PIPELINE DESIGN FOR WATER ENGINEERS

DEVELOPMENTS IN WATER SCIENCE, 6

aduisory editor

VEN TE CHOW

Professor of Hydraulic Engineering Hydrosystems Laboratory University of Illinois Urbana, I l l . , U.S.A.

FURTHER TITLES IN THIS SERIES

1 G. BUGLIARELLO AND F. GUNTHER COMPUTER SYSTEMS AND WATER RESOURCES

2 H.L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY

3 Y.Y. HAIMES, W.A. HALL AND H.T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF METHOD

4 J.J. FRIED GROUNDWATER POLLUTION

5 N. RAJARATNAM TURBULENT JETS

PIPELINE DESIGN FOR WATER ENGINEERS

DAVID STEPHENSON

ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxford - New York 1976

ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 211, Amsterdam, The Netherlands

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue New York, New York 10017

ISBN: 0-444-41417-7

Copyright @ 1976 by Elsevier Scientific Publishing Company, Amsterdam

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 Scientific Publishing Company, Jan van Galenstraat 335, Amsterdam

Printed in The Netherlands

V

P R E F A C E

Pipelines are being constructed in ever-increasing diameters, lengths and working pressures. Accurate and rational design bases are essential to achieve economic and safe designs. Engineers have for years resorted to semi- empirical design formulae. Much work has recently been done in an effort to rationalize the design of pipelines.

This book collates published material on rational design methods as

well as presenting some new techniques and data. Although retaining conventional approaches in many instances, the aim of the book is to bring

the most modern design techniques to the civil or hydraulic engineer. It is

suitable as an introduction to the subject but also contains data on the most advanced techniques in the field. Because of the sound theoretical

background the book will also be useful to under-graduate and post-graduate

students. Many of the subjects, such as mathematical optimization, are still in their infancy and the book may provide leads for further research.

The methods of solution proposed for many problems bear in mind the modern

acceptance of computers and calculators and many of the graphs in the book were prepared with the assistance of computers.

The first half of the book is concerned with hydraulics and planning of pipelines. In the second half, structural design and ancillary features are discussed. The book does not deal in detail with manufacture, laying and operation, nor should it replace design codes of practice from the engineer's desk. Emphasis is on the design of large pipelines as opposed to

industrial and domestic piping which are covered i n other publications. Although directed at the water engineer, this book will be of use to engin-

eers involved in the piping of many other fluids as well as solids and gases. It should be noted that some of the designs and techniques described

may be covered by patents. These include types of prestressed concrete pipes, methods of stiffening pipes and branches and various coatings.

The S.I. system of metric units is preferred in the book although imperial units are given in brackets in many instances. Most graphs and

equations are presented in universal dimensionless form. Worked examples are given for many problems and the reader is advised to work through these

as they often elaborate on ideas not highlighted in the text. The algebraic

symbols used in each chapter are summarized at the end of that chapter

together with specific and general references arranged in the order of the

subject matter in the chapter. The appendix gives further references and standards and other useful data.

v i

A C K N O W L E D G E M E N T S

The b a s i s f o r t h i s book w a s d e r i v e d from my expe r i ence and i n t h e

cour se of my d u t i e s w i t h t h e Rand Water Board and my p r e s e n t employers,

S t ewar t , S v i r i d o v 6, O l i v e r , Consu l t ing Engineers . The e x t e n s i v e knowledge

of Engineers i n t h e s e o r g a n i z a t i o n s may t h e r e f o r e be r e f l e c t e d h e r e i n

a l though I am s o l e l y t o blame f o r any i n a c c u r a c i e s o r misconcept ions .

I am g r a t e f u l t o my w i f e Les l ey , who, i n a d d i t i o n t o look ing a f t e r

t h e twins du r ing many a l o s t weekend, a s s i d u o u s l y typed t h e f i r s t d r a f t

of t h i s book.

David Stephenson

v i i

CONTENTS

1 ECONOMIC PLANNING, 1

I n t r o d u c t i o n , 1 P i p e l i n e economics, 2 Basics of economics, 6

Methods of a n a l y s i s , 8 Unce r t a in ty i n f o r e c a s t s , 8

Balancing s to rage , 11

2 HYDRAULICS, 13

The fundamental equa t ions of f l u i d flow, 13 Flow-head loss r e l a t i o n s h i p s , 14

Conventional f low formulae, 14 Rat iona l flow formulae, 16

Non-circular c r o s s s e c t i o n s , 20 P a r t l y f u l l p ipes , 22 Minor l o s s e s , 24

3 PIPELINE SYSTEM ANALYSIS AND DESIGW, 27

Network a n a l y s i s , 27 Equivalent p i p e s f o r p i p e s i n s e r i e s o r p a r a l l e l , 27 The loop method f o r ana lys ing networks, 28 The node method f o r ana lys ing networks, 29 A l t e r n a t i v e methods o f a n a l y s i s , 30

Dynamic programming f o r op t imiz ing compound p ipes , 34 Transpor t a t ion programming f o r l e a s t c o s t a l l o c a t i o n

Linear programming f o r design of l e a s t c o s t open

S t e e p e s t pa th a scen t technique f o r extending networks, 44 Design o f looped networks, 47

Optimizat ion of p i p e l i n e systems, 33

o f r e sources , 37

networks, 4 0

4 WATER HAMMER AND SURGE, 53

Rigid water column su rge theory, 53 E l a s t i c water hammer theory, 54

Method of a n a l y s i s , 56 E f f e c t of f r i c t i o n , 58

P r o t e c t i o n of pumping l i n e s , 6 0 Pump i n e r t i a , 6 2 Pump bypass r e f l u x va lve , 65 Surge tanks, 66 Discharge tanks, 67 A i r v e s s e l s , 73 I n - l i n e r e f l u x va lves , 77 Release v a l v e s , 78 Choice o f p r o t e c t i v e device, 8 0

v i i i

INTRODUCTION TO SOLIDS PIPING, 84

I n t r o d u c t i o n , 84 Flow reg imes , 84 Heterogeneous f low c o n d i t i o n s , 85

Head l o s s e s , 85 Minimum head l o s s v e l o c i t y , 8 7 C r i t i c a l d e p o s i t v e l o c i t y , 87 Drag c o e f f i c i e n t and e f f e c t o f p a r t i c l e s i z e , 88 T r a n s p o r t i n v e r t i c a l p i p e s , 89 T r a n s p o r t i n i n c l i n e d p i p e s , 90

Homogeneous s l u r r i e s , 90 P i p e a b r a s i o n and p a r t i c l e a t t r i t i o n , 9 1 Economics, 91 T r a n s p o r t i n c o n t a i n e r s , 93

EXTERNAL LOADS, 96

S o i l l oads , 96 Trench c o n d i t i o n s , 96 Embankment c o n d i t i o n s , 99

Superimposed loads , 102 T r a f f i c l oads , 103 Stress caused by p o i n t l o a d s , 103 Line loads , 104 Uniformly loaded a r e a s , 104 E f f e c t of r i g i d pavements, 105

E f f e c t of l a t e r a l s u p p o r t , 108 Bending and deformat ion o f c i r c u l a r p i p e s under load , 106

CONCRETE PIPES, 112

The e f f e c t o f bedding , 1 1 2 P r e s t r e s s e d c o n c r e t e p i p e s , 114

C i r c u m f e r e n t i a l p r e s t r e s s i n g , 11 5 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 l o s s e s , 116 C i rcumfe ren t i a l s tress under f i e l d p r e s s u r e , 1 1 7 Long i tud ina l p r e s t r e s s i n g , 118 Long i tud ina l s t r e s s e s a f t e r l o s s e s , 120 P r o p e r t i e s o f s t e e l and c o n c r e t e , 1 2 1

STEEL AND FLEXIBLE PIPE, 126

I n t e r n a l p r e s s u r e s , 126 S t i f f e n i n g r i n g s , 127

Tension r i n g s t o res i s t i n t e r n a l p r e s s u r e , 127 S t i f f e n i n g r i n g s t o res is t b u c k l i n g under e x t e r n a l l oad , 130 Allowance f o r c i r c u m f e r e n t i a l bending , 132

SECONDARY STRESSES, 139

S t r e s s e s a t b ranches , 139 Crotch p l a t e s , 139 I n t e r n a l b rac ing , 142

S t r e s s e s a t bends, 150

i x

The p i p e as a beam, 151 Long i tud ina l bending, 151 P ipe s t r e s s a t s a d d l e s , 152 Ring g i r d e r s , 153

Temperature s t r e s s e s , 153

10 PIPES, FITTINGS AND APPURTENANCES, 156

P ipe m a t e r i a l s , 156 S t e e l p ipe , 156 Cas t i r o n p ipe , 156 Asbes tos cement p ipe , 157 Concre te p ipe , 157 P l a s t i c p ipe , 157

S l u i c e va lves , 159 B u t t e r f l y va lves , 160 Globe va lves , 160 Needle and c o n t r o l va lves , 161 S p h e r i c a l v a l v e s , 162 Ref lux v a l v e s , 162

A i r v e n t va lves , 163 A i r r e l e a s e v a l v e s , 164

Line va lves , 158

A i r v a l v e s , 162

Thrus t b locks , 165 Flow measurement, 169

Ven tu r i me te r s , 169 Nozzles , 170 O r i f i c e s , 170 Bend me te r s , 171 Mechanical meters, 1 7 1 E 1 ec tromagne t i c i n d u c t ion , 1 7 2 Mass and volume measurement, 172

Telemet ry , 172

11 L A Y I N G AND PROTECTION, 175

S e l e c t i n g a r o u t e , 175 Laying and t r ench ing , 175 Thrus t bo res , 177 P ipe b r idges , 179 Underwater p i p e l i n e s , 179 J o i n t s and f l a n g e s , 180 Coat ings , 184 Linings , 185 Cathodic p r o t e c t i o n , 186

Galvanic co r ros ion , 186 S t r a y c u r r e n t e l e c t r o l y s i s , 189

Thermal i n s u l a t i o n , 190

1 2 SEWERS, 195

Design c r i t e r i a , 195 Cons t ruc t ion , 196 Access and v e n t i l a t i o n , 199 Corros ion , 200 Computer grad ing , 201

X

GENERAL REFERENCES AND STANDARDS, 202

APPENDIX, 210

Symbols f o r p i p e f i t t i n g s , 210 P r o p e r t i e s o f p i p e s h a p e s , 211 P r o p e r t i e s o f w a t e r , 212 P r o p e r t i e s o f p i p e materials, 213 Convers ion f a c t o r s , 214

AUTHOR I N D E X , 215

SUBJECT INDEX, 217

1

CHAPTER I

E C O N O M I C P L A N N I N G

INTRODUCTION

Pipes have been used for many centuries for transporting fluids. The

Chinese first used bamboo pipes thousands of years ago, and lead pipes were

unearthed at Pompeii. In later centuries wood-stave pipes were used in Eng-

land. It was only with the advent of cast iron, however, that pressure pipe-

lines were manufactured. Cast iron was used extensively in the 19th Century

and is still used. Steel pipes were first introduced towards the end of the

last century, facilitating construction of small and large bore pipelines.

