p l y Assn. C o n g r e s s , V i e n n a . Berthouex, P.M., 1971. Accommodating u n c e r t a i n forecasts. J. Am. W a t e r Works Assn., 66 ( 1 ) 14. I n s t n . o f C i v i l Engs., 1962. A n I n t r o d u c t i o n t o E n g i n e e r i n g Economics, London. and James, L.D., 1973. M a r g i n a l economics a p p l i e d t o Osborne, J.M. p i p e l i n e d e s i g n . P r o c . Am. SOC. C i v i l E n g s . , 99 ( T E 3 ) 637. W h i t e , J.E., 1969. Economics o f l a r g e d i a m e t e r l i q u i d p i p e l i n e s . P i p e l i n e News, N.J.

L I S T OF SYMBOLS

C

-

cost p e r u n i t o f t h r o u g h p u t

D

-

diameter

n

-

number of y e a r s

Q

-

throughput

r

-

i n t e r e s t r a t e o n s i n k i n g fund

R

-

i n t e r e s t r a t e on

loan

CHAPTER 2

HYDRAULICS

THE FUNDAMENTAL EQUATIONS OF FLUID FLOW

The

three

continuity For

most

steady, is

simply

the

flow

rate

is

with

meant

time.

stream

tube

The

that

there

there

states

that

one-dimensional

on

the

is

variation

no

lateral

stems in

forces

on

from

rate at

the

across

the

any

is

an

element in

energy of

energy.

the equation.

tion

and

for

this.

to

'steady point

along

a

boundaries

of

Newton's

fluid

basic

law

between

causing

of

two

the

motion sections

change.

For

flow t h i s i s

p i s the f l u i d

i s velocity

basic

equa-

section By

at

flow

the

i s irrotational.

momentum f l u x the

any

tube.

velocity

that

flow

the flow

the c o n t i n u i t y

stream in

implies

are

equation.

(2.1

i s the force,

The

from

no

flow

the

mechanics

t h e energy

pQAVx

flow r a t e , V

change

of

fluid

the flow

along

flow

change

steady,

F

sum

the

the

where

is

equation

equals

=

equating

section

I t also implies that

momentum

AFx

by

another

and

in

one-dimensional

'One-dimensional'

stream tubes.

and

obtained at

equations

the momentum equation a n d

incompressible,

tion

flow'

important

equation,

mass density,

Q

a n d s u b s c r i p t x r e f e r s to the

equation

fluid

by

is

i s the volumetric 'x'

direction.

d e r i v e d b y e q u a t i n g the work

gravitational

Mechanical

and

a n d heat energy

pressure transfer

In most systems there i s energy

1

forces

to

done the

a r e excluded

loss due to f r i c -

turbulence and a t e r m i s included i n the equation to account The

resulting

equation

for

steady

flow

of

incompressible

f l u i d s i s termed the B e r n o u l l i equation and i s conveniently w r i t t e n a s :

where V

=

mean velocity

V2/2g

=

velocity

at a section

9

=

g r a v i t a t iona I acce I e r a t ion

P

=

pressure

head ( u n i t s of

length)

17 p/y

=

Y

= u n i t weight of f l u i d

pressure head ( u n i t s of

Z

= e l e v a t i o n above an a r b i t r a r y datum

he

=

head

loss

due

to

length)

friction

or

turbulence

between

pressure

head p l u s e l e v a t i o n

sections 1 G 2 The

sum

the

of

velocity

head

plus

is

termed the total head. Strictly

the v e l o c i t y head should be m u l t i p l i e d b y a coefficient to

account f o r the v a r i a t i o n in v e l o c i t y across the section of the conduit. The average v a l u e of the coefficient f o r t u r b u l e n t flow laminar

flow

uniform

or

variation

it

2.0.

is

non-uniform in

the

Flow

through

depending

on

cross-sectional

a

conduit

whether

velocity

i s 1.06 and f o r termed

is

or

not

either

there

distribution

a

is

along

the

conduit. For i.e.

the

there

The flow should

Bernoulli should

be

equation

to

no change

apply in

i s assumed to be one-dimensional

be

incompressible,

although

the

gases w i t h reservations (Albertson et a l . , The tical

the

velocity

respective

heads

are

illustrated

flow at

should

any

be

and i r r o t a t i o n a l .

equation

steady,

p o i n t w i t h time.

may

be

The f l u i d applied

1960). i n Fig.

2.1.

For most p r a c -

cases the v e l o c i t y head i s small compared w i t h the other heads,

and i t may be neglected.

ENERGY LINE ENTRANCE LOSS FRICTION L O S S COtiTRACTION L O S S

FRICTION LOSS

I

PRESSURE H E A D P/ x

E L E \ A T ION

Fig.

to

2.1

Energy heads along a p i p e l i n e

18 F L O W HEAD LOSS RELAT I ONSH I PS

Empirical Flow Formulae The throughput

o r c a p a c i t y of

on the t o t a l head difference

a p i p e of

f i x e d dimensions depends

between the ends.

T h i s head

i s consumed

b y f r i c t i o n and other ( m i n o r ) losses. The

first

friction

f i e l d observations. waterworks developed.

head

These e m p i r i c a l

practice The

conventional

loss/flow

although

head

relationships

rational

formulae

,+-

formulae a n d a r e u s u a l l y

derived

stil I

relationships are

more

loss/flow

were

formulae

established

thus

i n an exponential

from

popular have are form

in

been termed of

the

type

v

= K

where V the

sY

R~

m

radius

perimeter,

q u a r t e r of per

= K*Q"/D~

i s the mean v e l o c i t y of flow,

hydraulic

wetted

s

or

and

of

K' a r e coefficients, R i s

area

for

pipe flowing

a

circular

the diameter) a n d S

length

K and

(cross-sectional

pipe).

Some of

of

flow

divided full,

i s the head g r a d i e n t

by

the

equals

one

( i n m head loss

the equations more f r e q u e n t l y a p p l i e d

a r e l i s t e d below:

SI units

Basic Equation Hazen -Wi I I iams Manning Chezy Darcy

S=K1 ( V / C w )

1.85

/D

1.33

1

units

K =3.03

(2.3)

K2=2.86

(2.4)

K =4.00

(2.5)

1

S=K2(nV)2/D

K2= 6.32

s=K~(v/c~)~/D

K 3=13.13

S= X V /~2 g ~

f.p.s.

= 6.84

1.167

3

DimensionI ess

(2.6)

Except f o r the Darcy formula the above equations a r e not u n i v e r s a l a n d the form of the equation depends on the u n i t s .

I t should be borne

i n m i n d t h a t the formulae were d e r i v e d f o r normal waterworks p r a c t i c e a n d take no account liquid. friction of p i p e .

of

variations

They a r e f o r t u r b u l e n t flow coefficients

vary

in gravity,

temperature o r type of

i n pipes over 50 mm diameter.

w i t h p i p e diameter,

type of

The

f i n i s h a n d age

19 The conventional do

not

involve

formulae a r e comparatively

fluid

viscosity.

They

simple to use as they

may be solved d i r e c t l y as they

do not r e q u i r e an i n i t i a l estimate of Reynolds number to determine the friction solved

factor

diameter rule,

or

for

friction

calculator,

paper. pipe

(see next

directly

The

The

Solution

head g r a d i e n t

computer,

equations

networks

section).

flow.

of

the

the

equations

formulae

i s simple w i t h

nomograph

are

where

rational

of

or

particular

flow/head

graphs use

for

loss

cannot

for

be

velocity,

the a i d of

a slide

p l o t t e d on

log-log

analysing

equations

flows

in

have

to

be

practice

is

the

i t e r a t i v e l y solved many times. The

most

popular

Hazen-Williams

flow

formula.

are tabulated

i n Table 2.1. the

waterworks

aid

in

of

organizations

for

I f the formula chart

a use

is

graphs

of

C decreases w i t h age,

type of

use in t h i s equation

i s to be used f r e q u e n t l y ,

the

a g a i n s t flow f o r v a r i o u s p i p e diameters, v a l u e of

waterworks

F r i c t i o n coefficients

solution

with

formula

most

efficient

loss

head

way.

gradient

a n d v a r i o u s C values.

Many plotted As the

p i p e a n d p r o p e r t i e s of water,

f i e l d tests a r e d e s i r a b l e f o r an accurate assessment of C.

TABLE 2.1

c

Hazen-Williams f r i c t i o n coefficients

New

Cond i t ion 25 years 50 years old old

150 150

140 130

140 120

130 100

150 130

130 110

100

60

90

50

120

100

80

Type of Pipe

PVC: Smooth concrete, AC: Steel, bitumen I ined, ga I vanized : Cast i r o n : Riveted steel, v i t r i f i e d woodstave:

For diameters less than 1 000 mm,

subtract 0.1

Badly corroded

45 ( 1 - Dmm

) C

1 000

Rational F l o w Formulae Although use

for

acceptance fic

the

many

conventional

years,

more

flow

amongst engineers.

b a s i s backed b y

formulae

rational

are

formulae

likely

are

to

remain

gradually

in

gaining

The new formulae have a sound scienti-

numerous measurements a n d they

are uni v ers al l y

20 applicable.

Any

consistent

units

of

measurements

may

be

used

and

I i q u i d s of v a r i o u s viscosities a n d temperatures conform to the proposed formulae. The for inal of

rational

flow

past

research

d a t a on

flow

formulae

bodies o r over was on

for

flat

smal I-bore

roughness f o r

flow

i n pipes a r e s i m i l a r

plates

to those

( S c h l i c h t i n g 1960). The o r i g -

pipes w i t h a r t i f i c i a l

roughness. Lack

l a r g e pipes has been one deterrent to use of

the r e l a t i o n s h i p s i n waterworks practice. The v e l o c i t y maximum

in

a boundary a

shear

in a full

the centre. layer

force

resistance

to

viscosity,

and

p i p e v a r i e s from zero on the boundary to a

Shear forces on the w a l l s oppose the flow and

i s established

onto

an

relative in

inner

w i t h each a n n u l u s of f l u i d i m p a r t i n g

neighbouring

motion

turbulent

of

flow

it

concentric

is

the

fluid

is

imparted

annulus.

termed

by

The

kinematic

turbulent

mixing

w i t h t r a n s f e r of p a r t i c l e s of d i f f e r e n t momentum between one l a y e r and the next. A this point

boundary

layer

layer gradually the f l o w

i s established expands

becomes uniform.

established flow

i s g i v e n by

at

until

it

The

the entrance to

a conduit and

reaches the centre.

length of

Beyond t h i s

pipe required for f u l l y

S c h l i c h t i n g , (1966).

X

D

= 0.7

The

Re'"for Reynolds

Re

number

i n c o r p o r a t i n g the f l u i d viscosity flow

formulae.

and becomes t u r b u i e n t

for

equation

shear

head

force

loss

over

a

=

f

nD2/4

L vz XE 29

v

is

of

is

a

dimensionless

which i s absent

higher Re

length

m u l t i p l i e d by the a r e a :

TnDL = v h

VD/V

=

Flow i n a p i p e i s l a m i n a r f o r

The

basic

(2.7)

t u r b u l e n t flow.

pipe

i n t h e conventional

low R e !less

( n o r m a l l y the case

derived equal

number

by

setting

to

the

loss

than

2 000)

i n practice!. the in

boundary pressure

Fig.

2.2

Moody r e s i s t a n c e d i a g r a m for u n i f o r m f l o w

i n conduits

21

22 where X is

the

=

( 4 7 / y ) ( V Z / 2 s ) ( r e f e r r e d to

shear

D

stress,

loss over

head

is

the

length L .

a

pipe

diameter

and

engineering

turbulent

ivterest

and

complex

hf

Re

is

the

friction

and

!he

relative

The and

X

transition

zone

undefined

but

= 64/Re

will

L a m i n a r flow

between

is

also

i.e. X

not occur laminar of

little

i n practice.

Turbulenr rough

practice.

is

flow

friction factor),

Poiseuille found that

i s independent of the r e l a t i v e roughness. i n normal

Darcy

the

X i s a f u n c t i o n of

F o r l a m l n a r flow,

roughness e/D.

as

flow

conditions

boundary.

conditions

are

distribution

The

derived

in

a

may

occur for

from

general

turbulent

the

boundary

either

with

equations

the

a

friction equation

layer,

smooth

factor for

which

is

or

for

the

a

both

velocity

from

derived

m i x i n g l e n g t h theory:

Integrating,

w i t h k = 0.4

where

the

v

is

velocity

hydrodynamical l y

/+=

a

distance

y‘= v / m s o

5.75

log Y

m

y is

from a

the

laminar

/%=V

+

= e/30

log

fl

-

where e i s the boundary roughness.

a n d 2.11

(2.11) and expressing v

D

~

2 log -

+

0.8 layer,

i n terms of

vdA w e get

(2.12) smooth b o u n d a r y ) a n d

1.14

(turbulent boundary layer,

i s affected and

+ 8.5

equations 2.10

( t u r b u l e n t boundary

Jf

and

(2.10)

5.5

i s r o u g h the l a m i n a r sub-layer

y’

5.75

Re-arranging

= 210g Re

sub-layer,

that

the average v e l o c i t y V b y means of the equation Q = 1

For a

was found e x p e r i m e n t a l l y .

the boundary

Nikuradse found that

J-T

boundary.

V

The constant 5.5

Thus

at

smooth b o u n d a r y there

Nikuradse found that

Where

a n d c o n v e r t i n g logs to base 10,

(2.13) rough boundary)

23 Notice tive

that

for

a

r o u g h n e s s e/D

smooth

A

boundary,

and for

a

very

is

independent of

rough boundary

it

is

the r e l a -

independent

o f t h e R e y n o l d s n u m b e r Re f o r a l l p r a c t i c a l p u r p o s e s .

Colebrook

and

White

combined

E q u a t i o n s 2.12

a n d 2.13

to p r o d u c e

a n e q u a t i o n c o v e r i n g b o t h smooth a n d r o u g h b o u n d a r i e s a s w e l l a s t h e t r a n s i t i o n zone;

(D e +Rx 9.35)

1 = 1.14 - 2 l o g

Their Equation

equation

reduces

2.13

rough

satisfactory

for

results

radse's original ness.

h!atural

roughness. TABLE

2.2

for

(2.14)

to E q u a t i o n 2.12 pipes.

This

various

experiments

for

smooth

semi-empirical

commercially

pipes,

and

equation

available

used s a n d as a r t i f i c i a l

to

yields

pipes.

Niku-

boundary

rough-

r o u g h n e s s i s e v a l u a t e d a c c o r d i n g to t h e e q u i v a l e n t s a n d

T a b l e 2.2

g i v e s v a l u e s of e f o r v a r i o u s s u r f a c e s .

Roughness of p i p e m a t e r i a l s

V a l u e o f e ~n mm f o r n e w ,

( H y d r a u l i c s Research station,

c l e a n surface

Finish:

1969)

unles? otherwise stated

Smooth

Rough

Average

_-__.

Glas?, d r a w n metals S t e r l , DVC or AC

0 0.015 Coat?(( strel 0.03 GaIvnr>cze-d, v i t r i f i e d c l a y 0.06 Cast i r c r o r cement l i n e d 0.15 Spun concrete or wood s t a v e 0 . 3 R i v e t e d steel 1.5 F o u l sewers, t u b e r c u l a t e d 6 water mains Ur,Iinrd rock, e a r t h 60

_____ 0.006

0.003 0.03 0.06 0.15 0.3 0.6

0.06 0.15 0.3 0.6 I .5

3

6

15

30

150

300

__-

Fortunately A increases constant

i s not v e r y

linearly

with

d e p e n d i n g on

sensitive

age

local

to t h e v a l u e o f e assumed.

water

for

conditions.

pipes, There

the

may

e

proportionality

also

be

reduction

i n c r o s s s e c t i o n w i t h age. The v a r i o u s r a t i o n a l f o m u l a e f o r ::J

Moody

this

and

viscosities

graph

water

of

at

is

were

A

presented

various

p l o t t e d on a Fig.

as

temperatures

2.2. are

single The

graph

kinematic

listed

the

in

Apprndix. The Moody d i a g r a m pipe

velocity

to d i r e c t

amenable head

loss,

velocity

to

and

i s useful for calculating manually

head

rate

is

flow

or

a

is

solution for trial

calculate

r e c a l c u l s t e v e l o city etc.

and

known. the

error

Reynolds

Unfortunately

convrse,

i.e.

approach

is

number,

Convergence

then

for

it

veloc:ity,

necessary read

loss

not

given

i.e.

off

i s f a i r l y r a p i d however.

A

if

very

guess

,

then

24 The Colebrook W h i t e e q u a t i o n i s e a s i e r to use i f h e a d loss i s g i v e n and velocity loss

Station

i s to b e c a l c u l a t e d .

given

X

or

at

equation

It

flow

Wallingford

Cclebrook-White graphs

or

velocity

i s not

so e a s y

however.

re-arranged produce

to

( H y d r s u l i c Research S t a t i o n ,

The the

simple

solve f o r

to

Hydraulics

in

variables explicit

head

Research the

flow/head

1055

1969): i n t h e form-

E q u a t i o n 2.14 may b e a r r a n g e d

(2.15) Thus e,

a

for

g r a p h may

graph for

has

any

water

plotted

f l u i d at a certain

b e p l o t t e d in terms of V,D at

step

15OC a n d e = 0.06.

similar

also a v a i l a b l e for a

graphs

further,

and

to

V,

the

R a n d S,

gravity.

for

non-circular

Colebrook-Whi te e q u a t i o n tional

Using

and S.

various sections,

conditions.

Research

dimensionless

but

including factors

this

form

resistance diagram

was

published

with

d e r i v e d on a s i m i l a r b a s i s

of

2.3

The

the

Station

charts.

(VVatson,

graphs

for

viscosity,

equation

they

Fig.

2.3

as

are

Going

re-wrote

parameters

the

propor-

roughness

produced

in dimensionless parameters.

their

i s such a

b y r e p l a c i n g D b y 4R.

in

terms o f

Fig.

The H y d r a u l i c Research S t a t i o n

Hydraulics

universal also

temperature a n d defined roughness

an

This

a

graph

example

is

1979).

Comparison of F r i c t i o n F o r m u l a e Diskin

(1960) p r e s e n t e d

a

useful

comparison of

the f r i c t i o n f a c t o r s

f r o m t h e Hazen-Will iams a n d D a r c y e q u a t i o n s : The D a r c y e q u a t i o n may b e w r i t t e n a s

or

v v

=JTprn

(2.16)

c Z &R

=

(2.17)

w h i c h i s termed t h e Chezy e q u a t i o n a n d t h e Chezy c o e f f i c i e n t i s

cz The

=

&p

(2.18)

Hazen-Wi I I iams

equation

may

be

rewritten

for

al I

practical

purposes in the f o l l o w i n g dimensionless form: S

=

5 1 5 ( V / C W ) * (Cw/Re)

By c o m p a r i n g be deduced t h a t

this with

0.15

(2.19)

/gD

t h e Darcy-Weisbach

equation

(2.16)

i t may

26 The

Hazen-Williams

Re

and 2.2).

and

It

values

will

be

Cw

coefficient

may

be

observed

from

Fig.

Hazen-Wi I I iams coefficient coincide

with

in the t r a n s i t i o n zone.

pipes

the

with

therefore

a

Moody

2.2

that

a

functionof

diagram lines

for

(see

X

Fig.

constant

the Colebrook-Whi te I ines o n l y

I n the completely t u r b u l e n t zone f o r non-smooth wi II

coefficient

i .e.

number

is

p l o t t e d on

actual l y

reduce

the

greater

the

Reynolds

cannot associate a c e r t a i n Hazen-Wil I iams coefficient

one

a p a r t i c u l a r p i p e as

i t v a r i e s depending on the flow

rate.

The

Hazen-Will iams equation should therefore be used w i t h c a u t i o n f o r h i g h Reynolds numbers and r o u g h pipes. of C w above approximately practice ( R The

It will

also be noted that values

155 a r e impossible to a t t a i n in water-works

6

around 10 ) .

Manning

p a r t f u l l pipes.

equation

is

widely

used

for

open

channel

flow

and

The equation i s

(2.21 where

K

is

1.00

in SI

P the wetted perimeter. for non-circular

TABLE 2.3

1.486

u n i t s and

where A

h y d r a u l i c r a d i u s A/P

R

One

is D/4

for

i s the

a c i r c u l a r p i p e , a n d i n general

Values of M a n n i n g ' s ' n '

method of

ventional

KV2/2g

and R

area of flow and

0.010

0.01 1 0.01 2 0.013 0.015 0.017

LOSSES

i n section

press

I b units,

sections, 4R may be substituted f o r D.

Smooth glass, p l a s t i c Concrete, steel ( b i tumen I i n e d ) galvanized Cast i r o n Slimy or greasy sewers Rivetted steel, v i t r i f i e d wood-stave Rough concrete

MINOR

in ft

i s the cross-sectional

expressing

i s the e q u i v a l e n t friction

loss

formulae

losses through f i t t i n g s where

K

is

the

head

loss

loss through f i t t i n g s and changes

l e n g t h method, are

used.

in terms of

coefficient.

often Modern

used when

the con-

p r a c t i c e i s to ex-

the v e l o c i t y head i.e. Table 2.4

gives typical

he = loss

27 coefficients plementary

although data

and

valve

loss

The v e l o c i t y V

opening.

to

manufacturers coefficients

K

may

which

also

will

provide vary

sup-

with

gate

use i s n o r m a l l y the mean t h r o u g h the f u l l

bore of the p i p e o r f i t t i n g .

TABLE 2.4

Loss coefficients f o r p i p e f i t t i n g s

Bends hBKBV2/ 2 g Bend a n g l e

Sharp

30" 45" 60 O 90 1 80° 90" w i t h g u i d e vanes r

=

2

6

0.07 0.10 0.12 0.13

0.06 0.08 0.08 0.08

if

the r a d i u s i s

r/D=l

0.16

0.07 0.13 0.18

0.32

0.68 1.27 2.2

0.22

0.2

r a d i u s of bend to centre of p i p e

A significant

reduction

in

bend

loss

i s possible

f l a t t e n e d i n the p l a n e o f the bend. Valves

hv = KvV2/2g

Type:

Opening:

SLuice Butterf I y Globe Needle Ref I u x

1/ 4

1 /2

3/4

Ful I

24 120

5.6 7.5

1 .o 1.2

0.2

0.6

0.5 1-2.5

0.3 10

4

1

Contractions a n d expansions in cross-section

Wal I-Wal I Angle ___

0 -

Contract ions :

Expansions

hc= KcVJ29

h = K V 2/2g c c l

A 2 4 0.2 0.4 0.6 0.8 1.0 - __ __

7.5" 150 30 O

180"

.5

.37

.25

.15

.07 0

Al/A2

0 .13 .32 .78 1 .O

0.2 0.4 0.6 0.8 1.0 .08 .24 .45 .64

.05 .15 .27 .36

.02 .08 .13 .17

0 0 .02 0 .03 0 -04 0

28 Entrance and e x i t

he = KeV2/2g

losses:

Protruding Sharp Bevel l e d Rounded

PRESSURE AND F L O W CONTROL

Entrance

Exit

0.8 0.5 0.25 0.05

1 .o 1 .o

0.5 0.2

IN PIPES

I ntroduct ion Valves the

and other

cause of

is

reduced

water

giving

rise

Knapp

et

upstream

fittings

vapour

to

vaporizes.

cavitation

1961).

al, and

in pipes

bubble formation The

bubbles

damage

The

downstream

fluid.

The

geometry

conditions

referred

to

velocity

increases

as

to

pressure,

r a t e affec? the c a v i t a t i o n p o t e n t i a l the

critical.

as

may

the

pipe

of

the

as

well

ratio

the

h e a d may

When

be

the

pressure

subsequently

collapse

or

valve

valve

the

damage

well

closure

vapour

cavitation

1970,

(Ball;

as

valve

as

which

cavitation

the

reduce

degree of

under

The

and

which

downstream.

as and

p r e s s u r e of

commences

increases

of

upstream

for

identifying

the flow

are

as

flow

downstream

to

p r e s s u r e increases. A of

number

operation

The

of

of

popular

reduction

empirical control

measure

ratio,

commercially

i.e.

to

work

of

the

potential

r a t i o of

valve types

influences of

s a f e zone

less

valves

the for

problem

is

the

pressure

u p s t r e a m to d o w n s t r e a m p r e s s u r e s .

valves

ratios

cavitation

are

than

said

3 or 4.

critical

operate

to

w i t h a c a v i t a t i o n index a s indicated

various

the

v a l v e s h a v e been p r o p o s e d b y v a l v e s u p p l i e r s . of

available

pressure reducing

relationships

It

for

i s h o w e v e r more l o g i c a l

Jater.

conditions,

pressure

Many

efficiently

reducing

Since t h e geometry the or

application

flow

control

of is

described.

TYPES OF VALVES

Valves

are

used

for

i s o l a t i n g o r flow control.

two

aistinct

purposes

in

pipelines,

namely

29 I s o l a t i n g Valves Isolating

valves

They

should

only.

They

be

are

and

there

damage not

close

off

in

the

fully

open

operated

the o b s t r u c t i o n to

is

to

the

flow

or

through

fully

a

closed

pipe.

position

h a v e poor flow c h a r a c t e r i s t i c s i n the p a r t l y open position.

i.e.

gate

used

is

often

shut.

the seat

to

to

i s l a r g e l y the exposed area of a gate

l i t t l e pressure reduction o r

practically

designed

the flow

and

At

that

stage

flow

control

cavitation

v a l v e body may r e s u l t .

is

until

the

possible

and

T h i s type of v a l v e i s

reduce the pressure a p p r e c i a b l y .

i.e.

it

i s designed

to pass the f u l l flow when f u l l y open w i t h minimal head loss. Such

valves

spherical

include

valves.

designed

to

These

close

operated b y

gate

as

valves,

valves

rapidly

spindles

with

are

as

screw

butterfly generally

possible. threads,

valves

and

operated

by

Although butterfly

gate

r o t a r y or hand

and

valves

are

valves and rot ary

v a l v e s can be operated b y t u r n i n g a h a n d l e t h r o u g h 90" to f u l l y close the

valve

seating these

from

the

fully

open

position.

I n order

pressures however a n d sometimes to control

valves

spindle

may

have

indirectly

from

reduction the

gear

boxes

handwheel.

in

to

overcome

high

the r a t e of closure order

Alternatively

to

drive

electrical

the and

pneumatic control systems can be used to operate these valves. Typical i!lustrated open

sl(iice in

bore area

non l i n e a r , area

valves,

10.1,

Figs.

a n d degree of

reduces

rapidly

i s reduced r a p i d l y rise

to

10.2

valves

and

10.4,

The

and

t u r n i n g of

the

I t can be observed i n the case of

in

u n f a v o u r a b l e w i t h respect

higher

reduction

butterfly

in

the to

final

water

d u r i n g the f i n a l

water

velocity

hammer in

the

pressures pipe.

handwheel

The

of

than

as

the

closure

and

tne

uniform

problems

of

are

between generally

is

valve

closure.

pressures

stages of

valves

relationship

the gate

stages

hammer

spherical

that

the

This

is

velocity

this

gives

rate

cavitation

of

are

g e n e r a l l y avoided w i t h such \ ~ a l \ / e sas they a r e not operated i n p a r t l y closed c o v d i t i o n s except i n emergencies.

Control Valves Various water

types of

v a l v e s h a v e been designed to control

i n pipelines b y

linearly

the flow of

r e d u c i n g the open a r e a a n d to c o n t a i n

the c a v i t a t i o n effect due to reduction of pressure through the v a l v e .

A

30 typical

valve

valve).

Diaphram type valves,

2.4)

in

common

and plunger

a v a r i a t i o n of slots.

any

( a needle

which

valves even

is

has a piston covering the

and

more

basket

type

appropriate

r e l y on a s i n g l e

high

for

duty

pressure

increase i n velocity

and

head d i s s i p a t i o n due to expansion whereas the l a t t e r types which

have

variable

pressure

resistance

trim

i n a number of

damage

as

a

result.

r a t i o i s l i m i t e d at It

10.3

Fig.

r a d i a l flow sleeve v a l v e s ( F i g .

valve

are

types

in

v a l v e s a r e also employed a n d there

type

2.5)

(Fig.

The former

illustrated

slotted

type control

expansion

valves

reducing.

is

the r a d i a l flow

Multiple

control

use

rings

or

layered

baskets

reduce

the

stages a n d a r e s a i d to s u f f e r less c a v i t a t i o n

This

is

largely

because

each stage b u t o v e r a l l

i s a l s o recognised that

pressure

reducing

may be r e l a t i v e l y h i g h .

it

these v a l v e s a r e

the

less noisy

when

reducing

h i g h pressures. The

needle v a l v e

was

as

it

at

a l l stages of opening.

or

the

COI

was

initial

valve) (e.g.

or

quality the

by

materials

pistons o r

1981).

Dvir,

water

as

workings.

valve

i s however expensive owing

i s now replaced to a

sophisticated

the

such

latter

type

of

elastometers

valve

other

hand

the

( t h e sleeve

its

diaphram

type

low pressure valves

i s not sensitive

f l e x i b l e d i a p h r a m prevents

the

to

l a r g e extent by simpler valves

as

diaphrams i n the case of

The

On

i n power s t a t i o n p r a c t i c e

v a l v e which could control flow

I t often h a d a p i l o t needle f o r f i n a l seating

The

opening.

struction and

using

once used widely

recognised as a streamlined

ingress of cannot

to

poor

dirt

into

accommodate

h i g h pressure d i f f e r e n t i a l s and s u f f e r s wear a f t e r prolonged operation. It

may

also

be

subject

to

vibration

and

instability

as

a

venturi

action i s caused d u r i n g the l a s t stages of closing. Piston is

type v a l v e s can b e accurately

towards

downstream or

the atmosphere

stream on

the

face of

system

and

of

of

the

i s supported

that

is off

also

be

operated

off

to

In

to

system.

some

types

the downstream

by a spring.

the

pressure

I n others the down-

f l u i d pressure depending Such p i l o t systems can be

the water

a r e often r e q u i r e d to prevent

can

piston.

i s exposed

a p i l o t control

hydraulically,

strainers

face

the p i s t o n

the p i s t o n can be subject

the operation

operated

exposed

face of

c o n t r o l l e d p r o v i d e d the flow

pressure.

the p i l o t

electrical

tubes

I n t h i s case blocking.

solenoids

or

The

from

a

31

Fig.

~

2.4

i 2 . 5~

F l e x f l o Sleeve v a l v e f o r f l o w control

, .i i g h d u t y control reduction

v a l v e n i t n t r i p l e aasi(ets

foi'

heao

p n e u m a t i c s u p p o r t system. The sure

control

to

system c a n

a n d t h e sensor a

be designed

a c e r t a i n maximum f i g u r e .

pilot

d o w n s t r e a m senses

valve held b y

means o f

to

limit

t h e downstream

pres-

As t h e u p s t r e a m p r e s s u r e b u i l d s UP an

i n c r e a s i n g p r e s s u r e i t w i l l open

in

a spring and this pilot valve will

t u r n p e r m i t a b i g g e r p r e s s u r e f r o m u p s t r e a m t o come b e h i n d t h e p i s t o n and

partly

close

control

flow

orifice

or

controls get

the

ports.

Alternatively

in

which

case

rate venturi

is

the

measured.

the

valve

p r e s s u r e on

The

pressure

may

b e u s e d to

either

side

of

in

difference

an turn

p i l o t v a l v e w h i c h w i l l e i t h e r p e r m i t a g r e a t e r p r e s s u r e to

the

behind

the

piston

to

close

the

valve

or

exhaust

the

pressure

b e h i n d t h e p i s t o n to open t h e v a l v e d e p e n d i n g on w h i c h i s r e q u i r e d .

CAVITATION

The

IN CONTROL VALVES

mechanism

control high

valve

is

velocity

turbulence from

by

jet

as

statical

whereby

flow

converting and

it

is

then

emerges

reduced o r the

the

into

pressure energy

p r e s s u r e i s reduced

pressure energy velocity

the

open

energy

pipe

upstream is

again.

to v e l o c i t y e n e r g y

in a

into a

dissipated The

in

conversion

i s obtained from

the

Bernoul I i e q u a t i o n ;

(2.22) Here

P

that

P/W

is

the is

datum a n d V velocity be V

upstream the

pressure,

pressure

i s the w a t e r

head,

w h e r e H = P/W.

This

replacing

h

by

ah.

coefficient

is

often

introduced

v = c where the the

d

i s often given

the the

Thus

if

unit

weight

elevation

2

of

is

the

To

affects

so

fixed

however n e g l e c t s u p s t r e a m v e l o c i t y the

account in

latter can for

the

be accounted f o r b y

upstream

equation

velocity

and

in

fact

head

a

the p i p e

in t h e f o l l o w i n g f o r m

(2.23) discharge

v a l v e opening. valve

a

i s a c o n s t a n t then t h e

JzsnF; Cd

water

above

t h e p r e s s u r e h e a d i s r e d u c e d to z e r o w o u l d

a n d downstream p r e s s u r e a l t h o u g h

velocity

is is

velocity.

p o s s i b l e assuming

=

W

Z

Apart the

coefficient

from

the f a c t

discharge

which that

coefficient,

may

vary

depending

on

t h e d e g r e e of o p e n i n g o f even

if

the

discharge

33 coefficient to v a r y

was separated from the degree of opening i t would be found

owing

to the d i f f e r e n t contraction r a t i o as a d i f f e r e n t area i s

open. The only

actual

be

found

incipient

i.e.

valve

is

and a

from the

a n d choking

stream.

stage of

throttling

experiment.

p o i n t at

i.e.

at

appreciably

Cavitation

which

the stage a t

vapour

which

affected

which c a v i t a t i o n

by

has

commences can

been

classified

bubbles commenced to

as

form,

the d i s c h a r g e coefficient o f

vaporization

of

the

water

the

down-

The v a p o r i z a t i o n occurs because the pressure head i s reduced

the energy

low

i s transformed i n t o k i n e t i c energy which impinges i n t o

pressure

A

downstream.

cavitation

have

significant

factor

been

proposed

the

in

number

of

(Winn

indication

of

parameters and

for

Johnson,

cavitation

is

assessing

1970). the

The

cavitation

index.

(2.24) where

P

is

the

pressure a n d s u b s c r i p t d r e f e r s to downstream,

u to

upstream a n d v to vapour pressure.

For

streamlined

tation

index

cavitation

drops as

often

case i t w i l l

valves low

occurs

it

0.1

a

then

the the

index

occur

until

the

cavi-

p o o r l y designed v a l v e s

i.e.

nearer

1.0.

at normal o p e r a t i n g pressure r a t i o s .

pressure

can

pressure

v a l v e s i s much lower. pressure

not

whereas f o r

higher

i s seen from the c a v i t a t i o n

whereas

the

as 0.1

at

may

In

any

be found on inspection that most control v a l v e s experience

a degree of c a v i t a t i o n Pv

cavitation

be

index

if

the c r i t i c a l

by

a

factor

reduced

reduction

ratio

possible

with

For an average c r i t i c a l c a v i t a t i o n

reducing

ratio

Neglecting

that

i s from 2.24

neglecting

of

index nearly

is 10

unstreaml ined index of 0.33

vapour

pressure,

4.0. Various forms of 1970).

Very

little

the c a v i t a t i o n is

however

c a v i t a t i o n index and much work T h e method of Various velocity

index

known

about

owing

quantitatively

to

the

photographic

the

scaling

down

of

the

has to be done i n t h i s f i e l d yet. assessing c a v i t a t i o n i s a l s o d i f f i c u l t .

methods have been attempted i.e.

format ion,

have been proposed ( T u l l i s ,

increase

in

methods,

bulk

estimation of the increase of volume

pressure

caused

by

measurements

bubble due

to

collapse

of

the

bubbles

and

sonic- methods.

2.6

Fig.

shows

the

v a r i a t i o n i n noise measured on a r a d i a l flow sleeve v a l v e at d i f f e r e n t c a v i t a t i o n indices f o r a f i x e d degree of opening.

I n t e r a c t i o n between C a v i t a t i o n and Water Hammer Pressures

It

is

recognised

that

pressure subsequently the

accompanying

the

collapse

of

vapour

bubbles

when

increases may cause erosion of steel.

noise a r e often

the

symptoms

of

a

the

T h i s and

poorly

selected

valve. There hammer

is

however

wave the

closed so

an

celerity

interesting

and

the

b u b b l e formation

relationship

cavitation increases

between

potential.

As

the a

water

valve

is

a n d t h i s reduces the water

hammer wave c e l e r i t y downstream (Ch. 4 a n d 5 ) . If

the

which

free

in

turn

then

it

the

valve

gas is

fraction

a

can

f u n c t i o n of

discharge

with

the

equation

equation

and

wave c e l e r i t y . the

when

cavity the

It

in

pressure.

The

bubbles

are

solved

pressure

to

pressure This

increases.

bubbles

downstream

may

as

release

they

which r a p i d l y

are

therefore

released

not then

initially

turns

q u a n t i t i e s of due

collapse it

drop

will

be

rapidly

water hammer deceleration effect.

the

wave c e l e r i t y

promise f o r

reduces r a p i d l y

i s reduced so that

neglects

wave

the p o t e n t i a l of

equation index,

the flow

reflections

and

cavitation

to effect

so

to

into l i q u i d gas

may

the

drop

completely.

the

lowering

index

cavitation

to

to

study

index

however b y the fact that p a r t

Small

(iteratively) would

cavitation

form

closed due begin

the

cavitation

downstream pressure

downstream pressure a n d water ham-

i s made d i f f i c u l t

again

contained

equations

gas

i s not gas b u t vapour

pressure

the

to

the water hammer equation,

the

relating

f o r free gas volume downstream,

of

related

i s possible to solve sirnuttaneously

together

mer

be

the upstream or

found

as

the

be in

If

the

that

the

valve

is

Then as bubbles

and the consequent

reduces more s t e a d i l y . on

but

it

does

hold

better control of flow

as a v a l v e closes especially

d u r i n g the i n i t i a l stages of closure where

this

flow

theory

indicates

that

previous1 y considered the case.

reduction

is

more

rapid

than

was

35 REFERENCES

1960. F l u i d M e c h a n i c s A l b e r t s o n , M.L., B a r t o n , J.R. a n d Simons, D.B., f o r E n g i n e e r s . P r e n t i c e H a l l , N.J. B a l l , J.W., 1970. C a v i t a t i o n d e s i g n c r i t e r i a . I n T u l l i s (1970). Proc. I n s t . Colorado State Univ. D i s k i n , M.H., Nov. 1960. The l i m i t s o f a p p l i c a b i l i t y o f t h e HazenW i l l i a m s f o r m u l a e . L a H o u i l l e B l a n c h e , 6. D v i r , Y., 1981. P r e s s u r e r e g u l a t o r s in w a t e r s u p p l y systems. Water a n d I r r i g a t i o n Review, Water Works A s s o c i a t i o n o f I s r a e l . H y d r a u l i c s Research S t a t i o n , 1969. C h a r t s f o r t h e H y d r a u l i c Designs o f C h a n n e l s a n d PiDes. 3 r d E d n . . H.M.S.O., London. K n a p p , R., D a i l y , 'J.W. a n d Hammitt, F.G.',1961. C a v i t a t i o n . McGrawHill. S c h l i c h t i n g , H., 1960. B o u n d a r y L a y e r T h e o r y . 4 t h E d n . , McGraw-Hill N.Y. T u l l i s , J.P., 1970. Control o f f l o w i n c l o s e d c o n d u i t s . Proc. I n s t . Colorado State U n i v e r s i t y . Watson, M.D., J u l y 1979. A s i m p l i f i e d a p p r o a c h t o t h e s o l u t i o n o f p i p e f l o w p r o b l e m s u s i n g t h e Colebrook-White method. C i v i l Eng. in S.A., 2 1 ( 7 ) , p p 169-171. Winn, W.P. a n d Johnson, D.E. December 1970. C a v i t a t i o n p a r a m e t e r s f o r o u t l e t v a l v e s . Proc. ASCE, HY12. LIST O F SYMBOLS

A

-

cross-sectional

C

-

Hazen-Will iams f r i c t i o n f a c t o r

C'

friction factor

Cz

-

d

-

d e p t h o f water

D

-

diameter

e

N i k u r a d s e roughness

force

he

-

hf

-

f r i c t i o n h e a d loss

K

-

loss c o e f f i c i e n t

L

length of c o n d u i t

n

-

Manning friction factor

P

-

wetted p e r i m e t e r

p

-

pressure

f Fx g

a r e a of flow

Chezy f r i c t i o n f a c t o r

Darcy f r i c t i o n factor

gravitational

( e q u i v a l e n t to A )

acceleration

h e a d loss

36

Q

-

R

-

Re S

flow rate hydraulic radius Reynolds number h y d r a u l i c gradient

v

mean velocity

V

velocity a t a point

X

distance along conduit

Y

distance from boundary

across a section

2

elevation

Y

specific

P

mass density

T

shear stress

V

k i n e m a t i c viscosity

x

Darcy f r i c t i o n f a c t o r - ( f i n USA)

F i g . 2.6

weight

Noise output from a control

valve

37

CHAPTER 3

PIPELINE SYSTEM ANALYSIS AND DESIGN NETWORK ANALYSIS

The

flows

controlled points

through

by

and

steady-state

the the

a

system

of

interlinked

difference between residual

flow

pressure

pattern

will

pipes or

networks a r e

the pressure heads a t heads

at

the

be established

the

input

A

drawoff

points.

network

such that

in a

the f o l l o w i n g two c r i t e r i a a r e satisfied:-

(1)

The net

(2)

The

flow

towards

must equal outflow, net

j u n c t i o n or

node

i.e.,

i s zero,

inflow

and

loss

head

any

around

any

closed

loop

is

zero,

i.e.,

only

one head can e x i s t at any point a t any time. The

l i n e head losses a r e u s u a l l y

and

most

loss

relationships

h

methods

= K?Qn/Dm

of

analysis

for

where

pipes

h

is

are

are

the

the o n l y based

usually

loss,

head

significant

on

this

assumed

P

head Losses

assumption. to

be

i s the p i p e

of

the

length,

Head form €Ithe

flow and D the i n t e r n a l diameter of the pipe. The

calculations

are

simplified

if

the

friction

K

factor

can

be

assumed t h e same f o r a l l p i p e s i n the network.

E q u i v a l e n t Pipes f o r Pipes i n Series o r P a r a l l e l

It the in of

i s often

same series

or

to

loss

and

parallel.

know

the

e q u i v a l e n t p i p e which

flow

as

a

The

number

equivalent

pipe

of

would g i v e

interconnected pipes

may

be

used

in

place

the compound pipes to perform f u r t h e r flow c a l c u l a t i o n s .

The of

head

useful

equivalent

different

equating

diameter

diameters total

the

and

head

the e q u i v a l e n t p i p e of

of

a

compound

lengths

loss

for

length equal

in

any

p i p e composed of

series flow

to

may

be

sections

calculated

the head

loss

by

through

to the l e n g t h of compound pipe:-

z

K ( Z P ) Q ~ / D= ~ K~P Q " / D ~

..

De

=

(3.1)

38 ( m i s 5 i n the Darcy f o r m u l a a n d 4.85

Similarly,

in the Hazen-Williams f o r m u l a ) .

the e q u i v a l e n t diameter of a system of pipes i n p a r a l l e l

i s derived by equating the

total

flow through the equivalent

pipe ' e '

to the sum of the flows through the i n d i v i d u a l pipes ' i ' i n p a r a l l e l : Now h = h.

KP Q " / D ~ =~ K P ~ Q ~ ~ / D ~ ~ e

i.e.

Q.

So

=

and Q = So cance l i n g out Q , a n d b r i n g i n g D

and

P

e

to the

l e f t h a n d side,

P i s the same,

and i f each

De = [ C ( D i m / n ) ] n / m The

equivalent

loss c h a r t .

read

off

(3.2)

diameter

For pipes

the

flow

could

also

in parallel,

through

each

be d e r i v e d u s i n g a flow/head

assume a reasonable head loss and pipe

from

the

chart.

Read o f f

the

e q u i v a l e n t diameter which would g i v e the total flow a t the same head loss.

For p i p e s i n series,

assume a reasonable flow a n d c a l c u l a t e the

loss w i t h assistance of

the c h a r t .

total

head

pipe

diameter which would discharge

Read off

the equivalent

the assumed flow

with

the total

head loss across i t s length.

I t often speeds network analyses to s i m p l i f y p i p e networks as much as

possible

parallel. could this

using equivalent

Of

always is

in

course be

fact

the

used the

diameters

methods of to

for

network

a n a l y s e flows

minor

through

p r e f e r r e d method f o r

pipes

in

series or

a n a l y s i s described below compound

more complex

pipes and

systems

than

those discussed above.

Loop F l o w Correction Method

The both

loop

involve

method

and

successive

the node method of approximations

a n a l y s i n g p i p e networks

speeded

technique developed b y Hardy C r o s s (1936).

by

a

mathematical

39 The steps

i n ba ancing

the flows

i n a network b y the loop method

are: Draw all

the

pipe

inputs,

network

drawoffs,

schematically fixed

heads

to

a clear

and

booster

scale. pumps

Indicate (if

pres-

connect

pairs

ent). If

there

is

more

than

one

constant

head

node,

of constant head nodes or r e s e r v o i r s b y dummy pipes represented by

dashed

the

flow

lines.

Assume

corresponding

corrections,

omit

a

to

this

diameter

fixed

pipe

and

but

length

loss.

head

include

In

it

n

and

calculate

subsequent

flow

calculating

head

I osses. Imagine the network

To

speed

may

be

as a

convergence assumed

assuming

a

loops

are

as

to

series

pattern of

of

the

form

large

loops

of

needed

to

closed

solution

of

superimposed

side

ensure

loops

some

by

side.

that

each

i n any

the loops

Use

is

pipes

instead

only

pipe

order.

major

of

as

many

at

least

in

one loop. Starting

with

containing tracting flows

as

time,

many

and

on

assume

a

calculating

loops

other

a

pipe

pipe,

drawoffs

to

one a t

any

the

flows

if

flow.

to

other

unknown.

as

there

around

i n each

loops

Proceed

a s i m i l a r basis.

assumptions

Proceed

the flow

at

to

a

loop

pipe by

sub-

nodes.

Assume

n e i g h b o u r i n g loops

I t w i l l be necessary to make

are

loops.

The

more

accurate

each assumption the speedier w i l l be the solution. Calculate formula

the

such

head

loss

h =

as

( p r e f e r a b l e i f the a n a l y s i s Calculate around

the

the

arriving the

loop

ing

the

at

net

loop,

head

starting

i s not zero, following

pipe

or

in

use

a

any

loop

flow/head

using loss

a

chart

i s to be done b y h a n d ) .

loss

add head

the

each

in

Kk'Qn/Dm

around t h e

loop,

i.e.,

proceeding

losses and s u b t r a c t head g a i n s u n t i l point.

correct

increment

If

the

net

head

the flows

arol;nd

flow

the

in

in

the

same

loss

around

loop b y adddirection

that

head losses were c a l c u l a t e d :

(3.3) T h i s equation

i s the f i r s t

o r d e r approximation

to

the differen-

t i a l of the head loss equation and i s d e r i v e d as follows:-

40

h = KEQn/Dm

Since

dh

KtnQn-'dQ/Dm

=

(hn/Q)dQ

=

Now t h e t o t a l head loss around each loop should be zero,

i.e.,

c ( h + dh) = 0

x

C h +

(hn/Q)dQ = 0

The v a l u e of h/Q i s a l w a y s p o s i t i v e ( o r zero i f h a n d Q a r e z e r o ) .

(7)

I f there

is a

head from

booster pump

i n any

loop,

Ch before m a k i n g the flow

subtract

correction

the generated

u s i n g the above

equations.

(8)

The flow around each

(9)

Steps 5-8

(steps 5-71.

loop i n t u r n i s corrected thus

a r e repeated u n t i l the head arourld each

loop

balances

to a s a t i s f a c t o r y amount.

The Node Head Correction Method With loops,

the

node

initial

corrected b y flows

were

method,

heads

are

of

instead

assumed

at

assuming each

successive approximation

corrected

for

the

loop

initial

node.

flows

Heads a t

around

nodes a r e

i n a s i m i l a r manner to the way

The

method.

steps

in

an a n a l y s i s

a r e as follows:-

(1)

Draw all

(2)

the

pipe

inputs,

Assume at

network

drawoffs,

initial

that

arbitrary

node

assignments,

schematically

is

the

to

a clear

scale.

Indicate

f i x e d heads a n d booster pumps. heads at

fixed).

speedier

The

will

be

each node more the

(except i f the head

accurate

the

convergence of

initial

the

solu-

tion.

(3)

Calculate head

the

using

flow

the

in

each

formula

Q

pipe

=

to

any

(hDm/Kt)'/"

node or

with

using

a a

variable flow/head

loss c h a r t .

(4)

Calculate zero,

*"

the

net

inflow

to

the specific

node a n d

if

this

i s not

correct the head b y a d d i n g the am'ount

-Q

=

C(Q/nh)

(3.4)

41 T h i s equation i s d e r i v e d as follows:-

dQ = Qdh/nh

c ( Q + dQ)

We r e q u i r e

But

= 0

dH = -dh

so F l o w Q a n d head loss h a r e

considered

H i s the head a t the node.

node.

positive

if

towards

the

i n p u t s ( p o s i t i v e ) a n d drawoffs

( n e g a t i v e ) a t the node should be i n c l u d e d i n E Q .

(5)

Correct

(6)

the head a t

each variable-head

node in s i m i l a r manner,

repeat steps 3 a n d 4 f o r each node.

i.e.

Repeat

ends

of

degree of a

pipe

to 5 )

(steps 3

the procedure

sufficient

accuracy.

i s zero

at

until

a l l flows balance to a

I f the head difference between the any

stage,

omit

the

pipe

from

the

p a r t i c u l a r b a l a n c i n g operation.

A l t e r n a t i v e Methods of A n a l y s i s Both

the

loop

method

and

the

node method of

networks can

be done m a n u a l l y b u t

networks.

done

If

tables

or

space.

Fig.

even

manually,

on the pipework

3.1

a computer

calculations

b a l a n c i n g flows

i s preferred for

should

layout drawing

be if

set

there

out

in

large

well

in

i s sufficient

i s an example analysed m a n u a l l y b y the node method.

There a r e s t a n d a r d computer programs a v a i l a b l e f o r network a n a l y s i s , most of which use the loop method. The main advantage of the node method i s that more i t e r a t i o n s a r e required

than

especially

if

for

the

loop

the

system

method is

very

to

achieve

the same convergence,

unbalanced

to

start

with.

It

is

n o r m a l l y necessary f o r a l l pipes to h a v e the same o r d e r of head loss. There

are

include

a

number of

overcorrection

methods in

some

for

speeding

cases,

or

the convergence. using

a

second

These order

approximation to the d i f f e r e n t i a l s f o r ca I cu I at i n g correct ions. The node head correction

method

i s slow

to converge on account of

42 the fact

that

Also

time.

corrections

the

d i s s i p a t e through

head correction

equation

the network one p i p e a t a has

an

amplification

factor

( n ) a p p l i e d to the correction which causes overshoot.

3Oomm

60m h *26

!a

x

2OOOm

-

lQIh

1140 1 7

s/h 0.7

0,7 0.7 97

0 0

0.7

N

+

+5s 80 1.5 + 5 4 80 1.5 tS2.7 80 1,s 0 .1.7 *0,7

0

0

0

5,3 9

6

NOTES Heads i n metres, flows i n l i t r e s per second, diameters i n millimetres, lengths i n metres. Arrows i n d i c a t e p o s i t i v e d i r e c t i o n of h C Q ( a r b i t r a r y assumption). Blackened c i r c l e s i n d i c a t e nodes w i t h f i x e d heads, numbers in circles indicate order i n which nodes were corrected. Head losses evaluated from F i g . 2.3. H =

Fig.

3.1

1.85

Q in Q/ h

Example of node method of network flow a n a l y s i s

43 The

loop

flow

preparation. defined data

Flows must

indirectly

preparation

gence.

data

in

be

the

and

method

assembly.

has

assumed

assumed

Trial

disadvantage

loops,

flows.

in

data

and drawoffs

The

often offsets

relatively and

the

around pipe

interpretation

i s so because of

This

with

correction

are

added e f f o r t

in

the q u i c k e r conver-

low computing costs compared

error

design

is

also

cumbersome

if

loop flows h a v e to b e changed each time a new p i p e i s added.

Network A n a l y s i s b y L i n e a r Theory The ual in

Hardy

Cross methods of

methods of comparison

involve

the

solution

but

network

suffer

simul taneous

head

balance.

solution

and

very

few

iterations are required

with

the

number of

simul taneous

of

Simultaneous

iterations

head correction methods. of

drawbacks

in

w i t h computer o r i e n t a t e d numerical

flow

ber

a n a l y s i s a r e s u i t e d to man-

sets

of

the

effort required

methods.

The

equations

solution

has

the

latter

describing effect

that

to balance a network when compared

for

the

loop

flow

correction a n d node

O n the other h a n d solution of

equations,

even

if

rendered

a

l a r g e num-

linear,

requires

a l a r g e computer memory a n d many i t e r a t i o n s . Newton-Raphson linear

equations

problem

becomes

techniques

for

successive

a r e mathematically subordinate

to

approximation

sophisticated b u t

the

mathematics.

Charles (1972)

l i n e a r i z e d the head loss equation,

approximation

at

each

step

and

establishing

of

non-

the engineering Thus

Wood

and

i m p r o v i n g the l i n e a r equations

for

head

balance around loops. Isaac and M i l l s

(1980) s i m i l a r l y

l i n e a r i z e d the head loss equation

as follows f o r flow between nodes i and j :

8..

= C..+(H.-H.) IJ I J

//m

where

the

the

IJ

term

each i t e r a t i o n .

Substitute node :

in

square

(3.5)

root

sign

is

assumed

a

constant

for

If the Darcy f r i c t i o n equation is employed,

equation

3.5

into

the

equation

for

flow

balance

at

each

EQ..

= Q.

J

i IJ

Q.

where

i

is

J

node

to

the

drawoff

negative

j,

at

if

node j

from

and

i.

to

j

Q.. i s the flow from node iJ There i s one such equation

f o r each node. If has

each

a

be

set

of

solved

each

is

Q.. 'J

by

simultaneous

H

for

node

replaced

at

then

node.

i n 3.5,

l i n e a r i z e d expression

equations

each

initially,

the

(one for

The

new H ' s .

solve f o r

each

procedure

node)

is

to

one

which can

H

estimate

at

The procedure i s repeat-

ed u n t i l s a t i s f a c t o r y convergence i s obtained.

OPTIMIZATION O F PIPELINE

The in

pipe to

previous

pipe

network

fixed

be

to

demands

for

layout

a

and If

and

at

is

repeated

for

so d e r i v e d

the

various

at

of

would

be

This

another

would

flows

could

flows

trial

A

the to

layout.

Each

would

to

meet

would

be accep-

alternative

diameters

process of

have to b e costed a n d

calculated.

sufficient

layout

error

corresponding be

i t would be neces-

just

try

flows

particular

proposed layout

i s repeated u n t i l

and

the

any

pattern

were

necessary

possible

then

For

flow

possibilities.

corresponding

hand.

calculating

nodes

pressures were s a t i s f a c t o r y , it

for

loops.

to meet c e r t a i n drawoffs,

sizes a n d a n a l y s i s of

solution

closed

diameters,

number

if

methods

without

inputs

network a

not,

pipe

described or

and

or

new

compare

analysed

table.

with

drawoffs

To design sary

section

networks

SYSTEMS

the

that

a satisfactory would final

then

be

networks

network w i t h

least

cost selected. A out

technique recourse

positive with

closed

trial

techniques and

error,

possible

head

losses

can

number of be

described

used

least-cost would

for

be

general

i s that

and

optimization

There a r e a

discussed

the

T h e problem

loops. flows,

and

is

mathematical

ships. tion

to

determining

technique

diameters, tine

of

costs

network desirable.

directly, No

optimization

with-

direct of

and

networks

the r e l a t i o n s h i p between p i p e i s not

techniques

linear

require

and most rou-

linear

relation-

s i t u a t i o n s where mathematical optirnizato optimize

below.

The

l a y o u t s a n d these cases a r e

cases

are

normally

confined

to

45 single mains or is

known.

To

tree-like

networks

optimize

for

network

a

which

with

the

closed

flow

in each b r a n c h

loops,

random

search

t e c h n i q u e s o r s u c c e s s i v e a p p r o x i m a t i o n t e c h n i q u e s a r e needed. Mathematical analysis design

techniques

techniques

techniques

techniques

(which not

(again

mathematical

a

optimization

techniques

is

analysis

a

not

also

as

as as

will

be

steepest

systems they

operations

or

retained

are

research

descriptive).

( o r mathematical

such

known

nomenclature

really

techniques

include simulation technique

are

incorrect

techniques),

name

optimization

selection

an

The

name

here.

Such

modelling) coupled with

path

ascent

or

random

searching. The which

direct

optimization

is useful for optimizing

ation

programming,

and

methods

linear

Dantzig,

Linear

dynamic

programming,

a series of events o r things,

for

transport-

for a l l o c a t i n g s o u r c e s to d e m a n d s

is u s e f u l

programming,

1963).

computer,

which

include

inequalities

programming

(van

usually

der

Veen,1967

requires

the

use

and of

a

b u t there a r e standard optimization programs available.

D y n a m i c P r o g r a m m i n g f o r O p t i m i z i n g Compound P i p e s One o f can

the simplest optimization

normally

gramming.

be used without

the

an optimum

technique

mathematics.

The

diameters

a

length trunk

of

main

along

the

line.

a

trunk The

indeed one which i s dynamic pro-

may

be

used

pipe which

and

number main

problem

way of selecting

s e r i e s of e v e n t s a n d does n o t

pressures

supplying

of

the

on

techniques,

i n fact o n l y a systematic

a

technique

compound

depending

diameters

is

program from

and

recourse to computers,

of

may is

may

to

select

vary

consumers

t h e most economic

in diameter

For

flows.

instance,

from

a

select

along

its

consider

reservoir.

be reduced as drawoff

to

involve any

a

The

takes place

t h e most economic d i a m e t e r

f o r each section of p i p e . A

simple

example

and

the

should point.

head at the

demonstrates

3.2.

the p i p e l i n e in Fig.

the

each d r a w o f f p o i n t

hydraulic

The e l e v a t i o n s

pipe are indicated.

grade of

u s e of

the

technique.

Consider

Two c o n s u m e r s d r a w w a t e r f r o m t h e p i p e l i n e ,

line

i s n o t to d r o p b e l o w 5 m,

drop

below

neither

the p i p e p r o f i l e a t

any

each p o i n t a n d the l e n g t h s of each section of

T h e c o s t o f p i p e i s EO.l

p e r mm d i a m e t e r

per m of

46 pipe.

(In

pressure

this head,

variation). pipe

case

although

it

is is

The a n a l y s i s w i l l

(point A ) .

residual

cost

the

assumed simple

5 m,

take

to

account

arrangement w i l l

point A.

at

independent

The head,

will

we

only

the

such

a

consider

three

be w i t h minimum at p o i n t B may be

H,

a n y t h i n g between 13 m a n d 31 m above the datum, analysis,

of

of

be s t a r t e d at the downstream end of the

The most economic

head i.e.

be

to

b u t to s i m p l i f y the

possible

5

m

corresponding

to

heads

with

increments between them a t p o i n t s B a n d C.

Fig.

3.2

The

P r o f i l e of p i p e l i n e optimized b y dynamic programming

diameter

each o f

D

of

the

pipe

A

between

B,

and

the three allowed heads may be determined from a head loss

c h a r t such as F i g .

2.3

and

i s i n d i c a t e d i n Table 3.1

along w i t h

(1)

the corresponding cost. We

also

will

number

of

3 =

but

9,

regarded. possible

one

In

HC =

the

diameter

Now B,

19

then

this

cost

only

three

hydraulic these 3.1

is

the

possible

grade

lines

heads

between

at B

at

an

adverse g r a d i e n t

(I I) a

set

of

grade

required

The cost of to

of

Table

hydraulic

and

2.3).

consider

possible

line

between

hydraulic for

a

flow

figures

is

6 and

C.

gradient of

from

110 P / s x

must

the

be

added the cost of

(I)).

x

may be dis-

Thus to B

if

H

B

i s 0.006

310 rnm

(from

310 x 1 000

t h i s p i p e l i n e w o u l d be 0.1

i n t h i s case f60 000 (from Table 3.1

so

The

is 3

C

presented f o r each

C

is

point C.

and

=

=

13 and

Fig.

f31000.

p i p e between A a n d

For each possible head

47 TABLE 3.1

I

D y n a m i c p r o g r a m m i n g o p t i m i z a t i o n of

1 1: 1 1 I 23

.004

300 1 6 0 0 0 0

.0065

260

52000

.009

250

50000

111

19

1

-006

I

310

62000

- ._

L

24

29

.0035

.001

340

430

68000

8 3000 __I. 5 1000:'f

86000 79000 165000

I --i

a compound p i p e

48

HC

there

with

an

which

In

i s one

minimum

asterisk.

need

this

be

total

cost

In

example,

the

when

next

pipe

and

for

proceeding

section

the

least

260,

total

cost cost

340

and

tively.

It

which

case

may

section

as

sections;

with

are This

added

000

mm

for

to

one w i t h

proceed the

next

course many would

increased

each s e c t i o n .

The cost o f

booster

station

pump

i t s cost

by

B

to

each

last

and

level.

t h e c o s t s of

optimum

Thus

pipe

at C,

and

the minimum diameters

C and C

-

pipe.

corresponding

economic

standard

could

-

are

D respecin

diameters

b e selected

length could

for

each

be made up

l a r g e r s t a n d a r d diameter

a n d one

b u t w i t h t h e same t o t a l

head

p i p e c o u l d b e c o n s i d e r e d and t h e

considering

more

possible

heads

at

the p i p e s c o u l d be v a r i e d w i t h presssures.

could

be considered a t a n y

a n d c a p i t a l i z e d p o w e r cost

c o m p u t e r may

the

To

the

(I1 I).

of

result.

more s e c t i o n s o f

be

is

and

of

most

pipes

or

D

p o s s i b l e head

3.1

- B,

t h e n e x t smal l e r s t a n d a r d d i a m e t e r ,

Of

costs

the

keep

and

indicated.

s t a n d a r d diameter

the c a l c u l a t i o n s

loss a s the theoretical

accuracy

marked

section

the r e s e r v o i r

each

Table

A

next

gradients

the

and

Sections

desirable

nearest

C

D are

i s done f o r

S151

the

namely

-

selected from is

be

the

of

two

C.

to

C

to

between

D,

hydraulic

Section

section

up

total

310

the

costs f o r

this

arrangement

possible

(It I)

3.1

Table

a n d C,

I t i s t h i s cost a n d t h e c o r r e s p o n d i n g d i a m e t e r s o n l y

recalled

t h e r e i s o n l y one p o s s i b l e h e a d a t

diameters

p i p e between A

of

p r ove useful

if

many

should

point.

be added

in

A

i n w h i c h case the tables.

A

p o s s i b i l i t i e s a r e to b e c o n s i d e r e d ,

and there a r e s t a n d a r d dynamic programming programs a v a i lable. It the

will

b e seen t h a t

at

each step.

describe a p p l i c a t i o n s of

Transportation not

required

programming the

use

of

is a

computer.

technique The

such the

a

resource

least-cost along

system each

is

route

which

technique

of

delivering

t h e least-cost

(1969)

a l l o c a t i o n of resources

another

a l l o c a t i n g t h e y i e l d o f a n u m b e r of that

reduces

t h e t e c h n i q u e to s i m i l a r a n d o t h e r p r o b l e m s .

p r i m a r i l y for consumers

programming

(1969) a n d B u r a s a n d Schweig

Kaliy

T r a n s p o r t a t i o n programming f o r least-cost

does

dynamic

p o s s i b i l i t i e s to be c o n s i d e r e d b y s e l e c t i n g

n u m b e r of

arrangement

t h e t e c h n i q u e of

sources

achieved. should

normally is

to

a

The be

of

use

number cost

of

linearly

49 proportional the

to

technique

the is

throughput

probably

of

along no

that

use

in

route

and

selecting

I t i s of use, however, i n selecting a least-cost

sizes.

through head

an

is

existing

small

in

pipe

distribution

comparison

with

system,

static

for

the

t h i s reason

optimum

pumping p a t t e r n

provided

head,

or

pipe

for

the

friction

obtaining

a

p l a n n i n g g u i d e before demands a r e accurately known.

CONSUMER REQUIREMENT

M

N

10 L l S

15 L I S

SOURCE

Fi9.

An

3.3

YIELD

12 L I S

Least-cost example

a l l o c a t i o n p a t t e r n f o r t r a n s p o r t a t i o n programming

example serves

to

a r e two sources o f

i l l u s t r a t e the

water,

and €3 could d e l i v e r

A and B,

1 2 a n d 20 e / s

10 a n d 15 4 / s respectively. of

pumping along

technique.

I n t h i s example,

a n d two consumers,

and N.

i s a s u r p l u s of water.

routes A - M, A - N,

I3

-

M

A

a n d M and N r e q u i r e

respectively

thus t h e r e

M

there

The cost

and B - N a r e 5,

7, 6

a n d 9 p/1 000 l i t r e s respectively. The

data

(I).

Each

unit

cost

corner

of

are

row of

the

set

out

represents

delivery

in a

along

corresponding

tabular source

form

and

each

route

block

in

for

each

solution column

a

i s indicated i n

the table.

in

Tabel

demand. the

The f i r s t

3.2 The

top

right

step

i s to

50 make

an

that

each

block

arbitrary yield

of

the

satisfies bottom

the f i r s t first

the

d e m a n d of

is

i s 2,

maximum

column

i s completed,

of

M

Starting

possible the

and

resources

the top

allocation is

left

10.

is

written

This in

P r o c e e d i n g to t h e n e x t c o l u m n ,

the

since

in the

the maximum possible a l l o c a t i o n

which satisfies the y i e l d of

So t h e n e x t b l o c k

r o w A.

Proceed

through the

the maximum possible assignment a t each stage u n t i l a l l

a r e allocated

allocation

with

amount

t o b e c o n s i d e r e d i s i n r o w 0 , n a m e l y c o l u m n N. table making

in such a manner

resources

satisfied.

c o r n e r o f b l o c k AM.

column

row

assignment

demand

table,

the

left

initial

and

the

is

13

i f to

(even in

the

the slack

second

row,

column).

then

t,he

Thus

7

the next the

in

third

column.

mi

T r a n s p o r t a t i o n p r o g r a m m i n g o p t i m i z a t i o n o f an a l l o c a i o n

TABLE 3.2

( 1 )

CONSUMER : M

SOURCE Y I E L D

REQUIREMENT 5 :

12

A

10

N

SURPLUS

EVALUAT ON

15

7

NUMBER : 0

2

---J

20

B

7 13

5

EVALUATION NUMBER:

-2

7

( 1 1 ) 0 A

12

B

20

~~

2 10

7

7

-2

the

figures

4 Once

initial

an

methodically would

be

until

the

allocation a

most

least-cost

is

made

d i s t r i b u t i o n emerges.

p r o f i t a b l e arrangement,

assign

a

are

re-arranged

To d e c i d e w h i c h relative evalua-

t i o n n u m b e r t o e a c h r o w and c o l u m n a s f o l l o w s : Assign numbers

the

such

evaluation

value that

number

0

to

row

t h e sum o f is

equal

to

1

and

work

out

the

the row evaluation the

cost

coefficient

other

evaluation

number a n d column for

any

occupied

51 block.

2,

The v a l u e f o r

and

numbers sum

on.

so

if

beneath bigger

is 5,

column M

Now

write

the cost coefficient

than

the cost

sum

and

BM.

The

determined b y

BN,

the

added

allocation

BM,

it

If

this

would p a y to

T h i s i s not easy to see

The

would b e to

amount

which

biggest

and

introduce

can

be

i n fact

an amount

put

in

block

in into

BM

is

( I ) ) . Now f o r each u n i t which i s

i n Table 3.2

AN

is

BM

T h e biggest possible

a closed loop u s i n g occupied blocks as corners

would

and

requirements

to

coefficient.

one u n i t

block

to

and

cost

maximum drawing

to block

yields

the block,

i n t o t h e block.

improvement

(see the dotted c i r c u i t added

of

unoccupied block

i s i n d i c a t e d b y the biggest difference between the

case the o n l y ,

Block

each

f o r row B i s

stems from the method of determining each e v a l u a t i o n

improvement

evaluation our

of

N i s 7,

and column e v a l u a t i o n

the cost coefficients of occupied blocks.

sum from r a t e of

but

column

the row

coefficient

introduce a resource a l l o c a t i o n immediately,

for

sum of

the

subtracted

consistent.

10,

maximum r e - d i s t r i b u t i o n

have to be subtracted

since

this

AM

block

this

would

10 i s made,

i.e.

from

In

case

from block to

keep

the

evacuate

AM.

block

and the amount

the

maximum The

i n the block

a t each corner of the closed loop adjusted by 10 to s a t i s f y y i e l d s a n d requirements.

Only one r e - d i s t r i b u t i o n

of

resources should

be done at

a time. After making number to

and

the

until the

most

fact

is

that in

best

new

sums

no

distribution

in

in

and

further

there

any

allocation, as

p r o f i t a b l e block

there

ficient

the

evaluation

repeat

possible

two

In

our

steps,

more

the e v a l u a t i o n

Allocate

improvement,

sum g r e a t e r

example

but

(11).

the r e - d i s t r i b u t i o n

cost

i s no e v a l u a t i o n

block.

re-calculate

Table 3.2

we

than

arrived

compl icated

resource procedure

indicated the cost

at

the

patterns

by

coef-

optimum involving

more sources and consumers may need many more attempts. The example can o n l y serve to introduce the subject of t r a n s p o r t a tion

programming.

with

in

niques

textbooks such

as

There on

the

subject

der

Veen

serves

in

happened

table

as

an to

blocks could

be allocated

Computat ions

then

at the end.

many

Van

example o n l y the

are

of

other

a

proceed as

and

evacuated

very

which

are

dealt

mathematical optimization

(1967)

introduction. be

conditions

small

before and

Dantzig

For

(1963)

instance,

if

simultaneously,

quantity

two one

denoted by

the q u a n t i t y

'e'

techn-

and

this

blocks of

'e'

the say.

disregarded

52 L i n e a r Programming f o r Design of Least-Cost Open Networks

i s one of

L i n e a r programming niques.

The

systems,

use

of

although

a

the

computer simple

The technique may o n l y is

linear,

cannot

so

be

closed

loops to

design

each

branch

loss,

diameter

used

in the

to

resort

known.

render

the

of

p i p e of

different

or

Since

cost

are

system

pre-selected

given

design

pipe

relationships the

is

tech-

complex

by

hand.

programming

networks

For each

where

with

can

be

the flow

in

It

between

following

flow,

head

technique

the

treated as the v a r i a b l e .

length of

will

indicate

that

some

thereby

i n effect e l i m i n a t i n g them.

can

treated

be

pressure

at

in

the

certain

I inear

inequality

certain

range.

points

of

The

analysis. in

diameter

pipes

The head losses

h a v e zero

Any other type of I t may

length o f

lengths,

linear constraint

above a f i x e d

the greater-than-or-equal-to-type)

total

The

be r e q u i r e d to m a i n t a i n

the network

p i p e of

a

each

and costs a r e l i n e a r l y p r o p o r t i o n a l to the respective p i p e lengths. program

is

b r a n c h o r main p i p e ,

diameters a r e allowed and

diameter

for

done

Linear

of

networks

non-l inear,

linear:

is

approximations.

tree-like the

essential here

application.

successive

to

mains

normally

optimization

the r e l a t i o n s h i p between v a r i a b l e s

restricted

and

number

if

optimizing

trunk

is

be used

is

without

used

is

example

for

it

used

the most powerful

minimum

or

within

the (a a

a c e r t a i n diameter may be

r e s t r i c t e d because there i s i n s u f f i c i e n t p i p e a v a i l a b l e .

DIAMETER m m UNKNOWN L E N t T H m OPTIMUM L E N G T H m HEAD LOSS m

Fig.

3.4

Least-cost

250

200

2 OD

150

X,

x3

50

x2 I50

XL L 00

012

3.2

t r u n k main b y

The example concerns

a

trunk

0 0

1 - 6 6 TOTAL 5 . 0

l i n e a r programming

main w i t h

two

drawoff

points

(Fig.

53 3.4). and

The p e r m i s s i b l e diameters of of

the

second

v a r i a b I es , X,, ferent

X2,

diameters.

comparison but

loss,

linear

200

and

and X4

This

the

of

leg,

X3

simple

costs

of

the f i r s t 150

leg a r e 250 a n d 200 mm,

alternatives

is

used

here

are

lengths of

example could

all

programming

There

mm.

which are the

thus

be optimized b y giving to

four

pipe of

the

manual

correct

demonstrate

dif-

the

head tech-

nique. losses per 100 m of p i p e a n d costs p e r m for the v a r i o u s

The head

pipes a r e i n d i c a t e d below:-

Head loss @ 40 e / ’ s m/1OO m

D iameter mm

250 200 150 linear

below

5

(I).

Table 3.3

0.1 0.42

constraints

and

the

on

the

coefficients

system

are

the

of

4 3 expressed

equations

x1

Objective Function: The computations

a, at

b

x4

= 4

+ 0.71X2 + 0.1X3 + 0.42X4 5x1 + 4x2 + 4x3 + 3xq

= 5

proceed

i s necessary to

and

very

to

satisfy c

in

the

Table

by

introduce

setting

indicate

the by

u n i t o f X, To

amount

of

introducing

real

slack

i n the program b y

to zero,

variables variables

into

each

The slack

v a r i a b l e s a r e designated

and

cost

(I),

particular the one

To

whether

it

is

are

set

the slack

4 and 5 respective-

(I)),

l i n e of

of

coefficients

i n i t i a t e the solution,

the values 5,

program unit

their

the

variable the

worthwhile

any other v a r i a b l e ,

main body of the t a b l e which

column

would displace 1 u n i t of a and 0.25

determine

= m i n i mum

equality.

l y (see the t h i r d column of Table 3.3

placed

+

3.3

b a n d c a r e assigned

The numbers i n any

all

artificial

h i g h values designated m.

variables a,

in

5

X2

+

x3

equation

equation

tabulated

0.25X1

Head L o s s :

it

are

in

Lengths a r e expressed i n h u n d r e d metres.

Lengths

so

cost i~100/100 m

m/1OO m

0.25 0.71

The form

@ 14 e/s

would

variable.

be Thus

disone

u n i t s of c. replacing

any

variable

a number known as the oppor-

54 T A B L E 3.3

P-qg.

L i n e a r P r o g r a m m i n g S o l u t i o n of P i p e P r o b l e m s

co>t

V c r able _ C o e_ ff.

Avo unt _ _

a

m

5

b

m

4

C

m

5

OPPORTUNI T Y

:l

x2 4

x3 4

x4 3

0.25

0.71

0.10

0.42

2

x3 4

x4 3

a

b

c

Red.

m

m

m

s

1

5/.71

5

b

m

4

C

m

1.45

i

x1

a

5

4

1

1

I l l

x2 b

4

5

2

m

0.55

x4

3

3.45

5

4

1

1

1

0.1

0.42-

x3 4

I

l 4

1 1 .71m-4

0

0

a

b

c

m

m

m

x4 3

j

1

0 3.28-0.76m

1.69 1

-1.69

0

1.1-0.69m

0.5"

-2.38 2.38 3.38-8.2

IV

x4

4

4.5

5

0.5

3

-0.69

0

2.16

1.52

0.69

1

4

0

1

1

0.31

0

( N o f u r t h e r improvement posc iD l e 1

-2.16 I

m-

3.45'

5

1

0.24

-1.1

b

-0.71

0.76

1.I

11-1.lm

1

4 - 1 . l m 3-1.42m

0

x1

a

1

-0.46 lt0.46

x2 x1

,

dALUE:

4

x2

!-

rn-

m-

55 tunity

number

i s calculated

introduced,

then

-

x

m

)

(0.25

the

cost

m)),

the

opportunity

value

entries

that

of

the

total

program

thus

for

from

column.

each

column

by

in

the

the

the

(5 -

column

coefficient

coefficients

subtracting

the column

introduce would be X2, unit

X2

The is

column

that

is

which

may

per

the

lowest

since i t shows

( o r negative o p p o r t u n i t y

designated

the

key

opportunity

The

column. value

the

variable.

value).

key

column

( i n the cost

mini-

Only one v a r i a b l e may be introduced a t a time.

determine

which

now

shows

mization case). To

reduction

i.e.

multiplying

the

cost

(0 x

value,

by

and

of

1

-

m)

cost

The most p r o f i t a b l e v a r i a b l e t o greatest

x

calculated

corresponding

second

cost

(1

the o p p o r t u n i t y

is

X

I f one u n i t of

increase b y

i s designated

column

variable

formed

each

would

which

the

in

for

be

the

maximum

introduced,

amount

calculate

the

of

the

key

column

replacement

ratios

variable for

each

row as follows:Divide

the

corresponding ment be

ratio

is

amount number

lowest

the

number

selected

positive at

the

program

variable

i n the key column.

introduced without

the

the

of

as

that

is

the

v i o l a t i n g any

replacement

ratio of

each

maximum amount

is

the

designated

key

row

by

the

lowest p o s i t i v e replace-

the constraints.

of

intersection

The

for

column

which

could

The row

the key and key

with

row a n d the

row,

key number. After

introducing

( T a b l e 3.3

(11))

so

a

new

variable,

that

the

replacement

program v a r i a b l e and by

the

new

the

and

its

cost

ratios

is

remain

rearranged correct.

The

in the key row a r e replaced

i t s cost coefficient

variable

matrix

coefficient.

The

amount

column

as well as the body of t h e t a b l e a r e r e v i s e d as follows:Each number i n the key row i s d i v i d e d b y the key number. From each

number

number

i n the key

in

key

the

The ratios

procedure and

negative all

a

non-key

row,

row m u l t i p l i e d b y

column

g i v e n as T a b l e 3.3

in

divided

by

the

the

key

subtract

the

corresponding

r a t i o of the o l d row number number.

new

The

tableau

is

(11).

of

revising

opportunity

studying the

table

value.

positive opportunity

opportunity is

In

values

repeated the

so

values until

example

the

there

Table

least-cost

and is

3.3

solution

replacement no

further

(IV)

i s at

shows hand

56 (indicated

by

the

current

program v a r i a b l e s

and

t h e i r corresponding

values). The ming

reader

(e.9.

cription

should

Van

of

der

the

r e f e r to a s t a n d a r d textbook on

(1967) a n d D a n t z i g

Veen,

technique.

There

are

l i n e a r program-

(1963)) f o r

many

other

a

cases

full

des-

which

can

type

and

be mentioned below:the

If

constraints

not

just

are

introduced

are

equations,

equations.

into

The

the 5 (less-than-or-equal-to)

of

slack the

variables

1.h.s.

artificial

of

slack

with

each

zero

cost

constraint

variables

with

coefficients

to make

high

them

cost

coef-

f i c i e n t s a r e then omitted.

If

the

constraints

of

are

introduce

artificial

slack

into

the

1.h.s.

the

with

zero

of

cost

the 2

(greater-than-or-equal-to)

variables

constraint

coefficients

from

high

with

and

cost

subtract

each

slack

inequality

type,

coefficients variables

to

make

them

equations. If

the

objective

value

with

function

function

the to

is

highest be

is

to

be

negative

maximized,

minimized,

value

the

is

the

opportunity

selected,

opportunity

if

the

with

the

but

value

h i g h e s t p o s i t i v e v a l u e i s selected. The

opportunity

ponding

variables

one u n i t of If

two

the when very

the

that

i.e.

amount the

of

represent they

shadow

indicate

the

values

of

the

corres-

value

of

introducing

row

i s selected,

v a r i a b l e i n t o the program.

replacement

ratios

program

matrix

is

a r e equal,

variable

in

rearranged.

whichever the

other

Merely

row

will

assume

it

to

first

the

be

zero

have

a

s m a l l v a l u e a n d proceed as before.

example then

values

of

pipe

programming

two

stage

diameters

and

search

optimization, is

given

methods

in are

namely

Stephenson also

Lam (1973) described a g r a d i e n t o p t i m i z a t i o n

(1964).

discussed. method.

layout

and

Non-linear

For

instance

57 REFERENCES Buras, N. and S c h w e i g , Z., 1969. A q u e d u c t r o u t e o p t i m i z a t i o n b y d y n a m i c p r o g r a m m i n g . Proc. Am. SOC. C i v i l E n g r s . 95 ( H Y 5 ) . Cross, H., 1936. A n a l y s i s o f f l o w i n n e t w o r k s of c o n d u i t s o r c o n d u c t o r s . U n i v e r s i t y of I I I i n o i s B u l l e t in 286. D a n t z i g , G.B., 1963. L i n e a r P r o g r a m m i n g and E x t e n s i o n s , P r i n c e t o n U n i v e r s i t y Press, Princeton. Isaacs, L.T. a n d M i l l s , K.G., 1980. L i n e a r t h e o r y m e t h o d s f o r p i p e n e t w o r k s a n a l y s i s . P r o c . Am. SOC. C i v i l E n g r s . 106 ( H Y 7 ) . Kal l y , E., 1969. P i p e l i n e p l a n n i n g b y d y n a m i c computer programm i n g . J . Am. W a t e r W o r k s Assn., (3). Lam, C.F., 1973. D i s c r e t e g r a d i e n t o p t i m i z a t i o n of w a t e r systems. P r o c . Am. SOC. C i v i l E n g r s . , 99 ( H Y 6 ) . S t e p h e n s o n , D., 1984. P i p e f l o w A n a l y s i s , E l s e v i e r , 204 pp. V a n d e r Veen, B., 1967. I n t r o d u c t i o n t o t h e T h e o r y of O p e r a t i o n a l R e s e a r c h . C l e a v e r Hurne, L o n d o n . Wood, D.J. and C h a r l e s , O.A., 1972. N e t w o r k a n a l y s i s u s i n g l i n e a r t h e o r y . ? r o c . A X E , 98 (HY7) p 1157 - 1170.

L I S T O F SYMBOLS

C

-

cost

D

-

diameter

h

-

head loss

H

-

head

K

-

const a n t

e

-

Iengt h

Q

-

flow

58

CHAPTER 4

AND SURGE

WATER HAMMER

R I G I D WATER COLUMN SURGE THEORY

Transient are

often

pressures

the

cause

caused

by

bursts.

of

a

change of

The

r a t e in conduits

flow

pressure fluctuations

associated

w i t h s u d d e n f l o w s t o p p a g e c a n b e s e v e r a l h u n d r e d metres head. Transients

in

gories:

Slow

surge,

whereas

of

the

or

fluid

small

closed

motion

rapid

and

conduits are normally

mass

in

change

conduit

changes

is

flow

in

of

oscillation flow

referred

rate

or

the

classed

into

two cate-

fluid

is

referred

accompanied

by

elastic

to

as w a t e r

pressure

hammer.

the

two

to

a

strain

For

slow

theories

yield

t h e same r e s u l ts. It

i s normally

(whenever

the

r i g i d column pressible the

easier

theory

theory

mass.

column

relating

A

umq

in

the

an

head

system b y

applicable)

than

difference

instantaneous

conduit

i s d e r i v e d from Newton's b a s i c

to

r i g i d column elastic

applied

the

theory

theory.

With

i s t r e a t e d a s a n incomacross

acceleration.

d i f f e r e n c e between

bore

by

in the c o n d u i t

pressure

uniform

a

is

the w a t e r

produces

tion

to a n a l y s e a

t h e e n d s of rate

the

The

the

in

of

equa-

water

change

of

ends

basic

col-

velocity

a n d is

l a w o f motion,

(4.1 where

is

h

conduit

tion and t The with

a

may

the

is

head

in

flow

useful

deceleration

the

or

is

between

velocity,

g

ends,

L i s the

gravitational

accelera-

the is

two

i s time.

water

starting

hydro-electric

tion

difference v

equation

slow

lating trip

the

length,

be

graphical ly.

of

level up

in

for

a

in

column.

in

variations a

pumping

installation

solved

calculating

water

steps

The f o l l o w i n g

of s o l u t i o n o f t h e e q u a t i o n : -

fed

by

of

At

the

a

surge

line,

or

a

head

I t may

shaft power

pressure

associated

in

calcu-

following load

pipeline.

b y computer,

e x a m p l e demonstrates

rise

be used f o r

power

changes The

tabular

the numerical

in

equa-

form o r method

59 Example

A

100 m l o n g penstock w i t h a c r o s s s e c t i o n a l a r e a ,

is protected against bine,

with

a

water

cross

unrestricted orifice. and

there

Calculate

is

a

the

hammer b y

sectional The

sudden

rise

load water

of

2

and

m2

an

i n t h e c o n d u i t i s 1 m/s

velocity

in

o f 1 m2

surge s h a f t a t the tur-

A2,

area,

initial complete

maximum

a

A,,

rejection

at

level

the

in

the t u r b i n e . surge

shaft

neglecting friction. Take -0.098h.

At

=

1 sec.

Then from Equ.

By c o n t i n u i t y ,

0.1 1-2 2-3 3-4 4-5 5-6 6-7

h

0.5 0.476 0.428 0.359 0.272 0.172 0.064

or

method

Av

Another water

column

the

and

of

a

cause

be the

Following

pumping

line,

vaporization

column

r i g i d column

may

stop

water

4.16,

o f 2.26

m.

taking

The a c c u r a c y o f t h e

smaller

time

intervals A h

include

a p p l i c a t i o n of t h e r i g i d w a t e r column e q u a t i o n i s w i t h

water

distance

pumps the

to

the

4.1

0.951 0.855 0.717 0.544 0.345 0.129 0.094

The method c a n r e a d i i y b e e x t e n d e d to

separation.

end

sufficiently

slowly

V

a n d i s calculator-orientated.

useful

upstream

cases

= 0.5~.

w h i c h may b e compared w i t h t h e a n a l y -

c o u l d be i m p r o v e d b y

respectively.

h e a d losses,

Equ.

m,

100 =

t h e mean v a n d h o v e r t h e t i m e i n t e r v a l s to c a l c u l a t e

taking

and

= -9.8h/

-0.049 -0.096 -0.138 -0.173 -0.199 -0.216 -0.223

s o l u t i o n o b t a i n e d from Equ.

numerical

= -ghAt/L

A v = -0.098h

0.5 0.976 1.404 1.763 2.035 2.207 2.271"

The maximum r i s e i s 2.27 tical

Av

A h = A1vAt/A2 = l v / 2

Ah = 0 . 5 ~

t

4.1,

beyond theory

water

column

column o f

i n i t i a l flow velocity.

the

stopping

of

the

pressure

frequently

peaks

along

vapour

the

with

will

length t w i l l

of

the

pump

time

before the 0

the

In

such

decelerate analysis.

to d e t e r m i n e

stopping.

vapour

be Q = A t v * / 2 9 h

t

at

drops

line. will

accurate for

r e s p e c t to

travel

volume

a

pocket

is s u f f i c i e n t l y

integrated twice

instantaneously

at

the

If

the

pocket

behind

where v

i s the

0

60 MECHAN I CS O F WATER HAMMER Water

hammer

occurs,

as

would

indicate an

name

implies,

(or accelerated).

water i s r a p i d l y decelerated theory

the

infinitely

The

a

column

r i g i d water

l a r g e head r i s e

p i p e l i n e c a r r y i n g l i q u i d were slammed s h u t however,

when

if

a

a c e r t a i n amount o f r e l i e f u n d e r these c o n d i t i o n s ,

e l a s t i c i t y of the f l u i d a n d of

the p i p e

d i s c h a r g e end of

is

a

pipeline

will

pack

against

the

will

rise

sufficiently

gate, to

The amount o f

the

water

of

the

causing

stop

momentum l a w . amount

shut

the

itself.

a

fluid

in

rise.

stopped p e r u n i t

to

the

required

replace

There

a is,

due

to

the

at

the

of

the

The

accordance

water

in

valve

upstream

pressure

liquid

if a

Thus

column

valve

instantaneously.

of

gate

pressure with

the

time depends o n

volume

by

created

the

compression of w a t e r a n d e x p a n s i o n o f t h e p i p e . I t can thus

be shown

that

the r e d u c t i o n in v e l o c i t y

Ah,

front

the

relationship

between

the

head r i s e

a n d the r a t e of p r o g r e s s of

A V

the

i s A h = -CAV/g

(4.2a)

where c i s t h e w a v e c e l e r i t y a n d g i s g r a v i t a t i o n a l

acceleration.

equation

It

is

often

wave

referred

to a s J o u k o w s k y ’ s

law.

can

This

further

be

p r o v e d from a mass b a l a n c e t h a t

c

=

1/ If

y,)

3 .iw

kd EY the p i p e h a s a

(4.3a) rigid

and r i g i d side f i l l

( e l a s t i c m o d u l u s E2 a n d

lining

( m o d u l u s Es)

thickness

t h e e q u a t i o n to use i s

(4.3b)

where w

i s the u n i t

p i p e diameter, k

a

factor

about 0.9).

w e i g h t of

or

a s ZOO

which

depends

on

the

y the p i p e w a l l

end

fixity

when

for

free a i r

thin wall i s present

of

for

steel p i p e s .

the

a

d

pipe

51,

the

(normally

r i g i d walled

I n t h e case o f

(see C h a p t e r

is

thickness a n d

c may

tunnel plastic

be a s

low

m/s.

The earlier,

pressure thus

illustrated reservoir the

i s i t s b u l k modulus,

K

c may b e a s h i g h a s 1370 m/s

o r as low a s 850 m/s

pipe,

liquid,

E i s i t s e l a s t i c modulus,

in

end,

reservoir,

wave

travels Fig. the so

caused

upstream, 4.1(1).

pressure that

the

by

valve

closure

superimposed on

When in

the

the

wave

the

front

the

pipe

forces

velocity

now

reverses

b a c k to s t a t i c ( r e s e r v o i r ) p r e s s u r e a g a i n .

referred

static

reaches

water and

head, the

backwards

to as

open into

pressure drops

61 PRE5WRE NAVE

4.1

Fig.

A

negative

That in

wave

the

The

entire

sucks

up

the

water

+v

and

(2)

wave

front

negative

back

to

Water hammer wave at d i f f e r e n t stages

will

in

thus

is

pipeline

-v

wave

of

head

pipe

to

the

into the

the head

w i l l repeat i t s e l f

travels

t u r n reach

pipe,

downstream from

the closed end.

where

0

v

amplitude

reservoir.

so

reverts

that to

was

0

Cv,/g Upon

the

static

the

below

velocity

the velocity

original

reaching

head.

the r e s e r v o i r .

Now

velocity.

static, the

travels

reservoir

in the p i p e r e v e r t s

The sequence of

i n d e f i n i t e l y unless damped b y f r i c t i o n .

---+---1

Hz g

‘with

0

no

friction

2 L/c

I

Fig. 4.2

Head f l u c t u a t i o n s at v a l v e end.

3 L/c

it

4L7c

I

t

waves

62 The 4.2.

variation

In

the

case

in

head

of

pumping

conditions

i s normally

stream

the

of

reduction charge wave

in

pump

(Fig.

returns

the v a l v e lines,

associated is

pressure.

end

at

The

4.3),

towards

be as

decelerated,

the

pumps.

wave

trip.

change

resulting

in

a

towards

reverses

pressure

i n flow

The water

travels

velocity

The

indicated i n Fig.

most violent

a pump

negative

where

the

the

with

suddenly

will

at

and

the

downsudden

the

a

dis-

positive

pump

alter-

nates from a pressure drop to a pressure r i s e .

Wave front

snit i d l y

Vapour pressure Initial V

F i g . 4.3

The tion,

Water hammer head drop a f t e r pump t r i p

wave

via

in

stages

in

air

valves

be

seen it

water

line friction, flow

will

Fig. up

behind

4.4

which

wave

are

Thus

and

air

in

cross

if

there

will

be

is

f r i c t i o n can

illustrates

wave

a

a rapidly

not

quite

shut

at

seca

drawn

The effect of

the p i p e l i n e from the

changes

variation. vaporize

and from solution.

travels

heads

by

gradual

the

from

most

different

valve.

horizontal,

due

The to

' p a c k i n g ' effect causing some flow across the wave f r o n t . The

effect of

one

equation.

tion

changes i n section The

2Ah'A./c. I

= A1/c,

2

wave

or b r a n c h pipes can be

head change

Ah'

i s r e l a t e d to the o r i g i n a l head change Ah'

Ah'

I

or

pressure,

as

pressure the

i s complicated

vaporization,

negative

readily

= Vo

Ah

/(1

+

after

r e a c h i n g a junc-

by the equation

I

A 2'=2

+

A3/C3 +

+ A 2 / A 1 + A3/A1

+

included

i4.4a)

...

.. .

where A . i s the cross sectional area of p i p e i and c

(4.4b) 1

. c2

c3-

63

~

1

3 "Distance from valve" 2

Xz- F 'x

cvo 9

F i g . 4.4

Head a t p o i n t s a l o n g along p i p e l i n e w i t h f r i c t i o n , instantaneous stoppage

When a

valve

i n a g r a v i t y main i s closed g r a d u a l l y ,

analogous to a series of system

could

form

of

be

analysed

analysis

simultaneous

to

the

is

solution

hammer equation. lar

minute waves emanating

of

the

In fact

so-called

numerical ly

useful

for

or

graphically.

the sines

discharge

the effect i s

from the v a l v e .

demonstrating

valve

for

The

The g r a p h i c a l

the

principles

equation

and

of

the water

d r a w n on a g r a p h a r e v e r y simi-

characteristic

lines

adopted

in

numerical

a

pipeline

solu-

t ions.

4.5

Fig. valve at to

4

illustrates

a

L/c

and

degrees of

the

valve

analysis

The v a l v e

discharge

compute

closure

one

a n d S.

Thus

the

applies

head the

of

i s closed over

characteristics

closure a r e p l o t t e d on the g r a p h .

head loss to p i p e velocity

To

graphical

the discharge end.

for

with

a

a p e r i o d equal four

different

The l i n e r e l a t i n g f r i c t i o n

i s also indicated. at water

the

valve

hammer

at

time

equation

L/c 4.2

after

initiating

between p o i n t s R

64 Av - A h

-(c/g)

A h

where

i s t h e d i f f e r e n c e i n f r c t i o n h e a d b e t w e e n R a n d S.

Ahf

Computations

at

+

+ (c/g) Av

the

Similarly

t h e w a t e r hammer e q u a t i o n f r o m S t o R ,

applying A h

(4.2b)

f

waves point

thus

hf

(4.2~)

proceed a l o n g

peak a n d

S occurs

then

at

t

l i n e s of

die out

4

=

L/c

slope

+

or

d u e to f r i c t i o n . and

is

- c/g.

Ultimately

The m a x i m u m h e a d

182 m e t r e s a c c o r d i n g

to t h e

4.5.

computations on F i g .

ELAST I C WATER HAMMER THEORY

The

fundamental

velocity

a

in

differential

conduit

may

wave

equations

b e d e r i v e d from

relating

pressure

consideration of

l a w o f m o t i o n a n d t h e c o n s e r v a t i o n of mass r e s p e c t i v e l y ,

to

Newton's

and are:

(4.5) ah + -c 2 - =a vo at

ax

g

The

last

t o obey tion

term

of

a

strictly

steady-state

correct.

are higher normal

Tests

accounts for

with

friction

which

a constant f r i c t i o n factor.

friction indicate

factor that

for

head

transient

i s assumed The assump-

conditions

i s not

losses d u r i n g t r a n s i e n t

flow

t h a n those p r e d i c t e d u s i n g t h e f r i c t i o n f a c t o r a p p l i c a b l e to

flow

reversals

4.5

in Equ.

Darcy's equation

conditions.

Energy

when t h e v e l o c i t y

is

probably

absorbed

during

flow

i s low and t h e f r i c t i o n f a c t o r c o n s e q u e n t l y

relatively high.

Methods of A n a l y s i s

A

common

pressures a chart for zero

no

method

used for

over

analysis

of

pipe

(Lupton,

systems

1953).

for

water

hammer

4.6

i s such

Fig.

the maximum a n d minimum heads a t the downstream v a l v e

line friction.

constant.

of

to be g r a p h i c a l l y

time

T

The v a l v e and

the

To u s e t h e c h a r t

area

valve

i s assumed discharge

to r e d u c e

coefficient

linearly is

to

assumed

c a l c u l a t e t h e v a l v e c l o s u r e p a r a m e t e r cT/L

65

H ( m 1 20c

6'

- 200

::m 55

R 5

S'cVALVE

SHUT

_/--

-. /---

R 2

RI

55 23

t'

51

t d t = L/c

_/-_ / - -

R, -R

F i g . 4.5

Graphical

x -t

CHART

water hammer a n a l y s i s f o r

closure with f r i c t i o n

S

' 0

slow valve

66 and

valve

axis

head

parameter

loss

the maximum

head parameter -h/(cvo/g) answers

are

the

below s t a t i c

of

lines).

Multiply

head.

Note t h a t

the c h a r t

is for

sluice

water

and

hammer

butterfly

valves

the e n d of heads

T h e most economical for

usually

are

the

are

such

the

graphical

in

intervals. set

than

to

in

finite

pipeline at

time

form

The x

By

expressing

of

the

flow

the result t h a t

by

A

the c h a r t .

important I ines.

computer.

(Streeter

Solution

and Lai,

is

1963 a n d

water and

hammer

solved

for

equations

are

successive

time

i n t e r v a l s a n d At

is

- t g r i d on which s o l u t i o n takes p l a c e i s

Starting

from

known

conditions

along

the

one proceeds to c a l c u l a t e the h e a d a n d velocity a t

each p o i n t a l o n g the l i n e at time t

adding,

most

with

i s d i v i d e d i n t o a number of

4.7. t,

in a r e a

t h e w a t e r hammer e q u a -

digital

differential

difference

ax/c. Fig.

by

a n d the

above a n d

The c h a r a c t e r i s -

predicted

solution of

is

characteristics

The

The c o n d u i t

equal

cvo/g

1950) w h i c h d i f f e r s l i t t l e i n p r i n c i p l e f r o m t h e o l d

method.

expressed

systems

method o f

Streeter a n d Wylie,

depicted

by

that

valve stroke,

higher

method of

particular

by

the vertical

linear reduction

more a c c u r a t e a n a l y s i s i s t h e r e f o r e n e c e s s a r y f o r

tions

off on

( f u l l lines) a n d minimum

maximum a n d minimum heads respectively

reduction occurs a t the

(dashed

Read

/g)

a c o n d i t i o n r a r e l y encountered in p r a c t i c e .

w i t h time, tics

he/(cvo/g).

head parameter h ' / ( c v

equations

4.5

+

and

At. 4.6

as

one gets two simultaneous equations

total

differentials

and

i n v o l v i n g dh a n d d v :

(4.7b) Equs. time

4.7a

t

+

At

and

4.7b

may

i n t e r m s of

be

known

and v ' at point p at P P h a n d v a t two other points q a n d r

solved

for

h'

at time t : (4.8a)

(4.8b)

67 Valve closure factor C T/L

F i g . 4.6

At

Maximum a n d minimum head a t downstream v a l v e f o r l i n e a r c Io s u r e

the terminal

either

h

speed.

The

is

fixed, correct

the known condition computat ions

points, v

or

Equ.

a n additional condition i s usually

imposed;

is

or

4.7a

a

function or

4.7b

of

t o e v a l u a t e t h e new

commence

at

known

a

gate

solved

is

opening

simultaneously

h a n d v at time t

conditions

and

are

pump with

+ A t . The

terminated

when

the pressure f l u c t u a t i o n s a r e s u f f i c i e n t l y damped b y f r i c t i o n . Where t h e n Equs.

a

branch 4.8a

pipe

a n d 4.8b

s occurs or

there

is a

change

snoulci b e r e p l a c e d b y E q u s . 4.9

i n diameter,

a n o 4.10:

68

TIME

A

/ Fig.

x - t G r i d for water method.

4.7

hammer a n a l y s i s b y c h a r a c t e r i s t i c

Effect of F r i c t i o n Fluid along

friction

the

gradually

damps

If

conduit. die

away

the

there

and

water is

the

no

hammer

exciting

waves

as

influence

pressure along

they

the

the conduit

travel

waves will

will

tend

to

s t a t i c pressure. The

characteristic

dicts

the

wave.

At

method the

of

wave

water

sharp

friction

wave

provided

pumping

over-pressures friction

head

there

computer accurately is

no discontinuity

i s necessary an analytical

(1950)

demonstrated

travelling stoppage

xxv

cates

it

by

back is

solution

along

in

the

t o r e s o r t to some o t h e r

that

indicated

pre-

a by

is feasible

at

t h e a m p l i t u d e of line the

with

a

friction

hyperbolic func-

4.4): tanh

-

In

solution

Fortunately

Ludwig

instantaneous

(see F i g .

of

wave front,

analysis. front.

hammer

fol lowing tion

of

effect a

method

lines at

is

minimum

following

the greater

and

cv

--./p)] 4

( 4gd '

pump than

maximum

sudden

will

(4.11) pump

exceed

the

approximately head

stopping

0.7cv / g .

envelopes

the

pumping 0

along

maximum

head

Fig.

if

4.8

pipelines

the indiwith

69 various stream plot

friction end.

heads

To

use

following

the

multiply

the maximum a n d minimum

factor

0

when

chart

stoppage

ordinates

is

should

only

valid

provided

at

no

point

fall

It

has,

however,

heads

with

column separation

profile.

at

by

the

up-

cv / g

and

head envelopes f o r the correct f r i c t i o n

there

is

no

column

the negative wave t r a v e l s up the p i p e l i n e i.e.

envelope

mum

the

above o r below s t a t i c on a p i p e p r o f i l e d r a w i n g .

hf/(cv / g ) ,

The

instantaneous

chart,

10

below

the minimum head below

m

separation

the

been found from experience that a r e often s i m i l a r

to

pipeline

the maxi-

those without

separation.

4.8 may also be used to determine the maximum and minimum

Fig.

heads along the

pipelines w i th fri cti o n

downstream

end b y

following

a closing valve.

sudden flow

Turn

the c h a r t

stoppage a t upside down

a n d r e a d o f f maximum envelope instead of minimum a n d vice-

versa.

PROTECT I ON OF PUMP I NG L I NES (Stephenson, 1972)

The

pressure

transients

fol lowing

power

failure

to

electric

motor

d r i v e n pumps a r e u s u a l l y the most extreme t h a t a pumping system w i l l experience. should often

Nevertheless,

also

be

the over-pressures

checked.

caused b y s t a r t i n g pumps

Pumps

with

steep

induce h i g h over-pressures

when

t h e power i s switched on.

i s because the flow so

a wave

By

partly

with

a

i s small

( o r zero)

head equal

closing

the

to

pump

when

head/flow

the pump

the closed v a l v e

delivery

characteristics

valves

This

i s switched on

head i s generated.

during

starting,

the

over-pressures can be reduced. The

over-pressures

caused b y

closing

l i ne valves or

scour

valves

should also be considered. the

If

denly,

pumps

the

flow

close

to

the

water

column

atmospheric

i s vapour thus

occur

supplying will

pressure.

at

r e t u r n e d as a

peaks

If

the

grade

line,

the

pressure

cause

pressure.

unprotected p i p e l i n e a r e stopped sud-

a l s o stop.

hydraulic may

an

The

lowest

pipeline

the

value

to

profile is relatively

sudden drop

deceleration

to

to which

a

value

of

the

less than

pressure could drop

Vaporization or even water column separation may along

p o s i t i v e wave

the

pipeline.

When

the

the water columns w i l l

to water hammer over-pressures.

pressure wave

is

rejoin g i v i n g rise

70

Fig.

4.8

M a x i m u m and m i n i m u m h e a d e n v e l o p e s f o l l o w i n g instantaneous pump stopping in pipelines w i t h friction

71 Unless pumping

some

method

pipeline

of

system

water will

hammer

normally

water hammer overhead equal to cv /g.

comparison

most

lines

with

where

the

economic

pumping

solution,

i n s t a l l e d i t may

water

hammer

head.

and

have

to

I n fact

0

high-pressure

protection be

is

designed

for

a a

t h i s i s often done w i t h

heads

may

be

small

this m a y

For

short

lines

if

water

hammer

even

installed,

in

be the

protection

is

be prudent to check that the u l t i m a t e s t r e n g t h of t h e

p i p e l i n e i s s u f f i c i e n t should the p r o t e c t i v e device f a i l . The

philosophy

against

water

reduce The

the

surge

is

is

the

design

similar.

of

The

most

methods of

object

in

downsurge i n the p i p e l i n e caused b y

upsurge

entirely

behind

hammer

will

then

eliminated. to

feed

be correspondingly

The

water

most into

common

the

pipe

most

stopping

reduced,

method

of

or

protection

cases

to

the pumps.

may

limiting

is

even

the

be

down-

as soon as the pressure tends

to drop.

Fig.

Pipeline p ro fi l e i I lustrat ing suitable devices f o r water hammer protection.

4.9

Suitable i n Fig.

4.9.

locations Most of

for

various

the systems

protective

locat ions f o r

devices

are

various

i I lustrated

i n v o l v e feeding water i n t o the pipe.

Observe

that

column

beyond

phenomenon of

the

gather pump

If

to

flux

in

the

The

valve

of

would

and

flow

then

allow

the

into

stopped

no

by

would

the

the

water

energy,

energy,

the entrance

between

against

however,

potential

kinetic

be

Part

column

impact

consequently

may

device a t

heads

in

water

If,

maximum

and

hammer

The w a t e r c o l u m n g r a d u a l -

direction

minimum

water

water

i s converted

difference the

of

the

phenomenon.

column

decelerate

which

there

will

water

hammer

installing

be

a

re-

the discharge t a n k

to

A small o r i f i c e b y p a s s to the re-

the pipeline.

in

water

the

elastic

surge

over-pressures.

point

reverse

or

to

momentum

throttling

vessel,

of

reverse

its

point

the

e l a s t i c energy.

hammer

in

so

motion

the

effect

the

at

change

of

allowed

water

with

v a l v e or

air

the

was

slow

energy

instead of

it

over-pressure.

or

a

arrested

sudden

flux

prevented

to

under

cause is

coincides no

is

momentum

column

momentum c h a n g e of

sudden

tank

energy

ends.

the

converted kinetic

decelerates

the

cases

the

is

original

potential ly

all

in

pressures on

e i t h e r s i d e to g r a d u a l -

l y equalize. Fortunately and

for

hammer theory

analysis may

be

i n some cases,

If

the

pump

of

bypass

is

of

similar

a

or

bypass In

the

design

effects,

necessary.

analysis

of

elastic may

reflux these

water

be

systems

protective by

incorporates

of

so

air

vessels

a

that

Rigid

surge

hammer

done are

in-line

water

water

tank

trial

device and

envisaged,

valves

may

instances

method of

oped,

by

a

solution,

varying

the

such

error.

column

action,

and

as

a

a

analysis

a

or,

or

a

is

usually

a

number

if

computer

discharge

location

similarly

design

valves

program

s i z e and d i s c h a r g e c h a r a c -

as

The

computer

reflux

graphically

Normal ly the location,

economical and

the

for

inertia

normally

an

analysis

of

determined

pump

for

system

b e developed.

teristics

not

valve,

be

available the

of d i s c h a r g e tanks.

The

solutions

trial.

are of

employed

pipeline

necessary.

could

charts

investigation

to

have

program

general

and

is

be

tank size

methodically,

to

in-line

determined

usually

program could

parameters

have

of

the

by most

be develan

opti-

reducing

water

mum s o l u t i o n a r r i v e d a t . The

following

sections

hammer in p u m p i n g l i n e s ,

describe various

methods o f

a n d offer design aids.

73 Pump Inertia If to

the r o t a t i o n a l

rotate

pressure

the

pump

the

of

for

transients

e n t r a i n e d water on

i n e r t i a of

may

will

delivery

a

while

be

side,

after

reduced.

continue

the water column.

a centrifugal

thereby

power

The

feed

to

pump and motor continue

rotating

water

water

pump,

hammer

motor

and

i n t o the p o t e n t i a l vacuum

a1 l e v i a t i n g

The effect

failure,

the

sudden

deceleration

i s most noticeable on low-head,

short

pipelines. After

the

gradually the

power

slow

delivery

higher pump

in

delivery gather

turbine

the

the

the

a

until

it

can

no

suction

at

side

direction,

of

the

these

will

it

with

there pump.

in

the

no

The

reverse

conditions.

If

The

of

the

off,

the

deliver

force

pump

still

reflux

pump

or

will

reverse

against

through

spinning

rapidly

of

will

head is s t i l l

control

and

speed

pump

water

water

direction,

i t reaches r u n a w a y speed.

deceleration

cut

the d e l i v e r y

then

the

is

is

longer

the time.

head,

provided

momentum

under

rapid

motor

reverse

increase u n t i l is

the

down

direction,

and

to

head e x i s t i n g

than

forward

supply

in

the the

valve

on

decelerate

will

act

as

the

pump

a

will

Under these conditions there

reverse flow

a n d water

hammer over-

pressures w i I I resu I t. I f there i s a r e f l u x v a l v e reverse still

wi II

flow

The

occur.

be

on

arrested,

pressure

the

but

changes

delivery water

at

ing

the

work

done

after

the wil I

1963) o r b y computer.

a time increment A t

i n decelerating

the pump,

overpressures

t h e pump f o l l o w i n g power f a i l -

u r e may be c a l c u l a t e d g r a p h i c a l l y (Parmakian, The pump speed N

side of

hammer

i s obtained b y equat-

the pump to the energy

transferred

to the water:

Solving f o r N

1

- N2

= L N we get

900wHQ A t A N = n' MNFN where

M

is

(4.12) the

N

is the speed

of

the time

head

above

moment of in

rprn,

interval, suction

w

inertia

of

the

pump

impeller

and

motor,

FN i s the pump e f f i c i e n c y a t the b e g i n n i n g i s the u n i t weight of f l u i d ,

head a n d

q

i s the discharge.

H i s the pumping As

an approxima-

74 tion,

FN = (N/No)Fo where s u b s c r i p t o r e f e r s to i n i t i a l conditions.

The

head/discharge

accurately

characteristics

of

the

pump

can

be

fairly

represented b y the equation

t i = aN2 + bNQ - cQ2 a,

b

and

c

on

the

points

are

(4.13)

constants

which

can

head/discharge/speed

be

evaluated

characteristics

provided

of

the

three

pump

are

known. Substituting pipe,

and

q

vA

=

A

where

4.8,

u s i n g Equ.

is

the

cross

two equations

sectional

result

area

involving

of

the

two

un-

knowns which may be solved f o r h and v at the pump. Fortunately summarized

the

in

r e s u l t s of

chart

form.

pipeline

P.

The

times

of

parameter and

the

various

for

presented

upsurge at

values

charts flow

l a r g e number of

Charts

the maximum downsurge and the

a

also

of

by

Parmakian

indicate

the pumps and midway

pump

indicate

reversal,

analyses have been

zero

inertia

the

and

maximum

pump

speed

the

pipeline

reverse

and

along

speed

maximum

re-

verse speed. Kinno

and

which

include

speed

of

ing

the effect

the pump

maximum

Kinno

flow

of

have

prepared

that

through

if

comprehensive

charts

l i n e f r i c t i o n and pump e f f i c i e n c y ; specific

i s also discussed.

upsurge

suggests

reverse

(1965)

Kennedy

reverse

Of

rotation

of

could

be

the

upsurge

the

pump was

i s a chart

interest the

pump

reduced

permitted,

but

is

indicat-

prevented.

considerably

with

if

reverse r o t a -

tion prevented.

4.10

Fig. will

occur

ues

for

the

maximum

summarizes

in

various minimum

head

the

head

values

around 85 p e r cent.

t

wave

will

of

the

subsequen

mitted,

are

also presented

ues

the

magnitude

prevented maximum

with

of

the

are

and

systems

from

to cover

be observed r e t u r n s to

upsurge in

at

the pump

upsurge which

reflux

valve.

headrise

above

operating

equal to the lowest head-drop

that

If

Kinno's

paper,

and

Valthe

pump efficiencies

( t = 2L/c).

if

flow

reversal

These values

would occur reversal

head

Ho

.

which

the minimum head often

flow

below H

pressures

power f a i l u r e .

typical

the pump

the c h a r t .

a

minimum

following

taken

a r e adjusted

occurs when

the f i r s t

maximum

pumping

if

would

The v a l i s per-

are lower than

reverse flow

was be

prevented

was the

approximately

75 The the

author

pump

inertia

pressures. the pump

has

If

a

simple

will

the

have

inertia

r u l e of an

thumb

effect

parameter

in

I

=

Here M

i s the moment of

i s the volume of water i n the pipe.

from

Fig.