The increasing use of high grade steels and large rolling mills has enabled

pipelines with diameters over 3 metres and working at pressures over 10

Newtons per square millimetre to be manufactured. Welding techniques have

been perfected enabling longitudinally and circumferentially welded or

spiral-welded pipes to be manufactured. Pipelines are now also made in re-

inforced concrete, pre-stressed concrete, asbestos cement, plastics and claywares, to suit varying conditions. Reliable flow formulae became avail-

able for the design of pipelines this century, thereby also promoting the

use of pipes.

Prior to this century water and sewage were practically the only fluids

transported by pipeline. Nowadays pipelines are the most common means for

transporting gases and o i l s over long distances. Liquid chemicals and solids in slurry form or in containers are also being pumped through pipelines on

ever increasing scales. There are now over two million kilometres of pipe-

lines in service throughout the world. The global expenditure on pipelines

in 1974 was probably over f5 000 million.

There are many advantages o f pipeline transport compared with other

forms of transport such as road, rail, waterway and air:-

(1) Pipelines are often the most economic form of transport (considering either capital costs, running costs or overall costs).

( 2 ) Pipelining costs are not very susceptible to fluctuations in prices,

since the major cost is the capital outlay and subsequent operating costs are relatively small.

( 3 ) Operations are not susceptible to labour disputes as little attendance is required. Many modern systems operate automatically.

Being hidden beneath the ground a pipeline will not mar the natural

environment.

A buried pipeline is reasonably secure against sabotage.

A pipeline is independent of external influences such as traffic con-

gestion and the weather.

There is normally no problem of returning empty containers to the

source.

It is relatively easy to increase the capacity of a pipeline by in-

stalling a booster pump.

A buried pipeline will not disturb surface traffic and services.

Wayleaves for pipelines are usually easier to obtain than for roads

and railways.

The accident rate per ton - km is considerably lower than for other forms of transport.

A pipeline can cross rugged terrain difficult for vehicles to cross.

There are of course disadvantages associated with pipeline systems:-

(1) The initial capital expenditure is often large, so if there is any uncertainty in the demand some degree of speculation may be necessary.

(2) There is often a high cost involved in filling a pipeline (especially l o n g fuel lines).

( 3 ) Pipelines cannot he used for more than one material at a time (although there are multi-product pipelines operating on batch bases).

( 4 ) There are operating problems associated with the pumping of solids,

such as blockages on stoppage.

(5) It is often difficult to locate leaks or blockages.

P I P E L I N E ECONOMICS

The main cost of a pipeline system is usually that of the pipeline it-

self. The pipeline cost is in fact practically the only cost for gravity

systems hut as the adverse head increases so the power and pumping station

costs increase.

Table 1.1 indicates some relative costs for typical installed pipelines.

With the economic instability and rates of inflation prevailing at the

time of writing pipeline costs may increase by 20% or more per year, and

relative costs for different materials will vary. In particular the cost of

petro-chemical materials such as PVC may increase faster than those of

concrete for instance, so these figures should be inspected with caution.

3

TABLE 1.1 RELATIVE PIPELINE COSTS

P i p e M a t e r i a l _-- PVC

A s b e s t o s cement

R e i n f o r c e d c o n c r e t e

P r e s t r e s s e d c o n c r e t e

M i l d s t e e l

H i g h t e n s i l e s t e e l

Cast i r o n

150

6

7

-

- 10 11

25

Bore mm

4 5 0 1 500

23 - 23 -

23 80

33 9 0 - 150

28 100 - 180 25 9 0 - 120

75

;,< 1 1 - 1 1 i n d i c a t e s n o t r e a d i l y a v a i l a b l e . 1 u n i t = 1 f / m e t r e i n 1 9 7 4 u n d e r a v e r a g e c o n d i t i o n s .

The components making u p t h e c o s t o f a p i p e l i n e v a r y w i d e l y f r o m s i t u -

a t i o n t o s i t u a t i o n b u t f o r w a t e r p i p e l i n e s i n o p e n c o u n t r y a n d t y p i c a l con-

d i t i o n s a re a s f o l l o w s : -

S u p p l y o f p i p e -

E x c a v a t i o n -

L a y i n g a n d j o i n t i n g - F i t t i n g s a n d s p e c i a l s - C o a t i n g a n d w r a p p i n g - S t r u c t u r e s ( v a l v e c h a m b e r s , a n c h o r s ) - W a t e r hammer p r o t e c t i o n - Land a c q u i s i t i o n , a c c e s s r o a d s ,

c a t h o d i c p r o t e c t i o n , s e c u r i t y

s t r u c t u r e s , f e n c e s

E n g i n e e r i n g a n d s u r v e y c o s t s - A d m i n i s t r a t i v e c o s t s - J - n t e r e s t d u r i n g c o n s t r u c t i o n -

55% (may r e d u c e as new m a t e r i a l s

a r e d e v e l o p e d )

20% ( d e p e n d s on t e r r a i n , may r e d u c e

a s m e c h a n i c a l e x c a v a t i o n t e c h -

n i q u e s i m p r o v e )

5% (may i n c r e a s e w i t h l a b o u r c o s t s )

5 7-

2 %-

2 "/,

1 "/,

1 v

Many f a c t o r s h a v e t o b e c o n s i d e r e d i n s i z i n g a p i p e l i n e : F o r w a t e r

pumping m a i n s t h e f l o w v e l o c i t y a t t h e optimum d i a m e t e r v a r i e s f r o m 0 . 7 m / s t o 2 m/s, d e p e n d i n g on f l o w a n d w o r k i n g p r e s s u r e , It i s a b o u t 1 m / s f o r low p r e s s u r e p i p e s a t a f l o w o f 100 !/s i n c r e a s i n g t o 2 m / s f o r a f l o w of 1 000

! / s a n d p r e s s u r e h e a d s o f a b o u t 400 m o f w a t e r , a n d may b e e v e n h i g h e r f o r

4

higher pressures. The capacity factor and power cost structures also in- fluence the optimum flow velocity or conversely the diameter for any par-

ticular flow. Fig. 1.1 illustrates the optimum diameter of water mains for typical conditions.

In planning a pipeline system it should be borne in mind that the scale

of operation of a pipeline has considerable effect on the unit costs. By

doubling the diameter of the pipe, other factors such as head remaining con-

stant, the capacity increases six-fold. On the other hand the cost approxi- mately doubles so that the cost per unit delivered decreases to 1/3 o f the original. It is this scale effect which justifies multi-product lines. Whether it is, i n fact, economical to install a large diameter main at the outset depends on the following factors as well as scale:-

(1) Rate of growth in demand (it may be uneconomical to operate at low capacity factors during initial years). (Capacity factor is the ratio of actual average discharge to design capacity).

( 2 ) Operating factor (the ratio of average throughput at any time to maximum throughput during the same time period), which will depend on the rate of draw-off and can be improved by installing storage at the consumer's end.

(3) Reduced power costs due to low friction losses while the pipeline is not operating at f u l l capacity.

( 4 ) Certainty of future demands. ( 5 ) Varying costs with time (both capital and operating). ( 6 ) Rates of interest and capital availability. ( 7 ) Physical difficulties in the construction of a second pipeline if re-

quired. The optimum design period of pipelines depends on a number of factors,

not least being the rate of interest on capital loans and the rate of cost

inflation, in addition to the rate of growth, scale and certainty of future demands.

In waterworks practice' it has been found economic to size pipelines for demands up to 10 to 30 years hence. For large throughput and high growth

rates, technical capabilities may limit the size of the pipeline, so that supplementation may be required within 10 years. Longer planning stages are normally justified for small bores and low pressures.

It may not always be economic to lay a uniform bore pipeline. Where

pressures are high it is economic to reduce the diameter and consequently the wall thickness.

.- Flow- l i t res/ sec

FIG. 1.1 Optimum pumping main d i a m e t e r s f o r a p a r t i c u l a r \ s e t o f c o n d i t i o n s .

6

In planning a trunk main with progressive decrease in diameter there

may be a number of possible combinations of diameters. Alternative layouts should be compared before deciding on the most economic. Systems analysis

techniques such as linear programming and dynamic programming are ideally

suited for such studies. Booster pump stations may be installed along lines instead of pumping -

to a high pressure head at the input end and maintaining a high pressure along the entire line. By providing for intermediate booster pumps at the

design stage instead of pumping to a high head at the input end, the pressure heads and consequently the pipe wall thicknesses may be minimized. There may be a saving in overall cost, even though additional pumping stations are re- quired. The booster stations may not be required for some time.

The capacity of a pipeline may often be increased by installing booster

pumps at a later stage although it should be realised that this is not always

economic. The friction losses along a pipeline increase approximately with

the square of the flow, consequently power losses increase considerably for

higher flows. Fig. 1.2 illustrates some important points to be considered in planning

a pumping pipeline:-

(1) At any particular throughput Q, there is a certain diameter at which overall costs will be a minimum (in this case D ) . At this diameter the cost per ton of throughput could be reduced further if throughput was increased. Costs would be a minimum at some

throughput Q,. Thus a pipeline's optimum throughput is not the same as

the throughput for which it is the optimum diameter.

If Q , were increased by an amount Q, so that total throughput Q, = Q, + Q,, it may be economic not to install a second pipeline (with optimum diameter D ) but to increase the flow through the pipe with diameter D 2 , i.e. Q4C4 is less than Q,C, + Q3C3.

The power cost per unit of additional throughput decreases with in-

2 (2)

( 3 )

3 ,

creasing pipe diameter so the corresponding likelihood of it being most economic to increase throughput through an existing line increases (Ref.l.1).