4.10,

and

i n e r t i a of

it

was

the water

MNZ/wALHoZ

the pump,

assumed

a s c e r t a i n i n g whether

reducing

i n e r t i a may reduce the downsurge b y

and AL ed

for

at

least

hammer

exceeds

0.01,

10 per cent.

N i s the speed

i n rpm

The expression was d e r i v that

c/g

is

approximately

e q u a l to 100.

F i g . 4.10

M a x i m u m and rninimiJm h e a d s at p u m p a f t e r power f a i l u r e

76 Some i n s t a l l a t i o n s have a fly-wheel the

moment

of

inertia.

impractically

be

In

heavy,

most

also

f i t t e d to the pump to increase

cases

flywheel

the

would

have

i t should be borne i n mind that

to

start-

i n g c u r r e n t s may thereby be increased.

In

subsequent

sections

the

effect

and the pumps a r e assumed to stop

of

pump

inertia

is

neglected

instantaneously.

Pump Bypass Reflux V a l v e

One

the

the

of

against

water

pump

(Fig.

charge o n l y ing

in

conditions

head

and

the

simples hammer 4.11).

arrangements is

a

reflux

of

through to

the the

the

line

parallel

with

non-return

pumping

pressure

head

would

difference

bypass

be

would

head,

valve.

pressure

in The

less

the

suction

reflux

valve

The head may water

would

friction

than the

the head i n the d e l i v e r y

case

pressure

any

dis-

Under normal pump-

higher

cvo/g.

which

v a l v e would

maintain

On stopping the pumps,

suction

suction

pumping

in

the pumps.

i n a closed position.

below

a

installed

or

Pipe would tend to drop b y the amount to

protecting

valve

the same d i r e c t i o n as the

The

reflux

loss

would

be

drop drawn

therefore o n l y in

the

bypass.

drop The

r e t u r n wave overpressure would be reduced a c c o r d i n g l y . REFLJX VALUE

J S U C T I O N PIPE

F i g . 4.11

This cases, tion er

Pump w i t h bypass r e f l u x valve.

method

as

pressure. hammer

sure).

of

water

the d e l i v e r y

The

hammer

I n other cases

overpressure method

protection

pressure w i l l

be

used

below

in

all

the suc-

there may s t i l l b e an a p p r e c i a b l e wat-

(equal

really

cannot

often never drop

has

i n v a l u e to use

only

the

when

initial the

drop

pumping

i n preshead

is

77 considerably

less

sure

along

the

tion

reservoir

than

entire level

cv /g.

should

s t i l l be column separation Normally reservoir. fairly

the

and

also

pipes

there

water

length be

draw

may

be

above,

prevent

initial

be

the

suction

drop

i n pres-

tolerable.

relatively

high

or

The

suc-

there

may

line.

directly cases

from

where

a

in a similar

pressure

constant

the

intake

hammer could be a problem i n

cases a bypass r e f l u x v a l v e would, ed

the

should

i n the d e l i v e r y

intake

However,

long

I n addition,

pipeline

it

way

exceeding

head

pipe

too.

is

I n these

to that describ-

the d e l i v e r y

pres-

sure.

It

should

be

noted

pump

during

the

period

head,

if

especially

speeds,

as

is

bypass r e f l u x

the

that

water

that

the

case

the

may

machine

with

also

delivery was

drawn

In

the

the suction

high

for

pumps.

be omitted,

through

i s below

designed

through-flow

v a l v e could even

be

head

specific

some cases

the

a l t h o u g h there i s normal-

l y a f a i r l y h i g h head loss through a s t a t i o n a r y pump.

Surge Tanks The

water

pressure, tank

in

as

of

of

a

grade

in

l i n e of

to be used. in

the

surge end

tank

of

treated

is

the

in

case

rise.

the

of

a pumping

tank tank

for

a

Surge

line

is

i s open

the

flow

head

drop are

although

there

It

profile

and water

relatively

the

i t could

pipeline

surge

the

and

tank

between

other

be treated

independent p i p e l i n e shorter

or

this

filling the

they

can

hydraulic

at the

means.

as

may

at

to enable an open a

some

length,

pipe,

principally

that

The

which

a r e cases where

pipeline

by

atmospheric

the p i p e l i n e .

the

seldom

is

hammer

large,

in

low enough

protect

to

to

variations

used

be possible to construct

against

exposed

is

tanks

systems.

intermediate

as an

surge

tank

penstocks,

I t may

tank

a

bottom of

pumping

pipeline

the

in

balancing

head

turbine

applied

and

the

a

discharging

case

head be

while

acts

occur,

surface

tank

a peak pumps If

the

the discharge

section

could

be

in

length than the o r i g i n -

The f l u c t u a t i o n s of the water surface level

i n a surge tank follow-

a l pipeline.

ing ween

power the

failure pumps

may and

be

studied

surge

tank

analytically. are

The

high-frequency

transients water

bet-

hammer

78 phenomena ably.

which

The

will

analysis

not

affect

concerns

the

the

water

slow

level

motion

in

column between the surge tank and d e l i v e r y end of R i g i d water column water

hammer

will

not

pass

the

h

surge very

is

the

tank end.

height

P

and The

the

of

open

delivery

head

h

gradually

since e l a s t i c

tank.

The

rate

of

(4.14)

above

increases as

water

the line.

i s the p i p e l i n e length between

head

notice-

the

= -gh/P

deceleration of the water column i s d v / d t where

of

may b e used in the s t u d y ,

theory

waves

the tank

surges

the

level

the tank

the

level

in

of

the

and delithe tank

drops (see F i g . 4 . 9 ) . Another equation r e l a t i n g h a n d v

may be d e r i v e d b y considering

c o n t i n u i t y of flow a t the tank o u t l e t :

(4.15)

A v = Atdh/t P

Ap

where

and At

respectively.

are

the cross sectional

Note that

the flow

from

areas of

the p i p e a n d tank

the pump side

i s assumed to be

zero. Solving

Equs.

0 and v = v

4.14

and 4.15

that at t = 0,

a n d u s i n g the fact

to e v a l u a t e the constants o f

integration,

h =

one o b t a i n s an

express ion f o r h :

(4.16) From surge

in

t h i s equation

it

is

v

the

tank

i s apparent ,

-

surge

is

(II

/ 2 ) J AtP/A

P

g:

m r

that

and

the amplitude of the down-

the

time

till

down-

the f i r s t

-

Time zero

i s assumed

be that

to

i n s t a n t at

which the water hammer wave reaches the surge t a n k . If pump the

there

i s a g r a d u a l deceleration of the water column between the

a n d surge t a n k , correct

deceleration

then

the above expressions w i l l

continuity

equation

will

to

have

be

and

solved

the

rigid

not

column

numerically

for

hold, and

equation

successive

for time

i n t e r v a I s. The

fluctuations

in

tank

level

may

damped

be

orifice,

I n t h i s case the pressure v a r i a t i o n s

extreme

than

for

the

unrestricted orifice.

in

with

a

throttling

the l i n e may be more

Parmakian

( 1 9 6 3 ) presents

c h a r t s i n d i c a t i n g the maximum upsurge f o r v a r i o u s t h r o t t l i n g losses. A have

number of been

sophisticated

proposed

(Rich,

variations

1963).

The

in

the design of surge tanks

differential

surge

tank

for

79 instance The

includes

tank

may

variations systems,

a

small-diameter

have a

are

more

as they

varying

riser

in

the

cross section o r

applicable

to

a r e useful f o r

hydro

m i d d l e of

the

tank.

m u l t i p l e shafts.

power

plant

than

Such

pumping

damping t h e surges i n cases o f r a p i d

load v a r i a t i o n on turbines. Discharge Tanks I n s i t u a t i o n s where the h y d r a u l i c g r a d e one

which

pipeline.

under

The

pressure b u t

the p i p e l i n e p r o f i l e i s considerably

l i n e i t may s t i l l be possible to use a

normal

tank

operating

water

would

lower than

surface

be below

conditions

is

tank,

but

from

the

isolated

be subjected to atmostpheric

would

the h y d r a u l i c g r a d e

line,

as opposed to

that of a surge tank.

Fig. 4.12

A

Discharge tank

discharge

tank

would

normally

be

situated

along

the p i p e l i n e a n d possibly on subsequent

rises.

The tank

the

nearer

should

be

discharge

will

the

level

connected from

below

the

valve

would

water be

the

should

be

Fig.

4.12

the the

surface held

bypass to

tank,

in to

the tank

small-bore

discharged.

b e more e f f i c i e n t

tank

is

pipeline

to

the h y d r a u l i c g r a d e

via

a

reflux

elevation

tank.

the

in

the

pressure

the r e f l u x v a l v e , installed depicts

to

a

first

rise

a n d successively h i g h e r

if

by

the

i n r e d u c i n g pressure v a r i a t i o n s

i n t o the p i p e l i n e

shut

on

fill

typical

valve

line.

It

i n s t a l l e d to

the p i p e l i n e head drops

in

Normally the

connected

to

the

slowly

tank

discharge

the

pumping a float

tank

reflux line.

A

valve

in

after

it

has

arrangement.

80 The

use

of

discharge

tanks

was

reported

in

detail

by

Stephenson

(1972). The f u n c t i o n of caused The

by

pump

water

pipeline action

a discharge stoppage,

column a

(or

of

subsequent head

the

necessary

between

i s to f i l l

any

low pressure zone separation.

the

end

tank

tank)

difference

prevent

to

cause water

tank

thus p r e v e n t i n g water column and

will

discharge

g r a d u a l l y decelerate

between

reverse motion of

hammer overpressures

the

the

two

ends.

the water

of

the

under

the

may

be

It

column which could

t h i s could be achieved by

-

instal-

l i n g a r e f l u x v a l v e i n the line.

A

discharge

the

lowest

following

first

tank

tank

will

only

to

which

the

level

drop

tanks

tank

head

pump stopping.

along

should

operate

a

line

be successively

i s considerably

in

Thus

should

the

the

water

surface

i s above

pipeline

would

otherwise

the normal o p e r a t i n g head on the

be

less

higher.

less than

if

than

c v /g,

I n cases

cv /g,

and

0

where

r i g i d water

subsequent

the head on the

column

theory

may

be used to c a l c u l a t e the discharge from the t a n k : Integrating expression f o r Multiplying yields the

twice

discharge

distance

sectional tank

respect

by

area

of

and

h

is

time,

to

column

travels

the cross sectional by the

the

the

P

obtains

the

the

the

pipeline

where A

0

is

an

before stopping.

area of

is

distance between the

the p i p e l i n e ,

head on

one

Q = APv 2/2gh,

tank;

pipe,

a n d the open end of

one,

is

with

the distance the water

this

the volume discharged

cross

there

4.1

Equ.

tank

or the next tank measured

below

if the

h y d r a u l i c g r a d e l i n e ( o r level of the next t a n k ) . Fig. a

4.13

fraction

depicts the

of

The coordinates using elastic the

head,

discharge

discharge

at

the

overpressures theory

tank

never

tank

accurately

hammer

a

indicated by

the g r a p h

column

overpressures

discharged from a

increase. exceeds

is

are

low. less

than by

r i g i d column equation. a computer

approximately

r i g i d column successively

For

applicable

I t appears, that

tank expressed as

analysis

I t i s i n t e r e s t i n g to note that when

less

predicted

becomes

the

were obtained from

water hammer theory.

h, is

of

the volume

and

however,

indicated

by

rigid

higher the

that

0.5

theory,

cvo/g,

water

heads

rigid

water

hammer

the discharge

water

column

p r o v i d e d the conditions discussed below a r e compl ied w i t h .

the

and

from

theory,

81 4.13

Fig. tank

i s only

i s assumed

should be an side)

of

a p p l i c a b l e f o r a s i n g l e tank along the line.

be a t

in-line

the

r e t u r n from

to

tank.

reflux valve

Without

the tank

immediately

this reflux

towards

upstream

valve,

the pumps.

would

I n a d d i t i o n to water hammer the discharge from the

would exceed that f o r the assumed case where i t o n l y discharged

downstream. along h'

(on the pump

a p o s i t i v e wave

overpressures which would r e s u l t a t the pumps, tank

The

the pump end of the p i p e l i n e , o r else there

The

maximum overpressures

the p i p e l i n e on

is

measured

fraction

above

the

If

cv / g .

of

indicated by

the c h a r t

the d i s c h a r g e side of the tank.

the

discharge

head,

upstream r e f l u x

and

occur

The overpessure

is

expressed

as

i s omitted F i g .

valve

a

4.14

should be used. For peaks, line.

very

long

pipelines

more

than

one

The

tanks

separation

is

should be likeliest.

p o i n t beyond a tank

with

a

discharge

lowest

of

may

i n s t a l l e d at

The

as

number

tank

successively

be

installed

peaks where

the

head

will

which

the downsurge t r a v e l s along

higher

along

the

water column occur

the

at

line

any

i s that

of the water surface elevation of the preceding t a n k .

4.15

Fig. along first

a tank

valve

is

be

tank

the

The

located

should

second of

indicates

pipeline.

at

the

installed

should

discharge

is

chart

also

from

also o n l y

pump end of immediately

have

a

two

the

line,

upstream

reflux

tanks

appiicable

valve

or

of

installed

provided else

the

the

a reflux

tank.

immediately

The

upstream

it.

4.15

Fig.

is

between

the

and

open

the

for

really

tanks

different

end

only

is

equal

of

the

length

applicable to

the

and

cases where

distance between

pipeline;

ratios

for

however,

4.15

Fig.

the distance

the second

the c h a r t s

could

be

tank

hardly vary

used a s a g u i d e

f o r other cases. in and

4.15

Fig.

second

second

tank

corresponding is

that

line.

and

is

and the

pipe

between

The

h,

tank,

the

and

the

relevant

discharge h

difference

head at

length

discharge

The

the

h2 i s

the

tank

that

from

e

the are

between

i n elevation

between

the

second

tank

is

tank

substituted i n

is

the

first

between

the p i p e l i n e . two

a n d the discharge

from the f i r s t

and

elevation

the discharge end of

t 1 is

second

in

difference

tanks

end of

2

the pipe-

broken

indicated

full

by

The

and J?

indicated by

the v a l u e

the

lines lines.

i n d i c a t e d on

the

82

tc

08

06

O L

9 h' c vo 0 2

0 10

08

Q ?gh __ A

LV,' 06

OL

0.2

0

0

Fig.

4.13

D i s c h a r g e f r o m t a n k a n d m a x i m u m head r i s e w i t h i n - l i n e r e f l u x v a l v e u p s t r e a m of t a n k .

83

F i g . 4.14

Discharge from tank a n d maximum head r i s e (no in-line r e f l u x v a l v e ) .

F i g . 4.15

Discharge from two t a n k s a n d maximum head r i s e ( e l = @ a n d r e f l u x v a l v e upstream of each tank).

2

85 v e r t i c a l scale to o b t a i n the respective discharge Q from each t a n k . The maximum overpressures i n d i c a t e d b y F i g . 4.15 I t is,

i n the downstream side of the second tank. limit

the

further the

lateral

reflux

pipeline.

water from

In

hammer the

water

extent of

valve

may

then

tank

hammer

the

last

tank

and

t h i s case the r e f l u x v a l v e on could

analysis

however, possible to

the overpressure b y j u d i c i o u s

between

tank

i n v a r i a b l y occur

possibly exceed

should

be

with.

indicated b y

carried

location of a

delivery

end of

the pump s i d e of

dispensed

that be

the

out

the

The discharge

the

chart,

based

on

and a

the

theory

out1 ined e a r l i e r i n the chapter. The best p o s i t i o n f o r selected

by

trial

and

discharge error

c h a r t s presented may suffice, or

major

pipelines

with

should

be c a r r i e d out,

cular,

a final

check

tanks

and

and i n - l i n e

experience.

For

b u t f o r complex

large

either

friction

ref l ux valves i s simple

cases

the

cases w i t h many peaks

heads,

graphically o r

a

complete

analysis

b y computer.

should be done f o r flows

In parti-

less than the maximum

design c a p a c i t y o f the p i p e l i n e . Even though a number of tanks may be i n s t a l l e d along a p i p e l i n e , vaporization

is

tanks.

Provided

steeply

between

water on

always there

possible

are

tanks,

no

this

hammer overpressures.

all

rising

vapour

sections.

along

local

rising

peaks,

sections

and

the

I imited v a p o r i z a t i o n However,

Cases

are

air

the

l i ne rises f a i r l y

should

not

lead

to

vessels should be i n s t a l led

known

bubbles t r a v e l l i n g along g e n t l y

between

where

vaporization

and

r i s i n g mains have r e s u l t e d i n

severe c a v i t a t i o n of a l i n e a l o n g the top of the pipe.

A i r Vessels If

the

p r o f i l e of discharge

tank

or

force

water

into

compressed

air

illustrated

in

the in

in

the

will

vessel

should

tend be

to

is

not

protect

vessel

4.16).

(a

The

to

line.

reverse.

the

air

to

may

it

use

be

a

surge

possible

to

wave b y means of

vessel

arrangement

i n the vessel

will

is

gradually

the pressure i n the vessel equals

this

However,

unrestricted,

enough

line,

low-pressure

pressure

At

high

the

typical

i s released u n t i l

adjacent

column

pipeline

p i p e b e h i n d the a

Fig.

decrease as water that

a tank

stage

the

whereas inlet

decelerating the outlet

should

be

of

water the a i r

throttled.

A

86 suitable reflux

arrangement valve,

smaf I - o r i f i c e

4.16

Fig.

2.5

pirical

vessel

the

the

design

ratio

of

2.5.

The

the

when

( 1 9 6 3 ) suggests that outlet

a n d recent of

have

water the

discharge

water

bypass would a l l o w the vessel

times

effect

to

shuts

out

column

through

a

A

reverses.

to r e f i l I slowly.

A i r vessel

Parrnakian be

is

which

inlet

author

the a i r

loss.

vessel

However,

inlet

head

loss should

t hi s relationship

i s em-

research w i t h the a i d of computers has enabled the

loss

head

charts

reverse

head

to

reproduced

flow has

head

loss

compiled

be by to

i n v e s t i g a t e d thoroughly. parrnakian forward

design

were compiled f o r

flow

charts

The a i r

head

loss

covering

an

the

equal

to

extended

i n l e t t h r o t t l i n g ( F i g s . 4.17

r a n g e of

p i p e l i n e s a n d v a r i o u s degrees of

to 4.19).

The c h a r t s a r e s u i t a b l e f o r checking water column separation

a l o n g the e n t i r e p i p e l i n e length, volume

and

degree of

inlet

l i m i t to the overpressures. The c a l c u l a t i o n s for er; lies

it

was

assumed that

between

adiabatic

a n d f o r r e a d i n g off the r e q u i r e d a i r

throttling

corresponding

to any

specified

Outflow t h r o t t l i n g i s neglected.

p l o t t i n g the c h a r t s were performed b y computthe expansion

(HS

1.4

-

law f o r

constant)

and

the a i r the

i n the vessel

isothermal

(HS

=

Fig.

4.17

Maximum a n d minimum pressure envelopes w i t h a i r vessel. 2 = 0.5.

F i g . 4.18

M a x i m u m and m i n i m u m p r e s s u r e e n v e l o p e s w i t h a i r vessel. P = 1.

88 constant).

The

relationship

adopted

further

assumed t h a t t h e a i r vessel

pumps,

t h a t the pumps would stop

flow

was

HS1'3

It

constant.

=

was i m m e d i a t e l y

was

d o w n s t r e a m of t h e

instantaneously and that no reverse

t h r o u g h the pumps was permitted.

The

dimensionless

parameters

associated

with

the

charts

are

as

fol l o w s :

Pipeline parameter

P = c v /gH

A i r vessel parameter

B = v 0'AL/gHoS

Throttling parameter

C = Z/Ho

where S

i s the

loss t h r o u g h

The

outlet

0

initial

air

t h e a i r vessel

head

loss

was

volume

in the

0

vessel,

and 2

i s the

head

i n l e t c o r r e s p o n d i n g to a l i n e v e l o c i t y -v assumed

to

be

negligible.

the absolute head ( i n c l u d i n g atmospheric head)

Note t h a t

0-

Ho i s

i n t h i s case.

0.2 I

Fig.

4.19

Maximum a n d minimum pressure envelopes P = 2.

with

a i r vessel.

The c h a r t s a r e u s e d a s f o l l o w s to d e s i g n a n a i r v e s s e l : line parameter,

P,

i s c a l c u l a t e d from the maximum l i k e l y

The p i p e -

line velocity

89 and

pumping

head,

4.17

to 4.19.

T h e p i p e l i n e p r o f i l e is p l o t t e d on the a p p l i c a b l e set of

minimum-head it

does

not

fall

to

below

The

inlet

the The

throttling

to

10 ( o r more). pumping

works

necessary

at

any

to

the

C.

a

of

of

l/lOth

thus

of

still

smaller

S,

air,

design

suppress

diameter

vortices

cause line

is

procedure

i s to

m,

the

main

the

cannot

value

of

2 m/s a n d

corresponding

pipe diameter).

be

equal

tolerated,

it

inlet I f the

will

B (corresponding to a

be

larger

i s known.

to ensure no a i r escapes volume.

is

usually

designed

to

The vessel

i n t o the pipe-

This

i s the volume

1/1.3

be about one-half

the

The outlet should be designed w i t h a bellmouth to

and

the water

in

to

the degree of

approximately

of

200

i s c a l c u l a t e d once B

be s u f f i c i e n t

main p i p e diameter.

dissolve

point selected

depend on

The normal

a n d should exceed the maximum a i r

outlet

such that

envelope a l o n g the p i p e l i n e from actually

d u r i n g minimum pressure conditions a n d i s S ( H /Hmin) The

Figs.

achieved b y s e t t i n g C approximately

order

be

from

than i n d i c a t e d b y the minimum pressure requirement.

volume

c a p a c i t y should line,

to

indicated

volume of a i r ) The

the

out

select

to

profile

maximum-head

of

selected

envelope selected

( F o r a p i p e l i n e velocity

head

chart

corresponding

overpressures

represented b y

a

overpressures

pipeline

B

of

minimize the overpressures,

diameter

corresponding

the

value

read off

same c h a r t .

the

the

curves a n d a minimum-head

vaporization. used

and

air

entrainment.

to some extent

The

air

and w i l l

in

the

vessel

will

h a v e to replenished b y

means of a compressor.

In-L ine R e f Iux V a Ives Normally not al

a

reflux

reduce water extent

of

positioning

the

of

shock.

reflux

hammer pressures. led

upstream of

the

effect

of

valve

i n s t a l l e d on

hammer pressures,

For the

the

In

valves

fact,

i t s own

although in

some

in

a

i t may

pipeline w i l l

limit

situations

the

later-

indiscriminate

in

a

l i n e could

instance

if

a pressure r e l i e f v a l v e was i n s t a l -

reflux

other

valve

valve.

It

the may

be detrimental

reflux

valve

also

amplify

would

to

water

counteract

reflections

from

b r a n c h pipes o r collapse of vapour pockets. In-line surge down, side

reflux

tanks, the of

tank

the

valves

discharge or

reflux

would tanks

vessel valve.

n o r m a l l y be used or

would This

air

vessels.

discharge would

water

alleviate

i n conjunction

Following into the

pump

with

shut-

the p i p e e i t h e r violent

pressure

90 drop

The reflux

a n d convert the phenomenon i n t o a slow motion effect.

valve

would

then

arrest

the

water

column

at

the

time

of

reversal,

which coincides w i t h the p o i n t of m i n i m u m k i n e t i c energy and maximum potential

energy

of

momentum change

the water

in

column.

There

would

therefore be

little

the water column when the r e f l u x v a l v e shut and

consequently n e g l i g i b l e water hammer pressure r i s e . There a r e s i t u a t i o n s where water column separation and the format i o n of

vapour

be

tolerable,

in

water

vapour at

the

would

pockets

i n the p i p e l i n e f o l l o w i n g

provided

the

vapour

hammer pressures.

pocket could

arrested

not collapse

the water

at

of

the

i t s p o i n t of

vapour

resulting

column beyond the

i n fact be prevented w i t h an i n - l i n e

downstream extremity be

pockets d i d

Reversal of

pump stoppage would

pocket.

reflux valve

the water

minimum momentum,

column

there would

so

be l i t t l e head rise. Vaporization water If

hammer

the

first

peaks,

the

of

extent theory.

along

vapour

peaks

wouJd

the

last

be

and

the

was

confined

could

head on

peak

in

the vapour

pipeline

pocket

The decelerating

rigid

water

column

be

pipeline

where

pressure of

the decelerating head,

In

the water.

than

subsequent

to

first

peak.

the

estimated

using

rigid

the

dispersion

the water column between two peaks

with

a vapour

would

be the d i f f e r -

respect pocket,

to

t,

Integrating

one o b t a i n s

an

k?AvoZ/2gh, where h i s

C i s the water column l e n g t h a n d A i s the p i p e

ensure

reflux

valve,

allowance should be made f o r

of the vapour pocket.

suitable dip to

The

column

area.

locating

lateral

the

higher

the discharge end

equation

the volume of

cross sect iona I

a

the

vaporization

expression f o r

at

at

in e l e v a t i o n p l u s the intermediate f r i c t i o n head loss.

ence

and

occur

pressure dropped to rise

the

o r between

the

would

i n the p i p e l i n e

proper

functioning

i n order

of

some

The v a l v e should be i n s t a l l e d to

trap

the vapour pocket

the v a l v e doors when

the water

column r e t u r n s .

A small-diameter permit

slow

occur on of

the

bypass

r e f i l l i n g of

restarting

order

of

to

the r e f l u x v a l v e should be i n s t a l l e d to

the vapour

the pumps.

one-tenth

of

pocket,

or

else overpressures may

The diameter of the bypass the

pipeline

diameter.

An

should be air

release

91 valve

be i n s t a l l e d in the p i p e l i n e a t

should

the peak

which would come out of solution d u r i n g the p e r i o d of

to release a i r

low pressure.

i s a common p r a c t i c e to i n s t a l l r e f l u x v a l v e s immediately down-

It

stream

of

the

pumps.

Such

hammer

pressures

in

through

the

and

pump

the

reflux

valves

pipeline.

prevent

They

water

would

not

merely

hammer

prevent

prevent

pressures

water

r e t u r n flow r e a c h i n g the

pumps. In

some pump

installations,

instead of r e f l u x valves,

(1968)

Kinno delivery to

be

valve.

the

Assuming he

closing

control

valves,

a r e i n s t a l l e d on the pump d e l i v e r y side.

studied

tolerable,

automatically

effect

a

of

limited

describes

how

c o n t r o l l e d closure return

water

flow

of

through

hammer

a

pump

the

pump

overpressures

can

be s l i g h t l y reduced b y c o n t r o l l i n g the r a t e of closure of the v a l v e .

Release Valves There

are

a

number of

sophisticated water

(often r e f e r r e d to as surge r e l i e f a b l e commercial l y . matically

open

The

valves

pipe

leading

in

the

effect are

suction

the

gradually

normally

the

v a l v e s o r surge suppressors) a v a i l -

close

the

needle

reservoir.

The

complete

range

the

valve

type, or

discharge

velocities

pump

valves,

have a gradual

closure.

Needle a n d

which

tripping.

discharge

else sleeve

v a l v e s must of

after

which

designed to minimize c a v i t a t i o n

high

release v a l v e s

have h y d r a u l i c actuators which auto-

to the suction r e s e r v o i r ,

suitably

with

then

are

over

These valves

hammer

into

a

mounted throttling

sleeve

valves

a n d corrosion associated

occur

d u r i n g the t h r o t t l i n g

process. The v a l v e s a r e u s u a l l y reflux

valves

should

not

and

i n s t a l l e d on

discharge

discharge

a i r through the throat,

into

directly

the

The returns on

should

the

the top of

maintaining

suction as

they

reservoir.

They

i n v a r i a b l y draw

by

an e l e c t r i c a l

fault or

b y a pres-

(as Fig. 4.20).

valve to

the

pipe

and t h i s could reach the pumps.

The v a l v e s may be actuated sure sensor

suction

the d e l i v e r y side of the pump

to

pumps

open as

the piston

the

pressure

a

fully

before

the

p o s i t i v e pressure

increases a g a i n within

negative wave.

pressure

As

wave

the pressure

the v a l v e g r a d u a l l y closes,

desired

limits.

The

closing

may be a d j u s t e d b y a p i l o t v a l v e i n t h e h y d r a u l i c c i r c u i t .

rate

92

Reflux Pump valve

Acc u mu lot or

Suction

reservoir 4.20

Fig.

The returns on

Water hammer release h y d r a u l i c actuator

valve to

should

the

open

pumps

as

a

the top of t h e p i s t o n

maintaining

the

fully

valve

arrangement

before

positive

the

pressure

within

negative

pressure

increases a g a i n

simp1 i f i e d

pressure

As

wave.

the v a l v e

wave

the pressure

g r a d u a l l y closes,

I i m i ts.

desired

with

The

closing

rate

may be a d j u s t e d by a p i l o t v a l v e i n the h y d r a u l i c c i r c u i t . If

no

the v a l v e to

the

water of

overpressure

operating hammer

the

llel.

higher

would be sized

pumps, They

head.

could

Where

likely

is

t w o or

be

than

the

to discharge

to

operating the f u l l

reliability

be

a

is

problem

more release valves set

operate

to

at

head

flow of

at

tolerable, head equal

importance,

during may

is a

be

and

if

partial

shutdown

installed

i n para-

successively

lower

delivery

throttled

delivery

heads. In

the

valves,

event

of

normal

pump

shutdown a g a i n s t

the release v a l v e s could be disengaged to prevent

t h e i r oper-

at ion. The types of ing

lines

use

is

This

control

v a l v e s a v a i l a b l e as release valves f o r pump-

n o r m a l l y cannot open

therefore

method

of

limited water

to

in

less than

pipelines

hammer

over

protection

is

about f i v e seconds. Their

two

kilometers

normally

most

in

length.

economical

93 for

cases

when

the

pumping

l a r g e r the pumping head, Since

there

hammer

is

analysis

no

head

greatly

exceeds

cv /g,

since

the

the smaller the v a l v e needed. protection

should

be

against

performed

underpressures, check

to

that

a

water

water column

separation i s not a problem.

A

less

above,

sophisticated

which

has

valve

been

used

spring-loaded

release v a l v e .

sure

a

reaches

prefixed

than on

the

small

The v a l v e

maximum.

control pump

i s set

valves

described

installations,

to open when

Some overpressure

is

the

the pres-

i s necessary to

open the v a l v e a n d to force water out.

Choice of Protective Device The will

best

method

depend

system.

on

The

of

the

water

hammer

hydraulic

and

accompanying

table

protection physical

for

a

pumping

characteristics

summarizes

the

ranges

line

of

over

the

which

v a r i o u s devices a r e s u i t a b l e .

The most i n f l u e n t i a l parameter i n selec-

ting

i s the p i p e l i n e parameter P = cvo/gHo.

the

When

bypassing

values

P

of

it

the pumping head Ho,

i s g r e a t e r than

the

pumps

may

becomes necessary

For

suffice. use

to

The protective

approximate

suitable

order

device,

of

a surge t a n k ,

a

smaller

discharge

a n a i r vessel, o r a

devices l i s t e d i n the t a b l e a r e a r r a n g e d

increasing

one checks

a reflux

successively

i n combination w i t h an i n - l i n e r e f l u x v a l v e ,

release v a l v e . in

protection

the head cvo/g

valve

tank

method of

down

cost,

the

thus,

table

to

until

select

the

the

most

variables

are

o r more p r o t e c t i v e devices on

the

w i t h i n the r e q u i r e d range. It

same omical In

may line.

be

to

in

use

two

p o s s i b i l i t y should

arrangement often

particular

effect

possible

This

the

reducing

the

be ignored as the most econ-

i n v o l v e s more than one method of protection.

rotational

comprehensive water

not

i n e r t i a of

required

hammer

the

capacity

analysis

of

pump a

often

tank

has

or a i r vessel.

would be necessary

of protection devices i n combination i s envisaged.

a slight A

i f a series

94 TABLE 4.1 Summary of methods of water hammer protection

Method of Protection ( i n approximate order of increasi n g cost)

I n e r t i a of pump

Required r a n g e of v a r i a b I es

MN*

Remarks

Approxi mate on I y

wALHo2 > 0 * 0 1 Some water may a l s o be d r a w n through pump

Pump bypass reflux valve

In-line reflux va I ve

__

cv

Surge tank

h small

> 1

0

gh

P i p e l i n e should be n e a r h y d r a u l ic g r a d e l i n e so height of tank i s Dractical P i p e l i n e p r o f i l e should be convex downwards. Water column separation l i k e l y .

Automatic release valve

D i scharge tanks

cv

0

>

h = pressure head a t t a n k , p i p e l i n e prof i l e should be convex upwards

'

gh

A i r vessel

Normally used i n conj u n c t i o n w i t h some other method of protection. Water column separation possible

cv

o < __ gHO

l

P i p e i ine p r o f i l e p r e f e r a b l y convex downwards

95 REFERENCES Kinno, H. and Kennedy, J.F., 1965. Water hammer charts for c e n t r i f u g a l p u m p systems, P r o c . Am. SOC. C i v i l E n g s . , 91 (HY3) 247-270. Kinno, H. ,1968 W a t e r h a m m e r c o n t r o l i n c e n t r i f u g a l p u m p s y s t e m s , P r o c . Am. SOC. C i v i l E n g s . , 94 (HY3) pp 619-639. L u d w i g , M. and Johnson, S.P., 1950 Prediction of surge pressures i n l o n g o i l t r a n s m i s s i o n l i n e s , P r o c . Am. P e t r o l e u m I n s t . , N.Y. 30

.

(5).

.

L u p t o n , H.R., 1953 G r a p h i c a l a n a l y s i s o f p r e s s u r e surges in pumpi n g s y s t e m s , J . ~ n s t . W a t e r E n g s . , 7. P a r m a k i a n , J . , 1963. W a t e r Hammer A n a l y s i s , D o v e r P u b l i c . I n c . , N.Y. R i c h , G.R., 1963. H y d r a u l i c t r a n s i e n t s , D o v e r p u b l i c s . I n c . , N.Y. and L a i , C., 1963. W a t e r h a m m e r a n a l y s i s i n c l u d i n g Streeter, V.L. f l u i d f r i c t i o n . P r o c . Am. SOC. C i v i l E n g s . , 88 (HY3) pp 79-112. Streeter, V.L. and Wyl ie, E.B., 1967. H y d r a u l i c T r a n s i e n t s , McGrawHill. D., 1966. W a t e r h a m m e r c h a r t s i n c l u d i n g f l u i d f r i c t i o n , Stephenson, P r o c . Am. SOC. C i v i l e n g s . , 92 (HY5) pp 71-94. Stephenson, D., 1972. D i s c h a r g e t a n k s f o r s u p p r e s s i n g w a t e r h a m m e r i n p u m p i n g l i n e s . P r o c . I n t n l . Conf. o n p r e s s u r e surges,B.H.R.A., C r a n f ie l d. Stephenson, D., 1972. Water hammer p r o t e c t i o n o f p u m p i n g l i n e s , t r a n s . S.A. I n s t n . C i v i l E n g s . , 14 ( 1 2 ) .

L I S T O F SYMBOLS

A

-

p i p e cross sectional area

B

-

a i r vessel

p a r a m e t e r v 'AL/(gHoS)

c

-

t h r o t t l i n g p a r a m e t e r Z/H

d

-

p i p e l i n e diameter

0

E

-

modulus of e l a s t i c i t y of p i p e w a l l m a t e r i a l

F

-

pump rated efficiency

f

-

Darcy friction factor

( e x p r e s s e d as a f r a c t i o n )

-

g r a v i t a t i o n a l acceleration

h

-

p r e s s u r e h e a d a t an i n t e r m e d i a t e s e c t i o n o f t h e p i p e l i n e

h'

-

water hammer h e a d r i s e measured above the d e l i v e r y h e a d

hf

-

f r i c t i o n h e a d loss

9

he -

h e a d loss t h r o u g h d o w n s t r e a m v a l v e f u l l y o p e n

H

head

-

(in

in

p i p e l i n e measured

c a s e of

air

vessel

above

design,

pump suction reservoir

take

H

as absolute,

atmospheric head) p u m p i n g head above suction r e s e r v o i r level

Ho I -

p u m p i n e r t i a p a r a m e t e r MN'/WALH

J

p u m p p a r a m e t e r F M N Z c / l 80wALv gH

-

0

0

'

0

i.e.

level plus

K

-

b u l k modulus of water

e

-

length of an intermediate p a r t of p i p e l i n e

L

-

pipeline length

M

-

moment

N

-

pump speed i n rpm

P

-

p i p e l i n e parameter cv /gh

Q

-

volume of water discharged from discharge tank

s

-

volume of a i r i n i t i a l l y

t

-

time

T

-

l i n e a r v a l v e closure time

v

-

water v e l o c i t y i n p i p e l i n e

v

-

i n i t i a l water velocity

of

inertia

t r a i n e d water

of

rotating

p a r t s of

motor

and

( = mass x r a d i u s of g y r a t i o n ' )

0

0

i n a i r vessel

in pipeline

w

-

weight of water per u n i t volume

x

-

distance a l o n g p i p e l i n e from pump

Y

-

wal I thickness o f p i p e

z -

pump,

head loss through a i r vessel Darcy f r i c t i o n factor

i n l e t f o r p i p e l i n e velocity

=

en-

97

CHAPTER 5

A I R IN PIPELINES

INTRODUCTION

It The flow or

is

recognised

that

air

a i r may be absorbed at at

the

entrance

i n free form

tively

large

they

will

the

air,

of

remain

in

The

in

likely

air

water

entrained

may

An

to

many

or

air

pockets.

The pockets may

either

present

line.

i n bubbles or

volume

section.

cross

to

is

free surfaces,

thus

pipelines.

in

be

pocket

turbulent

in

accumulate on top of

travel

along

equilibrium,

the

solution

implies a r e l a the pipe

l i n e to peaks.

be e n t r a i n e d by

There

the f l o w i n g

water or be released through a i r valves.

Air It

i n solution

i s only

air

losses the to

when

form

to

i s not

increase. to

form

those

in

partly

on

the

problems

bubbles

large full

drain

the system,

of

used

the to

pipe. release

l i n e a n d to draw

arise.

may

pockets.

coalesce

Flow pipes,

Air

valves

of

the

top

and

rise

to

conditions

then

become s i m i l a r

except

a

quantities

i n from

dissolved

b u l k s a n d head

that

in

slightly

pipeline

it

which collects

different

air

of

a

of

w i l l be pressurized.

used to e l i m i n a t e the a i r

large air

to permit

The water

i n c l u d i n g the free a i r ,

valves are frequently

top

also

that

The

pipe

Air

to present many engineering problems

the pressure reduces s u f f i c i e n t l y

bubbles

i s l i k e l y that

likely

during

design

the

are

filling

of

the atmosphere d u r i n g vacuum con-

d i t i o n s i n the line.

PROBLEMS O F A I R ENTRAINMENT

Air fects.

drawn

in

through

Minute a i r

bubbles o r

- the formation of lapse and erode lers

in

Air

vacuous

air

supply in

due

drawn

flow to be unsteady.

to in

the in

which

have can

a number of

ef-

promote c a v i t a t i o n

subsequently

rapidly

col-

T h i s effect occurs i n pump impel-

high

gulps

can

solution

cavities

the pump o r pipe.

particular

pressures.

pump

a

peripheral

can

cause

speed

which

vibration

by

lowers causing

98

Large quantities intermittent1

-

-

-__

a.

Low level sump

I-

i

I

Air e n t r a i n e d by a falling jet

o

* 1

+

'

To pump

Air enters intake b. Free fall into sump

__1-

____

Surface dimple barely detectable

-=-+

---+

Well define surface dimple

-.

Air drawn intermittent ly from bottom of vortex into intake

Air core extends into intake (fully developed entraining vortex)

c . Vortex formation

FIG.

5.1

Inlet arrangements conducive

to a i r

entrainment.

4

99 Air

i n a r i s i n g main,

emerge

Air

from

solution

if

whether ambient

in free form w i l l collect a t

pipes o r pocket

the

acceleration

in the pipe.

of

The

dispersed

pipeline w i l l

increase.

include surging,

by

latter

the water

b u b b l e form,

may

the top of

be washed along

air

or

temperature

may

i s reduced.

the p i p e l i n e and then r u n

Here i t w i l l e i t h e r escape through a i r v a l v e s o r

u p to h i g h e r points. vent

i n s o l u t i o n o r i n b u b b l e form, pressure

past

the velocity will

result

the a i r

i n the water,

of

the water past the

i n a head loss due to

pocket.

Even

if

air

is in

the f r i c t i o n head loss a l o n g the

Other problems caused b y a i r present i n pipes corrosion,

reduced pump e f f i c i e n c y , malfunction-

i n g o f v a l v e s or v i b r a t i o n s .

Air

may

water, size

in

present

bubbles,

1

from

velocity

be

is

mm/s)

that

rising

to

to

slow

they

the

concentration

micro

5 mm.

so

in

the

bubbles

Micro

in

surface.

or

pockets

in

bubble,

mm

suspension

I n fact

on

solution.

the

90 rnm/s,

for

a

top

Bubbles

be smaller,

turbulence

p r o f i l e i n the same way

sion i n f l o w i n g water.

of

bubbles may

1

(e.g.

stay

form

of

the

0.1

mm

considerable

bubble,

time

4

before

may create a n e q u i l i b r i u m

that

sediment

s t a y s i n suspen-

The r i s e velocity of small a i r bubbles i s

w = dgZ/18 v

(5.1

where the kinematic

in

range

and t hei r r i s e

v of water a t 20 C i s 1.1

viscosity

1

x 10-6m2/s, d

i s b u b b l e diameter a n d g i s g r a v i t a t i o n a l acceleration. For

air

bubbles

to

form

should be present.

Then

be

solution.

released

from

depends

on

the

2% a i r

by

volume

readily

temperature

nucleus

capacity

of

and pressure.

The

At

measured a t

f i g u r e v a r i e s from 3.2% at O°C

AIR

a

or

uneven

when conditions a r e correct,

standard

water

to

dissolve

air

20 C water may absorb

atmospheric

down to 1.2% a t

surface

air will rapidly

pressure.

This

looo.

INTAKE A T PUMP SUMPS

The major source of a i r i n pumping lines i s from the i n l e t sump o r forebay.

Here the water exposed to the a i r

depending the water.

on

its

temperature,

will

pressure a n d

absorb a i r a t a r a t e

degree

of

saturation

of

100

a.

Air

b.

Air

c.

pocket w i t h s u b c r i t i c a l

flow past.

pocket w i t h super-critical

A i r pocket

F i g . 5 .2

Air

flow past.

in e q u i l i b r i u m position. pockets in pipelines.

101 Air

in

flowing

form

free

water.

may

The

also

suction p i p e entrance w i l l entrance likely

velocity

to

be

partially carried

and

drawn

when

be

formation

dragged a

greater

This

free

in

vortex

entice a i r

the

in.

of

the conduit

to

or

drawdown

i n t o the conduit. the

turbulence,

air

may

later

the

conduit.

As

pressures

again

the the

The h i g h e r the

the

more

dissolve

air

is

wholy

or

I n any case i t i s

pressures increase beyond the pump.

along

by

outside

reduce,

it

may

be

released from solution. The the

configuration

tendency

draw

to

(1957), Prosser to

in air.

intake.

approaches

be1 lmouthed

the pump sump has an

and

important

b e a r i n g on

Extensive studies b y Denny a n d Young

(1977) and others have

minimize a i r

concentric

of

i n d i c a t e d how to design sumps

C i r c u l a t i o n i n the sump should be avoided b y and

preferably

straight facing

inflow.

The

downwards

inlet

should

be

Hoods ( s o l i d

upstream.

or p e r f o r a t e d ) above the i n t a k e minimize a i r intake. The

degree

entraining

of

submergence

drops . o r

hydraulic

should

be

jumps

are

as

great be

to

as

possible.

avoided.

Fig.

Air 5.1

indicates some i n l e t arrangements to be avoided.

AIR ABSORPTION AT FREE SURFACES

The

rate

of

diffusion

of

gas

across

a

liquid

interface

can

be

expressed i n the form

-dM_ dt

-

AK

(Cs-

C)

(5.2)

where M i s the mass r a t e of exposed, Cs

K

is a

transfer

l i q u i d f i l m constant,

is the concentration a t

v i s c o s i t y a n d turbulence.

saturation.

per

C

u n i t time

t,

A

i s the area

i s the gas concentration a n d

K is a f u n c t i o n of temperature,

T h i s equation i s often w r i t t e n i n the form

(5.3) where r i s the d e f i c i t r a t i o , Co i s the concentration a t time 0, i s the volume of water per surface area A ( V / A

= depth of water).

and V

102 HYDRAULIC REMOVAL OF A I R

Air

trapped

increase

the

in

air

a p i p e and allowed to pocket

The

size.

accumulate,

water

cross

some o r a l l of

the

Alternatively

water.

entraining

air

and

gradually

sectional

therefore d i m i n i s h a n d the velocity of the water w i l l some stage,

will

area

will

increase u n t i l at

the a i r w i l l be dragged along the l i n e b y a

hydraulic

carrying

i t away

jump

may

form

i n b u b b l e form,

in

as

the

pipe

depicted

in

F i g . 5.2 The pipe

relationship

diameter

between

has

been

a i r pocket volume,

investigated

by

a

washout

number of

these r e s u l t s a r e summarized by Wisner et a l

velocity

and

workers,

and

(1975).

Kal inske a n d B l i s s (1943) produced the f o l l o w i n g equation f o r r a t e of water flow Q at which removal commences:

Q2

5

gD

= 0.707

where D

e

tan

e

(5.4)

i s the p i p e slope angle.

equation energy

i s g r a v i a ional acceleration and

i s the p i p e diameter and g

is s i m i l a r diagrams

possibility critical.

that

The term on the l e f t h a n d side of K a l i n s k e ' s the

to

5.3

(Figs. the

air

in

parameter

is

the

specific

momentum a n d

In

fact

points

5.4).

and removed

if

the

water

it

depth

to

drops

the

below

Then a h y d r a u l i c jump forms downstream which could e n t r a i n

air. Wisner air

et

al

(1975) produced d a t a from which

pockets may be deduced.

independent of

slope and

r e l a t i v e volume of (see F i g .

Hydraulic A

air

They

is a

function

i s practically

of Reynolds number

vD/v

and

pocket 4U/71D3 where U i s the a i r pocket volume

5.5)

Jumps

hydraulic

jump

draws

in

air

in

the

bubbles e x h i b i t a s u r p r i s i n g l y low tendency free form f o r the water,

the r i s e velocity of

i n d i c a t e r i s e velocity

a

long time.

form

of

bubbles.

These

to coalesce a n d remain i n

A i r may be absorbed from

the bubbles by

b u t t h i s takes a long time a n d many of the bubbles r i s e to

the surface before they a r e absorbed.

103

104

1

>

1o2

Fig.

5.5

= VD/v

E q u i l i b r i u m velocity of water f o r at l o 0 to 70° to the h o r i z o n t a l .

Kalinske

and

Robertson

rate

air

entrainment

of

(1943) at

air

pockets

in

a

pipe

found from model experiments that a

hydraulic

jump

was

given

the

by

the

equation Qd/Q

where

= 0.0066(F1

Qd

is

the

- 1)le4

(5.5)

volumetric

r a t e of

air

entrainment,

Q i s the water I

flow

rate

F1

and

is

the

upstream

r e l a t i o n s h i p was d e r i v e d f o r free

surface

considerably

downstream. greater

air

a jump

Froude

vl//gyl.

That

i n a r e c t a n g u l a r channel w i t h a

Experiments entrainment

number

by

rate

the

author

i n closed

pipes

indicate

a

(see Fig.

5.6)

Free F a l l s

A

jet

falling

free

into

a

pool

of

water has a s i m i l a r

air

intake

105 effect was

to

a

low

jump.

Avery they

head

the

represents

found

aeration the

this

r a t e of

higher was

expended

air,

the base of

For an

a

for

efficiency

energy in

i n t a k e at

(1978).

a e r a t i o n at

decreasing

21% oxygen

Assuming

Oxygen

a n d Novak

losses,

loss

head

term

by

50%,

ciency of for m

hydraulic

studied

lost

or

1

free f a l l s

oxygen defi-

to 1.6

up

head

only

would

initial

k g 0 /kWh

2

losses,

e.g.

k g 02/kWh.

in

the

indicate

The

or

jump

aeration

1

at

kW

fall.

rates

for

a i r u p to 8 kg/kWh. Water

flowing

amount

inlets

with

a

fall

free

e.g.

morning

glory

o r even siphon s p i l l w a y s w i l l a l s o e n t r a i n a consider-

type s p i l l w a y s , able

into

of

air

1976).

(Ervine,

Gravity

pipelines

are

therefore

as much of a problem as pumping lines.

0

2

F:

6

4

~

v

J3A/B Fig.

5.6

A i r removal a t h y d r a u l i c jumps

in c i r c u l a r conduits.

A I R VALVES

Air valves. fices air

accumulating These

are is

up

are to

3

accumulated

in

a

pocket

normally mm

in

of

in

the

i n diameter). a

chamber

a

pipe

'small

may

be released b y

orifice'

type

(typical ori-

The v a l v e opens when to

permit

a

ball

to

air

sufficient

be

released

from a seat around the o r i f i c e . On

the other

hand a i r discharge

d u r i n g f i l l i n g operations

(before

106

This sectio parallel t o hydraulir gradient constitutes1 peak Section Of pipeline running parallel t o hydraulic gradient and constitutes peak

Section of pipeline forming peak with respect t o horizontal and 0150 t o hydraulic gradient and peok with respect t o hydraulic gradient only

No peok Horizontal Section Of Pipeline having d w n w a r d grade and point of increaseof downward grade

Datum

Section of pipeline having upward grade and point of decrease of upward grade

Scour

Horizontal Long descending section of pipeline

FIG.

5.7

P o s i t ~ o no f

Datum

Long ascending section of pipeline

a i r valves along pipelines

107 pressurization) may

be

leased pipe

and

through from

is

air

the

seat

Fig.

low.

i n t a k e d u r i n g vacuum conditions

'large orifice' around

10.6

a i r valves, the

orifice

illustrates a

the b a l l of when

double

in

pressure

air

valve

the p i p e

which

i s re-

inside

with

the

both

a

small and a l a r g e o r i f i c e . The sures

size a n d spacing of outside

the

pipe

a i r valves w i l l

and

permissible

depend on ambient pres-

pressures

inside,

the

size

of p i p e a n d water flow rates. The equations

for

the

discharge r a t e

the c o m p r e s s i b i l i t y of

air

and

are

as

through

follows

pressure beyond the o r i f i c e p 2 i s g r e a t e r absolute i.e.

orifice

(Marks,

than

gauge p l u s atmosphere a n d p,

an

involve

1951).

0 . 5 3 ~ ~( a l l

If

the

pressures

i s i n i t i a l p r e s s u r e ) , then

(5.6) For p 2 ' 0 . 5 3 ~ ~ then

the flow

becomes c r i t i c a l

a n d flow

r a t e i s inde-

pendent of p2. Then

(5.7) where W a

is

i s the mass r a t e of flow of a i r ,

the o r i f i c e area,

rn

is

the

mass

C i s a discharge coefficient,

density

of

air

and

letting

air

into

k

is

the

a d i a b a t i c constant. Normally or

out

of

whether it,

p2

for a i r a n d C = 0.5 Qa = 0.34 a

where second a

the

is

air

less

valve

than

is for

0 . 5 3 ~ ~ Then .

i n t o the last equation,

substituting

the

k

is

if

SI

pipe 1.405

i t s i m p l i f i e s to

/-

Qa

=

(5.8)

tne

volumetric

units are

i s the area of

of

water a n d S a Since a i r density

r a t e of

employed)

the o r i f i c e , is

the

h

air

measured

at

( i n c u b i c metres p e r the

initial

pressure,

i s the i n i t i a l absolute head i n metres

relative

i s proportional

flow of

density to

of

air

at

initial

the absolute head,

pressure.

t h i s simp i f i e s

f u r t h e r to

( 5 9)

Qa = lOOa where

Qa

is

the

air

volume

flow

absolute) in m 3 / s a n d a i s in m'.

r a t e at

initial

pressure

(+

O m

Thus

t o r e l e a s e 1% a i r f r o m a p i p e f l o w i n g a t a v e l o c i t y of 1 m/s,

a n d under

low

t h e n (5.8)

head (10 m absolute being the minimum f o r no vacuum)

r e d u c e s to ~ . _ _ _

~

1 m/s x A = 0 . 3 4 a J 9 . 8

0.01

x

i.e.

d = 0.006D

thus

10/1.15

x

lo-’

x

(5.10)

the o r i f i c e diameter should

1% o f

be about

the

p i p e diameter

to

r e l e a s e 1% a i r b y v o l u m e i n t h e p i p e . The

theory

of

(1950)

to

cavity

formed b y

derive

metres

of

d

is

the

velocity

h2

C

the

orifice flow

water for

the

is is a

was

through

used air

If initial

head

column

a

Parmakian

valves

to f i l l

a in

the

is

beyond

orifice,

g

discharge coefficient f o r the valve,

each

a i r and S

by

head (absolute)

D i s the p i p e diameter,

diameter,

a d i a b a t i c constant f o r h2 > 0 . 5 3

hl,

is

acceleration,

the

for

an

p a r t i n g water columns.

orifice of

through

equations

water

gravitational

flow

s i d e of

the

air

V

i s the r e l a t i v e

k

valve,

is

i s t h e r e l a t i v e d e n s i t y of a i r ,

the then

hl,

(5.11) a n d f o r H
2-

(5.12) air

For

at

300

m

above

sea

level

and

air

temperatures

of

24OC

3

t h e n Sa

p l u s 20% h u m i d i t y equivalent

t o hl

=

9.97

i s 1.15

m of

x 10-

water.

k

a n d atmospheric pressure i s

i s 1.405.

For a i r valves

C

is

a b o u t 0.5.

HEAD LOSSES I N P I P E L I N E S

Air

suspended

increase

in b u b b l e form o r

t h e s p e c i f i c volume.

in p o c k e t s i n f l o w i n g

T h e mean v e l o c i t y

water

will

i s consequently h i g h e r

109 to convey

a c e r t a i n volume of water per u n i t time.

increase i n accordance w i t h an equation such

The head loss w i l l

as the Darcy- Weisbach

equation (5.13) v to use w i l l be v = ( 1

The velocity where v

W

i s the velocity

of

+

f)v

(5.14)

W

p u r e water f l o w i n g and f

i s the volumet-

r i c concentration o f free a i r i n the pipe. The head loss around a s t a t i o n a r y a

of

loss

velocity

head.

If

h y d r a u l i c jump

forms,

I f the velocity

i s subcritical

difference

in

velocity

the

the head

head

a i r pocket

size of

air

i s due p r i m a r i l y to

pocket

is

such

that

loss may be e v a l u a t e d from F i g .

throughout, is

lost.

a

5.4.

i t may be assumed t h a t the could

It

be

established

more

closely u s i n g momentum p r i n c i p l e s though.

WATER HAMMER

The water

presence of hammer

free

air

considerably.

in

pipelines

Fox

(1977)

can

reduce the s e v e r i t y of

indicates

that

the

celerity

(speed) of an e l a s t i c wave w i t h free a i r i s

c =

/*

(5.15) (5.16)

For l a r g e a i r contents t h i s reduces to c =where pipe

p i s the mass density

diameter,

b

i t s thickness,

absolute pressure and f c

2% o f

of water,

K i s i t s b u l k modulus, D i s the

E i t s modulus of e l a s t i c i t y ,

p

i s the

i s the free gas f r a c t i o n b y volume.

i s reduced r e m a r k a b l y f o r even r e l a t i v e l y low gas contents. air

at

a

pressure

head of

50 m of

water

Thus

reduces the c e l e r i t y

from about 1100 m/s f o r a t y p i c a l p i p e l i n e to 160 m/s. The Joukowsky water hammer head i s A h = SAv

(5.17)

9

where

Av

reduction

is in

the Ah.

change

in

velocity

of

flow.

There

i s thus

a

large

I f the a i r collects a t the top of the p i p e there i s no

110 reason on

to see why

the other

partly full

the same equation

cannot a p p l y .

Stephenson (1967)

h a n d d e r i v e d a n equation f o r the c e l e r i t y of a bore i n a The c e l e r i t y d e r i v e d from momentum p r i n c i p l e s i s for

pipe.

smal I a i r proportions J gAh/f

c =

where

Ah

(5.18)

i s the head r i s e b e h i n d the bore.

of 158 m/s f o r f = 0.02 There valves

a n d Ah = 50m.

i s a school of

in

pipelines

pressures.

The

cessively

large

reduction

in

thought

as

a

intention

p r o a c h i n g columns.

head.

of

The

hammer

which f a v o u r s the i n s t a l l a t i o n of a i r

means

is

air idea

in

of

reducing

primarily

to

wi II ,

Calculations

volume

a l l e v i a t e water

This indicates a celerity

water

cushion

however,

the

hammer

over-

impact of

indicate that

ap-

an ex-

i s r e q u i r e d to produce any

significant

stems

vessels

from

pipelines.

the

I t will

use of

air

be r e a l i s e d that

air

to in

vessels i s under h i g h pressure i n i t i a l l y a n d therefore occupies a

air

smal I

relatively trip,

volume.

Upon

pressure

reduction

following

puk = constant where

U

pump

(5.19)

i s the volume of

necessary

volume

of

air.

air at

a n a l y s i s to be excessive f o r An

a

the a i r from an a i r vessel expands according to the equation

unusual

low

The size of

air

v a l v e s to draw i n the

(vacuum) pressures w i l l

be found on

l a r g e diameter pipelines.

problem due possibly

to a i r coming out of

solution

in

a r i s i n g main was reported b y Glass (1980). Here a t h i n stream of a i r along after

the a

top

pump

of

the

trip.

line

After

a

supposedly number of

collapsed on years

the

a

pressure

rise

p i p e b u r s t along a

l i n e a l o n g the s o f f i t .

REFERENCES

Avery, S.T. and Novak, P, 1978. Oxygen t r a n s f e r at h y d r a u l i c structures. Proc. ASCE, HY11, 14190, pp 1521-1540. 1957. The prevention o f vortices i n Denny, D.F. a n d Young, G.A.J., intakes. Proc., 7th Con. I n t . Ass. Hydr. Res., Lisbon E r v i n e , D.A., 1976. The entrainment of a i r i n water. Water Power and Dam Construction. pp 27-30. 1977. H y d r a u l i c A n a l y s i s o f Unsteady Flow i n Pipe NetFox, J . A . , works. Macmillan, London. Glass, W.L., 1980. C a v i t a t i o n of a pump p i p e l i n e . Proc. I n t . Conf. Pressure surges. BHRA, Canterbury.

111 K a l i n s k e , A.A. and B l i s s , P.H., 1943. Removal o f a i r f r o m p i p e l i n e s b y f l o w i n g w a t e r , C i v i l E n g i n e e r i n g , ASCE, 13, 10. p 480. K a l i n s k e , A.A. and R o b e r t s o n , J.M., 1943. C l o s e d c o n d u i t f l o w , T r a n s . ASCE. 108, 2205, pp 1453-1516. M a r k s , L.S., 1951. M e c h a n i c a l E n g i n e e r s H a n d b o o k , 5 t h Ed. McGraw H i l l , N.Y. 2235 pp. P a r m a k i a n , J., 1950. A i r i n l e t v a l v e s f o r s t e e l p i p e l i n e s . T r a n s . , ASCE, 1 1 5 , 2404., pp 438-444. P r o s s e r , M.J., 1977. T h e H y d r a u l i c D e s i g n of Pump Sumps and Int a k e s , BHRA, and C I R I A , L o n d o n . 48 pp. Stephenson, 1967. P r e v e n t i o n o f v a p o u r p o c k e t s c o l l a p s e i n a p u m p i n g l i n e . T r a n s . , S o u t h A f r i c a n I n s t . C i v i l E n g s . 9 , ( l o ) , pp 255-261. W i s n e r , P . , Mohsen, F.M. and Kouwen, N., 1975. Removal o f a i r f r o m w a t e r l i n e s b y h y d r a u l i c means. Proc., ASCE. 101, HY2, 11142, pp

243-257. LIST

OF SYMBOLS

a

-

orifice area

A

-

area

b

-

wal I thickness

c

-

wave celerity

c

-

d i s c h a r g e coef f i c i e n t

c

-

gas concent r a t ion

co -

i n i t i a l concentration c o n c e n t r a t i o n at s a t u r a t i o n

d

-

D

-

p i p e internal diameter

cs

b u b b l e diameter

E

-

modulus of e l a s t i c i t y

f

-

volumetric concentration of a i r

F

-

Froude number

9

-

g r a v i t a t i o n a l acceleration

h

-

head

k

-

a d i a b a t i c constant

k

-

I i q u i d f i Im c o n s t a n t

K

-

bulk m o d u l u s

m

-

mass d e n s i t y o f a i r

M

-

m a s s r a t e o f t r a n s f e r o r s p e c i f i c momentum

P

-

pressure

Q

-

water discharge r a t e

‘d

-

r

-

volumetric r a t e of air entrainment gas d e f i c i t

ratio

112

s

-

t

-

time

u

-

a i r pocket volume

r e l a t i v e density o r s p e c i f i c g r a v i t y

v

-

water v e l o c i t y

v

-

volume of water p e r surface area

w

-

r i s e velocity

w

o f bubbles

-

mass r a t e of flow of a i r

Y

-

depth of flow

x

-

Darcy f r i c t i o n factor

e

-

p i p e slope a n g l e

v

-

k i n e m a t i c viscosity

P

-

mass density of water

113

CHAPTER 6

EXTERNAL LOADS

Low large as

pressure

diameter

well

tion

as

with

pipes,

especially

pumping

internal vacuum

mains

loads.

sewers,

should

or

even

be designed f o r e x t e r n a l

loads

The v e r t i c a l

pressure

inside

the

gravity

soil

mains

load a c t i n g

p i p e could

in combina-

cause

the

pipe

to

collapse unless the p i p e i s adequately supported o r stiffened.

S O I L LOADS

The

load

transmitted

to

a

pipe

from

the

external

surroundings

depends on a number of factors: Rigidity sidefill

of

pipe:

The

more r i g i d

the more load i t w i l l

causing a

l a r g e p a r t of

a

take.

pipe

is

relative

The s i d e f i l l

the b a c k f i l l

to

tends to settle,

to rest on the pipe.

w i t h f l e x i b l e p i p e too to some extent,

the trench thus

T h i s occurs

as a p i p e i s supported l a t e r a l l y

b y the f i l l a n d w i l l not y i e l d as much as a free s t a n d i n g pipe. Type

trench o r f i l l :

of

lation

conditions

w i t h the w i d t h

for

Fig.

pipes.

a n d depth of

trench affects the r e s u l t a n t different

loads

to

a

pipe,

6.1

The

i l l u s t r a t e s v a r i o u s possible i n s t a l load

transmitted

to

the p i p e v a r i e s

trench since f r i c t i o n on the sides of the load.

Embankment f i l l s may

depending

on

the

relative

also t r a n s m i t settlement

of

s i d e f i l l and topfi I I . Young sive

and

equations

various

trench

Trott and and

(1984), a n d C l a r k e charts

for

evaluating

soil

embankment

conditions.

Although

assumptions a r e subject yet

no other

(1968) have developed

to question,

loads

on

many

exten-

pipes of

in

their

the f a c t remains t h a t there i s as

theory w i t h which to c a l c u l a t e s o i l loads on pipes so the

engineer must use t h i s theory w i t h d i s c r e t i o n

(ref.

Spangler,

1956).

Trench Conditions

The s o i l

load transmitted

t h e w i d t h a n d depth of

For

a

normal

to a

r i g i d pipe

in a

trench depends on

trench a n d the s o i l b a c k f i l l properties.

vertical-sided

trench

(Fig.

6.la),

fill

at

the sides of

114 the p i p e w i l l

settle more than

be neglected.

On

the sides of trench

fill

developed

and is

the

sides

of

the

pressure

the

trench

the downward forces,

load.

trench

is

to

the

the trench,

the

to

Equating to

vertical

the

proportional

a n d the sides of

horizontal

a n d the s i d e f i l l support can

h a n d the f r i c t i o n of

takes some of

therefore

between the f i l l active

the other

the trench

the pipe,

vertical

u p w a r d forces

on

any

the b a c k f i l l a g a i n s t The cohesion between

neglected.

The

friction

of

friction

coefficient

a n d the r a t i o K of pressure horizontal

in

the

slice

Marston e v a l u a t e d the total

the

soil. in

the

load on a

p i p e at depth H ( 1 9 1 3 ) : Upward

load

per

unit

length

depth h + ah below ground

Pipe a t

of

s u r f ace : W + dW = W

+

yBdh

2K t a d

-

Wdh/B

Solution g i v e s

W = CdyBZ -2K(tane )H/B where t h e load coefficient C

K

@

d

=

2Ktan6

=

r a t i o of

l a t e r a l s o i l pressure to v e r t i c a l

-

1 - s i n 9 f o r a c t i v e s o i l conditions 1 + sin 0

=

a n g l e of

load.

a n d cohesionless soil

i n t e r n a l f r i c t i o n of b a c k f i l l

e

=

a n g l e of f r i c t i o n between b a c k f i l l a n d sides of trench

H

=

h e i g h t of f i l l above p i p e

B

=

trench w i d t h

Y

=

u n i t w e i g h t of b a c k f i l l m a t e r i a l

Ktane

n o r m a l l y ranges from 0.11

coarse crushed stone. Ktan8.

l o ) , Cd

Note

that

For

approximately

i s given

f o r deep

approaches a

the s i d e f r i c t i o n trench.

Cd

soft c l a y s to 0.16

i n Fig.

trenches,

(H/B

6.2

for

greater

v a r i o u s values for than

shallow

equal to H/B,

takes more of or

wide

the

most of

approximately

load the deeper t h e

trenches

(H/B < l ) ,

Cd

is

so

W = yHB

i.e.

for sands and

l i m i t i n g v a l u e of 1/(2 K t a n e ) . T h i s implies that

i n the trench

very

for

(6.2)

the b a c k f i l l

load i s taken b y the pipe.

The trench equa-

115

ALTERNATIVE /-FORM

SIDE

b : WIDE TRENCH SHALLOW EMBANKMENT

o S R R O W TRENCH

dSHALLOW EMBANKMELT PLSlTlVE PROJECTION

c :

DEEP

EMBANKMENT PROJECTION

POSITIVE -__

TA2r - - - - - -- - - - --- --

g SHALLOW E M B A N K M E N T

N E GAT1VE PROJECT1 ON

h:

DEEP EMBANKMENT N E GAT IVE P R O JE CT ION

1 TUNNEL OR H E A D I N G OR THRUST BORE

Fig.

6.1

A l t e r n a t i v e backf i I I s

c

W I D E TRENCH D E E P EMBANKMENT

_-

f : EMBANKMENT NEUTRAL PROJECTION

NORMAL DENSITY FILL

----

__-_.__ ---/

i

EMBANKMENT INDUCED _ T R-E N C H

116 tion

indicates

the

trench

that

width

the

increases.

the equation

width

load on a r i g i d p i p e increases i n d e f i n i t e l y as

no

next section)

then

apply

evaluate

the

load

trying

and

embankment

the

This

i s not

the case and a t

some trench

longer a p p l i e s a n d embankment conditions (see F i g . both

6.lb

the

criterion

and 6 . 1 ~ ) . I t i s necessary to

trench

(using

(see

criterion

6 . 3 ) , and

Fig.

6.2)

(using Fig. select

to

that

load which i s least. In

the

case

the crown of

of

'V'

a

trench,

i s used since the shear

of

the

trench.

If

the

flexible

width

to

I n t h i s case @ instead of 0

the pipe.

Cd

relatively

the trench

plane

is

side-fill

then

the

is

load

use

i s that

i n the formula f o r

i n the f i l l not against well

on

compacted pipe

the

at

and

the sides

the

depends

on

pipe the

is

pipe

diameter a n d not to the same extent on the trench w i d t h : W

C YBD (flexible pipe) d

=

Note that f o r small H, W

(6.3)

C d = H/B

and i n t h i s case

YHD

=

Embankment Conditions

A p i p e bedded on a f i r m surface w i t h a wide embankment f i l l over it

is

pipe

said

to

be

projects

projection positive natural

under

above

(Figs.

6.ld)

projection

if

ground

the p i p e crown projection

or

embankment

the

level,

and

pipe

and

the

conditions. ground

6.le).

the

i s below

trench

original

It

is

is

rigid

embankment

(Figs.

6.lg

the it

referred and

the n a t u r a l ground

condition

If

level

is

to

lies

crown

the

a

positive

as

a

complete

completely

relatively

level

of

is

above

shallow.

If

one has a negative

and 6 . l h ) .

For very

deep

trenches u n d e r embankments t h e conditions may be the same as f o r the trench

condition

( r e f e r r e d to

If

crown

level

the

projection For tends to

drag

=

with

the n a t u r a l

complete

positive

projection

s e t t l e more than downwards

on

the load on the pipe. W

as the complete trench c o n d i t i o n ) . ground

level

it

i s a neutral

(Fig. 6.lf).

the to

is

here

CcyD2

the

directly fill

case the f i l l

above the pipe.

directly

above

beside the p i p e

The s i d e f i l l

the p i p e and

tends

increase

The r e s u l t i n g load per u n i t l e n g t h of p i p e i s

117 where the load coefficient

e cc =

2K [email protected] / D

-1

2Ktan@

LOAD COEFICIENT C,

F i g . 6.2

Load coefficient C d f o r trench conditions.

The l i n e

l a b e l l e d complete p r o j e c t i o n condition

v a l u e of

C

condition

.

for

Ktan@

= 0.19,

in F i g .

For deep embankments,

only

u

=

of

g i v e s the

a c e r t a i n h e i g h t of f i l l

the p i p e w i l l s e t t l e at a d i f f e r e n t r a t e to the s i d e f i l l . depends on the product He, settlement r a t i o s where

6.3

which i s the minimum adverse f r i c t i o n

the

pipe

projection

above

This height, ratio

projection of p i p e crown above o r i g i n a l ground level diameter of p i p e

u

and

the

118 s

settlement of adjacent f i l l o r i g i n a l l y at crown levelsettlement of p i p e crown reduction i n thickness of adjacent f i l l below crown level

=

The settlement ground

of

adjacent

fill

includes

level a n d the compaction of

p i p e crown

includes settlement of

the settlement of

the s i d e f i l l .

the bottom of

the o r i g i n a l

The settlement of the p i p e and v e r t i c a l

the

deflection of the pipe. The v a l u e of tion

soil

to

i s 0.7

value well

s n o r m a l l y v a r i e s from 0.3

1.0

for

compacted.

resulting

rock

b u t 0.5 The

product

soil

A

foundation.

common

is

su

for

used i n F i g .

6.3

to e v a l u a t e the

v a r i o u s embankment conditions.

I t i s not

.

necessary to e v a l u a t e H for

unyielding

soft y i e l d i n g founda-

i s recommended i f the foundation a n d s i d e f i l l a r e

load coefficient

The equation

or

for

for

C

deep embankments a n d

incomplete p o s i t i v e

projection conditions

i s more compl icated than f o r the shal low embank-

ment

condition,

it

(i.e.

minimum

but

was e v a l u a t e d by Spangler f o r

adverse

friction)

condition,

and

Ktan4

these

=

0.19

values

are

g i v e n i n F i g . 6.3. Fig. the

6.3

pipe

also

settles

Spangler

gives more

C

evaluated

friction)

for

label led

complete

this

values

than for

case, and

the

C

of

Ktanb

and

for

adjacent =

0.13

the

case

backfill (i.e.

when

(i.e.

minimum

the

values

are

incomplete

trench

conditions

the

top

negative

indicated

of 5).

favourable

in

since

6.3,

Fig. the

mech-

anics a r e s i m i l a r to those f o r trench conditions. For Fig.

w

the

6.19 =

where

negative

and 6 . l h )

projection

=

Cn

=

trench)

condition

(see

(6.6)

i s the same as C

i n Fig.

6.3 f o r

negative su b u t

with

I n t h i s case the projection r a t i o i s

depth of p i p e crown below n a t u r a l ground level w i d t h of trench

a n d the settlement

s

incomplete

CnYB2

replaced b y 6 i n a l l expressions. u

(or

w i t h the p i p e i n a trench under an embankment,

ratio i s

settlement of n a t u r a l ground level-settlement of level of top f i l l i n trench reduction in thickness of s o i l in trench above crown

D

119 1

Load coefficient C

F i g . 6.3

(Spangler,

f o r embankment conditions

1956).

The settlement of the level of the top of the f i l l i n c l u d e the settlement of the bottom of the pipe, a n d compaction

the f i l l

of

s may r a n g e from - 0.1

It

a

certain

replacing

negative

it

by

projection

r a n g e from -0.5 The

The

=

f o r u = 0.5,

load

amount

of

lightly

to - 1.0 f o r u = 2.0.

f o r u = 0.5 per

unit

compacted

compacted

condition

an embankment whether W

deflection o f the p i p e

i n the trench above the crown of the pipe.

i s possible to reduce the load on pipes under a n embankment b y

removing and

i n the trench may

(Fig.

to -2.0

l e n g t h on

fill

fill

6.li).

d i r e c t l y above the p i p e

directly, For

this

i.e.

inducing a

condition

may

f o r u = 2.0. a completely f l e x i b l e p i p e under

i n p o s i t i v e o r negative projection condition

YHD load on a

s

is

(6.7) r i g i d pipe i n a

p r o j e c t i o n c o n d i t i o n ) i s a l s o yHD.

trench up to

i t s crown

level

(neutral

120 C l a r k e suggests

lar

to

for

that

on a

that pipe

the

l o a d on

a

in a

trench,

w th

t h e cohesion of t h e m a t e r i a l above.

this

conclusion

are

not

or

tunnel a

thrust bore i s simi-

reduction

factor

allowing

The a s s u m p t i o n s i n a r r i v i n g a t

and

convincing

further

theory

would

be

extent

be

we1 come. Clarke

also

suggests

that

t r e a t e d a s embankments a n d preferable take

to

place

using

between

uniform

surcharges

the equivalen t

pure elastic

theory,

soil

different

fills

at

depth

as

of

large

determined.

some

load

densities

This

transfer

and

soil

is

must

masses

which settle differently. The pipe

soil

is

conditions the

density

the

are

water

which

saturated normally

pressure

would

(but

not

less

would

in

result

the

submerged)

severe

than

produce

a

for

the

uniform

l i k e l y to c r a c k a p i p e t h a n a p u r e v e r t i c a l

maximum

density.

load

The

saturated

radial

on

a

submerged case,

pressure

as less

load.

Example - N e g a t i v e p r o j e c t i o n case The

top

ground

of

a

pipe

level

trench.

and

in

2 m wide

a

there

is

7

a

trench

m

high

1

is

m below

embankment

natural

above

the

E v a l u a t e the load on the p i p e .

Projection r a t i o u =

1 2

- =

0.5

Settlement r a t i o s = -0.4 s a y .

H/B = 8 / 2 = 4 . From F i g .

6.3,

C

load coefficient

= 3.1

L o a d p e r m o f p i p e = CnyB2 = 3.1

x

18000 x 2*/1000 = 220KN/m

SUPER IMPOSED LOADS

It

is

customary

transmitted infinite,

the

surface

homogeneous,

The f a c t in

by

to

that

theory

transmitted

use

elastic

loads

to a

isotropic,

theory

to

evaluate

buried pipe and elastic

material

the

pressures

to assume a semi-

surrounds

the p i p e .

t h e s i d e f i l l may s e t t l e d i f f e r e n t l y to t h e p i p e i s i g n o r e d and suitable factors

load

if

the p i p e

s h o u l d be a p p l i e d

is flexible.

to

decrease

the

U n f o r t u n a t e l y no such f a c t o r s

a r e a v a i l a b l e yet. If the

t h e l o a d a p p l i e d o n t h e s u r f a c e i s of l i m i t e d l a t e r a l e x t e n t then induced

pressure

on

any

horizontal

plane

below

the

surface

121 decreases w i t h depth. consequently

the

The deeper

the p i p e ,

the w i d e r the spread and

lower the maximum pressure.

On the other

h a n d the

total v e r t i c a l force must remain equal to the superimposed load. Formulde point

load,

for or

evdluating

the

sections

The

impact factor

below

2.0

the

safe

varies

but

adopt

depth

the type of

from

full

due

a

to

surface

are given

in

1.3

load a n d depth of p i p e

for

slow

moving vehicles

i f the p i p e i s shallow.

greater

the amount of

the

loads

i n f i n i t e extent,

impact f a c t o r should a l s o be a p p l i e d .

with

moving vehicles cover

loads, to

vary

and

transmitted

limited or

suitable

will

surface,

reduces f o r point

A

below.

f o r fast

the

a pressure of

than

of

The impact factor

the p i p e diameter

reduction

value

the

and

i s debatable and

impact

to

factor

isolated

i t may be

until

conclusive

f i g u r e s a r e a v a i l a b l e (see also Compstan et a1 1978).

T r a f f i c Loads The

following

loads

Concrete Pipe Assn. (1)

I n fields

(2)

Under

are

useful

for

design

purposes

(see a l s o

1969).

: two wheel

loads each 30 kN spaced 0.9 m a p a r t .

light

roads

:

two

main

roads

and

wheel

loads each 73 kN spaced 0.9m

apart. (3)

Under eight

wheel

of four

heavy

traffic

153 type HB

(BS

load)

:

loads each 112 kN comprising two rows 1.8 m a p a r t

wheels each 0.9 m a p a r t

l a t e r a l l y , measured from i n n e r

a x l e to inner axle.

(4)

Airport

runways:

four

210

k N wheel

loads

spaced

at

1.67m x

0.66 m.

Pipelines

would n o r m a l l y be designed f o r f i e l d

ened under h e a v i e r t r a f f i c . design

loads e x t r a t o p f i l l

I f construction

should b e placed to spread the load d u r i n g

construct ion (Kennedy, 1971 )

Stress Caused by

Boussinesque's below p o i n t as

l o a d i n g a n d strength-

loads a r e l i k e l y to exceed

.

Point Loads

e l a s t i c theory

g i v e s the v e r t i c a l

load a n d a distance X

stress a t

depth H

h o r i z o n t a l l y from the p o i n t load P

122

w

3P H 3

=

27r(HZ + X ' ) The stress at the

stresses

occur

any due

(6.8)

5/2 point to

due to two o r

the

individual

d i r e c t l y under e i t h e r

load o r

more loads w i l l be the sum of

loads.

The

maximum stress could

somewhere between them,

and

the

worst case should be selected b y p l o t t i n g the stress between them. For the

large

pipe

Concrete for

pipe

diameter,

diameter Pipe

and

(l),

loads

average

value

a p p r e c i a b l y across

should

(1969) l i s t s the total

Association

surface wheel

stress w i I I v a r y

the

some

be

taken.

load i n kg/m

The

of

pipe

( 2 ) a n d (3) i n the p r e v i o u s section,

and

v a r i o u s p i p e diameters and depths.

Line Loads The v e r t i c a l

stress at

q per u n i t l e n g t h is,

w

depth H

any

below

a

l i n e load of i n t e n s i t y

b y e l a s t i c theory,

2q~3 n ( X Z + H 2) 2

=

Uniformly Loaded Areas The

stress

surface

could

intensity

load.

any

depth

beneath

be e v a l u a t e d b y assuming

p r i s e a number of

ma1

at

point

loads,

a

distributed

the

load to com-

i n f i n i tesi-

F o r t u n a t e l y Newmark has performed the i n t e g r a t i o n of point

loads

to

give

the

vertical

under

loaded

i n Table 6.1. To e v a l u a t e the v e r t i c a l

under

a

surface

with

one

rectangle

table.

the

particular rectangle. loaded

and

area

load

and

the

the

corner

influence

of

a

coefficients

stress at

loaded area

the p o i n t

any

point

into rectangles

i n question.

Calculate

a n d Y a r e the l e n g t h and b r e a d t h of

corresponding due

intensity

the p o i n t assume

the

stress

rectangle, If

above

where L

read the

Newmark's

d i v i d e the

directly

a n d Y/H

Evaluate

multiplying

loaded area,

corner

the r a t i o s L / H

area.

stress

uniformly

each

rectangular

the

a n d summating the stress due to each

are given

the

loaded area on

by

to the

summate

influence

each

the

factor

rectangle

influence stress

of

coefficient due

to

from

the

load

by

for

the

each

such

i n question f a l l s outside the boundaries of the loaded

area

extends

to above

s u b t r a c t the stress due to the i m a g i n a r y extensions.

the p o i n t and

123 TADLE 6.1

Influence factors f o r v e r t i c a l pressure under the corner of a uniformly loaded r e c t a n g u l a r area (Capper a n d Cassie, 1969).

o 0222 00435

Similar loaded

influence

circles

into

rectangles

also

influence

and

coefficients strips,

or

small

charts

but point

a r e available for

normally loads

available for

any

without

can

be

much e r r o r .

evaluation

shape loaded area (Capper a n d Cassie,

the stresses under

shape

the

intensity

dispersion.

of

the

surface

There

are

of stresses under any

1969).

The stress a t any depth under a uniform load of v e r y is

resolved

load

since

there

l a r g e extent

is

T h i s could also be deduced from Table 6.1

no

lateral

for large L and

Y.

Effect of R i g i d Pavements

A load

rigid

surface

load.

thickness h,

where

pavement

laterally

and The

above a p i p e has the effect of therefore

stress

at

reducing

the

a

H

depth

stress below

a

distributing

intensity rigid

due

the to

pavement

a of

is C_P/R2

(6.10)

W

=

R

-

E

=

modulus of e l a s t i c i t y

h

=

pavement thickness

= r a d i u s of

f o r concrete)

stiffness

(28 000 N/mrnz

or

4 000 000 p s i

124 U

=

Poisson's r a t i o (0.15 f o r concrete)

k

=

modulus

of

0.028N/mm3

subgrade for

reaction,

poor support

R

is

normally

function

of

H/R

s i n g l e point

A

concrete

and

three

X/R

times

and

the

slab

normally

Pressure

has

the

effect

of

coefficients

for

a

point

1970)

Pipe Assn.,

W

Pressure on p i p e = CrP/R2 Depth of Top of Pipe Below Pavement

P

Point Load

R

Radius o f Stiffness

X

Horizontal Distance from Point Load

-

-

20

24

020

054

032 03 1 03? 032

062 051

047

010

022

l!d2

026

o:?

24

043

036

llib

021

28

037

03 I

073

019

32 36

012

025

40 41

02%

013 0XJ 018

O? I 019 018

OIR

027

015

014

48

52 56

018 015 014

Olh 014

012 01 I

60

012

Ol!

013 012 cil? on9

63

011 010

ni;

00s

008

009

nux

uni

12 75

009

ons

007

037

OOR

'I-;

00-

[I!)&

gn

007

007

OOh

00'

n

08

1 2

l!3 101

ObX

061

12

089 076

16 ?0

03

08

68

020

~.

Cr

is

a

i n Table 6.2

for

a

thickness.

i s tabulated

__

(1

slab

H

00 ___

A typical

N/mm3

about

five

times

its

i n a t t e n u a t i n g a p o i n t load.

(Amer. Con.

H R

from

load P on the surface.

thickness of s o i l f i l l

TABLE 6.2

about

varies

to 0.14 N/mm'.

i s 0.084

v a i u e f o r good support

which

(Ifil

01:

-~~

-

U? i

02: 017

016 015

010 or19

28

load

on

a

pavement

125 REFERENCES

American Concrete Pipe Assn., 1970. Design Manual-Concrete Pipe, Arlington. Capper, P.L. a n d Cassie, W.E., 1969. The Mechanics of Engineering Soils, 5th Ed., Spon., London. 1968. B u r i e d p i p e l i n e s , MacLaren, London. Clarke, N.W.B., Concrete Pipe Assn., 1969. Loads on B u r i e d Concrete Pipes, Tech. B u l l . No. 2, Tonbridge. Cray, P., Schofield, A.N. a n d Shann, C.D., lc7C. Compston, D . G . , Design a n d Construction of B u r i e d T h i n Wall Pipes. CIRIA. Kennedy, H., 1971. E x t e r n a l loads and foundations f o r pipes, J . Arc!. Water Works Assn. March. Marston, A . and Anderson, A . O . , 1913. Theory of loads on pipes i n ditches and tests o f cement a n d c l a y d r a i n t i l e s a n d sewer p i p e . Iowa State Univ., Engg. Res. I n s t . B u l l . Spangler, M.G., 1956. Stresses i n pressure p i p e l i n e s a n d p r o t e c t i v e c a s i n g pipes, Proc. Am. SOC. C i v i l Engrs., 82 ( S T 5 ) , pp 1054-1115. Young, D.C. and Trott, J.J., 1984. B u r i e d R i g i d Pipes. Elsevier Applied Science, London. p p 230.

L I S T O F SYMBOLS

trench w i d t h load coefficient,

embankment

condj tion,

positive projection

case load coefficient,

trench condition

load coefficient,

embankment condition,

negative projection

case load coefficient,

r i g i d pavemerot

d i ameter effective modulus of e l a s t i c i t y of s o i l modulus of e l a s t i c i t y pavement thickness or v a r i a b l e depth depth of

cover

depth of f i l l below p l a n e of equal settlement momen t of

inert ia

modulus of subgrade reaction modulus of subgrade reaction r a t i o of

l a t e r a l to v e r t i c a l s o i l pressure

l e n g t h of

loaded area

bending moment bending and

y

or

refer

deflection to

top,

coefficient bottom

and

(subscripts side

t,

b,

moments a n d

zon t al and v e r t i ca I d i stort ions respective I y )

s,

x

hori-

126 pressure p o i n t load l i n e load per u n i t length radius r a d i u s of stiffness settlement r a t i o wal I thickness project ion r a t i o v e r t i c a l pressure W

l a t e r a l soil pressure

W

permissible r i n g load

W

permi ssi b l e v e r t i c a I load

w

vertical

load per u n i t length

horizon t a I distance w i d t h of

load area

a n g l e of bottom support def I ect ion u n i t weight of soil a n g l e of trench

friction

between

backfill

a n g l e of f r i c t i o n of b a c k f i l l m a t e r i a l Poisson's r a t i o

and

sides

of

127

CHAPTER 7

CONCRETE PIPES BEDDING

THE EFFECT OF

Non-pressure external the

design

pipes

are

external

load

per

drain

for main

self

at

are

the

crown,

weight

and

by

the

live

weight

of

a

pipe

to

a

bed.

The

load

loaded

thus

is

called

strength

practical

factors

determination

of

withstand

rigid

laboratory

of

water

the

and

vertical

concrete

per

the very

is

and

to

the

vertical

haunches

and

the

the

haunches.

The

and

horizontal

soil

pressure a n d

pipes).

supported

on

length r e q u i r e d to f r a c t u r e

the

strength.

load

reliable.

and

while

Although

theoretically,

theoretical

more

uncertain

load

laboratory

calculated the

load

line

unit

the

be

specify

r i g i d non-pressure pipes a r e normal-

vertical

influence

of is

could

due

( i n t e r n a l water

pressures a r e neglected f o r non-pressure

designed flat

bottom

loads,

withstand

classes

stresses at

to

standards

different

compressive stress

caused

designed

Various

The

the

run

Concrete pipes a n d the other ly

are

pipe

are

unit

pipes

pressures. of

stresses

transient

or

internal

reinforced accordingly.

stresses

loads,

not

loads

bending main

sewer

loads,

and

(The

effect

the

a

of

number

the of

experimental

tensile lateral

strength constraint

the supports may be a p p r e c i a b l e ) . Alternative

ed

in

that

Fig.

standard

7.1.

The

load which

inforced pipes,

will

concrete

or

testing

is

defined

produce a c r a c k

pipes

90% of

arrangements f o r

strength

the

60% of

or

1/10000

inch

(0.25

mm) f o r unre-

ultimate

load

for

1/100

the

(0.025

inch

i I lustrat-

pipes a r e

B r i t i s h Standard 556 as

in

mm)

wide

reinforced

crack

load

for

prestressed c y l inder type concrete pipes. The strength u s i n g the 3-edge in

excess

of

b e a r i n g test

that was

for

the

bearing

2-edge

standardized

and

test

bearing

is

test.

usually

Recently

5 the

to

10%

3-edge

t h i s i s now defined as the labora-

t o r y strength. Pipes wide higher

laid

length than

in

of the

trenches arc

by

are

the

laboratory

usually

bedding. strength

supported The

as

over

strength

the

bending

is

a

relatively

consequently moments

are

128 less.

F i g . 8.2

summarizes

the

theoretical

b e n d i n g moment c o e f f i c i e n t s

for pipes supported over various angles (defined i n Fig.

8 . 4 ) (CPA, 1962).

0 mm

15&brmm rubber

2 -EDGE

3-EDGE

3-EDGE FOR ANY OIA UP TO l8OOrnrn

Fig.

7.1

The the

S t a n d a r d c r u s h i n g test b e a r i n g s f o r r i g i d p i p e s

ratio

bedding

of

field

factor.

strength

Various

to

laboratory

types

of

bedding

b e d d i n g f a c t o r s a r e l i s t e d b e l o w (CPA,

TABLE 7.1

strength and

1967; ACPA,

is

defined

the corresponding

1970):-

Bedding factors

120" R.C. c o n c r e t e c r a d l e o r a r c h

3.4

Class A

120° p l a i n c o n c r e t e c r a d l e o r a r c h

2.6

Class B

Granu Ia r bedding

1.9

Class A

Class C

H a n d s h a p e d t r e n c h bottom

1.5

Class D

H a n d trimmed f l a t bottom t r e n c h

1.1

For

as

rigid

concrete

pipes

the

lateral

support

of

the

sidefill

in

the t r e n c h does n o t a d d n o t i c e a b l y to t h e s t r e n g t h .

A

factor

of

safety

forced concrete pipe.

K of

1.25

to

1.5

i s normally

When p i p e i s p r e s s u r i z e d ,

used w i t h u n r e i n -

i t s h o u l d be designed

so t h a t : -

external load f i e l d strength Unreinforced

+ or

internal pressure bursting pressure reinforced

(not

<

1

(7.1

K

prestressed)

concrete

pipes

are

129 normally

only

drains.

for

non-pressure

The

concrete

may

to

internal

pressures,

subjected certain

used

tensile

strength,

be

pipes

stressed

in

and

such

as

tension

even

if

though

c r a c k s may develop a n d

sewers the

and

pipe

concrete

leaks a r e

was

has

a

likely

if

the tensile stress i s maintained (Kennison, 1950). Concrete Special

pipes

normally

precautions

may

do

be

not

necessary

stance sewers a r e often made w i t h r a t e of corrosion of thus

require for

lining

certain

or

wrapping.

liquids,

for

in-

limestone aggregate to m a i n t a i n the

the aggregate a t

m a i n t a i n i n g an even surface.

the same r a t e as the

cement,

The f r i c t i o n coefficients of concrete

pipes, especially when c e n t r i f u g a l l y w i t h those of l i n e d steel pipes.

cast,

can

compare

favourably

PRESTRESSED CONCRETE PIPES

Prestressed pressure for

long

ses

than

that for

pipes

when

becoming

a popular

over

competes

medium for

800 mm

approximately

large-bore

economically

with

diameter.

It

steel

has

the

the prestressing steel can be stressed to h i g h e r stres-

plain-walled must

be

and d i s t o r t i o n even pressures.

is

Prestressed concrete

pipelines

advantage

steel

concrete

pipes.

pipes.

reasonably if

The

wall

thick

to

the

use

of

manufacturing plain-walled

p l a i n walled

buckling,

collapse

i s not r e q u i r e d to r e s i s t i n t e r n a l

the thickness

Consequently

thickness o f prevent

high-tensi le

steel

pipes,

steels

is

restricted

b u t t h i s i s not the case

w i t h prestressed concrete pipes which a r e more r i g i d than steel pipes. Prestressed wires core

onto

a

forming

concrete core.

pipes

The

helical

the b a r r e l of

each

i n a mould o r c e n t r i f u g a l l y then

steam

cured

to

the core may

mortar.

steel

cylinder

load

under

concentrated forcing.

Plain

stressed

longitudinally

winding

by

is

winding

subsequently

rapid

strength

improves

the

wire,

are

as well

pretensioned

increase

and

often

acts

as

lined with

longitudinal

The

is

winding.

distributes

reinforced with

as c i r c u m f e r e n t i a l l y .

vertically

The concrete before

a steel c y l i n d e r

watertightness,

The

coated.

p i p e may be concrete cast

be formed w i t h

concrete cores

reinforcing resists

formed

i n a h o r i z o n t a l position.

ensure

Alternatively The

are

bars or

the

reinpre-

longitudinal

l o n g i t u d i n a l bending i n the trench as well as local

130 bending

stress

longitudinal

i n the core w a l l s d u r i n g c i r c u m f e r e n t i a l

wires

are

pretensioned

and

released once

winding.

The

the

has

core

set.

Fig.

7.2

Prestressed concrete p i p e

The prestressing close

wire

to y i e l d p o i n t .

It

i s wound loses a

and s h r i n k a g e before the p i p e

h e l i c a l l y onto the core a t a stress

proportion i s put

of

i t s stress

due to creep

A concrete c o a t i n g

into service.

i s a p p l i e d as soon as possible a f t e r w i n d i n g to get as much stress as possible t r a n s f e r r e d to the c o a t i n g d u r i n g the s h r i n k a g e of the core. The

tensile

of

manufacture

inherent

do

on

some

seepage

cause

the

pipe of

Further

in

tees)

may

the

or

the

after

through

neglected

in

accounting

for

design of

soon

seal

h y d r o s t a t i c test

core

but

may

due is

to

before c o a t i n g

be

immediately

permitted

the

usually

at

and this

before dispatch of

and i n the field. joined with

The socket may

spalling

cracks A

winding

the

load tests may be done

insertions.

stressing

is

concrete.

immediately water

The pipes a r e u s u a l l y

needed

core

cracking,

t h e pipes from the works,

rubber

the

i f t r a n s i e n t stresses i n the f i e l d o r i n the process

h e a l i n g p r o p e r t i e s of

done

stage.

of

strength

f i e l d pressures b u t

of the

integral socket

be f a b r i c a t e d from steel

spigots a n d sockets sealed w i t h

be of

steel

o r concrete b u t care i s

concrete will

sockets

occur.

o r cast

iron,

to

Specials

ensure (bends

no and

and are f i t t ed with

matching

spigots

up

a

over

and

number

sockets.

Small

angle

joints

which

may

take

the prestressing

wires

of

bends

are

up

n o r m a l l y made

2"

to

deflection,

depending on the type of j o i n t .

C i rcurnferent i a l The

Prestressing

tensile

stress

wound

i s less than

wire

is

i s because the core deforms under the p r e s t r e s s i n g

instant

of

the

the core

immediately a f t e r

This

section

onto

in

prestressing

i s under

core

at

relaxation

of

any

section,

stress whereas

the

point

the

steel

of

the

the other

winding

stress a f t e r

the a p p l i e d tensile

is

pipe side

half

winding

to

its

final

wi II

stress.

force.

one

i s free.

side

the

At the of

the

The s t r a i n strain.

in The

therefore correspond

to h a l f the t o t a l s t r a i n of the concrete core and

Afs

i f

=

n co 1

i s the r a t i o of

where n,

the concrete a t

the

i s the steel stress, fso =

f

where f

.

so

i f

fco

elasticity

of

i s t h e core stress a f t e r

steel

to that of

w i n d i n g and f

hence

n co 1

i s the

51

winding. f

. -

SI

the modulus o f

time,

initial

steel

stress and f

so i s the steel stress a f t e r

Since f o r e q u i l i b r i u m of forces

s =

f

t co c

(7.2)

and

fSo

where S

=

(7.3)

fcotc'S

i s the steel cross sectional area per u n i t length o f p i p e , a n d

tc i s the core thickness.

Circumferential After static may

test

to

undergo

amount of

Prestress a f t e r

winding

the core,

detect has

cracks.

usually

concrete creep

Losses

the p i p e Any

taken

is usually

creep place

has occurred.

which by

this

subjected to a hydrothe

prestressing

time,

and

a

steel

certain

The concrete stress j u s t before

the test becomes f .

CI

=

fco

(7.4)

132 where

is

u

the

combined steel a n d concrete creep

winding and testing.

fsi The

stress

for

The corresponding steel stress i s :

(7.5) any

particular

test

loading

may

be

calculated

using

- 7.15 b u t r e p l a c i n g subscripts 2 b y 1 a n d s e t t i n g the core

7.9

thickness

tb equal

as q u i c k l y and

between

fCitC/S

=

Equs.

coefficient

The c o a t i n g should be a p p l i e d and cured

as possible a f t e r of

creep

to zero.

the

testing.

concrete core

This ensures t h a t the s h r i n k a g e

transfer

as

much compression

to

the

c o a t i n g as possible. The c o a t i n g adds to

the cross sectional

area

a n d reduces stresses

thereby a l s o l i m i t i n g the creep of the core. After up

to

the

pipe

has

been

three months a f t e r

cured and

i s r e a d y for

service

(usually

m a n u f a c t u r i n g ) the c i r c u m f e r e n t i a l tension

is

taken by the core and the coating: fS,SCf

(7.6)

t + f t c2 c b2 b

Equating

the

movement due to e l a s t i c compression,

creep

and s h r i n k -

age of the core a n d coatings to that of the steel: Deformation of the core:

w

-fc2-f + - c1 Ec2

c

Ec2

( E l a s t i c ) (Creep )

f .+f CI

c2

+ v c2

2

=

sl -fs2

(7.7)

ES

( E l ast ic-steel )

(Shrinkage)

Deformat ion of coating : Sl b2

b2

- f

b2

52

ES

( E l a s t i c ) (Creep) ( S h r i n k a g e ) ( E l a s t i c deformation of steel) Now

if

the properties of the concrete core and coating are s i m i l a r ,

the e l a s t i c modulus E

c2

-

Eb2

and the s h r i n k a g e coefficient Solving Equs. 7 . 5 - 7 . 8 f o r f

v

c2

= vb2 = v

sl - f52’

say.

133 2+ w v E c 2 ( t c + t 6 K b ) + tcfcl w c

fsl - fS2 =

(7.9)

S ( l + wc ) + (tc+tb2+wc)/n __ 2 2 2+Wb

Hence from 7.8

(7.10)

(7.11)

Circumferential There

are

Stress Under F i e l d Pressure

a

number

of

field

loading

conditions

which

should

be

examined i n c l u d i n g :-

In open

(1)

trench

internal

with

test

operating

or

pressure p l u s

self weight a n d weight of water.

(2)

I n b a c k f i l l e d trench

with

i n t e r n a l pressure p l u s l i v e load p l u s

self weight a n d weight of water.

(3)

I n backfilled

trench

with

live

load p l u s self

weight

and pipe

empty.

(4)

In

backfilled

trench

with

internal

pressure,

self

weight,

crown

(due to

weight o f water a n d t r a n s i e n t pressures. The

most

highly

prestressing sures), cal

at

plus

sections

bending

the haunches

load p l u s

sing

stressed

plus

plus

usually

external

at

loads

the plus

internal

pres-

( d u e to prestressing p l u s bending p l u s v e r t i -

i n t e r n a l pressure)

bending

are

under

vertical

a n d a t the support plus

horizontal

( d u e to prestres-

loads

plus

internal

pressure). Compressive area

per

unit

and

tensile forces a r e taken

length

of

wall

comprising

the

d i r e c t l y on

the effective

p i p e core

pl us coating

p l u s transformed steel section:

A = t

+ tb + ( n 2 - 1 ) S

Bending weight

of

moment

of

moments pipe

plus

inertia

I

due

(7.12)

weight per

soil

to

of

unit

loads,

water length

are of

external resisted wall.

The

live by

loads

the

effective

distance

c e n t r o i d of t h e effective section from the i n s i d e of the core i s :

and

of

the

134

(7.13) where d

i s the diameter of

the prestressing

wire.

The corresponding

distance to the outside of the c o a t i n g i s : e

= t c + t b - e

(7.14)

0

The moment of e3 I=____

+

e

i n e r t i a of the effective section i s :

+ ( t c + ds/2 - e I 2 ( n 2 - 1 ) S.

0

(7.15)

3

Long itud in a I Prestressing Some

prestressed

circumferential l y . tensile c r a c k s the

pipes

The

are

prestressed

longitudinal

prestressing

is

i n the concrete d u r i n g w i n d i n g of

longitudinal

bending moment i n service.

wires a r e p l a c e d i n the. mould f o r before

casting

causes

local

in

longitudinal

the

longitudinally

the

core.

When

the

assists

added

as

prevent

to

the core a n d due to l o n g i t u d i n a l b a r s or

The

the concrete core a n d pretensioned wire

bending stress and shear bars

we1 I

as

in

is

wound

onto

i n the core walls.

reducing

the

the

core,

it

The tension

resulting

principle

stresses b y i m p a r t i n g a compressive stress to the concrete core. As

for

the c i r c u m f e r e n t i a l

wires,

the

longitudinal

stress due to creep of the steel a n d concrete. ly

stressed

to

.

f

their

SlL’ core cross sectional area

uL,

total is A

bars

lose some

I f the b a r s a r e i n i t i a l -

cross sectional

i s AS,

area

the p i p e

and the creep r e l a x a t i o n coefficient

then immediately a f t e r release of the l o n g i t u d i n a l b a r s ,

is

the stress

i n the core becomes: u

L

f c o ~-

f . A SlL s + nlAS

(7.16)

a n d the l o n g i t u d i n a l steel stress becomes fsoL = fcoLAc’As The

core

release of

is

normal l y

the

of

prestressed circumferential l y

longitudinal bars.

end to the other. application

(7.17)

The h e l i c a l

wire

immediately

i s wound from one

The r a d i a l shear stress i n the core, the

winding,

is

derived

from

after

at the p o i n t of

elastic

pressurized c y l i n d e r s (Timoshenko and Woinowsky-Krieger,

theory

for

1959) and i s

135 approxima tel y q

0.54 S f S O / v

=

(7.18)

f

i s t h e circumferential steel so the e x t e r n a l diameter of the core. where

The

maximum

due to bending

local

longitudinal

stress a f t e r

tensile

and

winding,

and D

is

compressive stresses

i n the core w a l l d u r i n g w i n d i n g h a v e been determined

experimental l y to be approximately fcmL = 0.3fco

(7.19)

where fco i s the core c i r c u m f e r e n t i a l stress a f t e r winding. The effect of the l o n g i t u d i n a l steel on the cross sectional a r e a has been

neglected

neutral mum

in

Equs.

7.18

and

7.19

as

the

a x i s a n d the expressions a r e approximate

tensile

stress

in

the

core

wall

steel

is

near

the

anyway.

The rnaxi-

may be d e r i v e d w i t h

the assis-

tance of a Mohr d i a g r a m ( F i g . 7 . 3 ) : I f fcmL > fcoL, c t L =/qz

which i s u s u a l l y the case, - f

fcml

+

( 2

and

i f

""4

2-

then

-

+

COL

2

f c o L > fcmL

then f c t L =

h2

+ ( f C 0 L - FcmL 2

2

-

-

fcoL

cmL

2

I

SHEAR

STRESS

Fig.

7.3

( 7.20a )

Mohr c i r c l e f o r stress i n core d u r i n g w i n d i n g .

(7.20b)

136 It

will

be

longitudinal

seen

from

prestress,

Equs.

which

7.20

a

b

and

i s proportional

the p r i n c i p l e t e n s i l e stress in the core f

that

to

f

the

larger

the

the smaller

COL

is

ctL'

Longitudinal S t r e s s e s A f t e r L o s s e s

The

longitudinal

in the f i e l d .

pipe

points

the

in

creep

and

act

to

At

this

have occurred

stress

longitudinal neglected.

bedding.

will

compressive

bars

resist

longitudinal

bending

of

the

The p i p e a c t s a s a beam s p a n n i n g between uneven

will

stress

stage in

h a v e been

is

not

S h r i n k a g e of

high,

the

a

c e r t a i n amount of

concrete

and some

t r a n s f e r r e d to the

concrete

shrinkage

longitudinal

the c o a t i n g .

creep

can

As

the

usually

be

t h e c o r e does r e d u c e t h e t e n s i l e s t r e s s i n t h e

b a r s and t h e c o m p r e s s i v e s t r e s s i n t h e c o r e becomes:

u f A L siL s fc2L

=

Ac

Once added

(7.21 )

pipe

the

ESAS

n2AS

+

the

to

v

-

is

stress

in

service,

due

to

longitudinal

prestressing.

The

bending extreme

stresses fibre

are

bending

stresses a r e

-

MD 21

-

fc3L

-

where M

(7.22)

i s t h e b e n d i n g moment

t h e e f f e c t i v e moment o f

and

inertia of

the section i s

I

l

-64

=

(l D 4-

d

(d

4

+ (n2

-

+ 2t + d 5 ) *

1 ) AS

(7.23)

8

P r o p e r t i e s of Steel and C o n c r e t e The

steel

used

w i n d i n g prestressed concrete p i p e should h a v e

for

a s h i g h a y i e l d s t r e s s a s p o s s i b l e and a y i e l d s t r e s s o f 1650 N/rnmz (240 000 p s i ) shrinkage concrete it

is

i s n o t uncommon.

have

taken

is fairly

normal

to

place,

high.

This will

e n s u r e t h a t a f t e r c r e e p and

the r e m a i n i n g compressive stress

The steel stress w i l l d r o p a f t e r w i n d i n g ,

confine

working

stresses

to

less

than

50% o f

the

in

and

yield

stress. Steels work

with

w i t h and

high

yield

in f a c t

stresses

are often

b r i t t l e and

difficult

to

the u l t i m a t e s t r e n g t h may not be much h i g h e r

137 than

the

yield

stress.

The to

may creep s l i g h t l y

distinguish test,

stress

in

therefore

between concrete

so

the

necessary

in

of steel

selecting

the

i s approximately

(30 x 10

steel

works

is

T h e modulus of e l a s t i c i t y 6 psi).

prestressing w i r e . 200 000 N/mm2

Care

they

steel

are is

a f t e r prestressing a n d steel

usually

creep

considered

typically

around

but

it

is difficult

between w i n d i n g and

together.

5% a n d

This

occurs

loss

in

w i t h i n a few

hours a f t e r w i n d i n g . Concrete day

achieved.

are

the

under of

High e a r l y strength

cube

cracks

cast

strengths

vibration or

60 N/mm2

(e.9.

will

strength

at

the

most p r o b a b l y

time

centrifugally

and

28

( 8 700 p s i ) a r e f r e q u e n t l y

50 N/mm2 o r

w i n d i n g as e a r l y as possible.

able for of

cores

cube c r u s h i n g

7 200 p s i ) i s desir-

Compressive stresses u p to 50%

a r e permitted d u r i n g w i n d i n g

(any

h e a l ) . Working compressive stresses should

be confined to less than 1/3 of the 28 day cube strength. The less

bending

than

tensile

about

calculated

10%

bending

bending

in

the

strength

i n order

stress of

tensile

field

in

the

strength

stress

should

be

to a l l o w f o r

i t y of cracks developing.

the core d u r i n g w i n d i n g

cube

in

the

limited

at

that

core to

due

about

should be

age, to

and

the

longitudinal

5% of

the

cube

unknowns a n d to prevent any p o s s i b i l -

No c i r c u m f e r e n t i a l tensile stress i s permitted

i n the core i n the f i e l d f o r normal o p e r a t i n g conditions b u t tension i s sometimes the

p e r m i t t e d under

coating

(This

should

i s for

be

transient

less

b u r i e d pipe.

than

conditions. about

The

10% of

the

tensile stress cube

in

strength.

Exposed pipes may develop cracks

a n d the

tensile stress should be less t h a n t h i s . )

4625

BS

operating 0 . 7 4 7 m if 0.623

/

F

does

plus

not

permit

backfill

water

tensile

pressures,

stress but

in

permits

hammer pressure i s included,

during

works

hydrostatic

test.

the a

core

for

tensile

normal

stress

of

a n d a tensile stress of

(Where F

is

the

crushing

strength of 150 mm cubes a t 28 d a y s , a l I i n N/mm2 . ) The

modulus of

elasticity

of

and v a r i e s from 20 000 N/mm2 ( 6 000 000 p s i ) (BSCP 2007,

The creep coefficient, creep s t r a i n

=

AL/L

increases

( 3 000 000 p s i )

1970; Ferguson,

w,

= w

concrete

to

w i t h the s t r e n g t h

40 000 N/mm2

1958).

of concrete i s defined b y the equation

fc/Ec

138 where

fc

is

the

average

creep occurs a n d E

compressive

i s the f i n a l

stress

during

Creep i s high f o r

the r a t e of

time.

creep reduces w i t h

casting

(at

the

time

three

months a f t e r casting.

Hence f o r

test

be 30% h i g h e r f o r

I t may

' g r e e n ' concrete and

i s approximately 0.3

w

factory

that

testing

the

core)

the coating,

two days

and

is

1.3

the coefficient

1.3

b u t f o r the core, the creep coefficient f o r

i s used to c a l c u l a t e creep, the time between f a c t o r y

of

time

w varies with

modulus of e l a s t i c i t y .

time a n d the method of c u r i n g .

after

the

i s 1.3 - 0.3 = 1.0.

a n d f i e l d conditions

pipes not p r o p e r l y cured o r exposed to the

elements i n the f i e l d . Shrinkage ing,

typical

of

concrete also depends

l a r g e l y on the method of c u r -

values of s h r i n k a g e per u n i t length,

v,

v a r y i n g from

to Example

Calculate

circumferential

after

winding,

curing

concrete

and

and

under

field

steel

stresses

conditions

during for

the

prestressed concrete p i p e described below: Effective p i p e tc = 75 mrn,

15 mrn c/c. 8

length L = 4 m,

coating

=

S = 0.00131

diameter,

mm

tb

N/mmz.

Vertical

lateral

soil

loading

pressure 0.5 of

steel

m2/m of pipe.

A S = 0.00121

e f f e c t i v e l y over 30"

bore d i = 2 000 rnm,

25 mm,

due x

arc.

m2. to

winding

L o n g i t u d i n a l wires 24 N o .

Maximum

soil

vertical

=

core thickness

5 mm diameter at

internal

and

live

0.02

N/mmz.

pressure

load 0.04

x

0.8

N/mrn2,

Bottom support

Neglect selfweight a n d weight of water

inside. Steel: L o n g i t u d i n a l prestress = 400 N/mm2 .Winding prestress = 1000 N/mmz. ES = 200 000 N/mm2. Creep coefficient u = 0.95. Concrete:

E

= 30 000 N / m m 2 , CI from f a c t o r y test to f i e l d test. n

n

1

2

=

200 000/30 000

= 6.7

=

200 ooo/38 000

= 5.3

Ec2 = 38 000 N/mmz,

= 1.0

Wb

.

= 1.3,

Creep w 4 : v = 10 shrinkage

Stresses due to w i n d i n g -

fco

-

1 000 x 0.0131 0.075 i 0.00131 x 6 . 7 / 2

=

16.5

N/mm2

(7.2)

139

16.5

=

so

0.075 0.00131

= 880 N/mmz

(7.3)

Works t e s t o f c o r e : fcl fSl

=

0.95 x 16.5

=

15.7 N/mm2

=

0.95 x 880

=

840 N/mm2

(7.4)

a f t e r curing :

f

-f

=

s2

s2 fb2

2+1 .o 1 O-4x38x1 0 ' (0.075+0.025m)+O. 075x15.7xl.O 0.00131 (ld$)+(0.075+0.02*)/5.3 2+1 0

=

76 N/mmz

=

840-76 = 764 N/mmz 76 (200 000

= -

c2

38 Oo0 = 6.4 N/mmz 2+1 .3

764~0.00131-6.4x0.025 0.075

Under f i e l d

(7.9)

=

(7.10) (7.11)

11.2 N/mm2

loading:

Transformed section A =

0.075 + 0.025 + (5.3-1)0.00131 =

C e n t r o i d to inner surface e

0.105 m'/m

(7.12)

=

0.12/2+(5.3-1)(0.075+0.005/2)(0.00131) 0.1+0.00131(5.3-1)

=

0.0515

Cent r o id t o outer surface e

=

0.1

Moment o f

I = 0*05'53+0*04853 + (0.075+0.005/2-0.0515)2

inertia

-

(7.13) 0.0515

=

(7.14)

0.0485

3

x(5.3-1)(0.00131) Stresses a t b a s e B : ( t e n s i o n

-

=

85.9~10-~m~/m

(7.15)

v e f o r concrete stresses)

T e n s i o n d u e to n e t i n t e r n a l p r e s s u r e :

ft =

0.8 x 2 - 0.02 x 2.2

-

-7,4 N/mmz

2 x 0.105

B e n d i n g moment c o e f f i c i e n t s f r o m F i g . For vertical

load,

Nb =

8.2:

0.235, and f o r h o r i z o n t a l l o a d ,

NS

=

0.125 Net b e n d i n g moment = =

0.235 x 0.04 x 2.2/2 - 0.125 x 0.02 x 2.2/2 0.00755

140 Stress on o u t e r face f

Stress on i n n e r face f Steel s t r e s s :

e

=

b3

c3

0.075

- f

-

fb2

=

11.2-7.4-

t

+ -MeO I 0.00755 85.9

+ 0.005/2-0.0515

x 0.0515=

-IN/mm2

x

= 0.026

(on outer side of centroid)

= 764

+ 5.3

(7.4

0.00755

x 0.026)

=

791

N/mm2

85.9 x Although stresses

the stresses a t at

similarly,

other

points

t h e b a s e a r e u s u a l l y t h e most s e v e r e , on

the

circumference

should

the

be checked

and c h e c k s s h o u l d b e d o n e b o t h w i t h and w i t h o u t t r a n s -

ient winding,

t e s t i n g and i n t h e f i e l d .