BASICS OF ECONOMICS

Economics is used as a basis for comparing alternative schemes or de- signs. Different schemes may have different cash flows necessitating some rational form of comparison. The crux of all methods of economic comparison

7

FIG. 1.2 Cost - throughput curves for different diameter pipes.

is the discount rate, which may be in the form of the interest rate on loans or redemption funds. National projects may require a discount rate different from the prevailing interest rate, to reflect a time rate of preference, whereas private organisations will be more interested in the actual cash flows, and consequently use the real borrowing interest rate.

The cash flows, i.e. payments and returns, of one scheme may be compared with those of another by bringing them to a common time basis. Thus all cash flows may be discounted to their present value. For instance one pound re- ceived next year is the same as f1/1.05 (its present value) this year if it could earn 5% interest if invested this year. It is usual to meet capital expenditure from a loan over a definite period at a certain interest rate, Provision i s made for repaying the loan by paying into a sinking fund which

also collects interest. The annual repayments at the end of each year re-

quired to amount to 1 in n years is

r

(I+r) - 1 (1.1) where r is the interest rate on the payments into the sinking fund. If the interest rate on the 1oan.is R, then the total annual payment is

R (l+r) + r - R (1+r) - 1 (1.2)

Normally the interest rate on the loan is equal to the interest rate earned by the sinking fund so the annual payment on a loan of 1 is

r (l+r)n

Conversely t h e p re sen t va lue of a payment of E l a t t h e end of each yea r over

n yea r s i s

( l + r ) n - 1 r(1+rIn

The p resen t va lue o f a s i n g l e amount of E l i n n y e a r s t ime i s

1

( l + r ) "

( 1 . 4 )

( 1 . 5 )

I n t e r e s t t a b l e s a r e a v a i l a b l e f o r de t e rmin ing t h e annual payments on

l o a n s , and t h e p re sen t v a l u e s of annua l payments o r r e t u r n s , f o r v a r i o u s

i n t e r e s t r a t e s and redemption p e r i o d s (Ref . 1 . 2 ) .

Methods of Ana lys i s

D i f f e r e n t eng inee r ing schemes r e q u i r e d t o meet t h e same o b j e c t i v e s may

be compared economica l ly i n a number of ways. I f a l l payments and incomes

a s s o c i a t e d wi th a scheme a r e d i scoun ted t o t h e i r p r e s e n t va lue f o r comparison,

t h e a n a l y s i s i s termed a p r e s e n t va lue o r d i scoun ted cash f low a n a l y s i s . On

t h e o t h e r hand i f annual ne t incomes o f d i f f e r e n t schemes a r e compared, t h i s

i s termed t h e r a t e of r e t u r n method. The l a t t e r i s most f r e q u e n t l y used by

p r i v a t e o r g a n i s a t i o n s where t a x r e t u r n s and p r o f i t s f e a t u r e prominent ly . In

such cases i t i s sugges t ed t h a t t h e a s s i s t a n c e of q u a l i f i e d accoun tan t s i s

ob ta ined . P resen t va lue comparisons a r e most common f o r p u b l i c u t i l i t i e s .

A form of economic a n a l y s i s popular i n t h e Un i t ed S t a t e s i s b e n e f i t / c o s t

a n a l y s i s . An economic b e n e f i t i s a t t a c h e d t o a l l p roduc t s of a scheme, f o r

i n s t ance a c e r t a i n economic va lue i s a t t a c h e d t o wa te r s u p p l i e s , a l t hough

t h i s i s d i f f i c u l t t o e v a l u a t e i n t h e c a s e of domest ic s u p p l i e s . Those schemes

wi th t h e h i g h e s t b e n e f i t / c o s t v a l u e s a r e a t t a c h e d h i g h e s t p r i o r i t y . Where

schemes a r e mutua l ly e x c l u s i v e such a s i s u s u a l l y t h e c a s e w i t h p u b l i c

u t i l i t i e s t h e scheme w i t h t h e l a r g e s t p r e s e n t va lue of n e t b e n e f i t i s adop ted ,

I f t h e t o t a l water r equ i r emen t s of a town f o r i n s t a n c e were f i x e d , t h e l e a s t -

c o s t supply scheme would be s e l e c t e d f o r c o n s t r u c t i o n .

Unce r t a in ty i n F o r e c a s t s

Fo recas t s of demands, whether t hey be f o r w a t e r , o i l o r g a s , a r e i n -

v a r i a b l y clouded wi th u n c e r t a i n t y and r i s k . S t r i c t L y a p r o b a b i l i t y a n a l y s i s

i s r e q u i r e d f o r each p o s s i b l e scheme, i . e . t h e n e t b e n e f i t of any p a r t i c u l a r

scheme w i l l be t h e sum of t h e n e t b e n e f i t s m u l t i p l i e d by t h e i r p r o b a b i l i t y

f o r a number of p o s s i b l e demands. Berthouex (Ref . 1 . 3 ) recommends under -

des ign ing by 5 t o 10% f o r p i p e l i n e s t o a l l o w f o r u n c e r t a i n f o r e c a s t s , bu t

h i s a n a l y s i s does not account f o r c o s t i n f l a t i o n .

9

An a l t e r n a t i v e method of a l lowing f o r u n c e r t a i n t y i s t o a d j u s t t h e d i s -

count o r i n t e r e s t r a t e : i n c r e a s i n g t h e r a t e w i l l f avour a low c a p i t a l c o s t

scheme, which would be p r e f e r a b l e i f t h e f u t u r e demand were u n c e r t a i n .

Example :

A consumer r e q u i r e s 300 ! / s of water f o r 5 y e a r s t hen p l ans t o i n -

c r e a s e h i s consumption t o 600 ! / s f o r a f u r t h e r 25 y e a r s ( t h e economic

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

He draws f o r 75% o f t h e t ime eve ry day. Determine t h e most economic

d iameter and t h e number of p i p e l i n e s r e q u i r e d . The wa te r i s s u p p l i e d by

a p u b l i c body paying no t a x . Power c o s t s a f l a t 0 .5 p/kWhr, which i n -

c ludes an a l lowance f o r o p e r a t i n g and main tenance . The i n t e r e s t r a t e on

loans ( t a k e n over 20 y e a r s ) and on a s i n k i n g fund i s 10% p . a . , and t h e

r a t e of i n f l a t i o n i n c o s t of p i p e l i n e s , pumps and power i s 6% p . a . Pump

and pumpsta t ion c o s t s amount t o f300 pe r i nc remen ta l kW, ( i n c l u d i n g an

a l lowance f o r s tandby p l a n t ) and pump e f f i c i e n c y i s 7 0 % .

The e f f e c t i v e d i scoun t r a t e may be t aken as t h e i n t e r e s t r a t e l e s s

t h e r a t e of i n f l a t i o n , i . e . 4% p . a . , s i n c e fl t h i s yea r i s wor th

E l x 1.10/1.06 + 1 x 1.04 nex t y e a r . The supply cou ld be made through one l a r g e p i p e l i n e capab le of

hand l ing 600 l / s , o r two s m a l l e r p i p e l i n e s each d e l i v e r i n g 300 l/s, one i n s t a l l e d f i v e y e a r s a f t e r t h e o t h e r . A comparison of a l t e r n a t i v e

d i ame te r s i s made i n t h e t a b l e on p. 10 f o r a s i n g l e p i p e l i n e .

S i m i l a r l y an a n a l y s i s w a s made f o r two p i p e l i n e s each d e l i v e r i n g

300 l / s . Th i s i n d i c a t e d an optimum d iame te r o f 600 mm f o r each p i p e l i n e

and a t o t a l p r e s e n t va lue f o r bo th p i p e l i n e s of 7 500 per 100 m . Thus

one p i p e l i n e , 800 nun d i ame te r , w i l l be t h e most economic s o l u t i o n .

Note t h a t t h e a n a l y s i s i s independent of t h e l e n g t h of t h e p i p e l i n e ,

a l though i t was assumed t h a t p r e s s u r e was such t h a t a con t inuous low p r e s s u r e

p i p e was a l l t h a t was r e q u i r e d . Water hammer p r o t e c t i o n c o s t s a r e assumed

inco rpora t ed i n t h e p i p e c o s t h e r e . The a n a l y s i s i s a l s o independent of t h e

c a p i t a l loan p e r i o d , a l though t h e r e s u l t s would be s e n s i t i v e t o changes i n

t h e i n t e r e s t o r i n f l a t i o n r a t e s . Discount f a c t o r s were o b t a i n e d from p r e s e n t

va lue t a b l e s f o r 4% over 5 , 25 and 30 y e a r p e r i o d s . U n c e r t a i n t y w a s no t

a l lowed f o r bu t would f avour t h e two s m a l l e r 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 t h e a n a l y s i s : I f a 600 mm p ipe -

l i n e was i n s t a l l e d i n i t i a l l y , due t o a h igh u n c e r t a i n t y of t h e demand i n -

c r e a s i n g from 300 t o 600 ! /s, t hen i f t h e demand d i d i n c r e a s e , i t would be

10

S o l u t i o n :

1. I n s i d e D i a . mm

2. Flow I/s

3 .

4 .

5 .

6.

7

8

9 .