REFERENCES

C o n c r e t e P i p e A s s n . , 1970. D e s i g n M a n u a l - C o n c r e t e P i p e , A r l i n g ton. BSCP 2007 P a r t 2, 1970. R e i n f o r c e d and P r e s t r e s s e d C o n c r e t e S t r u c t u r e s , BSI, L o n d o n . C o n c r e t e P i p e Assn., 1962. L o a d s o n B u r i e d C o n c r e t e P i p e s , Tech. B u l l e t i n No.2, T o n b r i d g e . C o n c r e t e P i p e Assn., 1967. B e d d i n g and J o i n t i n g o f F l e x i b l y J o i n t e d C o n c r e t e P i p e s , Tech. B u l l e t i n No. 1 , T o n b r i d g e . F e r g u s o n , P.M., 1958. R e i n f o r c e d C o n c r e t e f u n d a m e n t a l s , Wiley,N.Y. 1950. D e s i g n of p r e s t r e s s e d c o n c r e t e c y l i n d e r p i p e , J. K e n n i s o n , H.F., Am. W a t e r W o r k s Assn., 42. T i m o s h e n k o , S.P. and W o i n o w s k y - K r i e g e r , S., 1959. T h e o r y o f P l a t e s and S h e l l s , 2 n d Edn., McGraw H i l l , N.Y.

Am.

L I S T O F SYMBOLS

A

-

d

-

inside diameter

D

-

outside diameter

e

-

distance from centre of g r a v i t y

E

-

modulus of e l a s t i c i t y

cross sectional area

f

-

stress

F

-

c r u s h i n g s t r e n g t h o f 150 mrn c u b e s a t 28 d a y s

(compressive o r tensile)

141 I

-

moment of

inertia

K

-

factor o f safety

L

-

length

M

-

bending moment

n

-

e l a s t i c modular r a t i o E

q

-

shear stress

s

-

steel area per u n i t length of p i p e

t

-

thickness

u

-

creep coefficient f o r steel and concrete before f a c t o r y test

v

-

s h r i n k a g e coefficient

w

-

creep coefficient

Subscripts b

-

coating

c

-

core

s

-

steel

m

-

bending

9

-

shear

t

-

tensile

L

-

longitudinal

I

-

initial

0

-

after winding

1

-

at time of f a c t o r y test

2

-

a t time of

3

-

under f i e l d pressure

laying

/E c

s

142

CHAPTER 8

STEEL AND FLEXIBLE P I P E 1 NTERNAL PRESSURES

The

highest

pressure

to

internal

fluid

due pipe

there

and

weight

are

effects.

pressure.

general

has

The

bending

no

The

pipe

a

to

resist

pressure

stresses

are

is

uniform

except

equations

for

normally

for

the

those

around

the

and

self

water

stresses

resulting

in a hollow cylinder are:

Circumferential

wal I stress

p.d.' I

Fw

-

I

where p is

Fr

i s pressure,

internal

diameter

f o r p l a i n stress, For

i.e.

-

doL

podo'

i s diameter at

and

d

-

) (8.1 )

d.'

1

d i z do2 ( P,-P

+

do2 - d.'

=

d

d i 2 doZ( P-,P

do2 - diL

=

p.d.' I 1

Radial stress

podo'

(8.2)

doZ - d i z

w h i c h the stress i s sought,

i s external

diameter.

a cylinder free to expand

d.

The e q u a t i o n s a r e

longitudinally.

the p a r t i c u l a r case of n o e x t e r n a l p r e s s u r e the c i r c u m f e r e n t i a l

s t r e s s i s a m a x i m u m on t h e i n n e r s u r f a c e a n d i s di2 F max

=

W

In with

practice

the

resist

where

di

+ do2 P + do 2t the

diameter,

wall

thickness

a n d the w a l l

is

normally

small

in

comparison

thickness of a steel p i p e designed to

i n t e r n a l pressure i s obtained from the formula

t is t h e w a l l t h i c k n e s s p

i s the i n t e r n a l pressure

d i s the e x t e r n a l diameter f

i s t h e y i e l d s t r e s s o f t h e steel

G i s the design factor

J i s the j o i n t factor. E q u . 8.4

of t/d.

overpredicts

On t h e o t h e r h a n d ,

the

true

stress

by

some

just u s i n g d instead of

3 r e d i c t s t r e s s b y 1% f o r e v e r y

1% t / d .

1%

(d-t)

for

every

1%

would under-

143 The

design

factor

depend on

will

have

been

a

many

the

burst.

over-pressure, are

the accuracy

assessed,

quences of

allows

G

a

safety

with

which

working

margin.

What

factor

to use

loads a n d t r a n s i e n t pressure

pressure,

economics and

the conse-

I f the p i p e l i n e i s protected a g a i n s t water hammer

unknown

o r even 0.7

of 0.6

a factor

loads,

use of

i s reasonable,

a factor

of

the order

b u t i f there to 0.5

of 0.3

would be wiser. The j o i n t varies

J allows

factor

0.85

from

for

for

furnace

imperfections

butt-welded

i n welded seams,

joints

1.0

to

and

for

seamless

resist

internal

pipes. Smal I-bore, pressures

the

only,

the

wal Is,

high

fluid

internal

but

more

the

pressure

pipe

pipes

larger

important

the

in

pressure

the

designed

diameter

also

be

be

the

loads.

possible,

considered

to

and

become the e x t e r n a l

may

should

are

acting

thinner

the

Vaporization

in

which

case

in

conjunction

of

this with

e x t e r n a l loads. If

the p i p e

magnitude

i s restrained

is

VF

induced

TENS I ON R I NGS TO RESIST

which

strengthened to

form

some

v

is

a

longitudinal

Poisson's

and

tension of

F

is

the

have

I NTERNAL PRESSURES

to

resist

very

high

internal

b y b i n d i n g w i t h hoops a n d s p i r a l s .

pipes

ratio

wal I tension.

circumferential

Pipes

longitudinally,

where

in

the

thick

h i g h pressures.

It

may be

i s often d i f f i c u l t

p l a t e s which would be r e q u i r e d to r e s i s t

the

On

pressures

other

hand

it

may be possible to form

the p i p e of t h i n n e r p l a t e than would be r e q u i r e d f o r an unstrengthened pipe,

a n d w i n d i t w i t h s t r a p s o r rods.

The r i n g s r e q u i r e d to r e s i s t same

function

against in

tension

resist the

as

external

stiffening

loads.

and

external

the

does load,

not

I n fact

the

need

which

to

should

through

pressures do not perform the

rings

The w i n d i n g

l o n g i t u d i n a l section

possible.

internal

required

to r e s i s t be

as

to

resist

internal

prominent

as

col lapse

pressures acts the

increase the moment of

rings

to

i n e r t i a of

the p i p e a n d therefore be as h i g h as

tension r i n g s , as they w i l l be c a l l e d ,

should be

144 a n d b r o a d as possible to keep the distance between them to a

as f l a t

minimum.

There

will

be

high

circumferential

stresses

are f a r apart.

I n comparison,

are

usually

also

be

spaced

used

to

in

a

the

number

strengthen

longitudinal pipe

wall

bending

between

stresses

rings

if

and

the r i n g s

s t i f f e n i n g r i n g s to r e s i s t e x t e r n a l loads of

old

diameters a p a r t . pipes

which

are

Tension r i n g s may to

be

subjected

to

h i g h e r pressures than o r i g i n a l l y designed for. Using

the

stress-strain

r e l a t i o n s h i p and

the d i f f e r e n t i a l

equations

equil i b r i u m f o r c y l i n d e r she1 I s under the action of r a d i a l pressure,

of

Timoshenko

(1959) d e r i v e d a n equation f o r

a

subjected

cylinder

to

a radial

the

radial

displacement of

pressure a n d w i t h tension r i n g s at

equi-spaci n g

.

Put

b4

=

12(1-v2) dZt'

(8.5)

and

a

=

bs

(8.6)

where v t

i s the P o i s s o n ' s r a t i o (0.3

i s the w a l l

Let

x1

thickness a n d s the centre-to-centre

r i n g spacing.

-

cosh a sinh a

+

cos a sin a

(8.7)

-

sinh a sinh a

-

sin a sin a

(8.8)

-

cosh a sinh a

-

x2

x3

for s t e e l ) , d i s the p i p e diameter,

Then the r i n g stress,

+ +

+

Fr,

cos a sin a

i s g i v e n by

(8.11) where A = cross sectional area of The

circumferential

pipe

wall

the r i n g .

stress,

Fw,

mid-way

between

rings

is

given by = FW

a n d the

@[l+ 2 ( F 2t - 1 ) X 4 ] 2t r Pd

longitudinal

i s given by

bending

stress

(8.12) in

the p i p e under

the

rings,

Fb

P

2

I

I -0 .

I

I < 0

1

I

N

d

I

”

g! r

>

i

i

a,

3

C

D

m L a,

m

c

[r

C .a, C

U

m

m In L a,

m

I

a .-C LL

D

m

a .-C L

145

m

a L

.-C

C

m

.-0 a,

C

-

5

r.-

U

E

D .-

m

m m

.-Um m

a, m

QJ

m

c

5

-

3 -

LL

.-a

m

-

m

i a,

vi

- .-5

C

L a, a, c

5 .-i V

m

146 (8.13) The

stresses

shown A)

that

are

indicated

by

Fig.

a s s i s d e c r e a s e d t h e r i n g s t r e s s Fr

w h i c h c o u l d h a v e been anticipated.

to equal

for

8.1,

the circumferential

f o r s m a l l s,

wall

v

0.3.

=

tends to

It

f

pds/

a p l a i n pipe,

the circumferential p i p e wall stress F

pd/2t.

tends to

W

+ A ) , w h i c h w o u l d b e e x p e c t e d , and t h e l o n g i t u d i n a l

be

(ts +

Fr tends

N o t e t h a t f o r s m a l l A,

stress of

may

Also,

4 pds/(ts

beding stress

Fb

tends to zero. For l a r g e r i n g spacing ( > approx.

pd

1

F tends to

1

+ 0.9lA/t&

pd

+ 0.91

tJtd/A

(8.14)

2t

1.65

Fb tends to

2-1,

(8.15)

2t

t e n d s to @ i.e. t h a t f o r a p i p e w i t h o u t r i n g s . 2t I t may be observed b y comparing F F r and Fb f r o m F i g s .

and F

W

8.la,

W’

8.lb in

and 8 . l c

fact

Fw,

reduced

if

t h a t t h e m a x i m u m s t r e s s f o r most p r a c t i c a l r i n g s i z e s i s the

circumferential

r i n g spacing should be area

A

pipe

wall

stress.

s / J y z i s l e s s t h a n a p p r o x i m a t e l y 2.0.

should

longitudinal

be of

cross

l e s s than

the

same

sectional

2 J l d

order

area

of

Also,

is

only the

the r i n g cross sectional

and

magnitude

between

Fw

I n other words,

rings,

as the p i p e w a l l to

ts,

enable

the

r i n g s t o b e o f use.

DEFORMATION O F CIRCULAR P I P E S UNDER EXTERNAL L O A D

large

For

diameter

sures,

the external

under

external

bending,

Spangler

critical occur

at

pipes

load is frequently

load

by

buckling,

under

low

t h e c r i t i c a l one.

overstressing

due

internal

pres-

Pipes may f a i l to

arching

or

excessive deflect ion. elastic

For

only,

flexible

and

rings

under

plane

stress

subjected

to

vertical

loads

(1956) e v a l u a t e d t h e b e n d i n g moments and d e f l e c t i o n a t

points around the crown,

the

the

circumference.

The

i n v e r t or t h e s i d e s .

worst

b e n d i n g moments

T h e b e n d i n g moments p e r

147

Angle of bottom

suwor? p degrees

Angle of tmttom support

Fig.

8.3

p

D e f l e c t i o n c o e f f i c i e n t s for

degrees

loaded pipe.

148 u n i t length a r e g i v e n b y a n equation of the form

M

R

where and

(8.16)

NWR

=

is

N

is

the a

pipe

respectively).

The

W

radius,

coefficient.

(Nt,

vertical

i s the v e r t i c a l

Nb

and

and

NS f o r

horizontal

load per top,

changes

unit

length

bottom and in

sides

diameter

are

p r a c t i c a l l y equal a n d opposite a n d a r e of the form A

NnWR3/EI

=

where N

(8.17)

i s a coefficient.

A

The moment of i n e r t i a I per u n i t length of p l a i n p i p e w a l l I

t3/12

=

Figs.

(8.18)

8.2

and

at

vertical

deflections

unit

the

8.3

moments

line

load or

a

length),

Collapse of

to

been

and

of

the

and

pipe

for

Nt,

sides

Nb a n d Ns f o r the

respectively,

diameter.

The

coefficients

across the w i d t h of

different

angles

of

and

N

bending

A

are

the p i p e

bottom

for

support 6

the

for a

( W per

,

as

8.4.

a steel p i p e w i l l p r o b a b l y not occur u n t i l the diameter

distorted

normally

linings,

bottom

load d i s t r i b u t i o n

5 percent of

should

g i v e values of

top,

indicated i n Fig.

has

is

the be

some 10 o r 20 percent. diameter limited

are

to

I n p r a c t i c e deflections

sometimes

about

2

tolerated.

up

The deflection

percent to prevent damage to

a n d for pipes w i t h mechanical j o i n t s .

The h y d r a u l i c properties,

i.e.

the cross sectional area a n d wetted

perimeter a r e not affected noticeably f o r normal d i s t o r t i o n s . LOAD W /UNIT LENGTH OF PIPE

UPTHRUST

F i g . 8.4

Pipe l o a d i n g and deflections

149 Effect of L a t e r a l Support The of

lateral

flexible

lateral

support

pipes

support

of

sidefill

in a

considerably

and

trench

increases the s t r e n g t h

reduces

deformations.

to a p i p e the r i n g bending stresses a t

Without

the s o f f i t a n d

haunches o r deflections would l i m i t the v e r t i c a l e x t e r n a l load the p i p e could

carry.

But

l a t e r a l l y as the

it

sidefill

be

may

a

is

in

the

established

compacted

sides

with

the

of

the

will

fill

deflect

outwards

i n c r e a s i n g the pressure of

pipe.

vertical

a r c h action as well

load

An

equilibrium

being

condition

transferred

as b y r i n g action.

The

to

the

stress due

i s compressive so that the load which the p i p e can

to the a r c h action

i s considerably h i g h e r than i f the p i p e were a c t i n g in bending.

carry the

extreme case,

the

lateral

stress a n d the p i p e wal I w i l l equal

a

loaded v e r t i c a l l y thereby

against

haunches b y

In

pipe

stress

will

equal

the v e r t i c a l

be i n p u r e compression,

load

w i t h the stress

to wd/2t

where If could

w

W/d

=

the

p i p e underwent

support

whatever

(8.19)

arch

would

no noticeable

be

strength

the sides of the pipe,

determined

i s given i.e.

by

lateral

distortion bending

the

to the p i p e b y

the

load

strength

it

plus

the s o i l pressure on

the permissible v e r t i c a l

load per u n i t area

on the p i p e i s

w

w

=

+ w

b

(8.20)

a

i s the permissible b bending stress or more l i k e l y

where

equal

w

to

lateral

servatively, greater

load

(limited

this

pressure on

as

lateral

indicated

dead

by

ring

i s the a r c h i n g load, a the sides of the p i p e , which, con-

in

subsequent

pressure i s approximately

pressure due to soil

either

by deflection). w

may be taken as the a c t i v e soil pressure,

than

the a c t i v e

soil

bending

load o n l y

one

but i s usually

equations.

For

sand,

t h i r d of the v e r t i c a l

and for clay

i t i s approximately

h a l f the v e r t i c a l pressure. I f the v e r t i c a l active and away

lateral

increase the from

load i s g r e a t e r than the sum of the r i n g load p l u s

soil

the

pressure,

lateral pipe

the

pressure.

and

pipe wall

will

deflect out

laterally

The h o r i z o n t a l stress w i I I decrease

Barnard

(19571,

using

elastic

theory,

150 suggests assuming a pressure equal wall

triangular

to total

stress d i s t r i b u t i o n w i t h the horizontal

vertical

pressure minus r i n g load a t the p i p e

decreasing l i n e a r l y to zero at 2.5d away from the p i p e w a l l .

corresponding AX/2

1.25

=

where E 1.25

i s the effective modulus of

pressure

factor

becomes 1.4

time,

a

e l a s t i c i t y of the s o i l .

The factor

increased as the l a t e r a l deflection increases, since the

radial

for

(8.21 )

d/ES

(w-w,)

should be

1.7

The

l a t e r a l deflection of each side of the p i p e i s

increases for

deflection

as

the

r a d i u s of

2 percent of

a deflection of

of

5 percent.

and an a d d i t i o n a l

c u r v a t u r e decreases.

The deflection

The

the diameter

also

and

increases w i t h

25 to 50 percent of the i n i t i a l deflection can

be expected e v e n t u a l l y . If

the

creep

decrease. e.g.

On

deflection

the

plastics,

other

of

the

hand

soil

pipe

can compensate f o r

large,

is

materials

this,

lateral

which

support

exhibit

can

creep,

a n d could even shed v e r t i c a l

load. The

relationship

determined modulus

from

of

degree of of

between stress and s t r a i n f o r

laboratory

elasticity

of

triaxial

soil

varies

for

N/mmz

for

fill

widely

compaction o r n a t u r a l density,

l o a d i n g and moisture content.

modulus

consol i d a t i o n

loose c l a y

is

or

approximately

as

it

loosely

density

Values h i g h e r than

be

The effective on

soil

c o n f i n i n g pressure, may be as

as 20 N/mm2 f o r

3 N/mmzfor

compacted to 90% Proctor

Proctor density.

depending

F o r example

high

the s o i l should tests.

type,

duration as 2

low

dense sands.

compacted f i l l ,

The

5 N/mrn2

a n d 7 N/mmz f o r f i l l

to 95%

100 have been recorded for moist

compacted sands. The r i n g

load on

indicated

by

as

ponds

to

bottom

the p i p e

Equ.

8.17.

support

i s p r o p o r t i o n a l to t h e e l a s t i c deflection

T a k i n g N,

over

30'

and

equal the

t o 0.108,

load over

which corresthe entire

pipe

width and p u t t i n g AY A -

d

AX

--

0. 108wd3 8E I +0.043ESd3

=

For p l a i n

a

A , then s o l v i n g f o r A/d

=

function

of

pipe

and 8.21, (8.22)

I = t3/12,

diameter,

from 8.17

so one has an equation f o r deflection

loading,

soil

modulus

and

the

ratio

as

wall

th i ckness/di ameter. A -

d

--

0.108~ 0.67E(t/d)'

+ 0.043Es

(8.23)

151 The r e l a t i o n s h i p between deflection a n d wal I thickness f o r p l a i n p i p e i s p l o t t e d i n F i g . 8.5. I t w i l l be observed that a steel p i p e wall

thickness

restrain

5MPa a n d soil Pipes trench

are

to

diameter.

as

low

distortion

as

to

4%

of

the

2% p r o v i d e d

diameter

the

soil

will

be

sufficient

modulus

is

greater

to

than

load i s less than 50 kPa. sometimes

increase The

the

lateral

strutted

internally

vertical

diameter

support

increases

during and

when

backfilling

reduce

the

of

the

horizontal

the s t r u t s a r e removed

a n d the p i p e tends to r e t u r n to the r o u n d shape.

The v e r t i c a l deflec-

tion a n d tendency to b u c k l e a r e consequently reduced considerably.

STRESS DUE TO C I RCUMFERENT I A L BEND I NG

It

i s possible to compute w a l l stresses due to bending a n d a r c h i n g

(Stephenson, the f u l l

1979).

w i d t h of

If

i t can be assumed that the load i s spread over

the p i p e

( a =

180')

and the bottom support i s over

( 6 = 60") f o r f l e x i b l e p i p e , then from (8.17) a n d F i g . 8.3,

60'

AY

=

0.103wbd

4

/8E1

(8.17b)

Now from (8.21), A X = 2.5(w-wb)d/ES Equating

AY a n d A X ,

the total

load,

w The

=

b

w I /d3 I/d3

bending

+

(8.21b)

a n d s o l v i n g for w b

the r i n g load i n terms of w

(8.24)

0.006ES/E

moment

in

the

wall,

M,

is

due

to

ring

load a n d

is a

maximum a t the base a n d from (8.16)

M

=

where

r

the

W

is

the

distance from

extreme f i b r e

i s taken

a

-

(8.25)

0 . 1 9 ~ d 2 / 2 hence b e n d i n g stress f = Mr/l b b the centre of

(t/2 for p l a i n wall

g r a v i t y of

pipe).

the section

The balance of

the

to

load

in a r c h action of the p i p e according to (8.20) so

0. 006wES/E

I /d'

(8.26)

+ 0. 006ES/E

The bottom w a l l hoop stress i s fa

=

wad/2a

where

a

is

the

plain wall pipe).

(8.27) cross

sectional

area

of

wall

per

unit

length

( t

for

152 The =

total

+

fb

lateral

fa,

permissible stress f

+

(l/d3 w

=

permissible

ES to steel

+

i n terms of

(8.28)

load w

The r e l a t i o n s h i p

i s a f u n c t i o n of

moduli of soi

a n d the r a t i o o

i s plotted

i n Fig.

the permis

f o r p l a i n wal

8.5

w i t h E = 210 000 N/rnmz, a n d f = 210 N/mrn2.

pipe,

For t h i c k as

the base i s f

0.003 dE /Ea

the r e l a t i v e thickness t / d

E.

at

is

Thus the permissible v e r t i c a l s i b l e stress,

i n the w a l l

l o a d i n g p e r u n i t area

0.006ES/E)f

r/d

0.1

compressive stress

therefore

more

load

can

deflection

becomes

the

is

pipe

the permissible load increases w i t h wal I thickness

pipes,

be

taken

large

i n r i n g bending. that

so

the

i n p u r e compression,

hoop stress.

Actually

side w a l l

so (8.27) p l u s (8.25)

soil

and

For t h i n n e r

side-thrust

the

limit

pipes,

increases

i s due

the

to

the

until wall

hoop stress exceeds that a t the base,

i s not the l i m i t i n g stress f o r h i g h a r c h i n g .

On the same c h a r t i s p l o t t e d b u c k l i n g load

The

(8.29a)

/32ESEl/d3

=

W

b u c k l i n g equation,

support. gated

This

by

proposed b y

ClRlA

(1978)

allows

for

lateral

may be compared w i t h an a l t e r n a t i v e equation

the

Transport

and

Road

Research

Laboratory

investi-

(which

is

found to o v e r p r e d i c t w ) : W

The

=

( 1 6 E s 2 E l / d 3 )1 / 3 g i v i n g a deflection of

load w d

on the c h a r t for

any

is

selected

(8.29b)

from

2% i n the diameter

The c h a r t

(8.23).

p a r t i c u l a r w a l l thickness r a t i o . in

each

case

from

the

The lowest permissible load w

chart

by

comparing

overstressing and b u c k l i n g lines f o r the r e l e v a n t t / d Similar

charts

should

be

i s also p l o t t e d

thus y i e l d s the l i m i t i n g c r i t e r i o n

p l o t t e d where

and E

stiffening

deflection,

.

rings

a r e used

a n d where a l t e r n a t i v e m a t e r i a l moduli and permissible stresses a p p l y . By

careful

design

it

is

c r i t e r i a a t the same load.

possible

to

reach

the

limit

in

two

or

more

T h i s i s c a l l e d balanced design.

More General Deflection Equations Spangler the

backfill

(1956) in

a

allowed more

for

lateral

theoretical

he d e r i v e d f o r v e r t i c a l deflection i s :

way

support than

to

the

Barnard.

pipe The

due

to

equation

153 UZWd3 8 E l + 0.06Esd”

A =

(8.30)

T h i s corresponds to ( 8 . 2 2 )

i s substituted for N

i f UZ

A

and a v a l u e of

0.15 i s assumed f o r N A i n the denominator ( i n s t e a d of 0.108 i n 8.22) s o i l consolidation time l a g factor

Here U = Z =

bedding

constant

(varies

from

( v a r i e s from 1.0

0.11

for

point

to 1.5). support

to

0.083 f o r b e d d i n g the f u l l w i d t h of p i p e ) , n o r m a l l y taken as 0.1. 1

moment of i n e r t i a of p i p e w a l l per u n i t length.

=

passive resistance modulus of s i d e f i l l .

E = The Due

to

pressure

inside a

pipe

the

that

vertical

fact

the

may

also

contribute

diameter

to

its

stiffness.

i s compressed to s l i g h t l y

less t h a n the h o r i z o n t a l diameter,

the v e r t i c a l u p t h r u s t due t o i n t e r n -

al

the s i d e t h r u s t b y an amount of

pressure becomes g r e a t e r

which

tends

to

return

the

than

p i p e t o a c i r c u l a r shape.

2@

A more general

expression f o r v e r t i c a l deflection thus becomes A

=

UZWd3

8 E I + 0.05ESd’

(8.31)

+ 2 UZpd3

W

Q

0.002

0.005

0.01

0.02

Relative thickness t / d

F i g . 8.5

Permissible load on p l a i n p i p e

0.03

154 ST

FFENING RINGS TO Morley

pipes

BUCKLING W I T H NO SIDE

RESIST

(1919) developed a

under

uniform

stiffening

ring

tice.

theory

The

external

bined internal

for

of

stiffened

often

indicates

i s considered necessary

in p r a c -

pressure.

spacings wider also

theory

than

neglects

the

the

SUPPORT

The

buckling theory

possibility

of

failure

a n d e x t e r n a l pressures and bending.

under

com-

The equations do,

however, y i e l d an i n d i c a t i o n of s t i f f e n i n g r i n g spacing. Using which

an

indicates

axial

the

she1 I

cylindrical no

analogy

a

take

without

and

assumes

comparison w i t h the diameter. w

2E

=

1

-

developed

external

b u c k l ing. the

wall

an

equation

w,

which a

pressure,

The equation a l lows f o r thickness

is

small

in

v i s Poisson's r a t i o , (8.32)

d7 I = t3/12, so

For p l a i n p i p e ,

w

Morley

El

24 1 - v 2

=

strut,

maximum v e r t i c a l

can

expansion,

with

t

(3)

v2

(8.33)

Experiments i n d i c a t e d a p e r m i s s i b l e stress 25% less than the theoretic a l , so f o r steel,

1.65 E

t

w

=

For

thick-wal led

stresses the w a l l

(8.34) tubes,

the

material

to

intermediate wal I thickness, and

elastic

yield.

An

col lapse i t s elastic

pressure limit

wi II

be

( w = 2ft/d)

that

which

whereas f o r

f a i l u r e w i I I be a combination of b u c k l i n g

empirical

formula

indicating

the

maximum

permissible pressure on a p i p e of intermediate thickness i s (8.35) where f

i s the y i e l d stress.

The e x t e r n a l resist w, s,

buckling.

load may be increased i f s t i f f e n i n g r i n g s a r e used to It

was

found

i s i n v e r s e l y p r o p o r t i o n a l to if s

by

experiment t h a t

the collapse

load,

the distance between s t i f f e n i n g r i n g s ,

i s less than a c e r t a i n c r i t i c a l length,

L.

155 From experiment L = 1 . 7 3 E

(8.36)

so i f w

=

2E ( t / d ) 3 for s t i f f e n i n g pipe, then f o r s t i f f e n e d p i p e ,

w

=

L2E ( t / d j 3 = 3.46

The

actual

permissible

(8.37)

stress

is

less

than

the theoretical

due to

imperfections i n the m a t e r i a l a n d shape of p i p e , so the p r a c t i c a l r i n g spacing i s g i v e n by

(8.38) If

the

full

vertical

elastic

external

r e n g t h of pressures

i.e.

he p i p e i s to be developed w

=

'Jt,

then

the

ring

o resist spacing

shou I d be

(8.39) t be of use i f w >1.65E ( - ) 3 . I t should also be ascerd 2f t tained that w z - , which i s the e l a s t i c y i e l d p o i n t . d = Pft/d, then r i n g s w i l l o n l y be of use i f If w

Rings w i l l o n l y

t/d
=

1/30

f o r m i l d steel.

(8.40) F u r t h e r studies of b u c k l i n g of s t i f f e n e d p i p e a r e g i v e n

b y Stephenson (1973) a n d Jacobsen (1974).

W

Y

Fig.

8.6

Pipe w i t h s t i f f e n i n g r i n g s

156 Example 1 :

Tension Rings A The

500 mm bore p i p e i s to take an maximum

N/rnm*,

permissible

and

the

working

maximum

wall

stress

in

thickness

mm.

Assume a

10 rnm diameter w i r e

and

calculate

the

required

internal

the

which

wall can

is av ai l abl e for

spacing

of

the

5 N/mm2.

pressure of is

be

to

be

rolled

100

is

10

b i n d i n g the p i p e

binding.

An

unstiffened

p i p e c o u l d take an i n t e r n a l pressure of

pd

“ow

=

2t

500 100

2 x

=

Tension r i n g area parameter By

inspection of Figs.

s o we r e q u i r e F/@ 2t

125,

1 82A r‘td

t

8.la

=

1 00 125

=

0.8

1 . 8 2 ~ ~ ~= 1o 0 . 2~

10 ~ ‘ 1 0 x 5 0 0

and 8.1b,

the r i n g spacing r e q u i r e d to

reduce both the c i r c u m f e r e n t i a l p i p e w a l l stress a n d the r i n g stress to the desired v a l u e i s g i v e n b y s

/

m = 0.45,

:.

Notice t h a t

s = 0.45

the

amount

x 500

J10 of

steel

=

32 mm.

required

i s s l i g h t l y more than

the p i p e were merely thickened to take the e x t r a pressure. becomes more noticeable the wider the r i n g spacing. it

is

stress

usually in

the

the

circumferential

system,

r i n g s i f the p i p e w a l l

there

so

wall

stress

i s l i t t l e point

if

T h i s effect

For wider spacing

which

is

the

i n using high-

limiting tensile

i s of m i l d steel.

Example 2:

St i f f e ning R i n g s

Design a 3 m diameter a

uniformly

distributed

steel

load of

pipe with stiffening 60

kN/m2

with

a

r i n g s to support

soil

modulus of

3

N/mm2. Maximum steel stress = 103 N/mm2, delection 2%. Select t = 10 mm Then s / d

=

(210 000/503) ( . 0 0 3 3 ) 3 / 2

(8.39)

157 0.39

=

Use

:.

s = 1177 mm; Select

100 mm h i g h r i n g s w i t h h/b

1200 mm.

= 10 m a x f o r

no b u c k l i n g

i.e.

ring

t h i c k n e s s b = 10 mm Centroid from c e n t r e w a l l c

I

Moment o f i n e r t i a =

=

(h+t)bh 2( bh+ts)

h3b + S ( t c ) ' -

=

+

hb

12s

=

t3 12

+

4.25

mm

tC2

(8.41) (8.42)

3104 mm3/mm

Extreme f i b r e d i s t a n c e r

=

h

+ (8.43)

100.8 mm

=

Deflection:

A/d

=

0.1~0.060~3~

(8.22b)

8x210 0 0 x 3 1 0 4 ~ 10-9+0.05x3x33 0.017

=

1.7%

=

Maximum w a l l s t r e s s

(0.1x0.10/3.0+0.003x3x3/(210000x0.011

f =

) ) 0.06

3 1 0 4 ~ 1 0 - ~ /+0.006x3/210000 3~ 1.7 N/mm2.

=

The tions,

section an

could

optimum

or

thus

be modified.

balanced

design

By

v a r y i n g the r i n g p r o p o r -

may

be

possible

i.e.

both

deflection l i m i t a n d stress l i m i t a r e attaine d simultaneously.

REFERENCES

B a r n a r d , R.E., 1957. D e s i g n a n d d e f l e c t i o n c o n t r o l o f b u r i e d steel p i p e s u p p o r t i n g e a r t h l o a d s a n d l i v e l o a d s , Proc. A m . SOC. f o r T e s t i n g M a t e r i a l s , 57. C l R i A ( C o n s t r u c t i o n I n d u s t r y Research a n d I n f o r m a t i o n A s s o c i a t i o n ) , 1978. D e s i g n a n d C o n s t r u c t i o n o f B u r i e d T h i n - d a l l P i p e s , Report 78, L o n d o n , 93 p p . Jacobsen, S., 1974. B u c k l i n g o f c i r c u l a r r i n g s and c y l i n d r i c a l t u b e s u n d e r e x t e r n a l p r e s s u r e , Water Power, 26 ( 1 2 ) . Morley, A . , 1919. S t r e n g t h o f M a t e r i a l s , L o n g m a n s Green E. Co., pp 326-333. S p a n g l e r , M.G., 1956. Stresses i n p r e s s u r e p i p e l i n e s and p r o t e c t i v e c a s i n g p i p e s . Proc. Am. SOC. C i v i l E n g s . , 82(ST5) 1054. 1973. S t i f f e n i n g r i n g s f o r p i p e s , P i p e s and P i p e l i n e s Stephenson, D., I n t l . , 18 ( 4 ) . Stephenson, D. ,1979. F l e x i b l e p i p e t h e o r y a p p l i e d to t h i n - w a l I nPVC p i p i n g , P i p e s a n d P i p e l i n e I n t l . , 24 ( 6 ) , 9-17.

158 Timoshenko, S.P. and Woinowsky-Krieger, S., 1959. a n d Shells, 2nd Ed., McGraw H i l l , N.Y., Ch. 15.

Theory

of

Plates

LIST OF SYMBOLS

a

-

bs, o r area per u n i t length

A

-

cross sectional a r e a of r i n g

b

-

C

-

distance from centre of w a l l to c e n t r o i d of w a l l section

d

-

p i p e diameter

d.

-

i n t e r n a l p i p e diameter e x t e r n a l p i p e diameter modulus of e l a s t i c i t y passive resistance modulus of s i d e f i I I permissible w a l l stress

F

-

wal I stress bending stress circumferential

stress

bending stress under r i n g r i n g stress,

o r r a d i a l stress i n p i p e w a l l

circumferential

wal I stress

design factor r i n g height depth of b a c k f i l l above pipe I

-

moment of

J

-

j o i n t factor

K

-

inertia

r a t i o of a c t i v e l a t e r a l pressure of f i l l to v e r t i c a l pressure c r i t i c a l r i n g spacing bending moment moment or deflection coefficient i n t e r n a l pressure pipe radius distance from c e n t r o i d to extreme f i b r e of beam r i n g spacing wal I thickness

159 deflection time l a g factor

V

-

W

-

vertical

U

Poisson's r a t i o pressure,

subscript

a

refers

bending a n d d to deflection

W

-

vertical

Y

-

r i n g thickness

z

-

bedding constant

A

load p e r u n i t length ( W = wd)

def I ect ion

to

arching,

b

to

160

CHAPTER 9

SECONDARY STRESSES

STRESSES AT BRANCHES

If in

a

the

stress

hole i s cut wall

on

will

the

in

the w a l l

increase on

horizontal

of

a pipe,

each

axis

at

side

the

S

where the a

+ 2

S(l

=

is

horizontal

the pipe

the

may

also

axis a

force

the w a l l

able

to

minimize

entrance

the main

pipe

forcing

may

angles

to

to

is

the

piece

the

collar

diameter

walls

are

on

a

each

side

of

plate

is

given

it

will

common

In

the bore o f

a

length.

is For

branch

hole o r

in be

branch

stress

The

If

9.1.

Fig.

on a

necessary

to

the

rein-

Reinforcing i s also desirespecially

with

low

the b r a n c h connection to m i n i -

such

necessary.

cases

the

the

size of

hole cut

in

the b r a n c h p i p e and r e i n -

For small

a s u i t a b l e method of onto

axis

the

practice

than

collar

plate,

in the p l a i n pipe.

that

loss.

a

the

hole o r

circular

significant

A

to

vertical

b r a n c h p i p e occurs.

assuming otherwise

out

of

thickness, t

the

that

the

a

head

by

cut

and

assuming

hole

i s consequently three times

larger

would

is

tangential

elliptical

b r a n c h pipes at r i g h t

strengthening

main

pipe

the p i p e con-

around

the

branch

The w i d t h of c o l l a r may be c a l c u l a t e d f o r any p a r t i -

thickness which

in

the

use s k i r t s at

weld

9.2).

of

times

all

a

is

applied

i s to

barrel

of

is

is

stress

hole

b

case

three

be especially

a

(Fig.

force

the

distortion.

mize

cular

and

at

where

pipework

pipe

the stress S '

to

is

pressure

nection

and

beside a stress

stress

b In

increase

wall

the

vertical

circumferential

pipe

The

small

1951),

length =

stress.

distribution

hole.

a

stress

(9.1

uniform

hole

applied

the of

a/b)

the

circular

of

side

i n a p l a t e i s (Tirnoshenko a n d Goodier,

S'

the circumferential

the

same,

the w'

the

col t a r

have

been

main

pipe,

the

thickness

of

permissible and

that

w i t h the b r a n c h p i p e diameter.

width, the

the

i.e.

D

wall

stress

the

takes taken

in

collar

the of

circumferential

across 2t'w'

the

tD,

=

branch the

main

the c o l l a r width

is

diameter where

t'

p i p e outside pipe.

and main small

This pipe

compared

161

s t

Fig.

9.1

Fig.

9.2

t

f

t

t

t

t

t

t

Stress d i s t r i b u t i o n beside a hole i n a p l a t e under a uniform stress S.

Collar reinforcing at a branch pipe

7-

lp-Sornetimes

used

LI Fig.

9.3

L a t e r a l w i t h e x t e r n a l crotch plates

162 For

l a r g e r diameter b r a n c h pipes and branches at

to the m a i n pipe,

v a r i o u s angles

crotch p l a t e s a r e p r e f e r a b l e to c o l l a r s ( B l a i r , 1946).

Crotch P l a t e s

Various

types of

ing laterals. of

crotch

s t i f f e n i n g p l a t e s have been proposed f o r r e i n f o r c -

The Swiss f i r m Sulzer Bros.

plates,

and

a design a i d

l a t e r p u b l i s h e d b y Swanson et a l plates

which

pipes.

The

cases.

Trial

and

would

be

welded

on use

sections were selected were

to 9.7

Swanson's in

plates

the j u n c t i o n of

nomograph was p r e p a r e d b y a n a l y s i n g a

deflections

9.3

a nomograph was

(1955). The design was f o r e x t e r n a l

i n t o the c r o t c h a t

compatible.

any

l a r g e number of

and r e v i s e d u n t i l forces balanced

Only

r e c t a n g u l a r sections f o r crotch

a r e design c h a r t s f o r e x t e r n a l crotch

results,

system

(normally

but

of

for

expressed

units. small

the

T sections may have g r e a t e r r i g i d i t y .

p l a t e s were considered although Figs.

(1941) pioneered the design

i n the form of

in

p l a t e s based

dimensionless

parameters f o r

The design c h a r t s a r e f o r s i n g l e c o l l a r diameter

laterals)

and

for

double crotch

p l a t e s ( f o r l a t e r a l diameter n e a r l y equal to the m a i n p i p e diameter). The method of design u s i n g the c h a r t s i s as follows: C a l c u l a t e Pd / f t knowing t h e maximum i n t e r n a l pressure P, the 1 main p i p e diameter dl, the permissible p l a t e stress f a n d an assumed p l a t e thickness t. Read

off

terms of plate

from

Fig.

9.5

the

crotch

the main p i p e diameter dl.

widths

for

a

90"

lateral

plate bl

with

width

and cl

expressed

in

a r e the crotch

the same diameter

as the

main pipe. If

the

are

branch

unequal,

is

at

an a n g l e to the b a r r e l ,

Kb

read

K

and

from

crotch p l a t e w i d t h b y m u l t i p l y i n g bl The

plate

Read

the

c/dl

and

widths

the

c a l c u l a t e a,

angle

whether

plate

thickness. the

now

depth of

or

9.6

the diameters

and

correct

b y K b a n d c1 b y Kc.

b=d (bl/d,) K a n d c=dl(c,/dl)K 1 b r a t i o a/dl corresponding to b/dl

the

the

lateral

from

Fig.

9.7

and

. or

hence

the top depth of each plate.

Check

increase

are

p l a t e top

Fig.

either plate width

plate

If

so,

the

thickness

i s g r e a t e r than 30 times the

plate and

is

try

liable again.

weight of p l a t e s could be optimized b y t r i a l .

to

buckle,

The

shape

so and

163

9.4

Fig.

The

E x t e r n a l crotch p l a t e s

shape

developing often

be

together

of

the

bent at

p l a t e should of

intersection

then of

be the

intersection

point,

and

if

b /d,

8

1.0

0.5

0.5

20

pd

-1

f t

Fig.

9.5

Single

plates

be f i l l e t

t h i n enough,

Crotch p l a t e side w i d t h

1.5

by may

welded

may a l s o be

the intersection of the b r a n c h a n d the b a r r e l .

c, dtJ,

0

drawn for fabrication pipes.

i n t o shape whereas double p l a t e s w i l l

the

bent to follow

the line

164

Fig.

9.6

Effect of

l a t e r a l diameter and angle on c r o t c h p l a t e s i z e .

165 For right

large

angles

installations the

to

design c h a r t s f o r

a

t h i r d p l a t e may

a x i s of

3-plate

the

main

pipe

(see F i g .

designs a r e not a v a i l a b l e ,

t h i r d p l a t e t o a 2-plate

merely a d d i n g a

be r e q u i r e d ,

design.

9.3).

placed at Although

Swanson suggests

Discretion should be

used i n r e d u c i n g the thicknesses o r widths of the two c r o t c h p l a t e s i n such

circumstances.

The

deflections

are

reduced

considerably

by

a d d i n g a t h i r d p l a t e though. Whether judgement.

crotch The

p l a t e s should

author

advises

be

using

design pressure P

i s g r e a t e r than

of

the

safety,

diameter

t,

of

is

Certain

the b r a n c h p i p e a n d

Wyss L t d . pipe

in

(Suss a n d Hassan,

'Y's.

material

(Stephenson,

disadvantages

External

inefficiently

at

all

crotch

i s also a matter of

plates

if

the

internal

y t a"/135Gd2 where G i s the factor 1 thickness of the main p i p e , d2 i s the

wall

the b r a n c h p i p e a n d b a r r e l . y

Internal Bracing

used

a i s the a n g l e between the a x i s of

i s y i e l d stress.

1971) external

crotch

plates

prompted

Escher

1957) to i n v e s t i g a t e i n t e r n a l b r a c i n g for

crotch

plates

and

require

act

in

more

p l a t e s which could be i n p u r e tension.

bending, steel

than

consequently internal

use

crotch

An

internal

web may also act

as a guide vane.

T h i s effect,

b i n e d w i t h a g r a d u a l taper of the pipes a t the j u n c t i o n , losses

for

through

some

two

flow

pipes

considerably

is

configurations.

For

to a confluence, less

than

the

the head

with

a

case of

water

loss w i t h

'T'

standard

com-

reduces head flowing

internal

webs

junction.

The

arrangement of the b r a n c h p i p e j o i n i n g the main p i p e a t a skew angle with

both

pipes

gradually

flared

at

the

confluence,

9.8)

i n t e r n a l b r a c i n g f o r the acute a n g l e (see F i g . pcmping into

station

two

delivery

branch

pipes.

pipes,

such

For

as

with

i s recommended f o r

water f l o w i n g from a main p i p e

pumping

penstocks f o r

hydro-electric

performed

determine the best angles of

to

together

stat ions,

station

suction

h y d r a u l i c model

pipes

and

tests should be

divergence a n d taper,

and

the arrangement of the i n t e r n a l b r a c i n g . The o v e r a l l external

weight

of

c r o t c h plates.

steel

The

for

internal bracing is

less than f o r

compact arrangement f a c i l i t a t e s

transport

a n d reduces excavation costs. Unless

the

confluence, ing

pipes

internal

increased

there w i l l

designed

bracing will

velocities

crotch p l a t e s o n l y .

diffusers

are

and

be a h i g h head rapidly

appears

to

be

flares

head

losses

than

h a n d i f the expansion

at

paths, with

the

caus-

external

i s too a b r u p t

loss.

The head loss coefficient f o r conical

with

the

between the w a i l a n d a x i s exceeds s u i t a b l e compromise f o r

cone-shaped

p r o t r u d e i n t o the flow

higher

On the other

increases

with

a n g l e of

flare

about 7; degrees

once

the

(Rouse,

angle

1961). A

the angles of f l a r e f o r a 45-degree confluence

74 degrees

to

the

axis

for

the

main

pipe

and

15

degrees to the a x i s f o r t h e b r a n c h p i p e . The

use

branch

and

of main

internal pipes

bracing to

a

confines

practical

the

range.

angle

For

between

the

confluence angles

approaching 90 degrees ( I T ' j u n c t i o n s ) the b r a c i n g web would obstruct the

flow

in

the

main

pipe.

hole cut

in

the

main

p i p e becomes l a r g e r a n d the b r a c i n g would be

impractically

heavy.

a

confluence,

45-degree

branch

pipe

would

before the f l a r e .

For

very

small

angles of confluence the

T h e design c h a r t s presented here a r e confined to and

have

Note that

to

for

any

other

be constructed the use of

angle with

a

of

approach

bend

the

immediately

i n t e r n a l b r a c i n g i s confined to

the

acute

obstruct

angle,

flow.

as

internal

The obtuse

bracing

a n g l e must

at be

the

obtuse

strengthened

angle with

would

external

crotch plates. To minimize head losses,

the diameters of the main incoming p i p e ,

the b r a n c h p i p e a n d the confluent

p i p e should be such t h a t the velo-

c i t i e s i n a l l pipes a r e approximately equal. If

all

branches

of

whose centre l i e s a t the surfaces w i l l flat. the

There may main

the

are

tangential

the intersection of t h e i r axes,

be i n planes.

flare

meets

resulting

continued

beyond

would

strong

be

confluence

to

a

sphere

the intersection of

T h i s means t h a t the b r a c i n g webs a r e

be a k i n k i n the obtuse a n g l e intersection l i n e where

pipe

eliminate

the

the

the

confluent

problems

point

of

where

should

it

enough to t r a n s f e r

pipe

(see

torsion,

Fig.

9.8).

To

'B'

may

be

plate

bend.

The

pipe

loads to t h i s p l a t e ,

the

walls

and the

r e s u l t i n g arrangement simp1 i f i e s f a b r i c a t i o n . To

determine

surfaces, and

the

the

p r o f i l e s of

the

lines of

intersection of

equations

were set

up r e l a t i n g the cone r a d i i a t any

horizontal

distance

along

the

that

the elevations of

intersection I ine, The

a n g l e of

the

the equation t a n confluence a x i s and The ceeding

point

i n t e r s e c t i n g p l a n e to the dis-

tance from the intersection of the axes along the main a x i s . fact

the cone

Using the

the surface of both cones a r e equal on the

the coordinates of the intersection I ine were derived. intersection

B

angle,

=

is

@

p l a n e to the main a x i s , 8 ,

e

) / s i n a cos

the

branch

(cos Q - cos a cos the

angle of

e ,

i s given by

where a i s the

cone surface to

its

e i s the a n g l e of the main cone surface to i t s a x i s .

intersection in

steps

profiles

along

the

w e r e obtained b y computer a n a l y s i s pro-

i n t e r s e c t i n g plane.

T h i s was done simul-

taneously w i t h the stress a n a l y s i s . Although simple must

pipe

it

i s possible to

'Y's

analytically,

be solved by computer,

intersection

line.

study

In

fact

the

stresses

the general

case

in

the

bracing for

i s v e r y complex a n d

proceeding i n small increments a l o n g the previous

studies

have

relied

on

model

studies to o b t a i n the stresses a t the intersecting plane. The

analysis

was

based on

can take no b e n d i n g stress. is

membrane

theory

i.e.

the

pipe walls

The c i r c u m f e r e n t i a l w a l l stress in a cone

PR/(cos 0 ) per u n i t w i d t h a n d the l o n g i t u d i n a l w a l l stress PR/(2

e

cos

)

per

unit

where P

width

l i q u i d pressure a n d R

i s the

i s the

r a d i u s of the cone p e r p e n d i c u l a r to the a x i s . The

forces

due

to

the

wall

stresses

results

indicated

force

intersection l i n e of

an

in

line,

action

the

that

the

plane,

was

of

element

of

the

p e r p e n d i c u l a r to

v e r t i c a l l y a n d h o r i z o n t a l l y in the plane. perpendicular

section p l a n e were i n a l l cases zero. resultant

on

I i n e were resolved i n t o three components,

intersection

the intersection plane, The

cone

to

the s i d e of

plotted g r a p h i c a l l y .

the r e s u l t a n t

forces on the

inter-

The p o s i t i o n and d i r e c t i o n of four

It

was

was constantly

points

the

along

the

observed that

the

i n s i d e the

intersection

p l a n e a n d a c t i n g i n tension on the i n t e r n a l b r a c i n g p l a t e . The

most

tension. centre

line

bracing

of

the

plates

crown,

is

Hence

economical

internal

the

plate.

indicated

twice

the

plate

Thus

of

the

stress,

be

force on found

that

impractically and

the

rower

plate

the

crown

to

plate

may

most

the

be

i n pure

from

+ F/tw,

where f

resultant

question

on

thickness.

A

to

in

terms

of

the

the r e s u l -

economical

the

a

bracing thicker

bending

I n some cases i t w i l l

from

and

plate

resultant for

thickness

p l a t e may

tension

force

at

extreme f i b r e

is

be selected,

the c h a r t s . and

p l a t e in p u r e tension.

the

equation

r e s u l t a n t force.

anywhere

l i n e of the cones.

i n combined

the

or

and position and magnitude of

interpolated

perpendicular

the

internal

centre,

calculated

shape

than

the

else

corresponding are

widest

at

derived

width

case

more steel

tude of

from

which

obtained /tw2

the

the

9.10

intersecting

thin,

of

to

in

plates

contain

the

width

9.8

and

knowing the p l a t e thickness

the

Figs.

in

distance

thickness

permissible

tant

b r a c i n g p l a t e would be one

T h i s s t a t e could be assured i f the r e s u l t a n t force f e l l on the

The n a r -

consequently

The w i d t h of

various

stress,

f

points 6F

=

i s permissible p l a t e tensile stress,

the was

(u-w/Z)

F i s magni-

force which acts a t a distance u from the p o i n t

the

intersection

similar

line,

analysis

was

w

is

plate

performed

width

for

and

t

is

the e x t e r n a l

in

plate crotch

p l a t e s b r a c i n g the obtuse angle. The design flare

charts, of

to

Figs.

crotch the

9.8

to

plates for

main

pipe

and

9.10, a

were compiled and f a c i l i t a t e the

45 degree ' Y ' branch

pipe.

and The

various

most

angles should be selected from h y d r a u l i c considerations. determine head losses could be conducted i f warranted.

angles

suitable

of

flare

Model tests to

169

I

-.---

6;

/

/

j

e

Lc=!23 tcn2 2 G.03303

..L

\

0 "

\

WITH R E C O M M E N D E D A N G L E S

b

TOP H A L F OF EXTERNALCROTCH PLATE B

c TOP HALF OF INTERNAL C R O T C H ~ P L ,TE

F i g . 9.8

Crotch p l a t e s f o r 7io/15O confluence.

4

170

b TOP H A L F OF EXTERNAL CROTCH PLATE 6

c TOP HALF OF INTERNAL CROTCH PLATE A Fig.

9.9

Crotch plates f o r p l a i n p i p e confluence

171

I

0.2

OL

b

0

02

0.L X / D

0.6

TOP HALF O F EXTERNAL CROTCH PLATE B

c TOP HALF OF INTERNAL CROTCH PLATE A Fig.

9.10

Crotch p l a t e s f o r confluence pipe.

plain

pipe

with

15"

taper

172 The design procedure f o r a

trial

plate

width,

p l a t e thickness, crotch

plate

t,

say

the crotch p l a t e s would then be to select

the

maximum

the

obtuse

angle

buck1 i n g under bending compression, than

w/30.

less

than

the

a g a i n s t wear.

and

from the corresponding formula.

for

less

indicated,

The

plate

is

the

To ensure that the

enough

withstand

to

the p l a t e thickness should not be

thicknesses the

rigid

compute

should also p r e f e r a b l y

main p i p e ,

not

be

to ensure d u r a b i l i t y

wall

thickness

of

After

selecting

the p l a t e thicknesses,

the corresponding

p l a t e shapes a r e interpolated from the a p p l i c a b l e c h a r t .

STRESSES AT BENDS

The than

wall

those

stresses

stresses

bending

induced

stresses wall,

are

and

as

the outside of

normally is

inside

the

caused

the

to

design the

of

bend by

or

cast

bend.

only

The the

for

The

iron

to

blocks

stresses

on

w a l l on the

worst

thickness

on

the

f

=

-PD

2t i.e. ratio

is

stress a n d stress

outside of

i s taken o n l y of

i.e.

on

the

it the

bend

circumferential

as on the i n s i d e of

assuming the p i p e w a l l

to

the t h r u s t

due to the

the i n s i d e segment of p i p e shown shaded i n F i g . (R-D/2)

i n s i d e i s less than

takes no bend-

the c i r c u m f e r e n t i a l force on a length of w a l l on the inside

the bend i s equated

ft

the

the bend

stresses a r e c a l c u l a t e d below.

Using membrane theory i n g stress,

local

against

inside of

circumferential

be reduced i f account

the b l o c k . Circumferential

out,

increase in c i r c u m f e r e n t i a l stress

i s safe to keep the same thickness

it

higher

Longitudinal

thrusting the

could a c t u a l l y but

are

straighten

stress,

of

bend

increase i n l o n g i t u d i n a l

the

wall

tends

anchor

l e n g t h of

the bend.

more severe than

safe

fabricated

circumferential

a r e m a g n i f i e d because on

a

i n a p l a i n p i p e w i t h the same w a l l thickness.

are

outside

in

0

=

P(R-D/4)

i n t e r n a l pressure on 9.11,

0D/2

(R-D/4)

(9.2)

(R-DJ2)

the normal (R-D/4)

/

w a l l thickness at the bend should be increased i n the (R-D/2)

the p i p e diameter.

where R

i s the r a d i u s of

the bend a n d D

is

173

9.11

Fig.

In

Stress concentration a t a bend

addition

etc.,

discussed

work

caused

variations designed pipeline

A

and

analysed bution,

so-called

piping in

pipe

using

systems.

if

in

the

form

and design

or

in pipe-

temperature

a n d should be or

at

large

thrust

anchors a r e

as a s t r u c t u r a l

system w i t h

movements.

The

techniques

such

standard

or

pipework

joints or

f i n i t e element

of

strains

plant

can be treated

structural

Crocker a n d K i n g ,

and

junctions

t e r t i a r y stresses

elastic

no movement

thrusts

bends,

These stresses a r e r e a l

chemical

system

slope-deflection

available

movements,

exposed

lateral

secondary stress at

there a r e sometimes

relative

interconnections

installed. axial

the

above,

by

in for

to

methods,

computer

system as

could

be

moment-distri-

many o f

programs.

which a r e (See

also

1967).

THE P I P E AS A BEAM

Longit udi na I Bend ing

Pipes should longitudinal settlement between

of

normally

direction the

supports,

be

even

if

designed they

bedding

could

Three

possible

are

cause

to to

r e s i s t some bending be

buried.

sections to c a n t i l e v e r

bending

patterns

i n the

Unevenness o r

together

o r span with

the

174 corresponding critical The

maximum

b e n d i n g moments a r e i n d i c a t e d i n F i g .

span

which

modate i s c a l c u l a t e d below.