Head l o s s m / 1 0 0 m

Power l o s s kW/ 100 m = ( 3 . ) x Ql70 Energy r e q u i r e - ment kW h r / y r / 100 m = ( 4 . ) x 8760 x 0 .75 Annual pumping c o s t f1100 m

E q u i v . c a p i t a l c o s t o f pumping o v e r 5 y r s . = ( 6 . ) x 4.452

E q u i v . c a p i t a l c o s t o f pumping o v e r 25 y r s . = ( 6 . ) x 15.622

P r e s e n t v a l u e o f pumping c o s t = ( 8 . ) / 1 . 1 7 0

_ _ ~

300

0 .14

0.60

3900

20

90

10. C o s t o f pumps e t c . Elloo m: = ( 4 . ) x 300 180

1 1 . P r e s e n t v a l u e o f pump c o s t = (10.) /1 .170 f o r s e c o n d s t a g e 180

f /100 m 3600

f/100 m f o r 300 6, 600 1 1 s 7 . +9. +11.+12. 7 140

1 2 . P i p e l i n e c o s t

13.TOTAL COST

6 00 7 00 800 9 00 -~

600 300 600 300 __ - . - .-. _ _ ~

600 300 600 ____ -

0 .55 0.06 0 . 2 4 0 . 0 3 -

4.72 0.26 2.06 0 . 1 3

31000 1700 13500 850

155

2430

2070

1410

1200

0 . 1 2 0.02 0.07

1.03 0.09 0 .60

6600

- 2 0

8 67 4

40

- 1060 -

9 00

80 620 40

86 530 40

33

520

440

590 3900

3 20

10

- 310

260

3 10 30 180

260 30 150

4200 4800 5400

5750 5560" 5850 ( l e a s t c o s t )

more economic t o b o o s t t h e pumping h e a d a n d pump t h e t o t a l f l o w t h r o u g h t h e

o n e e x i s t i n g 600 mm d i a m e t e r p i p e l i n e r a t h e r t h a n p r o v i d e a s e c o n d 600 nun

p i p e l i n e .

t h r o u g h o n e 600 mm l i n e ( f 7 140/100 m) w i t h t h e p r e s e n t v a l u e of pumping

t h r o u g h two 600 mm l i n e s ( f 7 500/100 m).

T h i s i s i n d i c a t e d by a c o m p a r i s o n o f t h e p r e s e n t v a l u e of pumping

11

BALANCING STORAGE

An aspect which deserves close attention in planning a pipeline system

is reservoir storage. Demands such as those for domestic and industrial water

fluctuate with the season, the day of the week and time of day. Peak-day de-

mands are sometimes in excess of twice the mean annual demand whereas peak

draw-off from reticulation systems may be six times the mean for a day. It would be uneconomic to provide pipeline capacity to meet the peak draw-off

rates, and balancing reservoirs are normally constructed at the consumer end

(at the head of the reticulation system) to meet these peaks. The storage capacity required varies inversely with the pipeline capacity.

The balancing storage requirement for any known draw-off pattern and

pipeline capacity may be determined with a mass flow diagram: Plot cumulative

draw-off over a period versus time, and below this curve plot a line with

slope equal t o the discharge capacity of the pipeline. Move this line up till it just touches the mass draw-off line at one point, which should be at the end of the peak draw-off period. Then the maximum ordinate between the two lines represents the balancing storage required.

An economic comparison is necessary to determine the optimum storage capacity for any particular system (Ref. 1.5). By adding the cost of reser- voirs and pipelines and capitalized running costs for different combinations

and comparing them, the system with least total cost is selected. It is found that the most economic storage capacity varies from one day's supply based on the mean annual rate for short pipelines to two days' supply for long pipe- lines (over 60 km). Slightly more storage may be economic for small-bore pipelines (less than 450 mm dia.). In addition a certain amount of emergency reserve storage should be provided; u p to 12 hours depending upon the avail-

ability of maintenance facilities.

REFERENCES

1.1 J.E. White, Economics of large diameter liquid pipelines, Pipe Line News, N.J., June, 1969.

1.2 Instn. of Civil Engs., An Introduction to Engineering Economics, London,

1962.

1.3 P.M. Berthouex, Accommodating uncertain forecasts, J.Am. Water Works A s s n . , 66 (1) (Jan. 1971) 14.

1.4 J.M. Osborne and L.D. James, Marginal economics app

design, Proc. Am. SOC. Civil Engrs., 99 (TE3) (Aug. ied to pipeline

1973) 637.

1 2

1.5 N . Abramov, Methods of r e d u c i n g power consumpt ion i n pumping water,

I n t . Water Supply Assn. Congress , Vienna , 1969.

LIST OF SYMBOLS

C - c o s t p e r u n i t o f t h r o u g h p u t

D - d i a m e t e r

n - number o f y e a r s

Q - t h r o u g h p u t

r - i n t e r e s t r a t e on s i n k i n g f u n d

R - i n t e r e s t r a t e on l o a n

13

CHAPTER 2

H Y D R A U L I C S

THE FUNDAMENTAL EQUATIONS OF FLUID FLOW

The t h r e e most impor tan t equa t ions i n f l u i d mechanics a r e t h e c o n t i n u i t y

equa t ion , t h e momentum equa t ion and t h e energy e q u a t i o n . For s t e a d y , incom-

p r e s s i b l e , one-dimensional f low t h e c o n t i n u i t y equa t ion i s simply o b t a i n e d

by equa t ing t h e f low r a t e a t any s e c t i o n t o t h e f low r a t e a t ano the r s e c t i o n

a long t h e s t r eam tube . By ' s t e a d y f low ' i s meant t h a t t h e r e i s no v a r i a t i o n

i n v e l o c i t y a t any p o i n t w i t h t ime . 'One-dimensional ' f low impl i e s t h a t t h e

f low i s a long a s t r eam tube and t h e r e i s no l a t e r a l f low a c r o s s t h e boundar ies

of s t r eam t u b e s . I t a l s o imp l i e s t h a t t h e f low i s i r r o t a t i o n a l .

The momentum equa t ion stems from Newton's b a s i c l a w of motion and s t a t e s

t h a t t h e change i n momentum flux between two s e c t i o n s equa l s t h e sum of t h e f o r c e s on t h e f l u i d caus ing t h e change. For s t e a d y , one-dimensional f low t h i s

i s

A Fx = pQAvx ( 2 . 1 )

where F i s t h e f o r c e , p

r a t e , V i s v e l o c i t y and s u b s c r i p t x r e f e r s t o t h e 'x' d i r e c t i o n .

i s t h e f l u i d mass d e n s i t y , Q i s t h e vo lumet r i c f low

The b a s i c energy equa t ion i s d e r i v e d by equa t ing t h e work done on an

element of f l u i d by g r a v i t a t i o n a l and p r e s s u r e f o r c e s t o t h e change i n energy .

Mechanical and h e a t energy t r a n s f e r a r e exc luded from t h e equa t ion . I n most

systems t h e r e i s energy l o s s due t o f r i c t i o n and t u r b u l e n c e and a term i s

inc luded i n t h e equa t ion t o account f o r t h i s . The r e s u l t i n g equa t ion f o r

s t eady f low of i ncompress ib l e f l u i d s i s termed t h e B e r n o u l l i equa t ion and i s

conven ien t ly w r i t t e n a s :

where V = mean v e l o c i t y a t a s e c t i o n

- v2 = v e l o c i t y head ( u n i t s of l e n g t h ) 2g g = g r a v i t a t i o n a l a c c e l e r a t i o n

p = p r e s s u r e

p/x = p r e s s u r e head ( u n i t s of l e n g t h )

X = u n i t weight of f l u i d Z = e l e v a t i o n above an a r b i t r a r y datum

he = head l o s s due t o f r i c t i o n o r t u r b u l e n c e between s e c t i o n s 1 and 2 .

14

The sum of the velocity head plus pressure head plus elevation is termed the

total head.

Strictly the velocity head should be multiplied by a coefficient to

account for the variation in velocity across the section of the conduit.

The average value of the coefficient for turbulent flow is 1.06 and for

laminar flow it is 2.0. Flow through a conduit is termed either uniform or

non-uniform depending on whether or not there is a variation in the cross-

sectional velocity distribution along the conduit.

For the Bernoulli equation to apply the flow should be steady, i.e.

there should be no change in velocity at any point with time. The flow is

assumed to be one dimensional and irrotational. The fluid should be incom-

pressible, although the equation may be applied t o gases with reservations.

The respective heads are illustrated in Fig. 2.1. For most practical

cases the velocity head is small compared with the other heads, and it may

be neglected.

E N E R G Y L I N E

G R A D E L I N E

DATUM 1

FIG. 2.1 Energy heads along a pipeline.

FLOW HEAD LOSS RELATIONSHIPS

E N T R A N C E L O S S

FRICTION LOSS CONTRACTION LOSS

F R I C T I O N L O S S

E L E VAT ION

Conventional flow formulae

The throughput o r capacity of a pipe of fixed dimensions depends on

the total head difference between the ends. This head is consumed by friction

and other (minor) losses.

15

The f i r s t f r i c t i o n head l o s s / f l o w r e l a t i o n s h i p s were d e r i v e d from f i e l d

o b s e r v a t i o n s . These e m p i r i c a l r e l a t i o n s h i p s a r e s t i l l popu la r i n waterworks

p r a c t i c e a l though more r a t i o n a l formulae have been developed . The head l o s s /

flow formulae e s t a b l i s h e d t h u s a r e termed conven t iona l formulae and a r e

u s u a l l y i n an exponen t i a l form o f t h e type

where V i s t h e mean v e l o c i t y of f low, K and K' a r e c o e f f i c i e n t s , R i s t h e

hydrau l i c r a d i u s ( c r o s s s e c t i o n a l a r e a o f f low d iv ided by t h e we t t ed pe r -

i m e t e r , and f o r a c i r c u l a r p i p e f lowing f u l l , e q u a l s one q u a r t e r o f t h e

d i ame te r ) and S i s t h e head g r a d i e n t ( i n m head l o s s p e r m l e n g t h of p i p e ) .

Some o f t h e equa t ions 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:

S . I . u n i t s f . p . s . u n i t s

( 2 . 3 ) Hazen - Wil l iams V=O. 849CR 0'63S0'54 V = l . 318CR

( 2 . 4 ) Chezy V=O . 552CZR0 ' 5S0 ' V=C R

( 2 . 5 ) Manning V=-R

Darcy V= (7) 8 g 1/2R1/2s1/2 v,(8g)1/2R1/2s1/2 f ( 2 . 6 )

0. 63s0. 54

0 .5s0 .5

1.49 2/3s1/2 V= -R 1 2/3s1/2

Except f o r t h e Darcy formula t h e above e q u a t i o n s a r e n o t u n i v e r s a l and

t h e form o f t h e equa t ion depends on t h e u n i t s . I t should be borne i n mind

t h a t t h e formulae were d e r i v e d f o r normal waterworks p r a c t i c e and t a k e no

account o f v a r i a t i o n s i n g r a v i t y , t empera tu re o r t ype o f l i q u i d . They a r e

f o r t u r b u l e n t flow i n p i p e s over 50 mm d i ame te r . The f r i c t i o n c o e f f i c i e n t s

va ry w i t h p i p e d i ame te r , t ype o f f i n i s h and age of p ipe .

The conven t iona l formulae a r e compara t ive ly s imple t o u s e a s t hey do

n o t i nvo lve f l u i d v i s c o s i t y . They may be so lved d i r e c t l y as they do n o t r e -

q u i r e an i n i t i a l e s t i m a t e o f Reynolds number t o de te rmine t h e f r i c t i o n

f a c t o r ( s e e n e x t s e c t i o n ) . The r a t i o n a l e q u a t i o n s cannot be so lved d i r e c t l y

f o r f low. S o l u t i o n o f t h e formulae f o r v e l o c i t y , d iameter o r f r i c t i o n head

g r a d i e n t i s s imple w i t h t h e a i d of a s l i d e - r u l e , c a l c u l a t o r , computer, nomo-

graph o r graphs p l o t t e d on log - log pape r . The e q u a t i o n s a r e o f p a r t i c u l a r

u s e f o r a n a l y s i n g f lows i n p i p e ne tworks where t h e flow/head l o s s e q u a t i o n s

have t o be i t e r a t i v e l y so lved many t imes .

The most popu la r f low formula i n waterworks p r a c t i c e i s t h e Hazen-

Wil l iams formula . F r i c t i o n c o e f f i c i e n t s f o r u s e i n t h i s equa t ion a r e tabu-

l a t e d i n Tab le 2 . 1 . I f t h e formula i s t o b e used f r e q u e n t l y , s o l u t i o n w i t h

t h e a i d of a c h a r t i s t h e most e f f i c i e n t way. Many waterworks o r g a n i z a t i o n s

u s e graphs o f head Loss g r a d i e n t p l o t t e d a g a i n s t f low f o r v a r i o u s p i p e

1 6

diameters, and various C values. As the value of C decreases with age, type

of pipe and properties of water, field tests are desirable for an accurate

assessment of C.

TABLE 2.1 HAZEN-WILLIAMS FRICTION COEFFICIENTS C

Type of Pipe Condition

New 25 Years 50 Years Badlv corroded

PVC : 150 140 140 Smooth concrete, AC: 150 130 120

130

100

Steel, bitumen lined, galvanized: 150 130 I00 60

Cast iron: 130 110 90 50

Riveted Steel, vitrified, woodstave: 120 ao 45

For diameters l e s s than 1 000 mm, subtract 0.1 (1 - x ) c 1 000

Rational flow formulae

Although the conventional flow formulae are likely to remain in use for

many years, more rational formulae are gradually gaining acceptance amongst

engineers. The new formulae have a sound scientific basis backed by numerous

measurements and they are universally applicable. Any consistent units of

measurements may be used and liquids of various visxosities and temperatures

conform to the proposed formulae.

The rational flow formulae for flow in pipes are similar to those for

flow past bodies or over flat plates (Ref. 2.1). The original research was o n small-bore pipes with artificial roughness. Lack of data on roughness

for large pipes has been one deterrent to u s e of the relationships in water-

works practice.

The velocity in a full pipe varies from zero on the boundary to a maxi-

mum in the centre. Shear forces on the walls oppose the flow and a boundary

layer is established with each annulus of fluid imparting a shear force onto

an inner neighbouring concentric annulus. The resistance to relative motion

of the fluid is termed kinematic viscosity, and in turbulent flow it is im-

parted by turbulent mixing with transfer of particles of different momentum

between one layer and the next.

A boundary layer is established at the entrance to a conduit and this

layer gradually expands until it reaches the centre, Beyond this point the

flow becomes uni form. The l e n g t h of p i p e r e q u i r e d f o r f u l l y e s t a b l i s h e d f low

i s g iven by

f o r t u r b u l e n t f low. ( 2 . 7 ) X 5 = 0 . 7 Re The Reynolds number Re=W/u i s a d imens ion le s s number i n c o r p o r a t i n g t h e

f l u i d v i s c o s i t y Y which i s absen t i n t h e conven t iona l f low formulae . Flow i n a

p i p e i s laminar f o r low R e ( < 2 000) and becomes t u r b u l e n t f o r h ighe r Re (no r -

ma l ly t h e c a s e i n p r a c t i c e ) . The b a s i c head l o s s equa t ion i s de r ived by set-

t i n g t h e boundary shear f o r c e over a l e n g t h o f p i p e equa l t o t h e l o s s i n

p r e s s u r e m u l t i p l i e d by t h e area:

ZlTDL = 3 hf TT D2/4

( 2 . 8 )

( t h e Darcy f r i c t i o n f a c t o r ) , r i s t h e s h e a r s t r e s s , D i s v2/2g where f = t h e p i p e d iameter and hf i s t h e f r i c t i o n head loss over a l e n g t h L .

f u n c t i o n o f R e and t h e r e l a t i v e roughness e / d . For laminar flow, P o i s e u i l l e

found t h a t f = 64/Re i . e . f i s independent o f t h e r e l a t i v e roughness . Laminar

flow w i l l n o t occur i n normal e n g i n e e r i n g p r a c t i c e . The t r a n s i t i o n zone be-

tween laminar and t u r b u l e n t f low i s complex and undef ined bu t i s a l s o o f

l i t t l e i n t e r e s t i n p r a c t i c e .

f i s a

Turbu len t f low c o n d i t i o n s may occur w i t h e i t h e r a smooth o r a rough

boundary. The e q u a t i o n s f o r t h e f r i c t i o n f a c t o r f o r bo th c o n d i t i o n s are

de r ived from t h e g e n e r a l equa t ion f o r t h e v e l o c i t y d i s t r i b u t i o n i n a tu rbu-

l e n t boundary l a y e r ,

2L- 5 .75 l o g < JF- Y ( 2 . 9 ) where v i s t h e v e l o c i t y a t a d i s t a n c e y from t h e boundary. For a hydro-

dynamica l ly smooth boundary t h e r e i s a laminar sub - l aye r , and Nikuradse found

t h a t y' = Y / m so t h a t

5 .75 l o g y J q + 5 .5 m- ( 2 . 1 0 ) The c o n s t a n t 5.5 was found expe r imen ta l ly .

Where t h e boundary i s rough t h e laminar sub - l aye r i s des t royed and

Nikuradse found t h a t y' = e l 3 0 where e i s t h e boundary roughness .

Thus &p = 5.75 l o g + 8 . 5 ( 2 . 1 1 )

18

Re-ar ranging equa t ions 2 .10 and 2 .11 and expres s ing v i n terms of t h e

average v e l o c i t y V by means of t h e equa t ion Q=/vdA we g e t

1 = 2 log Re JT -0 .8 ( 2 . 1 2 )

( t u r b u l e n t boundary l a y e r , smooth boundary) and

1 D = 2 log - + 1 .14 (2 .13 )

( t u r b u l e n t boundary l a y e r , rough boundary)

Not ice t h a t f o r a smooth boundary, f i s independent of t h e r e l a t i v e

roughness e/D and f o r a rough boundary i t i s independent of t h e Reynolds

number Re.

Colebrook and White combined Equat ions 2 .12 and 2 . 1 3 t o produce an

equa t ion cove r ing both smooth and rough boundar ies a s w e l l a s t h e t r a n s i t i o n

zone:

1 9 35 f i = 1.14 - 2 l o g (E + -) Re 6 ( 2 . 1 4 ) The i r equa t ion r educes t o Equ. 2.12 f o r smooth p i p e s , and t o Equ. 2 .13

f o r rough p i p e s . This semi-empir ica l e q u a t i o n y i e l d s s a t i s f a c t o r y r e s u l t s

f o r v a r i o u s commercially a v a i l a b l e p i p e s . N ikuradse ' s o r i g i n a l exper iments

u sed sand as a r t i f i c i a l boundary roughness . Na tu ra l roughness i s e v a l u a t e d

acco rd ing t o t h e e q u i v a l e n t sand roughness . Table 2.2 g i v e s v a l u e s o f e f o r

v a r i o u s s u r f a c e s .

TABLE 2 . 2 ROUGHNESS OF PIPE MATERIALS (from Ref . 2 .3)

Value of e i n mm f o r new, c l e a n s u r f a c e u n l e s s o the rwise s t a t e d .

F i n i s h : Smooth Average Rough

Glas s , drawn m e t a l s 0

S t e e l , PVC o r A C 0.015

Coated s t e e l 0 .03

Galvanized , v i t r i f i e d c l a y 0 .06

Cast i r o n o r cement l i n e d 0 .15

Spun conc re t e o r wood s t a v e 0 . 3

R ive ted s t e e l 1 .5

Foul sewers , t u b e r c u l a t e d wa te r mains 6

Unl ined rock , e a r t h 60

0.003

0 .03

0.06

0.15

0.3

0 . 6

3

15

150

0.006

0.06

0.15

0 . 3

0 . 6

1 . 5

6

30

300

F o r t u n a t e l y f i s no t ve ry s e n s i t i v e t o t h e va lue of e assumed. e i n -

c r e a s e s l i n e a r l y wi th age f o r wa te r p i p e s , t h e p r o p o r t i o n a l i t y c o n s t a n t

depending on l o c a l c o n d i t i o n s .

19

The various rational formulae for f were plotted on a single graph by

Moody and this graph is presented as Fig. 2.2.

of water at various temperatures are listed in the Appendix.

The kinematic viscosities

FIG. 2.2 Moody resistance diagram for uniform flow in conduits. (from ref. 2.2)

20

Unfortunately the Moody diagram is not very amenable to direct solution

for any variable for given values of the dependent variables, and a trial and

error analysis may be necessary to get the velocity for the Reynolds number if reasonable accuracy is required. The Hydraulics Research Station at

Wallingford re-arranged the variables in the Colebrook - White equation to

produce simple explicit flow/head l o s s graphs (Ref. 2.3): Equation 2.14 may be arranged in the form

(2.15)

Thus for any fluid at a certain temperature and defined roughness e, a graph

may be plotted in terms of V,D and S. Fig. 2.3 is such a graph for water at

1 5 O C and e = 0.06. The Hydraulics Research Station have plotted similar graphs for various conditions. The graphs are also available for non-circular

sections, by replacing D by 4R. Going a step further, the Hydraulics Research Station re-wrote the Colebrook - White equation in terms of dimensionless parameters proportional to V,R and S , but including factors for viscosity,

roughness and gravity. Using this form of the equation they produced a uni- versal resistance diagram in dimensionless parameters. This graph was also published with their charts.