Fb

simply

supported

pipe

.

9.12

could

accom-

The m a x i m u m f i b r e s t r e s s i s

M/Z

=

where

a

(9.3)

M is t h e b e n d i n g

p i p e whose w a l l

moment and Z

thickness,

t,

is t h e s e c t i o n m o d u l u s .

is small

For a

in comparison w i t h the i n t e r n a l

d i a m e t e r D,

Z

nD2 t / 4

=

So if M

Fb

and

WL2/8

=

(9.5)

W L Z / ( 2 n D 2t )

=

o r L2

(9.4)

2nD2 t Fb/W

=

if a

pipe of

specific weight

weight

yw

L

J8DtFb/(ywD

=

Thus f o r =

ys

(9.6)

D = 1 m,

+

4ySt)

t

=

80 000 N/m

is conveying

water

at a specific

(9.7)

0.012 m,

then

y

the

F b = 100 N/mm2,

maximum

yw

permissible

- 10 000 N/m simply

3 ,

supported

s p a n L = 26 m or 86 f t . For

low

internal

water

pressures,

b u c k l i n g of

the w a l l s

may

also

b e a p r o b l e m and t h e p o s s i b i l i t y s h o u l d b e i n v e s t i g a t e d ( e s p e c i a l l y a t t h e s u p p o r t s w h e r e t h e s h e a r s t r e s s is h i g h ) . See also P e a r s o n ,

1977.

P i p e S t r e s s at Saddles The m a x i m u m

local stress in a continuous o r s i m p l y supported p i p e

supported in a s a d d l e i s (Roark, F

(0.02 - 0.00012

=

where

Q

span,

Q

due

to

1954):

( B - 90°))(Q/t' )Loge (f)

i s the total saddle reaction.

(9.8)

Note t h a t f o r a s i m p l y s u p p o r t e d

i s the r e a c t i o n d u e to two ends r e s t i n g o n the support, the

weight

continuous pipe.

of

pipe

of

length

equal

to

the

span,

as

i.e.

for

If Q i s i n Newtons a n d t i n mm t h e n F i s i n N / m m 2 .

a

175

WlUNlT LENGTH

REACTION

WL12

WlUNlT LENGTH

Wll2

MAXIMUM BENDING MOMENT.

WLI2

WL

5 x 2 3 z

a.

El

Simply Supported

F i g . 9.12

Cantilevered

c.

Fixed Ends

b.

Beam moments and deflections

The a n g l e of or

width

0

WLl2

wL2/a

DEFLECTION

crete

WlUNlT LENGTH

strap

of

the

support

To t h i s

diameter.

,

bottom support, 6 saddles.

mum l o n g i t u d i n a l

i s n o r m a l l y 90"

The stress provided

it

local stress a t

is practically is

small

120'

or

for

con-

independent of

compared

with

the

the pipe

the saddle must be added the rnaxi-

bending stress a t the support f o r the p i p e a c t i n g as

a continuous beam,

unless a j o i n t occurs a t the support.

R i n g Girders Steel ground may

pipes etc.

may

be

B u c k l i n g of

laid

and

be

the

back

plain

the

vicinity

developed

under

the

a r e useful

rectangular

fixed

in

over

ground,

ravines,

of

all

in

cross

r o u n d the pipe.

there

i s no s i d e f i l l

the

Chapter

ring

8.

may The

be

Rigid r i n g

i n these circumstances. section, The

or

local

evaluated

longitudinal

marshy

i n these circumstances as

the p i p e saddles cause stress concentrations.

g i r d e r s at each support may

ground

t h e p i p e a n d stresses

be more severe than

support

above

The rings

T o r H shaped,

with

p i p e bending stress using

the

bending stress

ring

theory

i n the p i p e

w a l l under a r i n g is

1.65 0.91

Pd

+

h J%T/A

2h

in

(9.9)

176 A

where be

i s the cross sectional

added

any

longitudinal

area of

stress

in

the r i n g . the

pipe

To

t h i s stress must

due

to

beam

action

between supports. T h e r i n g stress i n the r i n g g i r d e r i s

1

1 0.91A/(hj hd)

+

In

addition

the

there

r i n g girder

Pd

(9.10)

2 7 will

be

local stresses

to the supports.

g i r d e r a n d rest on a s l i d i n g or of

the p i p e a n d contents

due to

t h e method of

Legs a r e securely

fixed

fixing

to the r i n g

r o l l e r b e a r i n g on a p i e r .

The weight

i s t r a n s f e r r e d to the bottom h a l f of the r i n g

g i r d e r a n d thence through the legs to the p i e r .

A

more comprehensive a n a l y s i s

presented

in

Crocker

and

King

of

the stresses

(1967).

See

i n r i n g girders

also

Cates

(1950)

is

and

( 1933).

Scharer

TEMPERATURE STRESSES

Pipes outside

with

will

high

differences

be subject

temperature

differences.

have

maximum

their

compressive the and

on

the

temperature on the

heat flow

the p i p e diameter,

in

to r a d i a l The

value

temperature

the

inner

the

inner

and

wall

is

above

I f the w a l l

l o n g i t u d i n a l stresses

a n d outer

i n n e r surface and tensile on

i s steady.

inside

a n d c i r c u m f e r e n t i a l stresses due to

circumferential and on

between

surfaces,

the outer

t h a t on

is thin

being

surface

the outer

if

wall,

i n comparison w i t h

the stresses may be approximated b y the f o l l o w i n g

equations: on the i n n e r surface F

a n d on the outer surface F where

a

i s the thermal

elasticity,

T

a ET

=

Fe c

=

F

(9.11)

-20

=

P

=

aET -2(1-u)

(9.12)

coefficient o f expansion,

i s the temperature difference,

E i s the modulus of

v i s Poisson’s r a t i o ,

c i r c u m f e r e n t i a l stress and Fe i s l o n g i t u d i n a l stress. i s compressive a n d

is

always

and l o n g i t u d i n a l stresses, For

normal

c I imatic

less

than

the

Fc i s

The r a d i a l stress

maximum c i r c u m f e r e n t i a l

b e i n g zero on both boundaries.

temperature differences

the stresses

are rela-

177 tively ‘C,

low. v =

A

Thus f o r

longitudinal

longitudinally

as

The m a g n i t u d e o f after

10°C, E = 200 000 N/mmz, a

T =

0.3 f o r s t e e l p i p e , t h e n F m a x

installation)

stress a

may

result

t h i s stress (see

Ch.

also

of

be

=

induced

temperature

i s aET

10 f o r

= 12 x

per

17 N/mmz. in

change

a

pipe

after

restrained

instal lation.

(tensile if the temperature drops an

analysis

including

effects

of

supports).

REFERENCES

B l a i r , J.S.,

1946. R e i n f o r e c e m e n t o f b r a n c h p i p e s . E n g i n e e r i n g , 162.

Cates, W.H., 1950. D e s i g n s t a n d a r d s f o r l a r g e d i a m e t e r s t e e l w a t e r p i p e , J. Am. W a t e r W o r k s Assn., 42. C r o c k e r , 5. and K i n g , R.C., 1967. P i p i n g H a n d b o o k , 5 t h E d . , M c G r a w H i l l , N.Y. P e a r s o n , F.H., 1977. Beam b e h a v i o u r o f b u r i e d r i g i d p i p e l i n e s . P r o c . Am. SOC. C i v i l E n g r s . , 108 ( E E 5 ) 767. 1954. F o r m u l a s f o r s t r e s s and s t r a i n , McGraw H i l l , N.Y. R o a r k , R.J., p . 418. Rouse, H., 1961. E n g i n e e r i n g H y d r a u l i c s , W i l e y , N.Y., S c h a r e r , H., 1933. D e s i g n of l a r g e p i p e l i n e s . T r a n s . Am. SOC. C i v i l Engs., 98 ( 1 0 ) . Stephenson, D., 1971. I n t e r n a l b r a c i n g f o r p i p e c o n f l u e n c e s . T r a n s . , S.A. I n s t n . C i v i l E n g r s . , 13 (11). Sulzer, 1941. P a t e n t e d s t i f f e n i n g c o l l a r s o n t h e b r a n c h e s o f highpressure p i p e l ines f o r hydroelectric power works. Sulzer Technical R e v i e w , 2 ,lo. Suss, A. and H a s s a n , D.R., 1957. R e d u c t i o n o f t h e w e i g h t and loss o f in d i s t r i b u t i o n p i p e s f o r h y d r a u l i c p o w e r l a n t s . Escher energy Wyss News, 30 (3) 25. Swanson, H . S . , C h a p t o n , H.J., W i l k i n s o n , W.J., King, C.L. and N e l 1955. D e s i g n o f w y e b r a n c h e s f o r s t e e l p i p e s . J. A m . son, E.D., W a t e r W o r k s A s s n . , 47 ( 6 ) 581. Timoshenko, S. and G o o d i e r , J.N., 1951. T h e o r y o f E l a s t i c i t y , McGraw H i l l , N.Y.

L I S T OF SYMBOLS

A

-

a r e a of r i n g g i r d e r

a,b

-

h o r i z o n t a l and v e r t i c a l

a,b,c

-

c r o t c h p l a t e t o p and s i d e w i d t h s

D or d-

a x e s o f an e l l i p s e

p i p e diameter

E

-

modulus of e l a s t i c i t y

f

-

permissible crotch p l a t e stress

F

-

r e s u l t a n t force on c r o t c h p l a t e a t a n y point,

G

-

wal I thickness

h

factor o f safety

o r stress

178 span o r length bending moment water pressure (same u n i t s as f ) saddle reaction radius stress a p p l i e d to a p l a t e crotch p l a t e thickness temperature difference distance from p i p e w a l l to F w i d t h of crotch p l a t e at any p o i n t load p e r u n i t l e n g t h horizontal

distance

from

crown

along

mid-plane

or

centre-I ine

of

crotch p l a t e

Y

-

vertical

distance

above

crown

of

crotch

plate section modulus a n g l e of confluence of b r a n c h p i p e w i t h main p i p e coefficient of temperature expansion a n g l e between crotch p l a t e and main p i p e a x i s , o r a n g l e of support

specific weight angle

of

flare,

or

taper,

of

main

pipe

a x i s to cone) a n g l e of f l a r e , Poisson's r a t i o

o r taper,

of b r a n c h p i p e

(measured from

CHAPTER 10

PIPES, FITTINGS AND APPURTENANCES

PIPE MATERIALS

Steel Pipe Steel

i s one of

i s d u c t i l e yet work,

the most

has a

high

versatile materials for tensile

It

strength.

pipe walls,

is

relatively

as

it

easy

to

i s the strongest

a n d the welded j o i n t f r e q u e n t l y used w i t h steel

type of j o i n t . Steel

grades

p i p e s i n the U.K.,

used f o r

and

their

corresponding

minimum y i e l d stresses a r e as follows:

BS 4360

Grade 40

230 N/mm2

:

Grade 50

355 N/mm2

Grade 55 US 572

450 N/mm2

:

290 N/mm2

Grade 4 2 Grade 50

345 N/mm2

Grade 60 Grade 65 The for

higher

grades

are

preferred

they

a r e easier

Small-bore seams,

but

welded

either

to

weld,

(less

a n d on

than

larye-bore

450

pipes

448 N/mm2 high-pressure

account

pipelines

but

horizontally or

asbestos cement

pressures,

and

it

of

steel

made

spirally. site.

pipes

from

are

steel

wall

thickness

rolled

plate,

Pipes a r e made Steel

and plastic

requires

the e x t r a

loads.

mm)

are

10 m and more a n d j o i n t e d on

than concrete, low

: for

low pressure p i p e l i n e s the lower grade steels a r e more economical,

they a r e more r e s i s t a n t to e x t e r n a l

to

414 N/mm2

:

pipe

pipe

coating

and

is

in

and

lengths

up

more expensive small

for

without

bent

sheathing

bores to

2nd

prevent

corrosion.

Cast Iron Pipe Cast

iron

is

more

expensive and more r i g i d . used f o r p l a i n p i p e , tapers

and

flanges.

corrosion-resistant I n fact

actual

steel,

sand c a s t i n g

a l t h o u g h i t i s used f o r Plain

than

'specials'

but

more

i s now r a r e l y such a s bends,

p i p e s a r e n o r m a l l y formed of

grey

iron

180 or

ductile

and

iron

by

dimensions

of

centrifugal ductile

spinning.

iron

pipes

Standard

are

pressure

specified

classes

BS 4772

in

and

i r o n p i p e s i n 854622.

of grey

Asbestos Cernent Pipe Asbestos cement is

able

to

cement

resist

is

p i p e i s made of relatively

relatively

high

cheap,

cement

tensile

strong

and asbestos f i b r e , stresses.

and

Although

which

asbestos

corrosion-resistant,

susceptible to shock damage and cannot be used f o r

it

is

'specials'.

Joints a r e made w i t h a sleeve f i t t e d w i t h r u b b e r sealing r i n g s .

Concrete Pipe Reinforced ical

for

loads

large

easier

and in

or

are

prestressed concrete pipes a r e s u i t a b l e and econom-

diameters. than

corrosion

jointing

and

They

steel

on

are

resistant.

making

able

account Their

later

to

resist

their

of

main

external

extra

wall

weaknesses

connections.

There

buckling thickness,

appear

i s as

to

be

yet not much

long-term experience w i t h prestressed concrete pipes.

Plastic Pipe Technological this

has

advances

been o f

special

cized p o l y v i n y l c h l o r i d e gases,

drains,

Kingdom for

and

small

UPVC

for

in

interest

Europe.

large

pipes

Glass

it

is

It

fibre is

Polyethylene although

diameters.

suitable

Although is still

a r e causing The

resin

for

some

were

engineering.

sewer

also

is

used

usually

density

r a p i d and Unplasti-

f o r chemicals,

pipes

i n the United

to

limited

a

more

extent

expensive

polyethylene

has

than

recently

l a r g e diameter p i p e l i n e s .

also

used

expensive

specialised

to a

but

l i m i t e d extent

used

applications

with as

an it

for

pipe-

inert

filler

i s corrosion-

l i g h t and strong.

pla sti c pipe not

and

is

it

High

is

comparatively

resistant, r i g i d ,

and

and

1960's

the

pipeline

irrigation

been used w i t h some success f o r

lines.

in

in

pipes a r e now used extensively

waterpipes,

bore

plastics

has

not

yet stood the r i g o r o u s test of

acceptable under many by-laws,

time

i t s many advantages

i t to g a i n r a p i d acceptance amongst engineers.

working

stress

of

plastic

i s less

14 N/mrnz

so p l a s t i c cannot

be used f o r limited

l a r g e diameter

strength

buckting

and

pipes. be

The

used

subject

deterioration w i t h

collapse.

can be used

high-pressure

in difficult

friction

as

to encrustation

to

I t i s also subject to

its flexibility

many

can cause

corrosive f l u i d s a n d

locations such as undersea o u t f a l l s

loss

beneficially

time a n d

i s resistant

It

mains.

a n d gas

i s lower than f o r

most m a t e r i a l s a n d

a

types of

l i n i n g to other

than other types of p i p e ,

pipes.

i t can

It

is

less

accommodates g r e a t e r

ground movements a n d i s easier a n d l i g h t e r to l a y . On the other h a n d it

has

a

loosened and

high

coefficient

of

thermal

expansion,

i f the p i p e contracts on cooling.

may

distort

or

even

collapse

joints

so

may

be

I t i s susceptible to damage

under

load.

Ribbed

and

stiffened

pipes a r e b e i n g developed to overcome these l i m i tations. P l a s t i c p i p i n g i s n o r m a l l y made of thermoplastic which i s r e l a t i v e ly

easy

to

extrude

As

PVC i s f l e x i b l e a n d e x h i b i t s creep,

plastics. in

an

when

heated.

u n p l a s t i c i z e d form is

ethylene

easier

to

were

with

developments

Polyethylene

(UPVC) f o r

work

than

and it

PVC

are

pipes.

the manufacture of

UPVC

polyethylene.

and

With

for

thermo-

i s generally

this

reason

improvement

used Poly-

earlier

i n production

techniques UPVC i s now cheaper than polyethylene a n d i t holds c e r t a i n other

advantages.

thermal

expansion

high-density lene for

is

stronger

than

and

has

polyethylene,

polyethylene

(HDPE)

has

a

smaller

although improved

coefficient

of

recent ly-developed

properties.

Polyethy-

i s more d u c t i l e than UPVC a n d f o r t h i s reason i s often p r e f e r r e d gas p i p i n g .

developed

in

It

the

for

Rubber toughened p l a s t i c s such as ABS have a l s o been

flexible

appendix.

pipework.

Because

of

Properties

the

low

of

plastics are tabulated

elasticity

of

plastics,

water

hammer pressures a r e less severe than f o r other m a t e r i a l s . Plastic

pipes

may

be j o i n t e d

though

these

methods

are

factory

fitted

flanges

or

by

difficult spigots

solvents for

and

large

sockets

of

fusion

welding,

diameters. with

rubber

al-

Screwed o r rings

are

also used. The

manufacture of moulded bends,

tees a n d branches

i n UPVC a n d even more so i n HDPE so steel o r C.I.

is difficult

f i t t i n g s a r e fre-

quen t I y used. The p r i c e of the 1960's,

PVC pipes

i n the U.K.

dropped b y more than

30%

in

whereas other p i p e m a t e r i a l s a r e c o n t i n u a l l y i n c r e a s i n g i n

182 cost.

In

the

U.K.,

cast

iron

than

1 N/mmz.

pipes

PVC

for

pipes

are

cheaper

diameters

less

than

than

asbestos cement

450 mm and

I t i s r a p i d l y r e p l a c i n g salt-glazed

and

pressures

less

and asbestos-cement

pipes f o r sewers a n d low pressure systems. least 5700 m i l l i o n was

At out

the

pipes

world.

are

The

used

figure

in

ever

may yet r e v o l u t i o n i z e BVMA,

1964; P a u l ,

spent

is

i n 1972 on p l a s t i c p i p i n g through-

likely

to

increasing

pipeline

r i s e considerably

proportions

engineering

and

(see

as

new

also

plastic

discoveries

Boucher,

1948;

1954).

LINE VALVES

There a r e many which

types of v a l v e s f o r use i n p i p e l i n e s , the choice of

depends on the d u t y .

depend on economics. the

pipe

diameter,

waterworks the

same

trapped, be

lined

weighed certain

level

and

as

up.

v a l v e s a n d the size w i l l

Normally v a l v e s a r e sized s l i g h t l y

practice i t

whereas

The spacing of

installed

with

a

reducer

on

soffit

the

I n choosing the

circumstances

either

side.

In

i s p r e f e r a b l e to keep the s o f f i t of the v a l v e at of

the

pipe,

to

prevent

in sewerage a n d sol i d s t r a n s p o r t ,

against

smaller than

cost it

the of

may

size, the

the cost of

head

loss

desirable

be

being

the v a l v e should be

through

to

air

the i n v e r t s should

it,

maintain

although the

full

in

pipe

bore ( t o prevent erosion o r b l o c k a g e ) . Isolating km,

valves

the

spacing

problems.

Sections

leaks

and

waste

would

the

valves

are

are

frequently

being of

a

function

the p i p e l i n e may

volume of

be

a

installed at

water

f u n c t i o n of

i n s t a l l e d at

which

of

intervals

economics

and

of

1

to

5

operating

have to be isolated to r e p a i r would

the spacing of

h a v e to

be d r a i n e d

i s o l a t i n g valves.

the bottom of each major d i p

to

Scour

i n the p i p e l i n e

profile. It around

i s sometimes a d v i s a b l e to in-line

facilitate

i n s t a l I smal I-diameter

by-pass

valves

v a l v e s to equalize pressures across the gate a n d thus

opening

( w h i c h may

be manual o r

by

means of

an e l e c t r i c

o r mechanical a c t u a t o r ) .

Sluice Valves Sluice v a l v e s ,

o r g a t e valves,

a r e the normal type of v a l v e s used

183 for

isolating or

scouring.

They

seal

well

under

high

pressures a n d

when f u l l y open o f f e r l i t t l e resistance to f l u i d flow. There spindle

are

which

handwheel,

two

types

is

attached

and

attachment

of

spindles to

the

a non-raising

gate

raising

and

the

does

s p i n d l e which

10.1).

i n the gate ( F i g .

for

A

gate:

rising

not r o t a t e w i t h the

i s rotated i n a

screwed

The r i s i n g s p i n d l e type i s easy to

I u b r ica t e. The

gate

be

may

para1 lel-sided

or

wedge-shaped.

seals best b u t may be damaged b y g r i t . gunmetal

sealing

the wedge-gate

For low pressures r e s i l i e n t o r

faces may be used b u t f o r

high pressures s t a i n l e s s

steel seals a r e preferred. Despite sometimes

sluice

valves'

troublesome

to

simplicity operate.

and

They

need

them a g a i n s t a high unbalanced pressure, minutes to t u r n open o r closed.

positive a

action,

big

force

they to

are

unseat

a n d l a r g e v a l v e s t a k e many

Some of the problems can be overcome

b y i n s t a l l i n g a v a l v e w i t h a smaller bore t h a n the p i p e l i n e diameter.

Fig.

10.1

Sluice v a l v e

Butterfly Valves ( F i g . 10.2) Butterfly a n d occupy for

sluice

fairly

valves less

are

space.

valves,

cheaper The

especially

h i g h resistance to flow

the thickness of

than

sealing at

valves

for

larger

sizes

i s sometimes not as e f f e c t i v e as

high

even

the d i s c obstructs

sluice

pressures.

i n the f u l l y the flow

They

also

open state,

even when

offer

a

because

i t i s rotated

184 90 degrees

to

the

the

gates

high

fully

open

position.

Butterfly

valves,

as

well

as

a r e not s u i t e d f o r operation in p a r t l y open positions as

s l u i c e valves, and

torques

seatings to

open

would them

erode

against

As

rapidly. high

both

pressure,

types

they

require

often

have

geared handwheels o r power d r i v e n actuators.

Fig.

10.2

Butterfly valve

Globe Valves Globe v a l v e s have a c i r c u l a r seal spindle pipe

and

axis.

handwheel. The

flow

The

seating

changes d i r e c t i o n

r e s u l t i n g i n h i g h head losses. bore pipework

connected a x i a l l y is

a n d as taps,

a

ring

through

to a v e r t i c a l

perpendicular

to

the

90 degrees twice thus

The v a l v e s a r e n o r m a l l y used i n small-

a l t h o u g h a v a r i a t i o n i s used as a control

valve.

Needle and Control Valves ( F i g . 10.3) Needle v a l v e s a r e more expensive than s l u i c e a n d b u t t e r f l y v a l v e s but are w e l l ing

action

l i t t l e flow

suited for

as

they

t h r o t t l i n g flow.

close

whereas

They

sluice

resistance u n t i l p r a c t i c a l l y shut

damage.

Needle

valves

springs,

accumulators

may or

be

used

actuators

butterfly

counterbalance

maintain

constant

conditions e i t h e r upstream o r downstream of the v a l v e , a constant

flow.

even

at

axial

needle o r

throttl-

valves

offer

and may suffer c a v i t a t i o n

with to

have a g r a d u a l

and

weights, pressure

o r to m a i n t a i n

They a r e streamlined i n design a n d r e s i s t a n t to wear

high flow

velocities.

spear-shaped

The cone

method of i n t o a seat.

sealing

is

to

push

an

There i s often a p i l o t

needle which operates f i r s t to balance the heads before opening.

A v a l v e w i t h a s i m i l a r h y d r a u l i c c h a r a c t e r i s t i c to the needle v a l v e

185 is

the

sleeve

valve

atmosphere or orifices spray,

around

the

thereby

an

electric for

valve.

cone

The

cone

tube

used

and

is

as

a

discharging

into

the

which s l i d e s over

i n a n umbrella-shaped

Needle a n d sleeve

valves are

hammer release v a l v e s when coupled

actuator

away

use

for

b y a sleeve

spread the flow

water

maintenance valve

seal

the energy.

as

rotates

suitable

They

hydraulic

or

dismantling

sealing

is

dissipating

a l s o occasionally to

which

an open b a y .

the

(see Chapter working

variation from

the

of

the

pipe

4). They

parts

are

needle v a l v e

axis

require

inside

instead

but of

the the

being

withdrawn a x i a l l y .

Fig.

10.3

Needle v a l v e

Spherical Valves

( F i g - 10.4)

Spherical

have

valves

it.

When the v a l v e

the

bore

i s equal

the sphere,

a

rotary

p l u g with

is fully

opened

there

to

that

a n d normally

of

the pipe.

an

axial

hole t h r o u g h

i s no resistance to flow

The v a l v e s close b y

have a n offset

r o t a t i n g the sphere into the open position.

action

as

rotating

to unseat them before

186

10.4

Fig.

Spherical

Reflux Valves

used

normal

to

flow

the flow

(Fig.

or

Reflux, are

flow

the

or

check

the gate

the horizontally

gate

valves

in

automatically

conditions

stops,

keep

10.5)

non-return,

stop

the a i d o f s p r i ngs. to

valve

as

the

they

i s k e p t open

by

A counterbalance can be f i t t e d to the gate s p i n d l e fully

open a t

practically

all

flows,

the flow

a l s o sometimes

u s e d to a s s i s t c l o s i n g .

Larger reflux

obstruct

the

in

which

flow,

to

flutter

at

case

and

Mounted in h o r i z o n t a l tend

Under

and when

the flow,

in r a p i d c l o s i n g when

multi-gates,

direction.

h i n g e d g a t e closes b y g r a v i t y o r w i t h

to assist

used

a r e also known,

reverse

the

swabbing

pipes,

low

thickness of

the

t h e g a t e s of

flows,

and

for

does

or

stop.

it

Springs are

v a l v e s may h a v e

of e a c h g a t e w i l l pipeline

could be

may

be

partially difficult.

some t y p e s o f r e f l u x v a l v e s this

reason

an

offset

hinge

w h i c h c l i c k s t h e g a t e open is sometimes used.

AIR

VALVES

Two One

types

is a

(see a l s o 1943). of

air

small-orifice

large-orifice

Lescovich,

release

1972;

valves

automatic

air

are

Parmakian,

normally

1950;

used

Sweeton,

i n pipelines.

release v a l v e a n d the other i s a

a i r vent v a l v e .

A i r Vent V a l v e s When a p i p e l i n e i s f i l l e d , profile,

thereby

increasing

a i r could be trapped a t peaks along the head

losses a n d r e d u c i n g

the capacity

of

187 the

Air

pipeline.

permit also

air

let

in

air

the into

pipeline.

Without

collapse,

or

It

is also

cause

valves

pipe the

normally

to escape when

installed

displaced

by

p i p e l i n e d u r i n g scouring o r

at

peaks

the f l u i d .

to

They

when emptying

the

them vacuum may occur a t peaks a n d the p i p e could

u n d e s i r a b l e to hammer

have a i r

pressure

T h e s e a l i n g element

full,

are

i t may not be possible to d r a i n the p i p e l i n e completely.

water

pipeline. is

vent

i s seated

pockets in the p i p e as they

fluctuations

during

operation

i s a buoyant b a l l which,

a g a i n s t a n opening a t

the top of

may

of

the

when the p i p e

the v a l v e .

When

t h e pressure i n s i d e the p i p e f a l l s below the e x t e r n a l pressure the b a l l

drops thereby p e r m i t t i n g a i r to be drawn i n t o the pipe. is

being f i l l e d

pipe and

the

valve

will

remain

l i f t s the b a l I a g a i n s t

open u n t i l

the seating.

When the p i p e

the water f i l l s

The v a l v e w i l l

the

not open

a g a i n u n t i I the pressure fal I s below the e x t e r n a l pressure.

10.5

Fig.

The

Reflux v a l v e

valve

will

tend

should be r e a l i s e d t h a t psi)

will

cause

sonic

Pressure differences close

some

air

vent

blow

to

closed

a differential velocity

of

0.027

around valves.

The

at

high

air

velocities.

pressure of o n l y 0.1

air N/mrn2

through

the

valve

N/mmz

It

(13

(300m/s).

( 4 p s i ) h a v e been found to

slamming

closed

may

damage

the

hollow b a l l seal. In

practice

a

diameter

approximately

l/lOth

of

the p i p e diameter

188 is

used,

although

preferable.

t i o n diameter, The

larger

The v a l v e

size

diameters

duplicate

installations

the size of

the

are

i n l e t connec-

a n d the o r i f i c e diameter i s u s a l l y s l i g h t l y smaller. of

air

vent

scouring

through

an o r i f i c e i s approximately

pressure

rate

valve w i l l

desired

same

and

i s r e f e r r e d to b y

of

the

difference,

depend on

pipe.

The

r a t e of

flow

of

air

40 times the flow of water f o r the

ful I

but

filling or

the r a t e of

volume

vacuum

pressure

should

not

be

a1 lowed to develop.

A i r vent relative

v a l v e s should be i n s t a l l e d a t peaks i n the p i p e l i n e ,

to

Various

the

possible

during

a

rule

is

the

of

During vent

free

the

air

fitting

as a l l

air

flow is

release air

the

equal

the a i r

valves area

most the

the

gradient.

including

reverse

gradients

They

normally

fitted

r a t e of

should

initial

difficult r a t e of

discussed

derived

m2

at

are

as

are in

the

both

hydraulic

valve,

in m 3 / s

air

operations,

will

to

gradients

orifice

of

relative

b e considered.

sizing

thumb

valves

taken

an

for

r a t e of

of

should

with

sec t ion. Equations

and

hydraulic

scouring,

Combination

flow

horizontal

to

evacuation the

the

in

Chapter

be

Q/lOO

pressure.

factor

filling

in

air

line.

next

5.

As

where

Q

The r a t e of

determine of

in

though.

through

all

Care should be

i s evacuated as the flow of water w i l l suddenly

be stopped a n d water

hammer

i s possible.

A i r Release Valves Air

i s e n t r a i n e d in water

suction r e s e r v o i r o r pockets there

or

velocity

the

by

vortices in the pump

suction

reservoir.

The

air

may

be

at air

released when

i n pressure,

e i t h e r a l o n g a r i s i n g main o r where the

i s increased through

a r e s t r i c t i o n such as a p a r t i a l l y closed

An

valve. from

in

i s a drop

in many ways;

merely b y absorption a t exposed surfaces,

increase i n temperature w i l l

solution.

Small

orifice a i r

also cause a i r

release

valves

are

to be released

i n s t a l l e d on

the

p i p e l i n e to bleed o f f the a i r which comes out of solution. Smal I size,

orifice

usually

air

valves

are

designated

12 to 50 mm diameter.

by

a i r release o r i f i c e size which may be from 1 larger size.

the

pressure

in

the

pipeline,

which

is

on

the

average

i n l e t connection

to 10 mm diameter.

the smaller

The volume of a i r to be released w i l l

entrained

their

This has n o t h i n g to d o w i t h the

2% of

be a f u n c t i o n of the

The

need be t h e o r i f i c e

volume of

the a i r

water

(at

189 atmospheric p r e s s u r e ) , (see Chapter 5 ) . The small

o r i f i c e release v a l v e s a r e sealed b y a f l o a t i n g b a l l , o r

needle which i s attached to a f l o a t . accumulated or

the

i n the connection on

needle

release v a l v e s

valve

will

a r e often

When a c e r t a i n amount of a i r has the p i p e ,

top of

open

and

release

combined w i t h

double

air

valve,

(see

Fig.

air.

drop

orifice

vent

valves

The arrangement i s c a l l e d

An

10.6).

will

Small

large orifice a i r

on a common connection on top of the pipe. a

the b a l l

the

isolating

sluice

valve

is

normally f i t t e d between the p i p e a n d the a i r valves.

n

LAREE ORIFICE AIR VENT VALVE

SMALL ORIFICE FIlTINQ

10.6

Fig.

Double a i r v a l v e

Double a i r

valves

should

be

installed

at

peaks

in

the

pipeline,

b o t h w i t h respect to the h o r i z o n t a l a n d the maximum h y d r a u l i c g r a d ient.

They

should

also

be

points

along

a

l e n g t h of

grade

line.

It

should

installed

be

borne

a l o n g i n the d i r e c t i o n of flow in

sections

Double a i r

falling

at

p i p e l i n e which

slowly

in

in

the

ends

is parallel

mind

that

air

and

intermediate

to the h y d r a u l i c may

be

dragged

i n the p i p e l i n e so may even accumulate relation

v a l v e s should be f i t t e d every

to

4

the

hydraulic

gradient.

to 1 km along descending

sections, especially a t p o i n t s where the p i p e d i p s steeply.

A i r release v a l v e s should a l s o be i n s t a l l e d on a l l

long ascending

190 lengths of due

to

pipeline

the

where

lowering

of

in g r a d i e n t .

decrease discharge

side

of

air

the

is

likely

pressure,

to

be released from solution

again

especially

at

p o i n t s of

Other places a i r v a l v e s a r e r e q u i r e d a r e on the

pumps

and

at

high

points

on

large

valves

and

upstream of o r i f i c e p l a t e s a n d r e d u c i n g tapers.

THRUST BLOCKS

1969)

!See also Morrison,

Unbalanced

thrust

results

at

a

bend

in

a

pipeline

due

to

two

due to the change i n d i r e c t i o n of flow.

The

act ions:

(1)

The dynamic force

thrust

any

in

direction

proportional

is

to

the

change

in

momentum in that d i r e c t i o n a n d is: Fxl

F

where

is

direction

x.

compared

force,

the

and A Vx

flow

(10.1)

oqAVx

=

is

p

the

the reduction of T h i s force

with

the

is,

force

fluid

mass density,

q

i s the

the component of velocity

i n the

under normal conditions,

due

to

the

internal

negligible

pressure

in

the

pipe.

(2)

The t h r u s t

i n the d i r e c t i o n of each

leg of the bend due to the

pressure i n the p i p e i s : Fx2

where p of

(10.2)

PA

=

i s the i n t e r n a l pressure,

p i p e flow a n d

10.7).

The

A i s the cross sectional area

0 i s the a n g l e of d e v i a t i o n o f the p i p e ( F i g .

resultant

outward

thrust

is

the

forces i n both d i r e c t i o n s of the p i p e a x i s , Fr

=

vector

sum of

the

and is

2pA s i n 0 / 2

(10.3)

The unbalanced t h r u s t may be counteracted b y l o n g i t u d i n a l tension in

an

all-welded

pipeline,

or

against

the foundation m a t e r i a l .

size of

the

addition and

to

block

a

concrete

thrust

block

bearing

I n the case of a j o i n t e d p i p e l i n e the

may be c a l c u l a t e d u s i n g s o i l mechanics theory.

frictional

resistance on

the circumference of

against

by

the outer face of

the

the

pipeline,

bottom there

the p i p e a n d block.

of

is

a

the

thrust

lateral

In

block

resistance

The maximum r e s i s t i n g

191 pressure

a

mass w i l l o f f e r

soi

f

1 + sin@ + ~ s 1h - s i n @

=

P

i s termed the passive resistance and

1969)

i s (Capper and Cassie,

1

+

sin<

2cJ 1 - s i n @

i s the r e s i s t i n g pressure a t depth h, y s i s the s o i l d e n s i t y , P the effective i n t e r n a l a n g l e of f r i c t i o n of the soil a n d c i s i t s

where f

@

is

cohesion.

If

the

thrust

'

HZ =

1

Ys-2

This thrust

maximum

block

extends from the surface to a depth H

a n d i t s l e n g t h i s L, the total r e s i s t i n g t h r u s t i s

below the surface,

P

block

-

+

+

sin6

sin6

kHLJ

possible

1 + sin$ 1 - sin

resistance

will

i s a b l e to move i n t o the soil

ponding

maximum

minimum

pressure

pressure,

which

soil

pressure

which may

be

is

termed

(10.5) only

be

developed

mass s l i g h t l y . the

passive

may occur on the t h r u s t block developed

if

the

thrust

block

if

the

The corres-

pressure.

The

i s the a c t i v e were

free

to

y i e l d away from the s o i l mass. This i s

Ifa

=

which

ysh 1

sin4 -

+ s i n @ "1

i s considerably

$-si+ nm sin4

(10.6)

less than the passive pressure and w i l l o n l y be

developed i f the force on which i t i s a c t i n g i s free to move away from the soi I e x e r t i n g the pressure.

Fig.

10.7

T h r u s t a t a bend

192

10.6

Fig.

Thrust block

I n p r a c t i c e the pressure which may be r e l i e d upon i s s l i g h t l y than

the

p a s s i v e pressure,

not permissible. the

soft

block

the

to

activate

excavation.

rest"

full

of

The

horizontal

used

These

in a l l o w i n g

internal

stress

soil

to 0.7

for

friction

i n Table 10.1.

to

for very

pressure

(CP

1972),

are

seldom

vertical

less

usually

least 2 should be used w i t h

at

i n w a r d movement of

passive

movements

the p i p e i s

varies

soil

p l a sti c clays.

stress

the top of from

where H

T h e "at varies

H/20

a

for

i s the depth

permissible,

coefficients a r e p r o b a b l y more r e a l i s t i c .

loose sands

of

safety

movement of

c l a y s to H/200 f o r dense sand,

"at of

of

passive resistance formula.

thrust

of

A factor

as f u l l

and the

rest" r a t i o

from

0.4 for

An impact factor should be

water hammer pressures. T y p i c a l values of a n g l e $ a n d cohesion

c

for

v a r i o u s soils a r e

indicated

193 Strength of Soils - t y p i c a l

TABLE 10.1 Type of soil

values

Angle of f r i c t i o n

4

Cohesion N/rnmz

Gravel

35 O

0

Sand

30

0

Silt

28 O

0.007

Dense c l a y

5"

0.035

Soft s a t u r a t e d c l a y

0

0.15

Exarnp I e Calculate pressures a t

the

size of

N/rnrnz.

block

to r e s i s t

the u n b a l a n c e d

L e n g t h of 1 p i p e l e n g t h = 10 r n ,

C = 0.005

@ = 30°,

thrust

i n a 1 000 rnrn d i a m e t e r p i p e o p e r a t i n g a t

a 45O b e n d

a p r e s s u r e o f 2.0

1 m,

a

N/mmz.

depth =

L a t e r a l f o r c e F = 2 x 2.0

3 x -If

4

s i n 2 2 i 0 = 1.20 MN. T r y a t h r u s t b l o c k 3 x 3 ~ 3rn: Weight of t h r u s t b l o c k 3 x 3 ~ 3 ~200x9.8/10 2

3

Weight of 10 rn l e n g t h of p i p e Weight of w a t e r

i n p i p e 0.785x1zx10x9.8x1

OOO/lO'

=

600kN

=

40kN

=

80kN

Weight o f s o i l a b o v e p i p e l x l x 7 x l 8O0x9.8/1O3

=

130kN

Total weight

=

850kN

F r i c t iona I resistance 0 . 3 ~ 8 5 0

=

250kN

=

889kN

=

640kN

L a t e r a l r e s i s t a n c e o f soi I a g a i n s t b l o c k :

1 + sin$ ~1

-

-

sin$

1 + 0.5 1 - 0.5

=

32 ~+32/ ~1 5 ~' v3 3~ 3 F = 18 0 0 0 ~ ~ ~ 3O3 L a t e r a l resistance of soil a g a i n s t p r o j e c t i o n of p i p e : (18 000x1 . 5 x 3 / 1 0 3 x 2 x 5 j 3 ) Total

1x6.5

l a t e r a l resistance

= 1770kN

F a c t o r o f s a f e t y 1.77/1.20

=

1.48

A t h r u s t b l o c k s h o u l d b e d e s i g n e d so t h a t t h e l i n e o f a c t i o n o f the resultant the

pipe.

may

best

of

the

resisting

This w i l l

forces

prevent

be done g r a p h i c a l l y

of the p i p e .

coincides w i t h

overturning or or by

taking

the

l i n e of

t h r u s t of

u n b a l a n c e d stresses. moments a b o u t

This

the centre

194 Thrust

blocks

horizontal not

be

alignment able

coupl ings.

to

needed

of

the

not

only

pipeline,

changes

in

vertical

or

b u t also a t f i t t i n g s which may flexible

V i k i n g Johnson o r s i m i l a r coupl i n g s a r e frequently

i n s t a l led

removal

transmit

at

as

on one side of and

are

a valve

of

the

longitudinal

forces,

i n a v a l v e chamber,

valve.

The opposite

such

to f a c i l i t a t e i n s t a l l a t i o n

wall

of

the

valve

chamber

would thus h a v e to be designed as a t h r u s t block.

-li;

10.9

Fig

s t a n d a r d med i urn-rad i us 900 f a b r i ca t ed

Segment geometry bend.

FORCES INDUCED BY SUPPORTS

The There

i n t e r a c t i o n of are

involved. above a

many The

ground,

tendency

of

effects on the

circumferent i a l l y tial

a p i p e and

secondary

temperatures,

are

plinths pipe

under

and

i t s supports tertiary

general ly or

to

the

secondary

by

effects

greatest

other

expand of

strains

i s a complex subject.

forces

or

isolated contract

and in

displacements

pipes

supports.

There

longitudinally

temperature change, due

supported is

and

differen-

to the Poisson r a t i o effect,

195 and

partial

pipe

or

f i x i t y of

total

vertically,

supports which

longitudinally

and

impose

forces on the

laterally.

iio

completely r i g i d a n d the more f l e x i b l e the support

the

support

is

less force or

reaction i t imposes on the pipe. The stresses on the p i p e may be loosely c l a s s i f i e d as follows: Primary;

circumferential

longitudinal Secondary;

due

to

due to

internal

internal

pressure

bending a t elbows,

across

supports.

Temperature

strains

at

angle

right

to

pressure across diameter; across

branches, effects.

primary

the

cross

section.

changes i n section a n d Poisson

strain.

ratio

Due

to

counter-

reaction

of

supports. Tertiary;

l a t e r a l deflection due to l o n g i t u d i n a l expansion.

Al though

the

secondary

accounted f o r i n p i p e design,

and

tertiary

stresses

are

often

not

they can be very s i g n i f i c a n t .

Longitudinal Stress If

the end of

a p i p e i s b l a n k e d o f f a n d u n h e l d the l o n g i t u d i n a l

stress i n the p i p e w a l l

i .e.

4

The

Poisson

and

the circumferential

if

Wpd/2t

is

the and

modulus,

stress.

ratio

effects

ends

are

tries

held

corresponding

t

the

wall

r a t i o (about 0.3

reduce

to

the

wall

force

is

thickness,

e

the

tensile

length

Vp-irdz/2 where the

strain

by

stress E

and

vLpd/2tE

becomes is

w

the is

Ee

=

elastic

Poisson's

for s t e e l ) .

Temperature stresses A pipe w i l l

expand or

contract

by aAT where a i s the coefficient

of expansion a n d LT the temperature change i n time.

A where

pipe the

restrained coefficient

"C a n d for p l a s t i c , If

there

Circumferential

is

a

longitudinally of

expansion

will for

undergo

steel,

a

a

, is

stress 12 x

aEAT

-6

10

per

50 x temperature

gradient

across

the

stress i s created equal to +_ aEAT/2(1-v).

pipe

wall

a

196 Forces at Bends The force due to water is

PA.

The

where

is

8

with

net

a

pressure along each a x i s at

resultant

force

the bend angle.

thrust

block

it

acting

outwards

i s 2pA

I f the r e s u l t a n t force

creates

a

longitudinal

a p i p e bend sin

(8/2)

i s not countered

force

in

the

pipes

equal to 2pA s i n 2 ( 8 / 2 ) .

Lateral Movement A

straight

outwards

a

a t u r e etc.,

pipe

of

relatively

length big

x

fixed

amount

at

under

both

ends

expansion

will

due

to

buckle temper-

b y an amount (10.8)

dy since dy2

+ (x/2)*

(x/2 + d ~ / 2 ) ~

=

dy = x =

loom,

dy =

400 x

e.9.

I f the

pipe

-6 d x = xaAT = 100 x 12 x 10 0.02/2

supports

or

to

required

to

that

t i o n of

= lm.

restrained

is

between

x 20 = 0.02m

twist

laterally

supports

prevent

a simple supported

the

or

it

could

exert

buckling.

a

try

lateral

For example,

( l a t e r a l l y ) p i p e beam,

dy,

to

buckle

force

equal

the deflec-

under a force

F is

-

dy =

1

=

:.

F

FL3/EI

48

=

f o r above example. 1 x 210 x

= 40

lo9

1 13x.01/(1003x-~

2

48

8

kN

The l a t e r a l deflection

is

referred

pipes to fa1 I o f f the side of supports,

A

pipe

fixed

at

both

ends

with

to

as

snaking

and

can

cause

or bend the supports. an

elbow

will

bend outwards

under expansion by an amount: dy = d x / ( y / a where

dx

is

+ y/b)

the

net

expansion

in

length

( a + b ) = c due to temperature, pressure etc.

of

pipe

with

length

197 Forces on Supports The equal

forces and

on

the

opposite

support

to

the

blocks or

forces

p i l l a r s or

exerted

by

hangers w i l l

the

supports

on

be the

pipe. Free

to

Slide:

The

h e l d p i p e equals IJ

is

the

the

Fixed force pipe

on

force

of

the f r i c t i o n force on

f r i c t i o n coefficient

pipe

snaking

longitudinal

the

block.

W

and

There

is

a freely

the block.

sliding or i s pW,

This

i s the normal force also

a

lateral

the

smaller

highly where

( w e i g h t ) of

force

to

resist

.

to

Block:

which and

The

resulting

force

is

can

be

imposed on block

block,

or

(ii)

ance of

support a t

tudinal

force

in

sliding

the other end, pipe

due

to

or

f r i c ion between

b y p i p e e.g.

resistance of

i ) maximum

of

block,

( i v ) maximum net

temperature

( i i i ) resist-

or

change,

total

longi-

Poisson

ratio

effect a n d bends. There i s thus a d i s t i n c t i o n between the f r e e to s l i d e case a n d the fixed

case

a n d case

in

that

( i ) or

(ii) will

apply

to a

"free

to s l i d e "

pipe

( i i i ) a n d ( i v ) w i l l g e n e r a l l y a p p l y to a p i p e r i g i d l y f i x e d

to the support. Thus

be

the force

i s computed as above,

as

above etc.

no

force

at

that end,

F o r the f i r s t

a n d then case ( i i i ) or For case

10.9) then there w i l l

a p i p e i s free (see F i g .

i f one end of

obviously

few

a n d beyond the f i r s t

a n d beyond the second support also lengths case

( i v ) w i l l take over

( i v ) the "force a t

support

( i ) or

as one

the other end"

may

( i i ) may

of

these

i n fact

is

apply less.

be that due

to other support blocks.

Unba Ianced Forces Generally balance out.

it

can

That

be

a n d opposite to those at however bends

there

may

expected

i s forces

be

i n one

that

the other end.

bending

there a r e net forces of

to t r y to s t r a i g h t e n the bend.

forces

direction

moments.

in

welded

pipework

a t one end a r e equal

Along the length of the p i p e

For

instance a t

l i q u i d on the p i p e w a l l

elbows

or

inside a c t i n g

198 During

transient

unbalanced instance

forces

pressure

direction. length

unbalanced

could

closure at

Although

and

conditions

which

following

hammer

flow

one the

eventually forces

which

of

end

there

pull

a of

pressure

line

wave

re-balance can

however

pipe

off

pocket

in

the

vapour a

may

may

force

may

travel there

cause s i g n i f i c a n t

large

supports. a

pipe

the

forces,

be

pipe

along are

For water

in

the

one pipe

momentary

movement s t a r t i n g at

bends.

4

Support resistance length x + dx

l a t e r a l movement dy Fig.

10.10

L a t e r a l movement due to temperature expansion

total force F1 + F2 + F3

Temperature Movement

5" E L T

loose c o u p l i n g

I

f r i c t i o n on supports

Fig.

10.11

L o n g i t u d i n a l forces i n p i p e w a l l created b y f r i c t i o n on supports

FLOW MEASUREMENT

Venturi Meters The with

Venturi

minimum

pipelines.

(Fig.

10.12)

meter

offers

an

accurate

head

loss.

It

is

The v e n t u r i

throat

used

method of mainly

has a converging

flow

for

measurement

large-diameter

section and g r a d u a l l y

d i v e r g i n g section to minimize head losses. Flow

measurement

r e a r r a n g e d as

is

based

on

the

Bernoulli

equation

which

is

199 v22

29

--29V 1 2

-

h , - h 2

h

=

(10.9)

Solving f o r discharge

,/=

Q = CdA2

(10.10)

(10.11 )

where d

i s the diameter and C

for

separation

flow

and

downstream

calibrated pipe

in

at

conditions

place

diameters f o r

i s a velocity

V

the throat in

the

double

bends,

smal I t h r o a t / p i p e

is

required

100 p i p e

lengths

10.12

meter

should

l e n g t h of a t

diameter r a t i o s ,

l a r g e r throats,

downstream

impart

be

least 5

increasing

o r from branches

s w i r l i n g motion may

before a

v e n t u r i meter i s o n l y approximately O.1V

Fig.

venturi

(BS 1042). F u l l bore branches, p a r t l y

closed valves a n d f i t t i n g s which to

A

pipe.

a n d should have a s t r a i g h t very

to 10 o r even 20 p i p e diameters f o r or

coefficient which allows

a n d includes a f a c t o r f o r upstream

meter.

The

head

need up loss

in a

‘/2g. 2

Venturi meter

Nozz 1 es A contracted form of v e n t u r i a short

meter

i s the nozzle meter,

which has

bellmouth rounded i n l e t a n d a b r u p t expansion beyond i t . The

discharge

equation

is similar

to

that

for

the

venturi

meter

but

the

v e l o c i t y head recovery beyond i s very small.

O r if ices An

o r i f i c e consisting

of

a

thin

plate with

a central ori f i c e i s a

200 popular

method

appreciable 0.61.

of

measuring

orifice

contraction

flow

in

large

coefficient

which

The discharge formula i s s i m i l a r to Equ.

The

velocity

straight

pipe.

head it

recovery

i s short

The

is

low

may

There be

as

is

an

low

as

10.10 where

-

a d v a n t a g e that

pipes.

but

the

orifice

has

the

a n d can be i n s t a l l e d i n a short l e n g t h of

lengths of

straight

the o r i f i c e a r e s i m i l a r to those f o r

p i p e r e q u i r e d on each side of

v e n t u r i meters.

The o r i f i c e should

be c a l i b r a t e d i n place f o r accurate results.

Fig.

10.13

O r i f i c e meter

Bend Meters

A g r e a t e r pressure i s exerted on the outside of a bend i n a p i p e than can

on be

the used

inside,

due to c e n t r i f u g a l

to measure the flow.

force.

The meter should be c a l i b r a t e d

place, b u t the discharge coefficient f o r E q u .

Cd

=

The pressure difference in

10.10 i s approximately (10.13)

r/2d

where r i s the r a d i u s of c u r v a t u r e of the centreline. Mechanical Meters Flow

to

meters which

water

consumers

is

invariably

i n t e g r a t e the flow over

has a r o t a t i n g t i l t e d disc w i t h types have r o t a t i n g wheels,

a

measured

a p e r i o d of time.

by

mechanical

A common type

s p i n d l e which rotates a d i a l .

lobes o r propellers.

Other

201 There

also

is

deflected vane

a

type

immersed

of

meter

which

i n the flow.

reacts

to

the

drag

on

a

The meter records f o r p a r t l y f u l l

conduits as well. The

rotometer consists of

has a taper

in

the

that

increasing

upward

flow

the d r a g on

a

diameter

in

through

it

calibrated

the

vertical

A

upwards.

tube.

The

tube

which

is suspended

float

float

( w h i c h depends on the flow

glass

positions

itself

so

p a s t i t a n d the tube

diameter) e q u a l s the submerged weight. The object may be e l t h e r a sphere o r to make i t r o t a t e a n d centre i t s e l f

Electromagnetic

By

motive

and

force

advantage including nique

there

an

and no

is

can

be

does

around

can

not

l o s s of

obstruct

shock

pipe

measured.

of

head

The

and

a

non-conducting

this

means.

the

flow

and

and

i s also

A

has

the

liquids

of

similar

tech-

i s based on sonic

being

downstream

an electro-

method

variety

by

wave,

upstream

a

i n s e r t i n g electrodes,

be

measured

impressed

between

field

I i q u i d by

the

induced

also

of

difference

magnetic

ionizing

sewage

which

velocity

a

is

that

i n the tube.

Induction

creating

material

a tapered shape w i t h vanes

developed.

sonic

velocity

The is

measured.

Mass and Vol ume Measurement The or

most

accurate

volumetrically.

over

a

volume

certain flow

To

time

may

be

methods

of

measure

the

the

flow

measurement mass

or

i s diverted

measured

from

the

of

flow

weight

of

are

i n t o a weighing volume

filled

by

fluid

in

tank. a

mass

flowing The

certain

t ime.

TELEMETRY

Automatic systems. ling

which

methods. Data

data

Telemetry is

transmission

a n d control

systems a r e tending

more

expensive

and

i s often used on p i p e l i n e

to replace manual less

reliable

d a t a hand-

than

automatic

Cables may be l a i d simultaneously w i t h the p i p e l i n e . which

pressures,

may

flows,

be

required

opening

of

include

valves,

water

levels

temperature,

in

reservoirs,

qua1 i t y

or

pro-

202 perties

of

the

fluid,

mation

is

read

by

pump a

speed

standard

or

power

consumption.

instrument

linked

by

The

a

infor-

mechanical

device to a local d i a l or c h a r t , or to a coder a n d transmitter. S i g n a l s may uous are

analogue usually

expensive uously

than

by

the

systems

relays,

numbers. as

private

sent

a

The

transmitter

are

are

signals

or

multiplex

cables.

common

to

termed

binary

contacts.

in

of

electric

possible

series

and

via

of

level

binary

in

groups

telephone

conductors

number

less

scanning.

code

Every

transmitted

pair

A

also

a d e f i n i t e combination of

be

One

feed.

messages

but

repeat messages contin-

intervals,

transmitted may

Analogue

accuracy)

converted

cores

i n the form of contin-

digits.

may

regular

i s represented by

a

of

2 percent

to

at

data

Signals

plus

series

(up

The

bits.

bit,

in

magnetic

measurand

known

each

from a t r a n s m i t t e r

interrogated

coded

or

or

digital.

be

digital

pulses

waves

out

less accurate

or

With

of

be sent

is

u n i t s of

lines

needed for

data

may

be

in r o t a t i o n v i a the same conductors.

Alternatively over

short

hydraulic

distances

by

or

pneumatic

pipe.

Radio

signals

may

transmission

be

conveyed

becomes

economic

f o r systems w i th long transmi ssion distances. The linked from

transmission to

a

information the

extreme a or

decoder

analogue

play

to

to

a

data

of

diagram

dials

at

the

will

digital

or

digital

readings

values

mimic

system

and

bank

or

stored may

the

measurand.

(a

for be

on

p a p e r or directly

decision-making programmed

pumps which to

information

be fed

which would

close

may

or

would

valves

r e l i e f mechanisms.

or

means or

The

of

be

fed

of

discs

later

stage

manipulat i n g

result

the

system, in

or

the

The

hydraulic

data or to

sound

decoded

feeds

the

alarms

at

displayed

on

system

with

tights

These diagrams

storage

cards. a

data.

for

minimum

power demand. operate

a

to

tapes,

a

a

be

rooms.

also

at

receiver

decoder

frequently

is

picture)

a

a servo motor can dis-

lights

Data

to

S i g n a l s can

the u n i t being observed.

magnetic or

messages

versa.

charts

diagramatic

to optimize

match

by

dials,

p o s i t i o n of

the

comparater. vice

on

a r e mounted i n c e n t r a l control The

send

system

The

computer

information to

be

L a r g e computers

instance b y selecting pumping

cost,

computer could machinery

or

and

or

send water

used can those

turbines signals hammer

203 Mini of

digital

computers

but

They

be

can

automatically which or

computers

telemetered

they

systems.

are

considerably programmed but

are

cannot

not

now

They

the

are

most

popular

sI ight ly

cheaper

cheaper to

be

perform used

programmed.

than

larger

certain

easily

They

can

for

other

of control

than

digital

tasks

be

form

or

analogue computers.

computat ions

computations

increased

for

in c a p a c i t y

l i n k e d to l a r g e r computers as t h e need a r i s e s .