NON-CIRCULAR CROSS SECTIONS

A circular pipe is normally the most economic if it is to be designed to resist internal pressures. A circular shape has the shortest circumference

per unit of cross sectional area, consequently it requires least material, as

well as being easy to manufacture.

Elliptical or horseshoe shapes are often adopted for sewers or drains.

They have different strength and hKdraulic characteristics to circular pipes.

Vertical elliptical pipes (major axis vertical) have smaller wetted perimeters when running partly full with low flows, consequently the velocity is higher than for a circular pipe, which assists in flushing. The load on a vertical elliptical pipe is less than on a circular pipe with the same cross

sectional area, and the strength is greater because the curvature is sharper

at the top.

Horizontal elliptical pipes (major axis horizontal) are sometimes used where vertical loads are low or clearance is limited. Running partly full they will discharge relatively high flows at small depths of flow which may be an advantage if head is limited.

L L

FIG. 2.3 Flow/head loss chart (from ref. 2 . 3 )

Arch shapes with flat bottoms have similar hydraulic characteristics

to horizontal elliptical shapes for low flow under partly full conditions. The arch shape is usually the most practical shape in tunnelling.

Provided the cross-sectional shape does not differ much from circular

i.e. could be elliptical or even rectangular, the Moody resistance diagram

L L

can be used to calculate head loss relationships for non-circular shapes. The hydraulic radius R=A/P is substituted for D/4. Thus the Reynolds number becomes Re=&RV/u and the head loss equation is:-

fL f hf = 4R 2g. ( 2 . 1 6 ) The equation is inaccurate for low Reynolds numbers and awkward shapes,

and a more popular equation is the Manning equ'ation. Manning developed his empirical equation for flow in channels. The metric form of the equation is

1 213 s1/2 (2.17a) V = - R

It will be observed from the equation that n is not dimensionless and a constant must be introduced if f.p.s. units are used:

(2.17b) 1.49 213 s1/2 V = - R

A table with values of n for different materials is given in the follow-

ing section.

PARTLY FULL PIPES

The most frequently used equation for solving head l o s s relationships in partly full conduits is the Manning equation, given in the preceding section. This equation was used in the preparation of Fig. 2.4.

The values of flow, velocity, area and hydraulic radius expressed as a proportion of the full flow conditions for circular sections are given in

Fig. 2 .4 . Using Figs. 2.3 and 2 .4 , given any three of the five variables Q, D, S , V and d/D, the other two may be determined. The flow conditions for full bore flow (d/D = 1) are yielded simultaneously. Designate Q' = flow at full bore and V' = velocity at full bore. Now assume the flow, pipe diameter and slope (Q, D and S ) ars known, and d/D and V are to be de- termined. Read Q' and V' from Fig. 2.3, and using the ratio Q/Q' , read d/D from Fig. 2 . 4 . Hence also read V/V' from Fig. 2.4 and calculate V knowing V'.

As another example, given say Q = 50 ! I s , S = 0.005 and d/D = 0.25, find the necessary diameter D and corresponding velocity V: From Fig. 2.4, Q/Q' = 0.135, s o Q' = 370 11s and from Fig. 2.3, D = 0.5 m and V' = 1.85 m / s . Now from Fig. 2.4, V/V' = 0.7 hence V = 1.3 m/s.

An interesting fact is illustrated in Fig. 2.4. The flow for a partly full pipe is greater than the flow through a fully charged pipe if the depth of flow is between 82% and 100% of the diameter. The reason for this is that

the wetted perimeter increases rapidly but the area does not, as the pipe

23

FIG. 2.4 R e l a t i v e v e l o c i t y and f low i n c i r c u l a r p i p e f o r any depth of flow (from r e f . 6 .3 )

f i l l s up over t h e l a s t p o r t i o n . The a d d i t i o n a l f low should, however, n o t b e

r e l i e d upon, f o r t h e s l i g h t e s t i r r e g u l a r i t y i n t h e p i p e may cause t h e p i p e

t o flow f u l l .

E s c r i t t (Re f . 2.4) proposes t h a t one h a l f of t h e f r e e s u r f a c e width i s

24

added to the wetted perimeter for the purposes of calculating flows. This

procedure brings the calculated flows through partly filled pipes in line with measured flows.

TABLE 2.3 VALUES OF MANNING'S In'

Smooth glass, plastic f). 010

Concrete, steel (bitumen lined), galvanised 0.011 Cast Iron 0.012

Slimy or greasy sewers 0.013 Rivetted steel, vitrified, wood-stave 0.015

Rough concrete 0.017

M I N O R LOSSES

One method of expressing head loss through fittings and changes in

section is the equivalent length method, often used when the conventional

friction loss formulae are used. Modern practice is to express losses through fittings in terms of the velocity head i.e. h =KV2/2g where K is the loss co- efficient. Table 2.4 gives typical loss coefficients although valve manufac- turers may also provide supplementary data and loss coefficients K which will vary with gate opening. The velocity V to use is normally the mean through

the full bore of the pipe or fitting.

1

TABLE 2.4 LOSS COEFFICIENTS FOR PIPE FITTINGS

Bends hg = KB V2/2g - Bend angle Sharp r/D = 1 2 6 30' 0.16 0.07 0.07 0.06

45O 0.32 0.13 0.10 0.08

60' 0.68 0.18 0.12 0.08

goo 1.27 0.22 0.13 0.08 180' 2.2

90' with guide vanes 0.2

r = radius of bend to centre of pipe. A significant reduction in bend loss is possible if the radius is flattened in the plane of the bend.

25

Valves hv = Kv +12g

Type Opening: 114 112 314 F u l l

Sluice 24 5.6 1.0 0.2 Butterfly 120 7.5 1.2 0.3

Globe 10

Needle 4 1 0.6 0.5 Ref l ux 1 - 2.5

Contractions and expansions in cross section Contract ions: Expansions:

hc = KcV2'/2g hc = KcV12/2g

A 2 4 A1 /A2 Wall - wall angle 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

7.5O 0 .13 .08 .05 .02 0 0

15' 0 .32 .24 .15 .08 . 0 2 0

30' 0 .78 .45 .27 .13 .03 0

180' . 5 .37 ,25 .15 .07 0 1.0 .64 .36 .17 .04 0

Entrance and exit losses: h = K, V2/2g

Entrance Exit

Protruding 0.8 1.0 Sharp 0.5 1.0

Bevelled 0.25 0.5

Rounded 0.05 0.2

REFERENCES

2.1 H. Schlichling, Boundary Layer Theory, 4th Edn., McGraw Hill, N.Y., 1960.

2.2 M.L. Albertson, J.R. Barton and D.B. Sirnons, Fluid Mechanics for Engin-

eers, Prentice Hall, N.J., 1960. 2.3 Hydraulics Research Station, Charts for the Hydraulic Designs of Chan-

nels and Pipes, 3rd Edn., H.M.S.O., London, 1969.

2 . 4 L.B. Escritt, Sewerage and Sewage Disposal, Macdonald and Evans, London, 1972.

26

L I S T OF SYMBOLS

A

C

C'

c z d

D

e

f

Fx

he hf

g

K

L

n

P

P

Q R

R e

S

V

V

X

Y z x P 7

V

c r o s s s e c t i o n a l a r e a of flow

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

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

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

depth of wa te r

d iameter

Nikuradse roughness

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

f o r c e

g r a v i t a t i o n a l a c c e l e r a t i o n

head l o s s

f r i c t i o n head l o s s

l o s s c o e f f i c i e n t

l e n g t h o f condu i t

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

we t t ed p e r i m e t e r

p r e s s u r e

flow r a t e

h y d r a u l i c r a d i u s

Reynolds number

h y d r a u l i c g r a d i e n t

mean v e l o c i t y a c r o s s t h e s e c t i o n

v e l o c i t y a t a p o i n t

d i s t a n c e a long condu i t

d i s t a n c e from boundary

e l e v a t i o n

s p e c i f i c weight

mass d e n s i t y

shea r s t r e s s

k inemat i c v i s c o s i t y

27

CHAPTER 3

PIPELINE S Y S T E M ANALYSIS AND DESIGN

NETWORK ANALYSIS

The flows through a system of interlinked pipes or networks are con-

trolled by the difference between the pressure heads at the input points and

the residual pressure heads at the drawoff points. A steady-state flow

pattern will be established in a network such that the following two criteria

are satisfied:-

(1) The net flow towards any junction or node is zero, i.e., inflow must equal outflow, and

( 2 ) The net head loss around any closed loop is zero, i.e., only one head

can exist at any point at any time.

The line head losses are usually the only significant head losses and

most methods of analysis are based on this assumption. Head loss relation-

ships for pipes are usually assumed to be of the form h = KB Qn/Dm (3.1) where h is the head l o s s , i is the pipe length, Q the flow and D the internal diameter of the pipe.

The calculations are simplified if the friction factor ti can be assumed

the same for all pipes in the network.

Equivalent Pipes for Pipes in Series or Parallel

It is often useful to know the equivalent pipe which would give the

same head loss and flow as a number of interconnected pipes in series or

parallel. The equivalent pipe may be used in place of the compound pipes to

perform further flow calculations.

The equivalent diameter of a compound pipe composed of sections of

different diameters and lengths in series may be calculated by equating the

total head loss for any flow to the head loss through the equivalent pipe

of length equal to the length of compound pipe:-

(m is 5 in the Darcy formula and 4 . 8 5 in the Hazen - Williams formula.) Similarly, the equivalent diameter of a system of pipes in parallel is

derived by equating the total flow through the equivalent pipe 'e' to the

sum of the flows through the individual pipes 'i' in parallel:

28

Now h = h .

i . e . K ~ ~ Q ~ / D ~ ~ = K P ~ Q ~ ~ / D ~ ~ .

so Qi = ( e e / ni) l/n(Di/De)m/nQ

and Q = 1 Qi = z C ( e e / t i ) So c a n c e l l i n g o u t Q, and b r i n g i n g D and 1 t o t h e l e f t hand s i d e ,

(Di/De)m/nQ].

and i f each P i s t h e same,

The e q u i v a l e n t d iameter could a l s o b e d e r i v e d u s i n g a flow/head loss

c h a r t . For p i p e s i n p a r a l l e l , assume a r easonab le head l o s s and r e a d o f f

t h e flow through each p i p e from t h e c h a r t . Read o f f t h e e q u i v a l e n t d iameter

which would g i v e t h e t o t a l f low a t t h e s a m e head l o s s . For p i p e s i n s e r i e s ,

assume a r e a s o n a b l e f low and c a l c u l a t e t h e t o t a l head l o s s w i t h a s s i s t a n c e

of t h e c h a r t . Read o f f t h e e q u i v a l e n t p i p e d iameter which would d i s c h a r g e

t h e assumed f low wi th t h e t o t a l head l o s s a c r o s s i t s l e n g t h .

It o f t e n speeds network a n a l y s e s t o s i m p l i f y p i p e ne tworks as much as

p o s s i b l e u s i n g e q u i v a l e n t d i ame te r s f o r minor p i p e s i n series o r p a r a l l e l .

Of cour se t h e methods o f network a n a l y s i s d e s c r i b e d below could always b e

used t o a n a l y s e f lows through compound p i p e s and t h i s i s i n f a c t t h e p re -

f e r r e d method f o r more complex systems than those d i s c u s s e d above.

The Loop Method f o r Analys ing Networks

The loop method and t h e node method o f a n a l y s i n g p i p e ne tworks both

invo lve s u c c e s s i v e approximat ions speeded by a mathemat ica l t echn ique de-

veloped by Hardy Cross (Ref . 3 .1) .

The s t e p s i n ba l anc ing t h e f lows i n a network by t h e loop method a r e

a s fo l lows: -

( 1 ) Draw t h e p i p e network s c h e m a t i c a l l y t o a c l e a r s c a l e . I n d i c a t e a l l i n -

p u t s , d rawoffs , f i x e d heads and b o o s t e r pumps ( i f p r e s e n t ) .

( 2 ) I f t h e r e i s more than one cons t an t -head node, connec t p a i r s of cons t an t -

head nodes o r r e s e r v o i r s by dummy p i p e s r e p r e s e n t e d by dashed l i n e s .

Imagine t h e network a s a p a t t e r n o f c l o s e d loops i n any o r d e r . To speed

convergence o f t h e s o l u t i o n some o f t h e major p i p e s may b e assumed t o

form l a r g e superimposed loops i n s t e a d o f assuming a s e r i e s o f l oops s i d e by s i d e . Use o n l y as many loops as are needed t o ensu re t h a t each

p i p e i s i n a t l e a s t one loop ,

( 3 )

29

S t a r t i n g w i t h any l o o p a s c r i b e a r b i t r a r y i n i t i a l f l o w s a r o u n d t h e l o o p ,

c o n s i s t e n t w i t h i n p u t s and d r a w o f f s f rom n o d e s . The more a c c u r a t e t h e

i n i t i a l f l o w a s s u m p t i o n , t h e s p e e d i e r w i l l be t h e s o l u t i o n . Once t h e

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

a u t o m a t i c a l l y , s i n c e t h e f l o w i n a t l e a s t one l e g w i l l be known.

C a l c u l a t e t h e head l o s s i n e a c h p i p e u s i n g a f o r m u l a s u c h as h =K!Q / D

o r u s e a f l o w / h e a d l o s s c h a r t ( p r e f e r a b l e i f t h e a n a l y s i s i s t o b e done

by h a n d ) .

C a l c u l a t e t h e n e t h e a d l o s s a round any l o o p , i . e . , p r o c e e d i n g around t h e

l o o p , add head l o s s e s and s u b t r a c t head g a i n s u n t i l a r r i v i n g a t t h e

s t a r t i n g p o i n t . I f t h e n e t head l o s s a r o u n d t h e l o o p i s n o t z e r o , c o r r e c t

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

t h e same d i r e c t i o n t h a t h e a d l o s s e s were c a l c u l a t e d :

n m

A Q = - x h m ( 3 . 4 ) T h i s e q u a t i o n i s t h e f i r s t o r d e r a p p r o x i m a t i o n t o t h e d i f f e r e n t i a l o f

t h e head loss e q u a t i o n and i s d e r i v e d a s f o l l o w s : -

S i n c e h = KIQn/Dm

dh = KenQn-'dQ/Dm = (hn/Q)dQ .

Now t h e t o t a l head loss a r o u n d e a c h l o o p s h o u l d be z e r o , i . e . ,

The v a l u e of hn/Q i s a l w a y s p o s i t i v e e x c e p t f o r dummy p i p e s between

c o n s t a n t - h e a d r e s e r v o i r s , when i t i s t a k e n as z e r o .

I f t h e r e i s a b o o s t e r pump i n any l o o p , s u b t r a c t t h e g e n e r a t e d h e a d

from t h b e f o r e making t h e f l o w c o r r e c t i o n u s i n g t h e above e q u a t i o n s .

The f l o w around e a c h l o o p i n t u r n i s c o r r e c t e d t h u s .

The p r o c e s s i s r e p e a t e d u n t i l t h e head around e a c h l o o p b a l a n c e s t o a

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

The Node Method f o r A n a l y s i n g Networks

With t h e node method, i n s t e a d o f assuming i n i t i a l f l o w s around l o o p s ,

i n i t i a l h e a d s a r e assumed a t e a c h node . Heads a t nodes a r e c o r r e c t e d by

s u c c e s s i v e a p p r o x i m a t i o n i n a s i m i l a r manner t o t h e way f l o w s were c o r r e c t e d

f o r t h e l o o p method. The s t e p s i n an a n a l y s i s a r e as f o l l o w s : -

3 0

(1) Draw t h e p i p e ne twork s c h e m a t i c a l l y t o a c l e a r s c a l e . I n d i c a t e a l l i n -

p u t s , d r a w o f f s , f i x e d h e a d s and b o o s t e r pumps.

( 2 ) A s c r i b e i n i t i a l a r b i t r a r y h e a d s t o e a c h node ( e x c e p t i f t h e h e a d a t

t h a t node i s f i x e d ) . The more a c c u r a t e t h e i n i t i a l a s s i g n m e n t s , t h e

s p e e d i e r w i l l be t h e c o n v e r g e n c e o f t h e s o l u t i o n .

( 3 ) C a l c u l a t e t h e f l o w i n e a c h p i p e t o any node w i t h a v a r i a b l e head u s i n g

t h e f o r m u l a Q = (hDm/K!)l/n o r u s i n g a f l o w / h e a d loss c h a r t . (4) C a l c u l a t e t h e n e t i n f l o w t o t h e s p e c i f i c node and i f t h i s i s n o t z e r o ,

c o r r e c t t h e h e a d by a d d i n g t h e amount

( 3 . 5 )

T h i s e q u a t i o n i s d e r i v e d a s f o l l o w s : -

S i n c e Q = (hDm/Kl)l/n

dQ = Qdh/nh

We r e q u i r e I ( Q + dQ) = 0 EQ + Qdh = 0 ix

But dH = -dh

s o

Flow Q and h e a d loss a r e c o n s i d e r e d p o s i t i v e i f towards t h e node . H i s

t h e h e a d a t t h e node . I n p u t s ( p o s i t i v e ) and d r a w o f f s ( n e g a t i v e ) a t t h e

node s h o u l d be i n c l u d e d i n ZQ. ( 5 ) C o r r e c t t h e head a t e a c h v a r i a b l e - h e a d node i n s i m i l a r manner . i . e .

r e p e a t s t e p s 3 and 4 f o r e a c h node .

( 6 ) R e p e a t t h e p r o c e d u r e ( s t e p s 3 t o 5 ) u n t i l a l l f l o w s b a l a n c e t o a

s u f f i c i e n t d e g r e e of a c c u r a c y . I f t h e h e a d d i f f e r e n c e between t h e e n d s

of a p i p e i s z e r o a t any s t a g e , omi t t h e p i p e f rom t h e p a r t i c u l a r

b a l a n c i n g o p e r a t i o n .

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

Both t h e l o o p method and t h e node method of b a l a n c i n g f l o w s i n ne tworks

can be done m a n u a l l y b u t a computer i s p r e f e r r e d f o r l a r g e n e t w o r k s . I f done

m a n u a l l y , c a l c u l a t i o n s s h o u l d be s e t o u t w e l l i n t a b l e s o r even o n t h e p i p e -

work l a y o u t drawing i f t h e r e i s s u f f i c i e n t s p a c e . F i g . 3 .1 i s a n example

a n a l y s e d m a n u a l l y by t h e node method. T h e r e 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 o f which u s e t h e l o o p method.

The main d i s a d v a n t a g e of t h e node method i s t h a t more i t e r a t i o n s a r e

r e q u i r e d t h a n f o r t h e l o o p method t o a c h i e v e t h e same c o n v e r g e n c e , e s p e c i a l l y

31

35;1 -4 11.1 34.4 -4 7 3 33,4 I 1

+ 20 47,s +13,8 + 10,7 *10,5

,om Q

140 130 110 100 100

- F 7 7.2

33.21

% a,o 26 6,4 IS

-1.7 + 5 4 80 I S 0.5 +52,71 801 1.5

NOTES : HEADS IN METERS, FLOWS IN L/S, DIAMETERS IN MILLIMETERS, LENGTHS IN M. ARROWS INDICATE POSITIVE DIRECTION O F h_& Q BLACKENED CIRCLES I N D I C A m O E S WITH FIX!D HEADS. NUMBERS IN CIRCLES INDICATE ORDER IN WHICH NODES WERE CORRECTED HEAD LOSSES EVALUATED FROM Fiq. 2.3 AH = 1,85 Z Q in

ARBITARY ASSUMPTION).

EQ/h

FIG. 3 . 1 Example of node method of network flow ana lys is .

i f the system i s very unbalanced t o s t a r t with. It i s normally necessary for

a l l p ipes to have the same order of head l o s s . Barlow and Markland (Ref. 3 . 2 )

suggest methods f o r speeding the convergence. These include overcorrect ion

i n some cases , o r using a second order approximation t o the d i f f e r e n t i a l s for

ca lcu la t ing cor rec t ions .

32

Reasons for the slow convergence of the node method are discussed below.

A flow correction to a loop is distributed over a number of nodes, where- as with the node method corrections proceed only one pipe length per cycle. Also , since head loss is proportional to flow to a power greater than unity,

with the loop method quite large head loss revisions result from a small flow

correction. The reverse is the case with the node method.

With modern generation computers, the actual computation time is insig- nificant compared with data input/output time, and the number of iterations required is of small consequence. The versatility of the node iteration method has been found to more than offset the convergence disadvantage for

practical networks.