REFERENCES Boucher, P . L . , 1948. Choosing v a l v e s , K i Imarnock. B r i t i s h V a l v e M a n u f a c t u r e r s Assn. 1964. V a l v e s f o r the C o n t r o l of Fluids. BS 1042, Flow measurement, BSI, London. C a p p e r , P.L. a n d Cassie, W.F., 1969. The M e c h a n i c s o f E n g i n e e r i n g Soils, 5 t h E d . , Spon, London. CP 1972. L a t e r a l S u p p o r t i n S u r f a c e E x c a v a t i o n s , S.A. Instn. C i v i l Engs. L e s c o v i c h , J.E., 1972. L o c a t i n g a n d s i z i n g a i r r e l e a s e v a l v e s . J. Am. Water Works Assn., 64 ( 7 ) . M o r r i s o n , E.B., 1969. Nomograph f o r the d e s i g n of t h r u s t b l o c k s , C i v i l Engg., Am. SOC. C i v i l Eng. P a u l , L . , 1954. Selection of v a l v e s f o r w a t e r s e r v i c e s . J. Am. Water Works Assn., 46 ( 1 1 ) 1057. P a r m a k i a n , J., 1950. A i r i n l e t v a l v e s f o r steel p i p e l i n e s . T r a n s . Am. SOC. C i v i l Engs., 115 ( 4 3 8 ) . Sweeten, A.E., 1943. A i r i n l e t v a l v e d e s i g n f o r p i p e l i n e s . E n g g . News Record, 122 ( 3 7 ) .

L I S T OF SYMBOLS A

-

C

-

soi I cohesion

-

con t r a c t i o n c o e f f i c i e n t

-

d i sc h a r g e coef f i c i en t

-

velocity

d

-

i n t e r n a l diameter

e

-

strain

E

-

e l a s t i c modulus

f

-

stress

F

-

force

9

-

gravitational

h o r H

-

d e p t h below s u r f a c e

cC

‘d

cross sectional a r e a

coefficient

cV

acceleration

f l u i d pressure flow r a t e r a d i u s of c u r v a t u r e wal I

thickness

temperature velocity normal force length direct ions

I a t e r a I d i s p I acement temperat u r e c o e f f i c i e n t soil density f r i c t i o n coefficient Poisson's r a t i o s o i l mass d e n s i t y a n g l e of

i n t e r n a l f r i c t i o n of s o i l

205 CHAPTER 1 1

LAYING AND PROTECTION

SELECTING A ROUTE

The

selection of

bearing is

on

the

selected

on-site

a suitable route for

capital

from

inspections

obstacles

and

have

the

profile

conditions

should

hydraulic

gradient

above

the

obviously

pumps

may

result

i n water

of

level grade

the

be

to

is

data

In

selecting

considered. the

i s flattest line

and

Care

a

pipeline

on

route,

the

the

grade as

low

flows).

the

input

If

be

p o i n t s of

(Low-flow

r o u t e should

possible

to

On

as

the

were

a peak heads,

t h i s peak.

separation

the other

be kept

as

discharge

p o s s i b l e water column

hammer overpressures.

and

to ensure

rates,

there

and

terrain,

line.

peak

route plans,

costs

should be taken

well

important

cadastral

available

as

for

between

A

costs.

hydraulic

considered

pipeline as

other

below

be

operating

topographical

would have to be designed to pump over

also

lines

any

services.

grade

Peaks

and

photos,

and

local

prcticability ground

cost

aerial

a p i p e l i n e has an

which

h a n d the general

near

to the h y d r a u l i c

minimize pressures a n d consequently

pipe

costs. Once

a

surveyor. regular tions. in

Pegs

Offset

for

is

2

in

along

changes

laying.

and

depth

in

the

it

is

pegged

centre-line

out

10

at

i n g r a d e a n d at

The

l e v e l s of

pegs

the

drawing

office,

at

peg.

indicated

identify

may

selected,

to

horizontal

by

a

100 m deflec-

5 m from the centre l i n e pegs a r e also put

to

plotted

tests

is

at

then

the r o u t e to

Strength

route

put

pegs,

use d u r i n g

levels

along

are

intervals and

profile bed

preliminary

each

a r e observed a n d the with Holes

ground may

levels,

be augered

m a t e r i a l s which w i l l have to be excavated.

be done on s o i l

samples to decide whether

trench

s h o r i n g w i l l be necessary. Underground at and

this

stage

and or

underground

at

overhead least

cables

services

before

may

excavation

have

a heading c o u l d be hand-excavated

should

to

be

to

be

accurately

avoid

supported

under them).

them. on

located (Drains

bridges,

or

206 L A Y I NG AND TRENCH I NG

type of

The

of

p i p e to be used for

economics,

although

materials

and

delivered

pipes

rigidity,

size of

pipes

are

be

supplied

and

to

in

i n t o the

For

small

may

be

lines

for

at

trench.

varies

small

pipe

Concrete

pipes may

p l a s t i c p i p e s may

handled

carefully

They

w a l l e d welded pipes,

side of

the trench,

The

working

width

the

method

laying

is

pipes. and

widths

of the

to

should

avoid

be

Pipes may

stored be

low-

b e l t s l i n g s from mechanical booms or t r i p o d s .

thick

laying

w h i l e steel

walled

pipe

weight,

jointing.

sandbags.

or

the

lengths

the

method of

be

of

The

pipe,

distortion.

the

diameter

working

or

of

thin

should

padded c r a d l e s

with

and

excavation, ally,

on

diameter

excavation

Pipes

or

life

decision.

m lengths,

and

l a r g e l y be a matter

laying,

type and

to 4

long,

will

job

for the

the

vehicles

m

trench by

done

i n t o the

of

rolls.

coatings

transported

ered

20

even

on

in 2

often s u p p l i e d

over

affect

depend

transport

any

facilities

factors

will

10 or

be

damage

other

the

done On

to

up

wrapping 'snaked'

required for

of

excavation

by

hand,

the

possibly

20 to 30 m a r e common,

j o i n t i n g and

a n d the p i p e then

construction of pipeand

hand for

other

m may

be

be

If

wide

large

even s i t e coating

50

laying.

3 m may

the

enough

mains where

i s done mechanic-

required.

Reserve

widths

f o r one or two a d d i t i o n a l pipe-

a n d allow

l i n e s at a l a t e r stage. Once the

the

trench

trench with

i s excavated,

the

centre

sight

of

line

w i t h the assistance of these (BSCP, Pipes to

m

is

may case

are

avoid

necessary

in

be reduced a

concrete

0.5

normally widening

300

normally damage

mm

and

m

the

with

U.K.

cover to

0.8

which

a

to

minimum

may m

be

more

for are

loads.

prevent

steep

i n rocky o r

pipe

be

set

marked

up across

thereon.

The

1970).

superimposed

deepening

diameter

jointing).

laid

by

the

may

above the bed a n d the pipe l e v e l l e d

frames a r e set a d e f i n i t e height

m

rails

frost

cover

A

0.9

of

minimum

m

penetration.

than

joints. difficult

the

(0.8 to

The pipe

m

is

width

of

diameter, used

straddle

for

while

of

1

The depth

ground to minimize costs

necessary.

1.2

to

cover

i n which trench

is

with

local

pipes

over

laying

and

207 The

sides of

hesive

or

shoring

wet

open

the

soils

and

for

the sides of

angle.

should

deep

trenches,

Spoil At

least

be

one

the trench may

kept

trench

r e q u i r e d i n the bottom of dry

h a v e to be supported

will

for

safety

should be designed to r e s i s t the a c t i v e

Alternatively

slips.

trench

while working,

to

enable

be

open

the

to

away

trench

is

advisable.

i n wet

to

and

ground

lead

away

to a safe

in

water.

a

In

to a v o i d

D r a i n s may be

to keep the trench

of embankment s l i p s a n d

be compacted p r o p e r l y . bedded

The

l a t e r a l s o i l pressure.

away from the excavation

depth a

reasons.

be b a t t e r e d back

to reduce the p o s s i b i l i t y

backfill

jointed

cross-drains

well

i n unco-

stone

very

D r a i n p i p e s should

filter

wet

with

ground

frequent

cutoff

drains

beside the trench may be necessary. Pipes

may

bedding reduce

is

be

damage

shaped

to

sandy

fit

to

or

may

haunches of settlement

the

and

flat

minimize pipes.

profile

crushed

the

on

to

rigid

the

material

gravel

bedded

preferable

of

The the

pipe,

up

b e d of

pipe.

20

to

a r e often

a r e easy

should

bottom, of

the

In

but

trench

rocky

to

used,

compact.

be continuous

in

mm

as

be a

to

the

do not e x h i b i t much

they

beds

and

haunches o r

low-pressure r i g i d p i p e s a n d sewers.

under

joints

but

haunches should stop

11.1

typical

bedding

bedding.

pre-

bed of

up

Fig.

of

to

Beds of sand,

brought

short of j o i n t s to f a c i l i t a t e r e p l a c i n g damaged pipes. types

proper

may

ground

size,

Concrete

a

f l e x i b l e pipes o r

be b u i l t up a n d trimmed to shape. stone

surrounds a r e sometimes used f o r Bedding

trench

deflections

(Refer

to

Chapter

associated w i t h d i f f e r e n t types of b e d d i n g )

7

to

(CPA, 1967; ACPA,

shows factors

1970).

The best method of b a c k f i l l i n g a n d compaction depends on the type of

soil.

(PI), with

Soil properties

liquid low

may

For

stampers o r

vibrators.

the

compact

Proctor

high

should

LL

often soils,

be

dynamic

Density.

the bed w i t h a w i d t h of at

be up

100

compacted

index

( P I = LL-PL).

satisfactorily

compacted

compaction

B a c k f i l l densities may be

Standard

the p i p e to 95 to

density

be described in terms of the p l a s t i c l i m i t

(LL) and plasticity

limit

PI'S fill

inundation.

of

may

% in

For

pipes

required

by i.e.

specified i n terms it

is

practice

to

least h a l f the p i p e diameter under

Proctor density. 100 to

large

is

For soils merely

150 mm

The soil f i l l around the p i p e l a y e r s to 90 or

95% Proctor

to haunch level or to 3 / 4 of the depth of the pipe.

I t may

208 be necessary to s t r u t f l e x i b l e p i p e i n s i d e to prevent d i s t o r t i o n d u r i n g t h i s operation.

fill

The

around

the

top

half

of

the pipe,

to 300

up

mm above the crown should be compacted to 85 to 90% Proctor density. For b a c k f i I1 under roads, ed

90-95%

to

Proctor

85%

quired.

is

the top

density,

normally

l a y e r s may have to be compact-

while

for

sufficient

in

as low as 80% a r e obtained i f control

open

h a v e stones

width of

to

in

it.

compensate

topsoi I

and

Surplus for

spoil

and

fill

around

re-

densities

the p i p e should

may be spread over

settlement

vegetation,

country

be

i s b a d ( R e i t r , 1950).

To a v o i d damage to p i p e o r coatings, not

110% may

runways

carted

or

p r o v i s i o n of

land

away.

the

trench

Reinstatement

drainage,

and

fencing

1956).

o f f of w o r k i n g areas should be considered (see also Sowers,

THRUST BORES

To

avoid

traffic soil

on

for

be

a

thrust

r o a d over

short

block

jack or

Small

timber

pipes

shoe

the

d i r e c t i o n of

the

pipe.

the

may

pipe

is

inserted The

in

stick.

by

the

then

With

auger

the

force

to 1 200 tons,

pipes

if

an

in

auger

end of

the

i t s back

may

the

pipes

and

length to

have

of

a

the

the

soil

bore a

the

sharp

deviate

lubricated stop

for

or

the

pipe The

r e t r a c t e d a n d another

pipe

and

a

ahead of

with

long

pipe.

pipe

i s used,

a

load a n d main-

is

never

removed

the j a c k

push

pipe

should

the

against

up on r a i l s i n front

diameter)

of

Pushing

to

disrupting

the s l i g h t e s t obstacle w i l l

outside

hand

each

to ease the j a c k i n g

larger

or

required

but

at

avoid

be j a c k e d through

hole w i t h

rnrn

as

pushed i n f u r t h e r ,

behind

one

600

than

reduce f r i c t i o n .

jacking

i s dug

up

to

or

the p i p e may

hole

the p i p e ,

Occasionally to

out

pipe,

f i t t e d to the f r o n t

bentonite pipe

excavations,

a n d the p i p e set

(less

tain

dug

a

i s set

head or

is

deep

A

lengths.

A

jacked.

jack.

excessively

through

the

with

the

process a

shoe

100 ton j a c k

repeated.

may

be up

may suffice.

PIPE BRIDGES

Pipes f r e q u e n t l y rigid

jointed

pipes

h a v e to span r i v e r s o r may

have

sufficient

gorges.

strength

For short spans, to

support

them-

209

Class A bedding Reinforced A,: RamforcedA,

Dt 200 v m mln

M= 4

8

0 4%M. 3

4

I 0% :

M.28

PI","

0 IZ5H 150 mm l l i "

0 25 d I O O m m mm Plain or Reinforced

bncr.1e 1 5 0 m k mi Compocted Gronulor t4 Y

CONCRETE ARCH

Class B bedding M=

19

Dens@ Compacted Bochf~l1

Class C bedding M : I ~

01251'

Lightly tO"'p0ct. Bocuftll

GRANULAR IFOUNDATION CiKANULAK WUNUAII~

Class D bedding

NOTE For rock or other incompresrible malert316,the trench should be ovaexcovoled 0 minimum of 150mm and

refi1l.d

wrlb gronulor mat.rm

M-bedding

Fig.

11.1

support from

Types of beddings

plus

selves

an

factor

the

the pipe

existing

fluid.

For

larger

spans,

it

may

on trusses or concrete bridges, traffic

bridge.

If

a

truss

be or

bridge

necessary

to

h a n g the

pipe

is

con-

to

be

210 structed for

the

the bottom, of

or

bridge

is

Two

cables

the

hangers

pipe.

as

one

with

attached

an

ing,

wind

the

at

but

on

stays

an

preferably

could

The

be

along

the

tension member at

A n a t t r a c t i v e form

from may

suspension

could

also

support to

act

erected.

would

together

In

the

all

pipe

the

wind forces

be

designed

to

be

necessary

-

w i t h cross brac-

cases

should

p i p e to prevent

and

p l a n e through

the p i p e against

pipe

cables.

be spaced a p a r t

the v e r t i c a l

lateral

bridge,

the

T h i s means there w i l l

steel.

joint

they

to

designed

independent

welded

expansion

as

to support

again be

as

the top.

supported

angle

helps

p i p e s could

cable

or

pipe

an

act

member a t

vibrations.

arch,

two

ported

p i p e could

a r e p r e f e r a b l e to one,

reduces

either

of

the

T h i s arrangement

and act

pipe,

the compression

except

be

if

sup-

r i g i d jointed;

h a v e to be some form thermal

stresses

devel-

oping.

UNDERWATER P I PEL I NES

The

often

is

it

laying or

is

and

through depend

In

common.

is

very

addition Gas

and

which

on

pipes.

conditions

The

such

under-water

Floating

to

river lines

as

currents,

crossings, from

undersea

undersea

a r e also many

or

sludge

into

underwater

wave

In

calm

water

out

and

sunk.

thus.

Care

is

needed

on

nevertheless

height,

beds

industthe

sea

laying

will

type

of

bed

pipe:

floated

laid

winches

There

effluent

method used for

a n d Sinking:

shore,

been

discharge

expensive,

oil

t a n k e r b e r t h s a r e frequent.

towns

undersea

on

underwater

inescapable.

a n d length of (1)

pipes

becoming

off-shore

ries

of

laying

barges

should

be

the p i p e may

Lengths in

up

sinking

positioned

be j o i n t e d

500

to

the

along

have

rn

pipe

the

and

line

to

assist.

(2)

Bottom be

Towing:

assembled

towed

along

Lengths

on

land

the

bed

are

jointed

bouyed to

each

Strings

sea.

is

pipe,

i n strings,

to

sea

at

the

up

the

The to

pipe

of

anchored pipe

of

up

by

air

reduce f r i c t i o n

in on

by

the the

a

few

up a

kilometres

may

long,

and

km

1

to

winch

together pipe sea

or

a

barge

on

the

shore.

by

buoys

of

bed.

If

the

tied pipe

211 i s f i l l e d with

air

crete

to

coating

would

be

line.

get

susceptible

The stresses

caused The

i t may

by

the to

correct side

buoyancy.

currents

i n the p i p e due

underwater

lateral

h a v e to be weighted down

drag

per

currents unit

to

the

or

length

and

light

may

to

a cona

bend

tension

waves

due

Too

by

pipe

out

of

and bending

may

waves

be

critical.

and

currents

is ( u + v ) ~D/2g

Cow

where C weight

pipe),

(about 0 . 9 ) ,

i s the d r a g coefficient

D of

current

(11.1)

the

water,

velocity

D

is

u

(or

the

is

the

their

wave

components

outside diameter of

i s the s p e c i f i c

w

velocity

and

v

perpendicular

the p i p e a n d g

is

the

to

the

is gravi-

t a t i o n a l acceleration.

(3)

Lay

Barge:

jointed an

on

arm

Pipes a

may

special

(or

be

taken

lay-barge

stringer)

into

to

sea

in

a

pipe

barge,

a n d lowered continuously

place.

The

barges

down

proceed f o r w a r d

as l a y i n g proceeds.

(4)

Underwater pipe

has

Jointing:

to

be

Underwater

accurately

jointing

aligned.

which l i m i t s the method to shallow

It

i s usually

currents. the

to b u r y

is

trench

before

fluidizer

the

pipe

is

depths (Reynolds,

passes of

the f l u i d i z e r

long length.

a

The

trench

High-pressure

a r e r e q u i r e d so that

Too

deep

.

1970)

to

or be washed into

normally

jets

of

has

water

to

be

a

fluidize

A number of

the bed under the pipes a n d the p i p e s i n k s into the bed.

over

each

l a y i n g the p i p e i n soft beds i s w i t h a

1971).

(Schwartz,

as

be r e q u i r e d ,

the p i p e to a v o i d damage due

laid.

A novel way of

difficult

diver w i l l

Dredging i s d i f f i c u l t as sediment may f a l l

over-excavated. bed

necessary

A

the p i p e s i n k s g r a d u a l l y

trench may cause bending stresses i n

the p i p e at the head of the trench. Most joints

underwater

should

be

pipes

well

d e s i r a b l e under the sea.

are

cwted

steel and

with

lined

welded and

joints.

cathodic

Pipes

protection

and is

212 JOINTS AND FLANGES

The of

type

pipe,

joint,

of

the

joint

to

use on

facilities

strength

and

flexibility

of j o i n t s a r e used ( E E U A ,

(I)

Butt-welded:

1968)

The

approximately

joint

is

clamped

right

around

walls

may

of

good

depand

cost,

on

the

type

watertightness

The

steel

back,

following

in

more

runs.

a

weld

the

pipe

leaving

tacked

or

require

make

the

30'

one

will

site,

requirements.

and/or

with

also

to

pipeline on

of

types

:-

ends

bevelled

difficult

a

available

places

and

root.

then

pipes

inside,

lining

trued

mm

and

Large

pass

internal

are

2

a

The

welded

with

thick

although

it

in

pipes

less

may

be

flared

is

than

600 mm bore. (2)

Sleeve to

welded:

form

a

barrel

of

welded

inside

and

One

socket the

the

end

and

other too.

of

a

the pipe.

If

lining,

if

pipe edge

possible

11.2).

(see F i g .

internal

steel

leading

fillet the

welded

joint

The e x t e r n a l

possible,

should

Two

methods

welding

and

essential

to

A

thorough

metallic

testing

and

to

including

the

air

should

weld

are

pockets

ultra-sonic

tion

and

and

useful

pipes,

the

or

a

film

a

single

is

but

For

from

may

is

X-ray,

are

Gamma-ray

around smaller

may the

where

of

double,

to

the

pipe

but

Strips

a n d bent over and

Normally

etched field

for

tests

hi gh defini-

With in

to

magnetic

i s more

corners.

is

welding.

necessary.

p i p e to receive

pipes,

the

Gamma-ray,

inserted

gas

cavities.

possible

preferred

be

good

welding

lengths be

equipment

inaccessible

made

avoid

ground

inclusions.

X-rays

and

the p i p e w a l l

weld

be

Oxy-acetylene

accompany

short

e i t h e r by

equipment

wrapped wall.

on tests

The

otherwise

X-ray

should

should

off.

Top-class

strength

confined

slag

methods.

contrast for

be

cut

w i I I be non-destructive: or

welding.

welds

radius.

available:

weld

destructive

predetermined

reveal

arc

high

sample

field

are

programme

testing

occasional

a

welding

achieve

Destructive factory

of

the

p i p e coating

be

at the weld a f t e r a l l s l a g has been wire-brushed

out

to

large-bore

the the

portable

p i p e and

image from

overlapping

expo-

213

o m i t t e d for small

bore pipes

Fig.

11.2

Sleeve-welded

Fig.

11.3

Push-in

sures

are

made

joint

type j o i n t

from

outside,

thereby

exaggerating

defects

i n the near w a i l . Welding that

for

of

high

low-tensile

and post-heating

(3)

Screwed:

tensile steels.

steels

require

Pre-heating

better of

the

craftmanship pipe

ends,

f o r stress r e l i e f a r e n o r m a l l y desirable.

Threaded

and

used on small steel pipes.

screwed

ends

and

sockets

are

only

(4)

Spigot

and

quently

socket:

Concrete,

have one

male end socket

has

is

end

a

rubber

forced

cast

socketed sealing

over

the

is

to

caulk

to

and

p l a s t i c pipes fre-

the other

gasket

ring.

shaped r u b b e r r i n g s a r e used.

resin.

iron

and

end

plain.

f i t t e d over

(see

i t a n d the

11.3).

Fig.

The

Various

Another way of sealing the j o i n t

i t w i t h bitumen compound,

cement

mortar,

lead or

Joints f i t t e d w i t h r u b b e r r i n g s a r e often c a u l k e d as well

hold

getting rings

the

ring

into

the

can

in

place,

joint.

improve s e a l i n g

Spigot

normally

and

socket

accommodate

one

and

prevent

joints

or

with

two

dirt

rubber

degrees

of

available

for

def I ect ion.

(5)

Clamp-on

joints:

jointing a

plain

rubber

to

as

a

illustrates

fixed

to

tension. in

the

If

the

the

tension

the

of

The

and

as

brackets

type

and

diameter

parallel

ing

stresses all

resist

four of

the

to

the

are

for

this

the

for

transferring coupling

the

(also

referred

These

designed

joints

to

take

longitudinal

or

tension,

tiebolts

brackets

even

bolts

barrel

shaped

pipe

axis.

to

may take

In

in

fact

like

tie-bolt

used f o r

the

plan

to

take

bolts

are

it

be the

to

of

the

shear

are the as The legs

legs of

a n d bend-

i s impossible to e l i m barrel

flanges,

iron

the

with

length

pipe

load

cast

bolts

minimize e c c e n t r i c i t y .

The

the

tie

selected

that

longitudinal

on

rings,

two

is

so

to

"U"

stresses

reason

sleeve.

bending a n d tension stresses

designed

acceptable.

bending

be

clamping

cover

joint

pipes

by

or

should be such that

and

Victaulic

to

normal l y

a

a n d may cause damage unless care-

brackets

possible

bracket

pipe

of

may

pipe

involve

and

coupling).

between

to

are

normally

Dresser

each

Normally

the

are

the

inate

is

of

joints

barrel

The b r a c k e t s cause local

sufficient.

welded

such or

easily,

joint

designed.

close

pipe

deflection

barrel

walls

fully

one

movement of

They

each

Johnson

degrees

movement.

proprietry

pipes.

between

Viking

accommodate several

ended

seal

11.4

Fig.

Various

may

the

pipes,

with be

pipe

brackets

preferable wai I.

The

uses l i p s on the

ends of the pipes to t r a n s f e r tension to the coupling.

215

Fig.

(6)

11.4

Slip-on

Flanged: ends,

Steel

drilled

to

cast

if

Faces

rubber

rnrn

and

especially

frequently.

type coupling.

or

iron

p i p e s or are

pipes

machined

other

insertion

standard

are

often

fittings are

patterns

and

bolted

gasket

flanged

likely

together

between.

according

to

at

the

to be removed

the

with

3

Flanges

are

diameter

and

working pressure. Ful I used

face

for

flanges. than

to

over

pipe

the

pipe

but

end

around

the

front

around

the

pipe

inside

edges

of

method

The

to

of

tend

method

use

to

leave

of

flanges

dish

metal less

attaching

the

pipe

and

at

the

Alternatively

flange high

10

may

be

pressure

back

both

is

flanges

extending

then of

bevelled work

carried

the one

for to

bolts

the f l a n g e

mm

or

a

rings.

the

the

is

thick

have

joint

when

weld

are

slightly

with

approximately fillet

barrel.

bore, soft

simplest method i s to s l i p

A

for

or

iron

Some

for

pipe.

the

a n d the

diameter

pipes.

circle

the

of

preferred

cast

outside

rings

The

bolts

the b o l t holes a r e used f o r

bolt

though.

the

an of

joint

varies.

beyond

A

the

with

tightened the

with

for

and

low-pressure

inside

used

holes

joints

diameter

and

face

Flanges are

rings

inside

flanges

raised

with

high-pressure Joint

the

steel

gaskets,

flange of

the

welding.

have

a

lip

216 on

the

flange

projecting

Jointing

and

standard

thrust

codes of

the

end

of

the

pipe

that

so

w i l l be f l u s h a f t e r welding.

the bore at the j o i n t

the

over

flanges

are

fairly

practice,

and

must

thick,

as

indicated

be securely

welded

by

to the

p i p e or cast i n t e g r a l l y . Two other types of f l a n g e s a r e common:Puddle

(1)

flanges

of

water

is

to

as

are

retaining

prevent

thick

tudinal

used

of

flanges

pipe

pipes

structures.

leakage

as

on

water

which

stress,

are

The

object

and

they

are

although

which

of

flange

in

these

are

designed

the

cast

not

to

walls

flanges

necessarily

transfer

should

be

longi-

securely

attached to the p i p e w a l l .

(2)

Blank

flanges

ends.

Their

moments

at

clamped occurs where and

t

at

the

is

the

name

should

edges

implies,

be

the

the the

the

perimeter

the

is

clamped

as

fluid

bolts,

flange

thickness.

maximum

stress

to

resist

For a c i r c u l a r

maximum

radial

and

pressure,

i n s t a l l e d at

calculated

the edges and centre.

at

p

are,

thickness

d

is

blank

bending

disc

rigidity

bending

equal

to

3

stress

pdZ/16t2

i s the p i t c h c i r c l e diameter

If

the

would

edges were not

occur

at

the

rigidly

centre

and

would be 50% i n excess of t h i s amount.

COAT I NGS

Buried

steel

the p i p e during

pipes

i s coated.

transport

are

laying

attack,

should

h a r d enough

to

stones

to

the

in

pipe

the wall

to

Coatings should

and

biological be

subject

electric to

prevent yet

pipe, and

to

damage

moisture,

temperature

damage

sufficiently

flexible

and

unless

i d e a l l y be r e s i s t a n t to s c r a t c h i n g

the

currents

trench, and

of

corrosion

enough

to

during

chemical

variations.

and They

h a n d l i n g a n d due

adhesive

to

withstand

adhere

the

well

flexing

of

the p i p e w a l l ( P P l , 1975; Cates, 1953). The

most

followed outer

by

common a

coating

wrapping.

brushing, applied

The

sandblasting by

spray,

coating

for

reinforced pipe or

brush

pipes with

surface

is

acid pickling, or

dipping

is

fibres

a

thin

and

initially

adhesive

then

possibly

cleaned

a n d the prime coat

the

pipe

in

a

bath.

by is

coat an wire then

Bitumen

217 or

coal

coat

t a r enamel

the

woven

pipe

glass

pipe

bare

to

matting.

The coating or

jointing

in

may

to 6 mm t h i c k ,

cement be

primed

and

in

in

in the

followed tar.

by

felt

or

paper

or

The p i p e i s then

in

the f i e l d a f t e r

welding

which case the ends a r e field.

The

total

thickness

left of

tar

an asbestos f i b r e bitumen mastic 3

include

pitch,

epoxy

paints,

PVC or polythene tapes

bedded on an adhesive),

zinc

applied

painted

with

by

resins or

galvanizing.

bitumen

mortar

coatings

offer

the case

of

l a r g e bore t h i n

usually

12

to

Finished means of metal

the p r i m a r y

based

plastics,

Exposed p i p e s may

or

aluminium

enamel

1962).

Cement

are

After

impregnated

least 5 mm.

coal

and

be a p p l i e d

coated

adhesive o r

mortar

(AWWA,

with

i s sometimes

factory,

Other types of coatings

self

This

the

and

coating should be at

(either

wrapped

in detecting damage a n d to s h i e l d the c o a t i n g

assist

joints,

for

spirally

impregnated w i t h bitumen or coal

from the sun. the

be

fibre

asbestos f e l t whitewashed

i s the p r e f e r r e d prime coating.

may

Holiday

brushes

or

($

mm

coatings

a

electrical

20

may

rolling

potential

is

3/4

An

springs

applied

for

is run

across

(AWWA,

flaws,

electrical

pin

conductor

along

the

to

buck1 i n g

Cement mortar coatings

inch) thick

checked

detector.

resistance

walled pipes.

to

be

additional

1954,

1962).

holes etc. in

the p i p e coating.

coating and

by

the form of

a

current

An is

observed when f l a w s in the c o a t i n g a r e de ected.

LININGS

Steel

pipes a r e

friction

losses.

substances

in

particular, corrosion.

l i n e d to r e s i s t

Unlined the

the

steel

fluid.

oxide

Corrosion

In is

of

interna

pipe

the

rapidly

by

be

case of

scraped

adding

oxidized

by

solids-conveying

water-conveying

m a i n t a i n i n g a h i g h pH e.g.

corrosion and minimize the

may

off

pipes

leading

may

be

corrosive systems to

in

further

inhibited

by

lime.

Lime on the other h a n d

bitumen

(3

to 5 mm t h i c k ) o r coal

Bitumen,

which

could cause carbonate s c a l i n g . The tar

most

enamel

petroleum, Before

popular (2

to

3

linings mm

are

thick).

is

a

by-product

of

i s t h e cheaper of the two. applying

the

lining

the

pipe

wall

is

cleaned

by

sand-

218 b l a s t i n g or spray,

other

dipping

in particular resistant

to

moisture

the

to

tar

bitumen,

damage

enamels

i s then a p p l i e d by

lining

a smooth surface.

smooth f i n i s h .

than

subject coal

and

s p i n n i n g to o b t a i n

provides a

consequently Plasticised

methods

or

Coal

tar

although

by

impact

have,

enamel is

it

and

however,

i s also more

more b r i t t l e

flexing

now

brush,

Spun enamel

of

been

the

and pipe.

developed

to

overcome the problems of b r i t t l e n e s s . Epoxy

paints

are

careful

appl ication

to

other.

each

the in

type o f

2

portable

The

but

layers.

water

used

it

Coal

tar

as

they

pipes

Cement mortar

6

also to

used

12 mm

gally

by

large

( 1 / 4 to

successfully cleaned b y

lining,

for

3

to pipes

machine

thickness

for

of

the

lining

adhere

varies

with

the

should

water.

be

Lead

avoided

for

based p r i m e r s

a r e toxic.

pipes. thick.

the f i e l d . or

although

coats

the o r d e r of 0 . 3 mm a p p l i e d

linings

taint

I inings,

successive

applied centrifugally

inch)

in

ensure

epoxy

bore

wire-brushing

a

to

i s n o r m a l l y of

should also be avoided as they

is

successful l y

necessary

recommended

paint,

4

to

also

is

The

spinning

Mortar

lining

the p i p e , is

usually

l i n i n g s have been a p p l i e d

The o l d surface i s f i r s t thoroughly

sand b l a s t i n g , propelled

drawn or

by

applied

and then coated c e n t r i f u -

slowly

along

the

line

(Cole,

1956). On made

pipes good

over at

about

field

600

joints

mm

bore,

manually

most

or

by

factory

linings

mechanical

may

be

applicators.

CATHOD I C PROTECT I ON

Despite of

steel

fittings.

the

pipes

use

Corrosion

protective

of

often

occurs

coatings,

through

i s due p r i m a r i ly

flaws,

corrosion pin-holes

to two causes;

and stray current electrolysis (Uhlig,

of

the at

or

walls

exposed

g a l v a n i c corrosion

1948).

Ga I vanic Corrosion When lyte,

two

current

electric

dissimi l a r may

current

electrolyte.

flow

flows

Particles

materials

are

connected

from one m a t e r i a l from or

the

ions,

T h e cathode i s not attacked.

anode leave

to to

the

through

the other. the

cathode

anode

an

electro-

The r e s u l t i n g through

causing

the

corrosion.

219 Such

action

may

take

place

when

two

dissimilar

contact

in

material

to the p i p e ) . Another more frequent

osion

conductive

occurs

marked

c e n t r a t ions sulphur.

w i t h pipes

soi I w i t h

in

or

type of

has

been to

i n corrosive

water

is a

lime

instance

high

w i th

form

the

successfully

trench

in an e l e c t r o l y t e may also

a

are

in

different

of

gal v ani c corr-

The effect

is particularly

differential

chemical

oxygen con-

,

content

biochemical

action

especial l y in

the s o i l

a c t i o n r e s u l t i n g from b a c t e r i a . A method

galvanic

used

metals

of

fittings

soils.

i s alSo caused by

which add

(for

v a r y i n g characteristics,

in

Corrosion

which

to

soil

to

backfill.

counteract

soil

corrosion

Stress concentration

lead to corrosion.

was

in the steel

The amount of corrosion

due to the l a s t named two influences i s u s u a l l y small. Potential

corrosive areas may

pipe - soil potential If

the s o i l

ivity

resistivity

using

associated

a

bridge

circuit

drop.

Standard

volt

measurements,

or

survey

but

any

major

or

by

highly

conductive

soil

may

The

potential in

difference

measured

by

connecting

electrode

in

contact

frequently tial If the

of the

used

soil,

is

are

should

at

currents are

current

done

by

taken

and

the

for

these

an

exper-

have a

resistivity

500 ohm cm,

of

more than 10 000 ohm cm. a

corrosive

buried

voltmeter

between

A

soil.

pipe

conditions.

the

to 0.7

a

a

available be

and

The

the

copper

the electrode under normal

as

a p i p e i s 0.5 pipe

a

with

Readings a r e n o r m a l l y

1952).

between

evaluating

Soi I r e s i s t -

conducted.

measuring

surveys

a n d a poorly conductive s o i l ,

important

be

probes

ienced corrosion engineer (Schneider,

A

in the pipe.

as a s t a n d a r d p r a c t i c e f o r major should

i n ohm - centimetres.

i s measured

in-situ

be detected by s o i l resistance tests,

measurement of the c u r r e n t

i s a t a l l suspect,

soi I

pipelines, a

tests or

pipe

the

soil

potential and

sulphate

is

a special

half

conditions.

is

cell

is

The poten-

v o l t s below that of the s u r r o u n d i n g s o i l .

higher

voltage

likely

to flow

than

0.85

volts

below

from p i p e to s o i l ,

that

thereby

of

cor-

r o d i n g the pipe. To by

a

(Fig.

prevent

t h i s corrosion,

conductor 11.5).

preferably

to

the

a

pipe

sacrificial

in

the

anode

vicinity

of

may

be connected

possible

corrosion

The s a c r i f i c i a l anode i s b u r i e d i n a conductive s u r r o u n d , below

the

water

table

and

currents

will

tend

to

leave

220 from to

the

the

pipe be

anode

anode.

to

instead If

cause

applied.

fact,

magnesium,

zinc

nesium

the

electro-chemical

may

the

Magnesium

anodes

Common

ions

are

normally

per

the

further

give

of

1

to

designed

potential

need

installed

in

anodes metals.

iron,

has

k i logram

are Mag-

a

high

of

material) hydrogen

thereby

This

is

actively

termed

polar-

200 ampere hours per for

the

( A l a y e r of

anodes,

attack.

200

steel

these

from

polarization.

leaving

against anodes

alloys

hours

be

corrosion the

sacrificial

difference

(ampere

the from

electrical

sometimes

or

anodic

limiting

dissimilar

external may

potential

to

the

metal

Magnesium

no

aluminium,

resistant

thereby

sufficiently

current.

greatest

replace

insulating ization).

the

and

pipe,

is

resistor

equivalent

is fairly

ions

the

action

a

I imi t

to

has

of

anode

galvanic

In

connection

and

the

about

10

a

year

kg. life

b u t zinc anodes often l a s t 20 or 30 years.

Junction

BOI

Wilh

R o a l ~ t o ri f Required

Fig.

11.5

S a c r i f i c i a l anode i n s t a l l a t i o n f o r cathodic protection

T h e spacing requirements volts The

below

the

spacing

of

ductive

soil

soil

provided

imating pipe

by

the

a n d size of

to

bring soil

poorly

the p i p e required of

anodes pipe

potential

sacrificial

and

means

the

a

will

to

a

along

the

anodes w i l l coated

i s well

pipe

is

ground

bed

to

potential

entire

vary to

coated.

current

be determined by the c u r r e n t safe

from 3 m

30

in

least

of

the

d.c.

r e s u l t i n g p i p e to s o i l p o t e n t i a l a l o n g the pipe.

drain source

current and

0.85

pipe).

poorly

in

highly

con-

conductive

The most r e l i a b l e way

actually

and

m

(at

length

of

est-

from

the

measure

the

221 As

a

rule

i

where

is

and

a

pipe

may

the

factor

the

i s i = 10/r

current of

be

thumb,

of

a g a i n s t corrosion

as

low

amps,

in

safety

3

of

0.5

as

current

required

to

protect

a

pipe

per square metre of b a r e d p i p e surface,

i s the s o i l

r

is

resistance

incorporated.

per

cent

for

a

The

good

i n ohm cm

area

of

exposed

coating,

r i s i n g to

20 p e r cent f o r a poor coating. Once the

the

total

anode

(less

than

soils

with

a

required size

500

ohm

lower

of

number

draw-off may

be

cm)

current

calculated.

large

anodes

In

(greater

should

km

than

be

of

pipe

highly

(20 k g )

anodes

conductivities

smaller

per

used

known,

conductive

may

be

about

1

to

is

used

soils

but

in

000 ohm cm)

ensure

an

adequate

c u r r e n t output.

If use a

an

an

electricity

impressed

sacrificial

current

A

anode.

v i d e the necessary With

supply

impressed

type

of

it

i s usually

protection

transformer-rectifier

current

Scrap

iron,

frequently

used.

Steel

soils

available,

(Fig.

may

11.6)

be

cheaper

to

instead

of

i n s t a l led to pro-

dc c u r r e n t .

corrosive.

ductive

is

and

i n s t a l l a t ions,

or

graphite

anodes

for

this

will

reason

the anode need not

rods

buried

quickly

in

corrode

high s i l i c o n

cast

coke in

be selffill,

are

highly

con-

i r o n anodes a r e

preferred. The

type

of

omics.

Impressed

in

long

the

lations

are

impressed applied,

run,

anode the

have

general

to in

potential

be

installations

3

to

immediate volts

be

at

1

qua1 i t y

of

the p i p e coating.

ferable

to

sacrificial

anodes

about 3 000 to 5 000 ohm cm.

are

protect

a

vicinity

long

potential

100

high

point

km

spacing

the

soil

case

of

may

be

pipe

a The

than

a

off

rapidly

high

voltage

pipe-to-soil

should

b e i n g below

coatings.

instal-

the

fa1 I s

length.

Impressed c u r r e n t if

of

econ-

cheapest

fewer

voltage

length

the

the

In

consequently

of

damage

and

potential

though,

to

low

anodes.

longer

pipe-soil

long-term

frequently

relatively a

(the pipe may

costs

a

The

applied

are

sacrificial

voltage

voltages may

for

protects

could.

applied

the

exceed Larger

maintainance than

turn

depend on the

i n s t a l Iations

installations

in

away

soi I ) .

as

current which

sacrificial

may

current

required

from

to use w i l l

protection

not,

that

of

in

the

Impressed

current

depending

on

the

installations are pre-

resistivity

is

higher

than

222 Considerable only

"hot

economic

spots",

occasional

savings

points

or

shutdowns

can

are

subject

be

often to

tolerated

achieved

aggressive this

by

protecting

attack,

should

be

and

if

considered.

Stray Current E l e c t r o l y s i s The

most

currents

severe

leaving

form

a

pipe.

r e t u r n c u r r e n t s through as a steer

p i p e nearby,

the conductor steel

will

ampere

of

instead of

be

corroded

current.

measurements recordings

The

should

be

corrosion

Railways

rails,

but

is

often

and

if

caused

other

there

users

by

of

stray

dc

dc

current

i s another conductor such

a p r o p o r t i o n of the c u r r e n t may flow through the r a i l . at

a

Where the c u r r e n t

rate

current

taken

of

may

pipe/soi 1

from

or

of

9

a

Ibs)

be detected

potent iat

over

(20

kg

day,

leaves the p i p e , per

by

actual

measurements. the

as

year

per

current

Continuous

currents

may

fluc-

t u a t e w i t h time. B o r e d or S u b m r r g e d Hod8 Cothodic Rolotlv8 t o E l w . t r o l y t 8

Structure

Iron or grophite 1

Fig.

11.6

The

Impressed c u r r e n t corrosion protection.

corrosion

by

connecting

to

the

may would

associated

the

destination

leave have

sufficiently

the to low

pipe of

pipe be to

at the

along

with the

point

current, some

impressed prevent

stray

on

currents

where with

length, the

current

pipe

the

a

may

current

conductor.

in to

escaping.

be

which

A

leaves

The

case

maintain

it,

current

a

current

the

voltage

ground

transformer r e c t i f i e r may be r e q u i r e d f o r t h i s protection.

prevented

bed

and

223 When the

a

pipe

mechanical

across

them

i s cathodically joints

if

are

protected,

electrically

necessary.

Branch

from the main p i p e to control

c a r e should be taken

bonded,

pipes

may

by

welding

have

to

a

be

that cable

insulated

the currents.

THERMAL I NSULAT I ON

Fluid

the

in

surroundings.

pipeline

are

sometimes

the

interior

of

layer face

through

fluid

of

the

be

heated

or

cooled

by

the

serious

climates

conduction

often

i n a r c t i c climates a n d h e a t i n g i n t r o p i c a l

or

from

may

Ice formation

the

inside

pipe

a

of

an

pipe

the

and

by

problem.

exposed

pipe

wall

pipe and

wall,

a

is

transferred

number of

wrapping

by

convection

Heat

in

and

ways:

a

to by

boundary

r a d i a t i o n from the e x t e r n a l

and

wind

currents

in

the

air

s u r r o u n d i n g the pipe.

TABLE 1 1 . 1

Thermal c o n d u c t i v i t i e s

Therma I c o n d u c t i v i t y

k cal

Mater ia I

m sec

Water

0.000

C"

sq f t hr F o

4

14

Air

0.000 005

0.15

Steel

0.014

420

B i tum i n ized Wrapping

0.000

Concrete

0.0002

Slag wool

0.000 01

EEUA, The is

B t u in

0.3

01

6

0. 3

1968. heat

loss

proportional

to

by the

conduction temperature

through

an

gradient

homogeneous

across

the

pipe

wall

wall

a n d the

heat t r a n s f e r r e d per u n i t area of p i p e w a l l per u n i t time i s -Q - -

AT

k

T

A0

(11.2)

224 Q

where is

the

temperature k

is

the

is

area

the

of

amount pipe

difference

thermal

of

heat

across

conducted

T

surface,

is

wal I ,

the

conductivity,

the

kilocalories or

in

duration

time, wal I

t

is

the

tabulated

in

Table

et

at

Btu,

A0

is

thickness

11.1

for

A the

and

various

materials. An heat

equation transfer

equation

has

was

developed by

through

the

been modified

wall to

Riddick of

agree

a

pipe

with

(1950) for

conveying

relationships

the total

water.

The

i n d i c a t e d by

EEUA (1964) a n d i s converted to metric u n i t s here: Rate of heat loss of water kcal/kg/sec

(0. I - ' o ) / 250D _- _ _ _ t, + t o + 1 +

1 (11.3)

and since

the

temperature k c a I/kg/sec

in

,

specific

heat

degrees

of

water

Centigrade

is

per

unity, sec

the

equals

r a t e of the

drop

in

loss

in

heat

where

temperature of water C " ambient temperature outside C " diameter m thickness of p i p e m c o n d u c t i v i t y of p i p e w a l l

kcal/m sec C o

thickness of coating m c o n d u c t i v i t y of c o a t i n g kcal/m sec C o heat loss through boundary 0.34(1

+

0.020i)v0'8/D0'2

water velocity

layer =

kcal/m2 sec C"

m/s

heat loss by r a d i a t i o n = 0.054 E('O

+

100

273 1 3

kcal/m2 sec C o e m i s s i v i t y factor

=

0.9 f o r b l a c k a s p h a l t a n d concrete 0.7 for cast i r o n a n d steel 0.4

for aluminium p a i n t

heat loss by convection = 0.000 kcal/m2 sec C"

v

=

w i n d velocity

km/hr

34f1

+ 0.78V

(-

'i

-90 0.25

D

1

225 The h e a t

generation

although i t theoretically

by

fluid

friction

is

usually

negligible

i n c r e a s e s the t e m p e r a t u r e of t h e f l u i d .

Example

An

uncoated

thick

steel

water

at

300

walls,

an

mm

diameter

exposed

initial

to

a

1

pipeline 10

t e m p e r a t u r e of

km/hr 10°C.

000 m

wind,

long

5 mm

with

conveys

100 C / s

of

The a i r t e m p e r a t u r e i s 30°C.

Determine t h e e n d t e m p e r a t u r e of t h e w a t e r .

E

=

0.7

V

=

0.1/0.785

=

0.34

=

0.054

=

0.005 ~-

Kf Kr

tl -

x 0.32

= 1.42

x 1.42°'8/0.30.2 x 0.7

( 30

+

m/s

= 0.67

10-3 = 0.001

27313

- o.35

0.014

kl

(30 - 1 0 ) / ( 2 5 0 x 0 . 3 ) 0.35 + 1/0.67 + (0.001

=

+

= 0.00103°C/sec

0.0029)

T e m p e r a t u r e r i s e o v e r 1 000 m = 1 000 x 0.00103 1.42 The

insulation

erature fluids

of

industrial

i s a subject

on

pipework i t s own.

= 0.7"C

h i g h or

carrying The cost of

low

temp-

t h e heat t r a n s f e r

s h o u l d be b a l a n c e d a g a i n s t t h e cost of t h e l a g g i n g . Heat erature heat heat the used,

transfer of

the

transfer. transfer

to

from

surroundings, The

may

temperature

or

of

buried which

temperature vary the

with

turn

in

gradient

time

and

surroundings

o m i t t i n g the term l / ( K

pipelines

is

and

are is

depends

the

influenced

consequently

difficult

known,

on

to

Equ.

temp-

by rate

evaluate. 11.3

may

the of If be

+ Kc).

REFERENCES

American Concrete P i p e Assn., 1970. Concrete P i p e Design M a n u a l , Arlington. AWWA S t a n d a r d C602, 1954. Cement M o r t a r L i n i n g of Water P i p e l i n e s in P l a c e , N.Y.

226 AWWA S t a n d a r d C203, 1962. C o a l T a r Enamel P r o t e c t i v e C o a t i n g s f o r Steel W a t e r P i p e 30 Ins. and O v e r , N.Y. AWWA S t a n d a r d C205, 1962. Cement M o r t a r P r o t e c t i o n L i n i n g and C o a t i n g f o r Steel W a t e r P i p e 30 I n s . and O v e r , N.Y. BSCP 2010, 1970. P a r t 2, D e s i g n and C o n s t r u c t i o n of Steel P i p e l i n e s in Land, B S I , L o n d o n . Cates, W.H., 1953. C o a t i n g f o r s t e e l w a t e r p i p e , J. Am. W a t e r W o r k s Assn. 4 5 ( 2 ) . 1956. D e s i g n o f s t e e l p i p e w i t h cement c o a t i n g and l i n Cole, E.S., ing, J . Am. W a t e r W o r k s Assn., 4 8 ( 2 ) . C o n c r e t e P i p e Assn., 1967. B e d d i n g and J o i n t i n g of F l e x i b l y J o i n t e d C o n c r e t e P i p e s , Techn. B u l . No. 10, T o n b r i d g e . Engineering E q u i p m e n t U s e r ' s Assn., 1964. Thermal I n s u l a t i o n of P i p e s and Vessels, H a n d b o o k No. 12, C o n s t a b l e , L o n d o n . Engineering E q u i p m e n t U s e r s Assn., 1968. P i p e J o i n t i n g Methods, H a n d b o o k No. 23, C o n s t a b l e , L o n d o n . 1975. P i p e l i n e p r o t e c t i o n r e v i e w , P i p e s and P i p e l i n e s I n t e r n a t i o n a l , 20(4). Reitz, H.M. 1950. S o i l m e c h a n i c s and b a c k f i l l i n g p r a c t i c e , J . Am. W a t e r W o r k s Assn., 4 2 ( 1 2 ) . R e y n o l d s , J.M., 1970. S u b m a r i n e p i p e l i n e s , P i p e s and M a n u a l . 3rd Ed. S c i e n t i f i c S u r v e y s , L o n d o n . and T o m a s s i , A., 1950. F r e e z i n g of Riddick, T.M., L i n d s a y , N.L. w a t e r in e x p o s e d p i p e l i n e s , J. Am. W a t e r W o r k s Assn., 42(11). S c h n e i d e r , W.R., 1952. C o r r o s i o n and c a t h o d i c p r o t e c t i o n o f p i p e l i n e s , J. Am. W a t e r Works Assn., 44(5). Schwartz, H. I., 1971. H y d r a u l i c t r e n c h i n g of s u b m a r i n e p i p e l i n e s , P r o c . Am.Soc. C i v i l E n g r s . 9 7 ( T E 4 ) Sowers, G.F., 1956. T r e n c h e x c a v a t i o n and b a c k f i l l i n g , J. Am. W a t e r W o r k s Assn., 4 8 ( 7 ) . 1948. The C o r r o s i o n H a n d b o o k , W i l e y , N.Y. U h l i g , H.H.,

L I S T OF SYMBOLS

A AS

B

-

area

-

a r e a of steel

-

drag c o e f f i c i e n t

w i d t h of t r e n c h

d

-

inside diameter

D

-

outside diameter

E

-

thermal emissivi ty factor

f

-

stress

9

-

gravitational acceleration

H

-

backfi I I a b o v e t o p o f p i p e

I

-

c u r r e n t in amps

227 K C

Kf Kr

kl k2

heat loss by convection heat loss through water f i l m heat loss b y r a d i a t i o n c o n d u c t i v i t y of p i p e wal I c o n d u c t i v i t y of coating

M

bedding factor

P

pressure

Q

amount of heat

r

resistance i n ohms

T

time

tl t2

thickness of p i p e w a l l t h i ckness of c o a t i n g

u

component of wave v e l o c i t y p e r p e n d i c u l a r to p i p e

V

mean

water

velocity,

or

component

p e r p e n d i c u l a r to p i p e

v

w i n d velocity

W

s p e c i f i c weight of water

Y

depth of bedding m a t e r i a l below p i p e

0.

temperature i n s i d e p i p e

0

temperature outside p i p e

0

of

current

velocity

228

CHAPTER 12

PUMP I NG I NSTALLAT IONS

INFLUENCE OF The

pipeline

engineer

pumping

lines.

pump

opposed

as

inter-related Pumes

must

head,

hence

of

the

pipe

That

with

i s frequently

is,

to

a

fluid

the

gravity

the

overcome

fed

pumping line

is

concerned forced

line.

The

equipment

friction

losses

The

engineer

The

pipe

Therefore

therefore

The system

engineers

selected

should

wall

their

(1969)

and The

have

considered

to

pump,

be i.e.

is

the

as

(1968).

aware

of

one

for

and

suction

the

in

thout

casing

electrical

a r e so s p e c i a l i z e d

that the following

It

various costs.

example, head

or

impeller

power that is

they

versa. static

function based

on

(which

considering

the

Pipeline

types of

pumps,

Stepanoff

(1957),

give

more

complete also

The d e s i g n o f

shape

material,

and

i s covered

the is

In fact

it

(1960),

K o v a t s (1964) and KSB

supply,

from

is

requirements w i l l

pumpstation design.

the manufacturer.

engineers.

the

pipe

hickness of the pipe. v,

and

(19551,

t h o s e b y K a r a s s i k and C a r t e r

electrical

view

be

size

optimum

Similarly

equipment by

design

a

net

turn

in

by

vice to

head a n d f l u i d velocity

characteristics

Addison pump

more the concern of such

the

and

addition

in

thickness

friction

cannot

therefore

information.

IWt

of

pipe

must be designed in a n integrated manner.

I i m i t a t ions,

their

the

design

a f f e c t s w a t e r hammer h e a d ) a l s o a f f e c t t h e w a l l

other.

the design of

with

through

t h e p u m p i n g h e a d a n d p o w e r r e q u l .ement a r e a diameter.

pressures.

interna I

DESIGN

PUMPS I N P I P E L I N E

motor design

in books

a n d control

should be selected o r designed

the

pipeline

engineer’s

point

of

still

used

in

i n f o r m a t i o n i s presented.

TYPES O F PUMPS

P o s i t i v e D i s p l a c e m e n t Types Least some

used

industrial

in

pipeline

practice

a p p l i c a t ions,

are

nowadays, positive

but

displacement

pumps.

229 The

reciprocating

to

move

to

to

enter

and

on

pump fro

a

A p i s t o n or

i s one such type.

i n a cylinder.

reverse

stroke

One-way

and

exit

ram

i s forced

v a l v e s p e r m i t the

into

the

receiving

liquid conduit

on a f o r w a r d stroke of the piston. The also

p i s t o n can

rotary

be

positive

replaced

by

displacement

a

flexible

pumps.

diaphragm.

These

There

include s pi ral

are

types

a n d intermeshing gears. Centrifugal Pumps Rotodynamic pumps move pressure like by

to

it.

through rotating

low

heads.

The

For an

latter

various

in

waterworks

may

high

may

the

form

is is

by

an

impeller

by

assembling

guide

centrifugal

the

most

engineering.

This

parallel is

are

limited

a number of

to

a

of

an

a

tabular

vanes

of

The

in

axial

the

casing

i s preferred.

the

pump

i s capable

head

pump

standard of

hand-

a n d to p e r m i t f l e x i b i l i t y , which

can

be

100 metres of

generated water,

but

i n series a wide r a n g e of heads

lcllcr

,nes

12.1

to

Here

centrifugal

practical ly

i s possible.

Fig.

flow

casing

l a r g e flows a n d

as

l i t t l e over

impellers

pump

and

accepted

For h i g h flows

common.

form

( r a d i a l ) c e n t r i f u g a l flow.

common,

type

and hence

along

i s most common f o r

universal ly

up to 20 cumecs.

in

the

forced

impeller creates an o u t w a r d

forms

pumps

in

is be

pump

heads

i m p a r t i n g a velocity

be

Liquid

There

T h i s type of

in

l i n g flows

action

propeller.

blades.

reduce s w i r l .

r o t a t i o n of

This

a

l i q u i d by

Sectional e l e v a t i o n of v o l u t e t y p e c e n t r i f u g a l pump (Webber, F l u i d Mechanics f o r C i v i l Engineers, 1971)

230 The

inlet

entrainment to

the

c o n d i t i o n must and

delivery

the

permissible

pressure of

instead of connecting of

be controlled

first

pumping

stage.

particularly

if

In

some

cases

i n series on a common shaft a number

independent pump u n i t s a r e i n s t a l l e d i n series.

flexibility

air

maximum suction pressure i s r e i a t e d

the

impellers

to a v o i d c a v i t a t i o n or

one

of

the

units

T h i s permits more be

to

is

driven

by

a

v a r i a b l e speed motor.

The

driving

device i n a c e n t r i f u g a l

pump

i s termed

the

impeller.

I t may be a series of

vanes shaped to f l i n g the water from a c e n t r a l

eye

the

outer

the

discharge

a

or

inlet

volute

type

of

motor

in

pump

units

large

pumping

towards

casing

into

may

be

together

sizes. solids

as

Pumping in

much

rarely

periphery.

90

as

installations can

for

have

it

The

percent

h a v e efficiences

suspension

Here

nozzle.

is

directed

efficiency

although

by

of

this

pump

and

above 80 percent except unusual

efficiencies

duties as

such

low

as

as 30

percent. Usually

the

inlet

to

the

impeller

eye

is

d r i v i n g shaft

to the motor on the other side of

s i t u a t i o n s an

i n l e t may

to

as

a

double e n t r y

be on each side pump.