Experience indicates that with the node iteration method data can be

read in and printed out in handy form. There are no imaginary loops to number,

or to revise whenever new pipes are added to the system. Node designations

need not be sequential, and could correspond to actual meter designations. Each pipe is identified by its end nodes, diameter and length. New pipes or nodes can readily be included by adding a line of data onto the end of the existing data set.

It is relatively simple to alter node conditions with the node method - for instance either the head could be fixed at a node (in which case drawoff would be an unknown) or drawoff could be fixed (in which case head would be an unknown).

The pipelines for the loop method are identified by two neighbouring

loop numbers. Node drawoffs are stated indirectly by arranging the initial flows in the pipelines in such a manner that node drawoffs are complied with. Thus the drawoffs are not simply tabulated as for the node iteration method.

In addition to the methods of successive approximation, head losses and

flows could be calculated by direct solution. A set of simultaneous equations

relating flows and head losses is set up, and solved by established procedures such as the Gauss elimination method.

Although computing time is reasonably short for this method, considerable

time is needed to set up or revise the equations, and a large computer memory capacity is needed for solving the resulting matrix.

Although analogue computers have been used to some extent in analysing networks, they are not as versatile as digital computers and have to be re- wired for each network. Since they cannot be used for many other purposes, it is often difficult to justify the expense of an analogue computer.

33

OPTIMIZATION OF PIPELINES SYSTEMS

The previous section described methods for calculating the flows in pipe networks with or without closed loops. For any particular pipe network layout and diameters, the flow pattern corresponding to fixed drawoffs or

inputs at various nodes could be calculated. To design a new network t o meet

certain drawoffs, it would be necessary to compare a number of possibilities. A proposed layout would be analysed and if the corresponding flows were just sufficient to meet demands and pressures were satisfactory, the layout would be acceptable. If not, it would be necessary to try alternative diameters for some or all pipes and to re-analyse the network. The process of adjusting pipe sizes and analysing flows is repeated until a satisfactory solution is

at hand. This trial and error process would then be repeated for another

possible layout. Each of the final networks so derived would then have to be

costed and that network with least cost selected. A technique of determining the least-cost network directly, without re-

course to trial and error, would be desirable, No direct and positive tech- nique is possible for general optimization of networks with closed loops.

The problem is that the relationships between pipe diameters, flows, head l o s s e s and costs is not linear and most routine mathematical optimization

techniques require linear relationships. There are a number of situations

where mathematical optimization techniques can be used to optimize layouts

and these cases are discussed and described below. The cases are normally

confined to single mains or tree-like networks for which the flow in each

branch is known. To optimise a network with closed loops, random search techniques or successive approximation techniques are needed.

Mathematical optimization techniques are also known as systems analysis techniques (which is an incorrect nomenclature as they are design techniques, not analysis techniques), or operations research techniques (again a name not really descriptive). The name mathematical optimization techniques will be retained here. Such techniques include simulation (or mathematical modelling) coupled with a selection technique such as steepest path ascent or random searching.

The direct optimization methods include dynamic programming, which is useful for optimizing a series of events or things, transportation programming,

which is useful for allocating sources to demands, and linear programming, for optimizing any system which can be described by a set of linear equations or inequalities (Refs. 3 . 3 and 3 . 4 ) . Linear programing usually requires the use of a computer, but there are standard optimization programs available.

3 4

Dynamic Programming f o r O p t i m i z i n g Compound P i p e s

One of t h e s i m p l e s t o p t i m i z a t i o n t e c h n i q u e s , and i n d e e d one which can

n o r m a l l y be u s e d w i t h o u t r e c o u r s e t o c o m p u t e r s , i s dynamic programming. The

t e c h n i q u e i s i n f a c t o n l y a s y s t e m a t i c way o f s e l e c t i n g a n optimum program

from a s e r i e s o f e v e n t s and d o e s n o t i n v o l v e any m a t h e m a t i c s . The t e c h n i q u e

may be used t o s e l e c t t h e most economic d i a m e t e r s o f a compound p i p e which

may v a r y i n d i a m e t e r a l o n g i t s l e n g t h d e p e n d i n g on p r e s s u r e s and f l o w s . F o r

i n s t a n c e , c o n s i d e r a t r u n k main s u p p l y i n g a number of consumers f rom a

r e s e r v o i r . The d i a m e t e r s of t h e t r u n k main may be r e d u c e d as drawoff t a k e s

p l a c e a l o n g t h e l i n e . The problem i s t o s e l e c t t h e most economic d i a m e t e r

f o r e a c h s e c t i o n of p i p e .

A B C ANSWER DIA I 260mm 310mm 340mrn

FIG. 3 . 2 P r o f i l e of p i p e l i n e o p t i m i z e d by dynamic programming

A s i m p l e example d e m o n s t r a t e s t h e u s e of t h e t e c h n i q u e : C o n s i d e r t h e

p i p e l i n e i n F i g . 3 . 2 . Two consumers draw w a t e r f rom t h e p i p e l i n e , and t h e

head a t each drawoff p o i n t i s n o t t o d r o p below 5 m , n e i t h e r s h o u l d t h e

h y d r a u l i c g r a d e l i n e d r o p below t h e p i p e p r o f i l e a t any p o i n t . The e l e v a t i o n s

of each p o i n t and t h e l e n g t h s of e a c h s e c t i o n of p i p e a r e i n d i c a t e d . The

c o s t of p i p e i s f O . l p e r mm d i a m e t e r p e r m of p i p e . ( I n t h i s c a s e t h e c o s t

i s assumed t o be i n d e p e n d e n t o f t h e p r e s s u r e h e a d , a l t h o u g h i t i s s i m p l e t o

t a k e a c c o u n t of s u c h a v a r i a t i o n ) . The a n a l y s i s w i l l b e s t a r t e d a t t h e down-

s t r e a m end of t h e p i p e ( p o i n t A ) . The most economic a r r a n g e m e n t w i l l be w i t h

minimum r e s i d u a l head i . e . 5 m , a t p o i n t A . The h e a d , H , a t p o i n t B may be a n y t h i n g between 13 m and 31 m above t h e datum, b u t t o s i m p l i f y t h e a n a l y s i s ,

we w i l l o n l y c o n s i d e r t h r e e p o s s i b l e h e a d s w i t h 5 m i n c r e m e n t s between them

a t p o i n t s B and C .

35

The diameter D of the pipe between A and B, corresponding to each of

the three allowed heads may be determined from a head loss chart such as

Fig. 2.3 and is indicated in Table 3.1 (I) along with the corresponding cost. We will also consider only three possible heads at point C. The number

of possible hydraulic grade lines between B and C is 3 x 3 = 9 , but one of

these is at an adverse gradient so may be disregarded. In Table 3 . 1 (11) a set of figures is presented for each possible hydraulic grade line between B and C. Thus if HB = 13 and HC = 19 then the hydraulic gradient from C to

B is 0.006 and the diameter required for a flow of 110 l / s is 310 mm (from Fig. 2.3). The cost of this pipeline would be 0.1 x 310 x 1 0 0 0 = f31 000. Now to this cost must be added the cost of the pipe between A and B, in this case f60 000 (from Table 3.1 (I)). For each possible head HC there is one minimum total cost of pipe between A and C, marked with an asterisk. It is

this cost and the corresponding diameters only which need be recalled when

proceeding to the next section of pipe. In this example, the next section

between C and D is the last and there is only one possible head at D, name-

ly the reservoir level. In Table 3.1 (111) the hydraulic gradients and corresponding diameters

and costs for Section C - D are indicated. To the costs of pipe for this section are added the costs of the optimum pipe arrangement up t o C. This is done for each possible head at C, and the least total cost selected from Table 3.1 (111). Thus the minimum possible total cost is f15l 000 and the most economic diameters are 260, 310 and 340 mm for Sections A - B, B - C and C - D respectively. It may be desirable to keep pipes to standard diameters in which case the nearest standard diameter could be selected for each section as the calculations proceed or each length could be made up of two sections; one with the next larger standard diameter and one with the next smaller standard diameter, but with the same total head l o s s as the

theoretical result. Of course many more sections of pipe could be considered and the

accuracy would be increased by considering more possible heads at each section. The cost of the pipes could be varied with pressures. A booster

pump station could be considered at any point, in which case its cost and capitalised power cost should be added in the tables. A computer may prove

useful if many possibilities are to be considered, and there are standard dynamic programming programs available.

It will be seen that the technique of dynamic programming reduces the

number of possibilities to be considered by selecting the least-cost arrange-

ment at each step. Refs. 3.5 and 3.6 describe applications of the technique

to similar and other problems.

36

n = C

HB

13

1 8

23

TABLE 3 . 1 DYNAMIC PROGRAMMING O P T I M I Z A T I O N OF A COMPOUND P I P E

-~ 19 24 29

COST DC-B f hC-B DC-B f hC-B DC-B COST f hCYB

.006 310 31000 .011 270 27000 .016 250 25000 6 o o o O 6 o o o O 60000 9 loo@ 87000 85000

.001 430 43000 .006 310 31000 .011 270 27000 5 x 0 52000 __. 52000 95000 83000* 7 9 0009c

.001 430 43000 .006 310 31000 50000 50000 - 93000 -0

I

I1

I11

1 :! 1 .004 1 300 1 60000 1 .0065 260 52000

.009 250 50000

1 29 1 .001 1 4301 86000 1 79000

165000

37

T r a n s p o r t a t i o n programming f o r l e a s t - c o s t a l l o c a t i o n of r e s o u r c e s

T r a n s p o r t a t i o n programming i s ano the r t echn ique which no rma l ly does

n o t r e q u i r e t h e u s e o f a computer. The t echn ique i s o f u s e p r i m a r i l y f o r

a l l o c a t i n g t h e y i e l d o f a number of sou rces t o a number of consumers such

t h a t a l e a s t - c o s t sys tem i s ach ieved . The c o s t of d e l i v e r i n g t h e r e s o u r c e

a long each r o u t e should b e l i n e a r l y p r o p o r t i o n a l t o t h e throughput a long

t h a t r o u t e and f o r t h i s r eason t h e t echn ique i s p robab ly of no u s e i n s e l e c t -

i n g t h e optimum p i p e s i z e s . I t i s of u se , however, i n s e l e c t i n g a l e a s t - c o s t

pumping p a t t e r n throug