This

on

one

side,

and

the impeller.

the

i n some

i n which case i t i s r e f e r r e d

involves

a

more expensive

casing

b u t i t r e s u l t s i n a balanced t h r u s t i n the d i r e c t i o n of the shaft.

Fig.

12.2 Section through a s i n g l e e n t r y horizontal pump

spindle centrifugal

231 Pump u n i t s may be mounted w i t h The

horizontal

the

maintenance

suction

well

preferable.

shaft

is

with

point

a

of

view.

required,

The motor

is

split

a

the s h a f t

casing

Where

is

vertical the

space

vertical

spindle

most

is

or

horizontal.

practical

limited,

or

arrangement

a

from deep

may

be

then mounted on a p l a t f o r m above the pump.

A N D DEF I N I T I ONS

TERMS

Head The water.

head It

in

i t s general

sense

comprises elevation

sure head p / w

p l u s velocity

is

above

the

a

energy

per

certain

datum,

excess

of

u n i t weight of z,

p l u s pres-

head v2/2g.

Total Head Total inlet

head

head

and

usually it

is

refers the

to

head

the

generated

by

12.3).

Fig.

12.3

discharge

the

D e f i n i t i o n of net p o s i t i v e suction head.

pumps

head

to

(see F i g .

232 Net Positive Suction Head In

order

to

avoid

in pumping efficiency The

net

cated The It

positive

by

a

cavitation,

suction

pump

requirement i s generally

head

referred

expressed

as

as

to

a

drop-off

the

can

function

the

net

absolute

u s u a l l y be

of

r a t e of

positive

head

head.

above

indi-

pumping.

suction

head.

vapour

pres-

I t is c a l c u l a t e d as follows:

( 2.1

= H r + H a - H s - H = head

P

and

- Pv/W

= Hp

where H

a

as

s u r e r e q u i r e d a t the pump inlet. NPSH

entrainment

(NPSH) r e q u i r e d

manufacturer is

air

a pump needs a c e r t a i n minimum suction

at

inlet,

f

- H

-

e

relative

( 2. 2)

PV/W to

centre

line

of

inlet,

equal to absolute pressure head p l u s velocity

head.

and

= vapour pressure

PV

= u n i t weight of water

W

= suction r e s e r v o i r head above datum

Hr

= atmospheric

Ha

pressure

head

10

(about

of

m

water

at sea l e v e l ) H S

Hf He

=

elevation

=

f r i c t i o n head loss between r e s e r v o i r a n d pump

=

turbulence head loss between r e s e r v o i r a n d pump

NPSH requirement drop-off

in

increases side of

figure

NPSH can tests

(Grist,

firstly at

at

al I

maximum impeller in

NPSH.

lishing at

discharge.

least

It

the head of should

be

1974).

flowrates

flowrates. erosion

the 3

At

an

due

NPSH.

NPSH

the f i r s t

the

NPSH

NPSH

to

stage of

at

above

i s not

this

datum

a sharp

value.

It

just

i s of

the

order

the pump b u t a more manufacturer's

the

tests.

c a v i t a t i o n o r from performance

i s reduced

less

than

the

cavitation

commences,

flowrate but eventually that

head

associated

generated

with

by

the

The head r a p i d l y decreases w i t h decrease in

NPSH r e q u i r e d 3%

there

level

requirement

from

cavitation

3% decrease

The the

suction

as

noise

away from best efficiency

l i m i t of

times

The

obtained

s t a r t s t o decrease.

A

figure

in

by observing

As

pump

i s p r e f e r a b l e to be on the conservative

NPSH.

be assessed

of

subjective increase

or

considering

10 percent of

re1 i a b l e

centreline

is a

efficiency

with

when

of

head

drop

head to

i s recommended for avoid

NPSH,

cavitation

though.

some degree of c a v i t a t i o n i s normally acceptable.

For

estab-

should this

be

reason

233 Specific Speed The

specific

pump's

Ns

speed

capabilities.

of

There

a

pump i s a useful i n d i c a t i o n of the

are

two

slightly

different

definitions

of

s p e c i f i c speed a n d the more general one i s g i v e n f i r s t : Specific

speed

to

relative

a

i s a c h a r a c t e r i s t i c velocity

characteristic

velocity

of

of

the

the r o t a t i n g element

water,

i.e.

it

is

the

v e l o c i t y of the impeller r e l a t i v e t o that of water. The

ND

speed

N

where will

(units velocity

Ns

is

the

be

periphery

impeller

introduced

i s proportional

is

vDb

the

of

b

is

the

the

rotational

later).

impeller speed

is

proportional

D

and

From B e r n o u l l i ' s

is

to

i t s diameter

theorum

the

water

to JgH where H i s the total head. Therefore

t o ND/ JgH.

proportional

where

of

width

But

of

the

p r o p o r t i o n b i s p r o p o r t i o n a l t o D,

Q

discharge

impeller.

is

proportional

For a n y

to

g i v e n geometric

so

Qa,&8D2

(12.3)

E l i m i n a t i n g D one a r r i v e s a t

/

NQf

O-

NS

This

(12.4)

is

a

dimensionless form of

are

consistent.

the

g

term

Unfortunately

and

express

r a d i a n s p e r second.

specific

most

N

in

speed p r o v i d e d the u n i t s

catalogues

revolutions

and per

text

books

minute

(12.5) units Q

imperial

is

u n i t s Q i s i n cumecs

S.1.

The

alternative

metrically similar (or

definition

gal.

per

min)

one

arrives

expression for N

of

Ns

the is

of

a

that

water

of

low

minute and H

per

NS i s the

of

such size that

against

the

1

metre

directly

former

impeller

associated

whereas

gallons

i n feet.

In

at

1

(or

the

speed

in

rpm

of

a

geo-

i t would d e l i v e r one cumec foot)

latter

head.

Then

dimensionally

by

sub-

dependent

.

reviewing size

in

( c u b i c metres p e r second) a n d H in metres.

pump of

stitution

By

not

One encounters the expression

NS = NQi/H3/4 In

omit

(rpm)

NS

with

t a n g e n t i a l component).

to

relative a

implies

(water

definition

thus

low a

pumping

head

high

has

a

NS,

of

and

head high

one head.

fast

and

radial

low

obtains In

fact

rotational

an

idea

a

high

speeds,

speed

relative

to

component

relative

to

234 Fig.

12.4

indicates

the

r e l a t i o n s h i p between

head per

stage

and

s p e c i f i c speed f o r a number of pumps selected from r e a l applications. i n r p m a n d w i n radians/sec

N is

(w

= 271

N/60).

Lo

7

0

Lo

\

W2 v u l t i - s t a g e 74-

t@

50

\-- n \ 7

Ln

0,

L

-

$20 E I

5

1

T e d flow

I

\ '

<

I

20

12.4

Fig.

\

50

Relationship

I

Ns

I

100

between

200 head

and

500

1000'

specific

speed

for

pumps

IMPELLER DYNAMICS

The shape of the

radiai

flow

A

acteristics. pump

the

impeller blades

pattern basic

which

in

understanding

i s therefore useful

i n a c e n t r i f u g a l pump influences

t u r n gives of

the pump c e r t a i n char-

relative

in selecting a type of

velocities

inside

the

pump f o r a p a r t i c u l a r

duty. Referring is

u,

the

to

Fig.

velocity

of

12.5 the

the absolute water velocity

the

circumferential

water

relative

i s v.

Torque T o n the water i s force x r a d i u s But force = change in momentum

to

speed the

of

the

impeller

is

impeller w

and

235 T h e r e f o r e i m p a r t e d T = c h a n g e i n moment of momentum = A(mvr) =

1 r2v2

Fig.

12.5

where

is

an

element

mass

of

discharge

per

unit

time,

where

P = A (mvr) w

(12.9)

.'.

P =

.*.

the

(12.8)

i s t h e r o t a t i o n a l speed

Now r w = u1 a n d r w = 1 2 P = 1 u 2 v 2 c o s a dm 2 A l s o P = pgHQ where Q

.'.

and

i s o v e r a l l the vanes.

Power P = T w

.'.

(12.7)

V e l o c i t y v e c t o r s a t i m p e l l e r of c e n t r i f u g a l pump.

dm

integral

(12.6) cos a dm 1

cos a dm - l r l v l 2

12.10)

u 2 /ulvl

cosa drn

12.1 1 )

i s the d i s c h a r g e r a t e

12.12)

gHJdm s i n c e p Q

1

12.13)

Jdm

=

12.14)

Head H = ( u v c o s a - UIVl cosal)/g 2 2 coscll i s n o r m a l l y r e l a t i v e l y s m a l l .

The term u,vl

cosa2 = u 2

Now v

-

t 2 cot

B2

(12.15)

where t 2 i s t h e r a d i a l component of v

.*. H

(u,

2 (12.16)

- t 2 cot B 2 ) / g

(12.17)

a n d Q = nDbt

...

2 A - BQ cot

H

where

A

density.

and The

0

B2 are

(12.18) constants,

blade angle

H-Q c h a r a c t e r i s t i c c u r v e ,

B2

g

is

gravity

therefore affects

as i l l u s t r a t e d

in Fig.

and

p is

water

t h e shape of 12.6.

mass

the pump

236

1

Q

Fig.

12.6

Effect of b l a d e a n g l e on pump c h a r a c t e r i s t i c s

PUMP CHARACTER I ST I C CURVES

The

head-discharge

usually

produced

pumping

system.

and

selection

of

by This

of

characteristic the

manufacturer

relationship

combinations

of

is

i s one

pumps and

used

useful

pumps

for

in

discussed l a t e r .

-BH

Fig.

12.7

Pump c h a r a c t e r i s t i c curves.

to

of

the

design

pipe

parallel

size as

curves

a

viable

selection will

be

237 Break and

horse

pump

Fig.

also

be

or

duty

H-Q

changed

shaft

power

any

head

is

proportional

to

the

i n d i c a t e d on the same d i a g r a m

c u r v e may a l t e r

on

required by

pump

with

by

pump (see

the d r i v e speed a n d

fitting

different

The BHP a n d E curves would also change then.

diameters. tant

The

(BHP)

E a r e often

efficiency

12.7).

could

power

impeller

The r e s u l -

NZ a n d D2, whereas the discharge

is

p r o p o r t i o n a l to N a n d D. The ally.

required

The

graph.

level and

A

for

and a

line

effect

pipeline

Thus

required

duty

a

Fig.

a

pump

12.8

illustrates

a's,

smooth

of

pipe,

between could

para1 lel i n g

the

assuming

more pessimistic f r i c t i o n somewhere

i s most e a s i l y

determined g r a p h i c -

c h a r a c t e r i s i t c s a r e p l o t t e d on a head - discharge

different new,

of

t w o or

(a)

and

(b)

factor be

head a

(static

friction)

h i g h suction sump a

low

suction

applicable

selected

more pumps

plus

can

for

to the

also

an

sump

level

older

pipe.

duty

line.

The

be observed

from

such a g r a p h .

I

2 in parallel

llow sump and old pipe

0flurves/

/

pipeline

a

\\

high sump level and new pipe

0

Fig.

12.8

P i p e l i n e a n d pump c h a r a c t e r i s t i c s .

water

238 Pumps a

pipeline line pump

more

pump

operate

within

are

the

a

or

lower

added one

discharge

difficult

station

increase c a p a c i t y

curves

parallel,

are

booster

to

to

series

in

however,

at

to

any

control. is

the

head

pressure. other, is

Frequently,

required

enable the f i r s t

design

above

to

12.9)

(Fig.

section of

In

for

i.e.

a

pipe-

such cases

whereas

added,

along

pumps

the in

the

abscissa

break

pressure

i s m u l t i p l i e d by the number of pumps i n operation. Booster

pumps

r e s e r v o i r or case one

a

certain

station.

reservoir

may

sump

The

level.

on

operate

in-line,

the

suction

side

balancing

volume

is

hydraulic In

the

grade

former

or of

required

l i n e must

case

with

the

water

I n the

in

of

also

12.9

Location of booster pump station.

case be

hammer

f a i l u r e i s more d i f f i c u l t to p r e d i c t a n d allow f o r .

Fig.

a

booster.

a

drawn due

to

latter

trip

at

down

to

a

power

239 MOTORS

The

driving

wheel

or

most

speed

of

the

current.

The

number of per or

be

a

an

electric

by

the

pump

speed

is

Slip

the e l e c t r i c a l

can

of

motor.

The r o t a t i o n a l of

N = 60Hz

alternating

divided by

the

frequency

in cycles

motors

per

second

to be made f o r s l i p between the r o t o r

be up t o 2 o r 5 percent

AC

water

where N i s the pump speed in r e v o l u t i o n s

i s a correction

types

turbine,

frequency

i s approximately

poles,

Hz

steam

a r e commonly

( f o r smaller motors).

used

for

driving

pumps:

s q u i r r e l cage induction motors (asynchronous) i n d u c t i o n motors (asynchronous)

synchronous i n d u c t i o n motors. Squirrel

require the

other

motor

cage motors

minimum

starting

two

with

types.

Their

However

starters.

it

i s usual

robust,

is

that

they

can be tolerated, offers

If

the

require

than large

a s q u i r r e l cage

most

direct-on-I ine

either star-delta

economic a n d

more e f f i c i e n t

economical

or

and

is

starting

not

auto-transformer

A l t e r n a t i v e s t a r t e r s compare as follows:-

S t a r t i n g Current ( p r o p o r t i o n of f u l l load)

and

full

induction h a v e the speed

0.5

to 2.5

1.2 t o 2.0

0.3

to 0.8

as r e q u i r e d

as r e q u i r e d

motors

are

started

a d v a n t a g e that

with

S t a r t i n g Torque ( f u l l load = 1 )

to 7.0

4.5

transformer

Slip-ring

to

this

to employ

Star - Delta

starters

design,

a r e general l y

disadvantage

starter

Direct - on

-

in

They

possible.

Type of Starter

Auto

simple

if

d i rect-on-1 ine

combination

permitted,

are

maintenance.

currents.

reliable

up

could

nowadays,

b ) wound r o t o r ( s l i p - r i n g ) c)

pump

controlled

is

p a i r s of

starter.

a

commonly

There

Three

of

pump

minute and Hertz.

and

a)

power

relatively

they

low

by

means

of

resistance

c a n be smoothly

starting

current.

brought

They

are

normal l y used when s q u i r r e l cage motors a r e impermissible. Synchronous the

motors

a d v a n t a g e of

a

are

high

used

load

on

factor.

large They

installations. may

They

be e i t h e r

have

induction

240 motors

or

speed.

Starting

ance o r

salient

pole

can

reactor

motors.

be

by

starters

They

run

at

direct-on-I ine,

depending

constant

synchronous

auto-transformer,

mainly

on

the

resist-

permitted s t a r t i n g

current. Variable

speed

motors

constant

speed

although

commutator

are

used

being can

as

well.

used

also

motors.

with

be

are Belt

motors,

for

on

motor

or

Thyristor savings

used

more expensive a n d drives

cost

in

and

for

two

less e f f i c i e n t units

are

v a r i a b l e speeds

small

motors.

speeds for

than

possible

resistances on s l i p - r i n g

starters

selecting

smaller

motors

are

now

Pole c h a n g i n g

squirrel

cage motors.

PUMPSTAT IONS

of

cost

the

machinery.

the as

of

The

pipeline will

he

the

structures While

engineer h a v e to

the

will

pump

with

control

will

effect

the

operation

hammer

protection

discussion The ant can

effect

its

of

reduce of

the

the

cost

details

cannot

be covered

here

concern

himself

with

The

the

is

and pump of

capacity

requirements

may

of of

may

A

be

more

layout to

sumps water

complete

al

(1974).

has an

import-

Twort

pipework

the

pipework

suction

there

by

inlet

of

the

stat ion.

given

Air

of

and

the

exceed

delivery

capacity

a pipeline.

efficiency. the

and

system, in

layouts

sump

operational

could

awareness

to

incorporated

the

house pumps

suction

valves.

pumpstation of

the

b e a r i n g on the capacity

turbulence The

of

design

need

install

each

to

design

et

Head losses i n t o the pump drawn the

the

in

by

pipeline entire

vortices

or

considerably.

pumping

system

therefore should be the d u t y of the p i p e l i n e engineer.

REFERENCES

H., 1955. C e n t r i f u g a l a n d Other Rotodynamic Pumps, 2nd Addison, Ed., Chapman a n d H a l l , London, pp 530. Grist, E . , 1974. Nett p o s i t i v e suction head requirements f o r avoidance of unacceptable c a v i t a t i o n erosion i n c e n t r i f u g a l pumps, Proc. C a v i t a t i o n conference, Instn. Mech. Engrs. London, p 153-162. I n s t i t u t i o n of Water Engineers, 1969. Manual of B r i t i s h Water Engineering Practice. V o l . I I , Engineering Practice, 4th ed. London, K a r a s s i k , I . J . a n d Carter, R., 1960. C e n t r i f u g a l Pumps, F.W. Dodge, N.Y. pp 488.

241 Kovats, A., 1964. Design a n d Performance of C e n t r i f u g a l a n d A x i a l Flow Pumps a n d Compressors, Pergamon Press, Oxford, p p 468. KSB, Pump Handbook, 1968. K l e i n , Schanzlin 0 Becker, F r a n k e n t h a l , p p 183. Stepanoff, A.J., 1957. C e n t r i f u g a l a n d A x i a l Flow Pumps, Theory, Design a n d Application, 2nd Ed., Wiley, N.Y., p p 462. Twort, A.C., Hoather, R.C. a n d Law, F.M., 1974. Water Supply, Edward Arnold, London, p p 478. 1971. F l u i d Mechanics f o r C i v i l Engineers, Chapman Webber, N.B., a n d H a l l , London, p p 340.

L I S T OF SYMBOLS

A,B

-

constants

BHP

-

break horsepower

b

-

width

D

-

di arneter

E

-

efficiency

9

-

g r a v i t a t ional acceleration

H

-

head ( s u b s c r i p t a - atmospheric, e - turbulence, f

-

friction,

p - inlet and s - elevation).

mass per u n i t time pump r o t a t i o n a l speed net p o s i t i v e suction head specific speed pressure power discharge r a t e radius torque r a d i a l component of velocity p e r i p h e r a l velocity velocity unit

weight

of

water

u n i t mass of water a n g I es

p 9.

(9800 Newtons per c u b i c metre)

242

GENERAL REFERENCES AND STANDARDS

BR I T I SH STANDARDS

Steel Pipes BS

534

Steel Pipes, F i t t i n g s and Specials

778

Steel Pipes a n d Joints

1387

Steel tubes and t u b u l a r s f o r screwing

1965Ptl

But-welding

2633

Class 1 a r c welding of f e r r i t i c steel pipes

pipe fittings.

291 0

Radiographic

ex.

Carbon steel

welded

circ.

butt

joints

in

steel

pipe

3601

Steel pipes a n d tubes - carbon steel

3602

Steel pipes and tubes for pressure purposes

3603

Steel pipes a n d tubes f o r pressure purposes

Carbon steel

- ordinary duties

: h i g h duties

Carbon a n d a l l o y steel pipes

3604

Low and medium a l l o y steel p i p e s

3605

Steel pipes a n d tubes for pressure purposes Austeni t i c stainless steel

Cast

I r o n Pipe

78-1

C

I spigot a n d socket pipes - p i p e s

78-2

C

I spigot a n d socket pipes - f i t t i n g s

143

Malleable C

41 6

C

I spigot a n d socket s o i l ,

437

C

I spigot a n d socket d r a i n p i p e s a n d f i t t i n g s

1 130

C I d r a i n f i t t i n g s - spigot a n d socket

121 1

Cast ( s p u n ) i r o n pressure p i p e s

I screwed pipes

I screwed pipes

1256

Malleable C

2035

C

4622

Grey i r o n p i p e s and f i t t i n g s

Wrought

788

waste a n d vent pipes

I f l a n g e d pipes a n d f i t t i n g s

I r o n Pipe Wrought

i r o n tubes a n d t u b u l a r s

243 1740

Wrought steel p i p e f i t t i n g s

D u c t i l e I r o n Pipe

L772

Ductile iron pipes a n d f i t t i n g s

Asbestos Cement Pipe

BS

L86

A C pressure p i p e s

582

A

201 OP t4

Design a n d construction of A

3656

A C sewer p i p e s a n d f i t t i n g s

C

soil,

waste

and

ventilating

pipes

and

fittings

C pressure pipes in l a n d

Concrete

556

Concrete pipes a n d f i t t i n g s

4101

Concrete unreinforced tubes a n d f i t t i n g s

4625

Prestressed concrete pipes ( i n c .

fittings)

Clay Pipe

65

Clay d r a i n a n d sewer pipes

539

Clay d r a i n a n d sewer pipes - f i t t i n g s

540

Clay d r a i n a n d sewer pipes

1143

Salt

glazed

ware

pipes

with

chemical l y

resistant

properties

1196

Clayware f i e l d d r a i n pipes

P l a s t i c a n d Other Pipe

1972

Polythene p i p e ( t y p e 3 2 ) for c o l d water services

1973

Spec.

3284

Polythene p i p e ( t y p e 50) f o r c o l d water services

polythene

pipe

(type

425)

for

general

3505

UPVC pipes ( t y p e 1420) f o r cold water supply

3506

UPVC p i p e f o r

3796

Polythene p i p e ( t y p e 5 0 )

3867

D i m s . of p i p e s O.D.

4346

Joints a n d f i t t i n g s for UPVC pressure p i p e

4514

UPVC s o i l a n d v e n t i l a t i n g p i p e

4660

UPVC underground a n d d r a i n p i p e

i n d u s t r i a l purposes

purposes

244 4728

Resistance t o c o n s t a n t

i n t e r n a l p r e s s u r e of

thermopl a s t i c p i p e 2760 CP312

Pitch f i b r e pipes and couplings P l a s t i c pipework

l n s u l at i o n 1334

Thermal

insulation

4508

Thermal l y i n s u l a t e d u n d e r g r o u n d p i p i n g systems

CP3009

Thermal l y i n s u l a t e d u n d e r g r o u n d p i p i n g systems

Valves

8s

1010

Taps a n d v a l v e s f o r water

1212Pt2

8 a l l valves - diaphragm type

1218 & 5163(m)

S I ui ce v a I ves

1415

V a l v e s f o r domestic p u r p o s e s

1952 & 51 54( m)

Copper a l l o y g a t e v a l v e s

1953

Copper a l loy check v a l v e s

2060

Copper a l l o y stop v a l v e s

2591

G l o s s a r y of v a l v e s

3464 & 5150(rn)

C

I wedge a n d d o u b l e d i s c v a l v e s

3948 & 5151 ( m )

C I parallel slide valves

3952 & 5155(m)

C

I butterfly valves

3961 & 5152(rn)

C I stop and check v a l v e s

4090 & 51 53( rn)

C

I check v a l v e s

4133 & 5157(rn)

F l a n g e d steel p a r a 1 l e l s l i d e v a l v e s

431 2

Stop a n d check v a l v e s

5156(m)

Screwdown d i a p h r a g m v a l v e s

5158(rn)

Plug valves

51 59(m)

Bal I valves

245 Jointing

10

Flanges a n d b o l t i n g f o r pipes etc.

21

Threads

1737

Jointing m a t e r i a l s a n d compounds

1821

Oxy-acetylene

welding of steel p i p e l i n e s

1965

B u t t we I d i ng

2494

Rubber j o i n t

3063

Dimensions of gaskets

4504

Flanges a n d b o l t i n g f o r pipes etc.

rings

Miscellaneous

1042

Flow measurement

1306

Non-ferrous pipes f o r steam

1553

Graphical symbols

1710

I d e n t i f i c a t i o n of p i p e l i n e s

205 1

Tube a n d p i p e f i t t i n g s

29’17

Graphical symbol s

3889

Non destructive testing of pipes a n d tubes

39 74

Pipe supports

4740

Control v a l v e c a p a c i t y

SOUTH AFRICAN BUREAU OF STANDARDS

Steel Pipe SABS 62

Cast

Steel pipes a n d f i t t i n g s up to 150 mm

71 9

E l e c t r i c a l welded low carbon steel p i p e s

720

Coated a n d l i n e d m i l d steel p i p e s

I r o n Pipe

50 9

Malleable C

746

C

81 5

Shouldered end pipes,

I soil,

I pipe f i t t i n g s

waste,

water a n d vent pipes f i t t i n g s and j o i n t s

Asbestos Cement Pipe

546

C

72 1

A C soil,

I

f i t t i n g s f o r A C pressure p i p e s waste and vent p i p e s a n d f i t t i n g s

246 946

A

C

pressure

pipe

286

A

C

pressure

pipes

819

A C sewer p i p e s

- constant - constant

internal

diameter

type

outside

diameter

type

Concrete Pipe 676

R C pressure p i p e s

677

Concrete non-pressure pipes

975

Prestressed concrete pipes

902

Structural

design

and

i n s t a l l a t ion of

precast

concrete

pipelines

Glazed Earthenware Pipe 559

Glazed earthenware d r a i n a n d sewer pipes a n d f i t t i n g s

P l a s t i c Pipe SABS 791

UPVC sewer a n d d r a i n p i p e f i t t i n g s

92 1

P i t c h impregnated f i b r e pipes

966

UPVC pressure p i p e

967

UPVC s o i l ,

997

UPVC pressure p i p e s f o r i r r i g a t i o n

waste and vent pipes

01 12

I n s t a l l a t i o n of PE a n d UPVC p i p e s

533

Black polyethylene p i p e s

Va I ves 144

C I s i n g l e door r e f l u x v a l v e s

191

Cast steel g a t e v a l v e s

664

C I gate v a l v e s

AMER I CAN WATER WORKS ASSOC I AT I ON

A W W AC20 1

F a b r i c a t i n g e l e c t r i c a l l y welded steel water p i p e

c202

M i l l type steel water p i p e

C203

Coal-tar

enamel

protective

coatings

for

steel

water

for

steel

water

pipe C205

Cement

mortar

p i p e of sizes 30"

protective a n d over

coatings

247 C206

F i e l d w e l d i n g o f steel water p i p e j o i n t s

C207

Steel p i p e f l a n g e s

C208

Dimensions f o r steel w a t e r p i p e f i t t i n g s

C300

R e i n f o r c e d c o n c r e t e w a t e r p i p e - Steel c y l i n d e r

C301

R e i n f o r c e d c o n c r e t e w a t e r p i p e - Steel c y l i n d e r

C302

R e i n f o r c e d c o n c r e t e w a t e r p i p e - Non-cyl i n d e r

type

-

not prestressed

t y p e - prestressed

type - not prestressed C600

-

54 T C602

I n s t a l l a t i o n of C I watermains Cement m o r t a r l i n i n g o f w a t e r p i p e l i n e s i n p l a c e

( 1 6"

and o v e r )

AMER ICAN PETROLEUM

API

INSTITUTE

T u b i n g and D r i l l P i p e

Std.5A

Spec.

f o r Casing,

Std.5AC

Spec.

f o r G r a d e C - 75 and C95 C a s i n g and T u b i n g

Std.5AX

Spec.

f o r H i g h S t r e n g t h C a s i n g and T u b i n g

Std.5L

Spec.

for Line Pipe

Std.5LA

Spec.

f o r S c h e d u l e 5 Alum.

Std.5LP

Spec.

for Thermoplastic L i n e Pipe

Std. 5LR

Spec.

for

Glass

Fibre

Alloy Line Pipe

Reinforced

Thermosetting

Resin

Line Pipe Std.5LS

Spec.

Std.5LX

Spec f o r H i g h Test L i n e P i p e

f o r S p i r a l Weld L i n e P i p e

RP5C I

C a r e and Use o f C a s i n g ,

RP5L I

R a i l r o a d t r a n s p o r t of L i n e Pipe

T u b i n g and D r i l l P i p e

RP5L2

I n t e r n a l C o a t i n g of L i n e P i p e f o r Gas T r a n s m i s s i o n

RP5L3

Conducting Drop Weight Tear Tests o n L i n e P i p e

Bul.5C2

Performance

Properties

of

Casing,

Pipe Bul.5T1

Non-destructive

Testing Terminology

Tubing

and

DrilI

248 AMER I CAN SOC I ETY FOR TEST I NG MATER I ALS

Concrete P i p e s ASTMC14 C76

Concrete Sewer, Storm D r a i n a n d C u l v e r t P i p e Reinforced

Concrete

Culvert,

Storm

Drain

and

Sewer

Pipe I r r i g a t i o n or Drainage

C118

Concrete P i p e f o r

C361

R e i n f o r c e d Concrete Low-Head

C412

Concrete Cjrain T i l e

c443

Joints

c444

P e r f o r a t e d Concrete P i p e

c497

Determining

for

Circular

Pressure Pipe

Concrete

Sewer

and

Culvert

Pipe

w i t h Rubber gaskets

Physical

Properties

of

Concrete

Pipe

or

T i le C505

Non-reinforced

Concrete

Irrigation

Pipe

and

Rubber

Gasket J o i n t s C506

Reinforced

Concrete

Arch

Culvert,

Concrete

El I i p t i c a l

Storm

Drain

and

Sewer P i p e C507

Reinforced

C655

Reinforced

Culvert,

Storm

Drain

a n d Sewer P i p e Concrete

D-load

Culvert,

Storm

Drain

and

Sewer P i p e

Steel P i p e s ASA

Cast

836.10

Steel p i p e s

i r o n Pipes A1 42

C

I Pipes

A377

C

I pipes

A121.1

C

I pipes

Asbestos Cement P i p e s C296

A C Pressure Pipes

C500

A C Pressure Pipes

C428

A

C

Pipes

and

Fittings

for

Sewerage

and

Drainage

249 Plastic Pipes

D2241

U n p l a s t i c i s e d PVC p i p e s

USDI BUREAU OF RECLAMATION S t a n d a r d Spec.

f o r Reinforced Concrete P r e s s u r e P i p e

250 BOOKS FOR FURTHER READING

Addison,

H.,

1964.

A

treatise on

Applied

Hydraulics,

Chapman H a l l ,

London Albertson,

M.L.,

Barton,

J.R.

a n i c s f o r E n g i n e e r s , P r e n t i c e - H a l I, American Pipe,

Concrete

Pipe

Simons,

and

D.B.,

F l u i d Mech-

1960.

Association,

1970.

Design

Manual-Concrete

Arlington.

Am.Soc.Civi I

Engs.

and W a t e r

Pol In.

Control

Federation,

and C o n s t r u c t i o n o f S a n i t a r y and Storm Sewers,

Am.Water

Works

M a n u a l M11, Bell,

1960.

Assn.,

1964.

Steel

Pipe

-

1970.

M a n u a l 37,

and

Design

Design

N.Y. Installation,

N.Y.

H.S.,

1963.

(Ed.),

Petroleum

Transportation

Handbook,

McGraw

H i l l , N.Y. Benedict,

R.P.,

1977.

F u n d a m e n t a l s of

Pipe Flow,

Wiley

Interscience,

531 pp. Bureau

of

Criteria

Public

for

Roads,

Structural

Was hi n g t o n ,

Of f i ce, Clarke,

N.W.B.,

1963.

Reinforced

and

Design

Concrete

Instal lation,

Proc.

State

Inst.,

-

Printing

Buried

Pipelines

-

Univ.,

1971.

Control

of

A

Manual

of

Structural

Closed

Conduits,

London. Flow

in

Fort Collins.

S.

Crocker,

Culverts

Govt.

DC. 1968.

D e s i g n a n d I n s t a l l a t i o n , M a c L a r e n and Sons, Colorado

Pipe

U.S.

and K i n g ,

R.C.,

1967.

Sorensen,

K.E.,

P i p i n g Handbook, 5 t h Ed.,

McGraw

Hill. Davis,

C.V.

raulics,

and

3 r d Ed.,

E.,

Holmes,

1973.

McGraw H i l l , Instn.

Nolte, cations, Rouse,

of

Applied

Hyd-

Handbook

of

Industrial

Pipework

Engineering,

Engs.,

1969.

Manual

of

British

Water

Engg.

Practice,

London.

Littleton, Martin,

Handbook

N.Y.

Water

4 t h Ed.,

1969.

McGraw H i l l , N.Y.

C.T.,

W.L., C.B.,

1962.

Industrial Piping,

1961.

H a n d b o o k of

1978.

Optimum

McGraw H i l l ,

349 pp

I n d u s t r i a l Pipework,

Pitman,

Pipe

Size

Selection,

Trans

297 p p . H.,

1961.

Engineering Hydraulics, Wiley,

N.Y.

London.

Tech

Publi-

251 Stephenson, Streeter, Hill,

D.,

1984.

V.L.,

Pipeflow Analysis,

1961.

(Ed),

Handbook

R.C.

and

E l s e v i e r , 204 pp. of

Fluid

Dynamics,

McGraw

N.Y.

Twort,

A.C.,

Hoather,

A. A r n o l d ,

2 n d Ed., Walski,

T.M.,

Walton,

J.H.,

Analysis

N.Y.

1970.

G.Z.,

pipelines. Young,

1984.

2 n d Ed.

D.C.

and

A p p l i e d Science,

Water

Supply,

of

Water

Distribution

Systems.

Van

275 pp.

Structural

P i p e Development Association, Watters,

1974.

London.

1984.

Nostrand Reinhold,

F.M.,

Law,

Design

of

Vitrified

Clay

Pipes,

Clay

London.

Analysis

and

control

of

unsteady

1984.

Buried

Rigid

flow

in

Butterworths. Trott,

London,

J.J., 230 pp.

Pipes.

Elsevier

252 APPEND I X

SYMBOLS FOR P I P E F I T T I N G S

1

GENERAL

Jo.

ELBOW SLEEVED

TEE

BEND JPtKETED FRONT VIEW OF TEE

1

BACK VIEW OF TEE

FILLET WELDEOTEE CROSS OVER

I

HANGER

I

BELLMOUTH

D

-

A-

YMPLE SUPWRT CHANGE IN D I A

TAPER JOINTS

E L E CT RICALLY BONDED

FLANGED

BMT W E L D

ELECTRIULLY INSULATED FLEXIBLE

SCREWED

SWIVEL

SOCKET

E X PANS ION

SPIGOT 0 SOCKET

--+_3_

-4_f_

D

END CAP

S L E E V E COUPLING

VALVES __

EUl TERFLY ISOLATING WEDGE

NEEDLE

GATE

GLOBE

REFLUX

DIAPHRAGM

ROTARY PLUG

RELIEF AIR

w

MISCELLANEOUS STRAINER

HYDRANT

SPRAY

FLOW INDICATOR

VENT

SURFACE BOX DRAIN

M 0 TOR

t

0

P L A T E BLIND HANDWHE E L

--+

J--

+ T

253

PROPERTIES OF P I P E SHAPES

Ring AREA

-

F(D2-d2kTDt

fa -11

I

MOMENT OF INERTIA ABOUT DIAMETER =

fi (0'-

d')

=TD"

MOMENT O F INERTIA ABOUT CENTRE g ( D b - d L ) Z s D a t

PROPERT I ES OF WATER

Temperature

Specific Mass

F"

kg/rn3

Ib/cu f t

0

32

1 000

62.4

10

50

1 000

62.4

20

68

999

30

86

997

C O

Kinematic Viscosity

B u l k Modulus

s q ft/sec

N/rnm2

psi

1.79~10-~

1 .93x 1 0-5

2 000

290 000

1 .31XI 0-6 6 1.01x10-

1.41~10-~

2 070

62.3

1.09~10-~

2 200

62.2

0.81 x10-6

0.87~1 0-5

2 240

325 000

rn2/ s

Vapour Pressure N/rnmz

psi

0.6~10-~

0.09

300 000

l.2~lO-~

0.18

318 000

2 . 3 ~ 1 0 ~ ~ 0.34 3 0.62 4.3~10-

PROPERTIES OF P I P E MATERIALS

Coef.

of

exp.per

5x10

Clay Concrete Asbestos cernen t Cast

Iron

M i l d Steet

O C

Density kg/m'

Modulus of elasticity N/mrn2

psi

Y i e l d Stress N/rnrn2

psi

Tensile Strength N/mm2

Poisson's

psi

ratio

-300

0.2

6

1ox10-6

2 600

14 000-40 000

8.5~ 1 0-6

2 500

24 000

8 . 5 ~ 0-6 1

2x106-6x106

6

-70

-10

000

-2.1

2 500

7 800

100 000

3.5~10 6 15x10

1 1 .9x10-6

7 850

210 000

3 1 ~ 1 0 ~

1 1 .9x10-6

7 850

210 000

31x10

900

138

20 000

3

955

240

35 000

22

3 200

32

4 600

8

1 100

17

2 500

0.5~10

40

+ 6 000

52

7 500

22 000

25

3 600

30

4 500

17 150

22 000

225

23 000

0.25

210

30 000

330

48 000

0.3

1 650

240 000

1730

250 000

400

14

2 000

High Tensile Steel Pitch Fibre Polyethylene

6

40x 1 0-6 1 60x 1 0-6

Po I y e t h y l e n e (high density)

2oox 1 o-6

PVC

50~10-~

1 300

UPVC

50~10-~

1 400

3 500

91 2

150

Po I y p r o p y I ene

1 8ox10-6

6

0.38

256 CONVERSION FACTORS

Length

1 i n c h = 25.4 mm

1

ft

=

0.3048 m

1 m i l e = 1.61 k m Area

1

644 mrn'

sq i n c h =

1 a c r e = 2.47 ha

Vol ume

0.0929 mz

1 sq ft

=

1 ha

lo4

=

mz

35.31 c u f t

1 m3

=

1 gal.(imperial)

4.54 I i t r e s

=

1 US g a l .

=

3.79 I i t r e s

1

=

42 US gal.

=

barrel

=

35 i m p . g a l .

159 I i t r e s 0.3048 m/s

Speed

1 ft/sec

Acceleration

32.2 f t / s e c 2

Discharge

1 mgd (imperial)

=

1 m p h = 1.61 k m / h r .

35.3 c u s e c

=

0.981 m/s' =

= g

1.86 cusec

im3/s

=

13.2 g p m ( i m p e r i a l ) = 1 P / s Mass

1 I b = 0.454 k g 32.2 I b = 1 s l u g (US)

Force

1 Ib. f o r c e = 4.45 N ( 1 Newton

1 k g x 9.81 m / s ' )

=

Pressure

145 p s i = 1 MPa

Energy

14.5 p s i = 1 b a r 778 f t 16. = 1 B t u 1 Btu

=

=

1 MN/m2

252 c a l o r i e s

1 c a l o r i e = 4.18 J o u l e s Power

550 f t

Ib/sec

=

1 HP

1 HP = 0.746 kW Kinematic Viscosity

1 sq ft/sec =

=

929 s t o k e s

0.0929 m z / s

Absolute o r dynamic viscosity

1 c e n t i p o i s e = 0.001 k g / m s

Temperature

F o = 32

+

1.8C"

=

1 N/mmz

257 A b s o l Ute temperature R " (Ranki ne)

= FO(Fahrenheit1

+ 460

= Co(Centigrade)

+ 273

A b s o l Ute temperature KO (Kelvin) 71

= 3.14159

e

=

2.71828

258

AUTHOR INDEX

Abrarnov, N. 15 A d d i s o n , H. 228, 241, 250 A l b e r t s o n , M . L . 17, 35, 250 A n d e r s o n , O.A. 125 A v e r y , S.J. 105, 110 B a l l , J.W. 28, 35 B a r n a r d , R.E. 149, 157 B a r t o n , J.R. 17, 35 B e l l , H.S. 250 B e n e d i c t , R.P. 250 B e r n o u l I i , 16 B e r t h o u e x , P.M. 10, 15 B l i s s , R.H. 101, 111 B o u c h e r , P . L . 182 203 B u r a s , N. 49, 57 C a p p e r , P . L . 123, 125, 191, 203 C a s s i e , W.E. 123, 125, 191, 203 C a t e s , W.H. 176, 77, 216, 226 C h a p t o n , H.J. 177 C h e z y , 18 C l a r k e , N.W.B. 113, 125, 250 Cole, E.S. 218, 226 Col e b r o o k , 23 C r o c k e r , S. 173, 177, 250 Cross, H. 38,57 D a n t z i g , G.B. 51, 57 D a r c y , 21 D a v i s , C.V. 250 D e n n y , 9 . F . 101, 110 D i s k i n , M.H. 24, 35 D v i r , Y . 30, 35 Ervine,

D.A.

105,

110

F o x , J.A.

109, 110 F e r g u s o n , P.M. 137,

J a c o b s o n , S. 155, Johnson, S.P. 95 60 Joukowsky

157

,

K a l i n s k e , A . A . 102, 104, K a l l y , E. 48, 57 K a r r a s s i k , I.J. 228, 240 K e n n e d y , H. 121, 125 K e n n e d y , J.F. 74, 95 K e n n i s o n , H.F. 129, 140 K i n g , C.L. 177 K i n g , R.C. 173, 177 K i n n o , H. 74,95 K n a p p , R. 28, 35 K o v a t s , A . 228, 241 L a i , C . 66, 95 L a m , C.F. 56, 57 L a w , F.M. 240, 250 L e s c o v i c h , J.E. 186, 203 L i t t l e t o n , C.J. 250 L u d w i g , H. 68, 95 L u p t o n , H.R. 64, 95

M a n n i n g , 18 M a r k s , L.S. 107, 1 1 1 Marston, A. 114, 125 M a r t i n , W . L . 250 M i l l s , K.G. 32, 43, 57 Moody, 21 M o r l e y , A . 154, 157 M o r r i s o n , E.B. 190, 203 N e l s o n , E.D. 177 N e w m a r k , 122 Newton, 16, 58 N i k u r a d s e , 21 N o l t e , C.B. 250

140 Osborne,

110 Glass, W.L. G o o d i e r , J.N. 177 G r i s t , E . 232, 240

J.M.

7,

15

H a s s a n , D . R . 165, 177 Hazen-VJ i I I i ams , 18 H o a t h e r , R.C. 240, 250 Holmes, E. 250

P a r m a k i a n , J. 73, 95, 186, 203 P e a r s o n , F.H. 177 P a u l , L. 182, 203 P o i s s o n , 176, 195 P r o c t o r , 150 P r o s s o r , 1vl.J. 101, 111

Isaacs,

R e i t z , H.M.

L.T.

43,

57

111

208,

226

108,

111,

259 R e y n o l d s , 20 R e y n o l d s , G.M. 211, 226 R i c h , G.R. 78, 95 R i d d i c k , T.M. 224, 226 R o a r k , R.J. 174, 177 R o b e r t s o n , J.M. 104, 111 Rouse, H. 166, 177, 250 S c h a r e r , H. 176, 177 S c h l i c h t i n g , H. 20, 35 S c h n e i d e r , W.R. 219, 226 S c h w a r t z , H . I . 211, 226 S c h w e i g , 2. 48, 57 Sirnons, D.B. 17, 250 Sorenson, K.E. 250 Sowers, G.F. 208, 226 S p a n g l e r , M.G. 113, 125, 146, 152, 157 S t e p a n o f f , A.J. 228, 241 Stephenson, D. 56, 57, 69, 80, 9 5 , 110, 1 1 1 , 151 157, 165, 177, 25 1 S t r e e t e r , V.L. 66, 95, 251 Suss, A. 165, 177 Swanson, H.S. 162, 177 Sweeton, A.E. 186, 203

TSSYJ,

Tirnoshenko, S . P . 134, 140, 158, T r o t t , J.J. 113, 125, 251 T u l l i s , J.P. 35 T w o r t , A.C. 240 Uhlig,

H.H.

218,

Van d e r Veen,

B.

226 51,

57

W a l s k i , T.M. 251 Walton, J.H. 250 Watson, M.D. 24, 35 W a t t e r s , G.Z. 251 Webber, N.B. 229, 241 Weisbach, 21, 109 W h i t e , 23 W h i t e , J.E. 9, 15 W i l k i n s o n , W.J. 177 W i n n , W.P. 33, 35 W i s n e r , P. 102, 111 W o i n o w s k i - K r i e g e r , S. 134, Wood, D.J. 43, 57 W y l i e , E.B. 66, 95 Young, Young,

160,

D.C. 113, 125, 251 G.A.J. 101, 110

140,

157

177

260

SUBJECT INDEX

A b s o r p t i o n 99 A c t i v e s o i l p r e s s u r e 114, 149, 191 A c t u a t o r 92, 185 A d i a b a t i c 86, 107 Aesthetics 2 A i r pocket 89, 100 v a l v e s 106, 189 vessel 87 American Concrete P i p e Assn. 122, 125, 140 Anode, s a c r i f i c i a l 219 A p p u r t e n a n c e s 182 Asbestos cement 180 A x i a l e x p a n s i o n 176, 195 B a c k f i l l 113, 192 B a r g e , 211 Beam 213 B e a r i n g t e s t 128 B e d d i n g 115. 127 B e d d i n g f a c t o r 128, 207 Bend loss 27 meter 200 s t r e s s 173 B e n d i n g 144, 196, 213 B i n a r y n u m b e r 202 B i t u m e n 217 B o n i n g 206 B o u n d a r y l a y e r 22 B r a c i n g 161 B r a n c h 62, 68, 161 B r i d g e 208 B u b b l e 99 Buck1 i n g 152, 168 B u t t e r f l y v a l v e 27, 32, 184 B y p a s s 76, 79, 182 Cant i l e v e r 174 Capacity factor 4 Cash f l o w 1 1 C a s i n g 217 Cast i r o n 180 C a t h o d i c p r o t e c t i o n 218 C a v i t a t i o n 30, 97 C e l e r i t y 60 C e n t r i f u g a l f l o w 200 f o r c e 190 pump 229 C h a r a c t e r i s t i c method 66 pump 70, 73, 229

Chemicals 1 C l R l A 152 Clamp 212 C l a y 255 C o a t i n g , bi tumen 21 6 c o a l t a r 217 epoxy 217 g l a s s f i b r e 217 m o r t a r 23, 217 t a p e 217 Cohesion 191 Col l a p s e 148 C o l l a r 143, 161 Concrete p i p e s 127, 180 p r e s t r e s s e d - 129 180 p r o p e r t i e s 137, 255 Compound p i p e s 37 Compressibi I i t y 16, 60 Computers, a n a l o g u e 202 d i g i t a l 41, 202 Concrete P i p e Ass. 122, 125 C o n d u c t i v i t y 219, 223 Conductor 219 Cone v a l v e 30, 184 Consol i d a t i o n 149 C o n s t r a i n t 52 Container 2 C o n t r a c t i o n c o e f f i c i e n t 200 Convect i o n 224 Conversion f a c t o r s 256 Core 129 C o r r o s i o n , g a l v a n i c 218 Cost, p i p e 3 power 4, 12 p u m p i n g 4, 8 C o u n t e r b a l a n c e 186 C o u p l i n g 230 C r a c k 127 Creep 130 Cross section 253 C r o t c h p l a t e 161 C r u s h i n g 137 Current, electric219 impressed21 9 stray220 C y l i n d e r 129 D e f l e c t i o n 146, 150 Design f a c t o r 142 O i ameter 4

261 D i s c h a r g e c o e f f i c i e n t 200 t a n k 79 Discount r a t e 9 D i s s o l v e 99 D r a i n 207 D y n a m i c p r o g r a m m i n g 45 Economics 2 E l a s t i c i t y 64, 255 E l e c t r o d e 220 E I ec tro-chemi c a l e q u i v a l e n t 220 E l e c t r o l y t e 218 E l e c t r o m a g n e t i c i n d u c t i o n 201 El l i p s e 253 Embankment 1 1 6 Ernrnissivity 224 E m p i r i c a l 18 E n e r g y 16 E n t r a n c e l o s s 27 Epoxy 217 E q u i v a l e n t d i a m e t e r 37 Escher Wyss 177 E x c a v a t i o n 206 E x t e r n a l l o a d 115, 146 F a b r i c a t e d b e n d 194 F a c t o r o f s a f e t y 143 F i e l d p r e s s u r e 133 F l a n g e 212 b l a n k 216 p u d d l e 216 F l e x i b l e p i p e 142 F l o a t i n g 210 F i t t i n g s 182 Flow measurement 198 b e n d meter 200 e l e c t r o m a g n e t i c 201 mass 201 m e c h a n i c a l 200 n o z z l e 199 o r i f i c e 199 v e n t u r i 198 volume 202 Flow r e v e r s a l 75, 91 F l y w h e e l 76 F r i c t i o n 18, 68, 74 F r o u d e n u m b e r 104 G a l v a n i c a c t i o n 218 Gasket 215 Gate v a l v e 182 Globe v a l v e 184 G r a p h i c a l a n a l y s i s 14, 65

H a u n c h 128, 148 Head, loss 16, 37, 199, 232 v e l o c i t y - 16, 232 w a t e r hammer- 62 H e a d i n g 208 Heat loss 224 History 1 Hole 161 Hol i d a y d e t e c t o r 21 7 H y d r a u l i c g r a d i e n t 18, 68, 238 H y d r a u l i c j u m p 104 I m p a c t f a c t o r 121 I m p e l l e r 229 I n e r t i a 73, 94 I n f l u e n c e c o e f f i c i e n t 122 I n s u l a t ion, t herrna I 224 I n t e r e s t 10 I sotherrnal 86 J o i n t , b u t t - w e l d e d 212 clamp-on 214 f a c t o r 142 f l a n g e d 215 screwed 213 sleeve w e l d e d 212 s p i g o t a n d socket 130, 214 K i n e t i c e n e r g y 72 L a g g i n g 224 L a m i n a r f l o w 20 L a t e r a l s u p p o r t 149 L a y i n g 205 L i n e l o a d 122 L i n e a r p r o g r a m m i n g 52 L i n i n g , b i t u m e n 217 c o a l t a r 217 epoxy 218 m o r t a r 21 8 L i v e l o a d 120 L o a d c o e f f i c i e n t 114 soi I 113 superimposed 120 Longitudinal stress 21 3 Loop 38 Loss c o e f f i c i e n t 27 Losses 26 Mass, c o n s e r v a t i o n of 16 M a t e r i a l s 255 M e c h a n i c a l meter 200

134,

198,

262 Membrane t h e o r y 167 Meter b e n d 200 rnkchan i c a l 200 n o z z l e 199 o r i f i c e 199 r o t 0 200 v a n e 200 v e n t u r i 198 Model 165 M o d u l u s of e l a s t i c i t y 137, 255 M o d u l u s of s u b q r a d e r e a c t i o n 124 Mohr d i a g r a m 136 Momentum 16, 90 Motor 239 Needle v a l v e 91, 184 Network 38 Net p o s i t i v e s u c t i o n h e a d 232 Node 39 Non-l i n e a r p r o g r a m m i n g 56 Normal t h r u s t 190 Nozzle 199 O b j e c t i v e f u n c t i o n 56 Ohm 219 Oil 1 Operating factor 4 O p t i m i z a t i o n 8, 45 O r i f i c e 86, 199 P a r t l y f u l l p i p e 100 P a s s i v e s o i l p r e s s u r e 191 Pavement 123 P e r i m e t e r , w e t t e d 102 P i p e , asbestos cement 180, 255 c a s t i r o n 179, 255 c o n c r e t e 180, 255 p l a s t i c 180, 255 p o l y e t h y l e n e 180, 255 PVC 180, 255 p r o p e r t i e s , 255 s a l t g a l v a n i z e d c l a y w a r e 255 steel 179, 255 UPVC 180, 255 Planning 1 P l a s t i c 180 P o i n t l o a d 121 P o i s s o n ' s r a t i o 144, 195, 255 P o l a r i z a t i o n 218 P o l y e t h y l e n e 180, 255 P o t e n t i a l e n e r g y 90 Power 236 Present v a l u e 10

P r e s s u r e , e x t e r n a l 110, 146 i n t e r n a l 142 w a t e r hammer 58 P r e s t r e s s e d c o n c r e t e p i p e 129 P r o c t o r d e n s i t y 150, 207 Profile, pipeline 72, 106, 238 P r o t e c t i o n , c a t h o d i c 218 w a t e r hammer 69 P u d d l e f l a n g e 216 Pump, b o o s t e r 2, 9 c e n t r i f u g a l 75, 229 i n e r t i a 73 power 5 t y p e s 228 P u m p i n g c o s t s 3, 12 R a d i a l s t r e s s 142 R a d i a t i o n 224 R a d i o 202 R a d i u s o f s t i f f n e s s 124 Rail 1 Reducer 169 R e f l u x v a l v e 71, 81, 186 R e i n f o r c e d c o n c r e t e 1 , 127 R e i n f o r c i n g 127, 143, 162 R e l a x a t i o n o f s t r e s s 131 Release v a l v e 91, 184 R e s e r v o i r 14 R e s i s t i v i t y 219 R e t i c u l a t i o n n e t w o r k 37 R e y n o l d ' s n u m b e r 20 R i b b e d p i p e 154 R i g i d pavement 123 p i p e 113 R i n g g i r d e r 175 l o a d 149 s t i f f e n i n g 154 s t r e s s 142 t e n s i o n 142 Road 1 Roughness 21 Route 205 S a c r i f i c i a l anode 219 S a d d l e 174 S a l t g l a z e d 255 S a n d 192 Scale 6 S c o u r i n g 210 Screwed j o i n t 213 Sea 210

205,

263 Secondary s t r e s s 160 S e c t i o n m o d u l u s 174 S e r v i c e s 205 Settlement 118 Shape 253 S h r i n k a g e 135 S i n k i n g fund 9 S l u i c e v a l v e s 182 Snake 206 Socket j o i n t 130, 214 S o i l l o a d 115, 146 m e c h a n i c s 193, 208 p r o p e r t i e s 113, 192 Span 213 S p e c i f i c mass 254 S p e c i f i c speed 223 S p h e r i c a l v a l v e 185 S p i n d l e 182 S t a n d a r d s 242 S t a r t e r s 239 Steady f l o w 16 Steel, high t e n s i l e 136, 255 p i p e 180 p r e s t r e s s i n g 136 p r o p e r t i e s 255 S t i f f e n i n g r i n g s 154 S t r a y c u r r e n t e l e c t r o l y s i s 219 S t r e n g t h , l a b o r a t o r y 137 Stress, b e n d i n g 196, 213 S t r e s s , c i r c u m f e r e n t i a l 133, 142 Stress, l o n g i t u d i n a l - 134, 198, 213 t e m p e r a t u r e - 21 5 S t r u t t i n g 149, 154 S u l z e r 177 Superimposed l o a d 120 S u p p o r t 128, 148, 194 S u r f a c e l o a d 120 S u r g e 58 - t a n k 77 Symbols 252 Systems a n a l y s i s 44 Te I erne t r y 202 T e m p e r a t u r e 195, 215 T e n s i o n r i n g s 143 T e s t i n g 206 T h e r m a l i n s u l a t i o n 224 T h e r m o p l a s t i c 180 T h i c k n e s s 142, 162, 224 T h r o t t l i n g 28, 184 T h r u s t b l o c k 192 b o r e 208 T i e b o l t 214 T o r q u e 235

T o w i n g 210 T r a f f i c l o a d 121 T r a n s i e n t p r e s s u r e s 60 Transport and Roads Laboratory 152 Transportation, a i r 1 p r o g r a m m i n g 45 rail 1 road 1 waterway 1 T r e n c h 113, 206 T u r b i n e 77

Research

Uncertainty 1 1 U n d e r w a t e r l a y i n g 210 U n i f o r m l o a d 122 Unp I a s t i c i z e d p o l y v i n y Ic h l o r i d e 180, 255 Vacuum 97, 153 V a l v e , a i r r e l e a s e 106, 188 a i r v e n t 186 b u t t e r f l y 183 cone 184 c o n t r o l 29, 184 g l o b e 184 n e e d l e 184 r e f l u x 71, 81, 186 r e l e a s e 91, 184 s p h e r i c a l 185 s l e e v e 185 s l u i c e 182 Vane, g u i d e 165 V a p o r i z a t i o n 90, 97, 153 V e l o c i t y 16, 58, 199 V e n t u r i m e t e r 198 V i c t a u l ic coup1 i n g 194 V i k i n g l o h n s o n c o u p l i n g 194 V i s c o s i t y 23, 255 V o l u t e 229 V o r t e x 89 W a l l t h i c k n e s s 142, 162, 224 Water hammer 60 p r o p e r t i e s 254 s u p p l y 14 Wave 60 Web 162 W e l d i n g 212 'Wetted p e r i m e t e r 148 W r a p p i n g 216 X-ray

217

Y i e l d s t r e s s 152, 255

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